Coupling ultracold matter to dynamical gauge fields in optical lattices: From flux attachment to ℤ₂ lattice gauge theories

Effective Hamiltonian

For Δ^f = U = ω and lattice modulations with a trivial phase ϕ_j1,j2 = 0, the effective Floquet Hamiltonian in Eq. 7 becomesH^eff2well=−tyaλy τ^〈j2,j1〉z(a^j2†a^j1+h.c.)−tyfΛ^τ^〈j2,j1〉x(12)with notations defined as follows. We describe the f-particle by a pseudo–spin-12τ^〈j2,j1〉z=n^j2f−n^j1f,  n^j2f+n^j1f=1(13)which becomes a link variable in a ℤ₂ LGT (see Fig. 1C). The Pauli matrix τ^〈j2,j1〉x=(f^j2†f^j1+h.c.) describes tunneling of the f-particle.

As shown in Fig. 2B, the interaction energy of the matter field changes by ±U in every tunneling event. As a result, the amplitude renormalization in Eq. 12 is λ^y = ∣ 𝒥₁(A_j2,j1/ω)∣ (see Eq. 8), and the phase of the restored tunnel couplings is eiφ^=τ^〈j2,j1〉z by Eqs. 10 and 11. Because the f-particle is subject to an additional potential offset Δ^f = U between the two sites, its energy can only change by 0 or 2U in a tunneling event. Hence, the phase of the restored tunneling in Eq. 12 is trivial, θ^=0 as in Eqs. 10 and 11, but the amplitude renormalizationΛ^=J0(Aj2,j1/ω)n^j1a+J2(Aj2,j1/ω)n^j2a(14)depends on the configuration of the a-particle in general.

The effective Hamiltonian (Eq. 12) realizes a minimal version of a ℤ₂ LGT: The link variable τ^〈j2,j1〉z≃eiπ𝒜^ provides a representation of the dynamical ℤ₂ gauge field 𝒜^, which is quantized to 0 and 1. The corresponding ℤ₂ electric field is given by the Pauli matrix τ^〈j2,j1〉x, defining electric field lines on the link. The ℤ₂ charges Q^ji, defined on the two sites j_i with i = 1, 2, are carried by the a-particle, Q^ji=exp(iπn^jia). These ingredients are summarized in Fig. 1C and justify our earlier notion that the a- and f-particles describe matter and gauge fields, respectively. The Hamiltonian in Eq. 12 realizes a minimal coupling (12) of the a-particles to the gauge field (see Fig. 2C).

Symmetries

Each of the two lattice sites j_i is associated with a ℤ₂ symmetry. The operators generating the ℤ₂ gauge group in the double-well systemg^i=Q^jiτ^〈j2,j1〉x,  i=1,2(15)both commute with the effective Hamiltonian in Eq. 12, [g^i,H^eff2well]=0 for i = 1,2. This statement is not entirely trivial for the first term in Eq. 12: While τ^〈j2,j1〉z and a^j2†a^j1 do not commute with τ^〈j2,j1〉x and Q^ji individually, their product commutes with g^i. The second term in Eq. 12 trivially commutes with g^i because [Λ^,Q^ji]=0 (see Eq. 14).

Physically, Eq. 15 establishes a relation between the ℤ₂ electric field lines, τ〈j2,j1〉x=−1, and the ℤ₂ charges from which they emanate (see Fig. 2D). Note that the eigenvalues of g^1 and g^2 are not independent, because g^1g^2=−1 for the considered case with a single a-particle tunneling in the double-well system.

The model in Eq. 12 is invariant under the gauge symmetries g^i for all values of the modulation strength A_j2,j1. In general, both terms in the effective Hamiltonian couple the ℤ₂ charge to the gauge field. An exception is obtained for lattice modulation strengths A_j2,j1/ω = x₀₂ for whichJ0(x02)=J2(x02)(16)

In this case, neither of the amplitude renormalizationsΛ^→Λ02=J0(x02)≈0.32(17)λy=λ02=J1(x02)≈0.58(18)is operator valued, and the second term in the Hamiltonian only involves the ℤ₂ gauge field. The weakest driving for which Eq. 16 is satisfied has x₀₂ ≈ 1.84.

