Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9709032 · Physical Review Letters

Evolution of Correlation Properties and Appearance of Broken Symmetry in the Process of Bose-Einstein Condensation

Yu. Kagan, Boris V. Svistunov

Published Nov 3, 1997• 142 citations
arXivHTMLPDF

Abstract

We consider the self-evolution of strongly nonequilibrium interacting Bose gas. Because of the mere fact of large (relative to unity) occupation numbers in the initial state the problem is directly reduced to the question of temporal evolution of the statistical matrix diagonal in the coherent-state representation. Strictly speaking, gauge symmetry is not destroyed even when the long-range ordering is completed, owing to the inevitable averaging over the ensemble. Actual symmetry breaking can occur only as a result of introducing a small term of the Hamiltonian violating conservation of particles, or quantum-mechanical measurement, also implying nonconservation of particles.

DOI
10.1103/physrevlett.79.3331

Related papers

Broken gauge symmetry in a Bose gas with constant particle number

Alexej Schelle

The interplay between spontaneously broken gauge symmetries and Bose-Einstein condensation has long been controversially discussed in science, since the equation of motions are invariant under phase transformations. Within the present model it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose-Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.

•Oct 26, 2014•0 citations

Bose-Einstein condensation and symmetry breaking

András Sütö

Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode $(\mathbf{k}=0)$ Hamiltonians and on the analysis of the associated microcanonical ensembles.

arXiv:cond-mat/0412440•Physical Review A•Feb 7, 2005•16 citations

Spontaneously Broken Gauge Symmetry in a Bose Gas with Constant Particle Number

Alexej Schelle

The interplay between spontaneously broken gauge symmetries and Bose–Einstein condensation has long been controversially discussed in science, since the equations of motion are invariant under phase transformations. Within the present model, it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose–Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.

arXiv:1410.7058•Fluctuation and Noise Letters•Jan 17, 2017•4 citations

Evolution of large-scale correlations in a strongly nonequilibrium Bose condensation process

Yu. Kagan, Boris V. Svistunov

The evolution of off-diagonal correlation functions (for the example of a single-particle density matrix) in the process of Bose condensation of an initially nonequilibrium interacting gas is discussed. Special attention is given to the character of the decay of the density matrix at distances much greater than the size of the quasicondensate region. Specifically, it is shown that the exponential decay of the density matrix necessarily presupposes the presence of a chaotic vortex structure — a tangle of vortex lines — in the system. When topological order is established but there is no off-diagonal long-range order, the density matrix decays with distance according to a power law.

Journal of Experimental and Theoretical Physics Letters•Apr 1, 1998•11 citations

Gauge-symmetry breaking and phase transitions in strongly and weakly interacting Bose systems

A. C. Olinto

A renormalizable parameter \ensuremath{\xi} (0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1) is introduced into the interacting Bose-system Hamiltonian such that the zero-mode amplitudes satisfy the commutation relation [${b}_{0}$,${b}_{0}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$]=1-\ensuremath{\xi}. The different phases of the system correspond to specific values of \ensuremath{\xi}. The normal component (phase I) is obtained when \ensuremath{\xi}=0 and there are two ordered states according to the strength of the interaction: Phase II' (0<\ensuremath{\xi}\ensuremath{\ll}1) is found in the strongly interacting system and is characterized by a broken gauge symmetry without Bose-Einstein condensation, while phase II'' (\ensuremath{\xi}=1), in the weakly interacting system, displays Bose-Einstein condensation. The continuity equation for the particle density holds in both ordered phases to within an infinitesimal term. A gauge transformation on the field operators shows that \ensuremath{\xi} ''twists'' the phase angle related to the zero mode, and a response to this twisting field is defined in terms of the generalized force \ensuremath{\chi}(T) associated with \ensuremath{\xi}. For the ideal Bose gas \ensuremath{\chi}(T) equals the condensate density and hence is simply related to the helicity modulus introduced by Fisher et al. A renormalization-group transformation shows that the dense system transition (I-II') gives rise to a two-thirds correlation-length critical exponent, whereas a unity crossover exponent appears in the dilute system transition (I-II''), in agreement with the experimental findings.

