Mott insulators of hardcore bosons in 1D: many-body orders, entanglement, edge modes

Mott insulators of hardcore bosons in 1D: many-body orders, entanglement, edge modes

Ioannis Kleftogiannis Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan    Ilias Amanatidis Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34051, Republic of Korea
July 14, 2019
Abstract

Many-body phenomena were always an integral part of physics comprising of collective behaviors through self-organization, in systems consisting of many components and degrees of freedom. We investigate the collective behaviors of strongly interacting particles confined in one dimension. We show that many-body orders with topological characteristics can be found at the Mott insulator limit for hardcore bosons, at different fillings, without considering the spin degree of freedom or long-range microscopic interactions. These orders have unique properties like weak or strong quantum correlations (entanglement), quantified by the entanglement entropy, edge excitations/modes and gapped energy spectrum with highly degenerate ground state, bearing resemblance to topologically ordered phases of matter.

I Introduction

Many-body systems often exhibit novel properties, due to the collective behavior of their many interacting components that self-organize in unique ways. As it is usually quoted, a many-body system is more than just a sum of its individual parts. This extra ingredient due to collectiveness that is difficult to extract reductively, can lead to extraordinary phenomena, such as, the fractionalization of the elementary electron charge in the fractional-quantum-Hall effect Tsui (); Laughlin (). Other celebrated examples where many-body interactions can lead to measurable consequences are, superconductivity and spin liquidsBardeen (); Savary (). During the last decades, it was realized that not all quantum phases of matter, due to many-body interactions, can be described by the Landau symmetry-breaking theory of phase transitions. For example, the superfluid phase of a spin liquid and the fractional-quantum-Hall phase, possess properties like long-range spatial correlations related to quantum entanglementkitaev1 (); amico (); horodecki (). These phases usually occur at the ground states of many-body systems, which are highly degenerate and require a topological measure for their characterization, that takes into account the overall spatial properties of the system, instead of local order parameters that describe different phases in the Landau theory. Entanglement is a key underlying ingredient and is strongly tied to topology in these quantum phases, which have been dubbed topological ordersGu (). In order to identify their properties several measures can been used, like the entanglement entropy and the entanglement spectrum Kitaev2 (); Popkov (); Levin (); Li (); Hamma (); Alba () which allow to quantify the strength of entanglement and relevant topological features.

Topological order has been primarily demonstrated theoretically in two-dimensional systems(2D) amico (); Gu (); Kitaev2 (); Kitaev3 (); Levin (); Li (). The existence of quantum phases with topological order characteristics is still ambiguous in one-dimension(1D) kitaev1 (); haldane0 (); AKLT (); Kitaev4 (); Calabrese (); Pollmann (); Chen (); kim1 (); wang2 (). One of the main issues is that gapped 1D phases can only contain states that have short-range entanglement that is protected by a symmetryChen (), unlike 2D topologically ordered phases where long-range entanglement can exist without symmetrieskim2 (). One such example of symmetry protected topological order (STP) in 1D, is the Haldane phase of integer spin chainsPollmann (). In many cases the spin acts as an essential degree of freedom for the identification of the topological orders and entanglement properties, while for computational efficiency half-filled systems might be considered. Some topological and entanglement aspects in systems with strongly interacting particles, have been investigated in Refs Ejima (); Raghu (); Yoshida (); Farias ().

In this paper we investigate the many-body orders in the ground state of strongly interacting spinless particles confined in 1D. We simulate our system by considering hard-core bosons in Hubbard chains with strong short-range interactions. At the Mott insulator limit when the particles become strongly localized at each site of the Hubbard chain, the ground state is determined by their spatial freedom according to the filling. The particles arrange in different configurations, that minimize the energy of the system, forming a quantum fluid. Thus many degenerate or nearly degenerate ground states that are energetically isolated from the other states of the system, emerge. Moreover, spatial quantum correlations are created when the system is in a superposition of these ground states, despite the strong localization of the particles. By splitting the system in two parts the reduced density matrix and the entanglement entropy of each can be calculated, which is way to quantify the many-body correlations amico (); horodecki (). Using this method, we show that the strength of entanglement varies according to the filling. In addition edge modes appear at the ends of the system that are entangled with each other.

