Susceptibility indicator for chiral topological orders emergent from correlated fermions

Introduction.– Chiral topological orders (TOs) breaking the time-reversal symmetry (TRS) have constituted a prominent topic over the last decades Kalmeyer ; chiraspinstate , and its discovery in fractional quantum Hall (FQH) systems xgwenc ; laughlin ; Tsui invokes some of the most fundamental concepts in modern condensed matter physics xgwen1 ; witten1 ; Arovas ; qniu ; kitaev1 . Chiral topological orders are usually formed in correlated fermion systems. For example, the FQH states are generated from correlated electron states in Landau levels xgwenc ; laughlin ; Tsui ; the chiral spin liquids (CSLs) are formed in frustrated magnets yzhou ; lbalents ; YChe or Mott insulators Bauer , which again originate from strongly-correlated electronic materials. Moreover, the chiral excitonic topological order (ETO) recently revealed in InAs/GaSb quantum wells also emerges from interacting electron-hole bilayers ruid . These facts suggest there might be a underlying mechanism of chiral TOs accounting for their emergence from interacting fermions, which is yet to be addressed.

A convenient starting point to study interacting fermions is the Fermi liquid landauFL . The instabilities of Fermi liquids provide a unified description of many long-range ordered states hvl ; Honerkamp ; cjwuu , including superconductors, charge density waves, and magnetic orders, which can be understood as the condensation of bosons in corresponding channels. However, chiral TOs are disordered and characterized by long-range quantum entanglement, in stark contrast with the ordered states xgwensc ; xchen ; mlevin . Therefore, it is of great challenge to find out their connections with the correlated fermion systems, which demands new developments beyond the conventional theory of Fermi liquid instability.

The two-dimensional (2D) interacting fermions is generally described by $H=H_{0}+H_{I}$, where $H_{0}=\sum_{\mathbf{r},\alpha}c^{\dagger}_{\mathbf{r}\alpha}h_{\alpha}(-i\nabla)c_{\mathbf{r},\alpha}$ and $h_{\alpha}(-i\nabla)=|\mathbf{k}|^{2}/2m_{\alpha}$ is the kinetic energy, with $\alpha=1,..,N$ being the band and spin indices. The fermion-fermion interaction reads as $H_{I}=\sum_{\mathbf{r},\mathbf{r}^{\prime},\alpha,\beta}V(\mathbf{r}-\mathbf{r}^{\prime})c^{\dagger}_{\mathbf{r},\alpha}c_{\mathbf{r},\beta}c^{\dagger}_{\mathbf{r}^{\prime},\beta}c_{\mathbf{r}^{\prime},\alpha}$, where $\alpha\neq\beta$ is allowed. $\mathbf{g}=\{m_{\alpha},\mu,...\}$ denotes the model parameters, including the mass $m_{\alpha}$ and the chemical potential $\mu$. The key quantity indicating possible instabilities is the susceptibility $\chi_{\mathbf{g},\alpha\beta}(q)=-\sum_{k}G_{0,\alpha}(k)G_{0,\beta}(k+q)$ hvl ; Honerkamp , with $G_{0,\alpha}(k)$ the bare Green’s function of fermions and $k=(\mathbf{k},i\omega_{n})$. At the random phase approximation level, the interaction is renormalized as $V^{\prime}(q)=V(\mathbf{q})/[1+V(\mathbf{q})\chi_{\mathbf{g},\alpha\beta}(q)]$. It is known that, when the condition $1+V(\mathbf{q})\chi_{\mathbf{g},\alpha\beta}(\mathbf{q},0)=0$ is satisfied at a single momentum point $\mathbf{q}=\mathbf{Q}$, the divergence indicates the formation of boson condensates or long-range orders.

