Kosterlitz–Thouless Physics: a Review of Key Issues Accepted for Publication 7 October 2015 Published 28 January 2016

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Rep. Prog. Phys. Rep. Prog. Phys. 79 (2016) 026001 (59pp) doi:10.1088/0034-4885/79/2/026001

79 Review

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J Michael Kosterlitz 026001 Department of Physics, Brown University, Providence, RI 02912, USA

J M Kosterlitz E-mail: [email protected]

Received 1 June 2015, revised 5 October 2015 Kosterlitz–Thouless physics: a review of key issues Accepted for publication 7 October 2015 Published 28 January 2016

Printed in the UK Abstract This article reviews, from a very personal point of view, the origins and the early work on ROP transitions driven by topological defects such as vortices in the two dimensional planar rotor model and in 4Helium films and dislocations and disclinations in 2D crystals. I cover the early papers with David Thouless and describe the important insights but also 10.1088/0034-4885/79/2/026001 the errors and oversights since corrected by other workers. I then describe some of the experimental verifications of the theory and some numerical simulations. Finally applications to superconducting arrays of Josephson junctions and to recent cold atom experiments are 0034-4885 described.

2 Keywords: two dimensions, topological defects, renormalization group, superfluid Helium, superconducting films, two dimensional crystals, cold atom arrays (Some figures may appear in colour only in the online journal)

1. Introduction films and domain walls in the Ising model of magnetism. This opened my mind to the world of condensed matter physics From my point of view, the discovery of defect mediated where, to my uneducated mind, there seemed to be a plethora phase transitions started in 1970 when I was a postdoc in of unsolved physical problems. This was a big contrast to high high energy physics at the Istituto di Fisica Teorica of Torino energy physics where there seemed to be very few interesting University in Italy. I was, and still am, a disorganized person problems with several very smart people working on every with a tendency to wait until the last possible moment or even problem. However, there was another difficulty which was later to do something important like submitting a job applica- my almost total ignorance of statistical mechanics which I tion. Consequently, in September 1970, I found myself in the had more or less ignored as a graduate student in high energy Department of Mathematical Physics, Birmingham University, physics. England instead of at CERN in Geneva, Switzerland where I had been warned by my colleagues that David could be dif- I had intended to go. During my first year at Birmingham, ficult because he was reputed to not suffer fools gladly. There I did a set of complicated calculations on a model which was a was little question that I was an ignorant fool as far as phase precursor to string theory, but I was always beaten by a group transitions were concerned so I was quite nervous listening in USA. At least twice, I was in the process of writing up my to David’s thoughts about some low dimensional problems in calculations when a preprint by my competitors appeared in phase transitions. He explained that he had been discussing the department. with Phil Anderson the problem of a phase transitions in the Somewhat discouraged, I began to look at other branches 1D Ising chain with long range interactions between spins fall- of physics in the hope of finding an interesting but tractable ing off as 1/r2. He had recently shown [1] that, in this model, problem and found myself talking with David Thouless about there was a phase transition with some peculiar properties. strange excitations such as vortices in superfluid 4Helium It was known that an Ising chain with shorter ranged interactions

0034-4885/16/026001+59$33.00 1 © 2016 IOP Publishing Ltd Printed in the UK Rep. Prog. Phys. 79 (2016) 026001 Review had no transition to an ordered state at any finite temperature for the domain wall at 0 xL well away from the ends [2] while the system with interactions decaying more slowly of the chain. Thus, the free energy ∆F of an isolated domain than 1/r2 had an ordered state at sufficiently low temperatures wall is [3] and, thus, a transition at a finite T > 0. Our early discussions of this and related problems intro- ∆=FT(),lLJ()−+kTB n/()La O()1. (3) duced me to more new concepts like topological defects The probability of a single domain wall in a system of length such as vortices in superfluid 4He and their role in disorder- La is ing a system. As I was operating from a position of extreme ignorance, these did not seem any stranger than other ideas e−∆FL(),/Tk()BT Pt(), L ∼ about phase transitions. In fact, they seemed more hopeful 1e+ −∆FL(),/Tk()BT than many others all of which had failed to make much of an 1 ⎧0, kTB < J, impression on this class of intractable problems. In an Ising ∼ Jk/1T − → ⎨ (4) 1/+ La ()B ⎩ 1, kTB > J. model, a topological defect is an easily visualized excitation () as it is just a domain wall between regions of opposite spin This implies that one can identify the critical temperature as orientation. It is not difficult to rewrite the Ising spin Hamilton kBcTJ= since there will be no isolated domain walls in the in terms of interacting domain walls which live on the lat- equilibrium system when TT< c so that the system will be tice dual to the original spin lattice. During these discussions, ordered. There will be finite domains of reversed spins below David gave me a set of papers by Anderson and coworkers Tc but still a finite magnetization. On the other hand, forTT > c, [4–8] and suggested I look at them as they may be useful. there is a finite probability that there will be some isolated free These were a major surprise because, as far as I was concerned, domain walls implying disorder and vanishing magnetization. their importance was that they solved the 1D Ising model with The original argument [1] went further to show that the mag- 1/r2 interactions and demonstrated that it is ordered at low netization has a discontinuity at Tc although the transition is temperatures and has a phase transition to a disordered state at continuous. This was later confirmed by more rigorous work a finiteT . I spent most of six months doing nothing but read- c [15–17]. ing and rereading these papers until I understood what they The free energy ∆F of equation (3) for an isolated domain were doing, especially the last in the series [6] which solved wall in the 1/r2 ferromagnetic Ising chain has important impli- the problem by a very strange method. I later realized that this cations which were to become relevant to later developments paper contained a very early version of the renormalization in the defect theory of phase transitions. Domain walls can group popularized by Wilson [9, 10] to formalize the scaling be regarded as topological defects in an Ising ferromagnet theory of Kadanoff [11] but, at the time, it just appeared to be and it is obvious that a finite concentration of these in thermal a very clever and unusual method of solving one particular equilibrium tend to disorder the system. Equation (4) shows example of a phase transition. that there will be no isolated domain walls in the system for kBTJ< and there will be long range ferromagnetic order. On 1.1. One dimensional Ising model the other hand, when kBTJ> , a finite density of these topological defects is present in thermal equilibrium thus disordering In 1969, Thouless was first exposed to the essential ideas the system. This argument is made more exact by Anderson, vital to theories of transitions driven by topological defects Yuval and Hamann [6, 8] who treated the 1/r2 Ising chain by during a visit to Bell Labs, one of the most important cen- a renormalization group method. Because this very early ver- ters of theoretical physics in the twentieth century. There he sion of a renormalization group treatment of a defect driven learned of the work by Anderson and colleagues on the trans- phase transition was so important to later developments by formation of the Kondo problem into a 1D Ising model with Thouless and myself [18 20], I review it here. 2 – 1/r interactions [4–8] and was asked if he knew anything The first step is to rewrite the partition function Z(K, y, h) about this intermediate case. It was known that the model of the 1D Ising model with Hamiltonian [6, 8], assuming peri- with interactions falling off more slowly than this have long odic boundary conditions range order [3], or finite magnetization, at low temperatures 2 and that systems with shorter ranged interactions are dis H 1 ssij− 1 KT () Assh2 s ordered [2] at all T > 0. = 0()∑∑2 +−()ii+1 − ∑ i kTB 2 ij> ()ij− 2 i i To answer this challenge, Thouless constructed an argu- (5) ment [1] based on that of Peierls [12, 13] and Landau and Lifshitz [14]. In it, one considers a system of length L with a in terms of a set of 2n alternating domain walls at r12, r n single domain wall between spins with s = +1 and spins with where ∆s =±2 across a wall. The partition function becomes s = −1. The entropy is just [6, 8]

∞ 2n rai+1− Sk=+Bln()La/1O() (1) dri Zy= 2n ∑∏∫ra+ and the energy of this isolated domain wall at x is n=0 i=1 i−1 a

xa− /2 L ⎡ ⎛ |−rr| ⎞ ⎤ ddyy12 ij ri EJ= dx ≈+JLln /1a O ×expl⎢Kq0 ∑∑ijq n,⎜ ⎟ − hqi ⎥ (6) ∫∫2 2 () () (2) a a 0 xa+ /2 ()yy12− ⎣⎢ ij> ⎝ ⎠ i ⎦⎥

2 Rep. Prog. Phys. 79 (2016) 026001 Review

i J A where qi =−(), KT0()= and y0 =−exp1−+KT0()()γ , kTB ()kTB with γ = Euler’s constant, is the domain wall fugacity, which is a measure of the defect density. As will become apparent, the value of the fugacity y0 affects only the value of the trans ition temperature Tc but not its nature. Note that A, the nearest neighbor coupling constant, affects only the the domain wall fugacity y and not the nature of the transition, provided A > 0. The idea is to proceed perturbatively [4–6, 8] and an expansion parameter is needed. A system of very dilute domain walls is amenable to an expansion in powers of the fugacity y0 which can be made small by choosing A 1 and this expansion is the basis of the renormalization group procedure. This −µ/kTB is a somewhat unusual expansion as y0 ∼ e is very singular at T = 0 so, although this expansion is most reasonable at low T, it is not a standard low temperature expansion. The partition function ZK()0,,yh0 defined by (6) is not analytically accessible but information can be obtained by a renormalization group procedure [5, 6, 8]. This consists of doing a partial trace by integrating over the domain wall posi- Figure 1. The critical temperature Tc as a function of y0. The δl solid line from (8) with C = 0 and the dashed line from (9) are tions in the interval ar<−ii+1 ra< e and rescaling the short indistinguishable for y0 < 0.3. distance cutoff or lattice spacing a to obtain a new partition function describing fluctuations on length scales larger than which is the dashed line in figure 1. The solid line which is l al()= ae . The rescaled Hamiltonian has the same form as the essentially indistinguishable from the dashed line is Tc from original Hamiltonian with cutoff a(l) provided the interaction (8) with C = 0. parameters K0,,yh0 are replaced by renormalized parameters The flows are plotted in figure 2 and the arrows denote K(l), y(l), h(l) which are solutions of the renormalization group the direction of increasing l. C()Ky0, 0 is a constant of inte- (RG) flow equations for K(l), y(l), h(l) and the free energy per gration which depends on K0B= Jk/()T and the fugacity y0, unit length f(l), [6, 8, 21] defined below equation (6). Since the flow line separating regions I and II flows into the pointK ()∞=1, y()∞=0, dK 24 =−4,Ky + O()y all systems with initial parameters K0, y0 lying on this sepa- dl ratrix can be interpreted as being at the critical temperature dy 3 −1 =−()Ky−+1,O()y kBcTy()0 /JK= c ()y0 . For TT=+c()1 t , C = −t(4y0 − t) to dl lowest order in t and y0. Systems in region I flow to some point dh 24 =−()12yh+ O()y , on the fixed line atK ()∞∞⩾(1, y ) = 0 which corresponds to dl a completely ordered system, while a system in region II or III df 242 flows to one where y(l) increases to O 1 . This equivalent sys- =+fy+ O()yh,. (7) () dl tem has a large density of domain walls which is interpreted as complete disorder or TT> . The RG flows are sketched in It was a major revelation in 1971 when I understood that, c figure 2 and the transition temperature as a function of y in by integrating these flow equations and by using a bit of physi- 0 figure 1 from (8) with C = 0. cal intuition, it was possible to extract the behavior of some One of the big surprises was that the zero field magnetiza- physically relevant quantities such as the magnetization, sus- tion m has a finite value asTT − so that there is a discon- ceptibility, specific heat and the correlation length. This was → c tinuity in mT()=−mT()cc()1 bT− T where b is a positive fortunate as calculating the partition function ZK()0,,yh0 was constant [8]. It is interesting to note that this discontinuity in well beyond my ability, but Anderson’s scaling ideas provided a more feasible method to obtain information. The flow equa- the spontaneous magnetization was first predicted by Thouless tions of (7) can be derived with some difficulty but within a [1] and derived from the RG flow equations (7) [8]. It was finite time and integrating these gives finally proved rigorously [16] that the model has a finite spontaneous magnetization at low T and later that the magnetiza- 2 2 tion m was discontinuous at the transition [17]. Kl()−−ln[(Kl)] 2lyl()=−KK00n2[]−=yC0 1/+ 2 (8) Since the form of the RG flow equations (7) turned out to where the integration constant C is also a renormalization be very similar to those of the 2D planar rotor model [20], group invariant. I rederive some of the important results of Anderson and Yuval The transition temperature Tc is determined by equation (8) [8] in more modern language. One of the most remarkable with C = 0 which gives features of the Ising ferromagnet of (5) is that the magnetization jumps discontinuously to zero at Tc. This follows from the kTBc 8 2 3 solution to the RG equations. Linearize (7) about K = 1 by =−12yy0 ++0 O()y0 (9) J 3 writing K = 1 + x so that to O()x , 3 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 2. Flows of the RG equations, (7) in regions I, II, III. The flows with increasing l are in the direction of the arrows. Figure 3. Flows from the linearized RG equations of (10). dx 242 =−4,yy+ O()xy , dl to TT> c()y . In region I when C()t ⩾ 0 and xy0 > 2 0, the solu- dy 32 tions to (10) are =−xy + O()yx,,y dl 1ea −2 Cl 22 TT− c + ⇒=Ct() xl()−=4wyl() here t . (10) xl()= C , T 1e− a −2 Cl c − Cl a e xC0 − 2 2 yl = C ,where a = . (12) Here C()tx=−0 42yx=−()0 yx0 ()0 + 2y0 , the small (xy0, 0) () −2 Cl 0 1e− a xC0 + expansion of C of equation (8), is a constant of integration which depends on the deviation t from the critical temper From this, yl()=∞ = 0 so that the scaled system at l =∞ ature Tc. The RG flows are a set of hyperbolae in the x, y has no defects which implies that, when TT⩽ c, one can define plane in figure 3. The transition is at x = y = 0 and the flows l∗ ∗ a correlation length ξ−()T = e where lC= 1/()2 , so that are towards the line xy()∞∞⩾(0, ) = 0 when C()t ⩾ 0 ⎛ ⎞ and xy0 ⩾ 2 0, which defines region I. When C(t) < 0, y(l) 1/ 2 C 1 T eexp⎜ ⎟ becomes relevant and flows to yl()= O()1 , defining region ξ−()==() ⎜ ⎟ ⎝ 24||ty()0+|t| ⎠ II. Finally, in region III where C()t ⩾ 0 and xy0 ⩽ − 2 0, the fixed line y = 0 is unstable and y(l) is relevant, increasing ⎧ ⎛ 1 ⎞ with l. In particular, one identifies the transition as C = 0 ⎪exp⎜ ⎟,4yt || 0, (13) ⎪ ⎜ ⎟ 0 ⩾ when x(l) 2y(l) and all points x(l) 2y(l) as TT since ⎪ 4 yt0|| = > < c = ⎨ ⎝ ⎠ the domain wall fugacity flows to the y = 0 line which corre- ⎪ ⎛ 1 ⎞ sponds to a completely ordered system with no spin flips. The ⎪ exp⎜ ⎟,1 ||ty4.0 ⎪ 2 t high temperature phase is identified with the region of the x, ⎩ ⎝ ||⎠ y plane which flows to the region where yl()= O()1 . This is Thus, the true critical region is t <=ty∗ 4 where one may interpreted as a system with a high concentration of domain 0 expect to see ξ T diverging as e1/()4 yt0|| but this region is walls corresponding to TT> c with spin disorder. Thus, one −() extremely small since the crossover temperature is propor- can identify xy0 −=2 0 −t so that −µ/()kTB tional to the defect fugacity y0 ∼ e which is exponen- Ct()=−ty()4.0 − t (11) tially small. Above this temperature but still in the region ||t 1, a gradual crossover will occur to the e1/()2||t behavior. The unstable separatrix C ==02xy0 + 0 corresponding This same phenomenon will be found in the 2D planar rotor to t = 4y0 or TT c is just part of the disordered phase, see model and related systems. The correlation length ξ−()T can figure 3. be interpreted as the mean separation between spin flips of The solutions to the flow equations (10) fall into three distinct opposite sign, or as the size of domains of reversed spins − regions of the x(l), y(l) plane: region I, C()tx⩾(0, ly)⩾2 ()l , which diverges as TT→ c . region II with C()ty<>0, 2 ()lx||()l and region III where In this Ising ferromagnet with 1/r2 interactions, one can use C ><00xl() with ||xl() ⩾(2yl). As will become apparent, the RG to show that the magnetization m(T ) is discontinuous region I corresponds to TT⩽(c y), regions II and III correspond at Tc [5]. From equation (7),

4 Rep. Prog. Phys. 79 (2016) 026001 Review

dm 24 =+2ym O()y dl

d 222 ⇒=()Km 00⇒=Km() ()0.Kl()ml() (14) dl This shows that K(l)m2(l) is a renormalization group invariant, 2 at least to O()y , so that one can obtain the value of the physical quantity K(0)m2(0) by a judicious choice of the length scale l. When TT⩽ c, y()∞=0 so that the effective system at this scale contains no defects which implies that mm()∞= sat = 1 and K()∞=11+=Ct+||+()4yt0 || so that

22 KT0()mT()=∞Km()()∞=14+|ty|+()0 ||tt,1|| ,

⎧ −1/2 ⎪KT0 ()c 14+|ty|+()0 ||tt/2,1− ⩽ 0, mT()= ⎨ ⎪ () ⎩ 00t > . (15) Thus, the zero field magnetization m(T ) has a discontinu-

−1/2 kTBc ous jump to mT>=Tc 0 of size K Tc = , as () 0 () J first argued by Thouless [1], derived by the RG method

[6, 8] and proved rigorously by Aizenman et al [17]. As men- 2 2 Figure 4. A plot of m as a function of T for various values of the tioned above, the combination K0m plays the role played by fugacity y . The straight dotted line is mm2 2 T− from (15). the renormalized stiffness constant in the planar rotor model 0 c = ()c 4 or in He in 2D as in figure 4. interactions. In the low temperature region I, ξ−()T may To this point, the discussion has been limited to region be interpreted as the mean separation of a pair of opposite I where yl()→ 0 as l → ∞ so that TT⩽ c. When TT> c domain walls or the size of a domain of reversed spins. In but t ⩽ 4y0, the system is in region II. As the length scale l region II, the fugacity y(l) decreases from its initial value y0 increases, the fugacity decreases but, at larger l, y(l) begins to some minimum but increases for larger l to some value to increase and continues to increase as l does. This is readily of O()1 which is outside the domain of validity of the renor- seen from the linearized flow equations of (10) when C < 0 malization group equations (7) and (10). Thus, choose l* in 22 ∗ ∗ and 4yx−=||C > 0, so that (17) so that xl()=−x0 and yl()==yx0 0 + O()C so that −−1 1 ∗ tan xC0/t||=−an xl //||C = π 2 and ξ+()T is dy 4 () (( )) =−xy + O()y , interpreted as the typical size of an ordered domain of spins dl when TT> c. dx 24 2 =−4,yy+=O() −+()xC|| dl The singular part of the free energy fs is obtained from the flow equations xl() ⎛ ⎛ ⎞ ⎛ ⎞⎞ d1x −−1 x0 1 xl() l =− = ⎜tant⎜ ⎟ − an ⎜ ⎟⎟, ∗ 2 ⎜ ⎟ ⎜ ⎟ l ∫x0 xC+| | ||C ⎜ ||C ||C ⎟ ∗−l∗ −l 2 ⎝ ⎝ ⎠ ⎝ ⎠⎠ fl()e0−=fl() ∫ de yl(). (18) ⎛ ⎞ 0 x0 ⎜ −1 ⎟ (16) * * xl()=|CC||cotc⎜ |+l ot ⎟. The trick is to choose a value of l such that the integral and f(l ) ⎝ ||C ⎠ ∗ are easily obtained and a suitable choice is lT= lnξ±(). Thus This is valid for any l, provided only that the RG equations (10) the the singular part of the physical free energy fs = f(l = 0) remain valid which requires that the renormalized fugacity is given by * ∗ y(l) < 1. Thus, a suitable choice of l is l where xl()=−x0, l∗ ∗ ∗ −1 ∗ −∗l −−ll2 yl = y with xC || and yC ||, so that ffs =−ed()lleeyl()∼∼ξ± ()T . (19) () 0 0 0 ∫0 2 x ππ l∗−= tan 1 0 = = . This result was first obtained by an RG procedure [8]. This ||C ||CC|| ty()4 0 − t means that the specific heat ch(t) has the same essential singu- ∗ larity at Tc as ξ T which is undetectable since all derivatives lyπ/40t ±() ξ+()tt==ee ,0< 4.y0 (17) of the specific heat are zero atT c [8].

When t > 4y0, the RG equations (10) must be used with C()ty,400=−tt()y > 0 in region III. Most of this region cor- 2. Two dimensional superfluids and crystals reponds to systems with TT c which is expected to display no critical behavior. While I was still trying to understand Anderson’s scaling papers, These results for the correlation lengths ξ±()t have a David Thouless drew my attention to some other problems physical interpretation for this 1D Ising model with 1/r2 which seemed to be closely related to the 1/r2 ferromagnetic

5 Rep. Prog. Phys. 79 (2016) 026001 Review Ising model, namely, the planar rotor model in 2D, films of + discontinuously at T c . This theory would also have to accom- 4 Helium and melting of 2D crystals. These systems provided a modate that the specific heat peak near the transition is finite major puzzle because of the argument of Peierls [12, 13] which and smooth at Tc [43] and the the low T susceptibility of the excludes long range translational order in 2D. The mean square 2D planar rotor model is infinite [44] with zero magnetiza- 2 deviation ⟨uR()⟩ of particles from their equilibrium positions tion. Even before we began to think about the problem, it was 2 R diverges with the system size L, ⟨uR()⟩ ∼ lnL. This means clear that the conventional wisdom about critical phenomena that there is no translational order in 2D. Using similar argu- was not very useful and some out of the box thought was ments, Mermin, Wagner and Wegner [22–24] proved rigor- needed. This was exactly the style of problem which suited us: ously that there is no long range order in the 2D magnets with Thouless liked strange and unusual challenges and Kosterlitz a continuous symmetry and short range interactions and there was so ignorant of statistical mechanics and phase transitions is no long range translational order in a 2D crystal. Hohenberg that this problem was no different from any other so ideas of [25] proved that there is no Bose–Einstein condensation in 2D topological defects seemed difficult but normal. This was one which apparently excludes superfluidity in 2D. At the time, the of the rare situations when ignorance of conventional wisdom generally accepted conclusion was that there is no transition to was a strength rather than a weakness. a more ordered state at any finite temperature in a 2D system with a continuous symmetry and short range interactions. This excludes a low temperature ordered state in the 2D Heisenberg 2.1. Topological order 4 and XY magnets, in a thin film of He or nematic liquid crystal For both an equilibrium film of 4He and a 2D planar rotor and in a 2D crystal, despite it being carefully pointed out that magnet, the Hamiltonian is the theorem only excludes long range order in 2D but not a transition to a distinct low temperature phase without true long H KT d2r =∇0() θ r 2 range order [22]. ∫ 2 (()) (20) kTB 2 a0 In the early 1970’s the situation regarding transitions in these 2D systems was somewhat confused from both theor where a0 is the lattice spacing and etical and experimental points of view. Numerical work on ⎧ J a system of hard disks in two dimensions [26–31] indicated ⎪ planar rotor, a transition between solid and fluid states which seemed to ⎪ kTB KT0()= ⎨ 2 0 contradict the rigorous result [24] which forbids true long ρ ()T ⎪ s superfluidfilm. ⎪ 2 range order in a 2D crystal. The natural conclusion from this ⎩ mkBT is that there is no transition from a crystal to a fluid in 2D which conflicts with the numerical study. A similar conflict Here, J is the exchange interaction between nearest neigh- arose between the rigorous result that long range order is for- bor unit length spins, sr()= ((cos,θθrr)(sin )), H[{s}] = bidden in a 2D magnetic system with spins s = ss,, , s J sr −=sr′′2 J 1c−−os rr. At ()12 n 2 ∑<>rr, ′′[()()] ∑<>rr, [(θθ() ( ))] where n ⩾ 2 interacting with short range interactions and low T, adjacent spins will be almost parallel so that cos()δθ ≈ series expansion results [32–34] which indicated that a phase 1/− ()δθ 2 giving (20). For a 4He film, transition does occur. The evidence for a transition was rather 1 d2r inconclusive but was stronger for the 2D n = 2 planar rotor HT= ρ0 vr2, s()∫ 2 s() (21) model than for the n = 3 Heisenberg model [34]. 2 a0 Some experimental work on 4Helium films indicated that where v r is the local superfluid velocity relative to the sub- there is a transition to a superfluid atT 0 with a discontinu- s() c > strate. Since vs = ∇θ where θ is the phase of the condensate ous jump to zero of the superfluid density [35–41] which was m i r interpreted as a first order transition [42], despite the lack of wave function ψ()rr=|ψ()|e θ(). Thus, one argues that (20) is a specific heat discontinuity [43]. The most convincing pieces a good description of a planar rotor ferromagnet and of a thin of experimental evidence for a transition to a superfluid were 2D 4He film and this form is extensively used. experiments by Kagiwada [35] where the velocity of third However, it is important to remember that this Hamiltonian sound, c3 ∝ ρs was measured as shown in figure 5 (right) and is periodic under θ()rr→(θπ) + 2 , which allows vortex exci- and by Chester [36, 37] where the change in the resonant fre- tations. Although these have large energy and are therefore quency of a quartz crystal due to the superfluid fraction of an improbable, vortices are the only excitations capable of dis adsorbed layer of 4He was studied as shown in figure 6. When ordering the system or, equivalently, of destroying a uniform we began to think about the transition in 2D superfluid films, superfluid flow. The drift of vortices across the flow to the the experimental situation could be summarized by figures 5 edges of the system unwinds the phase difference of the super- and 6. From these, we concluded that the areal superfluid den- fluid order parameterψ = eiθ between the ends of the system. sity drops very rapidly and even might vanish discontinuously This simple observation implies that these topological excita- at Tc in 2D. tions must be taken seriously and cannot be ignored because Thouless and I were faced with the task of construct- the are very improbable. Smooth fluctuations in the phase ing a plausible theory of superfluidity which would explain are the most probable excitations since these have arbitrar- − 4 why the superfluid density was finite atT c and vanished ily small energy and are third sound modes in He films and

6 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 5. Left: Temperature dependence of areal superfluid density for films of several coverages. The vertical axis is the angular momentum at the same angular velocity of the film. Reprinted from [41] with permission. Copyright 1972 American Physical Society. Right: Third sound velocity c3 ∝ σs against P0 − P, which is a measure of the chemical potential μ for several fixed temperatures. The third sound signal vanishes − at the hatched vertical lines indicating that σs()T c > 0. Reprinted from [35] with permission. Copyright 1973 American Physical Society.

Figure 6. Left: The horizontal axis measures the area mass density of the 4He film and the vertical axis is− ∆f , the reduction of the resonant frequency from the same film in the normal state. The deviation from the upper straight line is a measure of the superfluid mass density which decouples from the oscillating substrate. Reprinted from [36] with permission. Copyright 1969 American Physical Society. −8 Right: The horizontal axis measures the total mass density σ adsorbed and the vertical axis the superfluid mass densityσ s in units of 10 gm cm−2. Reprinted from [37] with permission. Copyright 1974 American Chemical Society.

