The eigenstate thermalization hypothesis in constrained Hilbert spaces: A case study in non-Abelian anyon chains

Many phases of matter, including superconductors, fractional quantum Hall fluids, and spin liquids, are described by gauge theories with constrained Hilbert spaces. However, thermalization and the applicability of quantum statistical mechanics has primarily been studied in unconstrained Hilbert spaces. In this paper, we investigate whether constrained Hilbert spaces permit local thermalization. Specifically, we explore whether the eigenstate thermalization hypothesis (ETH) holds in a pinned Fibonacci anyon chain, which serves as a representative case study. We first establish that the constrained Hilbert space admits a notion of locality by showing that the influence of a measurement decays exponentially in space. This suggests that the constraints are no impediment to thermalization. We then provide numerical evidence that ETH holds for the diagonal and off-diagonal matrix elements of various local observables in a generic disorder-free nonintegrable model. We also find that certain nonlocal observables obey ETH.

Figure 1

(a) A series of $N+1$ quasiparticles in a 2D anyon system arranged in a line. Local measurements include probing the fusion channel of adjacent pairs of anyons, indicated here by black circles. (b) The Hilbert space for the Fibonacci chain with $N$ bonds. Vertical legs represent the $N+1$ anyons depicted in (a), while each horizontal bond represents the net fusion channel of all anyons to its left.

Figure 2

(a) A natural choice of Hamiltonian consists of energetically favoring one of the two fusion channels $Y_i=1,\tau$ for each pair of neighboring anyons. (b) Another possible interaction measures the net fusion channel $Z_i=1,\tau$ of each triple of neighboring anyons. In both cases the dependence of the coefficients $\alpha$ on the labels $\left\{X_j\right\}$ is left implicit.

Figure 3

Braiding of distant anyons in the chain. (a) The local operation $R$, which exchanges a pair of neighboring anyons. (b) Performing this operation subsequently on two pairs of adjacent anyons moves the anyon $i$ to position $i+2$.

Figure 4

The density of states (DOS) normalized by its maximum value as a function of energy for the five largest system sizes $N$. Here $E_{\infty }=\mathrm{Tr}H/\mathrm{Tr}1$ is the energy corresponding to infinite temperature in the canonical ensemble. The normalized density of states is approximately constant over the middle $1/3$ of the spectrum.

Figure 5

Level statistics ratio $r_m$ [see Eq. (26)] for the Hamiltonian (8). The main figure shows the probability distribution of $r_m$ at the largest system size $N=23$ studied here; the red line shows the predicted distribution from Ref. [70] for GOE statistics. Inset: Mean of the level statistics ratio as a function of $N$ (red dots) compared to the mean of the ideal GOE distribution $\approx 0.530$ (dashed line).

Figure 6

Spectrum averaged expectation values of the operators ${\mathrm{\Pi }}_{i;\alpha }^1$ for $i=N/2,N/2+1$ and $\alpha =x,2,3$. The dashed line in each case represents the predicted infinite temperature thermal value, which fits extremely well with the numerical data for all system sizes.

Figure 7

Distribution of $|\mathrm{\Delta }{\mathrm{\Pi }}_{N/2;2}^1|$ [see Eq. (28)] for $N=18$ to 23. As predicted by ETH, the distribution becomes narrower and more sharply peaked about 0 with increasing $N$.

Figure 8

Mean of $\left|\mathrm{\Delta }\mathcal{O}\right|$ vs the system size $N$ for different operators. The dashed line indicates the predicted scaling behavior $\left[\right|\mathrm{\Delta }\mathcal{O}\left|\right]\sim {\phi }^{-N/2}$, which is seen to be a good fit to the data. Numerical least-squares best fit slopes of each operator are given in Table 1.

Figure 9

Histogram of $log|\langle m|\mathcal{O}|n\rangle |$ for $\mathcal{O}={\mathrm{\Pi }}_{N/2;x}^1$ (top) and $\mathcal{O}={\mathrm{\Pi }}_{N/2;2}^1$ (bottom) between hundred states at the center of the spectrum for $N=19,21$ and 23. The distribution shifts linearly to the left with increasing $N$, and the shape corresponds to the logarithm of the absolute value of a Gaussian distributed variable.

Figure 10

Exponential of $[log\left(I_n^{\mathcal{O}}\right)]$ for the observables indicated in the legend. The black dashed line indicates the predicted scaling behavior $I_n^{\mathcal{O}}\sim {\phi }^{-N}$, which is seen to be a good fit to the data. Numerical least-squares best fit slopes of each operator are given in Table 2.

Figure 11

The function ${\left|f\left(\omega \right)\right|}^2$ [see Eq. (29)] as a function of $\left|\omega \right|$ for $\mathcal{O}={\mathrm{\Pi }}_{N/2;x}^1$ (circles) and $\mathcal{O}={\mathrm{\Pi }}_{N/2;2}^1$ (triangles) at mean energy $\overline{E}$ corresponding to infinite temperature.

Figure 12

(a) Expectation of the modulus of the braid operator $\left[\right|\langle B\left(l\right)\rangle \left|\right]$ as a function of separation $l$, for $N=14$ to 21. The modulus falls off exponentially with $l$ for small $l$, reaching a plateau at approximately $l\ge 4$, and then exponentially increases as $l$ approaches $N$. (b) The height of the plateau for $4\le l\le N-5$ as a function of system size $N$. This height falls off as ${\phi }^{-N/2}$, as expected for a random positive quantity (dashed line).

Figure 13

Histogram of phases $\varphi$ of the braid operator $B\left(l\right)$ in the middle third of the eigenvectors as a function of separation $l$ for $N=21$. For $l=1$, the phases are sharply peaked about $\pi$. For $l=5,10$, the phases are relatively uniformly distributed in the interval $(0,2\pi )$. For $l=N$ the phase takes on only one value of $8\pi /5\approx 1.6\pi$; this value is fixed by the boundary conditions.

Figure 14

$\left[\right|\mathrm{\Delta }|\langle B(N/4)\rangle |\left|\right]$ as a function of system size $N$. The decrease is consistent with a scaling proportional to ${\phi }^{-N/2}$ (dashed line), exactly as for local observables.

Figure 15

$\left[\langle {\mathrm{\Pi }}_{N/2,x}^1\rangle \right]$ for 500 parity even random vectors (green, dashed line) and the central third of eigenvectors (solid, red line), as a function of system size $N$. Restricting to even parity gives a larger fraction of states with ${\mathrm{\Pi }}_{N/2,x}^1=1$ for $N$ odd than for $N$ even, as explained in the text; this is reflected in the larger value of $\langle {\mathrm{\Pi }}_{N/2,x}^1\rangle$ for $N$ odd in both eigenvectors and random vectors.