Intuition

We take this opportunity to explain in a mechanistic way the meaning of the ℤ₂ gauge field in the double-well system and its relation to more general LGTs. As a starting point, consider the situation when tyf=0 and the f-particle is localized. Depending on the position of the f atom (left or right), the restored tunneling amplitude of the a-atom between the two sites has a sign ±1. Formally, this corresponds to the appearance of the factor τ^〈j2,j1〉z=n^j2f−n^j1f in the first term on the left hand side of Eq. 12. At this point, we have realized a synthetic gauge field 𝒜_〈j2,j1〉 for the a-atoms, which depends on the density of the f-atoms. That is, the general tunneling matrix element of the a-particles, t∼yaeiπ𝒜〈j2,j1〉, has a phase π𝒜_〈j2,j1〉 depending on the f-density.

The two possible states of the link variable, corresponding to the two positions j₁ and j₂ of the f atom, define a 2D Hilbert space on the link 〈j₂, j₁〉. This Hilbert space is equivalent to the Hilbert space of a ℤ₂ lattice gauge field, with two orthogonal states on each link of the lattice. Using this language, we can identify the operator τ^〈j2,j1〉z with a ℤ₂ gauge field. It does not commute with the corresponding ℤ₂ electric field operator, τ^〈j2,j1〉x, which corresponds physically to the coherence of the f-atom between the two sites j₂ and j₁. This noncommutativity is related to the noncommutativity of the conjugate electric and magnetic fields e^ and B^ in quantum electrodynamics. Because of the small ℤ₂ gauge group, the ℤ₂ electric field only takes two possible quantized values: τ^〈j2,j1〉x=±1. The eigenstates correspond to even and odd superpositions of the f-atom on the two lattice sites: (f^j2†±f^j1†)∣0〉/2.

If we allow to add arbitrary perturbations to the tunneling Hamiltonian, e.g., terms δyτ^〈j2,j1〉y or δzτ^〈j2,j1〉z, we can introduce nontrivial dynamics of the synthetic gauge field. While this renders the density-dependent synthetic gauge field dynamical, it does not correspond to a ℤ₂ LGT in the strict sense, since local conservation laws are generically absent. These situations, without local conservation laws, lead to interesting physics nonetheless and have been studied for example in the context of the so-called ℤ₂ Bose-Hubbard model (52, 53).

A genuine ℤ₂ LGT is obtained if only terms are included in the Hamiltonian, which commute with the ℤ₂ gauge operators g^i in Eq. 15. In particular, this includes the term tyfΛ^τ^〈j2,j1〉x on the right hand side of Eq. 12 or simpler terms such as t∼yfτ^〈j2,j1〉x. These are the ℤ₂ analogs of terms ∼E² in the Hamiltonian of quantum electrodynamics, without the square due to the simpler nature of the ℤ₂ gauge group. In the double-well system, the ℤ₂ gauge symmetry leads to two decoupled sectors of the Hamiltonian with g₁ = −g₂ = 1 and g₁ = −g₂ = −1. As will be shown later, however, in extended systems, each lattice site is associated with its own conserved charge. This has important consequences for the possible many-body phases (12).

Effective Hamiltonian

For a properly designed configuration of lattice gradients and modulations, presented in detail later, we obtain an effective HamiltonianH^2leg=−∑〈i,j〉x(txaλ^〈i,j〉xxa^j†a^i+h.c.)−∑〈i,j〉y[tyaλy(a^j†a^iτ^〈i,j〉yz+h.c.)+tyfΛ^〈i,j〉yyτ^〈i,j〉yx](19)

Expressions for the amplitude renormalizations λ^y ∈ ℝ and λ^x, Λ^y are provided in section S2B.

