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dc.contributor.authorKelbert, Mark
dc.contributor.authorSuhov, Yurii
dc.date.accessioned2017-10-03T07:44:59Z
dc.date.available2017-10-03T07:44:59Z
dc.date.issued2013
dc.identifier.citationMark Kelbert and Yurii Suhov, “A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model,” Advances in Mathematical Physics, vol. 2013, Article ID 637375, 20 pages, 2013. doi:10.1155/2013/637375
dc.identifier.issn1687-9120
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/267614
dc.description.abstract<jats:p>This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>ℋ</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mtext>L</mml:mtext></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>),</mml:mo></mml:math>where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math>is a<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:math>-dimensional unit torus<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:msup><mml:mrow><mml:mi>ℤ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math>with a flat metric. The phase space of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>spins is<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:msub><mml:mrow><mml:mi>ℋ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mtext>L</mml:mtext></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow><mml:mrow><mml:mtext>s</mml:mtext><mml:mtext>y</mml:mtext><mml:mtext>m</mml:mtext></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>, the subspace of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:msub><mml:mrow><mml:mtext>L</mml:mtext></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>formed by functions symmetric under the permutations of the arguments. The Fock space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mi mathvariant="bold-script">H</mml:mi></mml:mrow></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0,1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo></mml:mrow></mml:msub></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℋ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>yields the phase space of a system of a varying (but finite) number of particles. We associate a space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mi mathvariant="bold-script">H</mml:mi><mml:mo>≃</mml:mo><mml:mi mathvariant="bold-script">H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>with each vertex<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mi>i</mml:mi><mml:mi>∈</mml:mi><mml:mi mathvariant="normal">Γ</mml:mi></mml:math>of a graph<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo>,</mml:mo><mml:mi>ℰ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>satisfying a special bidimensionality property. (Physically, vertex<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:math>represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M15"><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:math>, the minus a half of the Laplace operator on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M16"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math>, responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M17"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M18"><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>M</mml:mi></mml:math>, describing a field generated by a heavy atom, (b) two-body potentials<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M19"><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M20"><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>M</mml:mi></mml:math>, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M21"><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M22"><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>M</mml:mi></mml:math>, scaled along the graph distance<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M23"><mml:mi mathvariant="monospace">d</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>between vertices<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M24"><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mi>∈</mml:mi><mml:mi mathvariant="normal">Γ</mml:mi></mml:math>, which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M25"><mml:mrow><mml:mtext mathvariant="monospace">G</mml:mtext></mml:mrow></mml:math>acts on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M26"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math>, represented by a Euclidean space or torus of dimension<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M27"><mml:mi>d</mml:mi><mml:mi mathvariant="normal">'</mml:mi><mml:mo>≤</mml:mo><mml:mi>d</mml:mi></mml:math>, preserving the metric and the volume in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M28"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math>. Furthermore, we suppose that the potentials<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M29"><mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M30"><mml:mrow><mml:msup><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M31"><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:math>are<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M32"><mml:mrow><mml:mtext mathvariant="monospace">G</mml:mtext></mml:mrow></mml:math>-invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M33"><mml:mrow><mml:mtext mathvariant="monospace">G</mml:mtext></mml:mrow></mml:math>-invariant, provided that the thermodynamic variables (the fugacity<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M34"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:math>and the inverse temperature<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M35"><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:math>) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.</jats:p>
dc.publisherHindawi Limited
dc.rightsAll Rights Reserved
dc.rights.urihttps://www.rioxx.net/licenses/all-rights-reserved/
dc.titleA Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model
dc.typeArticle
dc.date.updated2017-07-13T08:33:22Z
dc.description.versionPeer Reviewed
dc.language.rfc3066en
dc.rights.holderCopyright © 2013 Mark Kelbert and Yurii Suhov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
prism.publicationNameAdvances in Mathematical Physics
dc.identifier.doi10.17863/CAM.13553
rioxxterms.versionofrecord10.1155/2013/637375
dc.identifier.eissn1687-9139


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