In this work, we focus on an intriguing question, i.e., what is the fate of bosons if the above condition is simultaneously satisfied by an infinite number of points on a momentum loop, i.e.,

given $S^{1}$ a 1D loop embedded in 2D momentum space with the radius $Q$, as indicated by Fig.1(a). Eq.(1) essentially implies that the bosons have an equal tendency to condense on each point of $S^{1}$, impying a strong frustration effect. We show that Eq.(1) generates a low-energy effective theory describing interacting bosons on a moat-shaped band sur ; Sedrakyan3 ; Sedrakyan1 ; Sedrakyan2 , as shown by Fig.1(b). The emergent physics mimics the flat band systems Regnault ; xwan ; zliu ; dnsheng along $S^{1}$ in a bosonic version, finally resulting in chiral TOs with TRS breaking. We apply the theory to study strongly-correlated semiconductors doped to the metallic phase, which satisfies the susceptibility condition in the particle-hole channel. A novel ETO state is revealed, which exhibits semionic anyons in the bulk and chiral excitonic edge state ruid . More interestingly, by applying the theory to frustrated quantum magnets, we predict a chiral spin liquid, which is numerically confirmed by our tensor network calculations. These results reveal the long-desired connections between chiral TOs and interacting fermions.

Emergence of Chiral TO under the susceptibility condition.– We assume Eq.(1) is satisfied at $\mathbf{g}=\mathbf{g}_{cri}$ and study the general fermion model. By introducing the auxiliary bosons $O_{\mathbf{r},\alpha\beta}$, $H_{I}$ can be decomposed into the boson-fermion interaction:

where the repeated indices are summed and “…” denotes the boson bilinear terms. Introducing the operator $b_{\mathbf{r},a}\equiv b_{\mathbf{r},\alpha\beta}=\sum_{\mathbf{r}^{\prime}}V(\mathbf{r}-\mathbf{r}^{\prime})O_{\mathbf{r}^{\prime},\alpha\beta}$, where $a$ is the boson flavor with $a=1,...,N^{2}$, and integrating out the fermions sup ; Melo , we obtain $Z=\int Db^{\star}Dbe^{-S_{eff}}$. The effective action of the bosons $S_{eff}$ reads as,

where $b_{\mathbf{r}}$ is the matrix with entries $b_{\mathbf{r},\alpha\beta}$. $G^{-1}(\tau,\mathbf{r})=G^{-1}_{0}(\tau,\mathbf{r})-\Sigma(\tau,\mathbf{r})$ is the renormalized Green’s function, with $G^{-1}_{0}(\tau,\mathbf{r})=[-\partial_{\tau}-h(-i\nabla)]$. $\Sigma(\tau,\mathbf{r})=b_{\mathbf{r}}+b^{\dagger}_{\mathbf{r}}$ is the self energy, which can be treated perturbatively at $\mathbf{g}=\mathbf{g}_{cri}$ Melo .

To the second order perturbation, the action of bosons is obtained as $S_{0,eff}=\sum_{a}S^{(a)}_{0,eff}$, and the $a$-flavor sector is described by

The saddle point equation, $\delta S^{0,(a)}_{eff}/\delta b_{q,a}=0$, can be derived, which exactly reproduces the identity in Eq.(1). Thus, Eq.(1) states that there are an infinite number of saddle point solutions and quantum fluctuations are non-negligible. Thus, we take into account the long-wave fluctuations around the saddle points. A generic momentum $\mathbf{q}$ can be measured in a local Cartesian coordinates with the origin $\mathbf{Q}$ and the unit vector tangential to $S^{1}$ (Fig.1(a)), i.e., $\mathbf{p}=\mathbf{q}-\mathbf{Q}$. Then, making expansion with respect to $|\mathbf{p}|$ and $i\nu_{n}$ leads to the low-energy effective theory sup :

where $\tilde{m}_{a}$, $\tilde{\mu}_{a}$ are the effective mass and chemical potential of the $a$-flavor bosons, which are dependent on $\mathbf{g}$. Interestingly, we observe from Eq.(5) that the kinetic energy of bosons is minimized for $|\mathbf{q}|=Q$, namely, on the loop $S^{1}$. Besides, quadratic dispersion takes place for $\mathbf{q}$ deviating from $S^{1}$, leading to the moat band shown in Fig.1(b). For $\mathbf{g}=\mathbf{g}_{cri}$ where Eq.(1) is satisfied, $\tilde{\mu}_{a}=0$ can be rigorously proved sup . This is a reflection of the fact that the bosons are about to condense at $\mathbf{g}=\mathbf{g}_{cri}$. However, as will be clear in the following, the condensation will be suppressed by quantum fluctuation.

