We consider a model in which positive and negative particles with equal densities diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. The model depends on two parameters. Analytic calculations using quadratic algebras, inhomogeneous solutions of the mean-field equations and Monte-Carlo simulations suggest that the model has three phases. A pure phase in which one has three pinned blocks of only positive, negative particles and vacancies and in which translational invariance is broken. A mixed phase in which the current has a linear dependence on one parameter but is independent of the other one and of the density of the charged particles. In this phase one has a bump and a fluid. The bump (condensate) contains positive and negative particles only, the fluid contains charged particles and vacancies uniformly distributed. The mixed phase is separated from the disordered phase by a second-order phase-transition which has many properties of the Bose-Einstein phase-transition observed in equilibrium. Various critical exponents are found.
This paper shows how mesoscopic nonequilibrium thermodynamics can be applied to condensation and evaporation. By extending the normal set of thermodynamic variables with two internal variables, we are able to give a new theoretical foundation for a mechanism of condensation that has been proposed from molecular simulation results. The flux does not follow a simple Arrhenius formula for small activation energies which are relevant here.
We consider a model in which positive and negative particles diffuse in an asymmetric, CP-invariant way on a ring. The positive particles hop clockwise, the negative counterclockwise and oppositely-charged adjacent particles may swap positions. Monte-Carlo simulations and analytic calculations suggest that the model has three phases; a "pure" phase in which one has three pinned blocks of only positive, negative particles and vacancies, and in which translational invariance is spontaneously broken, a "mixed" phase with a non-vanishing current in which the three blocks are positive, negative and…
Unbroken continuous translational invariance is often taken as a basic assumption in discussions of spontaneous symmetry breaking (SSB), which singles out SSB of translational invariance itself as an exceptional case. We present a framework which allows us to treat translational invariance on the same footing as other symmetries. It is shown that existing theorems on SSB can be straightforwardly extended to this general case. As a concrete application, we analyze the Nambu-Goldstone modes in a (ferromagnetic) supersolid. We prove on the ground of the general theorems that the Bogoliubov mode s…
The ferromagnetic transition in the Ising model is the paradigmatic example of ergodicity breaking accompanied by symmetry breaking. It is routinely assumed that the thermodynamic limit is taken with free or periodic boundary conditions. More exotic symmetry-preserving boundary conditions, like cylindrical antiperiodic, are less frequently used for special tasks, such as the study of phase coexistence or the roughening of an interface. Here we show, instead, that when the thermodynamic limit is taken with these boundary conditions, a novel type of transition takes place below $T_c$ (the usual …
We study the phase diagram of a topological string-net type lattice model in the presence of geometrically frustrated interactions. These interactions drive several phase transitions that reduce the topological order, leading to a rich phase diagram including both Abelian ($\mathbb{Z}_2$) and non-Abelian ($\text{Ising}\times \overline{\text{Ising}}$) topologically ordered phases, as well as phases with broken translational symmetry. Interestingly, one of these phases simultaneously exhibits (Abelian) topological order and long-ranged order due to translational symmetry breaking, with non-trivi…
The ground states of two types of distorted mixed diamond chains with spins 1 and 1/2 are investigated using exact diagonalization, DMRG, and mapping onto low-energy effective models. In the undistorted case, the ground state consists of an array of independent spin-1 clusters separated by singlet dimers. The lattice distortion induces an effective interaction between cluster spins. When this effective interaction is antiferromagnetic, several Haldane phases appear with or without spontaneous translational symmetry breakdown (STSB). The transition between the Haldane phase without STSB and tha…
Bose-Einstein Condensation, Fluctuations and Spontaneous Symmetry Breaking
Università di Roma Sapienza（罗马大学）；Istituto dei Sistemi Complessi - CNR（复杂系统研究所-意大利国家研究委员会）；Università della Campania “Luigi Vanvitelli”（坎帕尼亚路易吉·范维泰利大学）；Università di Salerno（萨莱诺大学）
After recalling briefly the connection between spontaneous symmetry breaking and off-diagonal long range order for models of magnets a general proof of spontaneous breaking of gauge symmetry as a consequence of Bose-Einstein Condensation is presented. The proof is based on a rigorous validation of Bogoliubov's $c$-number substitution for the ${\bf k}={\bf 0}$ mode operator $a_{\bf 0}$.