7 Rep. Prog. Phys. 79 (2016) 026001 Review

spin waves in magnets but have no effect on the rigidity of Here, μ and λ are Lamé coefficients and the strain tensor the 2D system. These cannot have anything to do with the 1 ∂ui ∂uj uij =+. Ignoring the periodicity under uu→ + a, phase transition. Very similar but technically more complex 2 ()∂rj ∂ri arguments hold for a 2D crystal in which phonons are the low where a is a primitive lattice vector, one obtains the Debye energy excitations which do not affect the transition to a fluid Waller factor which is a measure of translational order. iiGr⋅⋅Gu()r while the high energy topological excitations, dislocations, Defining the order parameterρ G()r ==ee with G a are responsible for the melting transition in 2D. reciprocal lattice vector, one has

Note that the Hamiltonian of (20) can be obtained from ∗ Crrr=∼ρρ0 −ηG()T a Ginsburg-Landau free energy density functional F []ψ [45] G() 〈(G )(G )〉 2 kTB G ()3µλ+ 1 22rT() u 4 ηG()T = (27) F [(ψψrr)] =|∇ ()|+ ||ψψ+||+ (22) 42πµ µλ+ 224 () Since the system is assumed to be well below the mean field This means that the long range translational order in the ground state is destroyed by the low energy phonon excita- transition temperature, one take r()T 0 and u 0. The tions at any T > 0 in a 2D crystal, in complete analogy to spin fluctuations in the amplitude|| ψ()r are negligible and can waves in the 2D planar rotor ferromagnet. be ignored, so that ||ψ 2 =|rT|/u which is a constant O 1 . () () The planar rotor and related models are special cases in Thus, in this phase only approximation, the Hamiltonian (20) which topological defects play a vital role [18, 19]. Minimizing results. H of (20), one obtains Ignoring effects of the 2π periodicity, at low T the obvi- 2 ous approximation is to regard the system as Gaussian with ∇=θ()r 0. (28) iθ −∞<θ()r <+∞ so that, with ssxy+=ie= ψ Because of the periodicity under θ()rr→(θπ) + 2 n, the solu- −1/2πKT0() tion to (28) obeys i rr′ ⎛ |−rr′| ⎞ 〈(ψψrr)(∗−′′)〉 =⋅〈(sr)(sr)〉 =∼〈〉e.((θθ)()) ⎜ ⎟ ⎝ a0 ⎠ (23) d2θπ==nn,0,1±±,2, (29) This result means that, as expected, there is no true long range ∮C order at any finite T > 0 in the planar rotor magnet [23, 46, where C encloses the vortex core of circulation n. It is easily 47], no Bose Einstein condensation with true long range phase seen that 4 coherence in 2D He films [48–50] and no long range order in 2D crystals [44, 51]. Also, the power law decay indicates that θφ()rr=+() Θ=()rrφ()+Θ∑ n()Rr(),,R D = 2 is the lower critical dimension. R

The question of the lower critical dimension in models −1 Yy− Θ=()rR,tan ,.nn()RR=−∑ iδ()Ri (30) with a continuous symmetry with a Hamiltonian of the form Xx− i

H Here, Θ()rR, is the phase at r = ()xy, due to a unit strength =−Ks∑ sr()⋅=sr()′ ,,ss()12ss,, n ,1||= , kTB <>r,r′ vortex at R = ()XY, on the dual lattice [19, 55, 56] corresp (24) onding to a local energy minimum, ni is the quantized circula- was finally settled by Polyakov [52] by a renormalization tion of the vortex at Ri and φ is the smooth deviation from the group method. The RG recursion relation in dimension d for local minimum configuration, corresponding to spin waves in 4 the temperature variable K−1 is a magnet and to third sound in a He film. After some tedious but straghtforward algebra, one obtains an expression for the −1 dK 12n − 2 23 energy of a configuration of phase angles in the presence of an =−()dK−+2 −−Kd+−O(( 2,))K− . dl 2π arbitrary set of vortices n()R [19], (25) This yields a critical fixed point at1/ Kd∗ 22/2n H Hsw Hv Hint =−π()()− =++ . which vanishes as d → 2. In exactly d = 2, the effective kTB kTBBkT kTB temperature K−1(l) increases with l implying that the system is Hsw 1 22 in a disordered phase at any finite K−1 > 0. On the other hand, = KT0()∫ d,rr((∇φ )) kTB 2 in exactly 2D, the n = 2 model is marginal with d/Kl−1 d0= −1 Hv 22 |−RR′| to all orders in K [53]. The correlation function of (23) =−πKT0 ddRR′′nnRRln ()∫|−RR′|>a ()() agrees with this result and is exact within the gaussian or spin kTB a 2 wave approximation. 22 2 L −+ln yn0 ∫ ddRR() πKT0() RRn() ln , The same considerations apply to a 2D crystal whose ()∫ a ground state is generally a triangular lattice with sites denoted 222 Hint =∇∫∫ddrrφφ()⋅∇Θ=()rr−∇()rrΘ=() 0. by r and deviations from these by ur() so that the elastic energy can be written in continuum notation as [54] (31) 1 Note that the spin wave and vortex contributions to H decou- 2 22 2 Hu=+d2rr[(µλij )(ukk r)]. (26) ple because ∇ Θ=r 0. 2 ∫ ()

8 Rep. Prog. Phys. 79 (2016) 026001 Review This decoupling of the spin wave and vortex degrees of one containing only two sets of excitations which are the low freedom is realized explicitly in the Villain model on a dis- energy spin waves and the disordering high energy vortices crete lattice [55, 56] where the decoupling of the spin wave [18–20]. All others are neglected and it is assumed that this and the vortex parts is made explicit. The partition function is description is adequate at low T and near Tc since the transition

2 is driven by the vortex excitations. π dθ r ⎛ ⎞ ZV=−()exp,⎜ θθrr′ ⎟ The first attempt [18] to solve the transition problem ∏∑∫0 ⎜ (()())⎟ rr2π ⎝<>,r′ ⎠ in the 2D planar rotor model was based on the free energy +∞ 1 of a single isolated vortex of unit circulation. From (31), V θ −−Kmθπ2 2 ee()= ∑ 2 (), m=−∞ ∆EL()/l()kTB0=+πKT()n/()La0 O()1 . The entropy is 2 2 ∆SL()/lkLB =+n/()a0 O()1 so that the free energy is ∆FL()/ mm()r, rr′′=+˜ (),r mm()rr−=()′′;,∑ mn˜ ()r, rR() <>r,rR′ ∈ ()kTBB=∆((EL)(−∆SL))/2()kT =−((πKT00))ln()La/1+ O(). 2 π dθ r Thus, in the thermodynamic limit La/ 0 → ∞, the vortex free ZK, y = ∏∑() e,A[]θ,m ()0 ∫0 energy ∆FL −∞ when πKT0 > 2 and ∆FL +∞ rr2π m(),r′ ()→ () ()→ so that the probability of an isolated vortex being present in K 2 2 A[]θθ, m =− ∑∑[(rr)(−−θπ′′)(2lmyr, rR)] + n 0 n () equilibrium is zero. On the other hand, when πKT0()< 2, 2 <>r,r′ R (32) ∆FL()→ −∞ so that the probability of an isolated vortex in where the integer variable m()r, r′ lies on the bond equilibrium is unity. Of course, there will be a non zero den- between nearest neighbor sites r and r′. One can write sity of finite clusters of vortices of zero total vorticity at all mm()r, rr′′=+˜ (),r mm()rr− ()′ , by making a discrete temperatures since the free energy of a finite size zero vor- gauge transformation, and then define the site variable as ticity cluster is finite. However, the transition temperatureT c φ()rr=+θπ() 2 m()r . This form of the interaction energy is determined by the renormalized stiffness KR()T such that V()θ is periodic under θ()rr→(θπ)(+ 2 n r) and approximates πKTRc()= 2, as discussed in the following section. K0()1c− osθ [55]. The difference between the Villain form David and I congratulated ourselves on finding impor- and (31) is quantitative but not qualitative and, invoking the tant new physics but our euphoria soon dissipated. We were concept of universality [57], one expects all models with informed that Berezinskii [47] had discussed the the vortex 2π periodicity and short range interactions to have identical driven transition in a superfluid film a year earlier than our behavior in the critical region. paper [18, 19]. Since neither of us knew any Russian we were One is now faced with computing the grand canonical par- blissfully unaware of this work while we were developing tition function ZK,eyHxp /kT. Here, L is the basic physics of the vortex driven transition. For some ()0 0 = ∑{(n R)} ()− B the linear size of the system and a is a short distance cutoff unknown reason, our work seems to have had much greater which is the lattice spacing in a magnet and the vortex core impact than that of Berezinskii. size in a superfluid so that La/1 . The last term in Hv of The careful reader will have realized that the discussion is 2 rather sloppy and the relation between the integer fieldn R equation (31) requires that d0RRn()= to ensure that the () ∫ and the vorticity is not obvious. Also, the relation between energy is finite in the thermodynamic limit L → ∞. The resulting expression is the energy of a neutral Coulomb plasma the topological charges q()R ==1/()2dπθ∮ 0, ±1 and the [18–20] since the interaction between charged particles in effect of boundary conditions are unclear. These issues 2D is qqijln()raij/ and the problem is reduced to the statistical have been addressed in [58] where the relation between the mechanics of the Coulomb plasma. topological charge q()R and the integer variables n()R is clari- The parameter y0 in (31) is the fugacity of a vortex since fied. This issue was first raised by Savit [59]. The effects of the probability of a vortex of circulation 2πn is proportional boundary conditions in the Coulomb gas language is also dis- n2 cussed in [58] which is important for the Coulomb gas at very to y . To proceed, assume the vortex fugacity y 1 so that 0 0 low temperature and for finding ground state configurations the system has a very small concentration of vortices. This [60 62]. 2 – implies that one might try a perturbation expansion to O()y0 −EkcB/ T where y0 = e . The vortex core energy Ec is the interaction energy in the region near the vortex center where the 2 2.2. Renormalization group for 2D Coulomb gas quadratic approximation (∇θ) is inadequate because ∇θ is large. This also defines the vortex core size a0 which is a few In this section, I discuss the planar rotor magnet described by lattice spacings in a magnet and a few interparticle spacings equation (31) at low T and fugacity y0 1. The first attempt in a superfluid. Near the center of a vortex,vr s ∼ 1/ which with David Thouless considered a neutral set of charges diverges as r → 0, which implies that the fluid is normal at the n()R =±1 interacting by a 2D Coulomb potential (31). Since vortex core. Whatever the microscopic structure of a vortex the fugacity y0 1, charges with larger n are ignored because core, there is a large energy density in the core region which they are less probable. A neutral pair of unit charges separated 2 is not described by (∇θ) and one assumes the contribution to by r interacting by a lnr Coulomb interaction will be screened the total energy from this region can be simply described by by the polarization of smaller pairs separated by less than r a core energy EcB kT. Furthermore, assume that the fugac- which are screened by even smaller pairs. This leads to a scale ity y0 1 and is a temperature independent constant. The dependent dielectric constant ε ()rK= 0/Kr() where the force description of the system by equation (31) is a very simplistic between the charges is 2πKr()/r. The energy of this pair is 9 Rep. Prog. Phys. 79 (2016) 026001 Review

r Kr()′ ⎛ r ⎞ The integrations over the vortex positions Ri are restricted by Er()=≡2dππr′ 2lfr()n,⎜⎟ ∫a r′ ⎝ a ⎠ the hard core repulsion |−RRij|>a where the short distance cut-off length scale a can be taken to be a lattice spacing or r dr′′⎛ r ⎞32− πfr()′ ⇒=Kr−1 Ky−1 + 4π3 2 . the diameter of a vortex core. It turns out that this is quite () 0 0 ∫ ⎜ ⎟ (33) a a ⎝ a ⎠ arbitrary, but it must be kept finite until the end. The infrared We were unable to solve this integral equation so we made cut-off is the system size L must also be kept finite during an unfortunate unnecessary approximation [19] by replacing the calculations to avoid unphysical divergences. It is of some fr()′ by f(r) in (33). This was justified by fr()′ − fr() 1, but interest to note that both cut-offs, L and a, have clear physical led to incorrect results. meanings so there is no temptation to take the limit L → ∞ In an important paper [63], Young demonstrated that our or a → 0 unless it is obvious that these limits make physical incorrect approximation was unnecessary and presented the sense. Actually, these appear in the combination L/a and the correct procedure by defining a scale dependent fugacity real physical limit is La/ → ∞. The original derivation of the RG flow equations was done 2−πfr ⎛ r ⎞ () ⎛ r Kr()′ ⎞ by an extremely involved method [20] based on the method yr() ==y0⎜⎟ yr0exp2⎜⎟ln()/dar− π ′ , ⎝ a ⎠ ⎝ ∫a r′ ⎠ pioneered by Anderson and coworkers [6]. This involves inte- r grating out the short distance degress of freedom in the par- −1 −1 32dr′ Kr()=+K0 4π yr()′ . (34) ZKy ∫a r′ tition function c0(), 0 of equation (36) and rescaling the l cut-off aa→ e resulting in the RG flow equations Differentiating these with respect to lr= ln yields −1 dK 32 4 =+4,π yyO() dyl() dl =−((2,πKl))yl() dl dy 2,Ky y3 −1 =−()π + O() dKl() 32 dl = 4.π yl() (35) dl df 24 =−22fyπ + O()y , (37) These are the recursion relations derived earlier [20] by a dl l renormalization group method. However, even if Thouless where the lattice spacing at length scale l is al()= ae . The and I had not made our incorrect approximation in our 1973 interaction is K(l) and the effective fugacity is y(l). The reduced paper, but obtained the correct results of (35), we would not free energy density fK()0, y0 and specific heatc ()Ky0, 0 are have known what to do with them as we knew nothing about F lnZK()0, y0 scale dependent coupling constants or renormalization groups fK()0, y0 ≡− = , at that time. kTB N

I was not happy with the self consistent results in [19], and ∂2f cK,.yk= K2 (38) realized that the scaling methods of Anderson and coworkers ()0 0 B 0 2 ∂K0 [6, 8] for the 1D Ising model might be applicable to the vortex problem of equation (31). The thinking was that in both the They can be integrated to give the renormalization group 1D Ising ferromagnet in the previous section and the 2D pla- invariant nar rotor model, the interactions between the important topo- ⎛ πKl()⎞ 2 22 Ct(), y0 logical point defects are logarithmic and the transitions might ln⎜⎟+−21π yl()=+ ⎝ 2 ⎠ πKl 8 be very similar. Of course, in the early 1970 s, the importance () ’ 32 2 of the spatial dimension and the range of the interactions [57] Ct(),8yt00=−()πyt−+O()tt,,yt0 y0 . (39) was not known to me so, motivated by Anderson’s solution The RG flows are shown in figure 7 for various values of C. of the Kondo problem [6, 8] the renormalization group equa- The critical line Kc()y is the separatrix between regions I and tions for the 2D Coulomb gas were derived [20] by working II in figure 7 which is given by equation (39) with C(t, y0) = 0. with the partition function ZK()0, y0 . This decouples into a Since y 1, product of a gaussian spin wave and a Coulomb gas partition functions, 16 22 3 ππKyc =+24++π yyO() (40) 3 ZZ= sw()KZ0c()Ky0,,0 and Kc()y from equations (39) and (40) is shown in figure 8 ∞ 2n 2n 2 ⎡ ⎤ y0 d Ri ⎛ |−|RRij⎞ These recursion relations were later derived by a techni- Zc =+expl⎢ πKq0 q n,⎥ ∑∏2 ∫|−RR|>a 2 ∑ ij ⎜ ⎟ n=0 na! ij i=≠1 ⎣⎢ ij ⎝ a ⎠⎦⎥ cally much simpler but more sophisticated method [56, 64] which starts with the Fourier transformed superfluid momen- ⎛ 1 22⎞ ZKsw =−Dφφexp⎜⎟0 d.rr((∇ )) (36) s ∫∫⎝ 2 ⎠ tum density g ()q correlation function ss ZKsw()0 is a trivial Gaussian partition function. In the CKαβ,0()qq,,yg0 =−⟨(αβ)(gAq)⟩ = ()q Coulomb gas partition function ZK, y , there are n q = +1 c0()0 qq ⎛ qq ⎞ charges and n q = −1 charges. These correspond to vortices αβ+−Bq δ αβ . 22()⎜ αβ ⎟ (41) of ±1 circulation which interact by a 2D Coulomb interaction. q ⎝ q ⎠

10 Rep. Prog. Phys. 79 (2016) 026001 Review Consider the renormalized stiffness constant proportional to R the renormalized or measured superfluid densityρ s ()T , which can be written as a correlation function [65]

2 R ρs ()T KTR()==2 lim((Aq)(− Bq)) mkBT q→0

2 2 〈(nnqq)(− )〉 =−KK0 4π 0 lim 2 q→0 q 2 2 22 =+KK0 π 0 ∫ d0rrrn〈()(n )〉. (42)

Now, the vorticity–vorticity correlation function in (42) can be calculated as a power series in the fugacity y0 to leading order to obtain

2 −2πK0 4 〈(nnr)(02)〉 =− yr0 + O()y0 ∞ 32− πK0 −−1 1 3 2 dr ⎛ r ⎞ 4 KKR =+0 4π y0 ⎜⎟ + O()y0 (43) ∫a a ⎝ a ⎠

Since the renormalized stiffness constant KR is independent Figure 7. RG flows for 2D planar rotor model from equation (39). l of the cut-off a, one can redefine the cut-offaa → e to obtain Arrows denote the direction of flow for increasing l. the renormalization group flows of equations (37) which is the same as the self consistent equation (33) which is correct 2 to O()y0 only. In principle, equation (43) can be calculated to any desired order. Since KR is independent of the cut-off, l by redefiningaa → e we can call K(l) and y(l) the effective coupling constat and fugacity at length scale l. This can be expressed as renormalization group equations (37) by standard methods. This derivation has several advantages over the self consistent one as it makes clear that the RG equations (35) and 2 (37) are valid to O()y only. Also, this derivation also has the advantage that it uses the perturbative expansion of KR()Ky, which yields the exact relation [64]

KKR0(),,yK0 = R((Kl)(yl)) (44) which is the Josephson relation [66] which, in d dimensions, is

()2−dl KKR0(),eyK0 = R((Kl)(,.yl)) (45) The first major physical prediction can be made from (37) − − Figure 8. Kc from equation (39) (solid line) and (40) (dashed line) and (44) at T c . From figure 8, one sees that K()T c is an irrelevant which are valid only to O y2 . variable which flows to the stable fixed pointK c()l =∞ = 2/π ()0 and y()∞=0. Thus, using (44), we have [64] accuracy. This is actually a remarkable result because this

2 R − theoretical prediction is a smoking gun prediction which is an ρs ()T c 2 2 ==KKR0(),,yK0 R((Ky∞∞)())==0 K()∞= , inescapable consequence of the theory. As an aside, note that mkBcT π the stiffness constant KR()T of (44) is sometimes called the ρR T− 2 s ()c 2mkB −−82−1 helicity modulus ϒ()T [74, 75] which is related to KR()T by ==2 3.491 × 10 gcmK . (46) Tc π 2 R ϒ=()Tm() /.ρs ()Tk= BRTK ()T (47) This prediction has been checked experimentally [67, 68] and The RG equations (37) have a fixed line y(l) = 0 which is the data from several different experiments [69 73] is pre- – stable for πKl 2 and unstable for πKl < 2 as shown in sented in figure 9. It is of interest to notice that the exper ()⩾ () figure 7. Thus, one interprets region I of figure 7 as the low imental data was obtained and plotted before the authors were temperature phase TT< , regions II and III as the disordered aware of the theoretical prediction of [64]. When the theor c TT> phase and the separatrix between regions I and II as the etical line was inserted, it fitted the plotted data to about 10% c critical line TT= c()y0 . All points in the K0()Ty, 0 plane on the

11 Rep. Prog. Phys. 79 (2016) 026001 Review separatrix and in region I flow to the stable portion of the fixed line y = 0, while all points in regions II and III flow to a region where the effective vortex fugacity y(l) is O()1 implying the system is disordered. To proceed further, it is convenient to concentrate on the region near yK==0, π 2 by definingπ Kl()=+2 xl() and writing the flow equations (37) to lowest non-trivial order as [20, 56, 64]

dx 22 24 =−16π yx+ O()yy, dl

dy 23 =−xy + O()xy, y (48) dl Integrating these leads to the renormalization group invariant

xl2216 yl2 Cy, t ()−=π () ()0 Figure 9. Results of third sound and torsional oscillator 2 − =−ty8,π −+ttO 32ty,.ty (49) experiments for superfluid density discontinuityρ s()T c as a function ()0 ()0 0 of temperature. Solid line is from equation (46) for the static theory. The flows are plotted in figure 10 which is nothing but a linear- Reprinted from [68] with permission. Copyright 1978 American ized version of figure 7. Note that the renormalization group Physical Society. equations and consequences are very similar to those for the 1D long range Ising magnet [6, 8]. The line y = 0 is identified as the spin wave phase since there are no free vortices when ⎛ 1 ⎞ T ee1/2 C xp⎜ ⎟ the fugacity y = 0. This phase has finite rigidity and can be ξ−()==⎜ ⎟ 28||ty()π 0+|t| identified as the low T quasi ordered phase. On the other hand, ⎝ ⎠ (52) when y > 0 there will be small density of free, mobile vortices ⎧ 1/ 28yt ⎪e8π 0|| ty, so that the rigidity vanishes at long length scales implying this = ⎨ ()|| π 0 ⎪ 1/()2||t is a high T disordered phase. ⎩ e8||ty π 0. The (x(l), y(l)) plane shown in figure 10 divides naturally The renormalized, or measured, stiffness constant for TT is into three separate regions; (I) C()ty,00 ⩾(,4xl)⩾ πyl()⩾ 0, ⩽ c (II) C()ty,00 <|,4xl()|< πyl(), and (III) C()ty,00 ><, xl() KT KKly,,lKKy 0 −4πyl(), which need separate solutions of equation (48). RR()==(()()) R((∞∞)())= In region (I) xl()⩾(40πyl) > and C()ty,00 ⩾ , so that 1 ⎛ ⎞ ==Kl()∞= ⎜⎟2 + C π ⎝ ⎠ dx =−16π22y , 1 ⎛ ⎞ dl =+⎜⎟28||ty()π 0 +|t| . (53) π ⎝ ⎠ dy 222 =−xy,1⇒−xl() 60π yl()=>C dl Equation (53) seems to give an experimentally measurable xl() R dx 2 1 ⎛ 11⎞ prediction for the superfluid densityρ ()T shown in figure 11 =−Cx⇒=l dx⎜ − ⎟ s dl 2 C ∫x0 ⎝ xC+ xC− ⎠ R 2 ρs ()T mkB −2lC =+28||ty()π 0 +|t| . (54) xC00++xC− e T π 2 () xl()= C (). (50) −2lC R xC00+−xC− e However, the superfluid densityρ s ()T is, unfortunately, not accessible experimentally as measurements on 4He films have () to performed with either, or both, the frequency ω ≠ 0 or the Now examine the behavior of x(l) in various limits by the superfluid velocityv s ≠ 0. Of course, because the system is of deviation from Tc small so that C()ty,10 and the initial finite size or because third sound waves are presentq ≠ 0, but value xC0 . From (50), this effect is not as important. The theoretical prediction here is the q ==ω vs = 0 component of the dynamical response ⎧ R x0 ⎛ ⎛ ⎞⎞ function ρ q,,ω v which, unfortunately, is inaccessible to ⎪ ⎜1f+O for2 C 1. KR()T should decrease linearly as T increases until about 10% ⎩ (()) below Tc but then vortex fluctuations will take over and in the This allows one to define a correlation length forTT ⩽ c by final approach toT c, the stiffness constant drops to the uni versal ∗ l * ∗ ξ−()T = e where l is defined by2 lC= 1, which gives value 2/π as ||t [64]. The theoretical form of ρs()T is shown in

12 Rep. Prog. Phys. 79 (2016) 026001 Review

The very reasonable assumption that ρs()t >=00 leads to R the universal jump in KR()T at Tc. One can show that ρs = 0 for TT> c by defining the generalized stiffness constant KR0()qK,,y0 at finite q by, following Nelson [77],

2 2 ⟨(nq)(nq− )⟩ KqR0(),,Ky0 =−KK0 4.π 0 (57) q2

Equation (44) gives

l KqR0(),,Ky0 = KqR((e,Kl)(, yl)) (58)

l∗ and we can evaluate this approximately at the scale e = ξ+ by a Debye–Hückel [78, 79] approximation 22∗ 4π Kl ∗∗ Kq,,Ky =−Kl∗ ()nqeellnq− R0()0 () l∗ ⟨( )( )⟩ (59) ()qe 2

where the average ⟨nq()ξξ++nq()− ⟩ must be calculated in the Coulomb gas ensemble with Hamiltonian

Figure 10. Linearized flows in x, y plane from equation (49). The Hl∗∗1 ⎛ 4π2Kl ⎞ V() d2k () Bl∗ nknk. arrows denote the flow direction as l increases. =+⎜ 2 ()⎟ ()()− (60) kTB 2 ∫ ⎝ k ⎠

figure 10 and an experimental measurement in figure 11. Note l∗ At the scale e = ξ+ the system has large density of vorti- that the experimental ρs()T does not drop discontinuously to ces with all integer values of n r , when a reasonable approx zero at Tc but does drop rapidly. This is because the measure- () ments [68] are performed at finite frequency while the theory imation is to integrate instead of summing over the n(q) to [64] is for ω = 0. A dynamical extension of the static theory is obtain needed [76] to make a more convincing interpretation of the 1 experiments on thin Helium films since the measurements are, nq nq ∗ . ⟨(ξξ++)(−=)⟩l 22∗ ∗ (61) 4/πξKl()()qB+ + ()l of necessity, performed with at least one of (ω, vs) non zero, while the static theory is for ω ==0 v only. The theoretical s Here, Bl()∗∗≈−ln yl()= O()1 , but its precise value is unim- behavior of ρs()T as a function of temperature T is shown in portant. Thus, one obtains figure 11 for different coverages. ∗∗ In region II, C(t, y0) < 0 in the small temperature region Bl Kl Kq,,Ty = ()() ∼ qξ 2, 0 < c as expected, which completes the argument for l ⎢tant−−11⎜ ()⎟ an ⎜ 0 ⎟⎥ , −= 2 = − − −−92−1 ∫x0 xC ⎜ ⎟ ⎜ ⎟⎥ the universal jump TT/3.491 10 gcmK [64]. +| | ||C ⎣⎢ ⎝ ||C ⎠ ⎝ ||C ⎠⎦ ρs()c c =× In the context of the renormalization group, the length ⎡ ⎤ xl x0 () =|cotc⎢ Cl|+ot−1 ⎥ , scale ξ−()t is a measure of when the RG trajectory, which ini- ||C ⎣⎢ ||C ⎦⎥ tially flows parallel to the critical separatrix, deviates signifi- ⎡ ⎤ cantly from the straight line. A more physical interpretation 4πyl() −1 x0 =|cosecc⎢ Cl|+ot ⎥ . (55) is that ξ−()t for t ⩽ 0 is the maximum size of neutral bound ||C ⎣⎢ ||C ⎦⎥ vortex pairs and there are no vortex pairs with separation

r > ξ−. Above Tc, t > 0 and ξ+()t is the largest separation of Here, the upper integration limit is chosen as xl =−x < 0 ()+ 0 bound vortex pairs. All pairs with separation r > ξ+()t must be and, since the deviation from Tc is very small when regarded as unbound free vortices which can move freely to

xC0 ||()ty, 0 , so that y(l+ ) = y(0) when lC+ =|π/ |. dissipate superfluid flow or to disorder the system. It is impor- This allows one to define another length scale when t > 0 tant to understand that ξ+()t <∞ for t > 0 and ξ+()t =∞ for t ⩽ 0. On the other hand, the shorter length scale ξ−()t <∞ lC+ ππ/2|| ξξ+()tt==ee = −() . (56) ()bt/ || both above and below Tc and diverges as e as ||t → 0. It is