Ii Model

In order to simulate our one dimensional many-body system we use a spin-less Hubbard chain Hamiltonian with short-range interactions,

(1)

where are the creation and annihilation operators for spin-less particles, is the number operator, the nearest-neighbor hopping and the next nearest-neighbor hopping. The particles interact with strength U only when they are occupying neighboring sites in the chain. We consider that the Hubbard chain terminates at sites 1 and M with hard-wall/open boundary conditions. As we shall show the spatial freedom of the particles, at different fillings, can give surprisingly complex behaviors, even for this simple short-range interacting model. We consider many-body wavefunctions that are symmetric under exchange of two particles, but the particles cannot occupy the same quantum state , that is, only one particle is allowed per site and therefore can be either 0 or 1. This is the case of the hardcore bosonsguo (); wang1 (); zhang (); Varney () that can be realized in cold atom and helium-4 systems experimentally islam (); goldman (); Bloch (). The hardcore bosons satisfy the commutation relation . We investigate the collective behaviors of the particles by examining the quantum correlations in the many-body wavefunctions via the density matrix, the entanglement entropy and the occupation probability at each site of the Hubbard chain.

In order to characterize the many-body states in the text we use 1(0) to denote occupied(unoccupied) sites of the Hubbard chain. Schematically we represent it with filled(empty) circles.

The interaction strength U can be either positive or negative for repulsive or attractive interaction, respectively. We consider strongly interacting particles at the Mott insulator limit where . At this limit the particles localize at each site of the Hubbard chain and the ground state can be obtained easily without diagonalizing the Hamiltonian Eq. (1), as we are going to demonstrate for half and lower fillings.

Iii Ground states

At the limit , the minimum energy of the system is achieved by arranging the particles in a way that there is always, at least a single unoccupied site between them. The particles form a quantum fluid mediated by charged density waves. Since there are different ways to achieve this type of configuration, the ground state of the system becomes degenerate. The degree of degeneracy depends on the ratio of the number of particles over the number of sites in the Hubbard chain , which is the filling , determining the spatial freedom of the particles. The number of allowed particle configurations/microstates can be expressed mathematically with factorials via combinatorics as

(2)

The simplest way to construct the respective many-body wavefunction, is to assume a linear superposition of the degenerate ground states, which are determined by the different microstates as,

(3)

We have while index denotes the occupied sites in the Hubbard chain. The list has length N and determines a Fock state . This type of superposition with equal amplitudes for each microstate is a reasonably good approximation when the microstates are degenerate or nearly degenerate. A similar wavefunction can be used for describing the states of ferromagnetic spin chainsPopkov (). If there are particles in a system of sites, for half filling , then from Eq. (3) the number of degenerate ground states is N+1. This can be seen in figure 1a where we show a schematic representation of the different microstates for a half-filled system with . Below half-filling , the particles are spatially less restricted and therefore the number of ground states increases. As we shall see this degree of spatial freedom affects the entanglement properties of the ground state.

The ground states will be separated by a large gap from the excited states, which will form separate energy bands containing different types of microstates. For example all the microstates in the first excited states will contain one pair of particles occupying neighboring sites in the Hubbard chain, contributing energy U in the respective many-body states, which is the value of the energy gap seperating them from the ground states. Therefore, the interaction between the particles splits the Hilbert space of the non-interacting system in subspaces that contain many-body orders according to the different particle configurations allowed.

According to the above, if we consider this system as a quantum fluid, then the encircled areas in figure 1a would be incompressible, in the sense that in order to bring two particles on neighboring sites we need to overcome the energy gap U. The half-filled system can be mapped to a XXZ chain of spins S=1/2spinhaldane (). By using this analogy we can see that the ground state of our model contains hidden anti-ferromagnetic order. This can be understood easily by replacing occupied(unoccupied) sites with spin up(down). Then the spin alternates between up and down, as in a chain containing anti-ferromagnetic order. This is essentially a consequence of the microscopic rule that there must be at least one unoccupied site between the particles. The idea can be applied to any filling, since we can condense successive unoccupied sites into one.

Figure 1: a) The different particle configurations/microstates for the ground state of a half-filled spinless Hubbard chain with nearest neighbor repulsive interaction U and four particles (N=4). Open(filled) circles denote unoccupied(occupied) sites. These microstates minimize the system energy forming N+1 degenerate ground states. The system can be split in two partitions A and B in order to calculate their quantum correlations (entanglement). b) First nearest neighbor hopping allows transition between the ground states lifting their degeneracy, unlike second nearest neighbor hopping which preserves it. c) The microstates for attractive interactions -U and . The stacking of all the particles minimizes their energy giving N+1 ground states.

