Results

The Ground State

$Figure 1: Within a flux sector, the fermion gap \Delta_f measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size L is defined \sqrt{N} where N is the number of plaquettes in the system. Error bars shown are 3 times the standard error of the mean. The lines shown are fits of \tfrac{\Delta_f}{J} = aL ^ b with fit parameters: Ground State: a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004 Random Flux Sector: a = 1.8 \pm 0.2, b = -2.21 \pm 0.03$

Figure 1: Within a flux sector, the fermion gap $\Delta_f$ measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size $L$ is defined $\sqrt{N}$ where N is the number of plaquettes in the system. Error bars shown are $3$ times the standard error of the mean. The lines shown are fits of $\tfrac{\Delta_f}{J} = aL ^ b$ with fit parameters: Ground State: $a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004$ Random Flux Sector: $a = 1.8 \pm 0.2, b = -2.21 \pm 0.03$

Ground State Phase Diagram

The Flux Gap

Conclusion

Discussion

Future Work

The Kitaev Honeycomb can be quite easily turned into a quantum error correcting code like this, the same idea applies to our model.

In contrast to the honeycomb case, the amorphous KSLs are gapless only along certain critical lines. These manifolds separate two gapped KSLs that are topologically differentiated by a local Chern number $\nu$^1,2 in analogy with the KSLs on the decorated honeycomb lattice³.

The $\nu=0$ phase is the amorphous analogue of the abelian toric-code QSL⁴, whereas the $\nu=\pm1$ KSLs is a non-Abelian chiral spin liquid (CSL).

We study two specific features of the latter liquid: topologically protected edge states and a thermal-induced Anderson transition to a thermal metal phase⁵.

Amorphous materials are glassy condensed matter systems characterised by short-range constraints in the absence of long-range crystalline order as first studied in amorphous semiconductors ^6,7. In general, the bonds of a whole range of covalent compounds enforce local constraints around each ion, e.g. a fixed coordination number $z$, which has enabled the prediction of energy gaps even in lattices without translational symmetry ^8,9, the most famous example being amorphous Ge and Si with $z=4$ ^10,11. Recently, following the discovery of topological insulators (TIs) it has been shown that similar phases can exist in amorphous systems characterized by protected edge states and topological bulk invariants ^2,12–17. However, research on electronic systems has been mostly focused on non-interacting systems with a few notable exceptions for understanding the occurrence of superconductivity ^18–20 $\textbf{J}$K: CECK OLD EMAIL WITH THEORY WORKS in amorphous materials and recently the effect of strong repulsion in amorphous TIs ²¹.

Magnetic phases in amorphous systems have been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems^22–25 and numerical methods ^26,27. Research focused on classical Heisenberg and Ising models which have been shown to account for observed behavior of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour ²⁸. However, the role of spin-anisotropic interactions and quantum effects has not been addressed. Similarly, it is an open question whether magnetic frustration in amorphous quantum magnets can give rise to long-range entangled quantum spin liquid (QSL) phases.

Two intentional simplifications of Andreev’s and Marchenko’s theory were the neglect of spin-orbit coupling induced anisotropies and the effects arising from the local structure of amorphous lattices. It is then expected that their theory is invalid for amorphous compounds generated from crystalline magnets with strong spin-orbit coupling with tight geometrical arrangements. Several instances of these magnets were synthesized in the last decade, among which we highlight the Kitaev materials^29–33. It was suggested (and later observed) that heavy-ion Mott insulators formed by edge-sharing octahedra could be good platforms for the celebrated Kitaev model on the honeycomb lattice²⁹, an exactly solvable model whose ground state is a quantum spin liquid (QSL) ^34–37 characterized by a static $\mathbb Z_2$ gauge field and Majorana fermion excitations³⁸. The model displays bond-dependent Ising-like exchanges that give rise to local symmetries, which are essential to its mapping onto a free fermion problem^39,40. Such a mapping is rigorously extendable to any three-coordinated graph in two or three dimensions satisfying a simple geometrical condition^3,41–43. Thus, it reasonable to suppose that the Kitaev model is also analytically treatable on certain amorphous lattices, therefore becoming a realistic starting point to study the overlooked possibility of QSLs in amorphous magnets.

