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Nobel Lecture, Aula Magna, Stockholm University, December 8, 2016

Topological Quantum Matter F. Duncan M. Haldane Princeton University

• The TKNN formula (on behalf of David Thouless) • The Chern and the birth of “topological insulators” • Quantum Chains and the “lost preprint” • In high school chemistry, we learn that bound to the nucleus of an move in closed around the nucleus , and quantum mechanics then fixes their energies to only be one of a discrete set of energy levels.

E 4p 4s 3d 3s 3p The rotational symmetry of the 2p spherical atom means that there 2s are some energy levels at which there are more than one state 1s • This picture (which follows from the Heisenberg uncertainty principle) is completed by the Pauli exclusion principle, which says that no two electrons can be in the same state or “orbital” E 4p 4s 3d ↑↓3s 3p ↑↓2↑p↓↑↓ An additional ingredient is that electrons ↑2↓s have an extra parameter called “spin” ↑↓1s which takes values “up” ( ↑) and “down” (↓) Occupied orbitals of the This allows two electrons (one ↑, one↓) Calcium atom to occupy each orbital (12 electrons) • If electrons which are not bound to are free to move on a two-dimensional surface, with a magnetic field normal to the surface, they also move in circular orbits because there a magnetic force at right angles to the direction in which they move • In high magnetic fields, all electrons have spin ↑ pointing in the direction of the magnetic field

magnetic field B force F = evB velocity v e F surface on which the x center of electrons circular move

As in atoms, the (kinetic) energy of the can only take one of a finite set of values, and now determines the radius of the orbit (larger radius = larger kinetic energy) • As with atoms, we can draw an energy-level diagram: (spin direction is fixed in each level)

E ↑ 2 unlike atoms, the number of orbitals in each ↓ E1 “Landau level” is huge! E ↑ 0 number of orbitals is proportional to area of surface!

Total magnetic flux through surface B area degeneracy of = = ⇥ Landau level (London) quantum of magnetic flux h/e • For a fixed density of electrons let’s choose the magnetic field B just right, so the lowest level is filled:

E2 empty E 1 energy gap E ● ● ● ● ● ● ● ● ● ● ● ● ● 0 filled independent of electron density if if n Landau eB details levels are exactly filled 2⇡~ • This appears to describe the integer quantum Hall states discovered by Klaus von Klitzing (Nobel Laureate 1985) • BUT: seems to need the magnetic field to be “fine-tuned”. • In fact, this is a “topological state” with extra physics at edges of the system that fix this problem • counter-propagating “one- way” edge states (Halperin) • confined system with edge must have edge states!

Fermi level ● bulk gap ● pinned at edge E0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● don’t need to fine-tune magnetic field K. von Klitzing

• The integer (1980) was the first “topological quantum state” to be experimentally discovered (Nobel Laureate1985, Klaus von Klitzing)

MAGNETI C FIELD ( T ) current • Hall conductance jx = E H y E~ e2 H = ⌫ 2⇡~ dissipationless current ⌫ = integer flows at right angles to electric field (number of filled Landau levels) • Von Klitzing’s system is much dirtier that the theoretical toy model and work in the early1980‘s focussed on difficult problems of disorder, random potentials and localized states • David Thouless had the idea to study the effect of a periodic potential in perturbing the flat Landau levels of the integer quantum Hall states: Bob Laughlin (Laureate 1998 for fractional QHE ) gave a clear argument V for quantization of the Hall conductance in this case edge edge flat potential in bulk

x V random potential in bulk (realistic but difficult)

x

V Toy model! periodic potential in bulk

smeared Landau level x The TKNN or TKN2 paper Quantized Conductance in a Two-Dimensional Periodic Potential D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs Physical Review Letters 49, 405 (1982)

• In 1982 David Thouless with three postdoc collaborators (TKNN) asked how the presence of a periodic potential would affect the integer quantum Hall effect of an electron moving in a uniform magnetic field • They found a remarkable formula ..... • David became particularly interested in an interesting “toy model”, of a crystal in a magnetic field, a family of models including the “Hofstadter Butterfly” 1 2 U0 ~!c H = p eA(r) + U(r) ⌧ 2m| | U 0 U(r)=U0(cos(2⇡x/a) + cos(2⇡y/a)) 0 EF • Harper’s equation (square -1 1 symmetry) or -2 2 • “Hofstadter’s Butterfly” splits the lowest Landau level into 0 bands separated by gaps.

