Abstract
The fractional quantum Hall effect is one of the most striking phenomena in condensed matter physics. It is described by a simple Laughlin wavefunction and has been thoroughly studied both theoretically and experimentally. In lattice systems, however, much less is currently known, and only few models and mechanisms leading to it have been identified. Here we propose a new way of constructing lattice Hamiltonians with local interactions and fractional quantum Hall like ground states. In particular, we obtain a spin 1/2 model with a bosonic Laughlinlike ground state, displaying a variety of topological features. We also demonstrate how such a model naturally emerges out of a Fermi–Hubbardlike model at half filling, in which the kinetic energy part possesses bands with nonzero Chern number, and we show how this model can be implemented in an optical lattice setup with present or planned technologies.
Introduction
The fractional quantum Hall (FQH) effect^{1} is one of the most fundamental phenomena in strongly correlated electronic systems and a paradigm of topological behaviour. It has been thoroughly studied and characterized in situations, where the solid surrounding the electrons has a modest effect on their properties^{2,3}. This was possible, in part, thanks to the successful description in terms of the celebrated Laughlin wavefunctions^{4}. The appearance of FQH behaviour when the lattice structure created by the solid becomes important is however not so deeply studied or understood. Nevertheless, the interest in lattice systems^{5,6,7,8,9,10,11,12} has been sparked by the possibility of observing such behaviour with ultracold atoms in optical lattices^{13,14,15,16,17,18}, obtaining hightemperature fractional quantum Hall states^{19,20,21,22}, or even its applications in quantum information processing^{23}. The relativistic FQH effect in graphene has also been in focus in recent years, and this effect has already been observed experimentally^{24,25}.
The standard route to the FQH effect in solids involves fractionally filled Landau levels (created by a strong, external magnetic field) and electron–electron interactions. This mechanism can be adapted to lattice systems by replacing the fractionally filled Landau level by a sufficiently flat and fractionally filled Chern band and adding nearneighbour interactions^{19,20,21,22,26,27} (the socalled flat band models). A Chern band can be obtained by a proper choice of complex hopping amplitudes^{5} and can be made flat with longer range hoppings^{19,28,29}.
Here we propose and investigate a different approach to obtain FQH states in lattice systems. The approach builds on conformal field theory (CFT) combined with deformations, and we use it to derive a model for spin 1/2 particles on a square lattice with some specific shortrange interactions. As we will show, the ground state of the system is extremely well described by the Kalmeyer–Laughlin (KL) wavefunction^{30,31} (the spin version of the bosonic Laughlin wavefunction with Landau level filling factor ν=1/2) and exhibits the topological behaviour expected for FQH states. In particular, we consider periodic boundary conditions and analyse the ground state (quasi) degeneracy, its response to twisting the boundary conditions, the manybody Chern number, the local indistinguishability of the ground states and the topological entanglement entropy. We also show that our model can be derived from a Fermi–Hubbardlike model, and we use this insight to propose an alternative mechanism for obtaining FQH states in lattices. Finally, we demonstrate how our model can be implemented with ultracold atoms in optical lattices. The temperatures, tunnelling amplitudes and interactions required to observe the exotic topological behaviour are the same as those required to observe the Néel antiferromagnetic order in the standard Fermi–Hubbard model, so that our predictions can be tested with present or planned technology^{32,33,34,35,36}.
Results
Two perspectives on the proposed model
The mechanism leading to the FQH properties can be viewed from two perspectives. The first perspective leans itself on the facts that states with interesting (topological) properties can be constructed from correlators of chiral conformal fields^{11,37,38} and that nonlocal parent Hamiltonians can be derived for a quite broad class of such states^{38}. We demonstrate here for the particular case of correlators from the SU(2)_{1} Wess–Zumino–Witten model that the nonlocal parent Hamiltonian can, in fact, be deformed into a local Hamiltonian without crossing a phase transition, and this leads to the proposed model. The second perspective considers spin 1/2 fermions moving on a lattice with (not necessarily flat) Chern bands and strong local interactions. When the lattice filling factor is 1/2 for both spin up and spin down, a Mott state is formed, and the spin state inherits the topological character of the Chern bands.