To the fourth order, similar derivations as above lead to the following effective action:

The coupling constant $U_{a}=4\sum_{k}G^{2}_{0,\alpha}(k)G^{2}_{0,\beta}(k)$ is generally positive. Collecting both Eq.(5) and Eq.(6), we arrive at the effective Hamiltonian of bosons

where the flavor $a$ is implicit. $\tilde{\mu}$ can be formally cancelled by shifting the zero of energy. Clearly, Eq.(7) describes interacting bosons on the moat band, as indicated by Fig.1(b).

We now examine the possible ground state of Eq.(7). Due to the flatness of the moat band along $S^{1}$, the interaction $U$ plays the dominant role. In this case, the system can lower the energy cost from $U$ by statistical transmutations via the flux attachment kyang ; kyangb ; alopez . Technically, we represent the bosons as the composite fermions (CFs) attached to an 1-flux quantum Jain ; Halperin ; SGK2 ; Ruia ; Tigrana ; ruib ; ruic , i.e., $\Psi_{b}(\mathbf{r}_{1},...,\mathbf{r}_{N})=\Psi_{f}(\mathbf{r}_{1},...,\mathbf{r}_{N})e^{i\sum_{i<j}\mathrm{arg}[\mathbf{r}_{i}-\mathbf{r}_{j}]}$. Although the fluxes cost the kinetic energy $\langle\Psi_{b}|H_{K}|\Psi_{b}\rangle$, where $H_{K}=(|\mathbf{q}|-Q)^{2}/2\tilde{m}$, the interaction energy from $U$, which plays the dominant role here, is significantly reduced in such a representation because of the antisymmetric nature of $\Psi_{f}(\mathbf{r}_{1},...,\mathbf{r}_{N})$. Hence, the fluxes that break TRS could be spontaneously generated, as they can further lower the system energy.

The next step is to look for the ground state wave function, $\Psi_{f}(\mathbf{r}_{1},...,\mathbf{r}_{N})$. Using $\Psi_{b}(\mathbf{r}_{1},...,\mathbf{r}_{N})=\Psi_{f}(\mathbf{r}_{1},...,\mathbf{r}_{N})e^{i\sum_{i<j}\mathrm{arg}[\mathbf{r}_{i}-\mathbf{r}_{j}]}$, the Hamiltonian in the fermionic basis describes composite fermions coupled to the Chern-Simons (CS) flux $B_{CS}=2\pi n$ ruic ; Ruia ; ruib ; Tigrana , where $n$ is the fermion density footnote4 . Consequently, Landau quantization is formed, and the ground state in terms of $\Psi_{f}(\mathbf{r}_{1},...,\mathbf{r}_{N})$ is obtained to be the lowest Landau level state. Therefore, we arrive at the bosonic ground state wave function sup ,

where $\chi^{l}_{m}(z_{j})=A_{l,m}(\frac{z}{l_{B}})^{m}e^{\frac{|z|^{2}}{4l^{2}_{B}}}L^{(m)}_{l}[\frac{|z|^{2}}{2l^{2}_{B}}]$ is the eigenstate of the $l$-th Landau level ($l$ determined by $n$ sup ) with the normalization factor $A_{l,m}$ and the complex coordinate $z$. Eq.(8) describes the lowest Landau level fully filled by composite fermions, which are bosons attached to 1-flux quanta.

The state in Eq.(8) is essentially a chiral bosonic topological order, as it can be equivalently understood as an $\nu=1/2$ bosonic FQH sup due to the following reason. Starting form an $\nu=1/2$ bosonic FQH under an intrinsic CS field $B_{\perp}$ and $\nu=\phi_{D}\rho_{0}/B_{\perp}=1/2$, where $\phi_{D}$ is the flux quantum and $\rho_{0}$ the particle number, and regarding the boson as the composite fermion attached to 1-flux quantum, then the effective field seen by the composite fermions is $B_{CS}=B_{\perp}-\phi_{D}\rho_{0}=B_{\perp}/2$. Hence, the fermion filling factor is $\nu_{eff}=\phi_{D}\rho_{0}/B_{eff}=1$, leading to the fully-filled Landau level state in Eq.(8).