Physical review. B, Condensed matter•Feb 1, 1986•6 citations

The Question of Spontaneous Symmetry Breaking in Condensates

David W. Snoke, Andrew J. Daley

The question of whether Bose-Einstein condensation involves spontaneous symmetry breaking is surprisingly controversial. We review the theory of spontaneous symmetry breaking in ferromagnets, compare it with the theory of symmetry breaking in condensates, and discuss the different viewpoints on the correspondence to experiments. These viewpoints include alternative perspectives in which we can treat condensates with fixed particle numbers, and where coherence arises from measurements. This question relates to whether condensates of quasiparticles such as polaritons can be viewed as “real” condensates. Introduction Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles. It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Mølmer [1]).

arXiv:1507.07456•Cambridge University Press eBooks•Oct 4, 2018•3 citations
We consider the self-evolution of strongly nonequilibrium interacting Bose gas. Because of the mere fact of large (relative to unity) occupation numbers in the initial state the problem is directly reduced to the question of temporal evolution of the statistical matrix diagonal in the coherent-state representation. Strictly speaking, gauge symmetry is not destroyed even when the long-range ordering is completed, owing to the inevitable averaging over the ensemble. Actual symmetry breaking can occur only as a result of introducing a small term of the Hamiltonian violating conservation of particles, or quantum-mechanical measurement, also implying nonconservation of particles.
The interplay between spontaneously broken gauge symmetries and Bose-Einstein condensation has long been controversially discussed in science, since the equation of motions are invariant under phase transformations. Within the present model it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose-Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.
Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode (k=0)(\mathbf{k}=0)(k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.
The interplay between spontaneously broken gauge symmetries and Bose–Einstein condensation has long been controversially discussed in science, since the equations of motion are invariant under phase transformations. Within the present model, it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose–Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.
The evolution of off-diagonal correlation functions (for the example of a single-particle density matrix) in the process of Bose condensation of an initially nonequilibrium interacting gas is discussed. Special attention is given to the character of the decay of the density matrix at distances much greater than the size of the quasicondensate region. Specifically, it is shown that the exponential decay of the density matrix necessarily presupposes the presence of a chaotic vortex structure — a tangle of vortex lines — in the system. When topological order is established but there is no off-diagonal long-range order, the density matrix decays with distance according to a power law.
A renormalizable parameter \ensuremath{\xi} (0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1) is introduced into the interacting Bose-system Hamiltonian such that the zero-mode amplitudes satisfy the commutation relation [b0{b}_{0}b0​,${b}_{0}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$]=1-\ensuremath{\xi}. The different phases of the system correspond to specific values of \ensuremath{\xi}. The normal component (phase I) is obtained when \ensuremath{\xi}=0 and there are two ordered states according to the strength of the interaction: Phase II' (0<\ensuremath{\xi}\ensuremath{\ll}1) is found in the strongly interacting system and is characterized by a broken gauge symmetry without Bose-Einstein condensation, while phase II'' (\ensuremath{\xi}=1), in the weakly interacting system, displays Bose-Einstein condensation. The continuity equation for the particle density holds in both ordered phases to within an infinitesimal term. A gauge transformation on the field operators shows that \ensuremath{\xi} ''twists'' the phase angle related to the zero mode, and a response to this twisting field is defined in terms of the generalized force \ensuremath{\chi}(T) associated with \ensuremath{\xi}. For the ideal Bose gas \ensuremath{\chi}(T) equals the condensate density and hence is simply related to the helicity modulus introduced by Fisher et al. A renormalization-group transformation shows that the dense system transition (I-II') gives rise to a two-thirds correlation-length critical exponent, whereas a unity crossover exponent appears in the dilute system transition (I-II''), in agreement with the experimental findings.
The question of whether Bose-Einstein condensation involves spontaneous symmetry breaking is surprisingly controversial. We review the theory of spontaneous symmetry breaking in ferromagnets, compare it with the theory of symmetry breaking in condensates, and discuss the different viewpoints on the correspondence to experiments. These viewpoints include alternative perspectives in which we can treat condensates with fixed particle numbers, and where coherence arises from measurements. This question relates to whether condensates of quasiparticles such as polaritons can be viewed as “real” condensates. Introduction Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles. It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Mølmer [1]).