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Fractionalization and topology in amorphous electronic solids

Band-topology is traditionally analyzed in terms of gauge-invariant observables associated with crystalline Bloch wavefunctions. Recent work has demonstrated that many of the free fermion topological characteristics survive even in an amorphous setting. In this work, we extend these studies to incorporate the effect of strong repulsive interactions on the fate of topology and other correlation induced phenomena. Using a parton-based approach, we obtain the interacting phase diagram for an electronic two-orbital model with tunable topology in a two dimensional amorphous network. In addition to the (non-)topological phases that are adiabatically connected to the free fermion limit, we find a number of strongly interacting amorphous analogs of crystalline Mott insulating phases with non-trivial chiral neutral edge modes, and a fractionalized Anderson insulating phase. The amorphous networks thus provide a new playground for studying a plethora of exotic states of matter, and their glassy dynamics, due to the combined effects of non-trivial topology, disorder, and strong interactions.

Topological band theory has made a profound impact on our fundamental understanding, and in the experimental search, of weakly correlated phases of crystalline electronic solids [1][2][3][4]. A useful starting point in this endeavor is to consider crystals with perfect translational symmetry, and to include perturbative corrections due to the effects of weak disorder [5][6][7][8][9][10][11]. However, translational symmetry is not a prerequisite for realizing noninteracting topological phases, even in the absence of a well-defined momentum, an associated electronic "bandstructure" and the very notion of a Brillouin zone [12][13][14][15][16][17][18]. Interestingly, experiments on a candidate topological material [19] and a photonic system [20] provide promising evidence for the existence of topological phases in amorphous settings. On the theoretical front, the study of realistic models and better computational techniques continue to advance our understanding of such amorphous topological phases of free fermions [21][22][23].
In this letter, we address a question of fundamental interest that has not received as much attention, except when in a crystalline setting [24][25][26][27][28][29][30][31], namely on the effects of strong local interactions on the fate of the amorphous topological phases. The combined effects of strong interactions, topology and disorder (manifest in the amorphous structure of the underlying network) can lead to a plethora of new phases. To study the interacting problem, we employ and adapt a parton meanfield theory [32,33] in this new amorphous setting and obtain the theoretical phase-diagram for an interacting two-dimensional electronic model as a function of the strength of interactions and tunable topology. At fixed electron-filling and as a function of increasing interaction strength, we find evidence of transitions into phases with deconfined fractionalized excitations, that can either support neutral edge modes (as in an amorphous Chern insulator), or display disorder-induced localization (as in an Anderson insulator) accompanied by a local inhomo- Theoretical phase diagram at half-filling as a function of M/t and U/t for system of size 16 × 16. The four distinct phases include a trivial electronic Mott insulator (MI), an amorphous Chern Insulator (aCI), a fractionalized Anderson insulator (AI * ), and a fractionalized amorphous Chern insulator (aCI * ), respectively. The inset denotes a typical amorphous network and the energy scales associated with hopping (t), orbital splitting (M ) and on-site repulsion (U ). geneous distribution of topological "puddles". We use a number of quantitative diagnostics for disentangling the effects of disorder and topology in order to sharpen the properties of the resulting phases and their regions of stability with increasing system sizes. While our study of a simplified theoretical model is not directly related to a realistic model of a quantum material, recent experimental work in twisted bilayer graphene [34] suggests that the notion of a "local" Chern number and a "mosaic" of distinct topological patches can emerge in the presence of long-wavelength inhomogeneities in a strongly correlated topological system. Our theoretical study highlights the incredibly rich playground provided by seemingly "disordered" systems as an exciting new platform for realizing topological and fractionalized phases of matter.