In this letter, we study Kitaev spin liquids (KSLs) stabilized by the $S=1/2$ Kitaev model³⁸ on coordination number $z=3$ random networks generated via Voronoi tessellation^2,13. On these lattices, the KSLs generically break time-reversal symmetry (TRS), as expected for any Majorana QSL in graphs containing odd-sided plaquettes^44–49. An extensive numerical study showed that the $\mathbb Z_2$ gauge fluxes on the ground state can be described by a conjecture consistent with Lieb’s theorem⁵⁰. In contrast to the honeycomb case, the amorphous KSLs are gapless only along certain critical lines. These manifolds separate two gapped KSLs that are topologically differentiated by a local Chern number $\nu$^1,2 in analogy with the KSLs on the decorated honeycomb lattice³. The $\nu=0$ phase is the amorphous analogue of the abelian toric-code QSL⁴, whereas the $\nu=\pm1$ KSLs is a non-Abelian chiral spin liquid (CSL). We study two specific features of the latter liquid: topologically protected edge states and a thermal-induced Anderson transition to a thermal metal phase⁵.

Once the three-edge colouring has been found, the Kitaev Hamiltonian is mapped onto eqn. [eqn:majorana_hamiltonian], which corresponds to the spin fractionalization in terms of a static $\mathbb Z_2$ gauge fields and $c$ matter as indicated in [fig:example_lattice](b)³⁹.

Strictly speaking, the Majorana system is equivalent to the original spin system after applying a projector operator^5,51,52, whose form is presented in 4.

Despite this caveat, one can still use eqn. [eqn:majorana_hamiltonian] to evaluate the expectation values of operators conserving $\hat u_{jk}$ in the thermodynamic limit^53,54. This type of operator is exemplified by the Hamiltonian itself, for which the ground state energy of a fixed sector is the sum of the negative eigenvalues of $iA/4$ in eqn. [eqn:majorana_hamiltonian], and whose excitations are extracted from the positive eigenvalues of the same matrix.

Fluxes and the Ground State

Let us now consider the conserved operators $W_p = \prod \sigma_j^{\alpha}\sigma_k^{\alpha}$ on amorphous lattices. When represented in the Majorana Hilbert space, these operators correspond to ordered products of $\hat u_{jk}$, and their fixed eigenvalues are written as \[\label{eqn:flux_definition} \phi_p = \prod_{(j,k) \in \partial p} (-iu_{jk}),\] where the pairs $j,k$ are crossed around the border $\partial p$ of the plaquette on the clockwise orientation.

In periodic boundaries there is an additional pair of global $\mathbb{Z}_2$ fluxes $\Phi_x$ and $\Phi_y$, which are calculated along an arbitrary closed path that wraps the torus in the $x$ and $y$ directions respectively. The energy difference between distinct flux sectors decays exponentially with system size, so that the ground state of any flux sector in the thermodynamic limit displays a fourfold topological degeneracy⁴.

Zero Temperature Phase Diagram

We numerically found that the amorphous KSLs are generally gapped, except along the critical lines displayed.

We believe that the $A_x, A_y, A_z$ phases remain Abelian as they are in the Kitaev model while the $B$ phase is non-Abelian. The B phase corresponds to the same extended kitaev honeycomb model.

Chern Number and Edge Modes

The QSLs separated by these lines are distinguished by a real-space analogue of the Chern number^55,56. A similar topological number was discussed by Kitaev on the honeycomb lattice³⁸ that we shall use here with a slight modification^1,2. For a choice of flux sector, we calculate the projector $P$ onto the negative energy eigenstates of the matrix $iA$ defined in eqn. [eqn:majorana_hamiltonian]. The local Chern number around a point $\textbf{R}$ in the bulk is given by \[\begin{aligned} \nu (\textbf{R}) = 4\pi \Im \mathrm{Tr}_{\mathrm{Bulk}} \left ( P\theta_{R_x} P \theta_{R_y} P \right ),\end{aligned}\] where $\theta_{R_x}$ is a step function in the $x$-direction, with the step located at $x = R_x$, $\theta_{R_y}$ is defined analogously. The trace is taken over a region around $\textbf{R}$ in the bulk of the material, where care must be taken not to include any points close to the edges. Provided that the point $\textbf{R}$ is sufficiently far from the edges, this quantity will be very close to quantised to the Chern number.

