[Paper Review] Thermodynamic Quantum Time Crystals

K. B. Efetov|arXiv (Cornell University)|Feb 20, 2019

Quantum many-body systems5 references4 citations

TL;DR

This paper proposes thermodynamic quantum time crystals (TQTCs) as macroscopically stable states in equilibrium, where the order parameter oscillates periodically in both real and imaginary time, leading to non-decaying oscillations in two-time correlation functions. Unlike previous non-equilibrium time crystals, TQTCs are shown to be thermodynamically stable in large systems via a fermionic model with competing attraction and repulsion, offering a candidate for the pseudogap state in cuprates.

ABSTRACT

Investigation of states with a periodic time dependence of physical quantities attracts a considerable interest now. Although it has been proposed initially that such states (coined Quantum Time Crystals) might be macroscopic and thermodynamically stable, results of a more careful study of the problem seemed to indicate that quantum time crystals could be realized only in systems out of equilibrium. Here we show that, in contrast to the general belief, thermodynamically stable macroscopic quantum time crystals can exist. The order parameter of this new state of matter is periodic in both real and imaginary time but its average over the phase of the oscillations equals zero. At the same time, correlation functions characterizing physical quantities oscillate periodically in time without any decay. An alternative interpretation of the results is based on a concept of an operator order parameter. Calculations are performed for a rather general microscopic model that may in particular be suitable for describing the pseudogap state in superconducting cuprates.

Motivation & Objective

• To resolve the long-standing debate on whether macroscopic, thermodynamically stable quantum time crystals can exist in equilibrium.
• To demonstrate that a phase transition into a time-periodic state is possible in a general interacting fermion model with competing inter-band interactions.
• To show that such a state exhibits non-decaying oscillations in two-time correlation functions, observable in scattering experiments.
• To propose that this state may correspond to the elusive pseudogap phase in underdoped cuprates.
• To extend the concept of long-range order to include time and space, generalizing the notion of space-time order.

Proposed method

• Formulating a Hamiltonian for two-band interacting fermions with both inter-band attraction (via Σ₂) and repulsion (via Σ₁), resembling a BCS-type model but with electron-hole pair interactions.
• Applying a mean-field approach to derive the order parameter B(t), which exhibits periodic oscillations in real time t and imaginary time τ, with period related to temperature T.
• Using the Schwinger-Keldysh formalism to compute two-time correlation functions, such as N(t), which remain periodic and non-decaying in time.
• Demonstrating that while single-time averages vanish (preserving time-reversal symmetry), two-time correlation functions exhibit persistent oscillations due to phase-averaged contributions.
• Establishing a mapping to a harmonic oscillator Hamiltonian H_TC to analytically confirm the periodicity and non-decaying nature of the correlation functions.
• Deriving the inelastic scattering response χ(ω, q) to predict experimental signatures, showing discrete peaks at harmonic frequencies 2nγ, distinct from the elastic peak of the static DDW state.

Experimental results

Research questions

• RQ1 Can thermodynamically stable, macroscopic quantum time crystals exist in equilibrium systems, contrary to previous 'no-go' theorems?
• RQ2 What is the role of competing inter-band attraction and repulsion in stabilizing time-periodic order in a fermionic system?
• RQ3 How do two-time correlation functions behave in such a state, and can they exhibit non-decaying oscillations?
• RQ4 Can this state be experimentally distinguished from conventional broken-symmetry states like the d-density wave?
• RQ5 Is this state a viable candidate for the pseudogap phase in underdoped cuprate superconductors?

Key findings

• A thermodynamically stable quantum time crystal (TQTC) state is shown to exist in a macroscopic system with a periodic order parameter in both real and imaginary time, with oscillation period in imaginary time τ given by (mT)⁻¹.
• The time-averaged value of the order parameter vanishes, preserving time-reversal symmetry, but two-time correlation functions such as N(t) exhibit persistent, non-decaying oscillations in real time.
• The correlation function N(t) is periodic and does not decay, even in the thermodynamic limit V → ∞, with oscillation frequency remaining finite.
• The system's response in inelastic scattering experiments is predicted to show discrete peaks at harmonic frequencies 2nγ, in contrast to the elastic peak of the static d-density wave (DDW) state.
• The model reproduces key features of the pseudogap phase—such as loop current order and broken time-reversal symmetry without net magnetic moments—suggesting a possible identification of TQTC with the pseudogap state.
• The non-decaying oscillations in correlation functions are robust and can be analytically confirmed via mapping to a harmonic oscillator Hamiltonian, validating the stability of the time-periodic order.