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arXiv:1007.4348

The symmetry breaking phenomenon in anharmonic oscillator model

Antonio Carlos Mastine, Paulo Laerte Natti, Érica Regina Takano Natti

Published Jul 25, 2010• 0 citations
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Abstract

In this article a non-perturbative time-dependent technique is used to treat the initial value problem, in Quantum Mechanics context, for a non-equilibrium self-interacting fermionic system in the presence of an external magnetic field. Particularly, in mean-field regime, we study the dynamical symmetry breaking phenomenon, identifying the physical processes associated.

DOI
10.48550/arxiv.1007.4348

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In this article a non-perturbative time-dependent technique is used to treat the initial value problem, in Quantum Mechanics context, for a non-equilibrium self-interacting fermionic system in the presence of an external magnetic field. Particularly, in mean-field regime, we study the dynamical symmetry breaking phenomenon, identifying the physical processes associated.
In many statistical models (but not all) one can introduce a local order parameter associated with a finite, discrete or continuous group of symmetries. The higher the temperature, the more important the fluctuations. One expects therefore in general to find pure phases at low temperature, with a reduced symmetry group, as a consequence of a nonvanishing expectation value of some order parameter. This situation is referred to as spontaneous symmetry breaking. A typical example is the classical Heisenberg model describing short-range interactions of an n -vector field ϕ, with an orthogonal O(n) symmetry group. For n = 1, 2, 3, this can account for a Curie transition from a ferromagnetic to a paramagnetic phase. In particle physics the σ-model of Gell-Mann and Levy involves a spontaneous symmetry breaking of chiral invariance, typical of massless spinor fields, accompanied by soft excitation modes, the so-called Goldstone modes, associated to a π-meson triplet and leading to a nonvanishing dynamical fermion mass. In a first and rather crude approximation, one can analyse the action itself or an effective one incorporating some fluctuation effects, and look for extrema as a function of field configurations, generally translationally invariant. The remaining fluctuations are then treated perturbatively. This mean field method is common to a great variety of domains, ranging from the Clausius–Mossotti formula for a polarizable medium, the Weiss molecular field in the theory of magnetism, Landau's effective action in various statistical contexts, the effective medium approximation in disordered systems, the Hartree–Fock method in atomic or many-body physics, to the semiclassical approximation in the study of quantum systems.
We analyze within a mean-field approach the stationary states of the kinetic Ising model described by the Glauber stochastic dynamics and subject to a time-dependent oscillating external field. We have found that the magnetization of the system oscillates in time around a certain value that is zero at high temperatures or large field amplitudes and nonzero at low temperatures and small field amplitudes. The transition from one regime to the other, which corresponds to a spontaneous symmetry breaking, is found to be continuous for sufficiently small values of the field amplitudes. For higher values the transition becomes discontinuous and the system exhibits a dynamical tricritical point.
The quantum Hall effect and the quantum anomalous Hall effect both require time-reversal invariance to be broken. We show that non-equilibrium effects can cause Hall physics to arise even when the system is weakly time-reversal symmetric and no magnetic field is applied. In our model, this occurs due to a fermionic subsystem breaking time-reversal invariance even if the system as a whole does not. The fermions receive a TRI-breaking self-energy, caused by interactions with bosonic degrees of freedom in the system and with an external reservoir. As a result, the fermions develop a non-quantized Hall conductance. We demonstrate that, unlike in the equilibrium case, the presence of a mass term is insufficient for the Hall conductance to appear, and wave-function renormalization effects have to be included.
A time-dependent projection technique is used to treat the initial-value problem for self-interacting fermionic fields. On the basis of the general dynamics of the fields, we derive formal equations of kinetic-type for the set of one-body dynamical variables. A nonperturbative mean-field expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Gaussian mean-field approximation, for a uniform system described by the chiral Gross-Neveu Hamiltonian. Standard stationary features of the model, such as dynamical mass generation due to chiral symmetry breaking and a phenomenon analogous to dimensional transmutation, are reobtained in this context. The mean-field time evolution of nonequilibrium initial states is discussed.
The theory of spontaneous symmetry breaking---one of the cornerstones of modern condensed-matter physics---underlies the connection between a classically ordered object in the thermodynamic limit and its microscopic quantum-mechanical constituents. However, a large, but not infinitely large, system requires a finite symmetry-breaking perturbation to stabilize a symmetry-broken state over the exact quantum-mechanical ground state, respecting the symmetry. Here, we use the example of a particular antiferromagnetic model system to show that no matter how slowly such a symmetry-breaking perturbation is driven, the adiabatic limit can never be reached. Dynamically induced collective excitations---``quantum defects''---preempt the symmetry-breaking phenomenon and trigger the appearance of a symmetric nonequilibrium state that recursively collapses into the classical equilibrium state, breaking the symmetry at punctured times. The presence of this state allows ``quantum-classical'' transitions to be investigated and controlled in mesoscopic devices by externally supplying a proper dynamical symmetry-breaking perturbation.