Model.-We define a two-dimensional (spinless) electronic model with local interactions, residing on an amorphous network with two orbitals (e.g. s and p) per site; see Fig. 1-inset. The Hamiltonian is of the form, where c † i,α (c i,α ) is an electron creation (annihilation) operator with orbital index α at site i and t αβ (r ij ) represents the tunneling matrix element between sites i, j separated by distance r = |r ij |. The on-site single-particle and interaction energies are given by αβ and U , respectively, whereN i = α c † i,α c i,α . We assume that the tunneling matrix elements have a statistical rotational symmetry with an exponential fall-off with distance over a characteristic scale, a B , and vanishes beyond r = R, i.e. t αβ (r) = t(r)T αβ (r) where t(r) = CΘ(R−r) exp(−r/a B ) and C is a constant chosen such that t(r = a B ) = 1. All other energy scales are scaled with respect to this unit scale. The orbital dependence of the single-particle energy and tunneling matrix element is given by, where θ is the angle subtended between a reference axis and the vectorr connecting sites i and j (see inset of Fig. 1), and M modulates the on-site energy of the orbitals. The sites that make up the amorphous graph are constructed randomly, but with a minimal exclusion distance, r min . In the non-interacting limit (U = 0), H belongs to class A of the ten-fold classification with broken timereversal symmetry, and where previous work has pointed out the existence of free fermion topological phases with robust edge states as a function of varying M, a B [12]. These phases exists over a large window of a B , r min [35], and it is straightforward to generalize these results to other classes of the ten-fold way. In the remainder of this paper, we obtain the phase diagram of H as a function of U/t, M/t, where the complex interplay of interactions, topology and the amorphous nature of the underlying graph reveals interesting phases of matter ( Fig. 1). In the remainder of our study, we set a B = 1, R = 1.5, and r min = 0.8, unless stated otherwise.
Results.-To analyze the fate of the Hamiltonian at large U , we use the parton construction for the electronic operators, c i,α = b i f i,α , in terms of a bosonic, electrically charged 'rotor' field (b i = e iθi ) and a neutral fermion (f i,α ) [32,33]. However, given the amorphous non-translationally invariant connectivity, we employ a new implementation of the two-site cluster mean field theory to obtain the phase-diagram. We will refer to the fermionic excitations, f i,α , as 'spinons' even though they are not tied to any underlying fractionalization associated with an actual spin degree of freedom.
In terms of the new fields, the Hamiltonian is given by where we have used the physical Hilbert space constraint L i + α n f i,α = 2 withL i = ∂ ∂θi and N i = α n f i,α . We include a chemical potential to maintain half-filling of electrons which imposes α n f i,α = 1 and L i = 1 at every site i for both spinons and rotors. We carry out a mean-field decoupling of H 0 → H b + H f , where nominally H f is described by a free Hamiltonian. However, the fermion hoppings are renormalized by the rotor correlators, b † i b j . As noted earlier, to solve for the interacting rotor-Hamiltonian, H b + H int , we use a two-site cluster mean field theory for every pair of connected sites on the amorphous graph. We calculate the superfluid order parameter on every site b i , and the fermionic correlators between pairs of sites f † i,α f j,β self-consistently until the results converge.
Our solution suggests that for U 9t, the rotors are condensed in a superfluid phase with b = i b i /N (∼ O(1)), which is consistent with the typical values even in non-topological and crystalline systems [32,33]. As a function of M/t, the spinon "bands" realize both topological and trivial phases, respectively (Fig. 1). In the rotor superfluid phase, the spinons are the same as the electrons upto an overall renormalization of their quasiparticle residue; the two phases described above correspond to an a trivial electronic insulator and an amorphous electronic Chern insulator which are adiabatically connected to the U = 0 limit [12].
To diagnose the topological properties for the spinons in the absence of translational invariance and a notion of a Brillouin zone, we evaluate their Bott index (B) which quantifies the non-commutativity of the position operators when projected to the lower "band" of the spinon system [12,36]. The index is given by where U = P exp (iΘ)P , W = P exp (iΦ)P and P = m |m m| is the projector made of all occupied spinonic wavefunctions |m . Θ, Φ are diagonal matrices whose elements correspond to the real space positions of every orbital (x iα , y iα ), but compactified into a torus (2πx iα /L, 2πy iα /L), where L is the linear dimension the system. B is thus a generalization of the usual Chern number, but in a setting where translational symmetry is completely absent.