The band are very narrow, and 2 • -2 the gaps wide, for low magnetic 1 -1 flux per cell (like Landau levels)

0 U color-coded Hall conductance 0 ½ 1 magnetic flux per unit cell simple Landau level limit colored “butterfly” courtesy of D. Osadchy and J. Avron • TKNN pointed out that Laughlin’s argument just EF 0 required a bulk gap at the Fermi energy for the Hall -1 1 conductance to be -2 2 quantized as integers

So it should work in gaps • between bands of the 2 -2 “butterfly” 1 -1 • so what was the integer? 0

magnetic flux per unit cell 0 ½ 1 simple Landau level limit colored “butterfly” courtesy of D. Osadchy and J. Avron • Bloch’s theorem for a particle in a periodic potential ik r kn(r)=un(k, r)e · periodic factor that varies over the unit cell of the potential • Starting from the fundamental Kubo formula for electrical conductivity, TKNN obtained a remarkable formula that does not depend in any way on the energy bands, but just on the Bloch wavefunctions:

2 ie 2 2 @un⇤ @un @un⇤ @un H = d k d r 2⇡h Brillouin @k @k @k @k n unit cell 1 2 2 1 X Z zone Z ✓ ◆ TKNN first form Sum over fully-occupied bands below the Fermi energy • Shortly after the TKNN paper was published, Michael Berry (1983) (Lorentz Medal, 2014) discovered his famous geometric phase in adiabatic quantum mechanics. • (The Berry phase is geometric, not topological, but many consider this extremely influential work a contender for a Nobel prize). • Berry’s example: a spin S aligned along an axis S ! direction of spin moves on closed path on unit sphere ⌦ˆ ei = eiS! Berry phase angle enclosed is ambiguous modulo 4π so 2S must be an integer • the mathematical physicist Barry Simon (1983) then recognized the TKNN expression as an integral over the (Berry) curvature associated with the Berry’s phase, on a compact manifold: the Brillouin zone. • This is mathematical extension of ’s* 1828 Theorema Egregium “remarkable theorem”

*foreign member of Royal Swedish Academy of Sciences • geometric properties (such as curvature) are local properties • but integrals over local geometric properties may characterize global topology! Gauss-Bonnet (for a closed surface) d2r(Gaussian curvature) = 4⇡(1 genus) Z product of 1 principal radii of =2⇡(Euler characteristic) cuvature R1R2 Risultato della ricerca immagini di Google per http://www.psdg... http://www.google.it/imgres?imgurl=http://www.psdgraphics.c...

Colorful 3D crystal balls www.psdgraphics.com 1 4⇡r2 = ⇥ r2 4⇡(1 0)

Le immagini potrebbero es • trivially true for a sphere, but non-trivially true for any compact 2D manifold

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1 of 3 7/3/13 11:19 PM 1 @u @u @u @u ab(k)= ddr n⇤ n n⇤ n Fn 2i @ @ @ @ Z ✓ ka kb ka kb ◆ unit cell Berry curvature an antisymmetric tensor in momentum space • The two-dimensional 1982 TKNN formula 2 2 ab e d k ab This is an integral over a H = 2 n (k) ~ BZ (2⇡) F “doughnut”: the torus define by a n Z X complete electronic band in 2D

Interestingly It emerged in 1999 that a (non-topological) 3D version of this form applied to the anomalous Hall effect in ferromagetic metals can be found in a 1954 paper by Karplus and Luttinger that was unjustly denounced as wrong at the time! • first form of the TKNN formula e2 d2k @ @ ab = ab(k) ab = H (2⇡)2 Fn n nb nb ~ n BZ F @ka A @kb A X Z Like a magnetic flux but in k-space Like a magnetic vector (the Brillouin zone) potential in k-space

Berry’s phase (defined modulo 2π) is eiB () =exp i dk a (k) like a Bohm-Aharonov phase in k- a n A space ✓ I ◆ • because the Berry phase is only defined up to a multiple of 2π, d2k (k)=2⇡ C Fn ⇥ n ZBZ TKNN formula form 2 Chern integer • TKNN give the formula as 2 ie 2 @un @un⇤ = dk d r u⇤ u⇤ H 4⇡ j n @k @k n ~ n BZB j j X I Z ✓ ◆ I learned from Marcel den Nijs, and Peter Nightingale that their memory is that the inclusion of this explicit general formula (in a single paragraph) was an “afterthought” while writing the paper, which was focussed on the specific values of the integers for the Hofstadter model!

