Mott insulators in strong electric fields
Subir Sachdev,∗ K. Sengupta,† and S. M. Girvin‡ Department of Physics, Yale University, P.O. Box 208120, New Haven CT 06520-8120 (Dated: May 8, 2002) Recent experiments on ultracold atomic gases in an optical lattice potential have produced a Mott insulating state of 87Rb atoms. This state is stable to a small applied potential gradient (an ‘electric’ field), but a resonant response was observed when the potential energy drop per lattice spacing (E), was close to the repulsive interaction energy (U) between two atoms in the same lattice potential well. We identify all states which are resonantly coupled to the Mott insulator for E ≈ U via an infinitesimal tunneling amplitude between neighboring potential wells. The strong correlation between these states is described by an effective Hamiltonian for the resonant subspace. This Hamiltonian exhibits quantum phase transitions associated with an Ising density wave order, and with the appearance of superfluidity in the directions transverse to the electric field. We suggest that the observed resonant response is related to these transitions, and propose experiments to directly detect the order parameters. The generalizations to electric fields applied in different directions, and to a variety of lattices, should allow study of numerous other correlated quantum phases.
I. INTRODUCTION
Recent experiments on ultracold trapped atomic gases have opened a new window onto the phases of quantum matter1,2. A gas of bosonic atoms has been reversibly tuned between superfluid and insulating ground states by varying the strength of a periodic potential produced by standing waves of laser light2. These experiments of- fer unprecedented control of the microscopic parameters, FIG. 1: Figures 1-4 contain schematic representations of the and allow exploration of parameter regimes not previ- Mott insulator, and of various states coupled to it. Shown ously available in analogous condensed matter systems. above is the Mott insulator with n0 = 2. Each well represents This paper focuses on one such “extreme” parameter a local minimum of the optical lattice potential - these we regime. Let w be the amplitude for an atom to tunnel number 1-5 from the left. The potential gradient leads to a between neighboring minima of the standing laser wave, uniform decrease in the on-site energy of atom as we move to the right. The grey circles are the b bosons of (1.2). The and U be the repulsive interaction energy between two i vertical direction represents increasing energy: the repulsive atoms in the same potential well. When w is smaller interaction energy between the atoms is realized by placing than a value of order U, the ground state is a Mott insu- atoms vertically within each well, so that each atom displaces lator for certain values of the atomic density or chemical the remaining atoms upwards along the energy axis. We have potential. In this state, the average number of atoms in chosen the diameter of the atoms to equal the potential en- each potential well must be an integer, n0 (see Fig 1). ergy drop between neighboring wells—this corresponds to the Now consider “tilting” this Mott insulator2 i.e. plac- condition U = E. Consequently, a resonant transition is one ing it under an external potential which decreases lin- in which the top atom in a well moves horizontally to the top early along a particular direction in space. Conceptually, of a nearest-neighbor well; motions either upwards or down- it is useful to imagine that the atoms carry a fictitious wards are non-resonant. ‘charge’, and then this potential gradient corresponds to applying a uniform ‘electric’ field, E (in practice this field ments of Greiner et al.2. More precisely, we shall discuss is applied by changing the position of the center of the the regime atomic trap2). We measure E in units of energy, defining E to be the maximal drop in potential energy of an atom |U − E|, w ¿ E, U, (1.1) moving between nearest-neighbor minima of the periodic potential (the potential energy drop depends upon the while allowing the ratio (U − E)/w to be arbitrary. choice of the nearest neighbor, and we choose the direc- We mention, in passing, another experimental system tion(s) along which the drop is the largest to define E). which has been studied under conditions analogous to In almost all Mott insulators consisting of electrons or (1.1). Electron transport has been investigated in ar- Cooper pairs, all reasonable electric fields that can be rays of GaAs quantum dots3, when the voltage drop be- achieved in the laboratory are small enough so that the tween neighboring quantum dots (the analog of E) is at relation E ¿ w, U is well satisfied. Remarkably, in the or above the charging energy required to make the transi- new atomic systems significantly larger ‘electric’ fields tion (the analog of U). However, in these systems the ex- are easily achievable: this paper shall discuss the regime cess electron energy can be dissipated away to the under- E ∼ U which has been explored in the recent experi- lying lattice, and so it appears that the threshold behav- 2 ior can be described by dissipative classical models4. In ` = m, and for large |` − m| its wavefunction decays as contrast, for the atomic systems of interest in the present · µ ¶¸ paper, there is essentially no dissipation over the time |` − m|E |ψm(`)| ∼ exp −|` − m| ln ; (1.7) scales of interest, and a fully quantum treatment must ew be undertaken. It useful to explicitly state our model Hamiltonian for the decay is faster than exponential, and is extremely 1.1). The reader should re- the Mott insulator for our subsequent discussion. We rapid under the conditions ( sist the temptation to imagine that a particle placed ini- will consider only Mott insulators of bosons, although tially at the site ` will eventually be accelerated by the the extension to fermionic Mott insulators is possible5. We label the minima of the periodic potential by lattice applied electric field out to infinity. Instead, the parti- sites, i, and assume that all bosons occupy a single band cle remains localized near its initial site, and undergoes Bloch oscillations with period h/E; indeed, as is clear of “tight-binding” orbitals centered on these sites. Let b† i from the simple form of (1.5), its wavefunction is exactly be the creation operator for a boson on site i. We will equal to its initial wavefunction at regular time intervals study the boson Hubbard model6,7,8 of h/E. The particle can escape to infinity only be a pro- X ³ ´ U X cess of Zener tunneling to higher bands not included in H = −w b†b + b†b + n (n − 1) i j j i 2 i i the single band tight-binding models in (1.4) and (1.2); hiji i the probability of such tunneling is negligibly small in X the experiments of interest here, and so will be ignored − E e · rini (1.2) i in our analysis. We now return our discussion to the full Hubbard where hiji represents pairs of nearest neighbor sites, model (1.2). As was the case in (1.5), the spectrum of this Hamiltonian is unbounded from below for E 6= 0, † ni ≡ bi bi, (1.3) and so it does not make sense to ask for its “ground state” for any density of particles. Rather, guided by r are the spatial co-ordinates of the lattice sites (the i the experimental situation of Ref. 2, we are interested in lattice spacing is unity), and e is a vector in the direction states which are accessible from the translationally in- of the applied electric field (e is not necessarily a unit variant Mott state (with an average of n particles on vector—its length is determined by the strength of the 0 every site) over the experimentally relevant time scales. electric field, the lattice structure, and our definition of The experiment2 begins at E = 0 with a Mott insula- E above). We will mainly consider simple cubic lattices, tor with n particles per site, rapidly ramps up E to a with the e oriented along one of the lattice directions and 0 value of order U, and detects the change in the state. For of unit length. Not shown in (1.2) is an implied chemical w ¿ U, and for most values of E, the experiments dis- potential term which is chosen so that the average density played little detectable change in the state of the system. of atoms per site is n . We will restrict our attention to 0 We can initially understand this by a simple extension the case where n is of order unity. 0 of the argument presented above for the non-interacting Some simple key points can be made by first consider- model H . Consider a ‘quasiparticle’ state of the Mott ing the non-interacting case, U = 0, and also by simpli- 0 insulator, created by adding a single additional particle fying to one spatial dimension9. For this special case, we on one site, as shown in Fig 2a. To leading order in w/U, can write H as the motion of this quasiparticle along the direction e is X ³ ´ † † † described by an effective Hamiltonian which is identical H0 = − wb`b`+1 + wb`+1b` + E`b`b` (1.4) in form to H0, but with the hopping matrix element w ` replaced w(n0 + 1). So any such quasiparticle states cre- where ` is an integer labelling the lattice sites. The exact ated above the Mott insulator will remain localized and single-particle eigenstates of H0 can be easily obtained: will not have the chance to extend across the system to the eigenenergies form a Wannier-Stark ladder, and the create large changes in the initial state. A similar local- most important property of the wavefunctions is that ization argument applies to the quasihole state shown in they are all localized. Specifically, the eigenstates can Fig 2b: it experiences an electric force in the opposite be labeled by an integer m which runs from −∞ to ∞, direction, the effective hopping matrix element is wn0, the exact eigenenergies are and all quasihole states are also all localized in the di- rection e. Indeed, it is not difficult to see that the same ²m = Em, (1.5) localization argument applies to all deformations of the Mott insulator which carry a net charge. and the corresponding exact and normalized wavefunc- The important exceptions to the above argument for tions can be expressed in terms of Bessel functions: the stability of the Mott state are deformations which carry no net charge. It is the primary purpose of this pa- ψm(`) = J`−m(2w/E); (1.6) per to describe the collective properties of such neutral for a derivation see e.g. Ref. 10 (their analysis is in a states. They will be shown to yield a resonantly strong different gauge). The m’th state is localized near the site effect on the Mott state when E ∼ U, which has been 3
