Universal sub-leading terms in ground state fidelity

TL;DR

This paper investigates sub-leading constant terms in ground state fidelity near critical points, revealing their dependence on universality class and topology, with analytical and numerical validation in 1D and 2D systems.

Contribution

It uncovers a universal sub-leading O(1) term in ground state fidelity along critical manifolds, linking it to conformal field theory and topological properties.

Findings

Sub-leading terms depend only on the system's universality class.

The sub-leading term encodes topological information similar to topological entanglement entropy.

Numerical validation performed on the XXZ chain.

Abstract

The study of the (logarithm of the) {\em fidelity} i.e., of the overlap amplitude, between ground states of Hamiltonians corresponding to different coupling constants, provides a valuable insight on critical phenomena. When the parameters are infinitesimally close, it is known that the leading term behaves as $O(L^\alpha)$ ($L$ system size) where $\alpha$ is equal to the spatial dimension $d$ for gapped systems, and otherwise depends on the critical exponents. Here we show that when parameters are changed along a critical manifold, a sub-leading O(1) term can appear. This term, somewhat similar to the topological entanglement entropy, depends only on the system's universality class and encodes non-trivial information about the topology of the system. We relate it to universal $g$ factors and partition functions of (boundary) conformal field theory in $d=1$ and $d=2$ dimensions. Numerical checks are presented on the simple example of the XXZ chain.