Exploring Interacting Topological Insulators with Ultracold Atoms: The Synthetic Creutz-Hubbard Model

Understanding the robustness of topological phases of matter in the presence of strong interactions and synthesizing novel strongly correlated topological materials lie among the most important and difficult challenges of modern theoretical and experimental physics. In this work, we present a complete theoretical analysis of the synthetic Creutz-Hubbard ladder, which is a paradigmatic model that provides a neat playground to address these challenges. We give special attention to the competition of correlated topological phases and orbital quantum magnetism in the regime of strong interactions. These results are, furthermore, confirmed and extended by extensive numerical simulations. Moreover, we propose how to experimentally realize this model in a synthetic ladder made of two internal states of ultracold fermionic atoms in a one-dimensional optical lattice. Our work paves the way towards quantum simulators of interacting topological insulators with cold atoms.

Popular Summary

Topological phases of matter and topological phase transitions are a hot topic, as shown by the 2016 Nobel Prize in Physics. The award recognized how the mathematical field of topology, which studies intrinsic properties of shapes, can elucidate strange behaviors seen in certain exotic materials. One of the most difficult and important challenges in this field is to understand how topological insulators and superconductors, whose surfaces have different electrical behavior than their interiors, behave in the presence of strong interactions and correlations among the particles. Since these effects are absent in the majority of topological materials explored so far, synthetic quantum matter engineered in systems of ultracold atoms appears to be a promising platform for testing our understanding. Using theoretical analysis and numerical simulations, we present a way of implementing such a test bed.

We study the imbalanced Creutz-Hubbard ladder, a variant of a type of one-dimensional topological insulator, which provides a simple and paradigmatic playground for exploring these challenges. We present a complete theoretical analysis of the model in all parameter regimes, paying special attention to the robustness of topological features (like edge states) versus more-standard quantum magnetic orderings. Moreover, we present a detailed proposal to realize this model using ultracold fermionic atoms in a one-dimensional optical lattice.

We believe that our work will pave the way toward the quantum simulation of interacting topological insulators with cold atoms and advance the current understanding of correlated topological phases of matter.

Figure 1

Phase diagram of the imbalanced Creutz-Hubbard ladder. Phase diagram displaying a topological insulator (TI) phase and a pair of nontopological phases: an orbital phase with long-range ferromagnetic Ising order (OFM) and an orbital paramagnetic phase (OPM). The horizontal axis represents the ratio of the interparticle interactions to the tunneling strength, whereas the vertical axis corresponds to the ratio of the energy imbalance to the tunneling strength. The dashed yellow line shows the transition points of the effective model in the strong-coupling effective (Ising) model. The dashed red line indicates the transition as obtained from the weak-coupling expansion. The red circle shows the transition point in the balanced model at intermediate interactions. Stars label numerical results, and the blue line is an extrapolation of the phase boundaries. The labels of the critical lines give the central charge of their underlying conformal field theory. Details on the different phases and transitions between them are provided in Sec. 3.

Figure 2

Standard and imbalanced Creutz ladder. Two-leg ladder where fermions tunnel along the black links enclosing a net flux $2\theta$ along a closed plaquete: (a) standard Creutz ladder Eq. (1) and (b) imbalanced Creutz ladder, which leads to Eq. (4) in the $\pi$-flux limit.

Figure 3

Aharonov-Bohm cages in the Creutz ladder. (a) Considering the tunneling paths in the Creutz ladder Eq. (4), one can identify two types of rhombic plaquettes that enclose a synthetic $\pi$ flux (left). Therefore, a particle trying to tunnel two sites apart (middle) will be subjected to a destructive Aharonov-Bohm interference that forbids this process. Accordingly, particles are confined to the so-called Aharonov-Bohm cages and cannot spread through the entire lattice. These cages correspond to square plaquettes, except at the edges (b), where destructive interference can also be found for a particle trying to tunnel one site apart (right). (c) Aharonov-Bohm cages with the relative amplitudes for a single-particle state in two possible flat bands and in the two possible zero-energy edge modes.

Figure 4

Exact density imbalance and mean-field magnetizations. In blue solid lines, we present the ground-state density imbalance $⟨\mathrm{\Delta }n⟩$ for systems with $V_v/\tilde{t}=1$, 2, 3, 4 (from bottom to top) as determined numerically by means of a matrix product state algorithm for a ladder with $L=256$ sites. In red dashed lines, we show the mean-field magnetizations $m_1(\mathrm{\Delta }ϵ,V_v,\tilde{t})=m_2(\mathrm{\Delta }ϵ,V_v,\tilde{t})$ of the coupled Ising chains. The agreement $⟨\mathrm{\Delta }n⟩\approx \frac{1}{2}[m_1(\mathrm{\Delta }ϵ,V_v,\tilde{t})+m_2(\mathrm{\Delta }ϵ,V_v,\tilde{t})]$ is reasonable even for considerably large interaction strengths. The green curve shows as a reference the magnetization of the noninteracting model $M(\mathrm{\Delta }ϵ/4\tilde{t})$.

