Universal Order Parameters and Quantum Phase Transitions: A Finite-Size Approach

TL;DR

This paper introduces a universal order parameter construction for quantum phase transitions in finite systems, linking fidelity measures to energy gaps, demonstrated on the Potts model using DMRG.

Contribution

It presents a novel finite-size scaling method for quantum phase transitions based on $H$-orthogonality and fidelity, applicable to various models.

Findings

Universal order parameter constructed from $H$-orthogonality.

Connection established between fidelity per site and energy gap.

Validated method on the quantum Potts model with DMRG.

Abstract

We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems. The method exploits the $H$-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms. An explicit connection is established between the fidelity per site between two $H$-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system. The physical information encoded in this gap arising from finite-size fluctuations clarifies the origin of the universal order parameter.We demonstrate the procedure for the one-dimensional quantum formulation of the $q$-state Potts model, for $q=2,3,4$ and 5, as prototypical examples, using finite-size data obtained from the density matrix renormalization group (DMRG) algorithm.