from the Mott state by U − E. So these states become degenerate at U = E, and an infinitesimal w leads to a resonant coupling between them. However, there are a large number of other states which are resonantly cou- pled to one of more of these states, and they also have to be treated on an equal footing. Indeed, it is suffi- cient for an given state to be resonantly coupled to any (a) one other state in the manifold of resonant states for it to be an equal member of the resonant family—it is not necessary to have a direct resonant coupling to the par- ent Mott insulator. The reader should already notice that multiple dipole deformations of the Mott insulator (like the state in Fig 3b) are part of the resonant family. (b) In dimensions greater than one, these dipole states are only a small fraction of the set of resonant states, as we will show below. We are now in a position to succinctly FIG. 2: Notation as in Fig. 1.(a) A quasiparticle on site 3; state the purpose of this paper: identify the complete set the motion of this quasiparticle is described by the localized wavefunctions (1.6) but with w replaced by 3w.(b) A quasi- of states resonantly coupled to the Mott state under the hole on site 3; the motion of this quasihole is also described conditions (1.1), obtain the effective Hamiltonian within by the localized wavefunctions (1.6) but with w replaced by the subspace of these states, and determine its spectrum 2w. and correlations. The results will allow us to address the strong response of the Mott insulator to an electric field E ∼ U observed by Greiner et al.2, and lead to some definite predictions which can be tested in future exper- iments. The first step in our program is a complete descrip- tion of the set of resonant states. We will do this first for one dimension in the Section IA, and for all higher dimensions in Section IB. The effective Hamiltonian in (a) the resonant subspace will be shown to contain strong correlations among its degrees of freedom, but we will demonstrate that these can be satisfactorily treated by available analytic and numerical methods in many body theory. Before embarking on a detailed description of our computation, the reader may find it useful to ex- amine Figs 3 and 4 for an understanding of the origin (b) of the strong correlations in the one dimensional case. Fig 3 contains only dipole states: notice that while res- FIG. 3: Notation as in Fig. 1.(a) A dipole on sites 2 and 3; onant dipole states can be created separately on nearest this state is resonantly coupled by an infinitesimal w to the neighbor links, it is not possible to create two dipoles Mott insulator in (a) when E = U.(b) Two dipoles between simultaneously on such links (as in Fig 4a) without vi- sites 2 and 3 and between 4 and 5; this state is connected via olating the resonant conditions. This implies an infinite multiple resonant transitions to the Mott insulator for E = U. repulsive interaction between nearest neighbor dipoles in the effective Hamiltonian. Two (or more) dipoles can be safely created when they are further apart, as shown in dramatically observed in the experiments of Greiner et Fig 3b. Thus the dipole resonances are not independent al.2. Indeed, Greiner et al. have already identified an im- of each other, and the wavefunction contains non-trivial portant neutral deformation of the Mott state—it is the ‘entanglements’ between them. dipole state consisting of a quasiparticle-quasihole pair on nearest neighbor sites, as shown in Fig 3a. A key con- sequence of our discussion above is that, for w ¿ E (a A. One dimension condition we assume throughout), we can safely neglect the independent motion of the quasiparticle and of the It is not difficult to see that, in one spatial dimen- quasihole along the direction of e. Only their paired mo- sion, the set of all nearest-neighbor dipole states consti- tion as dipoles will be important along e, although they tute the entire family of states resonantly coupled to the can move independently along directions orthogonal to Mott insulator in Fig 1 for U = E and an infinitesimal e. w. The only subtlety concerns states like those in Fig 4b, For w = 0, the dipole state in Fig 3a differs in energy which are not made up of nearest-neighbor dipoles. For 4
left edge of the dipole which actually resides on links between the lattice sites. Clearly, we cannot create more than one dipole resonantly on the same link: hence the dipoles satisfy an on-site hard core constraint
† d`d` ≤ 1. (1.10) (a) Moreover, we cannot create two dipoles simultaneously on nearest neighbor links—this leads to a non-resonant state like that in Fig 4a; such states are prohibited by a hard core repulsion between nearest neighbor sites
† † d`d`d`+1d`+1 = 0. (1.11) (b) The resonant family of states can now be completely specified as the set of all states of the boson d` which FIG. 4: Notation as in Fig. 1. Two states which are not part satisfy (1.10) and (1.11). A typical state is sketched be- of the resonant manifold. (a) An attempt to create dipoles low in Fig 5a. Notice that the dipole vacuum, |Mn0i is between sites 2 and 3 and also between sites 3 and 4; the result one of the allowed states. is a single dipole of length 2 which has energy U −2E relative It is now a simple matter to write down the effective to the Mott insulator, and so this long dipole is not part of the Hamiltonian, H for the d . It costs energy U − E to resonant family of states. (b) A state with energy 3(U − E) d ` create each dipole, and each dipole can be created or an- relative to the Mott insulator; this state is not part of the resonant family because its largest effective matrix element to nihilated with an amplitude of order w (this corresponds any state in the resonant family is of order w2/U (for U = E; to the horizontal motion of particles in Figs 1-4). So we see (1.8)). In contrast, all states within the resonant family have are connected to at least one other state also in the family by p X ³ ´ X † † a matrix element of order w. Hd = −w n0(n0 + 1) d` + d` + (U − E) d`d`. ` ` (1.12) w = 0, this state has energy 3(U − E) relative to that The Hamiltonian (1.12), along with the constraints in Fig 1. However reaching the state in Fig 4b from any (1.10,1.11), constitute one of the correlated many-body state in the resonant family requires a detour through a problems we shall analyze in this paper. The eigenstates non-resonant state. A simple second-order perturbation of Hd are characterized by n0 and the single dimension- theory calculation shows that the closest state from the less number resonant family connected to Fig 4b is a state with dipole between sites 3 and 4, and that the effective matrix ele- U − E λ ≡ , (1.13) ment between them is w p µ ¶ w2n n (n + 1) 1 1 0 0 0 + ; (1.8) and a description of their properties as λ ranges over all 2 U 2U − E real values is in Section II. (Strictlyp speaking, the eigen- states of H depend only λ/ n (n + 1), but λ and n this is negligibly small, under the conditions (1.1), com- d 0 0 0 do not combine into a single constant in higher dimen- pared to the non-zero matrix elements (= w) between sions.) states within the resonant family. Hence we can safely It is interesting to note that there is no explicit hopping neglect the state in Fig 4b. More completely, the argu- term for the d bosons in H : it appears that the bosons ment is that after we diagonalize the Hamiltonian within ` d only only allowed to be created from, and to disappear the resonant family, states coupled to that in Fig 4b will into, the vacuum by the first term in (1.12). However, differ from it by an energy of order w; the coupling in this is misleading: as we will see in Section II, the combi- (1.8) will then be too weak to induce a resonance. nation of the terms in (1.12) and the constraint (1.11) It is convenient now to introduce bosonic dipole cre- does generate a local hopping term for the d bosons ation operators, d†, to allow us to specify the resonant ` ` (see (2.1)). Additional dipole hopping terms also arise subspace and its effective Hamiltonian. Let |Mn0i be the from virtual processes of order w2/U in the underlying Mott insulator with n0 particles on every site (the state Hubbard model H; however, these are negligibly small in Fig 1 is |M2i). We identify this state with the dipole compared to those just mentioned and do not need to be vacuum |0i. Then the single dipole state is included in Hd. † 1 † We close this subsection by noting that the Hamilto- d |0i ≡ p b`b |Mn0i (1.9) ` n (n + 1) `+1 nian Hd in (1.12) and the constraints (1.10,1.11) can also 0 0 be written in the form of a quantum spin chain. We iden- Notice that we have placed the dipole operator on the tify the dipole present/absent configuration on a site ` as 5
z x,y,z a pseudospin σ` up/down (σ are the Pauli matrices). Then σz = 2d†d − 1 and ` ` ` (a) Xh p x z Hd = −w n0(n0 + 1)σ` + (U − E)(σ` + 1)/2 ` i z z + J(σ` + 1)(σ`+1 + 1) . (1.14)
The constraint (1.11) is implemented by taking the J → ∞ limit of the last term. The spinchain model so ob- tained is an S = 1/2 Ising spin chain in both transverse and longitudinal fields. This is known not to be inte- grable for finite J, but it does appear that the problem simplifies in the J → ∞ limit we consider here.