For the specific set of driving strengths x = x₀₂ that we encountered already in the double-well problem (see Eq. 16), we find that Λ^y only has a weak dependence on the ℤ₂ charges, Q^j=(−1)n^ja. Similarly, the amplitude renormalization λ^x depends weakly on the ℤ₂ magnetic field B^p only; hereB^p=∏〈i,j〉y∈∂pτ^〈i,j〉yz(20)

is defined as a product over all links 〈i, j〉_y on the rungs belonging to the edge ∂p of plaquette p. Hence, for these specific modulation strengths[Λ^〈i,j〉yy,Q^l]=[Λ^〈i,j〉yy,τ^〈k,l〉]=0(21)[λ^〈i,j〉xx,Q^p]=[λ^〈i,j〉xx,a^l(†)]=0(22)

Symmetries

Now, we discuss the symmetries of the effective Hamiltonian (Eq. 19) at the specific value of the driving strengths x₀₂. In the case of decoupled rungs, i.e., for txa=0, every double well commutes with g^i, i = 1, 2 from Eq. 15. These symmetries are no longer conserved for txa≠0; in this general case, a global ℤ₂ gauge symmetry remainsV^i=∏j=1Lxg^i(jex),  i=1,2(23)

with g^i(jex)=(−1)Q^jex+(i−1)eyτ^〈jex+ey,jex〉yx and for which V^i2=1. Using Eqs. 21 and 22, one readily confirms that [H^2leg,V^i]=0 for i = 1, 2.

In summary, the effective model is characterized by the global U(1) symmetry associated with the conservation of the number of a-particles and the global ℤ₂ symmetry V^1. Note that the second ℤ₂ symmetry, V^2, follows as a consequence of combining V^1 with the global U(1) symmetry: By performing the global U(1) gauge transformation a^j→−a^j for all sites j, V^2 is obtained from V^1. Thus, the overall symmetry is U(1) × ℤ₂.

Physical constituents

In the following, we will describe the physics of the ladder models using the ingredients of ℤ₂ LGTs (see Fig. 1C). The quantized excitations of the ℤ₂ lattice gauge field are vortices of the ℤ₂ (or Ising) lattice gauge field, so-called visons (13). They are defined on the plaquettes of the ladder: If the plaquette term in Eq. 20 is B_p = 1, there is no vison on p; the presence of an additional ℤ₂ flux, B_p = −1, corresponds to a vison excitation on plaquette p. Since the matter field a^ couples to the ℤ₂ gauge field, the resulting interactions with the visons determine the phase diagram of the many-body Hamiltonian.

Transition in the matter sector

First, we concentrate on the conceptually simpler phase transition taking place in the charge sector. When the tunneling along the legs is weak, t∼xa≲[(t∼yf)2+(t∼ya)2]1/2−t∼yf, and the number N_a of a-particles is tuned, we observe a pronounced rung-Mott phase (55) at the commensurate filling N_a = L_x, where L_x denotes the total number of rungs in the system. Similar to the analysis in (56–58), this phase can be characterized by the parity operatorOp(l)=〈exp [iπ∑j<l(n^jexa+n^jex+eya−NaLx)]〉(25)

In the limit l ≃ L_x and L_x → ∞, the observable O_p(l) remains finite only in the Mott insulating regime. Our results in Fig. 4A confirm that O_p takes large values with a weak size dependence for N_a = L_x. On the other hand, when N_a/L_x ≠ 1 is slightly increased or decreased, the parity O_p immediately becomes significantly smaller and a more pronounced L_x dependence is observed, consistent with a vanishing value in thermodynamic limit as expected for an SF phase.

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For larger values of t∼xa and N_a = L_x, no clear signatures of a rung-Mott phase are found (see section S4B): The parity operator O_p takes significantly smaller values, the calculated Mott gap becomes a small fraction of t∼xa, consistent with a finite-size gap, and we checked that the decay of two-point correlations follows a power-law at long distances until edge effects begin to play a role. These signatures are consistent with an SF phase.

Because of the global U(1) symmetry of the model, an SF-to-Mott transition at N_a = L_x in the quasi-1D ladder geometry would be of Berezinskii-Kosterlitz-Thouless (BKT) type (59). Hence, the gap is strongly suppressed and the correlation length is exponentially large close to the transition point, making it impossible to determine conclusively from our numerical results whether the ground state is a gapped Mott state in this regime. For hard-core bosons on a two-leg ladder, this scenario is realized: It has been shown by bosonization that an infinitesimal interleg coupling is sufficient to open up an exponentially small Mott gap (55, 60).