The local Chern marker distinguishes between an Abelian phase (A) with $\nu = 0$, and a non-Abelian (B) phase characterized by $\nu = \pm 1$. The (A) phase is equivalent to the toric code on an amorphous system⁴.

Since the (A) phase displays the "topological" degeneracy described above, I think that "topologically trivial" is not a good term to describe it. Another thing that I think it should be considered here. The abelian phase is expected to have 2x4 degeneracy, where the factor of 2 comes from time-reversal. On the other hand, the non-Abelian phase should display 2x3 degeneracy, as discussed by³. Did you get any evidence of this?

By contrast, the (B) phase is a chiral spin liquid, the magnetic analogue of the fractional quantum Hall state. Topologically protected edge modes are predicted to occur in these states on periodic boundary conditions following the bulk-boundary correspondence⁵⁷. The probability density of one such edge mode is given in 1 (a), where it is shown to be exponentially localised to the boundary of the system. The localization of these modes can be quantified by their inverse participation ratio (IPR), \[\mathrm{IPR} = \int d^2r|\psi(\mathbf{r})|^4 \propto L^{-\tau},\] where $L\sim\sqrt{N}$ is the characteristic linear dimension of the amorphous lattices and $\tau$ dimensional scaling exponent of IPR.

Finally, the CSL density of states in open boundary conditions indicates the low-energy modes within the gap of Majorana bands in 1 (b).

The phase diagram of the amorphous model in [fig:example_lattice](c) displays a reduced parameter space for the non-Abelian phase when compared to the honeycomb model. Interestingly, similar inward deformations of the critical lines were found on the Kitaev honeycomb model subject to disorder by proliferating flux vortices⁵⁸ or exchange disorder⁵⁴.

Anderson Transition to a Thermal Metal

An Ising non-Abelian anyon is formed by Majorana zero-modes bound to a topological defect⁵⁹. Interactions between anyons are modeled by pairwise projectors whose strength absolute value decays exponentially with the separation between the particles, and whose sign oscillates in analogy to RKKY exchanges^60–62. Disorder can induce a finite density of anyons whose hybridization lead to a macroscopically degenerate state known as thermal metal⁶⁰. One instance of this phase can be settled on the Kitaev CSL. In this case, the topological defects correspond to the $W_p \neq +1$ fluxes, which naturally emerge from thermal fluctuations at nonzero temperature⁵.

We demonstrated that the amorphous CSL undergoes the same form of Anderson transition by studying its properties as a function of disorder. Unfortunately, we could not perform a complete study of its properties as a function of the temperature as it was not feasible to evaluate an ever-present boundary condition dependent factor^51,52 for random networks. Instead, we evaluated the fermionic density of states (DOS) and the IPR as a function of the vortex density $\rho$ as a proxy for temperature. This approximation is exact in the limits $T = 0$ (corresponding to $\rho = 0$) and $T \to \infty$ (corresponding to $\rho = 0.5$). At intermediate temperatures the method neglects to include the influence of defect-defect correlations.

However, such an approximation is enough to show the onset of low-energy excitations for $\rho \sim 10^{-2}-10^{-1}$, as displayed on the top graphic of [fig:DOS_Oscillations](a). We characterized these gapless excitations using the dimensional scaling exponential $\tau$ of the IPR on the bottom graphic of the same figure. At small $\rho$, the states populating the gap possess $\tau\approx0$, indicating that they are localised states pinned to the defects, and the system remains insulating. At large $\rho$, the in-gap states merge with the bulk band and become extensive, closing the gap, and the system transitions to a metallic phase.

The thermal metal DOS displays a logarithmic divergence at zero energy and characteristic oscillations at small energies.^5,63. These features were indeed observed by the averaged density of states in the $\rho = 0.5$ case shown in [fig:DOS_Oscillations](b) for amorphous lattice. We emphasize that the CSL studied here emerges without an applied magnetic field as opposed to the CSL on the honeycomb lattice studied in Ref.⁵ I have the impression that [fig:DOS_Oscillations](b) on the top is very similar to Fig. 3 of⁵. Maybe a more instructive figure would be the DOS of the amorphous toric code at the infinite temperature limit. In this case, the lack of non-Abelian anyons would be reflected by a gap on the DOS, which would contrast nicely to the thermal metal phase