At larger values of U/t, we find two new correlationinduced phases (Fig. 2(a)-(b)). Depending on the value of M/t, we find distinct insulating regimes. For example, for M/t ∼ −1.84 and as a function of increasing U/t, the superfluid order b goes to zero across U = U c . Across the same transition, and within our numerical resolution, the Bott index jumps from 1 to 0. Interestingly, both of these changes across U ∼ U c occur over any typical configuration, such that configuration averaged quantities (∼ O(100) configurations) also reflect a relatively sharp transition. The latter phase is clearly a topologically trivial Mott insulator realized at half-filling due to strong correlation effect. Even while b goes to zero, there are rare regions in the system where SF order parameter is vanishingly small, yet finite, which results in intrinsic inhomogeneities in this system, which we discuss in detail later in the paper. In contrast, for M/t ∼ −1.94, there is an intermediate phase over U c1 < U < U c2 where the rotors are Mott insulating with b = 0, while the Bott index remains pinned at 1. This intermediate phase is a novel example of an electronic Mott insulator where the spinons inherit non-trivial topology in an amorphous setting. As we discuss below, when defined in an open system with boundaries, the phase is defined by protected electrically neutral chiral edge modes that can carry a finite energy. We dub this phase as the fractionalized amorphous Chern insulator (aCI * ). While the phase occurs over a rather small window of the parameter space, our calculations with increasing system size [35] suggest that they survive even in the thermodynamic limit.
A key element in our study is the inherently disordered nature of the amorphous network, which can lead to "glassy" physics and strong local inhomogeneities. At small U/t, we find that the disorder induced "rare-region" effects are relatively suppressed in the rotor superfluid and associated electronic trivial/Chern insulators; the distribution of site-resolved local superfluid order does not develop strong inhomogeneity. However, with increasing U/t, the amorphous network develops strong local inhomogeneities in the vicinity of the Mott transition. To characterize this phenomenology, we compute the fractal dimension for the rotors as [37], where i is the position index, L is the linear dimension and N is the total number of sites. D is useful for characterizing the effective dimensionality of the largest SF patch in the system. We note that when the rotors are condensed and the superfluid order is relatively homogeneous across the system, D approaches a value ∼ 2. On the other hand, across the Mott transition and regardless of whether the resulting insulator is (non-)topological, we find that D develops a characteristic kink showing the rotor SF-to-MI transition happens in a non-uniform way over the network. In contrast, for a crystalline system, a MI state when diagnosed on finite size lattices have an infinitesimally small but uniform SF order parameter which implies D remains ∼ 2 [35]. Here, the characteristic kink in D reflects the realization of a boseglass phase. That the transition is thermodynamically stable is reflected in the systematic system size scaling [35], where the kink remains even as system size is increased. Our discussion so far has focussed on the effects of inhomogeneities in the boson sector. Next we shift our attention to the fermions. In the aCI * phase, the f −fermions have a finite gap, ∆ pbc , at half-filling (with b = 0). Near M/t ∼ −2, the spinon hopping competes with disorder energy scales mixing the two orbitals, leading to three distinct regimes ( Fig. 3(a)): (i) for M −2.1t and M −1.9t, ∆ pbc = 0 and B = 0, (ii) for −2t M −1.9t, ∆ pbc = 0 but B = 1, and (iii) for −2.1t M −2t, ∆ pbc → 0 while B ∼ −1 (see below). Over the entire range of M/t values and large U/t considered above, b → 0 and D < 2, which signifies that all of the phases are electrical insulators affected strongly by the inhomogeneity in the amorphous network. We have already discussed the origin of the interesting behavior in regions (i) and (ii) above.
In region (iii), the "gapless" phase (∆ pbc → 0) is not tied to any itinerant fermions, but arises instead due to localization on the amorphous network, as can be seen by calculating the inverse participation ratio (IPR), and the density of states (DOS) near the Fermi-energy.
Here, ψ m,iα = iα|m denotes the spinon wavefunction (|m ) associated with orbital-α on the i th site, and has been calculated from an explicit diagonalization of the mean-field Hamiltonian, H f . We find a finite DOS at the Fermi level in the absence of a gap (see Fig. 3(b)); additionally, the IPR from the fermionic states is much larger than a typical delocalized state (∼ 1/2L 2 ) (inset of Fig. 3(b)). All of these observables are then consistent with a disorder induced localization of neutral spinons, and hence we dub it as a fractionalized Anderson insulator (AI * ). Interestingly, the seemingly non-trivial topological character associated with the large fluctuations of B near −1 is due to local topological "puddles", associated with different disorder realizations.
