Energy Correlations In Random Transverse Field Ising Spin Chains

TL;DR

This paper analyzes energy correlations in random transverse-field Ising chains using an advanced real-space RG method, revealing distinct decay behaviors at and away from the critical point, with implications for understanding quantum criticality.

Contribution

It extends existing RG techniques with unitary transformations to compute energy correlations, providing new insights into their decay behaviors at criticality.

Findings

Average correlations decay exponentially away from criticality.

Typical correlations decay with a square root dependence on the primary correlation length.

At criticality, average correlations decay sub-exponentially as e^{-const·L^{1/3}}.

Abstract

The end-to-end energy - energy correlations of random transverse-field quantum Ising spin chains are computed using a generalization of an asymptotically exact real-space renormalization group introduced previously. Away from the critical point, the average energy - energy correlations decay exponentially with a correlation length that is the same as that of the spin - spin correlations. The typical correlations, however, decay exponentially with a characteristic length proportional to the square root of the primary correlation length. At the quantum critical point, the average correlations decay sub-exponentially as $\bar{C_{L}}\sim e^{-const\cdot L^{1/3}}$, whereas the typical correlations decay faster, as $\sim e^{-K\sqrt{L}}$, with $K$ a random variable with a universal distribution. The critical energy-energy correlations behave very similarly to the smallest gap, computed previously; this is explained in terms of the RG flow and the excitation structure of the chain. In order to obtain the energy correlations, an extension of the previously used methods was needed; here this was carried out via RG transformations that involve a sequence of unitary transformations.