Results

The Ground State Flux Sector

Here I will discuss the numerical evidence that our guess for the ground state flux sector is correct, it relies on three key numerical observations arguments:

First we fully eumerate the flux sectors of ~25,000 periodic systems with a disordered unit cell of up to 16 plaquettes ($2^{16-1}$ sectors). Going to larger system sizes in impractical because of the exponential sclaling. However, as discussed earlier, finite size effects play a large role for small systems¹. To get around this we look at periodic systems with amorphous unit cells. This reduces the finite size effects but we can use Bloch’s theorem to diagonalise periodic systems with only a linear penalty in system area.

Looking at periodic systems comes at the expense of removing longer-range disorder from our lattices so we bolster this by comparing the behaviour of periodic lattice with amorphous to unit cells to fully amorphous lattice as we scale the size of the unit cell. We show that the energetic effect of introducing perodicity scales away as we go to larger system sizes.

From these two observations we argue that the results for small periodic systems generalise to large amorphous systems. We perform this analysis for both the isotropic point ($J^\alpha = 1$), as well as in the toric code phase ($J^x = J^y = 0.25, J^z = 1$).

In the isotropic case ($J^\alpha = 1$), our conjecture correctly predicted the ground state flux sector for all of the lattices we tested.

For the toric code phase ($J^x, J^y = 0.25, J^z = 1$) all but around ($\sim 0.5 \%$) lattices had ground states conforming to our conjecture. In these cases, the energy difference between the true ground state and our prediction was on the order of $10^{-6} J$. It is unclear whether this is a finite size effect or something else.

Spontaneous Chiral Symmetry Breaking

The spin Kitaev Hamiltonian is real and therefore has time reveral symmetry. However, the flux $\phi_p$ through any plaquette with an odd number of sides has imaginary eigenvalues $\pm i$. Further we have shown that the ground state sector induces a relatively regular pattern for the imaginary fluxes with only a global two-fold degeneracy.

Thus, states with a fixed flux sector spontaneously break time reversal symmetry. This was first described by Yao and Kivelson for a translation invariant Kitaev model with odd sided plaquettes².

Thus we have flux sectors that come in degenerate pairs, where time reversal is equivalent to inverting the flux through every odd plaquette, a general feature for lattices with odd plaquettes ^3,4. This spontaneously broken symmetry avoids the need to explicitly break TRS with a magnetic field term as is done in the original honeycomb model.

Ground State Phase Diagram

As previously discusssed, the standard Honeycomb model has a Abelian, gapped phase in the anisotropic region and is gapless in the isotropic region. The introduction of a magnetic field breaks the chiral symmetry, leading to the isotropic region becoming a gapped, non-Abelian phase.

Similar to the Kitaev Honeycomb model with a magnetic field, we find that this model is only gapless along critical lines, see ~1 (Left). Interestingly, the gap closing exists in only one of the four topological sectors, though this is certainly a finite size effect as the sectors must become degenerate in the thermodynamic limit.

In the honeycomb model, the phase boundaries are located on the straight line $|J^x| = |J^y| + |J^x|$ and permutations of $x,y,z$, shown as dotted line on ~1 (Right). We find that on the amorphous lattice these boundaries exhibit an inward curvature, similar to honeycomb Kitaev models with flux⁵ or bond⁶ disorder.

Figure 1: (Center) We choose an energy scale for the Hamiltonian by setting J_x + J_y + J_z = 1. This intersects a plane with the unit cube spanned by J_\alpha \in [0,1], giving a triangle with corners (1,0,0), (0,1,0), (0,0,1). To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them has a true gap closing. (Right) The Abelian A_\alpha phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the chern number \nu changes from 0 t0 \pm 1 from the A phases to the B phase. — Figure 1: (Center) We choose an energy scale for the Hamiltonian by setting $J_x + J_y + J_z = 1$. This intersects a plane with the unit cube spanned by $J_\alpha \in [0,1]$, giving a triangle with corners $(1,0,0), (0,1,0), (0,0,1)$. To compute critical lines efficiently in this space we evaluate the order parameter of interest along rays shooting from the corners. The ray highlighted in red defines the values of J used for the left figure. (Left) The fermion gap as a function of J for an amorphous system with 20 plaquettes, where the x axis is the position on the red line in the central figure from 0 to 1. For finite size systems the four topological sectors are not degenerate and only one of them has a true gap closing. (Right) The Abelian $A_\alpha$ phases of the model and the non-Abelian B phase separated by critical lines where the fermion gap closes. Later we will show that the chern number $\nu$ changes from $0$ t0 $\pm 1$ from the A phases to the B phase.

Is it Abelian or non-Abelian?

The two phases of the amorphous model are clearly gapped, though see later for a finite size scaling check on this.

The next question is: do these phases support Abelian or non-Abelian statistics? To answer that we turn to Chern numbers and markers. As discussed earlier the Chern number is a quantity intimately linked to both the topological properties and the anyonic statistics of a model. The Abelian/non-Abelian character of a model is linked to its Chern number citation. However the Chern number is only defined for the translation invariant case.

A family of generalisations to amorphous systems exist⁷ called local topological markers. We use the crosshair marker⁸ here to assess the Abelian/non-Abelian character of the phases.

