Extracting average properties of disordered spin chains with translationally invariant tensor networks

TL;DR

This paper introduces a tensor-network framework to compute disorder-averaged properties of random spin chains without explicit disorder sampling by representing the disorder-averaged density matrix as a translationally invariant MPO. The authors encode disorder with ancilla qudits, perform imaginary-time evolution, and maintain correct normalization via a variational MPO inversion of a normalization operator, enabling thermodynamic-limit calculations with manageable bond dimensions. Benchmarking on the infinite-randomness critical point of the random transverse-field Ising model demonstrates correct activated scaling, with the correlation length following $\xi \sim (\ln \beta)^2$ at criticality and spin correlations saturating to the infinite-randomness exponent. Additionally, they examine the distribution of correlation lengths through Lyapunov exponents, revealing heavy-tailed behavior characteristic of rare-region physics and highlighting the distinction between typical and average correlation lengths. Overall, the study provides a proof-of-principle for efficiently capturing disorder-averaged physics in random quantum spin chains using translationally invariant MPOs, with clear avenues for methodological refinements and ground-state extensions.

Abstract

We develop a tensor network-based method for calculating disorder-averaged expectation values in random spin chains without having to explicitly sample over disorder configurations. The algorithm exploits statistical translation invariance and works directly in the thermodynamic limit. We benchmark our method on the infinite-randomness critical point of the random transverse field Ising model.

Figures (8)

• Figure 1: (a) MPO representation of $\tilde{\rho}(\tau)$. Horizontal lines represent the virtual bonds. Vertical black full lines represent the physical spin indices. Red dashed lines act on the disorder qudits. (b) The MPO $\Lambda$ is obtained by tracing out the physical spin indices in $N(\tau)e^{-(\tau+\Delta \tau)H}$. (c) An ansatz MPO $\Lambda^{-1}_M$ (filled circles) is used to approximately invert $\Lambda$.
• Figure 2: Left: Correlation length of the average correlation function as a function of $(\ln\beta)^2$ for different step sizes $\Delta \tau$. The black line corresponds to a linear fit of the data for $\Delta\tau = 0.1$ and $\beta \in [10,30]$. Middle: Correlation length as a function of $(\ln\beta)^2$ for different bond dimensions. The black line represents a linear fit of the data for $D = 100$ and $\beta \in [30,50]$. Right: Correlation length as a function $(\ln\beta)^2$ for different numbers of disorder values $N_D = N^2$, obtained with $\Delta\tau = 0.05$. The black line represents a linear fit of the data with $N = 4$ and $\beta \in [20,30]$. In all figures the correlation length was extracted from the transfer matrix eigenvalues.
• Figure 3: Left: The correlation functions for different correlation lengths with $D = 80$ and $\Delta\tau = 0.05$. The data sets were obtained by taking $J_n$ and $h_n$ uniformly distributed between $[0.7,1.3]$ with $N_D=9$. The black line represents the algebraic decay of the infinite-randomness fixed point and the dashed line represents the algebraic decay of the clean critical point. Right: The correlation functions for different correlation lengths with $D = 80$ and $\Delta\tau = 0.05$. The data sets were obtained by taking $J_n$ and $h_n$ uniformly distributed between $[0.5,1.5]$ with $N_D=9$.
• Figure 4: Left: Distribution of the first lyapunov exponent. Middle: Distribution of the second lyapunov exponent. Right: Distribution of the correlation length. All the data was gathered by sampling the MPO for $\beta = 40$, $D=80$ and $\Delta\tau = 0.05$. To compute the lyapunov exponents we used $L=100$ and 20000 samples.
• Figure 5: (a) Operator entanglement spectrum of the disorder averaged density matrix $\rho(\beta)$. (b) Entanglement spectrum of the partition function MPO $\Lambda$. The data was acquired with $D=80$ and $\Delta\tau = 0.05$.