Discussion and Conclusions

We have studied an extension of the Kitaev honeycomb model to amorphous lattices with coordination number $z= 3$. We found that it is able to support two quantum spin liquid phases that can be distinguished using a real-space generalisation of the Chern number. The presence of odd-sided plaquettes on these lattices let to a spontaneous breaking of time reversal symmetry, leading to the emergence of a chiral spin liquid phase. Furthermore we found evidence that the amorphous system undergoes an Anderson transition to a thermal metal phase, driven by the proliferation of vortices with increasing temperature. The next step is to search for an experimental realisation in amorphous Kitaev materials, which can be created from crystalline ones using several methods^8,23,25.

Following the evidence for an induced chiral spin liquid phase in crystalline Kitaev materials^64–67, it would be interesting to investigate if a similar state is produced on its amorphous counterpart. Besides the usual half-quantized signature on thermal Hall effect^64–67, such a CSL could be also characterized using local probes such as spin-polarized scanning-tunneling microscopy^68–70. The same probes would also be useful to manipulate non-Abelian anyons⁷¹, thereby implementing elementary operations for topological quantum computation. Finally, the thermal metal phase can be diagnosed using bulk heat transport measurements⁵⁹.

This work can be generalized in several ways. Introduction of symmetry allowed perturbations on the model^72–76. Generalizations to higher-spin models in random networks with different coordination numbers^{40,41,44,48,49,53,77–81}

Probably one way to make this theory experimentally relevant is to do experiments on amorphous phases of Kitaev materials. These phases can be obtained by liquifying the material and cooling it fast. Apparently, most of crystalline magnets can be transformed into amorphous ones through this process.

Numerical Evidence for the Ground State Flux Sector

In this section we detail the numerical evidence collected to support the claim that, for an arbitrary lattice, a gapped ground state flux sector is found by setting the flux through each plaquette to $\phi_{\mathrm{g.s.}} = -(\pm i)^{n_{\mathrm{sides}}}$. This was done by generating a large number ($\sim$ 25,000) of lattices and exhaustively checking every possible flux sector to find the configuration with the lowest energy. We checked both the isotropic point ($J^\alpha = 1$), as well as in the toric code phase ($J^x = J^y = 0.25, J^z = 1$).

The argument has one complication: for a graph with $n_p$ plaquettes, there are $2^{n_p - 1}$ distinct flux sectors to search over, with an added factor of 4 when the global fluxes $\Phi_x$ and $\Phi_y$ are taken into account. Note that the $-1$ appears in this counting because fluxes can only be flipped in pairs. To be able to search over the entire flux space, one is necessarily restricted to looking at small system sizes – we were able to check all flux sectors for systems with $n_p \leq 16$ in a reasonable amount of time. However, at such small system size we find that finite size effects are substantial enough to destroy our results. In order to overcome these effects we tile the system and use Bloch’s theorem (a trick that we shall refer to as twist-averaging for reasons that shall become clear) to efficiently find the energy of a much larger (but periodic) lattice. Thus we are able to suppress finite size effects, at the expense of losing long-range disorder in the lattice.

Our argument has three parts: First we shall detail the techniques used to exhaustively search the flux space for a given lattice. Next, we discuss finite-size effects and explain the way that our methods are modified by the twist-averaging procedure. Finally, we demonstrate that as the size of the disordered system is increased, the effect of twist-averaging becomes negligible – suggesting that our conclusions still apply in the case of large disordered lattices.

Testing All Flux Sectors — For a given lattice and flux sector, defined by $\{ u_{jk}\}$, the fermionic ground state energy is calculated by taking the sum of the negative eigenvalues of the matrix \[\begin{aligned} M_{jk} = \frac{i}{2} J^{\alpha} u_{jk}.\end{aligned}\] The set of bond variables $u_{jk}$, which we are free to choose, determine the $\mathbb Z_2$ gauge field. However only the fluxes, defined for each plaquette according to eqn. [eqn:flux_definition], have any effect on the energies. Thus, there is enormous degeneracy in the $u_{jk}$ degrees of freedom. Flipping the bonds along any closed loop on the dual lattice has no effect on the fluxes, since each plaquette has had an even number of its constituent bonds flipped - as is shown in the following diagram:

where the flipped bonds are shown in red. In order to explore every possible flux sector using the $u_{jk}$ variables, we restrict ourselves to change only a subset of the bonds in the system. In particular, we construct a spanning tree on the dual lattice, which passes through every plaquette in the system, but contains no loops.