For length scales larger than ξ+, the effective fugacity important to note that both ξ+()t and ξ−()||t diverge exponen- y(l > l+ ) increases with l to large values outside the validity of tially for 80πyt0 >| | ⩾ as (56) our RG equations and the system is interpreted as completely disordered due to the large concentration of free unbound −1/2 b+ ξπ±()tb∼=expw()±t here 2. (63) vortices. b−

13 Rep. Prog. Phys. 79 (2016) 026001 Review

l −2 Cl ⎡1e− A ⎤ xC0 − ∫ dllx′′()lC=+l n ⎢ ⎥ where,A = 0 ⎣ 1 − A ⎦ xC0 + 1 ≈+Cl ln()1w+=xl0 hen l lnξ−()T , 2 C x0 1 ≈+Cl ln when l = lnξ−()T . 2 C 2 C (67) Thus, the correlation function for TT⩽ c is

−η()T 1/8 GT()r,l∼ rr()nfor 1, rTξ−() (68) −η()T GT()r,f∼ rTor 1, ξ−() r where the temperature dependent exponent η()T is

⎧ kTBc/2πJT,, T Figure 11. The superfluid densityρ s()T for different coverages 1 ⎪ () − η()T ==⎨ . showing the universal jump of ρs()TTc / c . Reprinted from [68] with ⎪ 2πKTR() 1/48−|ty|+()π 0 ||tt/8 ,1− ⩽ 0. permission. Copyright 1978 American Physical Society. ⎩ ()(69) This behavior is commonly taken to define the critical region A plot of η()T is shown in figure 12 where one can see the ||t of the 2D planar rotor model and of a superfluid film but approach, for ||ty 8π 0, to the asymptotic value η()Tc = 1/4. this region is limited to ||ty 8π and the vortex fugacity 0 Note that, for very large r ξ−()T , the correlation function −EkcB/ T y0 = e is extremely small since the core energy Ec ∼ J. has a pure power law decay with a temperature dependent The real critical region should be taken to be the range of t exponent, η()T , which comes from an effective Gaussian model with coupling constant K T . For r ξ T , G r, T for which ξ 1 which is the region ||Ct,1y where R() −() () ± ()0 has the same power law decay but modified by a universal ||Ct(),8yt00=| ()πyt−| 1 in (63). There is a discussion logarithmic correction in (68). of this crossover behavior of ξ and the behavior outside the ± The situation for G()r, T when TT> c is not so clear. Use critical region in [80]. equation (65) to obtain One can calculate other thermodynamic quantities from l renormalization group considerations, particularly for TT⩽ c −l ⎛ ⎞ Gr(),,Ky0 0 =−Gr((e,Kl)(,eyl)) xp⎜ d.ll′′η()⎟ (70) when the RG maps the physical system into a simple Gaussian ⎝ ∫0 ⎠ model when the vortex fugacity yl 0. The two point cor- ()→ r T , choose l ln but one still needs relation function G(r, K, y) was calculated by Kosterlitz [20] When ξ+() = ξ+ G rK/, ln ,ly n which is expected to be who obtained ((ξξ++)(ξ+)) O((exp/−r ξ+)) since the correlation function should decay i1((θθrr)(−−′)) /4 1/8 Gr(),,Ky =∼⟨⟩elrr()n, (64) exponentially. Thus, when r ξ+()T ,

when TT= c. I sketch the derivation [81] below by first noting ⎛ r ⎞ ⎛ lnξ+ ⎞ the scaling equation for d = 2, Gr(),eT ∼−xp⎜ ⎟expd⎜− ∫ llη()⎟ (71) ⎝ ξ+()T ⎠ ⎝ 0 ⎠ dGl() 1 1 xl() 2 ==ηη()lG()llwhere () =− + O()x , which was obtained explicitly by using a duality transforma- dl 2πKl 48 () l ⎛ l ⎞ tion [56]. On the other hand, when e ξ+, even when TT> c, GG()0e=−()llxp⎜ d,′′η()l ⎟ the correlation function G rK/1ξ ,,lyl = O 1 , so that ⎝ ∫0 ⎠ (⩽+ () ())() l Gl()= Gr((e,− Kl)(,.yl)) (65) ⎛ l ⎞ Gr(),,Ky0 0 =−expd⎜ ll′′η()⎟ (72) ⎝ ∫0 ⎠ To obtain the physical correlation function G()0,= Gr()Ky0, 0 in terms of the physical parameters K0 and y0, one needs where lr= ln . The integral can be carried out by using G(l) which is to be computed from the scaled Hamiltonian. η()lx=−1/4/()l 8, as before, but with x(l) from (55). The RG flow equations (48) are valid provided the fugacity

y(l) remains sufficiently small, so l in (65) is chosen so that l l ⎛ ⎞ −l l re1= and G(1, K(l), y(l)) is needed. This is the correlation dll′′ηψ()=−||Cl∫ dc′′ot⎜ ||Cl+ 0⎟, ∫0 4 0 ⎜ ⎟ function ⟨expi[(θθ()re+−(r))]⟩ ≈ 1 because, at low T, the ⎝ ⎠ correlation function of nearest neighbor spins of unit length C −1 x0 || is unity. Thus, the singular part of the correlation function is ψ0 = cot,≈ ||C x0 ⎛ lnr ⎞ Gr,,Ky =−expdllη . ⎛ ⎞ ()0 0 ⎜ ∫ ()⎟ (66) sin/lC||+|Cx| ⎝ 0 ⎠ l 1 ⎜ 0 ⎟ =−ln⎜ ()⎟. (73) To evaluate this, use η lx=−1/4/lx8 + O 2 l with x(l) 4 8 () () (()) ⎜ sin/||Cx0 ⎟ from equation (50) to obtain ⎝ ()⎠ 14 Rep. Prog. Phys. 79 (2016) 026001 Review Thus, we obtain

⎛ ⎞1/8 sin/lC||+|Cx| ⎛ l ⎞ ⎜ 0 ⎟ Gr expdll′′ e−l/4 (), () ξη+ =−⎜ ()⎟ = ⎜ ⎟ ⎝ ∫0 ⎠ ⎜ ⎟ ⎜ sin/||Cx0 ⎟ ⎝ ()⎠ −−ll/4 1/81/4 1/8 ==el()xl0 rr()n,rT= e. ξ−() (74)

When ξ+−>>r ξ , π >|Cl|>1/2 where the behavior of the correlation function G(r) of (74) is not so clear. The factor in the brackets does not vary much for ||Cl∼ O()1 and the cross over to the behavior for r ξ+ is unclear. The lowest order flow equations (37) yield a remarkable amount of information and appear to yield exact results for TT⩽ c because y(l) is an irrelevant variable so that the system becomes Gaussian as l → ∞. However, one sees that this mar- 1/8 ginally irrelevant variable is responsible for the (lnr) factor in the two point correlation function G(r). Do higher order terms in the RG flow equations give a noticeable correction to the expressions for G(r)? This calculation has been performed [82, 83] who derived flow equations to next order in y(l) and found the correlation function at T as c Figure 12. The exponent η()T as a function of temperature from equation (69). −1/41/8⎛ 1 ln lnr ⎞ Gr()∼+rr()ln ⎜⎟1 . (75) ⎝ 16 lnr ⎠ as in (38). Integrating the flow equation (37) for f(l) yields Note that these results imply the finite size scaling form for [86, 87] the susceptibility at criticality for a LL× system l −−2l 22l′ fK()0,eyf0 =−((Kl)(,2yl)) π dely′′()l . (77) 7/41/8 ∫0 χ()LL∼ ()lnL . (76) When TT⩽ c, one can take the upper limit l → ∞ because This has been verified to great accuracy by numerical simula- the RG equations (37) are valid for all 0 ⩽⩽l ∞ and tions on large systems up to L = 216 [84, 85]. e,−2lfKlyl 0 so that I now turn to thermal properties, in particular the specific (()()) → 4 ∞ heat of the planar rotor model in 2D and of a thin film of He. −22l fK()0,2yl0 = π de yl(). (78) In general, the specific heat is the most obvious quantity to ∫0 measure since it usually displays a singularity, albeit a rela- When TT> c, the fugacity y(l) is relevant and increases so that tively weak one, at a phase transition. For example, the 2D the RG equations become invalid for l > l*, where l* is chosen Ising ferromagnet has a specific heat singularity of the form judiciously [86, 87] and 1.75 c()Tt∼|ln | while the susceptibility χ()Tt∼||− . However, l∗ a measurement of the specific heat requires no special −∗2l∗ ∗−22l fK()0,eyf0 =+((Kl)(,2yl)) π dely()l , (79) knowledge of the particular system such as the form of the ∫0 order parameter or of the ordering field. For example, for an ∗ The idea is to use the RG flow equations (37) for ll⩽ antiferromagnet the uniform susceptibility, or the response ∗∗ and estimate fK((ly)(, l )) by an improved Debye–Hückel to a uniform magnetic field, has no dramatic divergence at approximation. Detailed discussions of the matching point T but just a minor kink. The dramatic effect is in the stag- ∗∗ c and calculations of fK((ly)(, l )) are found in [86, 87]. The gered susceptibility which is difficult to measure as it is the −1 result is shown in figure 13. At the transition at Kc , the spe- response to a staggered magnetic field conjugate to the stag- cific heat has an unobservable essential singularity of the gered magnetization. The specific heat of a ferromagnet and form c TT∼ ξ−2 where ξ T ∼ ebt/ || at the critical point an antiferromagnet are the same and are measured by the () + () +() −1 same methods which require no knowledge of the symmetry Kc with a non universal peak at much higher temperature. of the system or the nature of the order parameter. In gen- Since the specific heat is mainly due to the vortex degrees eral, if a transition exists, a specific heat measurement will of freedom, one can understand the peak as being due to the find it, except in cases such as the systems being considered entropy released by the unbinding of vortex pairs of decreas- here. ing size as the temperature increases. Since a very small density of very large pairs unbind at T , the specific heat is very One needs the free energy fK()0, y0 from which the spe- c cific heat is obtained by differentiation with respect to K0 small there. The density of vortex pairs increases as their size

15 Rep. Prog. Phys. 79 (2016) 026001 Review

decreases and these unbind at higher T with a corresponding (ii) y0 1 when the partition function becomes larger entropy release and larger specific heat. The maximum will be when the largest density of the smallest vortex pairs dφ()r ZK()0, yy00=+∫ ∏ [(12cos2πφ(r))] unbind, which is estimated to be 40% above Tc as shown in r 2π figure 13 [87]. ⎡ ⎛ ⎤ 1 2 ×−exp ⎢ ∑ ⎜φφ()rr− ()′ ⎥ , ⎣⎢ 2K0 <>r,r′ ⎦⎥ 2.3. Roughening of crystal facets ⎝ ⎡ dφ()r 1 2 Rather surprisingly, as pointed out by Chui and Weeks =−∫ ∏∑exp ⎢ ((φφrr)(− ′)) [88, 89], the 2D planar rotor model is intimately related to rr2π ⎣⎢ 2K0 <>,r′ models for the roughening of facets of crystals in thermody- ⎤ namic equilibrium as temperature is increased. We represent + 2cy0 ∑ os((2.πφ r))⎥ (82) the particle positions of an ideal flat facet as a 2D lattice of r ⎦⎥ sites labelled by r. Even if the equilibrium configuration of a particular crystal is a smooth planar facet which may appear To convert this to the sine-Gordon roughening model [88, 89, in a crystal grown at a particular temperature, but it also may 93], the height variable is defined byhb ()rr= φ() where h()r not because of competing out of equilibrium facets with faster is the local height at r in steps of spacing b between lattice growth rates. It is well known that a facet with screw disloca- planes. The sine-Gordon representation of the roughening tions terminating in it will grow very much faster than the problem has the Hamiltonian ideal equilibrium facet. There is some hope that a quantum 2 4 γ 22d rr⎛ 2πh()⎞ system such as a He crystal immersed in superfluid may H()γ, uH=+0 Hhu =−d rr((∇ )) u cos⎜⎟, 2 ∫∫a2 ⎝ b ⎠ equilibrate sufficiently fast to be seen. (83) The partition function of the 2D Coulomb gas representation of the planar rotor model of (31) is written as [56] where u = 2y0 when y0 1 and a is the short distance cut off. Flow equations for γ()l and u(l) [88, 93] are derived in a ⎛ ⎞ similar way to K(l) and y(l) in the vortex system by defining a 2 2 ZK()0,eyK0 =+∑∑xp⎜2,π 0 Gn()rr′′()rrny() ln 0 ∑ n ()r ⎟ renormalized stiffness γ T as n()rr⎝ ,rr′ ⎠ R() +∞ ⎛ 1 1 dφ()r 1 2 22 =−∑∏ exp⎜ ∑ φφrr− ′ FF()vv−=()0 Ω=γR()Thwith vrd,((∇ r)) ∫−∞ ⎜ (()()) 2 Ω ∫ n()rr 2π ⎝ 2K0 <>r,r′ (84) ⎛ 2 where F()v is the free energy of a surface of area Ω and v is the ++∑⎜2iπφny()rr() ln 0n ()r ), r ⎝ average slope of the surface. Definehh ()rv=⋅rr+ ′() with d2q e1iqr⋅−()r′ |−rr′| v = ()vv12, so that the sine-Gordon Hamiltonian is G()r, r′ ==∫ ln , (80) 2ππ2 qL2 2 () 1 2 HH()vv=Ωγ ++0[(hH′′rr)] u[(h )]+⋅vr. (85) with L the facet size. Consider two cases (i) when the vortex 2 2 fugacity y0 = 1, use the identity [90] Now expand F()vv=−kTBBlnTr exp/[(−Hk)]T to O()v to +∞ +∞ obtain 2in ∑∑e πφ=−δφ()h 2 n=−∞ h=−∞ 1 ∂ ⎛ 1 2 2 ⎞ γγR()T =+ ∑ ⎜〈〉Hu 0 −−[〈HHu〉〈0 u〉]0 ⎟ + to obtain the discrete Gaussian model [56, 88, 89, 93] 2 Ω αβ, ∂∂vvαβ⎝ 2kTB ⎠ 2 2 2 kTB ⎛ 2dππ⎞ 2 r ⎛ r ⎞ ⎛ 2 ⎞ +∞ ⎛ ⎞ =+γ ⎜⎟u ∫ ⎜⎟〈[cos⎜⎟hh′′()r − ()0,]〉0 1 2 8 ⎝ b ⎠ a2 ⎝ ab⎠ ⎝ ⎠ ZK()0 =−∑∑exp⎜ ((hhrr)(− ′)) ⎟. (81) 2K0 ∞ 32− πK h()rr=−∞ ⎝ <>,r′ ⎠ −1 −132 dr ⎛ r ⎞ KK=+π u ⎜⎟ , (86) R ∫a a ⎝ a ⎠ This is interpreted as a model of the facet in which the height of the facet at r, relative to a reference plane, is h r and 2 () where K ≡ kTB /()γb and we have used 〈cos2[( π/bh)( ′()r − neighboring columns of atoms differ by an integer num- hr′ 0/= a −2πK. ber of lattice spacings. The temperature of the roughening ( ))]〉0 () The last line of (86) yields renormalization group flow model is proportional to K0 while the temperature of the −1 equations exactly as for the planar rotor model by rescaling dual planar rotor is K . When K0 is small, it is clear that l 0 the cut off aa→ e , all columns tend to have the same height h()r and the facet −1 is smooth and flat. On the other hand, at higher temperature dKl() 32 4 =+π ul() O()u with K0 large, the column heights h r will vary with posi- dl () tion leading to a rough facet with the correlation function dul() 3 2 =−((2.πKl))ul()+ O()u (87) ⟨((hhr)(−∼0l)) ⟩ n r. dl

16 Rep. Prog. Phys. 79 (2016) 026001 Review

A flat facet with− xx00<<+x , in terms of reduced coordinates xx˜˜==/,Ly yL/ and hh˜ = /L, obeys ⎧ 3/2 ⎪hA˜00−|( xx˜˜|− ) ||xx˜˜> 0, hx˜ ,0 = ⎨ (90) (˜ ) ⎪ ⎩ hx˜00||˜˜< x .

Here, the reduced facet size xb˜0R∼∼/eξ+()TBxp((−−//TT))TR and hf˜0 = 0 /γ, with f0 the surface free energy density. For TT> R, the facet has vanished, x˜0 = 0, and the reduced height hx˜(˜,0) is

γ 2 hx˜(˜,0) =−h˜0 x˜ . (91) γR()T The universal jump in the stiffness constant translates, for a crystal facet, to the reduced curvature of the rough surface 2 + KRB()TL==()//RkTb()γπ2/ at TT= R which jumps dis- − continuously to zero at T R. Here, L is the facet size and R is Figure 13. Specific heat of 2D planar rotor model from theory. The the radius of curvature [91, 93]. Measuring the predicted jump full curve is from a renormalization group calculation and the dashed in the curvature of a disappearing facet is not easy and, to the (dotted) lines are exact low (high) temperature results. The transition is best of my knowledge, not all the universal predictions have −1 −1 at Kc ≈ 1.33 which is well below the rounded maximum at K ∼ 1.7 been verified experimentally, particularly that the facet size −1 and the asymptotic behavior as K → ∞ is ck= B/2. Reprinted from vanishes as −1 T [92, 94]. Attempts to measure the reduced [87] with permission. Copyright 1981 American Physical Society. ξ+ () curvature KR()T of (89) have been made on hcp crystals in This is of exactly the same form as the flow equations (37) contact with superfluid [95], but with mixed success. Despite except for the interpretation of the coupling constant K. For being the system most likely to be in equilibrium, the mea- 2 0 surements are consistent with theory but suffer seriously from the planar rotor model, K()Tm= () //ρs ()kTB while, for the 2 lack of equilibration. roughening model, K()Tk= BTb/()γ . The T dependence is inverted, as expected, because the models are duals of each other [56] and the low temperature phase of one is the high 2.4. Substrate effects temperature phase of the other. Thus, in the high T phase of the roughening model, the coefficient u(l) of cos2((πhbr))/ is One motivation for developing a theory of transitions in 2D irrelevant and the roughening Hamiltonian becomes a simple was the experimental results for very thin films of 4Helium Gaussian model at long length scales with adsorbed on a substrate [36, 37, 68]. This system is closely related to the 2D planar rotor model and the predictions from H 1 22 =∇KTR() d.rr((h )) (88) the theoretical model are directly applicable to the experi- kT 2 ∫ B ments provided one can argue that substrate effects are irrel- The integer spacing between the crystal planes becomes irrelevant in the renormalization group sense. Of course, the evant and the interface becomes rough in the sense that the substrate is essential as a fluid film a few atomic lengths thick, 2 interface width ⟨hL()r ⟩ ∼ ln in a size L system. a monolayer crystal or magnet cannot exist in the absence of One can immediately infer some exact results for the a substrate. About the only systems which can exist without roughening model by exploiting the relationship with the pla- a substrate are suspended liquid crystal films [96–103] and nar rotor model. At the roughening transition, the universal a monolayer of graphene [104]. Therefore, it is essential to relation of equation (46) becomes assess the effects of all possible perturbations due to a substrate on the ideal two dimensional system of section 2.2. kT 2 KT+ BR . Before discussing some of the substrate effects on 2D R()R ==+ 2 (89) γR()TbR π planar rotor systems, we can see why superfluidity in thin 4 This follows directly from the planar rotor result and the Helium films can be understood quantitatively by the duality relation and implies that facets with the largest lattice idealized 2D theory which ignores the effects of the far from perfect Mylar substrate. This can be modeled by a random spacing b will have the highest roughening temperature TR, 2 assuming the surface stiffness γ does not depend much on b. potential coupled to the local mass density ||ψ()r . The order However, to convert these theoretical predictions into exper parameter is the quantum mechanical condensate wave func- i r imentally measurable quantities, one has to remember that the tion ψ()rr=|ψ()|e θ() and the random substrate potential facets are the faces of a 3D real crystal, and the equilibrium cannot couple directly to the phase of the wave function but theory of a constant volume crystal is required. only to its amplitude and is irrelevant. This justifies using the The basic idea is that the crystal shape is determined by ideal planar rotor model for a superfluid Helium film on a minimizing the total surface free energy at fixed volume [91]. Mylar substrate.

17 Rep. Prog. Phys. 79 (2016) 026001 Review

Many other systems are influenced by the substrate part term ip ∑Rr, nm()Rr()Θ()Rr, . One can see immediately that icularly a planar rotor magnet in a symmetry breaking crys- equation (95) is self dual by relabeling n()Rr↔(m ), yy0 ↔ p 22 tal field such as a planar ferromagnet on a periodic substrate and K0 ↔(pK/4π 0). The partition function is unchanged by described by a Hamiltonian these transformations [56], so that

H 1 2 KT dd22rr hp2rrcos. ⎛ p ⎞ =∇() ((θθ)) − p (()) (92) ZK,,yy = Z ,,yy. (96) kTB 2 ∫∫ ()0 pp⎜ 2 0⎟ ⎝ 4π K0 ⎠ where h is the strength of the p-fold symmetry breaking p One can write down immediately the RG flow equations by field. When|| h 1, the free energy is minimized when l p rescaling the short distance cut off aa→ e as expansions in cos1θθ=+ ⇒=r 2/πnp for h > 0, and, when h < 0, () p p the fugacities y0 and yp to lowest order [56]. These are consist- θ()r =+()21npπ/ , with n =−0, 1, ,1p . These form the ent with the duality relation of equation (96) p-state clock models which correspond to the Ising ferromag- −1 net for p = 2 and to the 3-state Potts model for p = 3. dKl0 () 3 2 2 −22 =−4,ππyl() pK0 ()ly()l For p ⩾ 5, there are many possibilities since higher har- dl 0 p monics of the site anisotropy are possible and also of the inter- dyl0() =−((2,πKl0 ))yl0() action terms. The most general Hamiltonian on a lattice with dl sites r with p-fold symmetry breaking is 2 dylp() ⎛ p ⎞ =−2 yl. []p/2 ⎡ ⎜ ⎟ p() (97) H dl ⎝ 4πKl0()⎠ =−∑∑⎢ Knn [[1cos ((θθrr)(− ′))]] kTB n=<1 ⎣⎢ rr, ′> From these equations, when p > 4, there is a region of temper ⎛ ⎤ ature when both fugacities y0 and yp are irrelevant given by − ∑ hnnpcos.⎜ pθ()r ⎥ (93) r ⎝ ⎦⎥ 2 p2 ⩽(KTR )⩽ . (98) This Hamiltonian represents many statistical mechanical mod- ππ8 els in two dimensions such as Potts models [105], the Ashkin– In this range of T, the system is in a massless Gaussian phase. Teller model [106], the 8-vertex model [107], etc. These can −1 2 At low temperature K < 8/π p , yp(l ) is relevant and the sys- all be written as generalized Coulomb gases [108 114]. For 0 – tem is in the low T phase where there is long range order with example, the 4-state Potts model is obtained when p 4, = the system in one of the p ordered states. The phase diagram h ∞, h = 0, so that θ r = 0, ππ/2,,3/π 2, and K = 2K . 4 → 8 () 12is shown in figure 14 left for p = 6 and for p = 4 in figure 14 Then the Hamiltonian becomes Hk/2TK 1 with ()B2=− ()δss′ + right. When p 3, there is no intermediate massless phase as s 0, 1,3. ⩽ = one or both fugacities yy, are relevant for all K. This is inter- In the simplest case of n 1 only in (93), the symmetry 0 p = preted as a direct transition from the large K ordered state to breaking term in the partition function can be written as [55, a low K disordered state. For p = 2 and p = 3, at h =∞ it is 56] p easy to see that the Hamiltonian of (93) reduces to that of the

+∞ 2 Ising or the 3-state Potts model hppcosiθθpm m ee→ ∑ yp , (94) m=−∞ H K = ∑ ()1,−=δss()rr, ()′ sp()r 1, ,. (99) kTB p − 1 r,r with yhp ≈ p/2. The Coulomb gas representation of the planar <>′ rotor in a p-fold symmetry breaking field becomes [56] I have shown that the planar rotor model, Zp models

Hc and the discrete Gaussian model can be transformed into ZZ∝=c()Ky,,0 yp ∑∑e, nm()rR() Coulomb gases which is useful for analyzing critical behavior. The transformation can be done exactly for Villain models ⎛ |−RR′| ⎞ 2 Hc0= πKT()∑∑nn()RR()′ ln⎜ ⎟ + lnyn0 ()R [55, 108] because of the quadratic form of the Hamiltonian. ⎝ a ⎠ RR, ′ R However, many realistic models are not quadratic. For exam- 2 p ⎛ |−rr′| ⎞ 2 ple, the solid on solid model in which the step energy is linear + ∑∑mm()rr()′ ln⎜ ⎟ + lnymp ()r 4πK0 rr, ′ ⎝ a ⎠ r in |−hh()rr()′ | cannot be transformed simply into a Coulomb +Θi,pn∑ ()Rrm() ()Rr, gas model. However, the Hamiltonian can be expanded in Rr, powers of ∇θ, the leading term being quadratic. One can

−1⎛ Yy− ⎞ then perform a more standard renormalization group analy- Θ=()Rr,tan ⎜⎟. (95) ⎝ Xx− ⎠ sis by momentum shell integration [75] and all non gaussian terms with more than two gradients are irrelevant. These RG The integer vortex charges n()R lie on the sites R = ()XY, of arguments imply that SOS etc models with other than Villain the dual lattice and the symmetry breaking charges m()r lie interactions differ from a Coulomb gas by irrelevant opera- on the original lattice with sites r = ()xy, . Neutrality condi- tors only. Thus, provided one assumes that these models and tions ∑Rrnm()Rr==0 ∑ () are satisfied by both types of Coulomb gases can be connected by RG flows with no fixed charges. The two Coulomb gases are coupled by the angular points intervening, they are asymptotically equivalent. Given