The effect of weak first or second nearest hopping on the ground states can be understood perturbatively. Consider the half filled case. First nearest neighbor hopping will allow transitions between the ground states, creating a small dispersion that will lift their degeneracy. For example, acting with a hopping term on the ground state results in the state , as can be seen in figure 1b. The degenerate ground states can be thought as the different sites in a tight-binding chain. A hopping between the sites will create a dispersion in the energy spectrum of the chain, creating a band structure, which is equivalent to lifting the degeneracy of the ground states. We have verified this correspondence by applying degenerate first order pertubation theory. We have found that the energy of the perturbed system will be

(4)

where D is the degeneracy (Eq. (2)) and j is an integer taking values j=1,2..D, for each of the perturbed ground states. This is the energy dispersion of a tight-binding chain with D sites and hardwall boundary conditions. Moreover each of these perturbed ground states can be written as linear combination of the unpertubed ones. The amplitudes are given by the corresponding wavefunction for a state j in the tight-binding chain,

(5)

where is the lattice site, representing each of the unperturbed states in the linear combination Eq. (3), running over the degenerate space of dimension D. Both results above are well known solutions for a XXZ spin chain, to which our model can be mapped at half-fillingspinhaldane ().

Acting with a second nearest neighbor hopping on the ground states, for any i, will result only in excited states, which contain at least a pair of particles occupying neighboring sites, contributing energy U, the energy gap that separates them from the ground states. Therefore the degeneracy of the ground states will not be lifted when adding weak second nearest neighbor hopping. Also, the particle configurations for each ground state will be preserved. This result simplifies our analysis since we only need the positions of each localized particle in the Hubbard chain contained in Eq. (3) to analyze the properties of the ground states with weak second nearest neighbor hopping.

The ground states for a system with N=6 can be seen in figure 2a after diagonalizing the Hubbard Hamiltonian Eq. (1). As we have analyzed for half-filling () the degeneracy is preserved for while it is lifted for . Although the degeneracy is lifted for lower fillings (), the energy gap from the excited states does not close. Then we can still consider that the system still lies in a superposition of these nearly degenerate ground states (Eq. (3)), as long as hopping t is small compared to U.

Iv Entanglement

Even though the particles localize at each site of the Hubbard chain due to the strong interaction, spatial quantum correlations are created, when we consider a state of the system as a superposition of the different ground-states Eq. (3). A well established approach to quantify the correlations is to split a quantum system in two partitions, say A and B forming this way a composite system. Then the entanglement between these partitions can be estimated via the reduced density matrix of partition A, after tracing out the rest of the system, that is partition B. The elements of the reduced density matrix can be calculated via , where is the amplitude for each partitioned ground state , where i(k) is the corresponding microstate in A(B). Moreover the Von Neumann entanglement entropy can be calculated

(6)

The scaling of the entanglement entropy provides information about the strength of entanglement in 1D quantum systems. For critical 1D phases it has been shown that the entropy diverges logarithmically with increasing the partition size, while it saturates/converges for non-critical phaseskitaev1 (). The logarithmic divergence implies stronger entanglement than the converging case.

In the case of half-filling, the density matrix obtains a simple form that allows the calculation of the entanglement entropy analytically as follows. We start by noticing that each subsystem A or B is half the size of the full composite system, with particles distributed in sites. In order to calculate the element of the density matrix, we can fix the microstate inside A at and then count the different microstates in B. These are , as if B was an isolated system and we wanted to obtain the number of its ground states. Then we multiply by the square modulus of the normalization factor and obtain . All the other elements of the density matrix for are equal to , since the rest of the microstates in A appear only once in the ground state of the composite system. For convenience we define , . Then the only two non-zero eigenvalues of the reduced matrix are

(7)

where . The entropy of the subsystem A in term of the eigenvalues of the density matrix becomes

(8)

At the thermodynamic limit the above eigenvalues become , resulting in the entropy

(9)

This is the entanglement entropy value for a maximally entangled Bell state of two spins in the singlet state. The convergence at the thermodynamic limit implies semi-local correlations, that result in weak entanglement as in the non-critical phases of XY spin chains kitaev1 (). This result agrees with the numerical calculation of versus N using the ground state Eq. (3) , shown in figure 3 , where the dashed line is the analytical result Eq. (8).

We would like to note that the steps above are valid for even number of particles N, half of which go at each partition. Following a similar method we can derive that for odd N also.