The first perspective suggests natural generalizations to other models (for example, to Moore–Read^{37}like states), and in the present paper, we propose the CFT construction combined with the idea of deforming the nonlocal parent Hamiltonian as a general strategy to construct lattice models with interesting physical properties and only local interactions. A side advantage of this approach is that the ground state of the original Hamiltonian is known (analytically or expressed as a correlator), and this is a great help to determine the physical properties of the phase in question. The local model with a KLlike ground state that we derive below demonstrates the applicability of the approach. We choose, rather arbitrarily, to define the model on a square lattice, but we note that the CFT construction allows any lattice in two dimensions (2D) to be considered. We find that the deformation of the Hamiltonian, in fact, changes the ground state very little, and we can therefore use the ground state of the original Hamiltonian as an analytical approximation to the ground state of the local Hamiltonian. This is a significant advantage compared with models in which only the Hamiltonian is known analytically.
The second perspective arises because the considered model can also be derived from a Fermi–Hubbardlike model in a certain parameter regime, as we shall demonstrate. This provides additional insight and, in particular, enables us to compare our model with flat band models. The comparison shows that the properties of the Fermi–Hubbardlike model differ from those of the flat band models, which is interesting, because it opens new directions in the search for FQH states in lattices. We also use the second perspective for proposing a scheme to implement the model in ultracold atoms in optical lattices. Realizing a FQH state in optical lattices would give unique possibilities for detailed experimental investigations of the state under particularly wellcontrolled conditions and would, in addition, be a hallmark for quantum simulation.
Model
Referring back to the first of the two perspectives mentioned above, our starting point is a chiral CFT correlator
which we intend to interpret as a spin wavefunction of N spins sitting at fixed positions in the 2D plane. In this construction, the chiral conformal field (z_{n}) with spin j_{n} represents the nth spin at the position (Re(z_{n}),Im(z_{n})) with internal spin state s_{n} and spin quantum number j_{n}. s_{n} can take 2j_{n}+1 different values, and denotes the vacuum expectation value.
In the following, we shall take (z_{n}) to be the chiral spin 1/2 field of the SU(2)_{1} Wess–Zumino–Witten model and choose s_{n}/2=±1/2 to be the third component of the spin. In this case, (1) evaluates to^{39}
for N even, where δ_{s}=1 for and δ_{s}=0 otherwise and is the Marshall sign factor. The correlator is zero for N odd, and we shall thus take N to be even throughout. It has been found in a previous study^{11} that the state (2) is closely related to the KL state and reduces exactly to it in the case of a square lattice of infinite extent. In the following, we shall refer to (2) as the CFT state in the plane.
A nonlocal, exact parent Hamiltonian of has been derived in previous studies^{11,38} by utilizing the CFT properties of the state. The Hamiltonian consists of interactions between all pairs and triples of spins in the system and is simpler than the sixbody parent Hamiltonians of the exact KL state with periodic boundary conditions derived earlier^{7,9}. The longrange interactions can be challenging to achieve experimentally. One may argue however that deforming the Hamiltonian will not dramatically change the physical properties of the ground state as long as no phase transition is crossed. With this in mind and specializing to an L_{x} × L_{y} square lattice with L_{x} even, we investigate the local Hamiltonian that remains after removing all longrange interactions and making all coupling strengths position independent:
Here is the spin operator of the nth spin. As indicated in Fig. 1a, the first (second) twobody term is summed over all pairs of nearest (nextnearest) neighbour spins, and the threebody term, which breaks time reversal symmetry, is summed over all triangles of neighbouring spins (for each triangle only one term is included and n,m,p label the vertices of the triangle in the counter clockwise direction as indicated with the arrow). The Hamiltonian conserves the total spin and is hence SU(2) invariant. In the following, we shall parameterize the coupling strengths as J_{2}=Jcos(φ_{1})cos(φ_{2}), J′_{2}=Jsin(φ_{1})cos(φ_{2}) and J_{3}=Jsin(φ_{2}).
To test how much the deformation of the Hamiltonian affects the ground state, we show the overlap between the ground states before and after the deformation as a function of φ_{1} and φ_{2} in Fig. 2 and as a function of lattice size in Table 1. Even if the states are close, the overlap is expected to decrease exponentially with the total number of spins in the system due to the increase in Hilbert space dimension. This effect can be counteracted by considering the overlap per site, which is defined as the Nth root of the overlap, and we also provide this measure in the table. The figure reveals a region with overlaps close to unity and shows that, for example, φ_{1}=0.07 × 2π and φ_{2}=0.03 × 2π (marked with the white circle) is a reasonable choice of parameters. The overlaps per site in Table 1 are around 0.999 for all the considered lattice sizes and seem not to decay with lattice size. These results are a strong indication that the deformation of the Hamiltonian does not bring the system to a different phase. In fact, we may regard equation (2) as an analytical approximation to the ground state, which is extremely helpful to see the underlying physics and to perform computations for systems that are too large for exact diagonalization.