Similar considerations apply at the commensurate fillings N_a/L_x = 1/2, 3/2, where previous work on single-component bosons in a two-leg ladder (60) pointed out the possible emergence of an insulating charge density wave (CDW) for large enough t∼ya/t∼xa (hatched areas in Fig. 3B). Our numerical results indicate that these CDWs may exist in this regime also in our model (Eq. 24). But since the numerical analysis is plagued by a potentially even larger correlation length and a corresponding strongly suppressed Mott gap, it is difficult to pinpoint the exact location of this BKT transition.

The SF phase observed at incommensurate filling fractions is characterized by a power-law decay of the Green’s function 〈a^dex†a^0〉≃d−1/4K in the charge sector. The exponent is related to the Luttinger parameter K, which approaches K → 1/2 at a transition to the rung-Mott phase for commensurate filling. We confirm this behavior and obtain qualitatively identical results as in the case of a static gauge field (see section S4A for details) (55).

Transition in the gauge sector

In the gauge sector, described by f-particles, we observe a phase transition when the ratios of the tunnel couplings are tuned. In Fig. 4B, we tune t∼ya/t∼xa while keeping t∼yf/t∼xa fixed. We find a transition from a symmetric regime where 〈τ^〈i,j〉yz〉=0 to a region with a nonvanishing order parameter 〈τ^〈i,j〉yz〉≠0. Similar behavior is obtained when tuning t∼yf/t∼ya while keeping t∼xa/t∼ya fixed (see section S4E).

The observed transition is associated with a spontaneous breaking of the global ℤ₂ symmetry (Eq. 23) of the model. The f-particles go from a regime where they are equally distributed between the legs, 〈τ^〈i,j〉yz〉=0, to a twofold degenerate state with population imbalance, 〈τ^〈i,j〉yz〉≠0. This behavior occurs in the insulating and SF regimes of the charge sector, and it is only weakly affected by the filling value N_a/L_x (see Fig. 3B).

The two phases are easily understood in the limiting cases. When t∼ya=0, the ground state is an eigenstate of the ℤ₂ electric field τ^〈i,j〉yx on the rungs, with eigenvalues 1. The ℤ₂ magnetic field is strongly fluctuating, and no ℤ₂ electric flux loops exist. Thus, also the vison number is strongly fluctuating, and the state can be understood as a vison condensate. In the opposite limit, when t∼ya→∞, the kinetic energy of the matter field dominates. In this case, the ℤ₂ magnetic field is effectively static, and its configuration is chosen to minimize the kinetic energy of the a-particles. This is achieved when the effective Aharonov-Bohm phases on the plaquettes vanish, i.e., for B^p=1 (see Fig. 1C). In this case, vison excitations with B^p=−1 (see Fig. 4C) correspond to localized defects in the system, which cost a finite energy corresponding to the vison gap.

LGTs are characterized by Wilson loops (12). Their closest analogs in our two-leg ladder model are string operators of visonsW(d)=〈∏j=1dB^pj〉=〈τ^〈i,j〉yzτ^〈i+dex,j+dex〉yz〉(26)

see Fig. 4D. In the disordered phase (electric field dominates), we found numerically that W(d) → 0 when d → ∞. Sufficiently far from the transition, where finite-size effects are small, our data (see fig. S5) show the expected exponential decay. In the ordered phase (magnetic field dominates), W(d) → W_∞ quickly converges to the nonzero value W∞=〈τ^〈i,j〉z〉2>0 when d → ∞. See section S3 (D and E) for more details.

The qualitatively different behavior of the Wilson loop in the two phases is reminiscent of the phenomenology known from the ubiquitous confinement-deconfinement transitions found in (2 + 1) dimensional LGTs (12, 13, 26): There, visons are gapped in the deconfined phase and the Wilson loop decays only weakly exponentially with a perimeter law; in the confining phase, visons condense and the Wilson loop decays much faster with an exponential area law.

