Typicality at quantum-critical points

TL;DR

This paper explores the concept of typicality in quantum states at quantum-critical points, demonstrating how it can be used to efficiently identify critical behavior and universality classes using projector Monte Carlo simulations.

Contribution

It introduces a method to analyze quantum-critical points through typicality, optimizing simulation parameters to reduce finite-size effects and improve computational efficiency.

Findings

Critical points can be identified with increasing system size regardless of initial conditions.

Optimal projection time scaling can minimize finite-size corrections.

Typicality accelerates quantum criticality simulations across various numerical methods.

Abstract

We discuss the concept of typicality of quantum states at quantum-critical points, using projector Monte Carlo simulations of an $S=1/2$ bilayer Heisenberg antiferromagnet as an illustration. With the projection (imaginary) time $\tau$ scaled as $\tau =aL^z$, $L$ being the system length and $z$ the dynamic critical exponent (which takes the value $z=1$ in the bilayer model studied here), a critical point can be identified which asymptotically flows to the correct location and universality class with increasing $L$, independently of the prefactor $a$ and the initial state. Varying the proportionality factor $a$ and the initial state only changes the cross-over behavior into the asymptotic large-$L$ behavior. In some cases, choosing an optimal factor $a$ may also lead to the vanishing of the leading finite-size corrections. The observation of typicality can be used to speed up simulations of quantum criticality, not only within the Monte Carlo approach but also with other numerical methods where imaginary-time evolution is employed, e.g., tensor network states, as it is not necessary to evolve fully to the ground state but only for sufficiently long times to reach the typicality regime.