The model we have looked at so far is defined in the plane and hence has open boundary conditions. To investigate the topological properties of H and the CFT state, we shall also need to consider periodic boundary conditions. On the torus, that is, for periodic boundary conditions in both directions, there are two CFT states, and , and we find analytically that these are proportional to the KL states on the torus (see Methods for explicit expressions). To ensure that the states are eigenstates of the translation operators and of the operator that rotates the lattice by 90° for L_{x}=L_{y}, we form the linear combinations
for evenbyodd lattices and for L_{x}=L_{y}, respectively, where lε{0,1}. Table 2 shows that the large overlap between the ground states of H and the CFT states also hold on the torus and provides quantum numbers of the states.
Topological properties
FQH states are examples of topological states, and in the following we demonstrate that H and the CFT states exhibit topological properties that fit the properties of the ν=1/2 Laughlin state in the continuum. Characteristic features of topological states are a ground state degeneracy that depends on the topology of the surface on which the states are defined^{40}, the lack of ability to distinguish the topologically degenerate states through local measurements^{40,41} and a nonzero topological entanglement entropy^{42,43}. For FQH states in solids another important quantity is the Hall conductivity. The Hall conductivity is closely related to the Chern number^{44,45,46}, which can also be computed for spin systems. For Laughlin states the expected behaviour is a manybody Chern number of the ground state manifold on the torus that is unity and spectral flow of the ground states into each other for certain choices of the lattice size under continuous twisting of the boundary conditions^{22,47}.
We compute the spectrum on the torus through exact diagonalization of H (see, for example, the study by Läuchli^{48} for a description of the numerical approach), and the results are shown in Fig. 3a. The exponential increase of the dimension of the Hilbert space with the number of spins limits the system sizes we can consider. It is seen that the two lowest energy states approach each other as the system size is increased, and extrapolating the energy difference between the third lowest and the lowest energy state to the limit N→∞ points to the existence of a gap in the thermodynamic limit.
For finite size systems, the local indistinguishability of topologically degenerate states is not perfect, but the ability to distinguish the states decreases exponentially with the size of the system. In Fig. 4, we plot the deviation d between the correlators of local operators computed for the two states on the torus for both the exact ground states and the CFT states. In both cases, d decreases with increasing system size, and for the CFT states the decay is clearly exponential. We also observe that d is not small unless the lattice size is at least 6 × 5. This explains why the ground state degeneracy is more pronounced for the 6 × 5 lattice in Fig. 3a than it is for smaller lattices, and the general decrease of d in Fig. 4 provides further evidence that the two lowest energies will approach each other further for larger lattices.
In Fig. 5, we compute the topological entanglement entropy γ using the approach proposed in a previous study^{49}. For the ν=1/2 Laughlin state in the continuum, γ=ln(2)/2 (ref. 42), and the result we get for the CFT state is in perfect agreement with this value. For the exact ground state of H, we can again only consider rather small systems, but we observe that the results approximately follow those of the CFT state and also point to a nonzero value of γ.
We have computed the manybody Chern number of the exact ground states on the torus for a 4 × 5 lattice as explained in Methods, and the results are depicted as a function of φ_{1} and φ_{2} in Fig. 2. As the 4 × 5 lattice does not display a clear twofold ground state degeneracy for all values of the parameters, it may happen that the two lowest energy states are not the states that resemble the CFT states. We circumvent this problem by noting that the Hamiltonian is block diagonal, which allows us to select the right states based on quantum numbers as stated in the caption. The KL states in equation (11) on the square lattice can be generalized to twisted boundary conditions by modifying the states in the same way as the ν=1/2 Laughlin states on the torus are modified when subjected to twisted boundary conditions (see equations (5.1–5.5) of the study by Read and Rezayi^{50} for explicit expressions). From this we get that the manybody Chern number of the KL states on the torus for a 4 × 5 lattice is 1. Figure 3b demonstrates the flow of the ground states of H into each other when the boundary conditions are twisted, and for the KL states this property can be shown analytically. Again, our findings are thus in agreement with the properties of the ν=1/2 Laughlin state in the continuum.