B. Higher dimensions n
We consider here only hypercubic lattices in D spatial dimensions, with e oriented along one of the principal cubic axes and a lattice spacing in length (e.g. D = 3 and e = (1, 0, 0)). Other lattices, and other directions of e e, also allow for interesting correlated phases and these will be mentioned in Section IV. (b) Extension of our reasoning above quickly shows that the dipole states now constitute only a negligibly small FIG. 5: Typical states in the resonant subspace for (a) D = 1 fraction of the set of states in the resonant family. Once a and (b) D = 2. Black circles represent sites with quasipar- dipole has been created on a pair of sites separated by the ticles (these sites have ni = n0 + 1 (see (1.3)), grey circles vector e, its quasiparticle and quasihole constituents can represent quasiholes (these sites have ni = n0 − 1), while move freely and resonantly, with matrix elements of or- the remaining sites have ni = n0. Note that Q` in (1.17) is der w, in the (D−1) directions orthogonal to e. Allowing zero for each column i.e. the total number of quasiparticles in this process to occur repeatedly (while maintaining some every column equals the total number of quasiholes in the col- constraints discussed below), we can build up the set of umn to its immediate left. Only in D = 1 does this constraint imply that all states contain only nearest-neighbor dipoles. all resonantly coupled states. A typical resonant state in D = 2 is shown in Fig 5b. As in Section IA, it is use- ful to give an operator definition of the resonant family. particle or hole: To allow us to distinguish between the directions paral- lel and orthogonal to e, we replace the D-dimensional † p`,np`,n ≤ 1 site label i, by the composite label (`, n), where ` is an † integer measuring the co-ordinate along e (as in the one- h`,nh`,n ≤ 1 dimensional case), while n is a label for sites along the † † p p`,nh h`,n = 0. (1.16) (D − 1) transverse directions. Rather than using dipole `,n `,n operators, we now want to work with bosonic quasipar- Additionally, because of the manner in which these quasi- † † ticle (p`,n) and quasihole (h`,n) operators, which create particles and quasiholes appear from the Mott state, the states like those in Fig 2a and Fig 2b respectively. More total number of quasiparticles in the D − 1 dimensional precisely, we now identify |Mn0i with quasiparticle and layer with co-ordinate ` + 1 must equal the total number quasihole vacuum |0i, and so of quasiholes in layer `: X ³ ´ 1 Q ≡ p† p − h† h = 0. (1.17) p† |0i ≡ √ b† |Mn i ` `+1,n `+1,n `,n `,n `,n `,n 0 n n0 + 1 † 1 While the quasiparticles and quasiholes are allowed to h`,n|0i ≡ √ b`,n|Mn0i. (1.15) n0 move freely within each D−1 dimensional layer, they can- not move resonantly out of any layer on their own; this is, The set of resonant states can now be specified by a of course, related to the localization of the Wannier-Stark few simple constraints on these operators, which are the ladder states discussed earlier in this section. analogs of (1.10,1.11). First, there are the obvious on-site Continuing the analogy with Section IA, we can now hard-core constraints that no site can have more than one easily write down the effective Hamiltonian, Hph, for the 6 quasiparticles and quasiholes which acts on the set of present numerical evidence which strongly supports this states defined by (1.16) and (1.17). The terms in the first conclusion. two lines are the same as those already present in (1.12), Further analytic evidence for an Ising quantum crit- but expressed now in terms of the quasiparticle/hole op- ical point can be obtained by examining the excitation erators, while the last line is associated with motion along spectra for the limiting λ regimes, and noting their sim- the transverse D − 1 directions: ilarity to those on either side of the critical point in the 11 p X ³ ´ quantum Ising chain . † † Hph = −w n0(n0 + 1) p`+1,nh`,n + p`+1,nh`,n For λ → ∞, the lowest excited states are the single- † `,n dipoles: |`i = d`|0i; there are N such states (N is the (U − E) X ³ ´ number of sites), and, at λ = ∞, they are all degenerate + p† p + h† h (1.18) 2 `,n `,n `,n `,n at energy U −E. The degeneracy is lifted at second order `,n in a perturbation theory in 1/λ: by a standard approach X ³ ´ † † using canonical transformations, these corrections can be − w n0h h`,m + (n0 + 1)p p`,m + h.c. . `,n `,n described by an effective Hamiltonian, H , that acts `,hnmi d,eff entirely within the subspace of single dipole states. We Here hnmi represents a nearest neighbor pair of sites find h within a single (D−1) dimensional layer orthogonal to e. X Hd,eff = (U − E) |`ih`| Notice that all the Q` in (1.17) commute with Hph, as is ` required for the consistency of our approach. As was the i n0(n0 + 1) case in one dimension, the properties of Hph are deter- + (|`ih`| + |`ih` + 1| + |` + 1ih`|) (2.1) mined by the single dimensionless constant λ in (1.13); λ2 these will be described in Section III. Notice that, quite remarkably, a local dipole hopping It is worth reiterating explicitly here that upon special- term has appeared, as we promised earlier at the end of ization to the case of D = 1 (when the indices n, m only Section IA. The constraints (1.10,1.11) played a crucial have a single allowed value and the set hnmi is empty), role in the derivation of (2.1). Upon considering per- the Hamiltonian Hph above is exactly equivalent to the turbations to |`i from the first term in (1.12) it initially one dimensional dipole model Hd in (1.12). seems possible to obtain an effective matrix element be- 0 We note in passing that in a manner similar to Hd, tween any two states |`i and |` i. However this connec- Hph in (1.18) can also be written as a S = 1 spin model, tion can generally happen via two possible intermedi- with the empty/qausiparticle/quasihole states on a site † † 0 0 ate states, |`i → d`d`0 |0i → |` i and |`i → |0i → |` i, corresponding to spin states with Sz = 0, 1, −1. and the contributions of the two processes exactly can- The outline of the remainder of the paper is as follows. cel each other for most `, `0. Only when the constraints The properties the D = 1 model Hd will be described in (1.10,1.11) block the first of these processes is a residual Section II, while the D > 1 model Hph will be consid- matrix element possible, and these are shown in (2.1). ered in Section III. We discuss extensions of our results to It is a simple matter to diagonalize Hd,eff by going to other lattices and field directions in Section IV. Implica- momentum space, and we find a single band of dipole tions of our results for experiments appear in Section V. states. The lowest energy dipole state has momentum The appendices contain some technical discussion on the π: the softening of this state upon reducing λ is then nature of the quantum phase transitions found in the consistent with the appearance of density wave order of body of the paper. period 2. The higher excited states at large λ consist of multiparticle continua of this band of dipole states, just as in the Ising chain11. II. DIPOLE PHASES IN ONE DIMENSION A related analysis can be carried out for λ → −∞, and the results are very similar to those for the ordered This section will describe the spectrum of the one- state in the quantum Ising chain11. The lowest excited dimensional dipole Hamiltonian Hd in (1.12), subject to states are single band of domain walls between the two the constraints (1.10) and (1.11). filled dipole states, and above them are the corresponding An essential point becomes clear simply by looking at multiparticle continua. the limiting cases λ → ∞ and λ → −∞ (the coupling λ was defined in (1.13)). For λ → ∞ the ground state of Hd is the non-degenerate dipole vacuum |0i. In contrast, A. Exact diagonalization for λ → −∞ the ground state is doubly degenerate, be- cause there are two distinct states with maximal dipole We numerically determined the exact spectrum of Hd † † † † † † number: (··· d1d3d5 ··· )|0i and (··· d2d4d6 ··· )|0i. This for lattice sizes up to N = 18. As will be evident below, immediately suggests the existence of an Ising quantum these sizes were adequate to reliably extract the limiting critical point at some intermediate value of λ, associ- behavior of the N → ∞ limit. ated with an order parameter which is a density wave of The complete spectrum of Hd is shown in Fig 6 for dipoles of period two lattice spacings. We will shortly N = 8 and n0 = 1. We used periodic boundary con- 7
30
1. 6
20
1. 5 10 ∆ E/w N /w N=10 0 1. 4 N=12 N=14 N=16 -10 N=18 1. 3
-20 -1.90- 1.88 -1.86- λ 1.84 -1.82- 1.80
