Lee-Yang theory of Bose-Einstein condensation

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Lee-Yang theory of Bose-Einstein condensation

Fredrik Brange, Tuomas Pyhäranta, Eppu Heinonen, Kay Brandner, and Christian Flindt

Phys. Rev. A 107, 033324 – Published 23 March 2023

Abstract

Bose-Einstein condensation happens as a gas of bosons is cooled below its transition temperature, and the ground state becomes macroscopically occupied. The phase transition occurs in the thermodynamic limit of many particles. However, recent experimental progress has made it possible to assemble quantum many-body systems from the bottom up, for example, by adding single atoms to an optical lattice one at a time. Here, we show how one can predict the condensation temperature of a Bose gas from the energy fluctuations of a small number of bosons. To this end, we make use of recent advances in Lee-Yang theories of phase transitions, which allow us to determine the zeros and the poles of the partition function in the complex plane of the inverse temperature from the high cumulants of the energy fluctuations. By increasing the number of bosons in the trapping potential, we can predict the convergence point of the partition-function zeros in the thermodynamic limit, where they reach the inverse critical temperature on the real axis. Using fewer than 100 bosons, we can estimate the condensation temperature for a Bose gas in a harmonic potential in two and three dimensions, and we also find that there is no phase transition in one dimension as one would expect.

Received 2 February 2023
Accepted 1 March 2023

DOI:https://doi.org/10.1103/PhysRevA.107.033324

Physics Subject Headings (PhySH)

Bose-Einstein condensates Phase transitions

Trapped atoms

Lee-Yang & Fisher zeroes

Statistical Physics & ThermodynamicsAtomic, Molecular & Optical

Authors & Affiliations

Fredrik Brange ¹, Tuomas Pyhäranta ¹, Eppu Heinonen ¹, Kay Brandner², and Christian Flindt ¹

¹Department of Applied Physics, Aalto University, 00076 Aalto, Finland
²School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

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Vol. 107, Iss. 3 — March 2023

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Figure 1
Lee-Yang theory of Bose-Einstein condensation. (a) A two-dimensional harmonic potential $V (x, y)$ containing a gas of bosons. (b) Fraction of bosons in the ground state of a two-dimensional potential as a function of the temperature $T$ . Below the critical temperature $T_{c}$ , the ground state becomes macroscopically occupied. (c) Zeros of the canonical partition function in the complex plane of the inverse temperature, $β = 1 / (k_{B} T)$ , for different dimensions. The zeros correspond to $N = 20$ bosons and were obtained with a high-temperature expansion of the partition function. The red dots illustrate potential convergence points $β_{c}$ in the thermodynamic limit.
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Figure 2
Three-dimensional harmonic potential. (a) Partition-function zeros in the complex plane of the inverse temperature obtained numerically for $N = 2, ..., 20$ bosons. (b) Partition-function zeros closest to the inverse temperature $β = 1.15 β_{c}$ , marked with a cross, obtained with the cumulant method for $N = 2, ..., 100$ bosons. Here, we have used $m = 7$ different cumulant orders up to order 17. (c) The real and (negative) imaginary parts of the zeros as a function of $1 / N$ . The red dots are the extrapolated convergence points in the thermodynamic limit, which we determine based on the scaling ansatz in Eq. (25).
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Figure 3
Two-dimensional harmonic potential. (a) Partition-function zeros in the complex plane of the inverse temperature obtained numerically for $N = 2, ..., 20$ bosons. (b) Partition-function zeros closest to the inverse temperature $β = 1.15 β_{c}$ , marked with a cross, obtained with the cumulant method for $N = 2, ..., 100$ bosons. Here, we have used $m = 7$ different cumulant orders up to order 17. (c) The real and (negative) imaginary parts of the zeros as a function of $1 / N$ . The red dots are the extrapolated convergence points in the thermodynamic limit, which we determine based on the scaling ansatz in Eq. (25).
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Figure 4
One-dimensional harmonic potential. (a) Poles of the partition function according to Eq. (27) for $N = 6$ bosons with $β_{0} = 1 / (ℏ ω_{0})$ . (b) Poles closest to the origin obtained with the cumulant method for $N = 2, ..., 6$ bosons. Here, we have used $m = 6$ different cumulant orders up to order 25, with a tolerance threshold of 0.25. (c) The real and (negative) imaginary parts of the poles as a function of $1 / N$ together with a linear extrapolation, shown by a dashed line.
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Figure 5
Partition-function zeros from energy fluctuations. (a) Distributions $P_{N} (U)$ of the energy for a three-dimensional harmonic potential with $N = 10, 20, 30$ particles. (b) Energy cumulants of orders $n = 1, \dots, 14$ , obtained from $10^{7}$ Monte Carlo simulations. (c) Partition-function zeros obtained by averaging over 1000 sets of Monte Carlo simulations are shown by solid circles. The exact zeros are shown with open circles. For all of these calculations, the inverse temperature is $β = 1.15 β_{c}$ .
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Figure 6
Bose-Einstein condensation of a finite number of particles. We show the fraction of particles in the ground state as a function of the temperature and with different numbers of particles in the trapping potential. The three panels correspond to different dimensions, and the temperature $T_{c}$ depends on the number of particles according to Eqs. (5), (24), and (A4). The black lines are given by Eq. (A3), which holds in the thermodynamic limit. For all three dimensions, the fraction of particles in the ground state can become large with a finite number of particles. However, for the one-dimensional trapping potential, the transition temperature $T_{c}$ goes to zero in the thermodynamic limit, and there is no phase transition.
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