Finally, we note that we have done the tests above also for the parameters φ_{1}=0 and (the white square in Fig. 2), and the results are similar.
Connection to a Fermi–Hubbardlike model
As mentioned in the introduction, each spin in our model may represent the spin of a fermion sitting on a site in an optical lattice. Let us consider the Fermi–Hubbardlike Hamiltonian
on a square lattice, where annihilates a fermion with spin σ on lattice site n, U is a positive constant, H_{kin,σ} is defined in Fig. 1b, and we choose the total number of fermions to equal the number of lattice sites. H_{kin,σ} represents spinpreserving hopping of fermions between nearest and nextnearest neighbouring sites with complex hopping amplitudes, and the second term in H_{FH} represents interactions between fermions with opposite spin sitting on the same lattice site. We note that the hopping terms give rise to a staggered flux pattern with zero total flux through the unit cell as in the Haldane model^{5}.
When U is much larger than the hopping strengths t, t′, we are in the Mott insulating regime, where each site is occupied by a single fermion. In this limit, we can use the Schrieffer–Wolff transformation^{51,52} to derive an effective Hamiltonian H_{eff} acting in the space of states with only single occupancy on all sites in the same way as the Heisenberg model is derived from the standard Fermi–Hubbard model. Reexpressing H_{eff} in terms of spins, we get H_{eff}=H+constant to third order in t/U, where H is given by equation (3) with J_{2}=2t^{2}/U, J′_{2}=2t′^{2}/U, and J_{3}=6t^{2}t′/U^{2}. This is not by chance: the SU(2) invariance of H_{FH} is automatically inherited to H_{eff} and the chirality is built into the model through the complex hopping amplitudes. The choice φ_{1}=0.07 × 2π and φ_{2}=0.03 × 2π corresponds to t/U=0.10 and t′/U=0.07.
Overview of simulation scheme
The Fermi–Hubbardlike model can be simulated in fermions in optical lattices as follows. We encode the spin up and down states in four internal hyperfine levels as shown in Fig. 1d. By choosing the internal states appropriately and using polarized light, it is possible to construct the optical lattice in Fig. 1c, where the levels shown in blue (red) are trapped in a potential with minima located at the white (black) squares of a checkerboard.
In this setup, the atoms can tunnel through the potential barrier between nearestneighbour sites in the blue/red sublattice, which provides a contribution to the nextnearestneighbour hopping terms in the Hamiltonian. We propose to implement the nearestneighbour hopping terms by use of laserassisted tunnelling^{53,54}. This can be done with the three standing wave laser fields in Fig. 1b, which we refer to as r1, r2 and b3, respectively. As explained further below, r1 and b3 (r2 and b3) induce nearestneighbour hops in the horizontal (vertical) direction. A hop is accompanied by a Raman transition between internal states as illustrated in Fig. 1d. The alternating signs of the hopping amplitudes originate from the spatial oscillations of the amplitudes of r1 and r2 with a period that is twice the lattice spacing, and the i’s on the hopping amplitudes in the vertical direction appear because the amplitude of r2 is imaginary. Note that it is possible to choose the internal states in such a way that the energy difference between the blue and red spin up states is the same as the energy difference between the blue and red spin down states. r1, r2 and b3 therefore accomplish the hops for both spin up and spin down. A particularly convenient feature of the above model and implementation scheme is that only onsite interactions are needed. These occur naturally in the optical lattice, and so the Hamiltonian in equation (5) is implemented directly.
Optical lattice potential
The main idea behind the scheme we propose to realize the optical lattice potential in Fig. 1c is to encode the blue (red) states in internal levels that interact more strongly with right (left) circularly polarized light than with left (right) circularly polarized light and then create a checkerboard pattern of regions with highest intensity of right circularly polarized light and regions with highest intensity of left circularly polarized light. Since the blue and red sublattices are square lattices when viewed from the (x+y) or (x−y) directions, it is natural to let laser beams travel along these directions to create a standing wave pattern of intensity minima and maxima. In order to be able to control the strengths and phases of right and left circularly polarized light in the beams independently, however, the momentum vectors of the beams should have a nonzero zcomponent.
