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arXiv:1902.05836

Entanglement of boundary conditions giving rise to spontaneous symmetry breaking in quantum mechanics

A. Restuccia, A. Sotomayor, Vladimir Strauss

Published Feb 15, 2019• 0 citations
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Abstract

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 $NH_3$.

DOI
10.48550/arxiv.1902.05836

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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​.
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.
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 is a ubiquitous phenomenon in nature. One of the defining features of symmetry-broken phases is that the large system size limit and the vanishing external field limit do not commute. In this work, we study a family of extensions of the NNN-state clock model. We find that the exact symmetry and the ground state degeneracy under the periodic boundary condition heavily depend on the system size, although the model has the manifest translation symmetry. In particular, the ground state can be unique and all excitations are gapped even when the phase exhibits noncommutativity of the two limits. Our model hence poses a question on the standard understanding of spontaneous symmetry breaking.
We have recently shown that there is a limit to quantum coherence in many-particle spin qubits due to spontaneous symmetry breaking. These results were derived for the Lieb-Mattis spin model. Here we will show that the underlying mechanism of decoherence in systems with spontaneous symmetry breaking is in fact more general. We present here a generic route to finding the decoherence time associated with spontaneous symmetry breaking in many-particle qubits, and subsequently we apply this approach to two model systems, indicating how the continuous symmetries in these models are spontaneously broken and discussing the relation of this symmetry breaking to the thin spectrum. We then present in detail the calculations that lead to the limit to quantum coherence, which is due to energy shifts in the thin spectrum.
A spontaneous symmetry-breaking order is conventionally described by a tensor-product wavefunction of some few-body clusters; some standard examples include the simplest ferromagnets and valence bond solids. We discuss a type of symmetry-breaking orders, dubbed entanglement-enabled symmetry-breaking orders, which cannot be realized by any such tensor-product state. Given a symmetry breaking pattern, we propose a criterion to diagnose if the symmetry-breaking order is entanglement-enabled, by examining the compatibility between the symmetries and the tensor-product description. For concreteness, we present an infinite family of exactly solvable gapped models on one-dimensional lattices with nearest-neighbor interactions, whose ground states exhibit entanglement-enabled symmetry-breaking orders from a discrete symmetry breaking. In addition, these ground states have gapless edge modes protected by the unbroken symmetries. We also propose a construction to realize entanglement-enabled symmetry-breaking orders with spontaneously broken continuous symmetries. Under the unbroken symmetries, some of our examples can be viewed as symmetry-protected topological states that are beyond the conventional classifications.