Like the honeycomb model, the amorphous model retains an Abelian gapped phase in the anisotropic region with $\nu=0$. This phase is the amorphous analogue of the abelian toric-code quantum spin liquid⁹.

The isotropic region has $\nu=\pm1$ so is a non-Abelian chiral spin liquid (CSL) similar to that of the Yao-Kivelson model³. Hereafter we focus our attention on this phase.

Figure 2: (Center) The crosshair marker8, a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the center) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime J_\alpha = 1 in red has \nu = \pm 1 implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has \nu = \pm 0 implying it has Abelian statistics. (Right) Extending this analysis to the whole J_\alpha phase diagram with fixed r = 0.3 nicely confirms that the isotropic phase is non-Abelian. — Figure 2: (Center) The crosshair marker⁸, a local topological marker, evaluated on the Amorphous Kitaev Model. The marker is defined around a point, denoted by the dotted crosshair. Information about the local topological properties of the system are encoded within a region around that point. (Left) Summing these contributions up to some finite radius (dotted line here, dotted circle in the center) gives a generalised version of the Chern number for the system which becomes quantised in the thermodynamic limit. The radius must be chosen large enough to capture information about the local properties of the lattice while not so large as to include contributions from the edge states. The isotropic regime $J_\alpha = 1$ in red has $\nu = \pm 1$ implying it supports excitations with non-Abelian statistics, while the anisotropic regime in orange has $\nu = \pm 0$ implying it has Abelian statistics. (Right) Extending this analysis to the whole $J_\alpha$ phase diagram with fixed $r = 0.3$ nicely confirms that the isotropic phase is non-Abelian.

Chern Number and Edge Modes

The QSLs separated by these lines are distinguished by a real-space analogue of the Chern number^10,11. A similar topological number was discussed by Kitaev on the honeycomb lattice¹ that we shall use here with a slight modification^7,8. 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 system⁹.

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 correspondence¹². 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 vortices⁵ or exchange disorder⁶.

Anderson Transition to a Thermal Metal

$Figure 4: Within a flux sector, the fermion gap \Delta_f measures the energy between the fermionic ground state and the first excited state. This graph shows the fermion gap as a function of system size for the ground state flux sector and for a configuration of random fluxes. We see that the disorder induced by an putting the Kitaev model on an amorphous lattice does not close the gap in the ground state. The gap closes in the flux disordered limit is good evidence that the system transitions to a gapless thermal metal state at high temperature. Each point shows an average over 100 lattice realisations. System size L is defined \sqrt{N} where N is the number of plaquettes in the system. Error bars shown are 3 times the standard error of the mean. The lines shown are fits of \tfrac{\Delta_f}{J} = aL ^ b with fit parameters: Ground State: a = 0.138 \pm 0.002, b = -0.0972 \pm 0.004 Random Flux Sector: a = 1.8 \pm 0.2, b = -2.21 \pm 0.03$

Figure 4: 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: $a = 1.8 \pm 0.2, b = -2.21 \pm 0.03$

a thermal-induced Anderson transition to a thermal metal phase¹³.

An Ising non-Abelian anyon is formed by Majorana zero-modes bound to a topological defect¹⁴. 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^15–17. Disorder can induce a finite density of anyons whose hybridization lead to a macroscopically degenerate state known as thermal metal¹⁵. 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 temperature¹³.

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^18,19 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.^13,20. 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.¹³ I have the impression that [fig:DOS_Oscillations](b) on the top is very similar to Fig. 3 of¹³. 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

Conclusion

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.

Discussion

Failure of the ground state conjecture

We did find a small number of lattices for the ground state conjecture did not correctly predict the true ground state flux sector. I see two possibilities for what could cause this.

Firstly it could be a a finite size effect that is somehow amplfied by certain rare lattice configurations. It would be interesting to try to elucidate what lattice features are present when the ground state conjecture fails.

Alternatively, it might be telling that the ground state conjecture failed in the toric code phase where the couplings are anisotropic. Clearly the colouring does not matter much in the isotropic phase. However an avenue that I did not explore was whether the particular choice of colouring for a lattice affects the physical properties in the toris code phase. It is possible that some property of the particular colouring chosen is what leads to failure of the ground state conjecture here.

Full Monte Carlo

Outlook

Experimental Realisations and Signatures

The next step is to search for an experimental realisation in amorphous Kitaev materials, which can be created from crystalline ones using several methods^21–23.

Following the evidence for an induced chiral spin liquid phase in crystalline Kitaev materials^24–27, it would be interesting to investigate if a similar state is produced on its amorphous counterpart.

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.

Besides the usual half-quantized signature on thermal Hall effect^24–27, such a CSL could be also characterized using local probes such as spin-polarized scanning-tunneling microscopy^28–30. The same probes would also be useful to manipulate non-Abelian anyons³¹, thereby implementing elementary operations for topological quantum computation. Finally, the thermal metal phase can be diagnosed using bulk heat transport measurements¹⁴.

Generalisations

This work could be generalized in several ways.

Introduction of symmetry allowed perturbations on the model^32–36.

Generalizations to higher-spin models in random networks with different coordination numbers^2,37–46

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