Quantum phase transitions and ground-state correlations in BCS-like models

TL;DR

This paper provides exact analytical results for ground-state correlations and critical behavior in BCS-like lattice models of fermions, revealing direction-dependent correlation lengths and scaling near quantum-critical points in one- and two-dimensional systems.

Contribution

It offers explicit expressions for correlation functions and critical indices in two-dimensional BCS-like models, advancing understanding of quantum phase transitions beyond one dimension.

Findings

Exact phase diagrams with quantum-critical points are constructed.

Analytic asymptotic expressions for correlation functions are derived.

Direction-dependent correlation lengths and universal critical indices are identified.

Abstract

We study ground-state correlation functions in one- and two-dimensional lattice models of interacting spinful fermions - BCS-like models, which exhibit continuous quantum phase transitions. The considered models originate from a two-dimensional model of d-wave superconductivity proposed by Sachdev. Due to the exact diagonalizability of the considered models in any dimensionality, exact phase diagrams, with several kinds of quantum-critical points, are constructed and closed-form analytic expressions for two-point correlation functions are obtained. In one- and two-dimensional cases we provide analytic expressions for the asymptotic behavior of those correlation functions at large distances and in neighborhoods of quantum-critical points. The novelty of our results is that in two-dimensions explicit expressions for direction-dependent correlation lengths in terms of model parameters and the values of direction-dependent universal critical indices $\nu$, that characterize the divergence of correlation lengths on approaching critical points, are determined. Moreover, specific scaling properties of correlation functions with respect to parameters of underlying Hamiltonians are revealed. Besides enriching the knowledge of properties of lattice fermion systems exhibiting continuous quantum phase transitions, especially in two dimensions, our results open new possibilities of testing unconventional methods of studying quantum phase transitions, as the promising fidelity approach or the entanglement approach, beyond one-dimension and beyond the realm of paradigmatic XY and Ising chains in transverse magnetic fields.