The origins of the weak entanglement can be understood as follows. We can imagine the partitions A and B, as two isolated many-body systems. Each one is half the size of the full composite system and its ground states are determined by the different microstates that have at least one unoccupied site between all the particles. This microscopic rule plays a crucial role, since it restricts the combinations of the ground states of the isolated partitions that can form the ground states of the composite system. For example in Fig 1a, the combination has to be excluded, when we form the ground state of the full system. So in general, the ground state Hilbert space of the full system is not simply the tensor product of the individual spaces of A and B, that is . The two partitions become weakly entangled, due to the local particle interaction at their boundary.

In figure 3 we show the for lower fillings () calculated numerically using the ground state Eq. (3). For all these cases diverges logarithmically, as in the critical phases of XY spin chains kitaev1 (); Calabrese (), implying stronger entanglement than . This is an indication of the increased complexity of the particle configurations due to the larger spatial freedom compared to the half-filling. We remark that for , the particle number at each partition is not conserved. Therefore we cannot form the ground state Hilbert space of the full system as a tensor product of the individual spaces of two isolated partitions A and B, even if we remove the interaction at the boundary between them, unlike the half-filled case. This means that we cannot completely remove the entanglement with local operations, which is another indication of the strong entanglement, that induces long-range spatial correlations.

The different entanglements are related to the spatial freedom of the particles at the respective fillings. In a low filled system we can freely add an additional particle, as long as there is at least one unoccupied site between it and the rest of the particles. This way we can fill the empty space of the system with particles (for even M) that will go at the ground state without affecting its energy, or the gap from the excited states. In this sense, the low fillings in our model correspond to a transition towards a superfluid phasehen (), i.e. the system at low fillings lies in a critical regime. This could explain the logarithmic divergence of the entanglement entropy with the partition size, which occurs in general at the critical regime of 1D many-body systems. In the half-filled case on the other hand the system lies in an insulating phase, since in order to add an additional particle we have to excite the system. Therefore, the different scaling behavior of the entanglement entropy for the half-filled and low filled cases, is related to the different phases the system lies at these fillings.

We have found that most of the eigenvalues of the density matrix are doubly degenerate as in the Haldane phase of spin chains where string order and entanglement are presentLi (); Calabrese (); Pollmann ().

Another way to detect the entanglement is by measuring the purity of the quantum state with density matrix which can be quantified by islam (). When the state is mixed which means that it is not quantum mechanically fully consistent and contains statistical fluctuations. When this happens for the reduced density matrix of a partition of a quantum system, entanglement with the rest of the system is implied. We have found for all the fillings studied (), which is an additional indication of the entanglement present in our system.

It should be noted that the entanglement in the Mott insulator limit, when the particles become strongly localized, is governed by the different ways they organize to form the ground states. In essence, it could be said, that the quantum superposition of the enumerative combinatorics of the particles when viewed classically as macroscopic objects, is creating the correlations. This is also the reason that it is not necessary to diagonalize the Hamiltonian Eq. (1) in order to estimate the entanglement, at the Mott insulator limit.

As far as topological order is concerned, which is usually identified through its highly degenerate ground state and strong long-range entanglement properties, there are obvious similarities with our results. However we are not able to obtain both these properties simultaneously. For example the half-filled system with only second nearest neighbor hopping has a highly degenerate ground state but lacks the strong entanglement. On the other hand at lower filling the degeneracy is lifted but the entanglement of these nearly degenerate ground states, becomes stronger, as indicated by the logarithmic divergence of the entanglement entropy.

Figure 2: The energy levels for repulsive (U) and attractive (-U) interaction with first () or second () nearest neighbor hopping. In all cases the ground states are isolated from the excited states since they are separated by a large energy gap. For repulsive interaction U the ground state degeneracy is preserved only for .
Figure 3: a)The entanglement entropy for a partition A of the full system, at different fillings f vesrus the number of particles N. For half-filling the entropy converges at as two spins in a singlet maximally entangled Bell state. For lower fillings the entropy diverges logarithmically as , implying stronger entanglement. The last two points in the curve for and are calculated via extrapolation.b) The occupation probability (particle density) versus each site i in the Hubbard chain, for N=6,8,10 and different fillings. Edge modes can be seen for , with uniform bulk density and edge density that is a fraction . The edge modes are not affected by N.

V Edge modes

In order to obtain additional features of the ground states we calculate the occupation probability (particle density) for each site of the Hubbard chain. It can be defined as

(10)

where is the number operator at site i.