Figure 5

Nonstandard Hubbard terms in the Aharonov-Bohm cage basis. Virtual ladder structure to represent the relevant processes in the Creutz-Hubbard ladder. The two legs of the ladder correspond to the two possible flat bands at energies ${ϵ}_{\pm }=\pm 2\tilde{t}$, with sites representing the bulk Aharonov-Bohm cages Eq. (10). The two boundary sites represent the two zero-energy modes ${ϵ}_l={ϵ}_r=0$ and their corresponding edge cages Eq. (11). The imbalance of the original Creutz-Hubbard ladder Eq. (15) leads to standard intra- and interleg tunneling, but also to edge-bulk tunneling processes represented by curved arrows and with strengths proportional to $t_{\mathrm{imb}}$. The Hubbard interaction Eq. (35) can be decomposed into nearest-neighbor interactions of strength proportional to $V_v$ that are represented by wavy lines and involve two adjacent bulk cages in any possible leg, or adjacent edge-bulk ones. Nonstandard Hubbard terms involving a pair tunneling also arise, where pairs of fermions tunnel simultaneously either from the same leg (correlated tunneling) or from opposite legs (anticorrelated tunneling) of the ladder with amplitude $\tilde{J}$. Finally, we also describe a density-dependent interleg tunneling of amplitude amplitude $T_d$, where the fermion tunneling depends on the density of fermions populating the Aharonov-Bohm cage at the neighboring edge.

Figure 6

Paramagnetic susceptibility and energy gaps along the TI-OPM transition. Top: The divergence of the magnetization susceptibility ${\chi }_{\mathrm{\Delta }n}$ with growing system size indicates the critical point for a cut through the phase diagram at $V_v=4.0$ (left-hand inset: occupation imbalance $\mathrm{\Delta }n$; right-hand inset: fitted finite-size scaling of the susceptibility maxima assuming up to second-order corrections, $\mathrm{\Delta }{ϵ}_c(N)=\mathrm{\Delta }{ϵ}_c(1+a{N}^{-1}+b{N}^{-2})$). Bottom: The dashed lines show the finite-size results for the energy gap $\mathrm{\Delta }$, which is nonzero in both the TI and the OPM phase. The quantity $\delta$ (solid lines), on the contrary, is zero in the TI phase due to the presence of zero-energy modes, but achieves a nonvanishing value in the OPM phase. Blue, $N=8$; orange, $N=16$; green, $N=32$; red, $N=64$; violet, $N=128$; brown, $N=256$. The vertical line (black) indicates the transition point ($\mathrm{\Delta }{ϵ}_c=1.857$).

Figure 7

Ferromagnetic and paramagnetic magnetization along the OFM-OPM transition. Top: Paramagnetic magnetization susceptibility for a cut through the phase diagram at $V_v=16\tilde{t}$ and different system sizes [inset: fitted finite-size scaling of the susceptibility maxima assuming up to second-order corrections, $\mathrm{\Delta }{ϵ}_c(N)=\mathrm{\Delta }{ϵ}_c(1+a{N}^{-1}+b{N}^{-2})$]. Bottom: Ferromagnetic magnetization along the same line. Blue, $N=8$; orange, $N=16$; green, $N=32$; red, $N=64$; violet, $N=128$. The light blue vertical line (dashed) in the top figure indicates the critical point (here, $\mathrm{\Delta }{ϵ}_c/\tilde{t}=0.266$). The black dashed curves indicate the analytical predictions of an Ising model with the same critical point (and a saturation of the ferromagnetic magnetization $⟨T_i^y{⟩}_{\max }=0.48$). The effective Ising model in Eq. (31) suggests $\mathrm{\Delta }{ϵ}_c/\tilde{t}=0.25$ and $⟨T_i^y{⟩}_{\max }=0.5$.

Figure 8

Paramagnetic magnetization susceptibility along the TI-OFM transition. Blue, $N=8$; orange, $N=16$; green, $N=32$; red, $N=64$; violet, $N=128$; brown, $N=256$. The black dashed lines indicate the TD result in a TFIM model. Inset: The finite-size scaling of the maxima of the susceptibility yields $(V_v/\tilde{t}{)}_{c,\mathrm{num}}=8.003$ [here, fitted via $V_{v,c,\mathrm{num}}(N)=V_{v,c,\mathrm{num}}(1+a{N}^{-1})$], in good agreement with the analytical result $(V_v/\tilde{t}{)}_c=8$.