The energy of the state in Eq.(8) can then be evaluated via $\langle\Psi_{b}|H_{K}|\Psi_{b}\rangle$, leading to $E_{TO}=\langle\Psi_{b}|[\frac{(|\mathbf{q}|-Q)^{2}}{2\tilde{m}}]|\Psi_{b}\rangle=\frac{\pi^{2}n^{2}}{2\tilde{m}Q^{2}}\mathrm{log}^{2}\frac{4n}{Q^{2}}$ sup ; Sedrakyan3 . Remarkably, in the low-density regime $n\rightarrow 0$, $E_{TO}$ has lower energy than that of all the condensates proposed to date sup , including the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) ($E_{FF/LO}\propto n$ Fulde ; Larkin ) and the fragmented condensate ($E_{frag}\propto n^{4/3}$ Gopalakrishnan ). Notably, the energetics obtained here is confirmed by a recent Monte Carlo simulation cwei .

Last, we recall that $\tilde{\mu}=0$ has been proved in Eq.(5) for $\mathbf{g}=\mathbf{g}_{cri}$ sup and $n\propto 2\sqrt{\tilde{\mu}}$ ruid . Thus, $n\rightarrow 0$ is always satisfied for $\mathbf{g}\simeq\mathbf{g}_{cri}$. Therefore, we arrive at the key conclusion of this work, i.e., the chiral TO described by Eq.(8) will always emerge as the possible ground state, at least in a finite parameter region around $\mathbf{g}\simeq\mathbf{g}_{cri}$ where Eq.(1) is satisfied.

Appilication 1: excitonic topological order.– We now apply the above general theory to study strongly-correlated doped semiconductors. The Hamiltonian is given by $H=H_{0}+H_{I}$, where $H_{0}=\sum_{\mathbf{k},n,\sigma}\epsilon_{n}(\mathbf{k})c^{\dagger}_{n,\mathbf{k},\sigma}c_{n,\mathbf{k},\sigma}$ with spin $\sigma$. The conduction and valence bands are described by $\epsilon_{\pm}(\mathbf{k})=\pm k^{2}/2m\pm D/2-\mu_{0}$, where $\mu_{0}$ is the chemical potential and $D$ is the band offset. We are interested in the doped metallic state with the electron (or hole) Fermi surface, i.e., $0<D<2|\mu_{0}|$, as indicated by Fig.2(a). On top of $H_{0}$, we consider the short-range inter-band interaction between the electrons, i.e., $H_{I}=V\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}}\sum_{\sigma,\sigma^{\prime}}c^{\dagger}_{+,\mathbf{k}+\mathbf{q},\sigma}c_{+,\mathbf{k},\sigma}c^{\dagger}_{-,\mathbf{k}^{\prime}-\mathbf{q},\sigma^{\prime}}c_{-,\mathbf{k}^{\prime},\sigma^{\prime}}$. The intra-band interaction is negligible as it only modifies the band dispersion via the mass renormalization.

Even if at low temperatures, there are virtual processes where electrons are excited to the conduction band above the Fermi level, leaving hole states in the valence band, as indicated by the dashed arrows in Fig.2(a). For strong $V$, the electrons and the holes can form excitons. When the binding energy overcomes the excitation energy, such virtual processes will be relevant, leading to condensation of excitons, i.e., excitonic insulators (EIs). In conventional mean-field theory sup , we define the excitonic order parameter as $\Delta_{\tilde{\mathbf{q}}}=V\sum_{\mathbf{p}}\langle c^{\dagger}_{+,\mathbf{p},\sigma}c_{-,\mathbf{p}-\tilde{\mathbf{q}},\sigma}\rangle$ where $\tilde{\mathbf{q}}$ is the net exciton momentum. By solving the mean-field equations, the order parameters $\Delta_{\tilde{\mathbf{q}}}$ is self-consistently determined. The solution $\Delta_{\tilde{\mathbf{q}}}=0$ simply describes the electron (or hole) gas without ordering. For the solutions where $\Delta_{\tilde{\mathbf{q}}}$ is maximized at $\tilde{\mathbf{q}}=0$, the excitons intend to exhibit zero net momentum. Such condensate is referred to the zero-momentum EI, in contrary to the condensate at finite momentum, i.e., the FFLO state.

