Domain formation driven by the entropy of topological edge-modes

Introduction.— Topological phases of matter  [1, 2, 3] have been a subject of great interest in condensed matter physics ever since the discovery of the integer and fractional quantum Hall effects [4, 5]. In such systems, there exists a bulk invariant, e.g., the Chern number, distinguishing different phases of matter with the same symmetry, in contrast to Landau’s approach to phase transitions, predicated on symmetry-breaking [6]. In the Chern insulator [7] and topological insulator phases [8], one finds a remarkable feature: protected metallic edge-modes emerge at the boundary between topologically-distinct phases.

In materials where the electron-electron interactions are non-negligible, the phase diagram often features correlation-driven spontaneously-symmetry-broken phases. Prominent examples include quantum Hall ferromagnetism [9, 10], Wigner crystals [11], charge and spin density waves [12, 13], and metal-insulator transitions [14].

In this manuscript, we study how the underlying topological nature of the strongly-interacting electrons may affect their phase diagram. We focus our interest in systems where the broken symmetry is discrete, and the degenerate symmetry-broken states are topologically inequivalent. In the presence of disorder which couples to the Ising-like order parameter, the entropy of the metallic edge-modes at domain walls (DWs) drives a thermal phase transition from a uniform phase to a “domain phase” whose spatial shape is determined by the disorder landscape (see Fig. 1). Our theory may be relevant to quantum Hall plateau transitions [15], class-D superconductors [16], the fractional quantum Hall state at filling factor $\nu=5/2$ [17, 18, 19], quantum Hall ferromagnetism in graphene [20], and Chern insulators in graphene moiré heterostructures [21, 22, 23].

Employing field theoretical considerations, we characterize the dependence of this transition on temperature, disorder, and the topological properties of the symmetry-broken states. We demonstrate our findings using a concrete microscopic model of a two-flavor Chern insulator with strong on-site repulsion. The dependence of the phase transition on temperature and the strength of the disorder field in the microscopic model is found to be in remarkable agreement with the field-theoretical considerations, verifying the role of the edge-modes entropy at the DWs.

We propose that graphene moiré heterostructures [24, 25, 26, 27, 28, 29] may be a promising platform to observe and study this effect. These materials combine the unique topological properties of graphene multi-layers, and effectively large Coulomb repulsion in the flat bands, making them ideal candidates for the physics we discuss. Finally, we explain how the phase transition of interest can be experimentally measured with currently accessible high-resolution probes.

Theory.— We consider a spontaneous-symmetry-breaking phase transition described by the Ising-order Landau free-energy functional

where $\phi$ is a real scalar field, and $r$, $c$, and $u$ are positive. We denote $f_{0}=\frac{r^{2}}{16u}$ as the condensation energy density for the uniform solution, $\phi\left({\mathbf{x}}\right)=\pm\phi_{0}=\pm\sqrt{\frac{r}{4u}}$. Notice $F_{\rm L}$ possesses $\mathbb{Z}_{2}$ symmetry, $\phi\to-\phi$.

Let us examine a DW configuration along the $x$ direction with the ansatz $\phi\left({\mathbf{x}}\right)=\phi_{0}\tanh\frac{x}{\xi}$, where $\xi$ is a correlation length determined by minimization the DW free-energy. We may then find the DW energy per unit length $L$, or surface tension, $J_{\rm DW}=\frac{8}{3}f_{0}\xi$, and $\xi=\sqrt{2c/r}$.

We introduce disorder which couples linearly to $\phi$ in the spirit of Ref. [30], $F_{\rm L}\to F_{\rm L}+F_{\rm dis}$, with

and for concreteness, we consider Gaussian-correlated disorder, i.e., ${\cal H}$ has zero mean and its variance is

As a consequence, a judiciously chosen domain of linear size $L$ will benefit from an energy reduction of $F_{\rm dis}\approx-h\lambda L\sqrt{1-e^{-L^{2}/2\lambda^{2}}}$, which scales as $\sim-h\lambda L$ when the domain size is $\apprge\lambda$. Considering short-range disorder instead, our conclusions remain unchanged [31].

