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/0412440 · Physical Review A

Bose-Einstein condensation and symmetry breaking

András Sütö

Published Feb 7, 2005• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.71.023602

Related papers

Spontaneous U(1) symmetry breaking and Bose–Einstein condensation

Qian Feng, HUANG HONG-BIN, Qi Guan-Xiao +1 more

Based on Bogoliubov's truncated Hamiltonian H B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term V 1/2 (λ a 0 +λ * a 0 + ) to H B , we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose–Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schrödinger eigenvalue equation and using the self-consistent condition.

Chinese Physics•Jul 1, 2006•6 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

Physical aspects of symmetry breaking in Bose gases at thermal equilibrium

Alexej Schelle

The theory of non-interacting Bose gases is supplemented by a numerical quantum field description with a two-dimensional non-local order parameter that allows the modeling of wave-like atomic correlations and interference effects in the limit of low atomic densities. From the present model, it is possible to explain symmetry aspects of non-interacting and very weakly interacting Bose gases in the limit of fluctuating particle numbers, like the forward propagation of time and the relation to the breaking and preservation of phase gauge symmetry in solids. In the present formalism, the propagation of one-directional time arises from the pre-defined and equivalent convergence of independent quantum fields towards the Boltzmann equilibrium, and it is shown that Glauber coherent states are related to the definition of the quantized field. Coherently coupling condensate and non-condensate parts as a direct consequence of the increasing quantum coherence time between the different quantum field components in the Bose gas from cooling to below the critical temperature, the present model describes symmetry breaking, which is originally known from the definition of a specific gauge field from Elitzur's theorem for local gauge fields, as a global physical rather than a purely formal mathematical process.

arXiv:2410.10893•Oct 12, 2024•0 citations

Bose-Einstein condensation and gauge symmetry breaking

Vyacheslav I. Yukalov

The fundamental problem is analyzed, the relation between Bose-Einstein condensation and spontaneous gauge symmetry breaking. This relation is largerly misunderstood in physics community. Numerous articles and books contain the statement that, though gauge symmetry breaking helps for describing Bose-Einstein condensation, but the latter, in principle, does not require any symmetry breaking. This, however, is not correct. The analysis is based on the known mathematical theorems. But in order not to overcomplicate the presentation and to make it accessible to all readers, technical details are often omitted here. The emphasis is made on the following basic general facts: Spontaneous breaking of gauge symmetry is the necessary and sufficient condition for Bose-Einstein condensation. Condensate fluctuations, in thermodynamic limit, are negligible. Their catastrophic behavior can arise only as a result of incorrect calculations, when a Bose-condensed system is described without gauge symmetry breaking. It is crucially important to employ the representative statistical ensembles equipped with all conditions that are necessary for a unique and mathematically correct description of the given statistical system. Only then one is able to develop a self-consistent theory, free of paradoxes.

arXiv:0711.0848•Laser Physics Letters•Apr 4, 2007•80 citations

1 Bose-Einstein condensates in symmetry breaking states

Yvan Castin, Christopher P. Herzog

We consider two models of interacting Bose gases: a gas of spin one particles in the ground state of a cubic box and a one-dimension Bose gas with contact interactions. We show how to calculate exact eigenstates of the corresponding N-body Hamiltonians. Both models share the property of not leading to the formation of a Bose-Einstein condensate, even at zero temperature, in the strict sense of the existence of a single one-particle state with a macroscopic population. We show that a lot of physical insight can be gained on these two model systems by using the usual Hartree-Fock mean field approach: in this approximation, that we test against the exact result, everything happens as if a single realization of the system was a Bose-Einstein condensate in a state phi breaking the rotational or translational symmetry, and varying in a random way for any new experimental realization.