18 Rep. Prog. Phys. 79 (2016) 026001 Review this qualitative assumption, one can make quantitatively exact H 2 ⎛ 1 2 1 2 conclusions [83]. For the special case, p = 4, the multicritical =∇∫ d r⎜ KK6()θφ+∇1()+∇g∇⋅θφ kTB ⎝ 2 2 point, πK = 2 and yy==0, is the meeting point of three 04 ⎞ critical lines of continuously varying exponents separating a +−V()θφ⎟, high temperature disordered phase from two low T ordered ⎠ phases with two different spin orientations with θ()r = nπ/2 Vh()θφ−=−−61cos6((θφ)) −−h 2cos1((2,θφ)) + (101) or θ()r =+()21n π/4, where n = 0, 1, 2, 3. This point is identical to the critical point of the F model [115], and the three which is very similar to (100). In the very low temperature critical lines meeting there are in the universality class of the regime of interest, but above the crystallization temperature Baxter 8-vertex model [116]. Tm, one can write (101) as Other interesting models of experimental relevance are obtained by including both the first and second harmonics of H 22⎛ 1 1 2 ⎞ =∇d r⎜⎟KK++()θθ+∇−−()+ V()θ− , the p-fold anisotropy. The planar rotor model with both four kTB ∫ ⎝ 2 2 ⎠ and eight fold anisotropy Kg6 + θα+ =+θβφα,1=−β = , H KK61++2g =−K cos((θθrr)(− ′)) 2 kT ∑ KK16− g B <>r,r′ KK+−=+61Kg+=2; K . (102) K+ −−∑ [(hh48cos4θθ()rr)(cos8 ( ))] (100) r A detailed analysis of this model system is carried out [125] is an interesting theoretical model but can also describe the with results agreeing with the experimental observations by adsorption of hydrogen on the (1 0 0) surface of tungsten. adjusting the sign of h12 in (101). The phase diagram for this When the hydrogen coverage is changed, h8 does not change model is essentially identical to figure 15 with hh46→ and but h4 can be driven through zero when a structural transition hh81→ 2 [125]. of the tungsten surface occurs [117, 118]. At low T, ππK ⩾ /2, Dierker, Pindak and Meyer found a remarkable five-armed equation (100) has two ordered phases with finite magnetiza- star defect sketched in figure 16 in thin tilted hexatic liquid tion in one of the four directions θ()r = nπ/2 with n = 0, 1, 2, crystal films at low temperatures [126]. This can be described 3 when h4 > 0 and θ()r =+()21n π/4 when h4 < 0. The other by the Hamiltonian of (102) with h12 > 0 so that there is a first −1 anisotropy term h8cos8θ()r is irrelevant for K > π/8 and rel- order transition at h6 = 0 between a hexatic I phase for h6 > 0 −1 −1 evant for K ⩽ π/8. In the temperature range ππ/8 ⩽⩽K /2 and a hexatic F phase for h6 < 0 [125]. It is easy to see that the the ordered phases will be separated by a line of continuous Hamiltonian is minimized by θ−()r = 2/πn 6, n = 0, 1, ,5, −1 transitions at h4 = 0 At lower temperatures, K < π/8, h8 so that θ−()r can have a discontinuity of 2π/6. When a film becomes relevant which has important consequences. There is formed, it can have a vortex about which the tilt azimuthal are two possibilities depending on the sign of h8: (a) h8 > 0 angle φ()r goes through 2π. When the system is cooled into when the orderings favored by h4 and by h8 are compatible, the hexatic I phase, the bond orientational order increases and independent of the sign of h4. The transition between the two θ()r becomes locked to φ()r by the relevant h6 term, so that the ordered phases at h4 = 0 will be discontinuous first order 2π vortex in φ()r becomes a 2π disclination in θ()r . To reduce [117, 118]. In case (b) h8 < 0, the orderings favored by hh48, its energy, this defect breaks up into an N armed star structure are incompatible leading to an extra phase when ||h4 is small with energy with this phase separated from the other two ordered phases π/N ∞ by continuous transitions in the 2D Ising universality class. ∆E KN+ star 2 no star 2 =Θddrr()|∇θθ++|−|∇ |+εRN, kT 2 ∫∫−π/N a Scenario (i) is compatible with experimental data [119, 120] B (103) which seems to favor a first order transition at low temper atures. Figure 15 illustrates schematic phase diagrams for where R is the length of an arm, ε is the energy per unit length h8 > 0 (left) and h8 < 0 (right). in units of kBT. As shown in [125, 126], ∆Ek/()BT is obtained 2 Other systems where the first two harmonics have impor- by solving Laplace’s equation ∇ θ+()r,0Θ= in the wedge tant effects are tilted hexatic phases of liquid crystals. −ππ//NN<Θ<+ subject to the boundary conditions Experiments on the transitions between different tilted hex- ±−()πα/1NN()/6 for rR< atic phases in thermotropic liquid crystals have observed a θπ+()rN,/±= (104) /fNror R weakly first order transition from a hexatic-I to a hexatic-F { ±>π phase [121]. In an analogous layered lyotropic liquid crystal so that the L I and L F phases, which correspond to the hexatic-I and β β ∆E R R hexatic-F phases respectively, were seen to be separated by a =−NK+gN()ln + Nε where gN() kT a a third LβL phase with continuous Ising-like transitions between B απ the phases as in figure 15(b) [122]. These observations can be =−()12 αα− N . (105) explained by modeling the system in terms of coupled planar 36 rotor models for the bond angle, θ()r , and the tilt azimuthal For the equilibrium arm length, minimize this with respect to angle, φ()r , of the director with Hamiltonian [123–125] R so that [125]

19 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 14. Phase diagrams in the T, hp plane. Left: p = 6 showing the intermediate phase. Right: p = 4 with no intermediate phase but three critical lines with varying exponents. Asterisks denote lines and regions of continuously varying critical exponents. Reprinted from [56] with permission. Copyright 1977 American Physical Society.

6in σ R Kg+ ()N CC==Re e θ n = , 6n ⟨⟩ 6 a ε nnn 1. (108) σλn =+ ()− ∆E ⎛ Kg+ ()N ⎞ =−Ng()NK+⎜⎟ln − 1. (106) This has been verified in 3D [137] with λ ≈ 0.3 and λ = 0.8 kTB ⎝ ε ⎠ for a thin smectic-C film [133, 134]. However, the theoretical For reasonable physical values K61 Kg, , so that the value in 2D is λ = 1 which follows from a low T Gaussian parameters of (101) become αβ+=1 with αβ and theory for the planar rotor model [138]. Finally, agreement γ ∼ 103. For these parameters, the value N = 5 minimizes was found between theory and experiment on a two layer film the energy ∆Ek/ BT, in agreement with observation [126]. of smectic B 54COOBC which has bond but no herringbone N = 6 would relieve completely the bond orientational strain order [139–141]. Interestingly, early measurements claimed between the arms but the cost of the extra arm is prohibitive disagreement with KT theory because of a specific heat anom- [126]. aly at the Sm-A—Hex-B transition but later work [141] found The expression for the arm length R can be used to test exper a new Sm-A′ phase with a different density and that the spe- imentally one prediction of the KTHNY theory of 2D melting cific heat anomaly is at the Sm-A—Sm-A′ transition while the [125, 126]. The equilibrium arm length of the arms of the star Sm-A′—Hex-B—Cry-B phase transitions are consistent with defect is given by (106) and is proportional to the temperature the disclination/dislocation scenario. dependent coupling constant K+()TK=+61Kg+ 2 where R K6 is the bond orientational stiffness called K A()T in (153). 3. Melting of two dimensional crystals In the liquid crystal system of [126], the coupling K+ (T ) is dominated by the bond orientational stiffness KA()T whose In early papers [18, 19], Kosterlitz and Thouless suggested temperature dependence is sketched in figure 19. Although the that a two dimensional crystal melts by the unbinding of experiment of [126] is probably not in the asymptotic regime, bound pairs of dislocations. These are the appropriate topo- the data for the arm length R(T) close to the crystal-hexatic logical defects which can destroy the translational order in a transition was fitted to the form [126] periodic crystal. The crystal was pictured as a harmonic elastic solid with a small density of dislocations imposed, analo- −ν˜ RT()∼−exp.((bT Tm)) (107) gous to the vortices imposed on the gaussian background in This yields ν˜ =±0.38 0.15, which is consistent with the a 2D Helium film. It is now clear that these physical ideas theoretical prediction ν˜ = 0.3696 [127–130]. This is one of are correct but incomplete [127–129]. There are two types the earliest experiments to test the KTHNY theory of melt- of order in a translationally periodic crystal: (i) translational ing and the observations are completely consistent with all the order under ur()→(ur) + a where ur() is the displacement theoretical predictions. of the position of the particle whose equilibrium position is There have been a number of investigations of freely sus- the lattice site r and a is a lattice vector, and (ii) orientational pended thin liquid crystal films [100, 131–136] to look for order of the crystal axes. An obvious but fundamental ques- the BKTHNY melting scenario and the elusive hexatic phase. tion is: what excitations are responsible for destroying these This requires a material with the sequence of phases as T is orders, in what order are they destroyed as T is raised and lowered smectic-A—hexatic-B—crystal-B to avoid the extra what is the nature of the transitions? The theory of melting in complications due to tilted molecules. One of the aims was to 2D was worked out in seminal papers in 1978–79 by Halperin, investigate the predicted scaling relation between the harmon- Nelson and Young (HNY) [128–130] where they showed that, ics of the bond orientational order parameter [137–139] as T is raised, a 2D crystal melts to an anisotropic fluid atT m

20 Rep. Prog. Phys. 79 (2016) 026001 Review

−1 Figure 15. Left: Schematic phase diagram for the Hamiltonian of (100) in the hK4, plane with a fixed, compatible h8 > 0. For finite h4, −1 −1 −1 the two transition lines for KK> KT are in the universality class of the planar rotor in a 4-fold symmetry breaking field. BetweenK KT and −1 −1 −1 the multi critical point K m , there is a line of pure planar rotor transitions. At lower T, KK< m , the 8-fold symmetry breaking field h8 is −1 relevant and there is a line of first order transitions indicated by the dashed line at h4 = 0. The lower bound is K m ⩾ π/8 from (97). Right: −1 −1 The phase diagram for a competing h8 < 0. This differs from (a) only for KK< m when h8 is relevant. The first order line opens up into two lines of Ising transitions, which terminate at ±|4 h8|. Reprinted from [118] with permission. Copyright 1994 American Physical Society.

Figure 16. The orientation of the director (arrows) and of the bond orientation (crosses) about the 5-armed defect at temperature (a) TTm < TI. Reprinted from [126] with permission. Copyright 1986 American Physical Society.

and, at higher TT=>I Tm, there is a transition to the expected fluid with no intermediate anisotropic fluid, in contradiction isotropic fluid phase with exponentially decaying translational to theory [129]. Surprisingly, recent set of experiments on the and orientational order. melting of 2D colloidal crystals [143, 144] finally agree with The breakthrough of HNY was to realize the importance of theory but such experiments have been possible only over the the angular terms in the dislocation-dislocation interaction. In last decade. the original work [18, 19], the angular terms were ignored on The notion of a periodic crystal in 2D was quite contro- the grounds that they are subleading. By analyzing the system versial in 1972 when Thouless and I began to think about the carefully, it was realized that the dislocation unbinding trans melting problem because of a rigorous theorem that there ition at Tm did not yield an isotropic liquid, but an intermedi- is no long range crystalline order in 2D [24]. This theorem, ate anisotropic fluid with exponentially decaying translational although correct, misled most physicists at that time because order and algebraically decaying orientational order, due to it proves exactly what it claims but does not exclude finite remnants of the long range orientational order of the low elastic moduli in 2D, as was understood by the author of the temperature crystal phase. At the higher temperature, TI, the theorem. Thouless and I pictured a 2D crystal as a periodic orientational order is destroyed resulting in an isotropic fluid array of points, representing the particles, on an elastic sheet. with exponentially decaying translational and orientational We asked what would happen to this array when the sheet is correlations. Some experimental and numerical investigations elastically deformed without tearing? It is obvious that the lat- followed with little agreement with theory or with other tice does not change much locally, but the relative separation numerical simulations [142]. Early experiments and simu- of a pair of initially distant points can undergo a very large lations on 2D melting seemed to favor a discontinuous first change. The idea of a 2D crystal became much clearer with order transition directly from a periodic solid to an isotropic this pictorial representation. We realized that the essential

21 Rep. Prog. Phys. 79 (2016) 026001 Review

defining characteristic of a 2D crystal is not long range trans- 1 θ()rz=⋅((ˆ ∇ × ur)), (114) lational order, which is destroyed by thermal fluctuations, but 2 is the fact that the elastic moduli are finite. This picture is written mathematically by writing the local and the elastic free energy of a 2D crystal becomes number density r of particles as ρ() 1 2 22 Fu=+d2rr((µλij ))ukk . (115) 2 ∫ iGu⋅ ()r ρρ()rr=+0() ∑ ρGe, (109) G Here, the Lamé coefficients are given in terms of the constants A and B of (111) by [147] where r is a site of the periodic reference lattice, G a reciprocal lattice vector such that G ⋅=r 2πn and ur is the dis- 3 2 4 3 4 () µρ=+()ABGA0, λ =−()BG0. (116) placement of the particle from its equilibrium position r. The 4 0 4 most common crystal structure in 2D is the triangular lattice At this order, the elastic free energy is a quadratic func- where each particle has six nearest neighbors. This structure tional of the linearized symmetric strain tensor also has a six-fold rotational symmetry as the crystal axes are oriented at angles 2πn/6 relative to one another. The transla- 1 ⎛ ∂ui ∂uj ⎞ tional and orientational order parameters are [129] uij = ⎜ + ⎟. (117) 2 ⎝ ∂rj ∂ri ⎠ iGu⋅ ()rr6iθ() ρρGG()rr=| ()|=eand ψψ66()rr||()e, (110) In the same way as for the planar rotor model, the strain field s where θ r is the angle the local crystal axis makes with some can be decomposed as uuij()rr=+φij() ij()r where φij()r is () s arbitrary reference direction. Note that ρG()r is invariant under the smooth part of the strain uij()r and uij()r is the singular part ur()→(ur) + a with a a primitive lattice vector, and ψ6()r due to dislocations [148]. These are characterized by the value under θ()rr→(θπ) + 2/3. of the integral of the displacement ur() round a closed contour A natural way to proceed is to write down a Ginsburg- Landau-Wilson free energy functional in terms of the order d.ub==aa00()rr((nm)(erˆˆ12+ ))e (118) ∮C parameters ρG()r and ψ6()r which is invariant under the nec- essary symmetry transformations. This has been done by Here, br() is the dimensionless Burger’s vector, a0 is the crys- Halperin [145] following the treatment by de Gennes [146] tal lattice spacing, eˆ1,2 are unit lattice vectors of the underly- with the free energy density [147] ing triangular lattice and n, m are integers. Within continuum elasticity theory, one can show that [129] 6 ⎛ AB rT ⎜ 22() 2 2 F []ρθ, =|∑ GD⋅|αρρα +|GDαα×|αα+|ρ | s ⎛ 1 ∂ λ 2⎞ α=1⎝ 22 2 uij()rr= ⎜ εεik jl − δij∇ ⎟ ab0 ∑ mmG˜ (),,r′ ⎝ 24µ ∂∂rrkl µλ()+ µ ⎠ r′ K A 2 ⎞ 4 ∂ +|∇θ| ⎟ ∇=GK˜mm()rr,,′′− 0ε n δ()rr− 2 ⎠ ∂rn ++w ρρρρρρ + , 4µλ()+ µ ()135246 K0 = , 2µλ+ Dzαα≡−∇ i.(ˆ × Gr)(θ ) (111) ⎡ 01⎤ εε= ⇒=ε −δ . (119) ij ⎣⎢−10⎦⎥ ij jk ik Here, consider only the six smallest reciprocal lattice vectors Gα with ||Gα ==G0,1α ,2,, 6, of the underlying To solve the differential equation (119) for the Green’s triangular lattice and the corresponding Fourier components function G˜m()rr, ′ , one considers a crystal with free boundary r of the density. The unit vector z is normal to the conditions [129] which needs von Neumann boundary con- ρα = ρGα() ˆ (x, y) plane of the crystal and θ()r is the phase of the bond- ditions, ∇G˜m()rr,c′ = onstant, when r is on the boundary, orientational order parameter which gives In a mean field approximation, the presence of the third 2 ⎡ ⎤ order terms makes the transition first order [45]. K is a bare K0 ⎛ |−rr′| ⎞ A Gm()rr, ′ =− ∑ εmn()rrn − ′n ⎢ln⎜ ⎟ + C⎥ . (120) rotational stiffness constant like that of the planar rotor model 4π n=1 ⎣ ⎝ a ⎠ ⎦ (20). The gradient terms are fixed by the transformation prop- The constant C > 0 is a measure of the ratio of the dislocation erties under rotations rr+ which requires that θ()→(θθ) 0 core size to the short distance cut off a and can be absorbed

izGˆ⋅×()r θ0 into an effective core energy, as was done for a superfluid vor- ρρG()rr→(e,G ) (112) tex. Physically, this dislocation core can be thought of as the so that the free energy (111) is invariant under rotations. region near the center of the dislocation where the displace- Assume the transition occurs well below the mean field trans ment ur() is large so that linear elasticity breaks down. In exact ition temperature so that r()T 0 and ||ρρα = 0. In this phase analogy with the vortex fugacity y0, one absorbs this unknown only approximation [147], scale dependent dislocation core energy into an analogous dis- −EkcB/ T iGr⋅ location fugacity y0 = e . The end result is that the total ρρG()r = 0e. (113) elastic energy, HE, can be decomposed into a smooth part and The free energy of (111) is minimized when a singular part due to the dislocations as [128, 129]

22 Rep. Prog. Phys. 79 (2016) 026001 Review

HHE0=+HD, Unfortunately, this angle dependent behavior of the structure H 1 d2r function is also beyond experimental capabilities so the uni- 0 =+2,µφ22r λφ˜ ∫ 2 [ ˜ ij() kk] versal predictions for the structure function are unlikely to be kTB 2 a0 verified or falsified in the foreseeable future. HD K˜ ⎡ ⎛ |−rr′| ⎞ ()rr−−′′αβ()rr⎤ =− ∑ bbαβ()rr()′ ⎢ln⎜ ⎟δαβ − 2 ⎥ kTB 8π rr≠ ′ ⎣ ⎝ a ⎠ |−rr′| ⎦ 3.1. Renormalization group for melting Ec 2 +|br()| , kT∑ (121) B r Dislocations [148] are assumed to be responsible for the melting of a 2D crystal [18, 19] and, in analogy with vortices in the 2 2 2 where K˜ = Ka0 0/()kTB , µ˜ = ak0µ/()BT , and λ˜ = ak0λ/()BT . 2D planar rotor model, we want a correlation function which The dislocation density br() obeys a neutrality condition gives the renormalized elastic constants. Nelson and Halperin [128, 129] argued that inverse of the tensor of renormalized ∑r br = 0, because the energy of an isolated dislocation () R diverges as ln()La/ 0 where L is the linear size of the system. elastic constants Cijkl is One can immediately conclude that a low temperature CTR;ijkli=+µδ˜R()()kjδδliljδλki+ ˜R()T δδjkl, crystal phase will be one described by H0 of (121) with renormalized elastic constants T and ˜ T which are renor- −1 〈〉UUij kl µ˜R() λR() C = , R;ijkl 2 malized downwards from their bare values by the presence Ω a0 of bound pairs and triplets of elementary dislocations [127– 1 130]. In this phase, there will be no free, unbound disloca- Ulij =− d,(unijˆˆ+ unji) (124) 2 ∮P tions because the presence of mobile dislocations will cause where P is the perimeter of the crystal and n is a unit vector the shear modulus µ˜()T to vanish so that the system will have ˆ a fluid like response to a shear stress. This was worked out normal to the perimeter pointing outwards from the crystal. by these authors by a renormalization group treatment for Note that this definition of the renormalized compliance ten- −1 melting of a 2D triangular crystal on a smooth substrate using sor or inverse elastic tensor, CR;ijkl in terms of fluctuations at equation (121) but treating the angular term carefully. At low the perimeter is exactly how the compliances are measured. T, the 2D crystal is described by the Gaussian Hamiltonian H0 Applying Green’s theorem to the expression for Uij in (124), of equation (121) which gives the Debye–Waller factor CG()r one obtains [129] and the structure function S()q 2 1 Uuij =+d rrij bRijRRεεll+ bRjill, ∗ () ∑[()()] (125) −ηG()T ∫ 2 CrG()rr=∼〈(ρρG )(G 0,)〉0 R 2 iiqr⋅⋅qu[(ru)(− 0)] S()qq=|〈(ρ )〉|=0 ∑ ee〈〉0 where r denote the sites of the triangular crystal lattice and R r the sites of the hexagonal dual lattice on which live the dislo-

∼|qG−|−+2 ηG()T , cations. It is not hard to see that the dislocation contribution to 2 222 ∫ d rruij() vanishes, so that the singular part of Uij is kTB ||GG3µλ+ ||a0 3µλ˜ + ˜ ηG()T = = , (122) 4π µµ()24+ λπµµ2 + λ˜ 2 s ˜( ˜ ) UUij =+∫ d,rrφij() ij

where ⟨⟩0 means an average over H0. The algebraic decay of s 1 Uaij =+0 ∑[(bRijRR)(εεll bRji)]ll. (126) CG()r means that there is no true long range translational order 2 R in 2D because the smooth phonon excitations are enough to We have derived a suitable correlation function for make ⟨ρG()r ⟩ = 0, in agreement with the rigorous Mermin– Wagner theorem [22, 24], but the harmonic crystal has a finite the renormalized elastic constants, or response functions, from which we can deduce renormalization group flows, measured shear modulus µ˜()T > 0. Interestingly, the x-ray structure function S(q) diverges in exact analogy to the renormalized stiffness constant K TT= 2 R / mk2 T of equation (57). We have at small reciprocal lattice vectors G when ηG()T < 2 and R() ρs ()()B has finite cusp singularities whenη G()T > 2 as sketched in −−11−1 s s CCR;ijkl =+ijkliΩ 〈〉UUj kl , figure 17 for TT⩽ m. A more careful calculation of the struc- 1 ture function [149] gives the same power law decay but with C−1 = dd2rrφφ2rr ijkl 2 ∫∫11ij() 22kl() some angular dependence Ω a0

2−η 1 λ˜ ⎛ e−τθcos2 ⎞ G =+()δδik jl δδil jk − δδij kl. (127) S()q = ⎜ ⎟ , 4µ˜ 4µµ˜( ˜ + λ˜) ⎝ |−qG| ⎠ The correlation function UUs s is evaluated by a pertur- µλ+ ⟨ ij kl⟩ τ = ,; qq⋅−()Gq=| ||qG−|cos.θ (123) bation expansion in the dislocation fugacity y = e−EkcB/ T to 23()µλ+ obtain the RG flow equations [128–130]

23 Rep. Prog. Phys. 79 (2016) 026001 Review

d −1 l ⎛ Kl⎞ µ˜ () 2/Kl˜ ()8π ˜ () 3 =+3eπyl() I0⎜ ⎟ O()y , dl ⎝ 8π ⎠ dl ˜ l −1 ⎡ ⎛ Kl⎞ ⎛ Kl⎞⎤ (µλ˜()+ ()) 2/Kl˜ ()8π ˜ () ˜ () =−3eπyl() ⎢I01⎜ ⎟ I ⎜ ⎟⎥ dl ⎣ ⎝ 88ππ⎠ ⎝ ⎠⎦ 3 + O()y , dKl−1 3 ⎛ ⎛ Kl⎞ ⎛ Kl⎞⎞ ˜ () 2/Kl˜ ()8π ˜ () ˜ () =−πyl()e2⎜ I01⎜ ⎟ I ⎜ ⎟⎟ dl 4 ⎝ ⎝ 88ππ⎠ ⎝ ⎠⎠ + O()y3 , dyl ⎛ Kl⎞ ⎛ Kl⎞ () ˜ () 2/Kl˜ ()8π ˜ () =−⎜2 ⎟yl()+ 2eπyl() I0⎜ ⎟ dl 8π 8π ⎝ ⎠ ⎝ ⎠ Figure 17. A schematic sketch of the structure function S()q of + O y3 , a 2D crystal from equation (143). For TT⩽ m, peaks for small G () −+2 η diverge as |−qG| G but for larger G are finite cusps. ForTT > m, 4µµ˜()ll( ˜()+ λ˜()l ) 2−ηG Kl˜ ()= . (128) all peaks are finite with a maximum height∼ ξ+ . Reprinted from 2µλ˜()ll+ ˜() [238] with permission. Copyright 2002 Cambridge University Press.

Here, K˜ l is the coupling constant in the dislocation () irrelevant so that the system has no free dislocations at large Hamiltonian HkDB/()T of (121). The derivation of the flow equations (128) is identical in spirit to the vortex flows of (37) length scales and the stiffness but more complex because there are three different elementary µµ˜R( ˜,,λµ˜ yl) = lim ˜(), dislocations in a triangular lattice and only one elementary l→∞ 2 vortex in the planar rotor model. The origin of the term O()y λµ˜R( ˜,,λλ˜ yl) = lim ˜(), in the flow equation for the dislocation fugacity y(l) is that two l→∞ different elementary dislocations can combine to form a third KK˜R(), yK= lim ˜ ()l . (131) l ∞ −1 → elementary dislocation [128–130]. The RG flows forK ˜ ()l and y(l) are sketched in figure 18. For TT⩽ m, the values of µ˜R()T and λ˜R()T can, in principle, The recursion relations are very similar to those for the pla- be obtained by integrating (128) and they depend on the initial values µ0, λ0 and are non universal. However, the com- nar rotor model. Above a temperature Tm, K˜ ()l < 16π and the − fugacity y(l) increases with l, implying that the dislocations bination K˜R()T m = 16π does have a universal value from unbind and become free to move under any small applied stress (128), just like the universal stiffness of the 2D planar rotor so that the elastic moduli vanish and the crystal has melted. model [129],

The flow equations (128) up to a length scale ξ+()T [129], − 4µµ˜RR()TT( ˜ ()+ λ˜R()T ) KT˜R m = lim = 16π. (132) l∗ () − ′ −ν TTm 2 TT˜ ξν+()Tb==eexp()t with = 0.3696 . (129) → µλ˜R()+ R() To obtain more detailed behavior near the transition at with b′ a non universal constant. The scale ξ+()T can be interpreted as the scale below which dislocations are bound together 16πK−1 = 1, assume the dislocation fugacity y 1, write −1 in pairs or triplets of zero total Burger’s vector. Equivalently, 16πKl()=+1 xl() and expand the flow equations (128) to l∗ lowest order in x(l) to obtain [128–130] on length scales e ⩽(ξ+ T ), the system responds elastically, l∗ while on length scales e > ξ+()T , the system responds like dxl() 22 32 3 a fluid. =+12π Ay ()lyOO()=+Yl() ()Y , dl When TT⩽ m, K˜ ()l ⩾ 16π and the fugacity yl()→ 0 as Yl()= π 12Ay()l l → ∞ so that, at all length scales, the system responds elas- dYl() 23 tically. The renormalization group transformation is derived =+22xl()Yl() αYl()+ O()Y , dl from the perturbation expansion for the renormalized elastic 2 Ae=−((22II01)(2)) ≈ 21.94, constants µ T , λ˜R T and K˜R T so that [129] ˜R() () () 1 Be=≈I0()26.20, µµ˜RR( ˜,,λµ˜ yl) = ˜ (µλ˜(),,˜()ly()l ), B α = . (133) λµ˜RR( ˜,,λλ˜ yl) = ˜ (µλ˜(),,˜()ly()l ), 12A KK˜ ˜ ,,yK= ˜ Kl˜ yl . (130) RR( ) ( () ()) The flows in the x, Y plane are obtained from The Young’s modulus, K˜R()T is analogous to the renormal- 4 dY x ized stiffness KR()Ty, for a He film. At low T, corresponding =+22α, (134) to K˜ ()l ⩾ 16π, from (128), it is seen that the fugacity y(l) is dx Y

24 Rep. Prog. Phys. 79 (2016) 026001 Review which yields the renormalization group invariant

1/+−bb2 1/2 2 () 2 () Yl()−+αα21()++bx()lY()lb+−21()xl()