Let us define the ratio between the energy gain of a domain due to the disorder and the energy cost of the DW, as the dimensionless disorder strength,

Notice the expression for $\alpha$ contains both the disorder magnitude $h$ and the disorder correlation length $\lambda$.

One would normally expect that the condition for favoring the domains configuration is $\alpha>1$  ¹¹1When domain-roughening effects are considered [42], The critical value of $\alpha$ in a finite system is smaller than 1, see SM [31]. In this work we demonstrate how the topological properties of the system may alter this condition. Specifically, we show that at finite temperatures $\alpha<1$ may be sufficient for the system to transition into a domain-patterned phase, due to the edge-modes entropy.

Consider now the case where the two degenerate solutions with order parameter $\pm\phi_{0}$ are topological insulators with different Chern numbers, $C_{\pm}$ respectively. One expects from the bulk-edge correspondence that a DW between these phases hosts $N_{\rm ch}$ topologically-protected chiral edge-modes, with $N_{\rm ch}=\left|C_{+}-C_{-}\right|$. These modes will have an entropic contribution to the free-energy of the domains configuration at finite temperature, $F_{\rm ent}=-TsL$, where the entropy per unit length of each edge-mode with velocity $v$ is [31]

We now find the transition point from a uniform $\left|\phi\right|=\phi_{0}$ solution, to a configuration with domains following the topography of the disorder, with domain density $\propto\lambda^{-2}$. The transition is found by equating the total free-energy of both configurations, i.e., solving

from which we may recover the transition temperature $T^{*}$,

The temperature $T_{0}$ is an emergent energy scale, above which spontaneous DW formation is favored in a disorder-free system ($\alpha=0$). In order for such a transition to be observable in the absence of disorder, both energy scales $f_{0}\xi^{2}$ and $\hbar v/\xi$ must be sufficiently small as compared to the condensation temperature of $\phi$. However, in the presence of disorder the transition to the non-uniform case may occur at temperature much smaller than $T_{0}$ or the condensation temperature.

Interestingly, the transition temperature is driven down by a larger topological disparity between the phases, as it leads to more protected edge-modes which lower the free-energy of the domain phase via their entropy. Notice the separability of $T^{*}$, where one part of it is determined by the disorder strength $\alpha$ and independent of topology, whereas the other part $T_{0}$ is unaffected by disorder and is determined by the energetics and topology of the condensed phase.

The effect of an additional competing phase, which does not couple to the disorder field, and its possible relevance to magic-angle twisted bilayer graphene (MATBG), is explored in the supplementary materials (SM) [31].

Naturally, in a given system and disorder realization, the strength of the disorder field and size of the characteristic domains may vary around their nominal values of $h$ and $\lambda$, respectively. In that case, one expects the phase transition at $T^{*}$ to be broadened: beginning in the uniform $\pm\phi_{0}$ phase, increasing the temperature will not necessarily “flip” all the domains in the system at once. Examining Eq. (4), implies that larger stronger domains will flip first, followed by the weaker smaller ones. The width of the system-wide transition as a function of temperature will thus be determined by the variance of $h\lambda$ throughout the sample.

In the next section, we will focus on the transition of an individual domain in a microscopic model, and show how it relates to the results obtained above.

Microscopic model.— We consider two species of electrons on a square lattice, described by annihilation operators at site $i$, $c_{\sigma,i}$, with $\sigma=\pm$ for different species, with on-site interaction $U$,

and $n_{\sigma,i}\equiv c_{\sigma,i}^{\dagger}c_{\sigma,i}$. We use a model introduced in Ref. [33] for $t_{ij}^{\sigma}$, with nearest-neighbor-hopping strength set to $t_{1}=1$ (henceforth setting the energy units) and next-nearest-neighbor hopping $t_{2}=1/\sqrt{2}$. The non-interacting part features two energy bands per species, separated by a large gap, see Fig. 2a. The upper and lower bands have opposite Chern numbers $C=\pm 1$, and we employ $t_{ij}^{+}=\left(t_{ij}^{-}\right)^{*}$, which leads to opposite Chern numbers for the same band in different species. The Hamiltonian thus has a $\mathbb{Z}_{2}$ symmetry, corresponding to swapping of the species and complex conjugation (exchanging the Chern numbers).