•Jan 1, 2000•0 citations

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

Yu. Kagan, Boris V. Svistunov

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.

arXiv:cond-mat/9709032•Physical Review Letters•Nov 3, 1997•142 citations
Adding a gauge symmetry breaking field \ensuremath−\ensuremathνV(a0+a0∗)\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})\ensuremath−\ensuremathνV​(a0​+a0∗​) 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 limV\ensuremath→\ensuremath∞⟨a0⟩∕V=sgn\ensuremathνlimV\ensuremath→\ensuremath∞⟨a0∗a0⟩∕V{\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}limV\ensuremath→\ensuremath∞​⟨a0​⟩∕V​=sgn\ensuremathνlimV\ensuremath→\ensuremath∞​⟨a0∗​a0​⟩∕V​ where the averages are taken in the ground state or in thermal equilibrium states. Letting \ensuremathν\ensuremath{\nu}\ensuremathν 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.
Based on Bogoliubov's truncated Hamiltonian H B for a weakly interacting Bose system, and adding a U(1) symmetry breaking term V 1/2 (λ a 0 +λ * a 0 + ) to H B , we show by using the coherent state theory and the mean-field approximation rather than the c-number approximations, that the Bose–Einstein condensation(BEC) occurs if and only if the U(1) symmetry of the system is spontaneously broken. The real ground state energy and the justification of the Bogoliubov c-number substitution are given by solving the Schrödinger eigenvalue equation and using the self-consistent condition.
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 theory of non-interacting Bose gases is supplemented by a numerical quantum field description with a two-dimensional non-local order parameter that allows the modeling of wave-like atomic correlations and interference effects in the limit of low atomic densities. From the present model, it is possible to explain symmetry aspects of non-interacting and very weakly interacting Bose gases in the limit of fluctuating particle numbers, like the forward propagation of time and the relation to the breaking and preservation of phase gauge symmetry in solids. In the present formalism, the propagation of one-directional time arises from the pre-defined and equivalent convergence of independent quantum fields towards the Boltzmann equilibrium, and it is shown that Glauber coherent states are related to the definition of the quantized field. Coherently coupling condensate and non-condensate parts as a direct consequence of the increasing quantum coherence time between the different quantum field components in the Bose gas from cooling to below the critical temperature, the present model describes symmetry breaking, which is originally known from the definition of a specific gauge field from Elitzur's theorem for local gauge fields, as a global physical rather than a purely formal mathematical process.
The fundamental problem is analyzed, the relation between Bose-Einstein condensation and spontaneous gauge symmetry breaking. This relation is largerly misunderstood in physics community. Numerous articles and books contain the statement that, though gauge symmetry breaking helps for describing Bose-Einstein condensation, but the latter, in principle, does not require any symmetry breaking. This, however, is not correct. The analysis is based on the known mathematical theorems. But in order not to overcomplicate the presentation and to make it accessible to all readers, technical details are often omitted here. The emphasis is made on the following basic general facts: Spontaneous breaking of gauge symmetry is the necessary and sufficient condition for Bose-Einstein condensation. Condensate fluctuations, in thermodynamic limit, are negligible. Their catastrophic behavior can arise only as a result of incorrect calculations, when a Bose-condensed system is described without gauge symmetry breaking. It is crucially important to employ the representative statistical ensembles equipped with all conditions that are necessary for a unique and mathematically correct description of the given statistical system. Only then one is able to develop a self-consistent theory, free of paradoxes.
We consider two models of interacting Bose gases: a gas of spin one particles in the ground state of a cubic box and a one-dimension Bose gas with contact interactions. We show how to calculate exact eigenstates of the corresponding N-body Hamiltonians. Both models share the property of not leading to the formation of a Bose-Einstein condensate, even at zero temperature, in the strict sense of the existence of a single one-particle state with a macroscopic population. We show that a lot of physical insight can be gained on these two model systems by using the usual Hartree-Fock mean field approach: in this approximation, that we test against the exact result, everything happens as if a single realization of the system was a Bose-Einstein condensate in a state phi breaking the rotational or translational symmetry, and varying in a random way for any new experimental realization.
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.