We further study the stochastic model discussed in Ref.[2] in which positive and negative particles diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. We extend the analysis of this model to the case when the densities of the charged particles are not the same. The mean-field equations describing the model are coupled nonlinear differential equations that we call the two-component Burgers equations. We find roundabout weak solutions of these equations. These sol…
We investigate the steady-state and dynamical properties of a reciprocal many-body system consisting of self-propelled active particles with local alignment interactions that exists within a fan-shaped neighborhood of each particle. We find that the nonreciprocity can emerge in this reciprocal system once the spontaneous symmetry breaking is present, and the effective description of the system assumes a non-Hermitian structure that directly originates from the emergent nonreciprocity. This emergent nonreciprocity can impose strong influences on the properties the system. In particular, it can …
The fundamental problem is analized, 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 o…
We study a far-from-equilibrium system of interacting particles, hopping between sites of a 1d lattice with a rate which increases with the number of particles at interacting sites. We find that clusters of particles, which initially spontaneously form in the system, begin to move at increasing speed as they gain particles. Ultimately, they produce a moving condensate which comprises a finite fraction of the mass in the system. We show that, in contrast with previously studied models of condensation, the relaxation time to steady state decreases as an inverse power of ln L with system size L a…
We investigate what determines the shape of a particle condensate in situations when it emerges as a result of spontaneous breaking of translational symmetry. We consider a model with particles hopping between sites of a one-dimensional grid and interacting if they are at the same or at neighboring nodes. We predict the envelope of the condensate and the scaling of its width with the system size for various interaction potentials and show how to tune the shape from a delta-peak to a rectangular or a parabolic-like form.
We construct states describing Bose Einstein condensates at finite temperature for a relativistic massive complex scalar field with $|φ|^4$-interaction. We start with the linearised theory over a classical condensate and construct interacting fields by perturbation theory. Using the concept of thermal masses, equilibrium states at finite temperature can be constructed by the methods developed in arXiv:1306.6519 and arXiv:1502.02705. Here, the principle of perturbative agreement plays a crucial role. The apparent conflict with Goldstone's Theorem is resolved by the fact that the linearised theo…
The spontaneous breaking of a global discrete translational symmetry in the finite, lattice quantum sine-Gordon model is demonstrated by a density matrix renormalization group. A phase diagram in the coupling constant - inverse system size plane is obtained. Comparison of the phase diagram with a Woomany-Wyld finite-size scaling leads to an identification of the Berezinskii-Kosterlitz-Thouless transition in the quantum sine-Gordon model as the spontaneous symmetry breaking.
We discuss two different regimes of condensate formation in zero-range processes on networks: on a q-regular network, where the condensate is formed as a result of a spontaneous symmetry breaking, and on an irregular network, where the symmetry of the partition function is explicitly broken. In the latter case we consider a minimal irregularity of the q-regular network introduced by a single Q-node with degree Q>q. The statics and dynamics of the condensation depends on the parameter log(Q/q), which controls the exponential fall-off of the distribution of particles on regular nodes and the typ…
We suggest that the dynamical spontaneous symmetry breaking reported in a turbulent swirling flow at $Re=40~000$ by Cortet et al., Phys. Rev. Lett., 105, 214501 (2010) can be described through a continuous one parameter family transformation (amounting to a phase shift) of steady states and could be the analogue of the Goldstone mode of the vertical translational symmetry in an ideal system. We investigate a possible mechanism of emergence of such spontaneous symmetry breaking in a toy model of our out-equilibrium system, derived from its equilibrium counterpart. We show that the stationary st…
We analyze the role of the interplay between on-site interaction and inhomogeneous diffusion on the phenomenon of condensation in the zero-range process. We predict a universal phase diagram in the plane of two exponents, respectively characterizing the interactions and the diffusion disorder. The most prominent outcome is the existence of an extended condensed phase. In the latter phase, which originates as a result of the combined effects of strong enough interaction and weak enough disorder, a typical high-density configuration has a unique condensate on top of a critical background, but th…
We develop methods to probe the excitation spectrum of topological phases of matter in two spatial dimensions. Applying these to the Fibonacci string nets perturbed away from exact solvability, we analyze a topological phase transition driven by the condensation of non-Abelian anyons. Our numerical results illustrate how such phase transitions involve the spontaneous breaking of a topological symmetry, generalizing the traditional Landau paradigm. The main technical tool is the characterization of the ground states using tensor networks and the topological properties using matrix-product-opera…