Finally, to help clarify the topological character associated with the fractionalized phases, we also calculate a "local Chern marker" for a representative configuration in both the AI * and aCI * phases , respectively. The local Chern marker is defined as [38], where P is the projector, as defined earlier.
The results for C(r) for typical networks of size 24 × 24 are shown in Fig. 4 (a) and (b). In the aCI * phase, C(r) is approximately uniformly distributed in the bulk (Fig. 4(a)), while it exhibits stronger fluctuations in the AI * phase ( Fig. 4(b)) [35]. These observations are reflected in the location of the edge states in open systems. While the aCI * has spinon edge states (Fig. 4(c)), the distinction between bulk and edge states is unclear in the AI * phase (Fig. 4(d)). Thus even though B fluctuates strongly near −1 in the particular realization of the AI * phase in Fig. 4(b), there is no clear sign of an edge state, and the midgap states are localized sporadically throughout the system. In particular, often wavefunctions are localized inside the bulk but between two domains of differently oriented local Chern marker patches reflecting development of edge like states but on effective grain boundaries. How such patches determine the physical characteristics of a thermodynamic phase and its properties is an interesting direction for future work. Outlook.-In this work, we have analyzed the effects of strong correlations in the presence of non-trivial topology in an amorphous setting at the level of parton mean-field theory. None of the excitations we have obtained in the insulating bulk of these phases are itinerant. By including additional local degrees of freedom and by extending the analysis to higher dimensions, it will be interesting to analyze the possibility of hosting a diffusive spinon metal in a bulk electrical insulator coexisting with topological surface states. Going beyond the parton mean-field analysis, it is natural to address the effects of emergent gaugefield fluctuations, which are a defining feature of the fractionalized phases. In our simplified construction of the aCI * phase, the total Bott index for the spinons is B = 1; in the analogous crystalline setup with Chern number 1, the topological order and edge states can be gauged away [39]. While we leave a detailed analysis of such effects for future work, a clear direction to explore further is to construct amorphous models with higher B, where the edge states can not simply be gauged away. Similarly, in the AI * phase, the subtle interplay between disorder-induced localization and a tendency towards confinement in the presence of topological puddles is an interesting direction for future work. Finally, the platform considered here is ideally suited for studying the effects of non-trivial topology and fractionalization on their non-equilibrium glassy dynamics.
Supplementary material for "Fractionalization and topology in amorphous electronic solids" We present the Bott index in the non-interacting limit as a function of M/t and r min with the cutoff distance R = 1.5 and R = 2 in Fig. S1. The perfect amorphous case corresponds to r min = 0, in which the pair correlation function is uniform as a function of distance; we focused on the case of r min = 0.8 in the main text. These results suggest that a free fermion amorphous system hosts a stable topological phase over a wide range of r min .  Figure S2. System size dependence of configuration averaged physical properties (D, B, and ∆ pbc ) for U/t = 12 with periodic boundary conditions.
We present additional results at various system sizes L = 12, 16, 20 in Fig. S2. We find that the spinon gap and the plateau with B = 1 remains finite with increasing system size, indicating the robustness of the aCI * phase even in the thermodynamic limit. We present the full distribution of Chern markers for the fractionalized phases addressed in Fig. 4. For the aCI * phase ( Fig. S3(a)), the Chern marker distribution shows a sharp peak near C(r) = 1, pointing to an approximately homogeneous distribution of Chern marker values in the bulk. On the other hand, the AI * phase (Fig. S3(b)) shows a broad distribution between −1 ≤ C(r) 2. This implies that the Chern marker exhibits a strong inhomogeneity even in the bulk. We show the two-site cluster mean-field results on a square lattice with nearest neighbor hopping. There is a clear distinction between B = ±1 phases, and D ∼ 2 throughout the whole phase diagram.