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arXiv:1708.07097 · Physical review. E

Two-time correlation functions and the Lee-Yang zeros for an interacting Bose gas

Kh. P. Gnatenko, Alina Kargol, V. M. Tkachuk

Published Sep 11, 2017• 22 citations
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Abstract

Two-time correlation functions of a system of Bose particles are studied. We find a relation of zeros of the correlation functions with the Lee-Yang zeros of the partition function of the system. The obtained relation gives the possibility to observe the Lee-Yang zeros experimentally. A particular case of Bose particles on two levels is examined, and zeros of two-time correlation functions and Lee-Yang zeros of the partition function of the system are analyzed.

DOI
10.1103/physreve.96.032116

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Two-time correlation functions of a system of Bose particles are studied. We find a relation of zeros of the correlation functions with the Lee-Yang zeros of the partition function of the system. The obtained relation gives the possibility to observe the Lee-Yang zeros experimentally. A particular case of Bose particles on two levels is examined, and zeros of two-time correlation functions and Lee-Yang zeros of the partition function of the system are analyzed.
A systematic method to find the Yang-Lee partition function zeros of quantum many-body systems based on perturbation theory at finite temperatures was recently introduced in arXiv:2504.01880. This method identifies wave-vector and temperature-dependent complex virtual energies obtainable from the thermal electronic Greens function. The collection of virtual energies over all k⃗\vec{k}k yield the Yang-Lee zeroes. We apply this approach to the one-dimensional Hubbard model for different boundary conditions. We compare the results obtained by this method up to second-order in perturbation theory with the results found by exact diagonalization. We also propose a quantity that could be used for experimental detection of these zeros of a Hubbard ring. An example of the detection method is presented using exact diagonalization of the 8-site Hubbard ring.
We present a method for calculating the Yang-Lee partition function zeros of a translationally invariant model of lattice fermions, exemplified by the Hubbard model. The method rests on a theorem involving the usual single-electron self-energy ${\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i{\ensuremath{\omega}}_{n}|\ensuremath{\mu})$ with chemical potential $\ensuremath{\mu}$, in the imaginary time Matsubara formulation. The theorem maps the Yang-Lee zeros to a set of wave vector and spin labeled virtual energies ${\ensuremath{\xi}}_{\stackrel{P\vec}{k}\ensuremath{\sigma}}$. These, thermodynamically derived virtual energies, are solutions of the set of equations ${\ensuremath{\xi}}_{k\ensuremath{\sigma}}={\ensuremath{\varepsilon}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k})\ensuremath{-}\frac{1}{2}U+{\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i\frac{\ensuremath{\pi}}{\ensuremath{\beta}}|{\ensuremath{\xi}}_{k\ensuremath{\sigma}}+\frac{1}{2}U\ensuremath{-}i\frac{\ensuremath{\pi}}{\ensuremath{\beta}})=0$. Examples of the method in simplified situations are provided.
Zeros of partition functions, in particular Lee-Yang zeros, in a complex plane provide important information for understanding phase transitions. A recent discovery on the equivalence between the coherence of a central quantum system and the partition function of the environment in the complex plane enabled the experimental study of Lee-Yang zeros, with several pioneering experiments on spin systems. Lee-Yang zeros have not been observed in Bosonic systems. Here we propose an experimental scheme to demonstrate Lee-Yang zeros in Bosonic systems associated with a single trapped ion by introducing strong coupling between the spin and motion degrees of freedom, i.e. beyond the weak coupling Lamb-Dicke regime. Our scheme provides new possibilities for quantum simulation of the thermodynamics of Bosonic systems in the complex plane.