In figure 3b we show for different fillings and number of particles. In all cases, we observe fluctuations of the probability, that are larger at the two ends of the Hubbard chain, resembling charged density waves (CDW)Nishimoto (). For the lowest fillings the fluctuations smoothen out at the bulk of the Hubbard chain resulting in uniform density .

If we had ignored the edges of the system, by applying periodic boundary conditions for example, then it would be indeed reasonable to expect the probability to be uniformly distributed on every site, obtaining the corresponding value of the filling, with . This could be imagined as a fractionalization of each particle with the fractions distributed equally on all the sites of the Hubbard chain. However when edges are present a small percentage of the fractions distributes equally on them forming edge modes. This percentage is equal to the occupation probability at the edge. It can be obtained by the ratio between the microstates of the system that have one occupied site at the corresponding edge (site 1 or M) D(M-2,N-1), over the total number of microstates D(M,N). This is . By using Eq. (2) we find that , which becomes in the thermodynamic limit .

Even below the thermodynamic limit the edge density is very close to this value as shown figure 3b where is plotted for increasing particle number N=6,8,10. In addition, it changes only slightly with the size of the system. The dip of near the edge sites is due to the repulsive interaction U which reduces the probability to find a particle on the neighboring site to the edge, if the edge site is already occupied.

The edge modes could be considered as excitations of the particle density which remains uniform at the bulk of the Hubbard chain. Each edge is contained in one of the partitions A or B, which are quantum mechanically correlated(entangled), as we have shown. Therefore we can assume that the edge excitations at the opposite ends of the system are entangled with each other.

Vi Attractive interactions

We briefly analyze the case of negative U, that is, for attractive interactions between the particles. In this case the ground state is obtained by stacking all the particles together at neighboring sites, which minimizes their energy at . These states are separated by a large gap U from the first excited states. An example of the microstates can be seen in figure 1c. The degeneracy of these ground states is preserved for both first and nearest neighbor hopping as can be seen in figure 2, unlike the repulsive interaction case. For the half-filled case the entanglement entropy in figure 3 follows the limit where the number of the different particle configurations in subsystems A and B is equal to the corresponding number of microstates. At this limit the reduced density matrix is diagonal, resulting in maximum entropy which is the corresponding dashed curve in figure 3a. For lower fillings the entropy is reduced but still diverges logarithmically.

Vii Summary and Conclusions

To summarize, we have investigated the self-organization of strongly interacting spinless particles confined in one-dimension. We have consindered hardcore bosons at the Mott-insulator limit, modeled in a spinless Hubbard tight-binding chain, with first and second nearest neighbor hopping. The particles localize at each site of the Hubbard chain due to the strong interaction. However, the spatial freedom at different fillings allows the particles to organize in various configurations corresponding to different energy bands, that are separated by large gaps. These particle configurations/microstates result in many-body orders that contain quantum correlations, strong or weak entanglement between different parts of the system. At half-filling the corresponding ground state determined by these microstates, is highly degenerate when only second nearest neighbor in our Hubbard chain model is considered. When the system is split in two partitions then their entanglement resembles that of two spins in a singlet maximally entangled Bell state. At low fillings the degeneracy is lifted but the entanglement of the nearly degenerate ground states, becomes stronger as indicated by the logarithmic scaling of the entanglement entropy of each partition. In addition we found excitations at the ends of these Mott insulators, that are entangled with each other, due to the long-range correlations induced by the strong entanglement.

Our results show that certain Mott insulators made of spinless hardcore bosons can have spectrally isolated ground states with non-trivial many-body orders depending on the filling, despite the strong localization of the particles. These orders contain quantum correlations, that is, different parts of these Mott insulators are spatially entangled and therefore have topological characteristics.

An extension of the current analysis would be to investigate the many-body orders and the entanglement in the different excited states that are separated by energy gaps. Additionally, we expect richer orders for particles in two dimensions with more varied patterns of entanglement, due to the increased degrees of freedom.

As a final note, we would like to remark that our results contribute to the idea, that topological phenomena such as the edge states and topological orders can be created via the collective behaviors in many-body systems with relatively simple microscopic rules. Such well known examples are for instance, the Haldane phase of integer spin chainshaldane0 (); AKLT (), the majorana modes in the Kitaev chainKitaev4 () and the topological order in the toric codeKitaev3 (). Apart from its fundamental significance, this approach might be useful in the on-going research on the experimental realization of different topological phases in cold-atomic and photonic systems.

