[Paper Review] Odd-frequency Two Particle Bose-Einstein Condensate

A. V. Balatsky|arXiv (Cornell University)|Sep 17, 2014

Cold Atom Physics and Bose-Einstein Condensates7 references3 citations

TL;DR

This paper introduces a novel class of Bose-Einstein condensate (BEC) characterized by odd-frequency two-boson correlations, termed odd-frequency BEC, which exhibits long-range order despite vanishing single-particle expectation values. The authors establish its existence via symmetry classification and Matsubara time-ordered correlation functions, showing it breaks global U(1) gauge symmetry and behaves as a bosonic analog of odd-frequency superconductivity.

ABSTRACT

We introduce the concept of the {\em odd-frequency} Bose Einstein Condensate (BEC), characterized by the odd frequency/time two-boson expectation value. To illustrate the concept of odd frequency BEC we present simple classification of pair boson condensates that explicitly permits this state. We point qualitative differences of odd-frequency BEC with conventional BEC and introduce the order parameter and wave function for the odd-frequency BEC.

Motivation & Objective

• To propose a new class of Bose-Einstein condensate based on odd-frequency two-boson correlations, distinct from conventional single-particle BEC.
• To extend the concept of nematic order to bosonic systems by introducing an odd-frequency BEC nematic state.
• To establish the theoretical existence of such a state through symmetry analysis and time-ordered correlation functions in Matsubara formalism.
• To demonstrate that this state breaks global U(1) gauge symmetry and supports macroscopic coherence despite zero equal-time single-boson expectation.
• To explore potential realizations in multilayer systems and proximity-induced composite condensates, suggesting feasible experimental platforms.

Proposed method

• Uses Matsubara time-ordered two-boson correlation function $ D_{ab}({\bf r}, \tau) = \langle T_\tau b_a({\bf r}, \tau) b_b({\bf r}', \tau') \rangle $ to define the order parameter for the odd-frequency BEC.
• Classifies possible BEC states via discrete symmetries: parity (P), time reversal (T), and orbital permutation (O), identifying odd-frequency states as those with $ P = -, T = -, O = + $, satisfying $ PTO = +1 $.
• Defines the odd-frequency BEC as a state where $ D_{ab}({\bf r}, \tau) $ is odd in $ \tau $, leading to $ D_{ab}({\bf r} | \tau=0) = 0 $, yet the function exhibits long-range order in center-of-mass coordinates.
• Introduces a global U(1) gauge symmetry breaking: $ b_a \to e^{i\theta} b_a $, $ D_{ab} \to e^{2i\theta} D_{ab} $, confirming macroscopic coherence.
• Analyzes a two-layer BEC model with interlayer tunneling $ t $, showing that odd-frequency intralayer pairing arises from linear time dependence $ \partial_\tau b_a \sim t b_b $, leading to order parameter $ d_{aa}({\bf r}) \sim t \langle b_a b_b \rangle $.
• Demonstrates equivalence between odd-frequency intralayer BEC and even-frequency interlayer BEC, with wave function $ \Psi \sim \exp(\lambda \sum_{\bf r} b^\dagger_a({\bf r}) b^\dagger_b({\bf r})) $, confirming coherence.

Experimental results

Research questions

• RQ1 Can a Bose-Einstein condensate exist without a finite single-boson field expectation value?
• RQ2 What is the role of time-ordered two-body correlations in defining new types of macroscopic quantum order in bosonic systems?
• RQ3 How can odd-frequency symmetry in time lead to a stable, coherent many-body state despite vanishing equal-time correlators?
• RQ4 What are the symmetry constraints and classification rules for such odd-frequency BEC states?
• RQ5 Can odd-frequency two-boson BEC emerge in realistic systems such as multilayered or proximate BEC structures?

Key findings

• The paper establishes the theoretical existence of an odd-frequency two-boson BEC as a distinct quantum phase, characterized by a time-odd two-body correlation function $ D_{ab}({\bf r}, \tau) $, even when $ \langle b_a({\bf r}, \tau=0) \rangle = 0 $.
• The odd-frequency BEC breaks global U(1) gauge symmetry, transforming as $ D_{ab} \to e^{2i\theta} D_{ab} $, confirming its macroscopic coherence and condensate nature.
• A two-layer BEC model with interlayer tunneling $ t $ realizes the odd-frequency BEC via linear time dependence $ \partial_\tau b_a \sim t b_b $, leading to a non-zero order parameter $ d_{aa}({\bf r}) \sim t \langle b_a b_b \rangle $.
• The odd-frequency intralayer BEC is shown to be equivalent to an even-frequency interlayer BEC, with a coherent paired boson wave function $ \Psi \sim \exp(\lambda \sum_{\bf r} b^\dagger_a b^\dagger_b) $, confirming its stability.
• The study identifies proximity effects and interlayer hybridization as promising routes to realize such a state experimentally, particularly in tunable ultracold atomic systems with multiple bosonic species.
• The classification of BEC states via $ P, T, O $ symmetries identifies a new class of odd-frequency nematic BEC, extending the landscape of odd-frequency quantum orders beyond superconductors and spin nematics.