The Partition Function Zeroes of Quantum Critical Points

TL;DR

This paper introduces a new approach to analyze the zeros of the partition function in quantum systems, enabling the extraction of critical properties like the central charge through a novel zeros expansion and finite-size scaling analysis.

Contribution

It presents a new space-Euclidean time zeros expansion for quantum systems that avoids branch points and allows direct measurement of the central charge via entanglement entropy.

Findings

The new zeros approach reproduces Lee-Yang scaling.

It quantifies the RG flow to IR fixed points.

It enables direct extraction of the central charge.

Abstract

The Lee-Yang theorem for the zeroes of the partition function is not strictly applicable to quantum systems because the zeroes are defined in units of the fugacity $e^{h\Delta\tau}$, and the Euclidean-time lattice spacing $\Delta\tau$ can be divergent in the infrared (IR). We recently presented analytic arguments describing how a new space-Euclidean time zeroes expansion can be defined, which reproduces Lee and Yang's scaling but avoids the unresolved branch points associated with the breaking of nonlocal symmetries such as parity. We now present a first numerical analysis for this new zeros approach for a quantum spin chain system. We use our scheme to quantify the renormalization group flow of the physical lattice couplings to the IR fixed point of this system. We argue that the generic Finite-Size Scaling (FSS) function of our scheme is identically the entanglement entropy of the lattice partition function and, therefore, that we are able to directly extract the central charge, $c$, of the quantum spin chain system using conformal predictions for the scaling of the entanglement entropy.