Let us demonstrate explicitly how this can be done for fermions with one valence electron, when the ground state manifold is a ^{2}S_{1/2} orbital, and the light fields couple the ground states offresonantly to a ^{2}P_{1/2} orbital as in Fig. 6a, which can be achieved with alkali atoms. We first create a standing wave pattern in the (x+y) direction by adding up the fields labelled 1a and 1b in Table 3. Note that the intensity profile of the right (left) circularly polarized component of E_{1a}+E_{1b} is proportional to () and that the zpolarized components of the fields are fixed by the condition that the polarization vectors of the fields must be perpendicular to the wave vectors. As seen in Fig. 6a, the presence of the zpolarized component causes the trapping lasers to induce Raman transitions between the two ground states. This undesired effect can, however, be cancelled by adding the fields 2a and 2b in Table 3. The fields 3a, 3b, 4a and 4b do the same for the (x−y) direction.
Eliminating the excited states with a Schrieffer–Wolff transformation, we get the trapping potentials
for g_{−}› and g_{+}›, respectively, where V_{0} ∝ E^{2}/(β^{2}Δ), the detuning Δ, defined in Fig. 6a, and V_{0} are negative (positive) for red (blue) detuning, and
For −V_{0}(2β^{2}+α^{2}−1)>0 and −V_{0}(1+2α^{2}β^{2}−α^{2})>0, we thus conclude that g_{−}› (g_{+}›) is trapped around the minima of S (C), which gives the desired checkerboard pattern.
So far, we have considered the states at the level of fine structure. Assuming, as an example, that the nuclear spin is I=1, which is the case for ^{6}Li, the ground state manifold consists of six hyperfine states as depicted in Fig. 6b. The states 3/2,1/2› and 1/2,−1/2› see the potential 2V_{+}/3+V_{−}/3, and the states 3/2,−1/2› and 1/2,1/2› see the potential V_{+}/3+2V_{−}/3. For suitable parameters, for example, V_{0}<0 and
one can therefore obtain the desired checkerboard potential by choosing the blue and red spinup and down states as in Fig. 6b. The states and see the potentials V_{+} and V_{−}, respectively, and are hence, in general, shifted away in energy, so that they can be ignored.
The energy difference between the blue and red spinup states is the same as the energy difference between the blue and red spindown states for the above choice as desired, and this also holds when the trapping in the zdirection is taken into account. If this symmetry cannot be achieved in a given setup, one can compensate by choosing the right and left circularly polarized components of the field b3 in Fig. 1b to have different frequencies. Note also that the energy difference between a spin sitting on a blue site and a spin sitting on a red site must be nonzero for the implementation of the hopping terms to work. If this difference is not appropriate for a given set of parameters, it can be adjusted by adding an independent standing wave laser field along the z axis with right or left circular polarization.
Laserassisted hopping
The transitions induced by the fields r1, r2 and b3 in Fig. 1b can be understood by noting that the transition amplitudes are proportional to the spatial integral of the product of the Wannier functions at the two sites and the two field components that are involved in the transition. Since the Wannier functions decay rapidly, there are only transitions between nearby sites.
Let us first consider the hops between neighbouring blue and red lattice sites induced by r1 and b3. The transition amplitude for hops along the y axis vanishes because the Wannier functions and b3 are even functions and r1 is an odd function with respect to reverting the x axis around the x coordinate of the sites. Hops along the x axis are allowed, and since the sign of the amplitude of r1 between two neighbouring sites is alternatingly plus and minus (see Fig. 1b), the signs of the hopping amplitudes also alternate. r2 and b3 similarly induce hops in the y direction, but not in the x direction. For the choice of hyperfine levels considered above, it is the right (left) circularly polarized component of b3 that is involved when a fermion in the spin up (down) state hops. Therefore, one can make the hopping amplitudes for up and down spins equal by adjusting E_{±} appropriately, that is, by adjusting the polarization state of b3.
Let us also consider transitions involving one photon from r1 and one photon from r2. In this case, the fermion cannot change its internal state due to energy conservation, and it cannot hop to a site of different colour. If the fermion remains at the same site, the transition amplitude is zero due to symmetry. If the transition involves a hop to a nextnearestneighbour site, there is destructive interference between absorbing a photon from r1 and emitting a photon into r2 or absorbing a photon from r2 and emitting a photon into r1, and therefore no transitions occur.
r1, r2 and b3 also induce transitions, where a fermion absorbs and emits a photon from the same beam without changing its internal state and either remains at the same site or hops to a nextnearestneighbour site. The former case gives rise to the trapping in the z direction and also slightly modifies the potential in the xy plane. The modification can however be made insignificant by decreasing E_{r} and increasing E_{±} without affecting the nearestneighbour hopping amplitudes that are proportional to E_{r}E_{±}. The latter case gives rise to a contribution to the nextnearestneighbour hopping terms in the Hamiltonian. For a given choice of E_{±} it may happen that hops between blue lattice sites occur at a different rate than hops between red lattice sites. As mentioned above, however, one can lower the potential barriers between the sites for the sublattice with slow hops relative to the other to compensate.