The mean-field phase diagram with varying $|\mu_{0}|$ and $D$ is shown in Fig.2(b). As shown, in the strongly-doped regime with large $\mu_{0}$, the system remains as the electron (or hole) gas, because of the dominant excitation energy. For smaller $\mu_{0}$, the binding energy slightly overcomes the excitation energy. In this case, only the process indicated by the red dashed arrow in Fig.2(a) becomes relevant, as it has the lowest energy cost. Clearly, the excitons formed in this channel exhibit finite momenta, thus leading to the FFLO EI after condensation, as shown by Fig.2(b). With further lowering $|\mu_{0}|$, the binding energy becomes dominant, so that zero momentum excitons can be readily formed, as indicated by the $|\mathbf{q}|=0$ scattering channel in Fig.2(a). This leads to the zero-momentum EI in Fig.2(b).

We then examine the excitation energy as a function of the exciton momentum $\mathbf{q}$. From Fig.2(a), we observe that it decreases from $|\mathbf{q}|=0$, and reaches the minimum at $|\mathbf{q}|=Q$. This fact is further manifested by the exciton energy $E_{ex}$, which can evaluated by minimizing $W_{ex}=\langle FS|\Delta_{\mathbf{p}}[H-E_{ex}(\mathbf{p})]\Delta^{\dagger}_{\mathbf{p}}|FS\rangle$, where $\Delta^{\dagger}_{\mathbf{p}}=\sum_{\mathbf{k}}\phi_{\mathbf{p}}(\mathbf{k})c^{\dagger}_{+,\mathbf{k},\sigma}c_{-,\mathbf{p}-\mathbf{k},\sigma}$ with the variation parameter $\phi_{\mathbf{p}}(\mathbf{k})$, and $|FS\rangle$ denotes the Fermi sea sup . As shown by Fig.2(c), for the metallic states, the calculated exciton dispersion exhibits the minimum at finite momentum, leading to the moat-like band of excitons, as that in Fig.1(b). This indicates that there emerges a frustrated ordering tendency along $S^{1}$.

We further calculate the susceptibility $\chi(\mathbf{q},0)$ in the particle-hole channel. As shown by Fig.2(e), $\chi(\mathbf{q},0)$ exhibits the same maxima along a momentum loop, as indicated by the red dashed circle. Hence, for a proper parameter $\mathbf{g}=\mathbf{g}_{cri}$, $1+V(\mathbf{q})\chi(\mathbf{q},0)=0$ can be satisfied by all $\mathbf{q}\in S^{1}$, leading to correlated excitons on the moat band sup . Hence, a chiral TO formed by excitons, i.e., the ETO, should emerge as ground state for $\mathbf{g}\sim\mathbf{g}_{cri}$ according to our theory.

Comparison of the energetics between the boson condensates and the ETO leads to the phase diagram corresponding to Eq.(7), as shown in Fig.2(e). In terms of the original fermion model, we gradually decrease $\mu_{0}$ along the dashed line in Fig.2(f). For $\mu_{0}=\mu_{0,cri1}$, where the susceptibility condition is satisfied, $\tilde{\mu}\sim 0$ in Eq.(7), thus the density $n\sim 0$, as discussed above. With further decreasing $\mu_{0}$, $n$ increases and moves along the dashed trajectory in Fig.2(e). The system stays in the ETO state until the trajectory crosses the ETO-FFLO boundary at $\mu_{0}=\mu_{0,cri2}$. This determines the ETO region of the phase diagram in Fig.2(f) Thus, the quantum fluctuation results in a translational invariant, long-range quantum entangled chiral excitonic stateruid ; ldu1 ; ldu2 .