Our numerical results in Fig. 4B indicate that the phase transition in the gauge sector is continuous. Beyond this fact, its nature is difficult to determine. The interactions between the ℤ₂ link variables are mediated by the matter field, which has correlations extending over many sites following either a power-law (in the SF regime) or featuring exponential decay with a correlation length ξ ≫ 1 (in the considered Mott-insulating regions). On the one hand, this leads to relatively large finite-size effects in our numerical simulations, which explains the continuous onset of the transition in Fig. 4B. On the other hand, when nonlocal Ising interactions mediated by the gapless matter field compete with the transverse ℤ₂ electric field term ∝t∼yfτ^〈i,j〉x in the Hamiltonian, a rich set of critical exponents can be expected (61). For more details, see section S4E.

Interplay of matter and gauge fields

Last, we discuss the interplay of the observed phase transitions in the gauge and matter sectors. To this end, we find it convenient to consider the phase diagram in the μ−t∼yf plane, where μ denotes a chemical potential for the a-particles and t∼yf controls fluctuations of the ℤ₂ electric field. We collect our result in the schematic plots in Fig. 3C: Deep in the SF phase, realized for small μ and N_a/L_x < 1, t∼yf drives the transition in the gauge sector. Because of a particle-hole symmetry of the hard-core bosons in the model, similar results apply for large μ and N_a/L_x > 1. On the other hand, when t∼yf is small, permitting a sizable Mott gap at commensurate fillings, μ drives the SF-to-Mott transition.

More interesting physics can happen at the tip of the Mott lobe for commensurate fillings N_a = L_x. This corresponds to the hatched regime in Fig. 3B, where we cannot say conclusively if the system is in a gapped Mott phase. To obtain better understanding of the commensurate regime, we first argue that an SF cannot coexist with the ordered phase of the gauge field at commensurate fillings: In this regime, the ℤ₂ gauge field acquires a finite expectation value, 〈τ〈i,j〉yz〉≠0. This leads to a term in the Hamiltonian ∼−t∼ya〈τ〈i,j〉yz〉a^i†a^j, which is expected to open a finite Mott gap, following the arguments in (55, 60). Therefore, only the two scenarios shown in Fig. 3C are possible: In the first case, the Mott insulator coexists only with the ordered phase of the gauge field; in the second scenario, the Mott state coexists with the disordered phase of the gauge field.

To shed more light on this problem, we consider the case when the Mott gap Δ is much larger than the tunneling t∼xa. When t∼xa=0, every rung represents an effective localized spin-1/2 degree of freedom. As shown in section S4C, finite tunnelings t∼xa≪Δ introduce antiferromagnetic couplings between these localized moments, and in this limit, our system can be mapped to an XXZ chain. It has an Ising anisotropy, and the ground state has a spontaneously broken ℤ₂ symmetry everywhere, except when t∼yf/t∼ya→∞, where an isotropic Heisenberg model is obtained and the ground state has power-law correlations. The transition from the gapped Mott state, corresponding to the ordered phase of the ℤ₂ gauge field, to a symmetric state of two decoupled SFs with a disordered gauge field is of BKT type (62).

Our last argument demonstrates that scenario I in Fig. 3C is realized deep in the Mott phase. In this limit of small t∼xa and strong couplings t∼yf of the gauge field, our analysis proves that an intricate interplay of the phase transitions in the gauge and matter sectors exists. This behavior, characteristic for scenario I in Fig. 3C, is reminiscent of the phase diagram of the 2D ℤ₂ LGT (see Fig. 3D) (26, 28). In that case, the phase at weak couplings has topological order as in Kitaev’s toric code (28), and the disordered phases are continuously connected to each other at strong couplings.

On the other hand, a detailed analysis of the Luttinger-K parameter for larger values of t∼xa shows that the ground state at commensurate filling N_a = L_x is characterized by K = 1/2, in both the ℤ₂ symmetric and ℤ₂ broken regimes (see section S4, A and B). This behavior is indicative of scenario II in Fig. 3C, since the BKT transition is characterized by a value K = 1 of the Luttinger parameter (62).