= Ct(), α b = , α2 + 2 1/+−bb2 1/2 2 () 2 () Ct()=−Yb0 αα++21()XY0 0 ++21()− bX0 . (135) Note that, when the parameter B = 0, this reduces to the planar rotor problem since then (135) is just the square root of (49). The RG flows from (135) are shown in figure 18. One can extract some more detailed behavior of the system in the vicinity of the melting transition at TT= m by solving the recursion relations of equation (133) in detail [127, 129, 130]. To simplify the algebra, it is convenient to redefine the fugacity as Y = π 12Ay so that the recursion relations for x and Y become [130] dx = Y2, dl Figure 18. RG flows in the x, y plane from equation (135) for dY B b = 0.4. The arrows denote the flow direction for increasing l. =+22xY ααY2, = . (136) dl 12A equation (131). The Young s modulus K T has a universal 2 ’ ˜R() On the critical line, Y(l) = −m0x(l) with m0 =+αα2 − − − value at T m of K˜R()T m = 16π with [129] and ν KT˜R()=+16π()1,ct||˜ d||xl() 2 ||x0 mxlx2 , l . − ν˜ =− 0||⇒() ()=− 2 (137) µµ˜RR()TT=+˜ ()m ()1,ct|| dl 1+|xm0| 0l − ν˜ λλ˜RR()TT=+˜ ()m ()1 ct|| (141) For x0 slightly away from the critical separatrix, write [129, where c is a constant O 1 . This behavior has been verified in 130] Y(l) = −m0x(l) + D(l) where () detail by experiment and simulation [144]. The x-ray structure dDl 2 () 2 22/m0 function =−21xl()Dl()+⇒O()DD()lD=+00()||xm0l . dl (138) 2iqr i0quru S()qr= de⋅⋅⟨⟩e [()(− )] (142) The initial value Dt0m∝=()TT− /1Tm . When t < 0, the ∫ flow follows the critical separatrix for some distance and then where the Debye Waller factor for qG= is Y(l) breaks away and plunges rapidly to for l > l*. One can * ∗∗ i0Gu⋅−((ru)()) −ηG()T calculate l from the condition Dl()≈ Yl() which gives CrG()rr==⟨(ρρGG)(− 0e)⟩ ⟨⟩∼ 2 2 2 mx00|| ||G aT0(3µλ˜R()+ ˜R()T ) 22∗ /m0 Dx00()1 +| |≈ml0 , ηG()T = . (143) 2 ∗ 4 TT+ ˜ T 1+|xm0| 0l πµ˜RR()(µλ˜ () R()) 2 2 ∗ −+mm0 /2()0 −ν˜ The structure function for qG 1/a is the Fourier lD∼∼0 ||t . (139) |− | transform of the Debye–Waller factor CG()r so that From this we can define the length scaleξ T as the scale −() S qq∼| −|G ηG()T −2. This diverges for the smaller values of at which the deviation from the critical separatrix becomes () G when η < 2 as sketched in figure 17. Note that for a hex- significant [128–130] G agonal lattice with lattice spacing a0, the smallest reciprocal lb∗ ||t −ν˜ lattice vector so that ξ−()T ==ee. (140) ||G00= 4/π ()a 3

This length scale exists both above and below Tm and has the 4π 3µλ˜R()TT+ ˜R() ηG ()T = . (144) same interpretation as for the planar rotor model. When TTm, 0 ⩽ 3 µµ˜RR()TT(2 ˜ ()+ λ˜R()T ) ξ−()T is the separation of the largest bound pair of primitive For systems with long range repulsive interactions, such dislocations and there are no larger pairs. For TT> m, up to 4 this scale, the dislocations can be regarded as bound in neutral as electrons trapped above the surface of superfluid He [150] pairs and triplets while at larger length scales the dislocations and colloidal particles at a water/air interface [144], the ratio are unbound and freely mobile. λ˜R()TT/1µ˜R() so that Since the dislocation fugacity y(l) scales to zero for TT, ⩽ m 4π 1 TT−−4 . the system is elastic and the renormalized elastic constants µπ˜R()mm=⇒ηG0()==− (145) 3µ˜R()T m 3 µ˜R()T and λ˜R()T are finite but non universal and are given by

25 Rep. Prog. Phys. 79 (2016) 026001 Review

This is not a very good estimate for a detailed comparison with 2 −1 a0 qqij theory because, the initial value of λ =∞ is renormalized by KTA ()= lim ⟨(bbijqq)(− )⟩. (151) 0 q 0 2 −− → Ω q dislocations to a finite valueλ ˜R()TTm /1µ˜R()m ≈ 0 for colloidal particles interacting by a 1/r3 repulsive interaction [143] and is The dislocation correlation function is to be calculated from ≈ 23 for electrons on 4He [151]. The quantity measured by experi- the dislocation free energy of equation (121) written in Fourier 2 ment on the electron system is Γ()Tn==π s ek/ BT 137 ± 15 space as [128, 129] at TT= m [150] and a remarkably good fit has been obtained by ⎡ 2 ⎤ combining a molecular dynamics simulation with the renormal- HD 1 K˜ ⎛ qqij⎞ 2Eac 0 = ⎢ δδij −+ ij⎥ bbijqq− . ization group [151 154]. ∑ 22⎜ ⎟ () () – kTB 2 Ω q ⎣ q ⎝ q ⎠ kTB ⎦ Above Tm, the dislocation fugacity is relevant and, just as (152) in the planar rotor model, there is a correlation length ξ T +() At this point, one sees why the sign of the angular term in which is the length below which dislocations can be considered the dislocation interaction in (121) is so important and why as bound and above which they are free [129, 130] in analogy ignoring the apparently shorter ranged angular term leads to with vortices in the planar rotor model [19, 20]. The positional incorrect results [19, 127]. The existence of a finite Franck correlation function will decay exponentially for r ξ+ constant, KA()T , depends on the transverse projection opera- −rT/ξ+() 2 CG()r ∼ e, tor, Tqij()=−δij qqij/q , in (152). One can understand the −ν (146) bt ˜ behavior of the orientational stiffness constant K T by using ξ+()Tb∼=e,whereconstant > 0. A() the recursion relations (128) up to the limit of their validity, The scale dependent elastic constants µ˜R()ll,0λ˜R()= for lT∗ = lnξ , to obtain l l +() e > ξ+()T but are finite fore < ξ+()T . This is an example of a 2l system which behaves as an elastic solid at short length scales KTAA()==Ky(µλ˜()0,˜()0, 0)(e,KlA µλ˜() ˜()ly,.()l ) (153) el < ξ and as a fluid for longer scalese l > ξ . The x-ray struc- + + The scaling factor e2l arises from the two extra factors of q ture function in the definition ofK A in equation (151) compared to µ˜ and l∗ −+2 ηG()T ⎪⎧ |−qG| c , (147) tem and a sensible approximation is to regard the dislo- as sketched in figure 17. cation density br() as a continuous variable as all values At first sight, this phase where dislocations are all unbound of br() are present at scale l*. This Debye–Hückel approx and free can be interpreted as a conventional isotropic fluid imation allows the correlation function ⟨bbij()qq()− ⟩ in [18, 19, 127] as the translational order decays exponentially (151) to be evaluated because the transverse part of the dis- with distance. However, in their seminal work, Halperin and location interaction in equation (152) does not contribute. Nelson [128, 129] pointed out that this fluid phase is not the One obtains expected isotropic fluid but has remnants of the six-fold ori- kT KT−1 B 0 entational order of the underlying hexagonal lattice of the low A ()≈>2 (154) temperature solid phase. These possible orientational correla- 2Eac 0 tions may be described by a finite renormalized Frank con- which establishes that the fluid caused by free dislocations stant KA T of the anisotropic fluid () above Tm does have orientational stiffness [128, 129] and is 2 not isotropic as assumed by Kosterlitz and Thouless [19]. −1 q KTA ()= lim ⟨(θθqq)(− )⟩, (148) Now one is left with a problem of a system with short q 0 Ω → range exponentially decaying translational order and alge- where θ is the angle of a bond between nearest neighbor par- braic orientational order described by phenomenological ticles relative to an arbitrary reference direction. free energy of equation (149) with a finite bare orientational One expects that the distribution of bond angles θ()r are stiffness constant KA()T of (154). This is the planar rotor controlled by a free energy [128, 129], problem discussed earlier in detail, the only difference that the topological defects are π/3 disclinations which interact FA 1 22 = KTA() d rr((∇θ )) (149) logarithmically like vortices. The bond angle fieldθ ()r is kTB 2 ∫ decomposed into a smooth part φ()r and a singular part θs()r where KA()T must be finite for the existence of orientational which obeys correlations. In the underlying solid phase, the bond angle mπ θs()r due to a set of dislocations is dlθs()r ==m 01±±,2,, (155) ∮C 3 a0 br()′′⋅−()rr θs()r =− ∑ , (150) where m is the total disinclination strength enclosed by the 2π |−rr′|2 r′ contour C. Thus one obtains the disclination free energy from which one can write the stiffness constant as a correla- [128, 129] which is identical to the planar rotor problem of tion function, equation (31)

26 Rep. Prog. Phys. 79 (2016) 026001 Review

⎞ The most important, unexpected, and controversial pre- HD 1 22πKA ⎛ |−rr′| ⎞ =−∫ d rr⎟((∇φ )) ∑ mm()rr()′ ln⎜ ⎟ diction of the defect theory of melting is that, in 2D, the kTB 2 ⎠ 36 rr, ′ ⎝ a ⎠ melting process is not a first order transition from a peri- Ec 2 odic solid to an isotropic fluid but proceeds by two succes- + ∑ m ()r . (156) kTB r sive continuous transitions with an orientationally ordered hexatic fluid between the low temperature periodic crystal Here m()r =±0, 1, is a measure of the strength of the dis- and the high temperature isotropic fluid. In fact, this sce- inclination at r, a is the core size of a disinclination, Ec is the nario has been verified quantitatively by careful experiments disclination core energy and the disclinations are subject to on a monolayer of paramagnetic colloidal particles trapped at a water/air interface [155 158]. A recent very large scale the neutrality condition, ∑r m()r = 0. – Halperin and Nelson [128, 129] pointed out that one can simulation of hard disks [159–161] verified the two step immediately take over all the results for the superfluid film melting scenario with an intermediate hexatic phase with and showed there is a disinclination unbinding transition at a continuous melting transition but a first order hexatic to T = TI to the high temperature isotropic fluid. The renormal- isotropic fluid transition. However, in a very recent study ized orientational stiffness constant just below TI is given by [162] in the presence of pinning disorder, no signs of any first order character at either transition was observed. Since 36 1 T− . the original theoretical prediction of continuous two stage R − ==η6()I (157) 2πKTA()I 4 melting [129], there have been a number of experimental and numerical studies performed to test the heretical prediction R The renormalized Franck constant, K A()T , behaves precisely of continuous melting by two continuous transitions with a like the renormalized stiffness constant of a 2D superfluid film hexatic fluid intervening between the solid and the isotropic except that, slightly above the melting temperature, fluid. At the time, conventional wisdom was that melting is a first order transition just as in 3D. The theoretical behavior R2ct−ν˜ TT− m KTA()∼∼ξ ()Tte,= , c = nonuniversal of the orientational stiffness K T is sketched in figure 19 + T A() m which agrees fairly well with experiment. + constant,and jumpsdiscontinuously to zeroat,T I

R 72 ⎛ ⎞ TT− I KTA()=+⎜1,bt||⎟ t = ⩽ 0, 3.2. Substrate effects on melting π ⎝ ⎠ TI R (158) Most experiments on 2D melting are performed on mono KTA()=>0, t 0. layer of adsorbed molecules on a supporting substrate such as When T > TI, the orientational order is short ranged, graphite which has relatively large well oriented domains of binding sites arranged in a periodic array. One must therefore ∗ −rT/ξ6() ⟨(ψψ6 r0)(6 )⟩ ∼ e, (159) investigate the effect of such a periodic substrate on the possible states of the over layer and on the transitions from the most where the orientational correlation length ξ6()T diverges + ordered to the most disordered states of the combined system. exponentially as TT→ I , Halperin and Nelson investigated this in their seminal papers [128, 129], and the results are summarized here. Both the ⎛ b ⎞ T exp + . ξ6()= ⎜ 1/2 ⎟ (160) graphite substrate and the adsorbate overlayer have the same ⎝ ()TT/1I − ⎠ six-fold orientational symmetry but a generic substrate has a ∗ different periodicity to the natural periodicity of the adsorbed Assuming that the initial value of the bare stiffness K0A()l c overlayer. This leads to the elastic free energy [128, 129] is larger than the critical value K0A, the renormalized value of R the orientational stiffness will flow to a valueK A()T ⩾ 72/π. H 1 d2q =−uDii()qqjj()u ()q For T slightly above the melting temperature Tm, the theory kT 2kT∫ 2 2 ∗ BB()π predicts that K R TT∼=e2l ξ2 and, at T−, K R T− = 72/π A() +() m A()I iKr⋅ R ++∑ hKe1((i,Ku⋅+r)) [129]. Experimental measurements of K A()T [158] agree well rK, with these theoretical predictions. If, for some reason, the dis- 2 Dqij()q =+µδR ij ()µλR + R qqij. (161) location core energy Ec of equation (154) is very small, it is 0 ∗ possible that K A()lK< 0c and will flow immediately to zero. One has assumed that the substrate potential is weak and iKr⋅+((ur)) A hexatic phase will not exist, but this seems unlikely. It is can be written as V()r = ∑K hKe where {K} are the also possible that the hexatic phase exists only over a very reciprocal lattice vectors of the periodic substrate potential. small range of TTm < ⩽ TI. Below Tm, the translational cor- The {K} are assumed incommensurate with the reciprocal lat- R relation length ξ+()T =∞, so that K A()T =∞, and there is tice vectors {G} of the adsorbate, assumed to be in a floating true long range orientational order, as was first pointed out by solid phase. This can be described by a harmonic free energy Mermin [24]. The behavior of the renormalised orientational with renormalized elastic constants µR and λR. Redefine the stiffness KA()T is shown in figure 19. displacement field [129]

27 Rep. Prog. Phys. 79 (2016) 026001 Review

−1 uui()qq→(′i )(=+uDi qq)(i,ij ) ∑ hKKKj∆ ,q K ⎧1if KG=+q ∆=Kq, ⎨ ⎩0otherwise,

H 1 d2q ′ =−∫ 2 uD′()qqij()u j()q kTBB2kT ()2π Ω 21− − ∑ hKK iDKij ()K j, (162) 2 K where G is a reciprocal lattice vector of the adsorbed layer. The second term of equation (162) can be rewritten as

2 2 Ω⋅2 ⎛ KKεˆs()⎞ Ch()θ =− ∑∑ K⎜ ⎟ , (163) 2 K s=1 ⎝ ωs()K ⎠ Figure 19. A schematic plot of the orientational stiffness constant KTR as a function of temperature. Reprinted from [238] with where K is the sth polarization vector and K is the A() εˆs() ωs() permission. Copyright 2002 Cambridge University Press. eigenfrequency of the matrix Dij()K . For a hexagonal over layer with lattice vectors {G} on a To translate the coupling constants K1 and K2 into the lan- hexagonal substrate with incommensurate lattice vectors {K}, guage of [130] where the recursion relations were first worked perfect alignment of the crystal axes with the substrate cor- out for this general dislocation Hamiltonian, one has responds to θ = 0, and one can write the Fourier expansion KK= 8,π2 r of C as 1 ()θ 2 θ (167) KK2 = 8.π ∞ Cc()θθ=Ω∑ kcos6()k , (164) From the recursion relations of [130], it follows that k=0 dKl 3 ⎡ K K ⎤ 1() 2/K2 8π 2 2 ⎛ 2 ⎞ ⎛ 2 ⎞ and, for a hexagonal adsorbate on a square substrate =− πyKe ⎢()1 +−KI2 0⎜⎟KK12I1⎜⎟⎥ dl 4 ⎣ ⎝ 88ππ⎠ ⎝ ⎠⎦ ∞ 3 + O()y , Cc()θθ=Ω∑ kcos1()2.k (165) k=0 dKl2() 3 2 ⎡ ⎛ K2 ⎞ 1 2 2 ⎛ K2 ⎞⎤ =− πyK⎢2 12KI0⎜⎟−+()KK1 2 I1⎜⎟⎥ Assuming that the relative orientations of the substrate and dl 4 ⎣ ⎝ 8ππ⎠ 28⎝ ⎠⎦ 3 adsorbed layer vary slowly in space, θ is the bond angle + O()y , r =∂1 uy′′//∂−∂∂ux so that, for small deviations dyl K K θ() ()xy θ () ⎛ 1 ⎞ 2/K2 16π ⎛ 2 ⎞ 3 2 =−⎜⎟2 yy++2eπ I0⎜⎟O()y . (168) from perfect alignment, one obtains the effective elastic dl ⎝ 8π ⎠ ⎝ 8π ⎠ Hamiltonian [129, 130] From this, it follows that dislocation unbinding melting H 1 d2r E 22 ′′2 −EkcB/ T =+[2,µλ˜uu′′ij ˜ kk +∂γ˜()yuux −∂x y ] takes place when the fugacity y0 = e becomes relevant, kT 2 ∫ a B 0 which is controlled by K1. The renormalized stiffness coef- 1 d2r H − R − 22˜ 2 D ficient atT m is K1 ()T m = 16π and the value of the exponent ν˜ =+∫ 2 [2,µφ˜ ij λφkk +∂γφ˜()y x −∂xφy ] + R −−R 2 a0 kTB depends on K2 ()TKMM/ 1 ()T . From (168) it is clear that K2(l) H 1 ⎛ |−rr′| flows to some non-universal finite value atl =∞ and, when D =− bbrr′ K ln ∑ ij() ()⎜ 1 δij K2 = 0, ν˜ = 2/5 as found by Nelson [127] and melting on an kT 8π rr′ a B ≠ ⎝ incommensurate substrate is very similar to a smooth sub- ()rr−−′′ij()rr⎞ Ec 2 strate. As pointed out by Nelson [77], the situation above Tm − K2 2 ⎟ + ∑ br() |−rr′| ⎠ kTB r is rather different. The orientational bias can be modeled by adding a term hppcos θ to the hexatic free energy 2 1 d q ⎡ KK12+ 1 ⎛ qqij⎞ = ⎢ ⎜δij − ⎟ 2 ∫ 222 q22q FA 2 ⎛ KTA() 2 ⎞ ()π ⎣ ⎝ ⎠ =∇d r⎜⎟()θθ− hppcos. (169) kTB ∫ ⎝ 2 ⎠ 2 ⎤ KK12− qqij 2Eac 0 + 4 +−δij⎥ bbij()qq(), Clearly, for a hexagonal overlayer with orientational order 2 q kTB ⎦ 6iθ parameter ψ6 = e on a hexagonal substrate with p = 6, h6 4µµ˜( ˜ + λ˜) 4µγ˜ ˜ acts like a uniform magnetic field inducing long range orien- K1 = + , 2µλ˜ + ˜ µγ˜ + ˜ tational order for all values of TT> m, wiping out the hexatic to isotropic fluid transition on a smooth substrate at TI. 4µµ˜( ˜ + λ˜) 4µγ˜ ˜ K2 = − . (166) When the substrate and adsorbed layer reciprocal lattice 2µλ˜ + ˜ µγ˜ + ˜ vectors G and K have a set M in common, the over layer { } { } { }

28 Rep. Prog. Phys. 79 (2016) 026001 Review

− Figure 20. Correlation functions CG()r (left) and C6()r (right). The solid lines have slopes 1/3 (left) and 1/4 (right). At TT= m, data for −1/3 − −1/4 CG()r ∼ r and, at TT= I , C6()r ∼ r . Reprinted from [157] with permission. Copyright 1999 American Physical Society.

may become locked to the substrate as a commensurate phase. for hM0 → 0 so that dislocation melting is inappropriate at any This situation can be studied by considering the smallest com- T. On the other hand, for a fine substrate mesh, M0 is large

mon reciprocal lattice vector of length M00=|M | and writing and λM0()Tm < 0, so that the overlayer is unlocked from the the free energy as [129] substrate in a temperature range TTl < m. One performs an analysis for the = ∑∑hK e (171) kTB K r stiffness constant KA()T , defined in equation (151) in exactly the same way, using a Debye–Hückel [78] approximation and with ||K > M0 are less relevant in the renormalization group the dislocation Hamiltonian of equation (166) to obtain sense than those with ||K = M0. 2 The relevance of the hM is obtained from the correlation a qq KT−1 lim 0 ij bbqq function computed from the renormalized elastic free energy A ()= 2 〈(ij)(− )〉 q→0 Ω q Hk/ T with h 0, EB() M = 2 2q ⎛ qqij⎞ bbqq−= δ − e,i0Mu⋅−[(ru)()] r−ηM()T 〈(ij)( )〉 2 2 ⎜ ij 2 ⎟ ⟨⟩HE ∼ 4Eqc a0 q KK12++ ⎝ ⎠ 2 kTB ||M kTB 3µλTT++R γ T R() () R() 2 ηM()T = . 2q qqij 4π µγTT++2µλTTR + , ((R )(R ))( R() ()) 2 2 2 4Eqc a0 q (172) KK12−+ kTB From this, the renormalization group eigenvalue of ⎧ 0, ()KK12≠ λM0l 2 hlMM00()= h ()0e is −1 2q ⎪ KTA = lim = ⎨ kTB () 4Eq2a2 , KK= 1 q→0 KK−+c 0 ⎪ 2 ()12 12 kT 2Eac 0 ληM0()TT=−2 M () B ⎩ 2 0 (174) 2 Mk0 BT 3µλR()TT++R() γR()T This means that the orientational stiffness constant K T =∞ =−2 . A() 8π ((µγR TT)(++R ))(2µλR()TTR()) for an adsorbate on an incommensurate substrate, which, in (173) turn, means that there is true long range orientational order for

From (173), at sufficiently low T, all the hM are relevant so all TT> m and that there is no high temperature transition to the that the over layer is locked to the substrate, implying that a expected isotropic fluid. Thus, the experimental investigation lattice gas description is more appropriate [163]. If the sub- seems to be a very difficult proposition and there is an excellent discussion in the review article by Strandburg [142] with strate is sufficiently coarse,λ M0()Tm > 0, so that the adsorbate is locked to the substrate up to the melting temperature Tm, extensive references to many aspects of the melting problem.

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3.3. Experimental verification of the KTHNY melting scenario − − so that, at T m, ΓmηG0()T m = 1.496, as shown in figure 22. One of the main interests in melting in 2D is the order of This difficulty was finally overcome in 1999 [157] by trapping the transition. It is generally agreed that the solid/liquid trans a monolayer of colloidal particles at an air/water interface, ition in 3D is first order characterized by a latent heat. In fact, doped with paramagnetic ions in a magnetic field B normal to the melting transition in 3D is considered as the paradigm of the interface. The interaction between the particles is charac- a discontinuous first order transition. The KTHNY theory of 23/2 terized by Γ = ()µπ0/4 ()χπBn()/()kTB where n is the areal melting in 2D predicting successive continuous transitions number density and χ the magnetic susceptibility. This system from the low T solid to the high T isotropic liquid came as a is as close to the conditions of the theory as it is possible to heretical claim which needed to be demonstrated by experi- get as the substrate is completely isotropic and free of any ment to be false. However, it turned out to be very difficult to destructive perturbations. The particle positions are recorded either verify or falsify the KTHNY theory because there were by video camera so that all correlation functions can be read- no experimental systems available which obeyed the very ily calculated and compared with theory. The outcome is that, stringent conditions of the theory which requires the adsorb- in this system, measurements agree quantitatively with the ate layer to be constrained by a featureless substrate potential KTHNY theory in all respects with no adjustable parameters of the form V()xy,,zV=− 00δ()zz− with V0B kT which [155, 156]. restricts the adsorbate to a flat layer at z = z0 where the sub- One of the major experimental difficulties is equilibrating strate surface is at z = 0. The theory requires that the potential 5 the system, despite it being only about 10 particles. In the high V(x, y, z) is independent of the substrate coordinates (x, y) field crystalline phase, the equilibration time is several days and this can be realized only in two situations: (i) electrons [157] even with the system being jiggled by applying a small trapped in a layer above the surface of 4He [150, 153, 154] ac B|| in the plane of the system to anneal out the large out of and (ii) paramagnetic colloidal particles trapped at a water/air equilibrium concentration of defects such as grain boundaries, interface [157]. interstitials and vacancies. Eventually, the system consists of a Experiments on orientational order in the hexatic phase single domain with less than 0.1% of dislocations. To melt this have also been performed on a layer of paramagnetic colloi- −1 2D crystal, the temperature Γ is increased by decreasing B dal particles interacting by a 1/r3 repulsive energy [155, 157, in small steps equilibrating for about one hour after each step 158] with very good agreement with the theoretical predic- [157]. This procedure results in CG()r decaying as a power tions [129]. Excellent reviews summarizing the present status law for TT⩽ m and exponentially for TT> m, and similarly for of the transitions between the three phases of particles in two C6()r , as shown in figure 20 dimensions with 1/r and 1/r3 repulsive interactions are in [144, 156]. Unfortunately, orientational order and the hexatic phase −ηG()T ⎧ rT,,⩽ Tm CG()r ∼ ⎨ cannot be studied in the electrons on helium system because −rT/ξ+() ⎩e,TT> m measurements can be made only on the frequency spec- (175) −η6()T trum of resonances of the coupled electron ripplon modes ⎧ rT,,m < TT⩽ I — C6()r ∼ ⎨ , [95, 150, 151]. The comparison between theory and experiment −rT/ξ6() ⎩e,TT> I for the orientational stiffness K˜A()T is shown in figure 24. The theoretical renormalization group predictions of the KTHNY where ηG()T is given in (143), ξ+()T in (146), η6()T in (157) theory are the solid line and the experimental measurements and ξ6()T in (160). The experimental results for CG()r and are the points. To the author s eye, the agreement is very good, C6()r are shown in figure 20 and are consistent with the theor ’ considering that there are no adjustable parameters in the fits. etical predictions. Note that CG()r in the solid phase decays as −ηG()T The orientational correlation function G6 r is investigated a power law r with ηG()Tm = 1/3. This agrees with the- () −η ory because the measured power law is actually r G0 where and the results are summarized in figure 23 where it is seen that G rr∼ −η6()T for r/3a 0, which is limited by the sam- G0 is the magnitude of the smallest reciprocal lattice vector 6() ⩽ ple size. G6()r is consistent with the theoretical predictions for a hexagonal lattice G00= 4/π ()a 3 . with a power law decay with η T 1/4 for TT< T and Measured values of the Young’s modulus K˜R()T compared 6()⩽ m ⩽ I with theory are shown in the upper panel of figure 21 and for exponential decay for T > TI. The correlation length ξ6()T is shown in figure 24(a) and is consistent with the theoretical the elastic constants µ T and λ˜R TT+ 2µ in the lower ˜R() () ˜R() bt−1/2 panel which also agree well with theory. The experimental prediction ξ6()T ∼ e although the experimental points are systems are electrons on 4He [150] and paramagnetic colloi- scattered about the theoretical solid line. The orientational dal particles at an air/water interface [144] as these are the stiffness constant, K˜A()T , is shown in figure 24(b) where agree- R + R + −1/2 closest to the conditions of theory and are free from unwanted ment with theory, K˜ A()T I = 72/π and K˜ A()Tbm ∼ exp()t . substrate effects. As shown in (145), for an incompressible On balance, the agreement with theory verifies the KTHNY theory of melting at least for this particular system as there lattice, ηG0 = 1/3 which is close to the theoretical value for particles interacting by a 1/r3 repulsive potential. As shown in are no contradictions. However, it must be mentioned that the [144], the exponent actual critical region in which the exponential divergence of bt−ν˜ the correlation length ξ+()t ∼ e sets in is estimated to be λµ˜R()TT+ 3 ˜R() 0.985 T = − , 108 lattice spacings [164]. This is much larger than acces- ηG0() →TT= m 2 (176) O() πµΓ+˜R()TT(λµ˜R() 2 ˜R()T ) 4π sible experimental sizes which casts grave doubt about the