Discussion
In the present article, we have constructed a lattice model with local interactions and a KLlike ground state and proposed a scheme for implementing it experimentally. The observation that the Fermi–Hubbardlike Hamiltonian in equation (5) captures the physics of the investigated model for provides some insight into how our model compares with previous proposals for obtaining FQHlike behaviour in lattice systems. Specifically, H_{FH} can be seen as a sum of two free fermion models, H_{kin,↑} and H_{kin,↓}, and a local interaction term. Each of the free models has two bands of which the lowest is filled and the highest is empty. The Chern number of the filled band is plus or minus one for all nonzero t and t′, and the band flattening, that is, the ratio of the gap between the bands to the width of either of the bands, is moderate . Our results thus show that two copies of a fermionic Chern insulator with integer band filling plus onsite interactions can give rise to a bosonic FQH state. This behaviour is different from flat band models, where a fermionic (bosonic) Chern insulator with flattened and fractionally filled bands plus interactions give rise to a fermionic (bosonic) FQH state^{19,20,21}. We therefore conclude that our model provides a different mechanism to get the FQH effect in lattices.
The experimental requirements to observe the desired physics in the proposed setup are those needed to observe the Néel antiferromagnetic order in the Mott insulating regime of the standard Fermi–Hubbard model combined with the ability to perform laserassisted hopping in a checkerboard optical lattice. Laserassisted hopping has been demonstrated experimentally^{54}, and the Fermi–Hubbard model has also been investigated experimentally. The Mott insulating regime of the Fermi–Hubbard model has been previously realized^{33,34}, where measurements of compressibility and of the fraction of doubly occupied lattice sites were used to analyse the state. To observe the physics predicted by the effective Hamiltonian obtained by the Schrieffer–Wolff transformation, the temperature should be low compared with t^{2}/U. By using a local entropy redistribution scheme, this regime has been reached for a subset of lattice bonds in a recent experiment^{36}. Altogether, these results are promising for the implementation scheme proposed here.
We note that the condition is needed to ensure the validity of the Schrieffer–Wolff transformation. It is possible, however, that similar physics can be observed even if H_{FH} does not exactly reproduce the model in equation (3). This suggests that, for example, the choice t/U=1/2 and t′/U=1/6 (corresponding to the white triangle in Fig. 2), which leads to more moderate temperature requirements, may already give interesting results.
The method proposed previously^{38} to derive (nonlocal) parent Hamiltonians for states constructed from CFT correlators is quite general, and an interesting next step, for example, could be to apply the strategy proposed in the present paper to the Moore–Readlike state for which a parent Hamiltonian has already been derived^{38}.
Note added: it has been found recently in a preprint by Bauer et al.^{55} that a local threebody Hamiltonian on a kagome lattice can also give rise to a state having the same topological properties as the bosonic Laughlin state at half filling. This model is, in fact, another particular case of the CFT model presented in the present paper and can be derived following the lines above using a kagome lattice rather than a square lattice. Specifically, we find that the overlaps per site between the ground states of the Hamiltonian considered previously^{55} and the CFT states in equation (11) for a kagome lattice of 12 sites with periodic boundary conditions are >0.98 for both states and can be improved to values similar to those in Table 2 by adding nearestneighbour twobody interactions. The CFT framework thus provides an analytical approximation to the ground states found numerically in the study by Bauer et al.^{55} and puts their model^{55} into a broader context. In addition, it is encouraging for the method presented here to observe that similar results are obtained on different lattices.
Methods
Monte Carlo simulations
The Monte Carlo simulations are done using the Metropolis–Hastings algorithm. To compute correlation functions we write
and sample the probability distribution P(s_{1},…,s_{N})=〈s_{1},…,s_{N}ψ›^{2}/〈ψψ› as follows. We start from a random configuration s_{1},…,s_{N} fulfilling . In each step of the algorithm we choose a spin up and a spin down at random and flip the two spins if and only if , where is the configuration with the two spins flipped and q is a uniformly distributed random number between zero and one. Disregarding a warmup period, the correlation function is then the average of over the trajectory.