Simplified model

Now, we discuss a further simplification of the model in Eq. 19, leaving its symmetry group unchanged. We note that, even for the specific choice of the driving strengths Vωx/ω=Vωy/ω=x02, the renormalized tunnel couplings of the a-particles along x still depend explicitly on the ℤ₂ gauge fields on the adjacent rungs (see Eq. 27). This complication can be avoided, by simultaneously modulating the gradient along x at two frequencies, ω and 2ω, with amplitudes Vωx and V2ωx; i.e., we consider the following driving term in Eq. 5Vω(j,t)=[jxVωx+jyVωy]cos(ωt)+jxV2ωxcos(2ωt)(28)

Following (48), we obtain expressions for the restored tunnel couplings along x for an energy offset nω introduced by the Hubbard interactions U = ω between a- and f-particles; λn=∑ℓ=−∞∞Jn−2ℓ(x(1))Jℓ(x(2)/2), where x(1)=Vωx/ω and x(2)=V2ωx/ω (see section S2). By imposing the conditions λ₀ = λ₁ = λ₂, we obtain a simplified effective Hamiltonian where λ^〈i,j〉xx→λx∈ℝ is no longer operator valued, and thus completely independent of the ℤ₂ gauge field τ^z. The weakest driving strengths for which this condition is met are given byx(1)=x012(1)≈1.71,  x(2)=x012(2)≈1.05(29)

where λ^x = λ₀₁₂ ≈ 0.37. A similar approach can be used to make Λ^y independent of the ℤ₂ charges, which allows the implementation of H^2legsimp from Eq. 24.

Setup

We consider the setup shown in Fig. 6A in a layered 2D optical lattice, which is a particular type of brick-wall lattice. The a-particles tunnel vertically between the layers in the z direction, with coupling matrix element tza, and along the links indicated in the figure with tunnel couplings txa and tya. Every tube consisting of four lattice sites with coordinates x, y, and ne_z for n = 1, 2, 3, 4 defines a supersite j = xe_x + ye_y in the effective 2D lattice shown in Fig. 6B. The four links connecting every supersite to its nearest neighbors i : 〈i, j〉 are realized by double-well systems, with exactly one f-particle each, in different layers of the optical lattice. The f-particles are only allowed to tunnel between the sites of their respective double-wells in the x-y plane, with amplitudes txf and tyf, while tunneling along the z direction is suppressed, tzf=0.

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For the realization of the individual double-well systems, we consider a modulated potential gradient along x and y, seen equally by the matter and gauge fields. The modulation amplitudes Vωx/ω=Vωy/ω=x02 are chosen to simplify the amplitude renormalization of f-particle tunneling. As previously, we consider static potential gradients along x and y directions of Δxf=Δyf=U per site, seen only by the f-particles, and work in a regime where U=ω≫∣tμν∣, with μ = x, y, z and ν = a, f.

To realize a-particle tunneling along z, which is independent of the ℤ₂ gauge fields τ^z on the links in the x-y plane, we add a static potential gradient of Δza per site along the z direction. It is modulated by two frequency components ω and 2ω, with amplitudes Vωz and V2ωz. These driving strengths are chosen as in Eq. 29, i.e., Vωz/ω=x012(1) and V2ωz/ω=x012(2), such that the restored tunnel couplings with amplitude tzaλ012 become independent of the f-particle configuration.

Effective Hamiltonian

Combining our results from the previous section, we obtain the effective hopping Hamiltonian H^2DLGT for the setup described in Fig. 6H^2DLGT=−txyf∑〈i,j〉Λ^〈i,j〉τ^〈i,j〉x−txyaλ02∑〈i,j〉(τ^〈i,j〉za^i,m〈i,j〉†a^j,m〈i,j〉+h.c.)−tzaλ012∑j∑n=13(a^j,n+1†a^j,n+h.c.)(30)

using the same notation as introduced earlier. Here, we treat the z-coordinate ne_z, with n = 1, …,4, as an internal degree of freedom, while j is a site index in the 2D square lattice; m_〈i,j〉 ∈ {1, 2, 3, 4} denotes the z-coordinate corresponding to double well 〈i, j〉. For simplicity, we assumed that txa=tya=txya and txf=tyf=txyf. The amplitude renormalization for f-particles in the x-y plane depends on the ℤ₂ charges Q^j,nΛ^〈i,j〉=12[J0(x02)+J1(x02)]+Q^i,m〈i,j〉Q^j,m〈i,j〉12[J1(x02)−J0(x02)](31)

Using the multifrequency driving scheme explained around Eq. 28, a situation where Λ^〈i,j〉 becomes independent of the ℤ₂ charges can be realized.