-6 -4 -2 0 λ 2 4 6 FIG. 7: The spacing between the lowest two eigenvalues of Hd (= ∆) as a function λ for various system sizes and n0 = 1. FIG. 6: All the eigenvalues of Hd for N = 8 and n0 = 1. We used periodic boundary conditions for Hd. Note that the ground state is non-degenerate for positive λ, and there are two low-lying levels with an exponentially small 5.0 splitting for λ < 0 and |λ| large. N=10 N=12 N=14 ditions on the dipole Hamiltonian in (1.12). Note that N=16 4.0 these do not correspond to periodic boundary conditions for the original model (1.2); indeed, for (1.2) the pres- ence of the electric field implies that periodic boundary N∆/w conditions are not physically meaningful. Nevertheless, it is useful to apply periodic boundary conditions to the 3.0 translationally invariant effective model (1.12), merely as a mathematical tool for rapidly approaching the N → ∞ limit. Note that Fig 6 shows a unique ground state for λ → ∞ and a two-fold degenerate state for λ → −∞. 2.0 Above these lowest energy states, there is a finite energy gap, and the excited states have clearly split into bands corresponding to the various “particle” continua; these “particles” are dipoles for λ → ∞, and domain-walls be- 1.0 tween the two ground states for λ → −∞, as we discussed in the perturbative analysis above. We test for a quantum critical point at intermediate 0.0 values of λ by plotting the energy gap, ∆, in Fig 7. This -10.0 -5.0 0.0 5.0 10.0 gap is the spacing the between the lowest two of the N(λ-λ ) eigenvalues plotted in Fig 6 (for finite system sizes, these c low-lying levels are always non-degenerate). It becomes exponentially small in the system size as we approach FIG. 8: Scaling plot of the energy gap to test for (2.2). We the two degenerate ground states which are present for λ used λc = −1.850 and n0 = 1. sufficiently negative. In the opposite limit, ∆ approaches a finite non-zero value, which becomes U − E, for λ large and positive. If these two phases are separated by a quan- A second test of Ising criticality is provided by also tum critical point, we expect the energy gap to scale as rescaling the horizontal axis of Fig 7 with N. General −z finite size scaling arguments imply that the energy gap ∆ ∼ N at the critical point λ = λc, where z is the dynamic critical exponent. The Ising critical point has should obey the scaling form z = 1, and so Fig 7 plots N∆ as a function of λ. We ³ ´ −z 1/ν observe a clear crossing point at λc ≈ −1.850 which we ∆ = N φ N (λ − λc) (2.2) identify as the position of the Ising quantum phase tran- sition. Note that the critical point is shifted away from where φ is a universal scaling function, and ν is the cor- the naive value E = U (λ = 0) to E > U because of relation length exponent. We test for (2.2) in Fig 8 with quantum fluctuations associated with the hopping of the the Ising exponent ν = 1, and again find excellent agree- dipoles. ment. 8
0.220 it is not difficult to see that it pays to choose one of two regular arrangements, in which the occupation num- N= 8 Q Q 0.216 † † N=10 bers are independent of n: n ` even p`+1,nh`,n|0i or N=12 Q Q † † N=14 n ` odd p`+1,nh`,n|0i. So there is a two-fold degener- 0.212 N=16 ate ground state for λ < 0 and |λ| large, associated with 3/4 S π /N a broken translational symmetry and the development of 0.208 density wave order of period 2 in the longitudinal direc- tion, in both the quasiparticle and quasihole densities.
0.204 The excitation spectrum in the limiting ranges of λ can also be determined as in Section II. However, the computations are more involved and we limit ourselves 0.200 to an analysis of the λ → ∞ case in Section IIIA. We -1.90- 1.88 -1.86- λ 1.84 -1.82- 1.80 will investigate physics at intermediate values of λ in the subsequent subsections, where we will see that the pos- FIG. 9: Scaling plot of numerical results for the order param- sibilities are richer than the appearance of a single Ising eter structure factor, S , defined in (2.3). We used n = 1. π 0 quantum critical point between the states just discussed: Section IIIB will present a mean field theory, while Ap- pendices A and B will discuss continuum quantum field A final, and most sensitive, test for Ising criticality is theories which can describe long-wavelength fluctuations provided by a measurement of the anomalous dimension near the phase boundaries. of the order parameter. The order parameter is the den- sity of dipoles at momentum π, and so we computed its equal-time structure factor A. Excitations for λ large and positive *Ã !2+ 1 X S = (−1)`d†d . (2.3) There is a large manifold of lowest excited states, all π N ` ` ` of which have energy U − E, in the limit λ → ∞. These are the states with exactly one p quasiparticle and one Standard scaling arguments imply that this should scale h quasihole, with the particle on the D − 1 dimensional 2−z−η as N at λ = λc, where η is the anomalous di- layer ` + 1 and the hole on the layer `. We label these mension of the order parameter. Using the Ising expo- states by nent η = 1/4, we expect S ∼ N 3/4. This is tested π † † in Fig 9. Note that there is an excellent crossing point |`, n, mi ≡ p`+1,nh`,m|0i (3.1) at λc ≈ −1.853. This position of the crossing point is We break the degeneracy between these states by consid- completely consistent with the crossing point found in ering corrections in powers of 1/λ. At order 1/λ, the term Fig 7. Thus Fig 9 provides strong evidence for the ex- in the last line in (1.18) will allow the quasiparticle and pected Ising exponent η = 1/4. We have also examined a the quasihole to hop independently in their own layers, plot which scales the horizontal axis in Fig 9 as in Fig 8: but will not induce any couplings between states with the data collapse is again excellent. different values of `. The latter appear at order 1/λ2, when as in (2.1), a nearest neighbor dipole pair can hop longitudinally between neighboring layers; again, as in III. QUASIPARTICLE AND QUASIHOLE PHASES IN HIGHER DIMENSIONS D = 1, the constraints (1.16,1.17) play a crucial role in determining these perturbative corrections. These pro- cesses are described the following effective Hamiltonian This section will discuss the properties of the D > 1 di- for the manifold of excited states with energy ≈ (U −E): mensional model of the p`,n quasiparticles and h`,n quasi- " holes described the Hamiltonian Hph in (1.18), subject to X X the constraints (1.16,1.17). Hph,eff = (U − E) |`, n, mih`, n, m| As in Section II, it is instructive to first look at the two ` n,m ³ distinct limiting values of λ. The nature of the ground 1 X − n |`, n, kih`, m, k| states is very similar to those in D = 1 for these ranges λ 0 hnmi,k of λ. For λ → ∞, we have a unique ground state which ´ contains only small perturbations from the quasiparticle + (n0 + 1)|`, k, nih`, k, m| and quasihole vacuum |0i. For λ → −∞, it is clear that n (n + 1) X³ we want to maximize the total number of quasiparticles + 0 0 |`, n, nih`, n, n| and quasiholes in the ground state, subject to the con- λ2 n # straints (1.16,1.17). There are a very large number of ´ ways of doing this, but by considering perturbative cor- +|`, n, nih` + 1, n, n| + |` + 1, n, nih`, n, n| (3.2) rections to the ground state energy in powers of 1/|λ|, 9
Note that the first summation over n, m is unrestricted state, and just as in D = 1, it is a precursor to the ap- and ranges independently over the two variables, while pearance of longitudinal density wave order of period 2. the second is over nearest neighbor pairs hnmi. The appearance of this dipole bound state suggests that The Hamiltonian Hph,eff can be analyzed by the stan- the first quantum phase transition out of the featureless dard techniques of scattering theory. The terms within and gapped phase present for large positive λ is into a the first two summations in (3.2) lead to a “two-particle” state with Ising charge order; however, our discussion continuum of quasiparticle and quasihole states, while here is for a system with a well-developed gap to quasi- the terms within the last summation allow these states particle and quasihole states, it is yet not clear whether to scatter and possibly form a dipole bound state. We this approach continues to hold when the gap becomes first form states with total transverse momentum Q⊥, small—we will return to this question in Appendix A. and relative transverse momentum q⊥ (these momenta are D − 1 dimensional vectors) B. Mean field theory 1 X |`, Q , q i = eiq⊥·rn+i(Q⊥−q⊥)·rm |`, n, mi ⊥ ⊥ N ⊥ n,n This section will present the results of a mean field (3.3) analysis of Hph. The central idea of the mean field the- where