30 Rep. Prog. Phys. 79 (2016) 026001 Review

significance of the fitting of data to the theoretical form of system by MC [27] and by molecular dynamics (MD) [28–30] ξ+! In particular, because of the limited sample sizes, these agreed on the equation of state and on a first order transition experiments are unable to exclude weak first order transitions in 3D with small finite size effects. Later, a MD simulation of seen in large scale simulations of hard disks [159–161]. N = 870 hard disks in 2D was done [31] where a first order melting/freezing transition was observed. Since these early studies, there has been over fifty years of numerical study on 3.4. First order transitions in planar rotor and melting models the apparently simple hard disk system culminating in the mas- The transition the 2D planar rotor and melting models are sive simulations by Krauth and coworkers [159–161]. It was argued to be driven by vortex and dislocation unbinding [19] discovered that 2D melting is neither first order nor KTHNY which lead to continuous transitions. However, symmetry but has the low T crystal melts to a hexatic fluid by a con- arguments do not exclude the possibility that, in some sys- tinuous KTHNY transition followed by a very weak first order tems, these defect driven transitions are first order. The earli- transition between the hexatic and isotropic fluids. est study of melting in 2D which gave convincing evidence of Since Alder and Wainwright’s simulation of the hard disk a transition was carried out by Alder and Wainwright [31] and system [31], many simulations of 2D hard disk and Lennard- there have been many others since then. Jones systems have been performed, and excellent reviews of However, a recent very large scale simulation of a hard the situation before 1985 are by [142, 169]. The great majority disk system with N = 10242 disks [160, 161] obtains results of simulations concluded that melting in 2D is by a single first which agree with the theoretical predictions for the solid— order transition, with few exceptions e.g. [170] who pointed hexatic transition but show that the hexatic—isotropic fluid out that his simulations were unable to distinguish between a transition is weakly first order with a discontinuity in the den- single weak first order and the KTHNY scenario. It is of inter- sity of about 0.015. Although this scenario sounds unlikely, est to note that the existence of the hexatic fluid phase has it is entirely possible. It is known that the KT transition can been confirmed by simulation only very recently [159, 161]. be made first order by reducing the vortex core energyE c to Even relatively large simulations have not been able to yield an below some critical value [165–167]. In [165], a modified pla- unambiguous confirmation of the existence of a hexatic phase nar rotor Hamiltonian on a square lattice was proposed [171–178] and conclude that 2D melting is a single first order transition with the exception of [176] where the conclusion H 2 p2 was that 2D melting takes place by a single continuous trans =−KT0()∑ ([1cos ((θθij− ))/2 ]). (177) kTB <>ij ition. This is mainly due to size limitations, the largest system simulated was N = 214 disks, but also the underlying expecta- This is precisely the planar rotor model when p = 1 which has tion that 2D melting should be similar to 3D first order melt- a continuous transition driven by vortex unbinding while, for ing. Thus, usually only orientational order was followed and large p, V θ ≈ 2J for θ π/p and V θθ≈ 1 Jp22 for θ π/p translational order assumed to be slaved to this but, even when () ⩾ () 2 [166]. The physical argument for a first order transition for the two orders were considered, the hexatic phase was not seen unambiguously. In a simulation of 220 hard disks, Jaster [179] p 1 is that the Hamiltonian of (177) resembles that of a Potts model [105] with a large number of states which is known to obtained evidence of the hexatic phase by finite size scaling undergo a first order transition [107]. Vortex unbinding is at but the nature of the hexatic—isotropic fluid transition could not be determined [180]. This was finally settled very recently k TJ≈ p2 and vacancy condensation at k TV≈=π 2J. BKT BD () by Krauth and coworkers [159, 161], who found a very nar- For p sufficiently large, one expects thatTT < so that the DKT row hexatic phase reached from the solid by KTHNY melting continuous KT transition is preempted by a first order vacancy followed by a very weak first order transition to the isotropic condensation [166]. These physical arguments are verified by fluid. The first order nature is verified by the finite size scal- simulations [165–167] although the system sizes are some- −1/2 what limited and proved rigorously in [168]. Also, the vortex ing of the interface free energy ∆fN()∼ N for particle number N = 2n, n = 14, 16, 18, 20, provided the correlation core energy Ec is reduced as p increases which indicates that a defect description with an appropriate defect fugacity y0 is length ξ N [181]. The hard disk simulations [160, 161] sufficient to account for both types of transition. In particular, obey this criterion thus establishing the first order nature of this can explain the first order hexatic-isotropic fluid transition the hexatic-isotropic fluid transition. The crystal-hexatic trans observed in simulations of the hard disk system [159–161]. ition was also checked to be KTHNY by the behavior of the 20 −η It is worth noting that this simulation of up to N = 2 hard density–density correlation function, ⟨ρρGG00r ()0 ⟩ ∼ r in the discs is, in fact, larger than any 2D experimental system. Of solid phase and ∼ e−r/ξ in the hexatic phase with η = 1/3 at the course, it differs somewhat from a real particle system since hexatic-crystal transition [160, 161]. these have interactions beyond the hard core repulsion. An elastic solid can be described by interacting dislocations In some of the earliest Monte Carlo (MC) simulations of and disclinations, provided the concentration of point defects hard spheres and discs interacting by a Lennard-Jones potential such as vacancies and interstitials is not too large [19, 128, 12 6 V()rr=−4/ε[(σσ)(/r)] [26], it was found that a system 129]. These point defects are irrelevant in the renormaliza- of N = 56 particles in 2D and up to 256 in 3D with peri- tion group sense and can be absorbed into the elastic constants odic boundary conditions showed indications of a gas/liquid µ, λ. Thus, a natural way to study melting numerically is to transition in 3D but not in 2D. Studies of the 3D hard sphere simulate the dislocation free energy of (121) with parameters

31 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 22. The exponent ηG0()T from the measured elastic constants µR()T and λR()T . The theoretical prediction (solid line) compared with experiment and Monte Carlo for ηγ compared with experiment and simulation. Reprinted from [144] with permission. Copyright 2005 IOP Publishing.

Figure 21. Upper: Young’s modulus KTR(). Lower: Elastic constants, theory and experiment. Reprinted from [143, 144] with permission. Copyright 2005 IOP Publishing and copyright 2004 American Physical Society.

K0 and dislocation core energy Ec. When Ec → 0, dislocations are expected to form grain boundaries between crystallites of different orientation and Chui [182, 183] constructed a theory Figure 23. Orientational correlation function G6()r . The top three −η6()T of grain boundary melting which gives a first order transition curves have G6()r = constant, the next two G6()r ∼ r with −rT/ξ6() where both translational and orientational order vanish simul- η6()T ⩽ 1/4 and the bottom two G6()r ∼ e . Reprinted from taneously. Saito [184, 185] simulated a system of dislocations [158] with permission. Copyright 2007 American Physical Society. of (121) for different values of the core energy Ec. It is found ∗ 3 that, for large Ec >≈EKc 1.22 0 a continuous KYHNY melting introducing a ti = 0, 1 to represent a He atom by ti = 0 and a ∗ 4 transition occurs at KR = 16π [129] but when Ec < Ec, there He atom by ti = 1. The resulting Hamiltonian is [86] is first order melting directly to an isotropic fluid [184, 185]. When E < E∗, the dislocations organize themselves into grain H c c =−Kt12∑∑ijtK[(1cos(θθij−−))] ttij+∆∑ ti. boundary loops and first order melting results [182, 183]. Of kTB <>ij <>ij i course, these point defects are important for the dynamics of the collective defects like dislocations. Dislocation glide conserves This was studied on a triangular lattice by an approximate particle number but climb or motion parallel to the Burgers vec- Migdal [187] type bond moving renormalization group pro- tor requires diffusion of vacancies and interstitials and is there- cedure due to Kadanoff [188]. As expected, both continuous fore very slow [186]. Our purely elastic description does not and discontinuous transitions appeared. Although never car- take such effects into consideration and to do this would need ried out with impurities, a calculation along these lines has more of a microscopic description. been performed by Nelson [127]. It seems that this would be In the simpler system of superfluid 4He with some non- a relatively simple way to study the effects of point defects superfluid defects,3 He atoms, there is a continuous transition (vacancies and interstitials) on the continuous dislocation and for low impurity concentration which becomes first order disclination unbinding transitions and to explore possible melt- for larger concentrations [86]. In this work, the impurities ing scenarios. This goes beyond the purely elastic theories of are incorporated into the planar rotor model on a lattice by 2D melting which require the introduction of disclinations to

32 Rep. Prog. Phys. 79 (2016) 026001 Review

bt−1/2 Figure 24. (a) Orientational correlation length ξ6()T . The solid line is from theory ξ6()T ∼ e . (b) Orientational stiffness constant KT˜AA()= βF compared with theory. Reprinted from [158] with permission. Copyright 2007 American Physical Society. induce a first order transition. One outstanding problem with The KTHNY melting theory inspired a large experimental this approach is that the first order transition induced by dis- effort to verify or falsify the two continuous transition sce- clinations [189, 190] requires going beyond linear elasticity nario, summarized by Strandburg in a comprehensive review and causes the simultaneous destruction of both translational [142]. There have been several studies of liquid crystal films and orientational order. This seems rather restrictive and cer- [96–103, 121, 199–201]. In some of these, melting seems to tainly does not always happen in real systems. A typical phase proceed according to the KTHNY scenario [121] but these diagram from purely elastic theory is sketched in figure 25. freely suspended systems are much richer than theory sug- In a series of papers [189–196], Kleinert and coworkers gests with many different possible phases depending on developed and simulated a model for 2D melting including temperature and film thickness [201]. One interesting obser- an important additional parameter which allows for first order vation is that, as the film reaches its minimum thickness of melting within a theory based on elasticity. By allowing for two layers, a hexatic phase is observed which seems to behave higher order elastic effects [195] one can incorporate local as predicted by theory [123, 202], although the evidence for a rotations separate hexatic phase is not conclusive. Adsorbed gases on substrates provide a huge variety of H 2 22 2222 =+d2ru[ µλ˜ ij ˜ulkk ++(24µλ˜ ˜)(∂+ikulki) µω˜ ()∂ 2D systems but the interplay of the periodic elastic adsorbed kT ∫ ′ B layer with reciprocal lattice vectors G and the periodic sub- + ], (178) strate with reciprocal lattice vectors M complicates things 1 dramatically. The adsorbed over layer can be commensurate where uij is the strain tensor and ω =∂()12uu−∂21 is the 2 or incommensurate with the substrate, depending on the rela- local rotation field. The parameters l2 and l 2 appear in q ′ ω() tionship of G and M [128, 129]. An early but comprehensive relations as [195] theoretical review on commensurate/incommensurate trans 2 224 itions is [163] and one on experiments on rare gases on graph- ωµT =+()qlq + (179) 2 224 ite [203]. Graphite is one of the most widely used substrates ωµL =+()2 λ ()ql++′ q because it can provide fairly large atomically flat surfaces However, Kleinert [195, 196] has shown that the parameter l′ 2 of linear size ∼1600 Å. Favorite adsorbates are rare gases is irrelevant for melting and considers only l2, which is a mea- such as Ar, Kr, Ne and Xe which are believed to interact by a sure of rotational stiffness. A model combining both mech simple Lennard-Jones interaction and differ only in size and anisms leading to first order melting [192] and [86] might be monolayers of these have been extensively studied by diffrac- more realistic than existing models. tion of synchrotron radiation [204–206]. Xenon adsorbed on Most simulations of melting of an elastic solid have been graphite has been extensively studied [207] where both first performed in the dual roughening representation, which is an order melting of the incommensurate solid at lower coverage exact transformation of the interacting dislocations and discli- and continuous melting at higher coverage was seen. Krypton nations with long range interactions into a roughening model on graphite has been studied extensively [206, 208–214] and on the dual lattice with short range interactions [142, 193, a tongue of fluid found separating the commensurate and 197, 198]. These models have a phase diagram of figure 25 incommensurate phases as shown in figure 26. High resolu- with both continuous melting or a first order transition directly tion scattering experiments are consistent with a smooth con- to the isotropic fluid and have a definite computational advan- tinuous F-IC transition [208] but the resolution is insufficient tage of nearest neighbor interactions but incorporating local to make detailed fits to theory [215]. However, the scattering defects is more awkward. data seems to be inconsistent with KTHNY melting [208].

33 Rep. Prog. Phys. 79 (2016) 026001 Review Argon is very similar to Krypton except it has a slightly of this broadening. Thus, the broadening of the transition is smaller atomic size and always forms an incommensurate taken to be intrinsic to the overlayer melting by successive solid overlayer on graphite with a phase diagram of figure 27 continuous transitions but the resolution does not permit veri- left. Diffraction studies of the melting transition around fication that this is KTHNY melting [233]. monolayer coverage are consistent with KTHNY melting but the correlation length is equally well fitted by a conventional −ν power law ξ()tt= ξ0 [216]. It must be remembered that 3.5. Melting of arrays of anisotropic particles length scales are limited to ξ = O 103 . These results agree ()Å Another relevant system in 2D is a freely suspended film of with an early specific heat measurement [217] but disagree liquid crystal [96, 100, 132] on which measurements are pos- with a later specific heat measurement [218] and earlier dif- sible. There are a number of theoretical papers generalizing fraction experiments [204]. A later investigation [219] is also earlier work on melting of arrays of point particles to 2D consistent with KTHNY melting and also with the prediction arrays of anisotropic particles [123, 202, 234]. Ostlund and that the solid over layer is rotated from the substrate axes Halperin [202] discuss dislocation mediated melting of 2D [220, 221] but again detailed comparison with theory is dif- arrays of rods tilted from the normal to the layer. Note that, if ficult because of insufficient experimental resolution. A study the tilt angle is zero, the system can be described by an array using high resolution vapor pressure isotherms [222] yields of point particles so the theory is the same as the previous sec- evidence in favor of continuous melting by a two step process tion. However, for tilted rods as in figure 28(a), the projection which is constant with KTHNY melting. Although not con- of the rods into the plane is a set of directed arrows arranged clusive, this study does resolve some discrepancies between in a distorted triangular lattice. Two obvious physical cases the older measurements, which makes the KTHNY scenario considered are (i) the projection of the molecular axes tend more likely for monolayer Argon on graphite. to point along one of the six nearest neighbor bond directions Xenon adsorbed on graphite has a phase diagram of as sketched in figure 28(b) and (ii) tend to point intermediate figure 27 right and has also been studied by diffraction [207, between two bonds as in figure 28(c). A very important con- 219, 223 225] and by thermodynamic measurements [226 – – sequence of the molecular tilt is that, even if in the absence of 228]. The synchrotron diffraction experiments [207, 223, 224] tilt, the molecules would form a regular hexagonal lattice, the are consistent with first order melting for low coverages which molecules will form a uniaxial hexagonal lattice [123]. crosses over to KTHNY continuous melting at higher cover- To study the dislocation mediated melting of a 2D solid ages. Thermodynamic measurements of the compressibil- composed of rodlike molecules tilted at angle θ from the plane ity of the overlayer [227] are consistent with this. However, and the subsequent transitions which is relevant for freely sus- a high precision heat capacity and compressibility measure- pended films of smectic liquid crystals [96, 100, 121, 131, ments [228] are not consistent with a crossover to continuous 132], we must go through the same steps as for the melting of KTHNY melting and claim that melting is always first order. a hexagonal lattice. Unfortunately, even the first step is rather The scattering experiment of Rosenbaum et al [225] showed complicated because the energy of a set of dislocations in a that melting at high coverages is well described by KTHNY uniaxial hexagonal lattice was not known in the late 1970 s. melting in the presence of a weak h cos6θ substrate potential. ’ 6 The most explicit calculation is in [235] but, as pointed out by A study of melting of Xenon adsorbed on Ag(1 1 1) [229] Ostlund and Halperin [202] the result depended on the choice agrees with the KTHNY scenario. In fact, the 6-fold substrate of a cutoff and must be incorrect. In appendix A of [202] the potential of the silver substrate is extremely small [229] and dislocation energy is calculated without a cutoff and is more this model fits the scattering data very well. The observed hex- likely to be correct. In this review, I will try to focus on the atic order of the fluid seems entirely due to the Xenon over- salient points and leave all the gory technical details to the layer. The only effect of the substrate is to align the 6-fold references [202]. hexatic axes by steps in the substrate [229] and the hexatic The first complication is to compute the energy of a set of orientational order is intrinsic to the Xenon over layer. The dislocations embedded in the elastic medium with the uniaxial remaining rare gas, Neon, on graphite has also been studied symmetry of the underlying distorted hexagonal lattice shown [230 232] but in much less detail than the others. In view of – in figure 29. In such a system, there are four different nonzero the recent simulations of the hard disk system [159] it seems elastic constants or, equivalently, four distinct finite compli- that the conclusions from experiments must be viewed with ances. The compliance tensor S is the inverse of the elastic some caution because of the very limited system sizes and the ijkl tensor C or C S =+1 δδ δδ . The finite compli- difficulty of distinguishing continuous from weak first order ijkl ijklklmni2 ()mjninjm transitions. ances are S1111,,SS2222 1122, S1212 where the axes are shown in Many other adsorbates on graphite have been studied by figure 29. thermodynamic measurements with many complicated phase diagrams proposed. A neutron scattering study of melting of HH()rr1E n =+HD, ethylene adsorbed on graphite has yielded evidence of a con- H 1 d2r E = φφC , tinuous transition [233] and compared to methane on the same ∫ 2 ij ijkl kl kTB 2 a0 substrate. Both films melt from incommensurate solids which 6 rules out substrate heterogeneity being responsible for the HD 1 α β αβ c αα =−∑∑bbi j EEij()rr+ ijbbij, (180) apparent continuity in ethylene. Methane films show no signs kTB 2 αβ≠=1 α

34 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 25. Reprinted with permission from [193]. Copyright Figure 26. Phase diagram of Krypton adsorbed on graphite 1988 Elsevier. A schematic phase diagram for melting in 2D in a showing the fluid (F), the 33× commensurate phase (C), the purely elastic theory. The continuous solid/hexatic and hexatic/ reentrant fluid (RF) and the incommensurate solid (IC). The IC/RF isotropic fluid transitions are labelled KT. The dashed line marks the transition is the relevant melting transition here. Reproduced from 2 2 continuation of the solid/hexatic transition to ll< c where the sold [208] with permission. Copyright 1984 American Physical Society. line is a first order solid/isotropic fluid transition. See also [195, 196] dS 1111 =+yF2 22 2cyF2 os2φ 22, where αβ, denote one of the six dislocations of figure 29 dl I 1 II 0 2 in an anisotropic hexagonal lattice, i, j = 1, 2 are the two dS2222 α 2 2 11 independent directions in 2D and r is the position of the = 2syFin φ0 2 , dl II dislocation of type α. There are two independent Burger’s dS1122 2 12 vectors bI and bII. The interaction energy of a pair of ± type = 2syFII inφφ00cos,2 αα dl α dislocations separated by r is Kα = bbEijij()r , where dS1212 1 2 11 1 2 2 2 22 E r =+Krln /aVθ is the interaction energy which is =+yF1 yF(cossφφ0 + in 0 2 ij() ij () ij() dl 4 I 2 II a very, complicated expression given in appendix A of [202]. 12 The solid phase has dislocations which are tightly bound − 2sinφφ00cos,F2 ) in pairs and when the temperature is raised, the pairs unbind dyI ⎛ KI ⎞ 2 =−⎜⎟2 yA+ 2,23y and the crystalline order is destroyed. This melting occurs dl ⎝ 2 ⎠ I II when ∆FE=∆ −∆TS where ∆Sk= ln La22/ and B () dyII ⎛ KII ⎞ =−⎜⎟2 yA+ 2,12yy ∆EK= π ln()La/ where K is determined by the lattice spacing dl ⎝ 2 ⎠ II III and elastic constants, L is the size of the system and a the lattice 2π 2π αβ ij α 2 −Vα()θ bbijVij()θ constant [19]. The melting temperature is given by ∆F = 0 or wheredFrα ==()b θθˆˆijrAe,αβ de . ∫∫0 0 πKk= 2 BT. There are two types of dislocation and two values (181) K and K in an anisotropic solid, as shown in figure 29. Thus, I II An expression for Vij()θ is given in appendix A of [202]. The melting of the solid will be controlled by the smaller of KII, K I ij quantities Fα and Aαβ depend on the compliances which have which is, in turn, controlled by the shorter of the elemen- 2 finite fixed point values and differ from these byO ()y . Thus, tary Burger’s vectors bI and bII. Following the discussion of one can consider these as l-independent constants [202]. [123, 202, 234], one can study the melting of the anisotropic For type I melting, K > 4 and K ≈ 4 so that the fugacity solid by the renormalization group procedure of section 3.1. II I y 0 because KII −>20 and the flow equations involving Following [202], the compliance tensor Sijkl is the inverse of II → ( 2 ) 1 the elastic tensor C so that C S =+δδ δδ and, yI become ijkl ijklklmni2 ()mjninjm 2 to O()y , and Sijkl is given by dS1111 2 22 = yFI 1 , 3 2π dl 0 1 2 −Vα()θ αα dS 1 SSijkl =+ijkl ∑ ybα ∫ deθ ( i εεjmrbˆˆm + j imrm) 1212 2 11 4 0 = yF1 , α=1 dl 4 I ∞ 3−Kα αα dr ⎛ r ⎞ 3 dyI 1 ×+(brk εεlnˆˆn brl kn n)(⎜⎟ + O y ), =−()4.KyI I (182) ∫a a ⎝ a ⎠ dl 2

where the interaction energy of a ± pair of type α dislocations Near the melting temperature, the compliances S1122 and S2222 separated by r is are dominated by analytic terms since their RG flows are not controlled by the fugacity y . The compliances S , S are EKrr=+ln ra/.V I 1111 1212 αα() () α(ˆ) expanded as

The RG equations are derived from this by the standard ∗ δl Sl1111()=+Sm1111 ((ylI )(+ Dl)). (183) method of rescaling aa→ e to obtain [202]

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Figure 27. Phase diagrams of argon (left) and xenon (right) adsorbed on graphite. Reproduced from [216] (argon) and [224] (xenon) with permission. Copyright 1987 American Physical Society, and copyright 1983 Elsevier. Here, S∗ is the fixed point value of S at melting and is 1111 1111 S ⎛ 2 2E 2 ⎞ HD Bk2 ()λ x sξS 2 equal to its renormalized value at Tm. D(l) is the deviation from = d kk⎜ +|⎟ b | , ∫ ⎜ 2 2 4 ⎟ y() the incoming separatrix with Dt0 =|| and y is the value of kTB 2 ⎝ kkyx+ λ B ⎠ () I the fugacity on the incoming separatrix. ∗ −12 2 c ∗∗2 BS==()2222 ,2λξSES11 2222 = 2,ξSS2222 Below melting, TT⩽ m, after some further algebra [202], c ∗∗∗∗ EEs ==22()lySS−=ln II()lp()ll12+ . (186) one can derive the flow equations for ylI() and D(l) At first sight, it seems that at this length scale one can 22 dD F1 treat both type I and type II dislocations by a Debye–Hückel =− yD, ∗ I approximation but yl 1 so that the discrete nature of bII dl m− II()S must still be accounted for [234]. Thus, a further l∗ iterations dyI F1 2 3 = yI (184) are performed so that dl m− ∗∗ ∗ ∗∗∗−pl()12++ll3 −1 ylII()12++ll3 =⇒yII()0e = 1. (187) where the slope m− < 0 [202]. For ||t 1, ylI()∼ l for l 0. Also Dl()ylII()= Dy()00() which allows one to define a At this length scale, the system becomes a gas of free disloca- * ∗∗ * value l such that Dl()= ylI() such that for l > l , the fugac- tions of all types and one can use a Debye–Hückel approx ∗ 2 imation. At this scale, the Frank constants ity ylI()→ 0 so that one can write ||tD∝∝()0 ()l . In turn, this observation, allows one to define a length scaleξ −()t and 22 −1/2 Ktx ∼∼∼ξξSSexp2(), the singularities in the compliances 22()p+1 Ky ∼∼ξξNS . (188) −1/2 ξ−()tt=|exp,()| At length scale L with Lt< ξS(), all dislocations are R ∗ Stijkl()=+Sbijkl 1,||t ()(ijkl = 1111)(, 1212), bound and the system responds like a solid while, if () ξ < 0 and b′ > 0 are nonuniversal constants O 1 . Thus 4 () terms like (∇⋅n) in the Hamiltonian will renormalize Kxy, K for TT⩽ m, the compliances S1111 and S1212 have ||t singulari- 2 to equality for Lc>∼ξξI exp()S [236]. This melting scenario ties while S1122 and S2222 have undetectable essential singu- is called type I melting [202] with a phase diagram sketched in −2p larities ξS at Tm, since KII > 4 [202]. figure 27, left. When KII > KI, the smectic-like phase is absent Above melting, TT> m, the RG trajectories for ylI() follow and the phase diagram is sketched in figure 27, right. ∗ the incoming separatrix up to ll= 1, break away and join the ∗ ∗ outgoing separatrix ylI()=−mS+((1111 lS))1111 for another l2 ∗ 3.6. Superconductivity in two dimensions RG iterations. ylI()2 = O()1 but is still small. Thus, we see ∗∗ that the length scale ξS ∼+exp()ll12 is the typical separation Another system which, at first sight, seems that it should of free dislocations of type I. The fugacity ylII() is decreas- be a good example of experimentally accessible Kosterlitz ∗∗ – ing exponentially for ll⩽ 12+ l . Thus, we can integrate out Thouless physics is a thin film of superconductor. In this the type I dislocations by a Debye–Hückel approximation to system, it is clear that the physics will be controlled by vor- obtain an effective Hamiltonian for the type II dislocations at tices in the phase of the superconducting order parameter ∗∗∗ scale ll=+l , which is the Hamiltonian for a 2D smectic iφ r S 12 Ψ()rr=|Ψ|()e () where φ is the phase of the condensate liquid crystal [202, 234] wavefunction. This is exactly analogous to the superfluid

36 Rep. Prog. Phys. 79 (2016) 026001 Review

and e* = 2e. The essential difference between the neutral 4He superfluid and the charged superfluid of a superconductor is that a vortex in a superconductor has a circulating electric current producing an associated magnetic fieldB ()r, z which extends outside the film at( r,0). The behavior of a superconducting film was first discussed by Pearl [237] who found that the circulating current of a vortex at r = 0 falls off as 1/r2 instead of exponentially as in a bulk superconductor. Also, the screening length 2 in a film isλ eff = λ0/d, where d is the film thickness and ∗ 22∗ ∗ 4 ˚ λ0 ==mc /4((πnes )) O( 10 A) is the London penetration depth. The Ginzburg–Landau coherence length is 1 d ξ0 = ∗ (190) m 2||rT()

which is the length scale over which |Ψ()r | varies so that Ψ()r, z is constant over the film thickness. Since one is inter- ested in the temperature range over which phase fluctuations are important, it is clear that the parameter r()T 0 so that |Ψ()r |=O()1 since the phase is not defined otherwise. Thus, one can assume that the temperature TT 0, the Ginzburg– Landau critical temperature. The system is effectively two dimensional when the film thicknessd ξ0 since we can integrate (189) over z to obtain an effective 2D theory with 2 0 |Ψ0|=ns, the Cooper pair areal number density, which is taken to be constant except at vortex cores. The vortex core has radius O()ξ0 which is the smallest length scale of the system. When H 0, these considerations lead to the free energy Figure 28. (a) Tilted molecules in a Smectic C monolayer. = (b) Top view of monolayer of tilted molecules where arrows are F()θ, A upper ends. The molecular axes tend to point along one of the ⎧ 0 ∗ 2 ⎫ nearest neighbor directions so that arrows tend to form chains. The 2 ns ⎛ e ⎞ 1 2 F()θθ,dAr=∇dz⎨ ⎜⎟−+AA()∇ × ⎬ distorted hexagonal lattice formed by the centers of the molecules ∫ 2m∗ ⎝ c ⎠ 8π is shown by dotted lines. (c) Same as (b) except that the projections ⎩ ⎭ of the molecular axes lie halfway between nearest neighbor bonds. (191) Reprinted from [202], with permission. Copyright 1981 Elsevier. From the extremal equation δF/0δA = , we obtain

4 ∗ ∗ He situation. The main difference is that the superconduct- 4πJ 2 e ⎛ e ⎞ ∇∇××()AB=×∇∇==4,πθ|Ψ0|−⎜⎟A ing condensate is charged which implies that the all impor- c mc∗ ⎝ c ⎠ tant vortex vortex interaction is screened from the lnr form of ∗ – ∗ 2 ⎛ e ⎞ ⇒=Jr(),,ze|Ψ0|−⎜⎟∇θ A (192) the neutral superfluid to 1/r due to screening currents flowing m∗ ⎝ c ⎠ round the vortices [237, 238]. where e* = 2e and mm∗ = 2 are the charge and mass of a This is best discussed from the Ginzburg–Landau form of e the free energy for a charged condensate of Cooper pairs [239] Cooper pair. To see the effect of the fieldB ()r, z due to a set of vortices in the superconducting plane at z = 0, one follows ∗ 2 ⎪⎧ the analysis of Pearl [237] and Nelson [238] by assuming the 2 1 ⎛ e ⎞ F[]Ψ= ddrAz⎨ ⎜⎟−−i ∇ Ψ ∫ ⎪ ∗ superconducting current is only in the z = 0 plane so that ⎩ 2m ⎝ c ⎠

2 ⎫ cφ ⎛ 2π ⎞ 1 241 B HB⋅ Jr, z 0 rAr z , +|rT()Ψ| +|u Ψ| +− ⎬. (189) ()=−2 ⎜∇θ() 2d()⎟δ() (193) 2 48ππ4 ⎭ 8πλ⊥ ⎝ φ0 ⎠

Ψ()r is the coarse grained Cooper pair wave function which is where the flux quantumφ 0 = π ce/ , A2d()rA= ()r,0, and the 22 non zero only in the plane of the film z = 0, A()r, z is the vec- transverse penetration depth λ⊥ = mces/8()πne . Now one tor potential which is finite outside the plane of the film and can use the standard equations of magnetostatics to solve for B()rA, z =×∇ . The parameter r(T ) < 0 for T < T0 the BCS the current due to a vortex of unit strength at r = 0 so that temperature for the onset of the formation of Cooper pairs and 1 φ zr× u is positive and T independent. Here r = xy, is a point in 2Ar, z Ar 0 ˆ z , () −∇ ()+=2d() 2 δ() (194) ∗ λ 2πλ r the film in the z = 0 plane, the Cooper pair mass mm= 2 e ⊥⊥

37 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 29. Left: (a) Direct lattice vectors and elementary Burgers vectors for the lattice of figure 28(b) and (b) reciprocal lattice vectors. Right: the same but for figure 28(c). Reprinted from [202], with permission. Copyright 1981 Elsevier.