The tree contains $n_p - 1$ edges, shown in red, whose configuration space has a $1:1$ mapping onto the $2^{n_p - 1}$ distinct flux sectors. Each flux sector can be created in precisely one way by flipping edges only on the tree (provided all other bond variables not on the tree remain fixed). Thus, all possible flux sectors can be accessed by iterating over all configurations of edges on this spanning tree.

Finite Size Effects — In our numerical investigation, the objective was to test as many example lattices as possible. We aim for the largest lattice size that could be efficiently solved, requiring a balance between lattice size and cases tested. Each added plaquette doubles the number of flux sectors that must be checked. 25,000 lattices containing 16 plaquettes were used. However, in his numerical investigation of the honeycomb model, Kitaev demonstrated that finite size effects persist up to much larger lattice sizes than we were able to access³⁸.

In order to circumvent this problem, we treat the 16-plaquette amorphous lattice as a unit cell in an arbitrarily large periodic system. The bonds that originally connected across the periodic boundaries now connect adjacent unit cells. This infinite periodic Hamiltonian can then be solved using Bloch’s theorem, since the larger system is diagonalised by a plane wave ansatz. For a given crystal momentum $\textbf{q} \in [0,2\pi)^2$, we are left with a Bloch Hamiltonian, which is identical to the original Hamiltonian aside from an extra phase on edges that cross the periodic boundaries in the $x$ and $y$ directions, \[\begin{aligned} M_{jk}(\textbf{q}) = \frac{i}{2} J^{\alpha} u_{jk} e^{i q_{jk}},\end{aligned}\] where $q_{jk} = q_x$ for a bond that crosses the $x$-periodic boundary in the positive direction, with the analogous definition for $y$-crossing bonds. We also have $q_{jk} = -q_{kj}$. Finally $q_{jk} = 0$ if the edge does not cross any boundaries at all – in essence we are imposing twisted boundary conditions on our system. The total energy of the tiled system can be calculated by summing the energy of $M( \textbf{q})$ for every value of $\textbf{q}$. In practice we constructed a lattice of $50 \times 50$ values of $\textbf{q}$ spanning the Brillouin zone. The procedure is called twist averaging because the energy-per-unit cell is equivalent to the average energy over the full range of twisted boundary conditions.

Evidence for the Ground State Ansatz — For each lattice with 16 plaquettes, $2^{15} =$ 32,768 flux sectors are generated. In each case we find the energy (averaged over all twist values) and the size of the fermion gap, which is defined as the lowest energy excitation for any value of $ }$. We then check if the lowest energy flux sector aligns with our ansatz (eqn. [eqn:gnd_flux]) and whether this flux sector is gapped.

In the isotropic case ($J^\alpha = 1$), all 25,000 examples conformed to our guess for the ground state flux sector. A tiny minority ($\sim 10$) of the systems were found to be gapless. As we shall see shortly, the proportion of gapless systems vanishes as we increase the size of the amorphous lattice. An example of the energies and gaps for one of the systems tested is shown in fig. [fig:energy_gaps_example]. For the anisotropic phase (we used $J^x, J^y = 0.25, J^z = 1$) the overwhelming majority of cases adhered to our ansatz, however a small minority ($\sim 0.5 \%$) did not. In these cases, however, the energy difference between our ansatz and the ground state was at most of order $10^{-6}$. Further investigation would need to be undertaken to determine whether these anomalous systems are a finite size effect due to the small amorphous system sizes used or a genuine feature of the toric code phase on such lattices.

A Gapped Ground State — Now that we have collected sufficient evidence to support our guess for the ground state flux sector, we turn our attention to checking that this sector is gapped. We no longer need to exhaustively search over flux space for the ground state, so it is possible to go to much larger system size. We generate 40 sets of systems with plaquette numbers ranging from 9 to 1600. For each system size, 1000 distinct lattices are generated and the energy and gap size are calculated without phase twisting, since the effect is negligible for such large system sizes. As can be seen, for very small system size a small minority of gapless systems appear, however beyond around 20 plaquettes all systems had a stable fermion gap in the ground state.

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