N⊥ is the number of sites in each layer, and rn are ory is very simple: we treat the quantum fluctuations the spatial positions of the sites. In this basis of states along the longitudinal direction for all n by the exact Hph,eff Next, we also transform the single longitudinal numerical treatment developed in Section II for D = 1, co-ordinate, `, to a ‘dipole momentum’, qk: while the transverse couplings are treated in a mean field manner. One important benefit of this approach is that ¯ ® 1 X ¯q , Q , q = eiqk` |`, Q , q i (3.4) the important constraints (1.16) are treated exactly. k ⊥ ⊥ N ⊥ ⊥ k ` This approach also naturally suggests the appearance of additional phases which have no analog in the D = 1 In this basis of states, Hph,eff takes a form which makes case. In particular, the motion of single p and h bosons the mapping to standard scattering theory very explicit. in the transverse direction implies that superfluid order The total transverse momentum, Q⊥, and the longitudi- can develop along these D − 1 dimensions only. There is nal dipole momentum, qk, are conserved, while there is no possibility of superfluidity in the longitudinal direc- scattering between different values of q⊥: tion because motion along this direction can occur only X via charge neutral dipole pairs which appear in the first Hph,eff (Q⊥, qk) = [εp(q⊥) + εh(Q⊥ − q⊥)] term in (1.18). This transverse superfluid therefore has 12 q⊥ a ‘smectic’ character , and its existence implies that we ¯ ® ¯ ¯ ¯ have to allow for hpi and hhi condensates: these appear × qk, Q⊥, q⊥ qk, Q⊥, q⊥ 2 naturally in our mean field theory. w n0(n0 + 1)(1 + 2 cos q ) + k As in the mean field treatment of the zero field boson N⊥(U − E) Hubbard model6,7, the approximation involves a decou- X ¯ ® ¯ ¯ 0 ¯ pling of a hopping term. In particular, we only decouple × qk, Q⊥, q⊥ qk, Q⊥, q , (3.5) ⊥ the last transverse hopping term in (1.18), and obtain q ,q0 ⊥ ⊥ the following mean field Hamiltonian for a set of sites, where (for a hypercubic lattice) labelled by `, representing any chain along the longitudi- nal direction (U − E) X " εp(q⊥) = − 2w(n0 + 1) cos(q⊥α) (3.6) X ³ ´ 2 † ∗ α Hph,mf [hp`i, hh`i] = −wZn0 hh`ih` + hh`i h` ` and the summation over α extends over the D − 1 com- ³ ´ − wZ(n + 1) hp ip† + hp i∗p ponents of q⊥. The expression for εh(q⊥) is identical to 0 ` ` ` ` p ³ ´ (3.6) but with n0 +1 replaced by n0. The Hamiltonian in † † (3.5) is that of a particle moving in (D − 1) dimensions − w n0(n0 + 1) p`+1h` + p`+1h` with momentum q and dispersion ε (q ) + ε (Q − ³ ´ ⊥ p ⊥ h ⊥ (U − E) † † q⊥), scattering off a delta function potential at the ori- + p`p` + h`h` 2 2 # gin with strength w n0(n0 + 1)(1 + 2 cos qk)/(U − E). ³ ´ Its solution is well known: in addition to the scattering † † − µ` p p`+1 − h h` . (3.7) states, a bound state must be present in D − 1 = 1, 2 `+1 ` for any infinitesimal attractive potential, and for strong enough attraction for D − 1 > 2. So for the physically Here Z is the co-ordination number of any site along the relevant cases of D = 2, 3, a bound state must form for a D − 1 transverse directions, and the expectation values range of qk values near π. It is clear that the lowest en- hh`i and hp`i have to determined self-consistently from a ergy bound state has Q⊥ = 0 and qk = π: this is a dipole diagonalization of (3.7) subject to the constraints associ- 10 ated with (1.16), which now become
=0
=0
=0 even
=0 † † † † =0 is obeyed; note that these constraints are macroscopic, =0 =0 and so there is no approximation involved in using a = chemical potential to impose them. In practice, the di- agonalization of Hph,mf [hp`i, hh`i] must be carried out B C A 0.6 > en v e 0.4 + p e odd 0.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 λ FIG. 13: A typical state in the resonant subspace for a square FIG. 12: As in Fig 11. The values of hhodd + heveni are very lattice with e = (1, 1). Representation is as in Fig 5. The close, but not identical, to the p values shown above. quasiparticles and quasiholes occur only occur in dipoles ori- ented along the +x or +y directions. Note that it is possible for dipoles to undergo a ring-exchange around a plaquette, in which the configuration around plaquette a can become like transition at C2 involves the continuous vanishing of the p (h) condensate in the odd (even) layers in a superfluid- that around plaquette b; this process is contained in the reso- nant model H0 in (4.1) and (4.2), and does not require much insulator transition, in the presence of a background of d weaker virtual processes in the Hubbard model H (which are Ising density wave order. suppressed by powers of w/U). The final transition at the point B involves loss of all p and h condensates. There is long-range Ising density wave order at all λ to the left of B, and a gap to all ever, once such a dipole has been created, the quasipar- excitations. ticle and the quasihole cannot move resonantly to any The theory of fluctuations about these mean-field re- other sites (except by processes of order w2/U which sults is discussed in Appendices A and B. As we have al- we have consistently neglected here). So the resonant ready noted, these could be strong enough to also modify subspace can be described completely in terms of dipole the topology of the phase diagram in Fig 10a. One ex- states, just as in the D = 1 case discussed earlier. A treme possibility is that the transverse superfluid phases typical state is illustrated in Fig 13. The effective Hamil- could disappear entirely, and we are left only with two tonian of this space of dipole resonant states is identical insulating phases, one with Ising density wave order and in form to (1.12): the other without; the phase diagram is then as in D = 1. p X ¡ ¢ X However, we show in Appendix A that the value of a par- H0 = −w n (n + 1) d + d† + (U − E) d† d , ticular critical exponent determines that this is not the d 0 0 a a a a a a generic situation. (4.1) except now the label a extends over the links of the square lattice. There continues to be a hard-core con- IV. OTHER FIELD ORIENTATIONS AND † straint dada ≤ 1 like (1.10), but the possibility for new LATTICES physics arises from the complexity of the generalization of the constraint (1.11), which is now Our discussion has so far limited itself to hypercubic † † lattices, with the direction of the electric field, e, oriented dadadbdb = 0 for links a, b which share a common site. along one of the principal axes. Similar analyses can be (4.2) carried out for other lattices and for other directions of Note that each dipole blocks the occupancy of dipoles e. A large variety of correlated phases appear possible, on six neighboring links. It would be interesting to de- 0 including many not related to those already discussed. termine the properties of Hd subject to the constraint We will illustrate these possibilities by an example here, (4.2). but leave a more detailed discussion to future work. The possibility of rich physics becomes apparent in Consider a square lattice (in D = 2) but with e = thinking about the case λ < 0 and |λ| large. Here the (1, 1). In this case, the resonant transitions from the low-lying manifold of states corresponds to maximizing Mott insulator involve moving a bi boson by one lattice the number of dipoles, and these are in one-to-one corre- spacing, either along the +x or the +y direction. How- spondence with the close-packed dimer coverings of the 12 square lattice. A natural ring-exchange term of the dipole before measurement can be made much larger than the bosons also becomes apparent upon considering pertur- original lattice dimensions. In this limit the final spatial bative corrections in powers of 1/|λ|: this derivation is position at which an atom is detected determines the mo- similar in spirit to that in Section II (see Fig 13). We mentum at which the momentum distribution function is emphasize that the dominant ring exchange does not being measured. come from virtual higher order processes in the under- The momentum distribution for the boson Hubbard lying Hubbard model H (which are strongly suppressed model containing N sites is given by by factors of w/U), but is already contained within the X D E physics of the resonant subspace as described by (4.1) and 1 ~ † Π(q) = |f(q)|2 eiq·(rj −rk) b b , (5.1) (4.2). In analogy with other studies of quantum dimer N j k j,k models13,14,15 and boson ring exchange models, possibili- ties of bond-ordered phases open up. Fractionalized, and 16 where f(q) is the form factor for the tight-binding or- Bose metal phases are also possible, but these may be bitals associated with the lattice