• Another quote from Marcel: “the genius of David Thouless to choose the periodic potential generalization[to split the Landau level] not the random potential one was the essential step” • We finally arrive at the central TKNN result: Integral of the (Berry) curvature over the 2D Brillouin zone = 2π times an integer C

e2 = C Hall conductance: H h ⇥

Chern number QHE without Landau levels • The 1982 TKNN paper considered the effect of a periodic potential on Landau Levels due to a strong magnetic field • In 1988, I for reasons that are too long to describe here, I found that the Landau levels could be dispensed with altogether, provided some (broken time- reversal symmetry) was present. The 2D Chern insulator • This was a model for a “quantum Hall effect without Landau levels” (FDMH 1988), now variously known as the “quantum anomalous Hall effect” or “Chern insulator”. • It just involves particles hopping on a lattice (that looks like ) with some complex phases that break time reversal symmetry. • By removing the Landau level ingredient, replacing it with a more standard crystalline model the “topological insulators” were born Broken inversion

• gapless graphene “zig- zag” edge modes

Broken time-reversal (Chern insulator) Kane and Mele 2005 • Two conjugate copies of the 1988 spinless graphene model, one for spin-up, other for spin-down Zeeman coupling B=0 opens gap E At edge, spin-up moves one way, spin-down the other way

k k If the 2D plane is a plane of mirror symmetry, spin- orbit coupling preserves the two kind of spin. Occupied spin-up band has chern number +1, occupied spin-down band has chern-number -1. From the account of Marcel Den Nijs, the TKNN formula was found unexpectly “by accident” because David picked just the right “toy model” to study

In 1981, I made a similar “unexpected discovery” that may be the simplest example of “topological matter • Integer and half-integer quantum Antiferromagnetic Chains, “Quantum Kosterlitz-Thouless”and the “lost preprint”. • ILL preprint SP81-95 (unpublished) (now available on arXiv) • This original preprint was rejected by two journals and when the result was finally published, the connection to the Kosterlitz- Thouless work has been removed. • The original preprint was lost, but now is found! • Conventional magnetic ground states have long-range order, without significant entanglement (modeled by product states)

H = J S~ S~ H = J S~i S~j i j · neighbors · i,j i,j hXi hXi Ferromagnet Antiferromagnet

A B A B A B A B A B spin direction is spin direction is arbitrary, spin direction is arbitrary, but same but same for all spins on arbitrary, but same for all spins same sublattice, opposite for all spins lattice spins are antiparallel H = J S~ S~ i · j Has a conserved order i,j • hXi parameter direction Ferromagnet (conserved total spin angular momentum)

order parameter direction is H = J S~ S~ • i · j NOT conserved (zero total neighbors i,j hXi spin angular momentum) Antiferromagnet • quantum fluctuations destroy A B A B A B A B A B true long range order in one spatial direction • For a long time the conventional wisdom assumed the one-dimensional antiferromagnetic systems behaved like the ordered 3D systems, with a harmonic-oscillator treatment of small fluctuations around the ordered state. • This was partly due to a misinterpretation of a remarkable exact solution in 1931 of the S=1/2 chain by Hans Bethe* (before he moved on to nuclear physics!) The full understanding of Bethe’s solution required almost fifty years!

