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

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 algorithm.

Random matrix theory and critical phenomena in quantum spin chains

“…This is reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) topological transitions in two-dimensional classical systems [27,28]. Recently, the link between BKT transitions and quantum phase transitions (QPTs) has been the object of intensive studies [29][30][31][32]. It would be interesting to understand the relation between BKT transitions and QPTs for the models we consider in this paper in more depth and we intend to pursue this direction of research in a future publication.…”

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

“…The advantage of the universal order parameter over local order parameters is the former's universality in characterizing QPTs in quantum lattice many-body systems, in the sense that the universal order parameter is not model dependent, in contrast with model-dependent order parameters. Reference [18] extends the universal order parameter from one-dimensional systems of infinite size to onedimensional finite-size systems. Herein we further extend the use of the universal order parameter to two-dimensional quan-tum systems.…”