A simplified effective Hamiltonian, where the internal degrees of freedom are eliminated, can be obtained when U=ω≫tza and λ012tza≫txya; the first inequality is required by the proposed implementation scheme. In this limit, the tunneling of a-particles along z can be treated independently of the in-plane tunnelings txya. The ground state with a single a-particle tunneling along z at supersite j is a^j†∣0〉, where a^j†=∑n=14ϕna^j,n†, with ϕ1=ϕ4=(5+5)−1/2 and ϕ2=ϕ3=(1+1/5)1/2/2. It is separated by an energy gap Δε=λ012tza≫txya from the first excited state, which justifies our restriction to this lowest internal state.

The ground state energy ε_2a with two hard-core a-particles tunneling along z in the same supersite is larger than twice the energy ε_a of a single a-particle, by an amount U_eff, i.e., ε_2a = 2ε_a + U_eff. By solving the one- and two-particle problems exactly, we find Ueff=λ012tza. In the effective model restricted to the lowest internal state, this offset corresponds to a repulsive Hubbard interaction on the supersites j. Because Ueff≫txya, double occupancy of supersites is strongly suppressed, and we can treat the new operators a^j(†) as hard-core bosons.

By projecting the Hamiltonian (Eq. 30) to the lowest internal state on every supersite, we arrive at the following simplified modelH^2DLGTsimp=εa∑ja^j†a^j−txyf∑〈i,j〉Λ^〈i,j〉τ^〈i,j〉x−txyaλ02∣ϕ1∣2∑〈i,j〉∈E(τ^〈i,j〉za^i†a^j+h.c.)−txyaλ02∣ϕ2∣2∑〈i,j〉∈B(τ^〈i,j〉za^i†a^j+h.c.)(32)

Here, we distinguish between two sets of links, 〈i,j〉 ∈ E or B, which are realized in layers at the edge n = 1,4 (E) and in the bulk n = 2,3 (B) in the 3D implementation (see Fig. 6A). Because the internal state has different weights ∣ϕ₁∣² ≈ 0.14 and ∣ϕ₂∣² ≈ 0.36, they are associated with different tunneling amplitudes. This complication can be avoided by realizing bare tunnelings of a-particles with different strengths on E- and B-type bonds.

Symmetries

In contrast to the two-leg ladder (Eq. 19), the models in Eqs. 30 and 32 are both characterized by local ℤ₂ gauge symmetries. The ℤ₂ charge on a supersite is defined as Q^j=exp [iπ∑n=14n^j,na], which becomes Q^j=exp [iπa^j†a^j] when projected to the lowest internal state. The ℤ₂ gauge group is generated byG^j=Q^j∏i:〈j,i〉τ^〈j,i〉x(33)

where the product on the right includes all links 〈j, i〉 connected to site j.

It holds [H^2DLGT,G^j]=0 and [H^2DLGTsimp,G^j]=0 for all j, using the respective ℤ₂ charge operators. These results follow trivially for the first line of Eqs. 30 and 32, which contain only the operators τ^〈i,j〉x and n^j,na (n^j) (see also Eq. 31). For the last two lines in the effective Hamiltonians, it is confirmed by a straightforward calculation.

In addition to the local ℤ₂ gauge invariance, the models (Eqs. 30 and 32) have a global U(1) symmetry associated with the conservation of the a-particle number. Very similar Hamiltonians have been studied in the context of strongly correlated electrons, where fractionalized phases with topological order have been identified (35). When the a-particles condense, effective models without the global U(1) symmetry can also be realized. These are in the same symmetry class as Kitaev’s toric code (28).