* David Thouless’s Thesis Advisor at Cornell! • In the mid-1970’s, another piece of work from the 1930’s (the Jordan-Wigner transformation) provided another more standard way to analyze the spin-1/2 chain without Bethe’s method. spins: (spinless) fermions: fermion operators + S = c† 0 = 1 on different sites + | #i | "i | i | i anticommute! S =0 c† 1 =0 | "i | i

+ n S = ( 1) j c† i i j

1 + + z z H = 2 (Si Si+1 + SiSi+1)+Si Si+1 i X < 1 easy plane > 1 easy axis

1 + 1 1 H = (c c + c† c )+(n )(n ) 2 i i+1 i+1 i i 2 i+1 2 i X E =0 free fermions

⇡ ⇡ - a a 2⇡ 4kF = a (Bragg vector) “Umklapp processes”

Half-filled band (in zero magnetic field) • Converting to a field theory (Luther and Peschel:)

H = i dx( † @ † @ ) 4kF Umklapp? 0 R x R L x L R† R† L L = 0! H=H0 + R dx( † L † R) R L R Left-movers Right-movers E

⇡ ⇡ The forgotten Umklapp term! - a a H0 = dx( R† @x R† L@x L + h.c. R 2⇡ 4kF = a (Bragg vector) VOLUME 45, NUMBER 16 YSICA& REVIEW TERS 2p QCTgBER 198P . Vg in rinciple v~ should also be th (4), even in the det»&s y»).le . A detailed d'iscussion of the of the critical region — iven elsewhere. en g( —1, and ~= 2' "~1 pp process -op lin regions exp—(-2 )-1 stabilizes a 2k F pinned density wavve. yhe behav- ~ . W'l'th perturbation expansio lj uid parameeters near this

= — (- 2y) (a+ In —2 ) . Th t t ly interacti g g = —' 's ' near n= 2 i due to an m with chnrgr eg —,'~ andspid liton Fermi vec 1 k 8=2kF- ag. he limiting beh ior exp( 2 y) H' = WP dxe '""(( t&(, )(tc p'|(,), (4) 4 characterizes'zes the noninterac ting limit of such a pf spinless ferm o„'~~ Such a so].jton gas reci rocal-lattice vector. &he has previously be en redictedd fromm studies pf the ~ limi t pf the Heisenbergp»c Infprma- tion on the soliton mass, size, and interaction ems": For exp— cted"&. /he~h linnear decrease oof (-2 ) 2, exp(-2p) nt "nlte sollton densl'tyy indicates at- ~ . dependence on ik —~i, y =y, tractive sollton splitpn coup»gi hence rep&- slve spllton-an- ntisolitonl coupling, consistent with the absence ofo bound states in thee gap In a treatment offthemo del by linearization preted accordingn totheIu tting -li id he v is unrenormalize d" Fig. 2(c) exactly mirror the scaling-theory prpredictions for indicates the region of vali1'd'ti y of s ch a scheme.

n 1.0

0.5

' ~ \ I ' ' ' ' 0.0 I I -0.5 -1 -2 05 0 -0.5 Vs (b). v, • Around that time I developed the n 1.0 “Luttinger theory” (a fit of microscopic models to an effective .X" "" Tomonaga/Luttinger0.5 model),2.0 an 0.5-,