potential, and the mo- more likely on non-bipartite lattices. mentum q = mR/(~tex), where R is the distance from We close by noting that it is easily possible to orient the detection position to the center of the trap, m is the e so that only one direction is resonant. For a cubic lat- mass of the atoms, and tex is the time elapsed in the tice in D = 3 this can be done by choosing e = (1, a, b) expansion (this expression ignores the influence of grav- where a, b 6= 0, 1 are some arbitrary real numbers. Then ity, but an appropriate modification is straightforward). resonant transitions to dipole states can occur only along The development of off-diagonal long-range order peaks the x direction, and the resonant manifold separates into the momentum distribution at the values of q equal to decoupled one dimensional systems, each of which is sep- the reciprocal lattice vectors of the optical lattice poten- arately described by the one-dimensional dipole Hamil- tial, and has been used as an experimental signature of tonian Hd in (1.12). This may be a simple way of exper- the superfluid phase.1,2 imentally realizing the model Hd. Let us first consider the D = 1 case. A very important consequence of our restriction to the subspaceD E of resonant † V. IMPLICATIONS FOR EXPERIMENTS states is that the boson correlator b`b`0 vanishes for |`−`0| > 1. Hence Eq. (5.1) becomes (q is the component An important issue that must be faced at the outset is of q in the direction of the ‘electric’ field the extent to which the non-equilibrium time-dependent " p experiments can be described by the ground and low en- n0(n0 + 1) Π (q) = |f(q)|2 n + ergy states of the effective models that have been dis- 1D 0 2 cussed in this paper. In the experiments of Greiner et # 2 X n o al. , the ‘electric field’ (in practice, this is realized by × eiqhd†i + e−iqhd i ,(5.2) a magnetic field gradient) is turned from an initial zero ` ` ` value to E in a time of order h/w. In a system under the conditions (1.1), this may not allow easy access of where the lattice spacing has been taken to be unity, and the ground state. As an alternative, we suggest that E † d` is the dipole creation operator defined in (1.9). For be ramped up rapidly to a value to the right of the point the periodic boundary conditions we have used (as we A in Fig 10, and then slowly increased through the pos- noted earlier, such boundary conditions are not physi- sible critical points in Fig 10. This could produce states cal, but they should not modify the results in the limit with either the density wave Ising order, or the transverse of large system sizes), the values of hd`i depend only on superfluid order. the parity of ` (a very small ordering field is applied to Having produced such states, the next challenge is to lift the Ising symmetry, and choose one of the ground directly detect the quantum order parameters associated states in the region with spontaneous Ising order), and with the phases in Fig 10. We address two possible hence the overall amplitude of (5.2) is determined only probes in the subsections below. by hdeveni + hdoddi. We show our numerical results for these, and other related quantities, for the Hamiltonian Hd in (1.12) in Figure 14. There is a broad maximum A. Momentum Distribution in hdeveni + hdoddi near the Ising critical point, as this is the region with maximal dipole number fluctuations. One experimental quantity that is relatively easy to The critical singularity in this quantity at λ = λc is de- measure is the momentum distribution of the atoms con- termined by that of the energy operator of the Ising field tained in the optical lattice. This is done by shutting off theory: this singularity is weak and is essentially unob- the lattice potential and the trapping potential and allow- servable in Fig 14. The quantities sensitive to the Ising ing the atoms to freely expand until the resulting cloud order parameter (such as hdeveni − hdoddi), show more is large enough that its density profile can be spatially singular behavior in Fig 14 near λc determined by the resolved optically. The scale to which the cloud expands magnetization exponent β. However these observables 13 1.0 for φ is 1 X D E φ = (−1)` d†d (5.3) 0.8 N ` ` ` + + † V (x) ∝ −[cos2(Qx) + B2 cos2(Qx/2 + θ)]. (5.5) FIG. 14: Ground state expectation values of hd`i and hd` d`i for the D = 1 model Hd in (1.12). The results are for N = 16 sites and periodic boundary conditions. A very small ordering Adjusting the relative phase to θ = 0 or π/2 adds a field was applied to choose one of the degenerate Ising ground ‘staggered magnetic field’ term to the Ising Hamiltonian states present for sufficiently negative λ. We have chosen the gauge in which hdi are real. 2 HB ∝ ±B φ. (5.6) are not detectable by a measurement of the momentum A simpler experimental method for the case where the distribution function. trap confinement is strong in the directions transverse to the axis of the 1D lattice is the following. An additional In higher dimensions (D > 1) for the case where e is standing wave (derived from the same laser) but oriented aligned along one of the lattice directions, the dependence in the y direction (say) would yield of the distribution function on qk should be qualitatively similar to the q dependence in the D = 1 case discussed Φ(x, y = 0, t) ∝ [cos(Qx) cos(Qct) + B cos(Qct)] (5.7) above. However a much clearer signal of the transverse superfluidity should be visible. The presence of the hpi and hence a potential along the x axis of and hhi condensates imply that the correlator (5.1) has phase coherent contributions when r −r lies in the plane j k V (x, y = 0) ∝ −[cos2(Qx) + 2B cos(Qx) + B2] (5.8) perpendicular to the applied ‘electric’ field. This implies that in states with transverse superfluidity, there should which would also couple to the Ising order be Bragg peaks along lines in q space with values of q⊥ equal to the reciprocal lattice vectors of the D−1 dimen- sional lattice lying in the plane perpendicular to e. As the HB ∝ Bφ. (5.9) transverse dimensionality is D − 1 = 2, the superfluid or- der can only be quasi-long-range at nonzero temperature, In either case, such a perturbation could be used to break and hence the Bragg peaks are not true delta functions the Ising symmetry and selectively populate one of the in the infinite volume limit, but are power-law singular- two Ising states. In addition, it could be used to mea- ities. Experimental detection of these Bragg lines would sure the order parameter itself. The AC stark shift of be quite interesting. the atomic hyperfine levels would differ between adja- cent sites. The relative strengths of the split hyperfine absorption lines would then be a measure of the Ising order parameter.17 B. Ising Order Parameter Acknowledgments We have seen that the Ising order is not directly re- flected in the momentum distribution and hence can We thank Immanuel Bloch and Mark Kasevich for nu- not be measured in the free expansion method described merous valuable discussions of their experiments. This above. The properties of the Ising order parameter, φ, research was supported by US NSF Grants DMR 0098226 are discussed in Appendix B; one convenient definition and DMR 0196503. 14 APPENDIX A: FLUCTUATIONS AND layer. The values of µ` are to be determined at the end QUANTUM FIELD THEORIES: by the requirements SUPERFLUID-INSULATOR TRANSITIONS The mean field theory of Section IIIB can be used ∂ ln Zph as a starting point for a more sophisticated treatment = 0 (A4) ∂µ` of fluctuations. Such fluctuations will modify the mean- field exponents in the vicinity of the second order phase boundaries in Fig 10a, but could also change the topology of the phase diagram to that in Fig 10b. Further progress in describing the properties of Zph re- We analyze fluctuations about the mean field results quires some understanding of the structure of S1. This using a method very similar to that described in Chapter was already addressed to some extent in Section IIIB 10 of Ref. 11 for the Hubbard model. We decouple the where we explored the properties of the Hamiltonian intra-layer hopping terms in Hph (those in the last line Hph,mf . However, here we need to generalize that anal- of (1.18) only by Hubbard-Stratonovich transformations ysis to the case where its arguments are time-dependent using complex fields P`(r⊥, τ) and H`(r⊥, τ) where r⊥ is fields P`(r⊥, τ) H`(r⊥, τ). This is quite an involved task, a spatial co-ordinate for the D − 1 transverse directions, but we will only need some general constraints that are and τ is imaginary time. Then, after standard simplifica- placed on the