# Imaginary time crystal of thermal quantum matter

## TL;DR

This paper introduces the concept of an imaginary time crystal in thermal quantum matter, revealing a new phase characterized by crystalline order in imaginary time and unique temperature-dependent behaviors, supported by quantum Monte Carlo simulations.

## Contribution

It demonstrates the existence of an imaginary time crystal phase in thermal quantum systems, a novel state arising from bath-induced interactions and exhibiting distinctive properties.

## Key findings

- Presence of crystalline order in imaginary time space.
- Temperature-oscillating behavior of physical observables.
- Distinct ground-state and thermal properties from conventional systems.

## Abstract

Spontaneous symmetry breaking is responsible for rich quantum phenomena from crystalline structures to superconductivity. This concept was boldly extended to the breaking of time translation, opening an avenue to finding exotic phases of quantum matter with collective time modulation and correlation. Here we report that a thermally open quantum ensemble manifests in the dual space of imaginary time with crystalline ordering due to a bath-induced retarded interaction. Exact quantum Monte Carlo simulations are performed to show that this imaginary time crystal phase exhibits characteristic ground-state and thermal properties absent in conventional quantum manybody systems, especially the striking temperature-oscillating behavior of its physical observables.