.4 Abelian precursor to the later- —1.2— 1.0 g developed and more general ——0.5. I ~ I I I ~ I I I I 0.0 I 0.0 -2 -5-10-~ -0. -1 -5-10-~ 1 0.5 0 -0.5 1 0.5 0 5 conformal field theory, and applied it 25 RAPID COMMUNICATIONS 4927 (c). (d). exp(-2 p) = to this model: V) g (c)c thee renormalizeded Fermi problem, this time with P' = 2m g ', describing an — J2/J) DIME R BOUND STATES 2 ) that determines correrelationa exponents TheT e basal-plane spi- instability leading to a singlet pinned ground state t' PRESENT correlation exp onent 0 of e . The lo v commensurate with the external dimerizing potential, From thelattice numericalspacing. results using n when not at n = &. ona • ' with soliton excitations that now carry S'= +1. The vz, vq,v anand exp( —2p) acrossss thee line n=&, &1 d' t d. The remaining Luttinger-liquiu pa isotropic model here corresponds to a P'=2m SG Bethe’s{&/a)n. methodsSee also H,itef.f. the18. presence of system, and the scaling theory shows the dimer gap the till-then missed Umklapp term d. d opens as IgI' '." It is interesting to note that 1361 P'=2m is precisely that value where the SG has just was obvious, and driving a quantum two S'=0 breather excitations, ' with opposite parity, analog of the Kosterlitz-Thouless and where the lowest (even parity} breather is pre- cisely degenerate with the S'= +1 soliton doublet, transition, but with a “double vortex”RAPID COMMUNICATIONS forming an S = I triplet; the second (odd parity) breather is a singlet S =0 state with a gap J3b q. FIG. 2. (Schematic. ) Ground-state phase diagram in the ratherby athanBogoliubov a singletransfl~rmation, vortexand characterized by These two S =0, 1 states are the only elementary ex- (Jpl jt, I hl) plane. For I 5I 1, J2 J2 (5), the system is the correlation exponent vt: as In —n'I ( ( citations. in the gapless spin-j7uid phase with in-plane spin-correlation (S+S —(—1)"In —n'I ", (c„'c, —(—1)"In If the external dimerizing potential is applied to an ) ) exponent g (1 (lines of constant q are depicted). The um- '"+' 4'i' ' already spontaneously dimerized isotropic model with —n'I q = —, for free fermions, and the klapp coupling y2 vanishes along the broken line, and along its continuation separating the broken-symmetry dimer and J2 & J2, a similar spectrum results: The doublet solution of the Luttinger model when @2 =0 gives Neel with critical correlation exponent 1. In ground-state degeneracy is lifted, and there is now an = 't'. spin-correlation phases, q & g ( 4 +pi/2n ) Isotropy of the = these latter phases, the ground state is doublet, with a gap energy cost linear in the length of regions where the functions dictates that q approaches the value q I for excitation of pairs of S'=+ solitons (topological de- system is in the "wrong" ground state: This imposes in the isotropic limit =1.6 s When the 2 I4I F2=0, fects); the region where S'=0 breather bound states are a linear potential (a 1D potential) that con- Luttinger model approximation gives =0.82 (3) vi present in the gap is shown. The soluble model with fines the S = —, solitons (i.e., boundaries separating

