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Bulletin of the South Ural State University Series Mathematical Modelling Programming and Computer Software

On a Model of Spontaneous Symmetry Breaking in Quantum Mechanics

Universidad de Antofagasta, A. Restuccia, Universidad Simon Bolivar, A. Sotomayor, Universidad de Antofagasta, Vladimir Strauss, Universidad Simon Bolivar

Published Jan 1, 2020• 1 citation
Publisher

Abstract

Our goal is to find a model for the phenomenon of spontaneous symmetry breaking arising in one dimensional quantum mechanical problems.For this purpose we consider boundary value problems related with two interior points of the real line, symmetric with respect to the origin.This approach can be treated as a presence of singular potentials containing shifted Dirac delta functions and their derivatives.From mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator which has a domain containing smooth functions vanishing in two mentioned above points.We calculate the resolvent of corresponding extension and investigate its behavior if the interior points change their positions.The domain of these extensions can contain some functions that have non differentiability or discontinuity at the points mentioned above, the latter can be interpreted as an appearance of singular potentials centered at the same points.Next, broken-symmetry bound states are discovered.More precisely, for a particular entanglement of boundary conditions, there is a ground state, generating a spontaneous symmetry breaking, stable under the phenomenon of decoherence provoked from external fluctuations.We discuss the model in the context of the "chiral" broken-symmetry states of molecules like N H 3 .We show that within a Hilbert space approach a spontaneous symmetry breaking disappears if the distance between the mentioned above interior points tends to zero.

DOI
10.14529/mmp200301

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Our goal is to find a model for the phenomenon of spontaneous symmetry breaking arising in one dimensional quantum mechanical problems.For this purpose we consider boundary value problems related with two interior points of the real line, symmetric with respect to the origin.This approach can be treated as a presence of singular potentials containing shifted Dirac delta functions and their derivatives.From mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator which has a domain containing smooth functions vanishing in two mentioned above points.We calculate the resolvent of corresponding extension and investigate its behavior if the interior points change their positions.The domain of these extensions can contain some functions that have non differentiability or discontinuity at the points mentioned above, the latter can be interpreted as an appearance of singular potentials centered at the same points.Next, broken-symmetry bound states are discovered.More precisely, for a particular entanglement of boundary conditions, there is a ground state, generating a spontaneous symmetry breaking, stable under the phenomenon of decoherence provoked from external fluctuations.We discuss the model in the context of the "chiral" broken-symmetry states of molecules like N H 3 .We show that within a Hilbert space approach a spontaneous symmetry breaking disappears if the distance between the mentioned above interior points tends to zero.
We analyze, by means of singular potentials defined in terms of Dirac functions and their derivatives, a one dimensional symmetry breaking in quantum mechanics.From a mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator with a domain containing smooth functions which vanish at two inner points of the real line.As is well known, the latter leads to a two-point boundary problem.We compute the resolvent of the corresponding extension and investigate its behavior in the case in which the inner points change their positions.The domain of these extensions can contain some functions with non differentiability or discontinuity at the points mentioned before.This fact can be interpreted as a presence of singular potentials like shifted Dirac delta functions and/or their first derivative centered at the same points.Then, we study the existence of broken-symmetry bound states.For some given entanglement boundary conditions we can show the existence of a ground state, which leads to a spontaneous symmetry breaking.We also prove that within a frame of Pontryagin spaces this type of symmetry breaking is saved if the distance between the mentioned above interior points tends to zero and then we can reformulate this result in terms of a larger Hilbert space.
We study spontaneous symmetry breaking in one dimensional quantum mechanical problems in terms of two-point boundary problems which lead to singular potentials containing Dirac delta functions and its derivatives. We search for broken-symmetry bound states. For a particular entanglement of boundary conditions, we show the existence of a ground state, giving rise to a spontaneous symmetry breaking, stable under the phenomenon of decoherence generated from external fluctuations. We discuss the model in the context of the "chiral" broken-symmetry states of molecules like NH3NH_3NH3​.
Patterns of symmetry breaking induced by potentials at the boundary of free O(N) O ( N ) -models in d=3- \epsilon d = 3 − ϵ dimensions are studied. We show that the spontaneous symmetry breaking in these theories leads to a boundary RG flow ending with N - 1 N − 1 Neumann modes in the IR. The possibility of fluctuation-induced symmetry breaking is examined and we derive a general formula for computing one-loop effective potentials at the boundary. Using the \epsilon ϵ -expansion we test these ideas in an O(N)\oplus O(N) O ( N ) ⊕ O ( N ) -model with boundary interactions. We determine the RG flow diagram of this theory and find that it has an IR-stable critical point satisfying conformal boundary conditions. The leading correction to the effective potential is computed and we argue the existence of a phase boundary separating the region flowing to the symmetric fixed point from the region flowing to a symmetry-broken phase with a combination of Neumann and Dirchlet boundary conditions.
Spontaneous symmetry breaking (SSB) in quantum systems, such as ferromagnets, is normally described as (or as arising from) degeneracy of the ground state; however, it is well established that this degeneracy only occurs in spatially infinite systems, and even better established that ferromagnets are not spatially infinite. I review this well-known paradox, and consider a popular solution where the symmetry is explicitly broken by some external field which goes to zero in the infinite-volume limit; although this is formally satisfactory, I argue that it must be rejected as a physical explanation of SSB since it fails to reproduce some important features of the phenomenology. Motivated by considerations from the analogous classical system, I argue that SSB in finite systems should be understood in terms of the approximate decoupling of the system's state space into dynamically-isolated sectors, related by a symmetry transformation; I use the formalism of decoherent histories to make this more precise and to quantify the effect, showing that it is more than sufficient to explain SSB in realistic systems and that it goes over in a smooth and natural way to the infinite limit.
The basic problem how spontaneous symmetry breaks is studied rigorously for local fields in a finite domain. Another basic problem is to investigate whether or not there exist lower dimensions (of the domain to which a field H is applied) below which no spontaneous symmetry breaking (SSB) occurs. This is studied using scaling, renormalization group and mean-field theory for partial fields in an infinite domain. Our main results are the following: (i) The total magnetization is proportional to the square of the spontaneous magnetization when a field is applied only to a finite region (an extension of Yang's result [Phys. Rev. 85 (1952) 808], M = N m 2 s for a fixed single spin). A general formulation is given to calculate the proportionality coefficient which is shown to vanish as the field goes to zero. This indicates no SSB in this situation. (ii) On the other hand, if we apply partial fields to a d '-dimensional region and make them vanish, we find that SSB occurs, at least, for d '> d ' c =β/ν. In the region 0< d '≤ d ' c , the situation is subtle, as is exemplified by an exactly soluble model with such a field on one-side boundary as depends on the system-size.