We consider this system at quarter-filling, where we find that the on-site repulsion may lead to spontaneous breaking of the $\mathbb{Z}_{2}$ symmetry due to a mean-field Stoner-like instability, and the development of species polarization. If $U$ is large enough, a species-polarized Chern insulator forms, where the lower-energy band of one species is fully occupied, and all the other bands are completely empty [31].

We now introduce a Zeeman-like term which couples to the $\mathbb{Z}_{2}$ order parameter, $H\to H+H_{\rm dis}$, with

For simplicity we consider the case where ${\cal H}_{i}$ is negative in a single large domain and positive elsewhere with a zero mean $\sum_{i}{\cal H}_{i}=0$, see Fig. 2b. This may be thought of as a “zoom-in” on a particular domain of some specific disorder realization in a much larger system. The parameter $\lambda$ used above to characterize the disorder correlation length is approximately the diameter of the domain.

We now employ a self-consistent spatial-dependent mean-field approach, approximating $Un_{+,i}n_{-,i}\approx U\left\langle n_{+,i}\right\rangle n_{-,i}+Un_{+,i}\left\langle n_{-,i}\right\rangle-U\left\langle n_{+,i}\right\rangle\left\langle n_{-,i}\right\rangle$. We calculate $\left<n_{\sigma,i}\right>$ at different temperatures and magnitudes of $H_{\rm dis}$. We note that throughout this work our real-space calculations are done with periodic boundary conditions, in order to exclude the entropic effects of edge-modes on the system boundaries. For more details on the microscopic model and mean-field calculations, see SM [31].

In Fig. 2c we compare the free-energy $F$ for a self-consistent solution, where the initial $n_{\sigma,i}$ we plug into the iterative process are either (i) roughly uniform and the system globally has the same species polarization, or (ii) the polarization within the domain is opposite to that outside of it [31]. Both choices yield a stable self-consistent solution, reflecting the existence of local minima of the free-energy. In the non-uniform scenario, the Chern number inside the domain is opposite to the one outside, and one expects to find chiral edge-modes on the domain boundaries. The transition between the uniform and the non-uniform phases is the focus of our work.

As we expected from our theoretical considerations, at low temperatures the system prefers to avoid the exchange energy costs of having a DW, whereas at high enough temperatures the domain structure is preferred, as one gains entropic free-energy from the topological edge-modes. To show the existence of these metallic DWs, we compute the local compressibility for a temperature higher than the transition temperature $T^{*}$, cf. Fig. 2d. It features a clear compressible ring encircling the negative ${\cal H}_{i}$ region, while everywhere else the system is in an incompressible Chern insulator state.

By varying the strength of $H_{\rm dis}$ via its root-mean-square (RMS), $h=\sqrt{\frac{1}{\Omega}\sum_{i}{\cal H}_{i}^{2}}$, ($\Omega$ is the system volume) and monitoring the evolution of $T^{*}$, we can compare the microscopic model analysis to our field-theoretical considerations above. As shown in Fig. 3, at higher temperatures the critical dimensionless disorder strength is lowered as $\alpha_{c}\approx 1-\left(T/T_{0}\right)^{2}$, in agreement with the functional form predicted by Eq. (7). The $T^{2}$ dependence is evidence of the entropic contribution of the DW modes towards lowering the free-energy, as their entropy $s$ is approximately linear in $T$. In the absence of this topological effect one would expect the usual $\alpha_{c}=1$.

We note that at high enough temperatures the critical disorder strength begins to deviate from this theoretical formula. A possible explanation for that is the saturation of edge-mode entropy [31]. As the temperature increases and becomes comparable to the mean-field gap, the linear-in-$T$ form of the entropy [Eq. (5)] no longer holds. This form assumes a constant-in-energy density of states of the edge-modes, an assumption that loses its validity at energies comparable to the gap, where the edge-modes “merge” into the bulk dispersion.

Experimental consequences.— The ingredients needed to manifest the sort of phase transition discussed in this work are (i) topologically non-trivial bands, and (ii) an instability towards spontaneous symmetry breaking due to strong interactions. Natural candidates for systems having both ingredients are the multi-layer graphene moiré heterostructures [24, 25, 26, 27, 28, 29, 34]. In such materials, the interplay between the effectively large interaction due to the flat-band structure and the Berry curvature, may indeed result in phases where the ground state is a “flavor-polarized” Chern insulator.