when I in- when I hI =1 (or 71=1 hI =6J2/Ji =1.76), is marked with an asterisk. J2=-J~2 regions of the two now inequivalent dimer configura- dicating that renormalizations due to nonlinear terms tions) into bound S =0 or S = I pairs; the lowest- than quantitative corrections. '0 0 other y2 also give energy bound state is symmetric, with =1. = S This suggests that the special line J2(b, along which In the isotropic model I the fundamental ) FIG. 1. (See text. } Scaling( hItrajectoriesI), of (4}: %hen In the model of the spin-Peierls transition" the vanishes deviates from the value J2/Ji = —, IXI at excitations in the spontaneouslyvalues falldimerizedcriticalstatelinesareaa', y2 I hI =1, initial parameter on the "external" dimerizing potential arises spontaneously Sscaling= —, solitoneither tostates,the limitingcreatedcriticalonly gaplessin pairs;spin-fluidthe lowestpoint larger values of I 4I. The gapless, fluid character of because of lattice distortion; thus topological defects =0, it =1 J2 & J or to the dimer fixed point the Luttinger model suggests the term "spin Jluid" is excitationsy2 above{ thej)doubly degenerate (moments where g changes sign may be "frozen in. " For (J2 & J2 }. Lines bb' and cc' are the loci of initial values for an appropriate description of the y2 =0 spin chain. P =0, + m) ground states of an even-membered ring P'( 4a, the energy gain per unit length associated I hI & 1 and I hl &1, respectively. Systems with initial %hen y2 40, I note following Ref. 1 that the um- of spins are thus a continuum of degenerate S =0, 1 with the opening of the SG gap is finite, and an exact values close to (but not on} the unstable =fixed line F2 =0, klapp term can be treated by a scaling theory entirely pair states, with the gap minima at P 0, + m, the (Bethe ansatz) calculation gives it as —,tan( n tt) 1, are identified with the P2 Svr/=g sine-Gordon field 2 analogous to that used for the umklapp effects in the identificationq ) of the isotropic dimer state with the theory. x d, z/w„'s where v, is the spin-wave velocity in the spin- . " The term y2 leads to an instabil- P2 = 8m SG system rules out breathers or soliton- 2 limit g 0, and tt=(P2/8m)/[I —(P2/gm)] = antisoliton bound states in the near P =0, + m. 3 ity against a 2k' doubly degenerate density-wave gap P'= '2 Shastry—and Sutherland have recently reported such when 2m. Since e,/hq is also the characteristic state, with spontaneously broken symmetry. I (J2 Jq) ] for Jq & J2, where a is some numerical "healing length" for such a defect (which carries bound states in a region near the gap maxima (at identify y2 & 0 as leading to the Neel state, and constant controlled by the cutoff structure. This = = + for the special model with = in S —, ), the defect energy is of order d q itself. In the & 0 as leading to the dimer state. [Note that the Ptransition2 n) is very similar to that seen J2/Jtin spin-isotropic~, » absence of interchain coupling, dynamics canonical transformation i[i» exp( ipsr) ifi», which thissystemsrange suchlatticeaseffectsthe spin-are important,Fermi gasandwithit back-is out- 4 2 — would allow tunneling motion of the defect, as in re- side the scope of the long-wavelengthmodels'4 low-energy changes the sign of y2, changes g» to (i)»g» ]The. scattering'3 and Kondo as the coupling cent models solitons in "and 2 used The low-energy spectrum of polyacetylene, scaling equations are of a familiar form, "and in- SGchangesdescriptionsign. here. deduced here for the dimer state is in complete ac- features of the "spontaneously dimerized" spectrum volve» and the correlation exponent ri(yi): For Id I & 1, the system will remain in the gapless cord with Ref. 3. are recovered. spin-fluid state that characterizes the planar Heisen- To conclude. The analysis does not it is interesting to contrast spontaneously present explain ' — bergFinally,chain until J2 exceeds a critical coupling J2 (I), — =2(q 1)F2+0(y2)', dimerized states with those due to an externally im- one interesting feature of the special limit J2= 2J~ of d lnD when the trajectories will flow to the strong-coupling posed symmetry-breaking term X'=g (—1)" the isotropic model —that the correlation between di- dimer fixed point. The nature of the gtransition will 2 2 " mers vanishes. However, it places this state in a . 4 x S„~S„+~, as considered in the spin-Peierls problem. =2»+0{v,), v, =»», now be of "Kosterlitz-Thouless" type, with the or- As noted by Cross and Fisher, "when translated into continuum of spontaneously dimerized states for d(lnD) der parameter, etc. growing as exp[—b (&)/[J2 c 1 fermion variables, this term gives rise to a new SG J2) J2 = J]. —J2 (d ) ]' ] in the dimer region; the numerical con- 6 where D is an ultraviolet cutoff scale or effective stant b(A) diverges as I5I 1. For I5I 1, J2& J2 bandwidth. The familiar scaling trajectories of these ) there is a similar transition to the Neel state, as seen equations are shown in Fig. l. W'hen IAI = &, sym- ' in the anisotropic chain with J2 = 0. For I 5 I 1, the metry dictates that the starting point —and subsequent ~F. D, M. Haldane, Phys. Rev. Lett. 45, 1358 (1980).) 4C. K. Majumdar and D. K. two density-wave regions are separated by the gapless Ghosh, J. Phys. C 3, 911 evolution of the scalin—g trajectories must be identified 2J. L. Black and V. J. Emery, Phys. Rev. B 23, 429 (1981); (1970); J. Math. Phys. 10, 1388, 1399 (1969); P. M. van- line J2 (5) along which the umklapp term y2 van- the critical scaling trajectories q =1+y2. For M. P. M. den Nijs, ibid. 23, 6111 (1981). den Broek, Phys. Lett. 77A, 261 (1980). 3B.ishes. Along this line, the Neel and dimer correla- = the system is described by the stable S. Shastry and B. Sutherland, Phys. Rev. Lett. 47, 964 P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928). J2 & J2 6 Ji, (S„'S*. are the tions(1981). ) and ((S„S„+i)(S S,)) A. Luther and I. Peschel, Phys. Rev. B 12, 3908 (1975). trajectory scaling to the critical point y2 =0, q =1, dominant correlations at large separations, both fall- and the interaction J2 does not change the competing ing off as (—1)'" " 'In —n'I ""', as easily obtained character of the simple antiferromagnetic case J2 =0. from a Luttinger-model calculation following LP. For J2 & J2, the system must be identified with the ' The critical exponent q continuously decreases unstable critical trajectory, leading away from the belo~ 1 along the critical line, Close to this line, the point =0, = 1 to the strong-coupling dimer-state y2 q system behaves as a SG system with P'= 8+vi ', the fixed point. Note that systems ~here scaling starts ' principal elementary excitations are solitons carrying near the line of unstablefixed points q &1, F2=0, S*=2 —, (created in pairs), but in regions adjacent to can be identified with the sine-Gordon (SG) field S'= theory" with coupling parameter p'= Smq ". the iso the section of the critical line with q ( —, 0 tropic dimer state must thus be identified with the "breather" bound-state excitations will also be limiting case p2 Sm of the SG theory. The dimer present (p' & 4n SG spectrum). The predicted gap, order parameter g~, and inverse correlation ground-state phase diagram in the (J2/Ji, I&I) plane length will all initially grow as (J2 —J2)' 'exp[ —aJi/ is sketched in Fig. 2. • The topological Kosterlitz-Thouless transition occurs in a “classical” system in two dimensions at finite , but there is a well-know mapping from classical statistical mechanics in two spatial dimensions to quantum mechanics “(1+1) dimensions” (1D space + time) • One difference is that in classical mechanics the Boltzmann probability is always positive, while the quantum amplitude can be positive, negative or complex giving rise to interference effects. These spins rotate These spins rotate 1800 anticlockwise 1800 clockwise “vortex” in space-time= time winding-number * tunneling event