2 since ∇θπ()rz=×2/(ˆ r) r due to the vortex at r = 0 and In the absence of an external magnetic fieldH = 0, the condi- A2d()rA≡=()r,0z . This can be solved by taking Fourier tion for the onset of superconductivity is given by the ana- transforms, with the result [237, 238] logue of (46), [240, 241]

cφ φ2 J r, θ = 0 ˆ,,r λ kT T 0 , (197) 2d() 2 θ ⊥ Bcλ⊥()c = 2 8dπλ⊥r 32π

cφ0 where T is the temperature at which phase coherence first = θˆ,,r λ⊥ (195) c 4dπ22r appears From this discussion, we see that there can be no true which translates into the vortex–vortex interaction behaving 2 phase transition in a superconducting film because of the as lnr for r < λ⊥ and as 1/r for r λ⊥. It is very important to note that the onset of superconduc- finite penetration depthλ ⊥()T for TT⩽ c [19]. However, since −2 tivity in 2D is due to vortex type phase fluctuations which λ⊥()T can be O()10 m which is larger than most experimental requires a pre-existing condensate or pairing of electrons. This systems so that we can regard most superconducting films at finite temperature when quantum effects are unimportant as phenomenon is described by a Ginzburg–Landau free energy functional of (189) in which the condensate forms at the mean described by the class of Hamiltonians field temperature T0 when r(T0) = 0. Even at T < T0, although F()θ, A 2 ⎡ ⎛ 2e ⎞⎤ there is a finite condensate or BCS pairing,|Ψ |>2 0, there =−KT0() d1rr⎢ cos⎜⎟∇−θ() Ar()⎥ 0 kT ∫ ⎣ ⎝ c ⎠⎦ is not necessarily any superconductivity. This requires phase B 2 coherence or a finite renormalized superfluid density, just KT0() 2 ⎛ 2e ⎞ =∇d rr⎜⎟θ()− Ar() , as in the neutral superfluid 4He which is controlled by vor- 2 ∫ ⎝ c ⎠ tices. The onset of phase coherence or true superconductiv- 2 0 nTs() ity in the thermodynamic limit of infinite system size L ∞ KT0()= , (198) → 2mkBT occurs when the experimentally measurable kinetic induct- 2 ance LTKe()= mn/2()se which is related to the transverse 0 ◻ for a uniform 2D superconducting film, wheren s()T is the penetration depth bare superfluid number density and 2m the Cooper pair mass. mc2 LTc2 In a clean uniform film, the mean field Cooper pair forma- T e K◻() . λ⊥()==2 (196) tion temperature T0 and the phase coherence onset temper 84ππens ature TTc0< are rather close and effects due to these tend to

38 Rep. Prog. Phys. 79 (2016) 026001 Review

become mixed together. It is known that Cooper pairing alone where 0.4 ⩽⩽C 0.75 [252]. When TT> m, theory says that does not imply superconductivity which requires global phase there should be a hexatic fluid up to Ti but this is not relevant coherence. Thus, to study superconductivity in a thin film, as there is no known probe coupling to a vortex hexatic. Of it is important that T0 and Tc are well separated. This can be course, in any real system, there will be pinning of vortices by accomplished by a granular film with a large resistance RN in some random pinning sites [252] which will not be discussed the normal state [242–247] or by coupling the grains by weak in this review. However, it should be noted that the assumed links [248] as in arrays of Josephson junctions [249]. KTHNY melting of the vortex lattice is controversial. Even When the external magnetic field Hz = 0, the supercon- the existence of a 2D flux lattice has been questioned [254]. ducting film should be like a finite size 2D planar rotor model Later work [255–261] showed that, although numerical esti- with L = λ⊥. It was pointed out [240, 241] that the vortex mates of µR()T are consistent with the KTHNY scenario, vor- unbinding mechanism [19] could be very important in thin tex lattice melting proceeds by a weakly first order transition superconducting films provided the transverse penetration so that µR()T is very close to the KTHNY value. depth λ⊥ was very large which could be realized in very thin There has been much interest in trying to apply the the- films with a large normal sheet resistanceλ ⊥ ∝ R◻. When ory of 2D vortex lattice melting to thin films of YBCCO in a 3 R◻ >Ω10 , there is a reasonable separation between the pair transverse magnetic field. This has met with only partial suc- formation and KT temperatures TT0K> T which is essential for cess [262–264] as vortex lattice melting does not appear to be the experimental observation of vortex unbinding. They pro- the mechanism for the destruction of superconductivity [265,

posed [240] that TTKT ≈ 2.18 0cRR/ ◻ but, on a more fundamen- 266]. Vortex pinning by random impurities dominates [267] tal level [249, 250], so that the system is better described as a moving glass phase of a current driven moving elastic lattice [268–271]. However, 2 π icφ πE kT 0 J , the basic ideas of dislocations and disclinations in a 2D vortex BKT ≈=2 = (199) 84eL◻ 2 lattice have been verified by direct observation [272]. where the vortex fugacity is ignored, which fits the data much better. 3.7. Superconducting arrays In a uniform superconducting film in a finite magnetic field A regular array of superconducting grains coupled by almost B normal to the plane of the film there will be a hexagonal lat- identical SNS or SIS junctions can be made with up to about tice of vortices each containing one flux quantumφ 0 = ce/2 106 grains with modern lithographic deposition techniques which form a stable triangular lattice of lattice spacing a0 [273] and almost any desired geometry and coupling con- where [251, 252] stant arrangement can be constructed. Also, when an external magnetic field is applied, a frustrated XY model can be 3 a2 1 φ 0 ==0 , (200) obtained, described by Coulomb gas Hamiltonian [274] 2 nB 1 −1 where n is the areal vortex density due to the field. In the pres- Hq=−()i QCi ij ()qQj −+jjEAJ1((1c−−os θθ− ij)), 2 ∑ ence of arbitrarily weak pinning of vortices, the flux lattice ij (203) will be pinned for TT< m, the lattice melting temperature, and where the first term is the charging energy with Cij the capaci- vortices will be free to move when TT> m so that the onset tance matrix, qei = 2 ni with ni the number of Cooper pairs on of superconductivity can be identified with the vortex lattice ex grain i, and Qi ==2eqex ∑j CVij j is the charge on grain i melting temperature Tm [252]. induced by the external gate voltage. The capacitance matrix 2 Cij has diagonal elements Cii =+Cz0 C where z is the number aT0µµ()mm[(TT)(+ λ m)] = 4,π (201) of nearest neighbors and Cij = −C for i, j nearest neighbors kTBm2µλTTmm+ [()()] as shown in figure 32. EJ is the Josephson coupling energy j and µ()T = 0 when TT> m. In (201), the elastic constants between adjacent grains and Aij =⋅()2/ec ∫ Ald . The R R i should be interpreted as the renormalized µ ()T and λ ()T . magnetic frustration is the flux f penetrating a plaquette given The phase diagram of a 2D superconductor in the (H, T) plane by fA1/2 where means the sum over the bonds is sketched in figure 31 assuming that the transition is caused = ()π ∑P ij ∑P by vortex lattice melting [252]. round the plaquette in a counter clockwise direction. Quantum −2 −2 mechanical effects result from the non commuting operators In the most important field regime,ξ GL B/φλ0 ⊥ , the −3 [θδij,in ] = ij which become important at TK= O()10 ° when bare λ0 =∞ because of the long range vortex–vortex interac- the capacitances C , C are small so the grain charging energy tions and the bare shear modulus [253] 0 Ec is comparable to the Josephson energy EJ. φ B Many experiments are done on arrays of large capac- T 0 , µ0()= 2 ity superconducting grains on a lattice coupled either by the 32πλ⊥()T proximity effect [248] or arrays of Josephson junctions with φ2 kT = C 0 , (202) ECJ E [249]. These are best described by a lattice model of Bm 2 64 3 πλ⊥()Tm the form

39 Rep. Prog. Phys. 79 (2016) 026001 Review

⎡ ⎤ The problem was discussed in [283] where it was explained F()θ, A KT0() ⎛ 2π ⎞ =−∑ ⎢1cos⎜θθij−− Aij⎟⎥ , (204) that the renormalized helicity modulus in the XY ordered phase kTB 2 ⎢ φ ⎥ ⟨⟩ij ⎣ ⎝ 0 ⎠⎦ must obey ϒ()Tk/2BT ⩾ /π and will drop discontinuously to

rj zero at TT= KT with a jump to zero ∆ϒ/2kTBKT ⩾ /π. There where Aij =⋅drA()r and charging effects can be ignored. ∫ri are two possibilities: (i) ∆ϒ=/2kTBKT /π at some TTKT < I or − A junction array in a uniform magnetic field is a realization (ii) as TT→ I , there is a non-universal jump ∆ϒ>/2kTB /π. of an interesting class of models which, in the Coulomb gas At first sight, these alternatives could be readily distinguished representation is by simulation, but this hope was rapidly dashed because either H the system sizes were too small to resolve the two transitions =+πKT0()∑()nfi i Gnij()j + fj (205) or they could be resolved with TTKT < I but the jump in ϒ()T kTB ij, at TKT we larger than the universal value of 2/π. This situation where i labels the center of the ith plaquette on the led to the suggestion that the XY-Ising model [287] would be easier to simulate where the Hamiltonian is dual lattice, ni =±0, 1, and 0 ⩽⩽fi 1/2 the magn etic flux fi =⋅2/πφ0 ∮ dlA penetrating the ith pla- H =−∑ [(AB+−ssij)(cos,θθij)]+ Csijs (207) quette. The interaction Gij is the screened Coulomb kTB <>ij interaction, Gij ∼|ln()RRij−|/a for |−RRij|<λ⊥ and 22 where si =±1 represents the chirality of a plaquette. However, Gij ∼|()a / RRij−| when |−RRij|>λ⊥. Even in this classical limit, these systems provide realizations of a great variety in the end it turns out that despite the short range interactions of models in 2D and have been the subject of intense activity of the XY-Ising model, in the interesting region of C ⩽ 0 and A = B, simulations of the model of (207) turn out to be just as in the 1980’s. The models become even more interesting, and difficult, at very low T when quantum effects play a vital role difficult as the original representation. The derivation of (207) for frustrated systems was done [274–282]. The systems with a Hamiltonian of (205) represent a large in a series of papers [287, 288, 304–310] by a set of approxi- class of statistical mechanical models. When the frustration mate steps based on symmetry and renormalization group arguments and missed some of the short distance physics. fi = 0 in zero applied magnetic field, the system on a square lattice becomes a 2D planar rotor model when the screen- For example, for an isotropic fully frustrated junction array [309], A = B so that there is no phase stiffness across an ing length λ⊥ → ∞. In a junction array one can make this larger than the array size, so the junction array is a realiza- Ising domain wall with ssij=−1 when i, j represent sites on tion of the planar rotor model. In the phase representation the opposite sides of domains of opposite chirality. In the origi- Hamiltonian of an array is [283], nal system, described by (206) with fi = f = 1/2, there is a finite phase stiffness across domain walls. However, in an H important paper, Korshunov [311] showed that domain wall =−KA0 ∑ [(1cos θθij−−ij)] , kTB <>ij kinks unbind at TTKV< , the vortex unbinding or phase coherence temperature. It was also shown that TT< , the Ising AAij ++jk AAkl +=li 2.πf ()R (206) VDW critical temperature. These results were verified numerically Because of the arbitrariness in the vector potential, for the fully [312]. This establishes, in an isotropic fully frustrated system, frustrated case of f = 1/2, one can choose Aij = 0 on horizon- (i) phase coherence is destroyed at TTV < I and (ii) the domain tal bonds and Aij = π on alternating vertical bonds. The sys- walls roughen at TTKV< so that the XY-Ising model of (207) tem is mapped into one with ferromagnetic bonds of strength describes the long distance behavior near continuous trans J on horizontal bonds and alternating vertical strength J ferro- itions of (206). It is a reasonable model of a fully frustrated magnetic and strength ηJ antiferromagnetic bonds [284]. The isotropic junction array on a square or triangular lattice. This phase diagram in the η, T plane from MC simulations shows is contrary to the findings in [289] where a difference was that, away from the isotropic point η = 1, there is an Ising found between a square and triangular lattice when f = 1/2. transition followed by an XY transition at higher T. The situa- There is yet another representation obtained by taking the tion near the isotropic point η = 1 is unclear but they seemed dual of the XY part which yields the RSOS-Ising model on a to occur simultaneously [284]. square lattice with About the same time as this investigation was being done, there was much interest in the isotropic fully frustrated array Hh[], s =|Ah˜˜∑∑I −|hCJ + ssij, (208) with f = 1/2 [283–303]. It was well understood that the iso- kTB <>I,J <>ij, tropic fully frustrated system has both chiral order and phase coherence at sufficiently low temperature and that phase where the nearest neighbor sites are on the original coherence required chiral, Ising, long range order, but not square lattice and are the corresponding sites on the vice-versa. Thus the main question was whether the XY trans dual lattice. hI =±0, 1, si =±1 and A˜ and C˜ are given in ition happened at a lower temperature than the Ising trans terms of A and C of (207), [313]. There is a constraint on the ition or whether the two transitions happened simultaneously height variables hI such that a step in h is forbidden to cross and, if so, what is the nature of this transition? The problem is a domain wall in s [313]. Another duality transformation on analytically intractable and there have been many simulations the si gives exactly the coarse grained model for CsCl(001) over the years, mostly inconclusive with a few exceptions. facets [314]. The XY-Ising model of (207) describes the

40 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 30. Phase diagrams for 2D systems of elongated molecules in the L−1, T plane where L is the shorter of the experimental length

scale or ξ+()T . The dashed lines are cross-overs between different behaviors at L = ξi()T (see text). The left figure is for type I and the right is type II melting. Reprinted from [202] with permission. Copyright 1981 Elsevier. critical behavior of a surprising number of models, includ- There are few experiments on these fully frustrated systems ing the 4-state Potts model and also contains the much dis- because the XY and Ising transitions are too close to resolve cussed points where Ising and XY degrees of freedom are in an isotropic junction array. However, they become quite simultaneously critical at A −=Bx± c where the Ising and well separated in an anisotropic array [284, 305] but there are XY transition lines cross, as shown in figure 32, center. There some technical issues in fabricating an array with the required has been some discussion that new critical behavior might be anisotropy in the coupling strengths. The helicity modulus in found in isotropic systems at a transition in a new universal- an isotropic array with Hamiltonian ity class between a fully ordered to a disordered phase [296, H 309, 315–317]. This was later shown not to happen by Olsson =−∑()ABµµ++ss()rr()µµcos((θθrr)(−+)) by large scale simulations which were sufficiently accurate to kTB r,µ distinguish the two transitions [318, 319]. The phase diagrams −+Csµ ()rrs()µ , of figure 32 summarise my understanding of these fully frustrated planbar rotor models. where µ = xy, for a square lattice. The measurable helicity Finally, it seems that some issues still remain in the moduli are [305] near the low T Ising transition reflects the isotropic fully frustrated system despite all the effort Ising singularity [305] expended. In a relatively recent study of the FFXY model and the coupled Ising XY model (207) on the A = B line by γµ ∼−TTIIln|−TT|. 6 () (209) large scale simulations with up to 10 sites on a square lat- kTB tice [320], there is agreement with [313] for a wide range This very weak singularity was observed in a custom built junc- of the parameter A and C but disagreement when C < −5 tion array [325] by measuring the complex sheet impedance when the Ising and XY transition lines seem to merge into ZT()ωω=+RT(),iωωLT(), and observing a small peak in a single line at a multicritical point at a critical value C* the resistivity at TTI < KT well below the BKT temperature with −5 > C* > −7 [320]. In the RSOS Ising representa- 1 obtained from the universal relation LT− = 8/πφ kT . tion, the two critical lines do not merge into a single line, but ◻ ()KT ()0 BKT The custom built junction array and the peak in the resistivity just become very close together [313]. This disagreement is are shown in figure 33. now purely philosophical and might be resolved by studying Quantum effects become important at very low temper the full XYI model of (207) in the space of all parameters A, atures and when the charging and Josephson energies are B, C. It is unlikely, but not impossible, that a single multic- comparable E /1E = O . This requires very small junctions C A B cJ () ritical point will emerge for 0 with − → 0 but such as discussed in a comprehensive recent review [274]. The a scenario will require massive computations. The issue of partition function corresponding to the Hamiltonian of (203) systems with nearby Ising and XY transitions also arises in is [275] the competition between surface reconstruction and roughening in (1 1 0) facets of fcc crystals which are described ZS=ΠiDφτi()exp,([− φ]) by a two component body centered solid on solid model or ∫ β ⎡ ⎤ a staggered 6-vertex model [321 324]. Unfortunately, this C0 2 ⎛ C 2 ⎞ – S φτ=+d ⎢ φφ˙ ⎜⎟˙˙−−φφE cos,− φ ⎥ [] ∫ 2 ∑∑i 2 ()ij J ()ij is also insoluble but is amenable to accurate transfer matrix 0 ⎣⎢ 8e i <>ij ⎝ 8e ⎠⎦⎥ studies. (210)

41 Rep. Prog. Phys. 79 (2016) 026001 Review

β ⎡ 2 −1 Sq[],dve=+∫ ττ∑ ⎢2 qCi() ij qEj()τπJvGii()ττjjv () 0 ij ⎣ 1 ⎤ ++qG˙˙()ττijqqji() i Θijv˙j()τ ⎥ , 4πEJ ⎦ ⎛ yy− ⎞ −1 ij Θ=ij tan,⎜ ⎟ ⎝ xxij− ⎠ 1 Grij =− ln()ij/aE+ a (215) 2

where the charge qi()τ and the vorticity vi()τ are integer valued functions of imaginary time τ. Gij is the interaction between two vortices separated by rij/a lattice spacings with Ea the short distance contribution. Provided the capacitance to ground C0 C, the capacitance between neighboring grains, the interaction between charges has the same form as between Figure 31. Theoretical phase diagram in the (H, T ) plane for a vortices uniform thin film superconductor in a magnetic field H with vortex solid, hexatic and fluid phases. The flux lattice melting curve (TM) 2 −1 EC eCij = Gij, (216) is the solid line and the hexatic-fluid transition (TH) is the dashed π line. The H = 0 BKT transition is at Tc. The bulk upper critical field Hc2(T ) is sketched as the dotted line terminating at the bulk BCS so that, from (215), the charges and vortices are dual with a 2 temperature Tc0. The lower critical field Hc1(T ) is so small that, on self dual point at EJC/2E = /π which is exact when C0 = 0 the scale of the figure, it is indistinguishable from H = 0. Reprinted and in the absence of the spin wave term qG˙˙q in (215). This from [252] with permission. Copyright 1980 Elsevier. term is irrelevant for the statics but not for dynamics. These considerations lead to a phase diagram as shown in figure 34, ˙ where β = 1/kTB and φ = d/φτd . The first and second terms right, which should be compared with the experimental phase are charging energies in terms of the voltages at the lattice sites. diagram of figure 34, center. To include dissipative tunneling, the action includes a term There are some interesting predictions arising from these 1 β considerations related to the experimental observation that, at SFD[]φτ=−∫ ddτα′′∑ ()ττ((φτij )(− φτij ′)), (211) 2 0 <>ij the border of the superconducting -insulator transition, the 2D ∗ 2 where, for Ohmic dissipation system is metallic with a resistivity ρ ()Th=≈0/ρQ = ()2e , the quantum of resistance. An oversimplified argument is in π 1 . [326] based on vortex flow and duality. The Josephson rela- ατ()= 22 2 (212) 2eRβπN sin/()τβ tion for the voltage V across a junction and the current I are The dissipation mechanism might be normal electron tunnel- ˙ h ing by discrete charge transfer and V ==θ v˙, 2e 2e Ie= 2 q˙, ⎛ φτij()− φτij()′ ⎞ FQP[]φij =−1cos⎜ ⎟ (213) V h v˙ ⎝ 2 ⎠ ⇒=ρ = (217) I 4e2 q˙ which is 4π periodic. Other mechanisms which yield qua- At the self dual point of (215), we have qv= , which dratic forms of F φ [274] and may force the phase φ τ to ⟨ ⟩⟨⟩ [] () means that for every Cooper pair crossing a square, one vortex be continuous. If the dissipation is by discrete charge transfer, crosses perpendicular to the Cooper pair. Thus, at the T 0 SI the path integral implies a sum over winding numbers so that = 2 transition the system is metallic with resistivity ρ∗ = he/4(). 2π φπi0+2 mi Of course, the self duality is not exact but one might expect DDd, φφ→(∏∑i0 φτi ) (214) ∗ ∫∫0 ∫φ that , especially as the self dual point is a univer- i mi=±0, 1, i0 ρ = O()ρQ sal fixed point in the RG sense. Some explicit calculations because the charges on the grains are integer multiples of 2e of ρ∗ have been performed, mostly as a 1/N expansion [274]. [326–328]. In the exactly soluble N =∞ limit, the universal To obtain the dual Coulomb gas action for a Josephson ∗ resistivity of a superconducting array ρ ()08= RQ/π in zero junction array, one expresses the array partition function applied field [327] and, in the fully frustrated case of f = 1/2, in terms of a sum over charge and vortex configurations as ∗∗ ρ ()1/20==ρπ()/2 4/RQ [328]. Measurements of the T = 0 ZS= ∑qv, exp,[(− qv)] by [274] universal resistivity [329] are not easy.

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Figure 32. Left: Reprinted from [309] with permission. Copyright 1991 Elsevier. Showing the state of understanding of the FFXY model in 1991. Right: Phase diagram of the RSOS Ising model with A = B dual to the FFXY model. Note the sliver of intervening Ising ordered phase for all values of C. Reprinted from [313] with permission. Copyright 1997 Elsevier.

4. Dynamics dlnD =−4,π22y dl

As mentioned earlier, to interpret experiments like the tor- dC sional oscillator measurements of the superfluid density in = 0. (219) dl a 2D 4He film, one must extend the static theory discussed to finite frequencyω . This is because the quantity called the The renormalized vortex diffusion constant DTR()==Dl()∞ 1/2 superfluid density is theq ==ω 0 limit of the linear response =+DTRc()()1 bt with DTRc()> 0 so that DTR() has a cusp at T . As pointed out in [76] this has little effect so that function ρs()q, ω where gs()qq,,ωρ= s()ωωvqs(), is the super- c the dynamics is described by diffusive motion of the vorti- fluid momentum density andvq s(), ω the superfluid velocity. ces supplemented by using renormalized values of K(T). The response function ρs()q, ω happens to have the dimensions of mass per unit area but has to be measured dynamically by, Experimentally, the vortex diffusivity D does have some − for example, Bishop and Reppy’s torsional oscillator experi- dependence on T [337, 338] and seems to increase as TT→ c , ment [67, 68] which must be done at ω > 0 or a third sound which is the opposite of the theoretical prediction [330, 331]. To my knowledge, this discrepancy has not been resolved and measurement at q > 0 and ω > 0. The quantity ρs for which it is not known if it is significant. When the frequency ω and exact calculations can be done is ρs()q,0ω = which is inaccessible to experiment. To relate theory [19, 20, 56, 63] to the average superfluid velocityu s are small, one immediately experiment [36, 37, 67–69], the dynamical extension of the sees that the average superfluid velocity in a film of width W static theory is needed. and length L decays by the motion of vortices perpendicular This was done at the same time as the key Bishop–Reppy to us as experiment by a group at Cornell [76, 330, 331]. Following ν dus 2dπ r the early work of Hall and Vinen [332–336] by balancing =− ∑ nνzˆ × . (220) Magnus and drag forces, the position rν of the νth vortex of dtmLW ν dt strength n =±1 obeys [76] ν A uniform flow of superfluid relative to the substrate decays slowly to zero because superfluidity in 2D is only metastable ν D2 0 dr ννπρs νν =×n zvˆ ((ns−+vr)) C((vvns−+rv)) s()r and vortex pairs are driven apart over the free energy barrier dt mk T B to become free as if the system is above Tc. + ην()t , Ambegaokar et al [76] also noted that the motion of νµ vortices on an oscillating substrate is equivalent to the dif- ηηtt′′=−2,Dtδδij µνδ t 〈(ij)()〉 () 20 fusive motion of 2D charges ±q0 where qm0 = πρs / µ vrss()=×zrˆ ∇+∑ nGµ (),,ru()r between capacitor plates in an external electric field with m µ 0 qm0Ezext =×2/πρs() ˆ vn. In linear response theory, the the 2 |−rr′| time dependent behavior of the system, for v 1 and 1 ∇=GG()rr,2′′πδ()rr−⇒()rr,l′ ≈ n,(218) ||n ω a is encoded in a dielectric function ε ()ω as a function of frequency ω defined by ignoring boundaries. By writing the Fokker Planck equation for the N vortex −1 – uvsn()ωω=−((1 ε )) ()ω distribution function Γ rν, t corresponding to (218) [76, 330, N() 41πωPE=−ε −1 ωω. (221) 331] the scaling equations for the various parameters can be () (()) ext() 2 found. The scaling equations to O()y for K(l) and the fugac- Using this capacitor analogy, it is not difficult to see that, iωt ity y(l) are the same as for the statics and, for the diffusion when the substrate oscillates as vtnn()= v e , the power dis- constant D(l) and the parameter C(l) are sipated per unit area is

43 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 33. Left: a small piece of the anisotropic array with alternating weak (WB) and strong coupling (SB) rows. Right: The peak in R at TTI < KT. Reprinted from [325] with permission. Copyright 2002 Elsevier.