For example, the microscopic model we explored resembles a spin-polarized version of MATBG aligned with hexagonal Boron-Nitride (hBN) [21]. In that context, the $\pm$ species correspond to the two valleys, and the gap in the non-interacting band structure is a result of $C_{2}$ symmetry breaking by the hBN substrate. It is strongly believed that the mechanism that leads to a ferromagnetic Chern insulator in hBN-aligned MATBG [26, 27] is similar in nature to the mechanism present in the model we explored near quarter-filling [21].

Another, perhaps more promising candidate system is twisted monolayer-bilayer graphene, where Chern insulators have been observed at odd fillings [22]. The relevant band at the appropriate range of perpendicular electric field (which tunes both the topology of the bands and their width) has Chern number $C=\pm 2$. This higher Chern number may lead to $N_{\rm ch}=4$ in domains with opposite valley polarization, and thus an enhancement of the edge-modes entropy. Ultimately, enhancement of $N_{\rm ch}$ will contribute to expand the range of disorder regimes where thermal switching between a uniform and a domain-patterned phase can take place.

Yet, the effect we describe is difficult to uncover using global transport measurements. One possibility is to use spatially-resolved scanning tunneling microscopy to detect the compressible edge-modes [35]. Another relevant tool is the superconducting quantum interference device (SQUID) utilized in Refs. [36, 37, 38], which has spatial resolution of the order of $\sim$ 100 nm. In Ref. [37] the domain structure of a suitable ferromagnetic Chern insulator was in fact measured, revealing that the order parameter indeed couples to some spatial disorder field pinning the domains. We postulate that a similar measurement as a function of temperature, perhaps in several devices with varying disorder profiles and magnitudes, may reveal the topology-driven phase transition we discuss.

An alternative probing method is local compressibility measurements [39, 40]. At a particularly high resolution, one could map the compressibility of the system as a function of temperature and obtain an image resembling Fig. 2d. However, even at a realistic resolution where the DW cannot be fully resolved, one may detect the temperature-driven phase transition. At the transition, a previously insulating area, the DW, becomes compressible. Taking into consideration the mesoscopic charging energy of the DW of order $e^{2}/L$, it becomes an effective quantum dot. This will lead to discrete charging events [41] when, e.g., a gate-voltage is tuned, which disappear at the low-temperature uniform incompressible phase.

Detection of the one-dimensional chiral nature of the $T>T^{*}$ compressible state may be done by applying a small magnetic field perpendicular to the sample. This will shift the charging events in gate-voltage due to the spectral flow of the chiral modes encircling the insulating domain. The shift will then be periodic in flux penetrating the encompassed domain with the periodicity of the flux quantum, $\Phi_{0}=2\pi\hbar/e$.

Conclusions.— We have unveiled and studied a phase transition occurring in strongly-correlated topological materials with spontaneous discrete symmetry breaking. Due to the topological nature of the symmetry-broken phase, metallic edge-modes reside on DWs between topologically-distinct degenerate ground states. In the presence of disorder, the entropic contribution of the edge-modes to the thermodynamic free-energy may make the free-energy of the non-uniform state lower. The domain structure of this non-uniform phase will be determined by the topography of the disorder field.

Employing field theoretical considerations, we demonstrated the competition between the exchange energy cost of DWs on one hand, and the free-energy gain due to edge-modes entropy and coupling to disorder on the other. This enabled us to map out the phase boundary separating the uniform and non-uniform ordered phases as a function of both temperature and disorder strength. We have verified our predictions using a microscopic model of a Chern insulator with strong interactions. Our analysis of the microscopic model was able to reproduce the relevant phase boundary that was obtained from the phenomenological field theoretical treatment, emphasizing the role of the edge-mode entropy.

Finally, we discussed possible materials suitable for the realization of this unusual phase transition, with graphene-based moiré heterostructures showing the most promise. Experimental detection of the thermal uniform-to-non-uniform transition was proposed using currently available measurement schemes, e.g., scanning SQUID [36] and local compressibility probes [39, 40].