1d space These spins rotate These spins rotate 1800 anticlockwise 1800 clockwise time *

1d space • The tunneling events (vortices) occur on “bonds” that couple neighboring spins. • If the bonds are equal strength, and the vortex is moved one bond to the right, one spin that formerly rotated 1800 clockwise now rotates anticlockwise. • The difference is a 3600 rotation which gives phase factor of -1(and destructive quantum interference) if the spin is half-integral, +1 if not. • From this, it became clear that the progression from easy plane to easy axis was different for integer and half-integer spin antiferromagnets

easy-plane isotropic easy-axis (XY) (Heisenberg) (Ising) < 1 =1 > 1 half-integer S gapless, topologically-ordered gapped long-range Ising order with conserved winding number (two-fold degenerate ground state) double-vortex Kosterlitz- Thouless transition integer S non-degenerate gapped phase “Topological gapless, topologically-ordered gapped long-range Ising order with conserved winding number matter!” (two-fold degenerate ground state) single-vortex Kosterlitz- Ising transition Thouless transition • The new gapped phase in a “window” containing the integer-spin isotropic Heisenberg point turned out to be the first example of what is now called “topological matter” • The window is large for S=1, but gets very small for S =2, 3, ..... • The S=1 case is now classified as a “Symmetry-Protected Topological Phase” (the “protective symmetries” are time-reversal and spatial inversion) later developments were: • the identification of a topological “theta” term in the effective field theory of the Heisenberg antiferromagnet that distinguishes integer and half- integer spins. This perhaps started to popularize Lagrangian actions to complement Hamiltonian descriptions in condensed matter theory. • the identification (Affleck, Kennedy, Lieb, Tasaki) of the “AKLT model” that provides a very simple model state, which explicitly exhibits the remarkable topological edge states and entanglement of this phase • AKLT state (Affleck, Kennedy,Lieb,Tasaki) • regard a “spin-1”object as symmetrized product of two spin-1/2 spins, and pair one of these in a singlet state with “half” of the neighbor to the right, half with the neighbor to the left:

S = 1 1 2 S = 2

⇠ “half a spin” maximally left unpaired at entangled each free end! 0 (2) singlet state • The fragments of old work presented here may have seemed difficult for non-experts to understand, but mark the beginnings of what has turned into a completely now way to look at quantum properties of condensed matter • A large experimental and theoretical effort is underway to find and characterise such new materials, study entanglement, and dream of new “quantum information technolgies” • It has been a privilege to have contributed to these new ideas, and I thank the Royal Swedish Academy of Sciences for honoring us and our exciting field.