The entanglement entropy is similarly computed from the expression^{39,56}
where () is shorthand notation for the spins in part () of the system. The sampling is done as before, except that there are now two copies of the spins, s_{1},…,s_{N} and s′_{1},…,s′_{N}, and we therefore choose at random in each step whether we choose two spins from s_{1},…,s_{N} or two spins from s′_{1},…,s′_{N}.
Wavefunction on the cylinder
A cylinder with circumference L_{y} can be mapped to the plane via the coordinate transformation z=, where (l_{x},l_{y}) is a point on the cylinder and z is the corresponding point in the plane written as a complex number. The wavefunction for a square lattice on the cylinder is therefore easily obtained from by choosing z_{n}= for n=1,2,…,N, where l_{x,n}ε{−(L_{x}−1)/2,−(L_{x}−1)/2+1,…,(L_{x}−1)/2} and l_{y,n}ε{1,2,…,L_{y}}.
Wavefunctions on the torus
In the spin basis, the states and take the form
where kε{0,1/2}, δ_{s} and χ_{s} are defined below equation (2), ω_{1} and ω_{2} are the periods of the torus fulfilling Im(ω_{1})=0 and Im(ω_{2}/ω_{1})>0, and is the theta function. For the L_{x} × L_{y} square lattice, ω_{1}=L_{x}, ω_{2}=iL_{y}, and {z}_{p+{L}_{x}(q1)}=p−(L_{x}+1)/2+i(q−(L_{y}+1)/2) with pε{1,2,…,L_{x}} and qε{1,2,…,L_{y}}. Note that the states in (11) are not necessarily orthogonal, but the linear combinations in equation (4) for square lattices are.
Twisted boundary conditions and Chern number
Let S_{n,m} be the spin operator of the spin at position (n,m) on the lattice, and let . Twisted boundary conditions are then defined such that , , , and . We compute the manybody Chern number by following a previously described approach^{14,57}. In brief, this is done by first computing ψ′_{T0} and ψ′_{T1} (defined in Fig. 2) as functions of θ_{1} and θ_{2} in the interval θ_{1},θ_{2}ε[0,2π]. Let ψ(θ_{1},θ_{2})=(ψ′_{T0}(θ_{1},θ_{2})›,ψ′_{T1}(θ_{1},θ_{2})›) and choose two sets of reference angles (θ_{1a},θ_{2a}) and (θ_{1b},θ_{2b}). For each zero of det(ψ^{†}(θ_{1a},θ_{2a})ψ(θ_{1},θ_{2})ψ^{†}(θ_{1},θ_{2})ψ(θ_{1a},θ_{2a})), the increase in the phase of det(ψ^{†}(θ_{1a},θ_{2a})ψ(θ_{1},θ_{2})ψ^{†}(θ_{1},θ_{2})ψ(θ_{1b},θ_{2b})) when going in a small circle around the zero is computed. Adding up the contributions from all the zeros and dividing by 2π gives the manybody Chern number, which is necessarily an integer. The Chern number of the bands of the free fermion model H_{kin,σ} is computed using equation (12) of the study by Shi and Cirac^{58}.
Additional information
How to cite this article: Nielsen, A.E.B. et al. Local models of fractional quantum Hall states in lattices and physical implementation. Nat. Commun. 4:2864 doi: 10.1038/ncomms3864 (2013).
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Acknowledgements
We acknowledge discussions with Fabio Mezzacapo, HongHao Tu and Shuo Yang. This work has been supported by the EU project AQUTE, FIS201233642, QUITEMAD and the Severo Ochoa Program.
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All authors planned the project together and contributed to the ideas behind the work. A.E.B.N. and G.S. performed the analytical computations related to the CFT states. A.E.B.N. did all other analytical and numerical computations and drafted the manuscript. A.E.B.N. and J.I.C. developed the simulation protocol, and all authors contributed to the final version of the manuscript.
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Nielsen, A., Sierra, G. & Cirac, J. Local models of fractional quantum Hall states in lattices and physical implementation. Nat Commun 4, 2864 (2013). https://doi.org/10.1038/ncomms3864
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DOI: https://doi.org/10.1038/ncomms3864
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