Acknowledgements

We would like to thank Ara Go, Guang-Yu Guo and Alexander Schnell for useful comments. We acknowledge resources and financial support provided by the National Taiwan University, the Ministry of Science and Technology, the National Center for Theoretical Sciences of R.O.C. Taiwan and the Center for Theoretical Physics of Complex Systems in Daejeon Korea under the project IBS-R024-D1.

References

References

  • (1) Tsui D C, Stormer H L and Gossard A C 1982 Phys. Rev. Lett. 48 1559
  • (2) Laughlin R B 1983 Phys. Rev. Lett. 50 1395
  • (3) Bardeen J, Cooper L N and Schrieffer J R 1957 Phys. Rev. 108 1175
  • (4) Savary L and Balents L 2017 Rep. Prog. Phys. 80 016502
  • (5) Vidal G, Latorre J I, Rico E and Kitaev A 2003 Phys. Rev. Lett. 90 227902
  • (6) Amico L, Fazio R, Osterloh A and Vedral V 2008 Rev. Mod. Phys. 80 517
  • (7) Horodecki R, Horodecki P, Horodecki M and Horodecki K 2009 Rev. Mod. Phys. 81 865
  • (8) Chen X, Gu Z-C and Wen X-G 2010 Phys. Rev. B 82 155138
  • (9) Kitaev A and Preskill J 2006 Phys. Rev. Lett. 96 110404
  • (10) V. Popkov and M. Salerno 2005, Phys. Rev. A 71, 012301
  • (11) Kitaev A Y 2003 Ann. Phys. (NY) 303
  • (12) Levin M and Wen X-G 2006 Phys. Rev. Lett. 96 110405
  • (13) Li H and Haldane F D M 2008 Phys. Rev. Lett. 101 010504
  • (14) Hamma A, Ionicioiu R and Zanardi P 2005 Phys. Rev. A 71 022315
  • (15) Alba V, Haque M and L uchli A M 2013 Phys. Rev. Lett. 110 260403
  • (16) Haldane, F. D. M. 1983a Phys. Lett. A 93 464
  • (17) Affleck I, Kennedy T, Lieb E H and Tasaki H 1987 Phys. Rev. Lett. 59 799 802
  • (18) Kitaev A Y 2001 Phys. Usp. 44 131
  • (19) Calabrese P and Lefevre A 2008 Phys. Rev. A 78 032329
  • (20) Pollmann F , Turner A M, Berg E and Oshikawa M 2010 Phys. Rev. B 81 064439
  • (21) Chen X, Gu Z-C and Wen X-G 2011 Phys. Rev. B 83 035107
  • (22) Kim I H 2014 Phys. Rev. B 89 235120
  • (23) Wang Y, Gulden T, Kamenev A 2016 Phys. Rev. B 95, 075401
  • (24) Kim I H 2013 Phys. Rev. Lett. 111 080503
  • (25) Ejima S, Lange F and Fehske H 2014 Phys. Rev. Lett. 113 020401
  • (26) Raghu S, Qi X-L, Honerkamp C and Zhang S-C, 2008 Phys. Rev. Lett. 100 156401
  • (27) Yoshida, Peters R, Fujimoto S and Kawakami N 2014 Phys. Rev. Lett. 112 196404
  • (28) Costa Farias R J and Oliveira M C de 2010 J. Phys.: Condens. Matter 22 245603
  • (29) Guo H M 2012 Phys. Rev. A 86 055604
  • (30) Wang Y-F, Gu Z-C, Gong C-D and Sheng D N 2011 Phys. Rev. Lett. 107 146803
  • (31) Zhang N G and Henley C L 2003 Phys. Rev. B 68 014506
  • (32) Christopher N V, Sun K, Rigol M and Galitski 2010 Phys. Rev. B 82 115125
  • (33) Islam R, Ma R, Preiss P M, Tai M E, Lukin A, Rispoli M and Greiner M 2015 Nature 528 77
  • (34) Goldman RN, Budich J C and Zoller P 2016 Nat. Phys. 12 639
  • (35) Bloch I, Dalibard J and Nascimb ne S 2012 Nature Phys. 8 267 276
  • (36) Haldane F D M 1980 Phys. Rev. Lett. 45 1358
  • (37) Hen I and Rigol M 2009 Phys. Rev. B 8̱0 134508
  • (38) Nishimoto S, Ejima S and Fehske H 2013 Phys. Rev. B 87 045116
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