structure of S1 by the principles of gauge tions, we obtain an expression for the partition function invariance. In particular, associated with the conserva- Zph of Hph which has the following schematic form tion laws (1.17), we observe that Zph is invariant under Z the time and layer-dependent transformations generated Zph = DP`(r⊥, τ)DH`(r⊥, τ) by the arbitrary field φ`(τ) · Z ¸ D−1 exp − d r⊥ (S0 + S1) . (A1) iφ`(τ) p`+1 → p`+1e −iφ`(τ) The action S0 involves couplings only within a single h` → h`e layer `, but with different values of r iφ`(τ) ⊥ P`+1 → P`+1e Z X −iφ`(τ) £ 2 2 H` → H`e S0 ≡ dτ Kp|∇⊥P`(r⊥, τ)| + rp|P`(r⊥, τ)| ∂φ` ` µ` → µ` + i (A5) 2 2¤ ∂τ +Kh|∇⊥H`(r⊥, τ)| + rh|H`(r⊥, τ)| , (A2) and Kp,h, rp,h are coupling constants. Note that the We are interested here only in the case of time- factors of n0 and n0 + 1 in the last line of (1.18) break particle-hole symmetry and so there is no special sym- independent µ`, and so this transformation takes µ` into metry relation between these coupling constants. The an unphysical set of values; nevertheless, as we will see shortly, (A5) is still useful in placing constraints on S1 in action S1 couples different layers and times together for the physical regime. the same value of r⊥ Z First, we address the influence of fluctuations by ap- −S1 e ≡ Dp`(τ)Dh`(τ)P [p`(τ), h`(τ)] proaching the transition involving condensation of P`, H` " Z ( µ ¶ from the side of large and positive λ, i.e. we increase E X ∂p ∂h × exp − dτ p† ` + h† ` (and decrease λ) until mean-field theory indicates we are ` ∂τ ` ∂τ approaching a phase with transverse superfluidity at the ` )# point A in Fig 10. The ground state of Hph,mf is trans- lationally invariant in this region, and so we can safely +Hph,mf [P`(r⊥, τ),H`(r⊥, τ)] (A3) assume that all the coupling constants in S1 are also in- dependent of `. Similarly, we can assume that µ` is also with Hph,mf defined in (3.7), and P is a projection oper- independent of `. If we were to approach the condensa- ator which represents the constraints (3.8) (these could tion of P`, H` from the opposite side of negative λ, the be imposed formally in the functional integral by a very ground state of Hph,mf would have a broken Ising sym- strong on-site repulsive interaction among the p` and h` metry, and the following analysis would only need to be bosons). As in Section IIIB, we have imposed the con- modified by allowing all couplings, and µ, to depend upon straints (1.17) by time-independent Lagrange multipliers the ` sublattice. We describe the action S1 by expanding (“chemical potentials”) µ`: as we noted earlier, there is it in powers of the fields P`, H`, and in their temporal no approximation involved in neglecting the fluctuations gradients (the r⊥ and τ dependence of these fields is now of µ`, because there is only one constraint per layer and implicit); to second order in the fields and to first order there are a macroscopic number of particles within each in temporal gradients, the most general terms invariant 15 under (A5) are positive because of the repulsive interactions between the " microscopic bosonic degrees of freedom. The parameter X Z e ∗ ∂P` e ∗ ∂H` rψ tunes the system across the quantum phase transition S1 = dτ KpP` + KhH` ∂τ ∂τ at the point A in Fig 10 which resides at rψ = rψc; the ` µ ¶ transition is from the featureless, gapped phase at large ∂H ∂H∗ ` ∗ ` positive λ (rψ > rψc) to a phase with superfluidity in the + Keph P`+1 + P (A6) ∂τ `+1 ∂τ transverse D−1 dimensions as λ is decreased (r < r ); # ψ ψc the superfluidity is associated with the condensation of 2 2 ¡ ∗ ∗¢ + rep|P`| + reh|H`| + reph P`+1H` + P`+1H` Ψ`. Just as in the derivation of (A7), we can also exam- ine the consequences of time-dependent gauge transfor- Consistently requiring invariance of (A6) under the time- mations in (A5,A9) on (A10). This now leads to the dependent gauge transformations (A5) to the order we relationship have performed the expansion in S1 demands additional constraints on the coupling constants above; these are ∂r K = − ψ . (A11) ψ ∂µ ∂rep ∂reh ∂reph Kep = − ; Keh = ; Keph = . (A7) ∂µ ∂µ ∂µ Combined with (A4), the above result now yields a cru- cial result. Close to the quantum critical point, the singu- There are also a large number of permitted higher order lar free energy associated with Z is determined directly terms in S which we have not written down explicitly; ph 1 by r . For this singular term to obey (A4), we conclude some of these will play an important role below. ψ (as also argued in Chapter 10 of Ref 11) that ∂rψ/∂µ = 0 Armed with the low order terms in the action S0 + S1 at rψ = rψc;(A11) now implies controlling the fluctuations of P` and H` we can now use standard techniques to focus on the low energy exci- Kψ = 0 at rψ = rψc. (A12) tations. It is natural to diagonalize the quadratic form displayed in these actions: this will lead to two eigen- So we can neglect L1, and the critical theory is described modes with distinct eigenvalues. We focus attention on entirely by L0. Within each layer `, this theory has the the lower eigenmode, while integrating out the higher relativistic invariance of (D−1)+1 spacetime dimensions eigenmode. We identify the lower eigenmode by the field and dynamic critical exponent z = 1. Ψ`: this has the structure Before turning to an examination of the properties of ∗ (A10), we pause to discuss the modifications required Ψ`(r⊥, τ) = cpP`+1(r⊥, τ) + chH` (r⊥, τ) (A8) to describe the onset of transverse superfluidity with in- creasing λ in the region λ < 0 at the point B in Fig 10 (a for some constants c . Note that we are performing p,h similar reasoning can also be applied to the point C in the same ‘rotation’ in field space for all r and τ (and 2 ⊥ Fig 10b). Here, long-range Ising order is already present hence all frequencies). This ensures that Ψ has a simple ` in H for λ sufficiently negative. We can proceed behavior under (A5): ph,mf to a description of the superfluid transition as above, iφ`(τ) but as noted earlier, all couplings in (A10) will acquire Ψ` → Ψ`e . (A9) an ` dependence which modulates with period 2. The We integrate out the high energy mode orthogonal to tuning parameter rψ will also be different for even and (A8), and obtain our final effective action now expressed odd `. Consequently only Ψ` with ` even (say) will be- in terms of Ψ`: come critical near the transition, while Ψ` with ` odd " # remains non-critical and can be integrated out. The sim- Z Z X D−1 plest interlayer coupling between critical modes is now Zph = DΨ`(r⊥, τ) exp − d r⊥dτ (L0 + L1) 2 2 |Ψ`| |Ψ`+2| , but its co-efficient should be small and is ` ¯ ¯ likely to be attractive. ¯∂Ψ ¯2 u We now return to an examination of (A10) for the case L = |∇ Ψ |2 + ¯ ` ¯ + r |Ψ |2 + |Ψ |4 0 ⊥ ` ¯ ∂τ ¯ ψ ` 2 ` of `-independent couplings at the transition with U − E positive at the point A in Fig 10. It remains to exam- + v|Ψ |2|Ψ |2 ` `+1 ine the consequences of the interlayer coupling v on the ∗ ∂Ψ` standard theory of the superfluid-insulator transition. At L1 = KψΨ` . (A10) ∂τ v = 0, we have the standard ϕ4 field theory with O(2) symmetry in (D − 1) + 1 = D spacetime dimensions. As We have rescaled Ψ` and time to obtain unit coefficients a first step, we can compute the scaling dimension of v at for the first two terms in L0, and the r⊥ dependence of its critical point. A standard power-counting argument Ψ` is implicit. We have also written down a quartic non- linearity within a layer (u), and the simplest coupling shows that between neighboring layers (v) which preserves invari- 2 α dim[v] = − D = (A13) ance under (A9); we expect both these couplings to be ν ν 16 where ν and α are the standard correlation length and APPENDIX B: ISING PHASE TRANSITION “specific heat” exponents in D spacetime dimensions. In 18 D = 3, the O(2) fixed point has α = −0.015 < 0, and In Appendix A we completed a description of fluc- so we conclude that v is formally irrelevant. In D = 2, tuations near all the superfluid-insulator transitions in the very weak specific heat singularity at the Kosterlitz Fig 10. It remains to describe the second order Ising Thouless transition suggests the same conclusion. critical point C1 in Fig 10b; this we do in the present A more complete analysis of the influence of v can be appendix. obtained by considering a physical susceptiblity for or- As usual, we expect the Ising phase transition to be dering in the longitudinal direction. As we have seen in realized by a quantum field theory of a real scalar field Sections II and IIIB, the simplest allowed ordering is a φ(r, τ), where r = (`, r⊥) is a D dimensional spatial density wave of