Figure 9

Scaling of the entanglement entropy for critical points on the different transition lines. The prefactor $c$ in $S(l)=(c/6)\ln [(2N/\pi )\sin (\pi l/N)]+\text{const}$ identifies the central charge of a critical phase. Here, we show the entanglement entropy in systems with $N$ sites and fit the data for $N/4<l\le N/2$. The fitting results for the TI-OPM transition yield $c=1.003$ for $V_v/\tilde{t}=4.0$, $\mathrm{\Delta }ϵ/\tilde{t}=1.857$, $N=128$ (blue line), the OFM-OPM transition yield $c=0.524$ for $V_v/\tilde{t}=16$, $\mathrm{\Delta }ϵ=0.266$, $N=128$ (green line), and the TI-OFM transition yield $c=0.503$ for $V_v/\tilde{t}=0$, $\mathrm{\Delta }ϵ/\tilde{t}=8$, $N=256$ (yellow line). All these numerical fits agree considerably well with the model predictions $c\in \{1,1/2,1/2\}$.

Figure 10

Degeneracies of the entanglement spectrum for different phases. For a ladder of length $L=128$ and for a bipartition in the half chain, the 20 lower eigenvalues of the ES are depicted for the three different phases. The dots represent the degeneracy of the corresponding eigenvalue. In the TI phase, for $V_v/\tilde{t}=4$ and ${\mathrm{\Delta }}_{ϵ}/\tilde{t}=1.5$ the eigenvalues are all doubly degenerate. In the OPM phase, for $V_v/\tilde{t}=4$ and ${\mathrm{\Delta }}_{ϵ}/\tilde{t}=2$, and in the OFM phase for $V_v/\tilde{t}=9$ and ${\mathrm{\Delta }}_{ϵ}/\tilde{t}=0.2$, almost all the eigenvalues are not degenerate. We find the same behavior elsewhere in phase space.

Figure 11

Ladder scheme for a binary hyperfine mixture of ultracold atoms in an optical lattice. (a) Atoms in two hyperfine states, $|\uparrow ⟩=|F=I-\frac{1}{2},M⟩$ (blue circles) and $|\downarrow ⟩=|F^{\prime }=I+\frac{1}{2},M^{\prime }⟩$ (red circles), are trapped at the minima of an optical lattice. At low temperatures, the kinetic energy of the atoms can be described as tunneling of strength $-t$ between the lowest energy levels ${ϵ}_{\uparrow },{ϵ}_{\downarrow }$ of neighboring potential wells. Additionally, the $s$-wave scattering of the atoms leads to contact interactions of strength $U_{\uparrow \downarrow }$ whenever two fermionic atoms of different internal state meet on the same potential well. (b) Ladder representation of the binary mixture, where each hyperfine state corresponds to a different leg of the ladder. The tunneling (on-site energy) is represented by solid lines linking neighboring (same) sites within each leg, whereas the on-site interactions are represented by wavy lines among two neighboring sites that belong to different legs. (c) Creutz-Hubbard model obtained by including a Peierls phase in the intraleg tunnelings $-t\to \pm -t_h{e}^{\pm i\theta }$, and introducing interleg tunnelings $-t_d$ represented by solid lines along diagonal rungs of the ladder.

Figure 12

Raman-assisted tunneling scheme for a binary hyperfine mixture of ultracold atoms in an optical lattice. (a) Atoms in a given hyperfine state from the ground-state manifold $n{S}_{1/2}$ can tunnel to the neighboring potential well changing its hyperfine state by means of a Raman transition that virtually populates an excited state from the manifolds $n{P}_{1/2}$, $n{P}_{3/2}$. (b) The interleg tunneling of $|\uparrow ⟩$ (blue circles) to $|\downarrow ⟩$ (red circles) is mediated by a pair of lasers providing the energy necessary to overcome the offset ${\omega }_1-{\omega }_2=({ϵ}_{\downarrow }-{ϵ}_{\uparrow })+\mathrm{\Delta }$, and a recoil kick to assist the tunneling. (c) The interleg tunneling of $|\downarrow ⟩$ (red circles) to $|\uparrow ⟩$ (blue circles) is mediated by a pair of lasers providing the energy necessary to overcome the offset ${\omega }_1-{\omega }_3=({ϵ}_{\uparrow }-{ϵ}_{\downarrow })+\mathrm{\Delta }$, and a recoil kick to assist the tunneling.

Figure 13

Photon-assisted tunneling scheme for a binary hyperfine mixture of ultracold atoms in an optical lattice. (a) Atoms in a given hyperfine state can tunnel to the neighboring potential well preserving its hyperfine state by absorbing energy from an additional optical lattice whose intensity is periodically modulated. (b) The intraleg tunneling of $|\uparrow ⟩$ (blue circles) is activated by the weak modulated lattice with ${\omega }_{d,\uparrow }=\mathrm{\Delta }/2$, such that the tunneling picks the modulation phase $e^{-i2{\varphi }_{\uparrow }}$. (c) The intraleg tunneling of $|\downarrow ⟩$ (red circles) is activated by the weak modulated lattice with ${\omega }_{d,\downarrow }=\mathrm{\Delta }/2$, such that the tunneling picks the modulation phase $e^{-i2{\varphi }_{\downarrow }}$.