Figure 34. Reprinted from [274] with permission. Copyright 2001 Elsevier. The figure to the left is a sketch of a piece of an array with the parameter C0 and C shown, the middle figure is the measured phase diagram for f = 0 for a square (solid squares) and a triangular (solid triangles) showing the S-I transition for EcJ/E ∼ 1.7. The solid line is a guide to the eye and the dotted line on the superconducting side is from a calculation [280, 281].

r 1 02 −1 2 dr′ Pv0 =−ρωIm ε ω . (222) Ur =−2q qEEr⋅+2, s n [()] () 0 ∫ 0 c (224) 2 a0 rr′′ε˜ () and, provided the oscillating system has no dissipation other −1 where ε˜ ()rK= 0Kr(), defined in (33), accounts for the than in the 4He film of area A, the Q and the shift ∆P of the vortex pairs of separation less than r. The contribution of all oscillator period P are bound vortex pairs to the dielectric function ε ()ω for TT< c 0 when ξ−()Tr D = ()14D/ω , where D is the vortex diffu- 1 ρsA 1 Q−−=−Im[(ε ω)], sion constant, is [68, 76, 331] M

ρ0A ∞ dε r 14Dr−2 ∆P s −1 ε ω =+1dr ˜() , = Re[(ε ω)]. (223) b() ∫ −2 P 2M a0 dr 14Dr − iω

Thus, most important dynamical behavior is encoded in the Reεεb()ωω≈ ˜ 14D/, the dielectric function ε ()ω defined in (221) which can be esti- () mated in different regimes. π ⎛ dε˜()r ⎞ Imεb()ω ≈ ⎜⎟r . (225) In the Coulomb gas language, a vortex of unit circulation 4 ⎝ dr ⎠rD= 14 /ω 0 can be regarded as a particle with charge qm= 2/πρ 1/2 in − 0 ()s Nothing very dramatic happens as TT but, when TT> c, 0 → c an electric fieldE with qmEzext =×2/πρ vn. The energy 0 s()ˆ vortex pairs with separation r > ξ+()T are free and the indi- U(r) of a pair of vortices of circulation n =±1 and separation vidual vortices diffuse freely in the fieldE and (225), using r is [19, 63, 76] the Coulomb gas analogy,

44 Rep. Prog. Phys. 79 (2016) 026001 Review

+ 4πσ mt()rr,,=−[(hh0 tf)] /,g εε()ωω=+b () i , ω ⎡ ⎤ 2 ⎛ dρ¯ ⎞ ξ −2 gf=+⎢ρ h0⎜⎟ ⎥ , + + dε˜ ()r 14Dr ¯ ε ω =+1dr , ⎣ ⎝ dh ⎠hh= 0⎦ b () −2 (226) ∫a dr 14Dr − iω 2 ∂vs ⎛ ⎞ =−gm∇ zJˆ × v, 20F ∂t σπ==nDf qD0 2,ρs ⎜ ⎟ F = O()1. ⎝ mξ+ ⎠ ∂m 0 =⋅ρ ∇ vs. (229) ∂t s Here, the analogy of the motion of charges between capacitor plate is used [76] and the estimate of the free vortex density The last step is to absorb the effects of the bound vortex (charge) 0 − −2 pairs into the dielectric constant εb ≈=εc ρρs/ s()T c , so that the nf = Fξ+ where F is a constant O()1 [76]. Putting everything together, one arrives at expressions free vortex density Nfree()rr,,tN=−+−()tN()rr,,tN≡ δ ()t for ε ()ω [68, 339] in terms of experimentally measurable and quantities and a direct comparison of theory and experiment ∂vs [68, 339, 340] can be made as shown in figure 35. However, εc,=−gm∇ zJˆ × v free, ∂t other samples do not fit the predictions of theory as well as this ∂m 0 which is believed to be due to substrate inhomogeneity [339]. =⋅ρsg∇ Jv,free, The agreement is remarkable and the data also gives a value ∂t − εc∇ ×=vzs δN ˆ, for the static exponent η()T c =±0.25 15% which again agrees well with the theoretical value of 0.25. This gives some ∂δN =⋅∇ Jv,free. (230) confidence that the vortex theory [19, 20, 56] and the dynami- ∂t cal extension [76] is a good description of the physics of thin The last step is to use the Langevin equation for r˙i to express 4He and superconducting films, in the linear response regime. the free vortex current Jv,free in terms of vs and one finds [76], AHNS [76] discuss the effects of vortices on the third in a linear approximation sound modes in thin superfluid films , which are due to variations in the film thickness or mass density, [341–346], using Jzvs,free =−γδ0 ˆ ×−v DN∇ , the hydrodynamic equations of Bergman [347, 348] supple- 0 2 Dnρs f ⎛ 2π ⎞ −2 ⎜⎟ mented by equations for the vortex density N()r, t and vortex γ0 =∼ξ+ . (231) kTB ⎝ m ⎠ current Jv()r, t To find the frequencies of the modes, one takes Fourier 2π transforms of everything in (230) and write v qv, ω =+v , Nt()rr, =−∑ ntiiδ((r )), s() LT m i where vL,T are the longitudinal and transverse components of 2dπ ri vs. vL()q, ω couples to m()q, ω and the eigenfrequencies of the Jrv(), t =−∑ ni δ((rri t)), longitudinal modes are solutions of [76] m i dt ∇ ×=vrs(),,tN()rzt ˆ, 2 2 0 2 εcωγ+−i00ωρgqs = , ∂N =−∇ ⋅ Jv, 4ε gq0 2 ∂t iγγ0 0 c ρs ⇒=ω±()q −± − 1. (232) 22 2 ∂vs εεc c γ0 +×zJˆ v = ∇µ()r,,t ∂t Thus, for qg /2 ε 0 the eigenfrequencies are ∇∇µ()r,,tS=−¯ Tf∇h (227) γρ0 c s 2 −2 where S¯ is the film entropy per unit mass and f the van der ωγ+ =−i/0 εc +∼O()q −i,ξ+ 2 Waals constant. To simplify this rather complicated problem ω− =−i,Dqh but keeping the essential physics, one can ignore variations in Dg=∼ργ0 2/.ξ2 (233) temperature in the film and mass transport between film and h s 0 + vapor so that 2 The transverse part of vs relaxes as ω0()qD=−i/()γ0 εc +≈q −2 ∂(ρ¯h) 0 −i/γξ0 εc ∼−i + [76]. =−ρ ∇ ⋅ vs. (228) ∂t s These results can be used to define a measurable mass transport coefficientλ by [76] Now one can write down a complete set of equations for the film which are just like the Maxwell equations of electro- ∂ ρh ( ¯ ) =−λµ∇=22λfh∇ . (234) magnetism [76]. Defining ∂t

45 Rep. Prog. Phys. 79 (2016) 026001 Review

Comparing this with the relaxation rate ω− from (233) one sees One obtains R in the two limits rc ξ− and rc ξ− [76] 2 2 that λ =∼Dgh/ ξ . Remarkably, measurements of λ [70, 71, + ⎧ D ⎛ a ⎞4+xT() 340] agree that λ ∼ ξ2 as TT+ although the exponential ⎪ xT 2 0 rT, + → c 4 ()⎜ ⎟ c ξ−() ⎪ a ⎝ rc ⎠ dependence on TT− c differs somewhat from the torsional R ∝ ⎨ 0 (239) oscillator parameters [340]. When TTc, third sound modes ⎪ D ⩽ rTc ξ . ⎪ 4 2 −() will propagate and have a dispersion ω()qc= 3q where c3 is the ⎩ rrc()ln c 2 2 0 third sound velocity. From (232), for qg> γρ0/4()εc s , there are Since nf relaxes quickly it will be determined by the instanta- two propagating but damped modes and, when TT< c, γ0 = 0, neous value of the superfluid velocity so that so that there are propagating modes for all q. AHNS [76] argue that, for TT< , ε should be replaced by ε q, ωω= ε and xT c c b() () nt ()rt−πKTR(), f ()∼ 2 c() Jv,free ==0 Nfree so that a 0 du 220 2 s πKTR()+1 ε ωω −=gqρ 0, ∼|xT()uvsn−| ,.rTc ξ−() (240) b() s dt ω ωωImεb() ⇒=q±()ω ±+i , When one applies a uniform super current density Js there cc2Re 33()ω ()ωωεb() is dissipation caused by a flow of current induced vortices 0 00 cc3()ωω==3 Reεεb();/cg3 ρs c . (235) perpendicular to the direction of Js from (218). From the Josephson relation [352, 353] a mean electric field E is gener- As q → 0 with TT−

πKT()−1 ω±()qc=± 33()0i− Dq (236) dφ 2π E ==vn⊥ f , (241) 2etd 2e where D3 > 0 is a constant and K(T ) is the renormalized stiffness constant (53). When TT= c, we have where v⊥ is the mean velocity of the vortices normal to Juss∝ , the superfluid velocity. Thus one obtains [351] ⎛ π ⎞ ω qc=±01q − i . −1 2/+x 2 () 3()⎜ 2 2 ⎟ (237) Ej//s ≈ σn xj()s j0 , (242) ⎝ 4ln/((Dqca3()0 0)) ⎠ where

Thus, it follows that third sound propagates when TT< c for xT≡ max1/lnξ ,1/ln jj0 /.s (243) 02− [(− )()] any wavenumber q ≠ 0 and for qg>∼γρ0/2 εc ξ when ()s + α()T Thus, for infinitesmally small js, we have V ∼ I where TT> c [76]. There is an excellent discussion of thermal conduction and third sound in 4He films by Teitel [349] where he α()TT>=c 1 corresponding to the expected Ohmic behavior, attempts to fit the thermal resistance data using parameters α()Tc = 3 and απ()TT<=cR13+>KT() which agrees with from the simultaneous torsional oscillator dissipation data as experiment [247–249] as shown in figure 37, left. However, in shown in figure 36. a more recent study on a thin film of highT c superconductor Of course, it is very easy experimentally to get out of this [354] of figure 37, right, the IV relation is also power law but regime into a regime where non linear effects dominate. In this was used to argue for the absence of a KT transition in a torsional oscillator, large amplitude oscillations are all that the high Tc film because there is Ohmic behavior atTT c. is needed and a simpler situation is the decay of a uniform A commonly used criterion to estimate TKT from the IV rela- superfluid flow or of a persistent current in a superconduc- tions of a thin film is to defineT KT as the temperature at which α T = 3. However, a quick glance at (242) shows that, as js tor. In this situation, the superfluid velocityu s or current in a () superconductor decays by the unwinding of the phase of the increases, the IV characteristics all become power laws with α ≈ 3. Also, when j is small enough, all the IV curves become order parameter ψ rr=|ψ |eiθ()r by the motion of vortices s () () Ohmic because of the finite screening length in charged super- perpendicular to u . s fluids [18, 19, 355]. For large currents jj/ −1, from (242) The non linear effects have been studied by several workers s 0 ξ− 3 [76, 339, 350, 351] who conclude that the decay of the mean we see that E ∼ I which seems to agree with the data of fig- superfluid velocity is due to the escape of vortices over the ure 37 for both t < 0 and t > 0 [355]. This follows because, when jj0 / s < ξ−, (242) becomes energy barrier at pair separation rc given by Errccε˜ ()= 2q0.

AHNS [76] show that the rate of production of free vortices 2 −−1/2ln/()jj0 s 21/2 Ej//ss≈=()jj0 ()jj0 //s ()jjs 0 e , by escape over the barrier is 3 ⇒∝Ejs (244) dnf 2 =−Rvσc ⊥n f (238) Making a quick comparison of this with the IV characteristics dt in figure 37 it seems that they agree with (244) for the large where R is the rate of escape over the barrier, the capture cross values of jjs / 0 in the upper left corner where all the IV charac- section σs ∼ rc and v⊥ the velocity of the vortices normal to us. teristics appear to have almost the same slope of 3. 24l∗ The free vortex production rate R = 2eDy ()l∗− [76] where Early investigations of 2D superconductivity and the trans * l depends on the two large length scales rc and ξ−()T . ition to the normal state were first discussed in [240, 242–249].

46 Rep. Prog. Phys. 79 (2016) 026001 Review The fits of experimental data to theory is, at fist sight, very where the geometrical factor F(q) is determined by the geom- convincing but some of the fits are spurious. The fitting of the etry of the detector coil. For a circular pick up coil of N turns experimental IV relation for TT⩽ c is fine but the agreement with of radius r, thickness b and height h above the superconduct- the resistivity R ∼∼nbf exp/()− t [245, 247, 249, 356, 357] ing film is outside the true critical region where the exponential form −Nbq 2πrJ1()qr hq 1e− of the resistivity is not valid. Similar criticism can be made of Fq()= e− . (247) a number of studies of the onset of superconductivity in films q 1e− −bq of YBa23Cu O7 films [358–366] which claimed KT behavior This can be related to correction to the mutual impedance near onset, even for thick films. δZm()ω as measured by the two coil method More recent experiments on the dynamics of superconducting arrays of SNS coupled grains on a triangular lattice where 2 2 2 µω0 d q Fq()Fq˜() vortex pinning by impurities is minimized show that the vor- δωZm()=− . (248) 4 ∫ 2iπω2 −+Lq Zq, ω tex dynamics has some surprising behavior [367] where it was () s() ◻() found that the vortex mobility µv()ω vanished as the frequency Here, F˜()q is the geometrical factor for the drive coil, and, ω → 0 ignoring the difference between F(q) and F˜()q , one sees that S and Z are linearly related ⎛ 1 1 ⎞ ()ω δ m()ω f //2 i/2 , µωv()=+()πµ0⎜ ()π 2 ⎟ (245) ln / ln / 2kTB ⎝ ()ωω0 ()ωω0 ⎠ S Re Z . ()ω =− 2 [(δωm )] (249) 2 ω where µ = ωπ0/2 qnv with the vortex charge qE= π J and 0 v v Following the discussion in [370], when the height h of the 2 the areal vortex density. nv = 4/fa()3 coil is sufficiently large one can ignore the q dependence of This experimental observation agrees with the conjecture Zq, ω and write Zq,iωω=− LRωω+ so that, for by Minnhagen [350] later justified [368 370] who showed ◻() ◻◻() () ◻() – weak screening and Lq L , that, in the absence of any pinning by impurities, vortex s() ◻ dynamics become very sluggish as ω → 0 due to their cou- 2 R◻()ω pling to spin waves. However, the results were somewhat Sk()ωπ= µ BTY()hr/, 0 ωω2LR22+ ω puzzling until a study [371] demonstrated that both SIS ◻◻() () and SNS junction arrays should show similar behavior and ⎧ 1 ,,hr (250) apparent differences can be explained by different sizes of ⎪ 2 Yh/r ≈ the three regions. The dynamics of these superconducting ()⎨ 2 ⎪ r ⎪ ,,hr arrays is determined by the competing length scales ξ+()T ⎩16h2 and the probe length r()ωω=Γ0/ . The three regions are (i) which is consistent with [379]. When TT, L is almost r()ωξ + one recovers the expected Drude free vortex behav- ⩽ KT ◻()ω ior in the hydrodynamic region with the vortex ‘conductiv- constant, so that [370], 2 ity σ1 ωξ0 ∼ . (ii) At intermediate scales r ωξ∼ the πKTR()−−31 ’ (→) + () + ST()ωω∼∼ω , ≈ TKT (251) σ ∼ lnω behavior [350, 367 369] is recovered, while, at high 1 – which agrees with observation [380, 381]. In the strong frequencies, (iii) r()ωξ +, in a temperature regime from screening limit, LL()ω s()q and it can be shown that the TT> KT down to T = 0, one finds a scale dependent vortex ◻ 22 21− ω dependence depends very much on the geometry of the damping with σ1()ωω∼∼((r ))/ln ωω((lnω)) corresp onding to large vortex pairs moving in a viscous medium of experiment [370]. For r h, one findsS ()ωω∼ 1/ , the ubiq- f smaller pairs. A finite free density of free vortices when the uitous 1/ noise. However, it seems difficult to understand the observed 4 decade range of this 1/ω noise [380, 381]. Noise external field B = 0 is obtained when TT> KT and, for TT⩽ KT, −2 measurements in films of YBCO [382–384] also show that when B ≠ 0 when nv =∼2πξf where f = φφ/ is the num- 03 + 0 S()ωω∼ 1/ for 10 <<ω 10 which appears consistent with ber of flux quanta per plaquette. this discussion. One non invasive probe of a 2D superconducting film or Perhaps this is not the full story but this work [370] is more array is the two coil mutual inductance method [243, 247, successful in explaining experimental observations than most 372–379] which measures the voltage induced in the detec- others. To obtain the 1/ω behavior over 4 decades requires a tion coil by the currents in the film induced by the current in ratio r/1h ∼ 04 which is much larger than the experimental the drive coil. This method measures the complex impedance ratio. Thus, it seems that this aspect of the flux noise experi- per unit area at frequency ω, Z◻()ω . Another method is even ments is still awaiting explanation. less invasive which is the flux noise spectrumS ()ω which is related to the impedance Z◻ by [370]

2 2 5. Cold atoms and conclusion µ kTB d q ⎛ 1 ⎞ S ω = 0 Fq2 Re , () ∫ 2 () ⎜ ⎟ 2 ()2πω⎝ −+i,Lqs() Zq◻()ω ⎠ In the last decade the number of papers on cold atoms has µ0 exploded and there is renewed interest in topic such as Bose Lqs()= , 2q Einstein condensation (BEC) and superfluidity in two dimen- (246) sional systems [385–404]. From my simple minded viewpoint, 47 Rep. Prog. Phys. 79 (2016) 026001 Review definition of the condensate density which should be different to the definition of the superfluid density. Some years ago, Leggett [386, 387] defined the global superfluid density fraction by considering N Bosons enclosed by a rotating circular surface of radius R with angular speed ω [386]. He defined the superfluid fraction

ρs()T ⟨⟩L fTs ()==1 − lim , (254) ρωω→0 Ic where ⟨L⟩ is the expectation value of the mechanical angular 2 momentum and INc = mR/2 is the classical moment of iner- tia. Using rigorous but subtle arguments, he proved, among other things, that, in the thermodynamic limit, that fs(0) = 1 for interacting Bosons and fs(0) = 0 if the Bosons do not interact, thus establishing that BEC does not imply superfluidity. It is well known that the Bose Einstein onset temperature T0 is finite for N non interacting Bosons in a 2D harmonic trap of radius R is [390]

2n kTB0≈ (255) mNln where n is the areal number density of particles. In an interesting paper [396], the relation of T0 and Tc in a trapped gas was shown to be

−1/2 ⎛ 3mg 4 2 ⎞ Figure 35. −1 TT/1⎜⎟n 00.75 The fractional period shift ∆PP/ and dissipation Q for a c0=+32λ c() ≈ (256) torsional oscillator due to a film of 4He adsorbed on a mylar substrate. ⎝ π ⎠ The solid lines are from theory and the symbols are experimental. 2 The dashed curve is from the static theory. Reprinted from [67] with for typical experimental conditions where λ = 2/π mkBT permission. Copyright 1978 American Physical Society. is the 2D thermal wavelength so that TT0c> and the density nc at Tc is in 2D the superfluid transition is the onset of quasi long range 2 phase coherence for which a prerequisite is that the system is kTBc 2π = (257) in its ground state or very close to the ground state. The BEC nmc ln()C/α transition occurs at a lower T or density when one of the single where 2 C ≈±380 3 [388, 389]. particle states gets a macroscopic occupancy. It can therefore π Somewhat surprisingly, a BKT transition is observed in a be described by the quantum mechanical analogue of the clas- gas of 87Rb atoms trapped in a 2D optical trap [395]. It is sical free energy functional [392] expected that, when sufficiently cold, an isolated bosonic par- 2 ticle will condense into its ground state of zero momentum 2 ⎛ 22g 4 ⎞ F0()T =|d rr⎜ ∇|ψψ() +|V()rr()|+ ||ψ()r ⎟, which is well known from elementary quantum statistical ∫ ⎝ 22m ⎠ (252) mechanics, and in general a system of interacting system of where g is the coupling parameter for the particle interactions, bosons will condense into its ground state with wavefunction and V()r is the harmonic laser trapping potential [390]. To trap ψ()r . The experiment of [395] consists of optically trapping the gas as a 2D pancake, the laser trapping potential has the the bosons in two parallel planes at x1 and x2 separated by form d with about 105 atoms per plane. These 2D Bose Einstein condensates (BEC) are allowed to equilibrate independently m 2 2 2 2 1 2 222 Vz()r, =+()ωω0rzz =+mrωλ0()z (253) [393, 395]. The problem now is to detect the the algebraic 2 2 † η()T correlations aar0α ∼ ξ /r in each independent 222 ⟨ α() ()⟩(h ) where r =+xy and the anisotropy parameter layer α = 1, 2. λ = ωωz/10 . The solution is to turn off the trapping potentials and to The interaction parameter g has nothing to do with the allow the two independent correlated clouds of particles to trapping potential V()r which, at least for for small interac- expand and merge into a single 3D cloud [395]. The two gases tion strength g, is responsible for the condensate fraction expand most rapidly perpendicular to the planes and interfere 2 N00/1NT=−()/T where the BE condensation temper with each other. A probe laser beam parallel to the planes ature is given by kB0TN= ωζ0 /2() in the 2D limit λ =∞. (in the y direction) to examine the interference pattern formed However, to my naive thinking, the quantum mechanical cold when the clouds overlap. Before the expansion, each plane of trapped atom literature seems to be very confused about the atoms is highly correlated and it is reasonable to assume that

48 Rep. Prog. Phys. 79 (2016) 026001 Review

44 Figure 37. Left: IV characteristics of a 5.10−−× 1.10 m, 100 Å thick film of indium/indium-oxide at H = 0. This obeys the expected theory with a KT transition at 4.16 °K. Reprinted from [247] with permission. Copyright 1983 American Physical Society. Right: IV characteristics in a single unit cell film ofY Ba23Cu O7−δ at TT c showing Ohmic behavior at very small currents. The dashed line corresponds to V ∝ I. Reprinted from [354] with permission. Copyright 1996 Elsevier.

different planes are uncorrelated with each other since allow- of condensate i and Q1,2 =±mz()dt/2 / where t is the time ing the system to equilibrate ensures that the planes are inde- of flight during the free expansion of the atom clouds. After a pendent of each other. At sufficiently low T, individual planes bit of algebra, one finds that the correlation function in (258) will have quasi long range phase order which is encoded in the contains an oscillatory piece with Q = md/ t [394, 395] 3D cloud after expansion in the density fluctuations which are ρρzz =|2cAQ|−2 os Qz z , detected by observing the interference fringes in the density– ⟨(12)( )⟩int ⟨()⟩ [(12)] density correlation function as in figure 38 [394, 395] ||AQ 22= ddrr2 aa††rraarr, ⟨()⟩ ∫ 1 2⟨(2 11)(1)(1 22)(2)⟩ 2 LLxy ρρzz=−δρzz d rr † 2 ⟨(12)( )⟩ ()12∫ () = LLxy ddxy⟨(aar0)()⟩ (259) ∫∫00 2 2 †† + ∫ ddrr1 2⟨(aatof RR1)(tof 2t)(aaof RR1t)(of 2)⟩, when the two planes of condensate are independent before the (258) expansion occurs [394, 395]. The imaging area is Ω=LLxy where Rrjj= (), zj , r = ()xy, . The operators aztof()r, = and, since these length scales are not large, one must resort to i/Qz11()−−Qt 2mQi/22()zQtm2 aa1()rree+ 2() where ai is an operator finite size scaling to compare theory to observation.

49 Rep. Prog. Phys. 79 (2016) 026001 Review

Figure 38. Reprinted from [395] with permission. Copyright 2006 Nature Publishing Group. A sketch of the experimental setup of the interference experiment [395] with coordinate system of text. The imaging area LxyL of the individual condensates are the dark shaded areas in the xy planes. In the usual setup, all the physics is contained in the two ∗ ∗ where α = 0.50 for c0 < c0 or TT> c and α ⩽ 0.25 for c0 ⩾ c0. † point correlation ⟨aa()r0()⟩ which is where the BKT physics The measured values of the exponent α()c0 are very noisy with appears in the cold atom system. It is well known and verified large error bars but are certainly consistent with the predic- that the essential behavior in 2D at low T is tions of KT theory [395]. It is interesting to speculate how well the theory will be able to explain future more detailed η()T ⎛ ξh ⎞ and accurate observations on cold atom systems. ⟨(aa† r0)() ∼ ρ⎜ ⎟ , ⎝ r ⎠

mk T T B 1/4 6. Summary and acknowledgements η()= 2 ⩽ (260) 2πρ s()T In this review, I have attempted to summarize many areas of and one sees that, 2D physics where Kosterlitz–Thouless theory is relevant to 1/−η()T 2 experimental observations and, although I am naturally biased, 〈(||AQ)〉∼Ω ,0⩽(η TT)⩽1/4, ⩽ Tc, I am very pleased to discover that the theory still is relevant to ∼Ω1/2,.TT> c (261) several diverse bits of physics and that it can even be quanti Thus, the universal discontinuity of the 2D planar rotor model tively accurate in some situations, which is more than many appears in the context of cold atom physics and is measured in other theories are! Despite its venerable age of almost half [395] by fitting the density distribution with a century, occasionally new physics to which it is applicable appears. Very likely there are applications not mentioned in Fx,,zF=+xz 1ccx osQz + φ x () 0()[()( ()] (262) this review for which I take full responsibility and apologize where c(x) is the contrast or amplitude. The average central to those people whose work I either did not mention or have contrast c0 acts as an effective temperature and is effectively described incorrectly. varied by the chosen imaging length Lx [395]. Thus, the phys- I would like to acknowledge my many teachers, collabo- ics is encoded in the two measurable parameters c0, η which rators and colleagues especially D Thouless, L Kadanoff, can be varied by an experiment, sketched in figure 38, measur- J Reppy, M Fisher, D Fisher, M Fisher, P Anderson, D Mermin, ing the integrated contrast D Nelson, B Halperin, S Doniach, T Ala-Nissila, M Grant, E Granato, J Lee, M Simkin, N Akino, D Obeid and of course my 1 +Lx/2 2 † 2 −2α wife who unfailingly supported me during a very long period of 〈(CLx)〉 =∼∫ d,xa〈(xa00)(,0)〉 Lx , Lx −Lx/2 (263) neglect, chaos and irritability while this review was being written.

50 Rep. Prog. Phys. 79 (2016) 026001 Review

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