Application 2: chiral spin liquid and numerical evidences.– Our theory can also be applied to predict chiral spin liquids in frustrated magnets. We consider the spin-1/2 $J_{1}$-$J_{2}$-$J_{3}$ XY model on square lattice, which satisfies the susceptibility condition as will be shown below. The model is defined as $H=\sum_{\mathbf{r},\mathbf{r}^{\prime}}(J_{\mathbf{r},\mathbf{r}^{\prime}}/2)(S^{x}_{\mathbf{r}}S^{x}_{\mathbf{r}^{\prime}}+S^{y}_{\mathbf{r}}S^{y}_{\mathbf{r}^{\prime}})$ where the sum evolves the first, second, and third nearest neighbor bonds. We fermionize the quantum spin model using the 2D Jordan-Wigner transformation kyang ; kyangb ; alopez ; Ruia ; ruib ; Tigrana , which transform the spin operator into fermions and the lattice gauge field. The lattice gauge field has two effects. One is to generate fluxes that minimize the energy of fermions, and the other is to mediate fermion-fermion interactions. Focusing on $J_{3}=J_{2}/2$ and gradually increasing $J_{2}$, we plot the single-particle dispersion of the fermions in Fig.3(a)-(c). As shown, the Dirac fermion states emerge, and the chemical potential $\mu_{0}$ gradually moves away from the Dirac points with increasing $J_{2}$. For small $J_{2}$, the gauge-mediated fermion-fermion interaction has been shown to induce paring of fermions, whose condensation leads to Néel order Tigrana ; ruib . While for large $J_{2}$, another Fermi pocket emerges around $\Gamma$ point (Fig.3(c)). Then, the Fermi surface nesting would favor spin density waves SGK2 . Interestingly, for intermediate $J_{2}$ (Fig.3(b)), the same low-energy physics as that of the ETO example (Fig.2(a)) occurs, and the susceptibility condition is satisfied sup . Therefore, a chiral TO is expected to take place in between two magnetically ordered states.

We then use tensor network calculations PESS2014 ; SU1D2007 ; SU2D2008 ; FU2014 ; CTMRG1996 ; CTMRG2009 ; CTMRG2014 ; Cirac2011 ; Poil2016 ; Saeed2018 ; krylov ; Verstraete ; zyxieb to simulate the ground state. As shown by Fig.3(d), in between two in-plane magnetic orders, an intermediate phase ($0.33\lesssim J_{2}/J_{1}\lesssim 0.49$) occurs, which is completely free from any in-plane ordering. The chirality order also shows a significant enhancement in this region, clearly indicating the spontaneous breaking of TRS. Moreover, the entanglement spectrum exhibits the level counting, 1,1,2,3,5… (Fig.3(f)), in consistent with the $\mathrm{SU}(2)_{1}$ conformal field theory, implying the existence of the chiral edge state. These data offer strong evidences for the chiral spin liquid state, justifying our analytic predictions.

Summary and discussion.– The ETO is a chiral bosonic TO exhibiting semionic excitations in the bulk and chiral excitonic edge stated sup ; ruid . The experimental evidence was recently reported in the semimetal phase of InAs/GaSb quantum wells ruid ; ldu1 ; ldu2 with density imbalance kunyang3 . Here, we reveal that ETO can even be formed in the metallic phase of doped semiconductors. In this case, a strong interaction $V$ comparable to the band width $W$ is desired. The twisted TMD bilayers provide a promising platform, which can realize semiconductors with strong correlation and remarkably flat bands fuliang . Therefore, our theory could have intimate connections with the recently reported fractional quantum anomalous Hall states in the twisted Morié systems jcai ; park ; fxu ; yacoby .

The mechanism revealed here applies to correlated fermionic systems, in which the number of bosons depends on how many fermions are paired. In this case, the system can always lower its energy at the optimal density ruid where the lowest Landau level is fully filled. In contrast, for bosons with fixed density on a moat band, generic filling of the Landau level is likely. Consequently, metallic states with quasi-long-range order are expected sur . The proposed mechanism may also be used to predict other chiral TOs, such as fractional Chern insulators zliu ; Regnault . The generalization to TRS-preserving TOs and non-Abelian TOs is also an interesting direction.

Supplemental material for: Susceptibility indicator for chiral topological orders emergent from correlated fermions