period two. The tendency to this order- co-ordinate. The main subtlety here is that the Ising ing is measured by the static susceptibility χπ transition occurs in a background of transverse super- Z X fluid order, and corrections from superflow fluctuations 1 D−1 `+`0 χπ = d r⊥dτ(−1) can lead to anisotropic singular corrections to the criti- Nk `,`0 cal theory. A theory of an Ising order parameter coupled 2 2® |Ψ`(r⊥, τ)| |Ψ`0 (0, 0)| . (A14) to isotropic superflow fluctuations has been analyzed by Frey and Balents19; here, we will show that the partic- Note that this response function is similar to Sπ in (2.3), ular anisotropic nature of both the superfluid and Ising but we are considering here a zero frequency response, order leads to a more singular coupling between the two while (2.3) involved an equal time correlator. We can order parameters. compute (A14) in powers of v, and by a familiar Dyson- Any observable sensitive to the period 2 modulation type argument, write it as in the density of particles or holes can be used to define C the order parameter φ(r, τ). A convenient choice in our χ = (A15) π 1 − 2vC present continuum formulation is to take where C is an ‘irreducible’ correlator within a single layer ` 2 φ(`, r⊥, τ) ∼ (−1) |Ψ`(r⊥, τ)| . (B1) (it is irreducible with respect to cutting a v interaction line): An effective action, Sφ, for the Ising field φ can be gen- Z erated by using φ as a Hubbard-Stratonovich field to de- D−1 2 2® C = d r⊥dτ |Ψ`(r⊥, τ)| |Ψ`(0, 0)| . (A16) couple the v term in (A10). This leads to an action with the structure The computation leading to (A15,A16) is the field- Z " theoretic analog of the computations which lead to a 1 K S = dDrdτ (∂ φ)2 + ⊥ (∇ φ)2 dipole bound state induced by the interlayer coupling φ 2 τ 2 ⊥ in the strong-coupling analysis of Section IIIA. Ignoring # the influence of v on C, standard scaling arguments imply Kk ¡ ¢2 + ∇ φ + u φ4 that C has a singular part which behaves as 2 k I −α X Z C ∼ |rψ − rψc| . (A17) D−1 ` 2 − wI d r⊥dτ(−1) |Ψ`(r⊥, τ)| φ(`, r⊥, τ), If we had α > 0, then the denominator in (A15) would ` vanish at some rψ > rψc for any small v, and χπ then di- (B2) verges: this would imply the presence of an Ising density wave transition before the onset of superfluidity. How- where the fluctuations of Ψ` are described by (A10), and ever α < 0 in D = 3, and so this condition does not we have included the usual analytic terms present in the apply. Nevertheless, there is a significant (albeit finite) φ4 theory of an Ising quantum critical point. The last enhancement of the specific near the O(2) critical point term in (B2) represents a linear coupling between the in D = 3, and so the instability in χπ may well occur Ising order parameter and density fluctuations in the su- for a moderate value of v. If so, the mean-field phase perfluid state. In the isotropic case considered by Frey diagram would be modified, and the Ising ordered phase and Balents such a linear coupling was absent, and the would fully overlap and extend beyond the region with simplest allowed coupling was between φ2 and the density transverse superfluidity. Indeed, under suitable condi- fluctuations: this was because the Ising order parameter tions, the superfluid phase could also shrink to zero, and represented a density wave at a large wavevector, and we would then have only a single Ising transition be- they coupled linearly only to fluctuations of the super- tween two insulating phases. Alternatively, if the Ising fluid phase at the same wavevector, and the latter are ` fluctuations are weaker, χπ could diverge somewhere in quite high energy. In the present case also, the (−1) the superfluid phase to the left of A in Fig 10, and then factor in the last term in (B2), also shows that φ couples the mean-field phase diagram would be modified to the linearly to the superfluid phase fluctuations at a wavevec- structure in Fig 10b. tor qk = π. However, the key difference here is that the 17 superfluidity is present only along the transverse direc- to (A12) hold only at the superfluid-insulator transition. tion, and, to leading order, the superfluid phase fluctua- All the couplings in Sφ can be expected to be a smooth tions are independent of qk. function of µ, and the constraint is now expected to lead 20 The singular effect of the wI term in (B2) can be illus- only to a Fisher renormalization of exponents. An anal- trated by integrating out the Ψ` using the action (A10) ysis of Sφ with (B3) included requires a renormalization in a single loop approximation. To leading order in u, we group computation: this we leave to future work, as a are in the transverse superfluid state as long as rψ < 0, full discussion of the renormalization of the momentum and a simple calculation of the phase and amplitude fluc- dependence of the propagator requires a two loop analy- tuations of the superfluid order parameter shows that we sis. generate the following term in Sφ: X 2 2 In closing, we note that although Kψ 6= 0, in practice 1 2 |rψ|(q⊥ + ω ) |φ(qk, q⊥, ω)| 2 2 2 2 , the degree of particle-hole symmetry breaking is quite 2u |rψ|(q⊥ + ω ) + Kψω /2 qk,q⊥,ω small, as is indicated by the almost equal values of hpi (B3) and hhi in Figs 11 and 12. So Kψ can also be expected where ω is an imaginary frequency. Note that this is a to be quite small, and we should, therefore, also consider singular function of q⊥ and ω only when Kψ =6 0. We the case Kψ = 0. In this case, (B3) does not induce any do not expect Kψ = 0 near the Ising critical point, be- singular terms, and we have to consider terms induced by cause exact particle-hole symmetry is not present in the Ψ` fluctuations at higher orders in u, and also the term underlying Hamiltonian, and the arguments which lead included in Ref 19. ∗ Electronic address: [email protected]; URL: http: 6 M. P. A. Fisher, P. B. Weichman, G. Grinstein, and //pantheon.yale.edu/~subir D. S. Fisher, Phys. Rev. B 40, 546 (1989). † Electronic address: [email protected] 7 D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner,and ‡ Electronic address: [email protected]; URL: http: P. Zoller, Phys. Rev. Lett. 81, 3108 (1998). //pantheon.yale.edu/~smg47 8 T. D. Kuhner, S. R. White, and H. Monien, Phys. Rev. B 1 C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and 61, 12474 (2000). M. A. Kasevich, Science 291, 2386 (2001). 9 J. Zapata, A. M. Guzm´an,M. G. Moore, and P. Meystre, 2 M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, and Phys. Rev. A 63, 023607 (2001). I. Bloch, Nature 415, 39 (2002). 10 J. H. Davies and J. W. Wilkins, Phys. Rev. B 38, 1667 3 C. I. Duru¨oz,R. M. Clarke, C. M. Marcus, and J. S. Harris, (1998). Jr., Phys. Rev. Lett. 74, 3237 (1995). 11 S. Sachdev, Quantum Phase Transitions, Cambridge Uni- 4 A. A. Middleton and N. S. Wingreen, Phys. Rev. Lett. 71, versity Press, Cambridge (1999). 3198 (1993). 12 S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, 5 If we ignore the spin of the fermions (as may be the justified 550 (1998). under certain physical conditions, such as the presence of 13 D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, a magnetic field), then the generalization of our results to 2376 (1988). 14 fermionic Mott insulators (which must have n0 = 1) is rela- N. Read and S. Sachdev, Phys. Rev. B 42, 4568 (1990); tively straightforward. The D = 1 results apply unchanged S. Sachdev, Phys. Rev. B. 45, 12377 (1992). to fermionic Mott insulators. For D > 1, the Hamilto- 15 R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 nian Hph in (1.18) applies unchanged, but with p and h (2001). fermionic operators. The first two constraints in (1.16) are 16 A. Paramekanti, L. Balents, and M. P. A. Fisher, cond- now automatically accounted for by fermionic statistics, mat/0203171. while the last must be implemented by an infinite local 17 We are grateful to Mark Kasevich for this suggestion. repulsion between quasiparticles and quasiholes. Gapped 18 M. Campostrini, A. Pelissetto, P. Rossi, E. Vicari, Phys. phases appear for large |λ| (the region with λ ¿ 0 having Rev. B 61, 5905 (2000). Ising density wave order), but the particle-hole hopping 19 E. Frey and L. Balents, Phys. Rev. B 55, 1050 (1997). asymmetry allows this fermionic model to exhibit gapless 20 M. E. Fisher, Phys. Rev. 176, 257 (1968); M. Krech, Com- Fermi surfaces for motion in the transverse direction. The puter Simulation Studies in Condensed Matter Physics case of spinful fermions will have an even richer behav- XII , Eds. D. P. Landau, S. P. Lewis, and H. B. Schuettler, ior, driven by the antiferromagnetic coupling between the Springer Verlag, Heidelberg (1999), cond-mat/9903288. spins.