The Phase Diagram of the Quantum Curie-Weiss Model

Chayes, L.; Crawford, Nicholas; Ioffe, Dmitry; Levit, Anna

doi:10.1007/s10955-008-9608-x

Cited by 36 publications

(53 citation statements)

References 21 publications

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“…The quantum mean-field, or Curie-Weiss, model has been studied using large-deviation techniques in [16], see also [25]. A random-current representation of the quantum Ising model may be found in [28], and, as explained in Remark 1.3 and [17], this is intimately related to that discussed and exploited in the next section.…”

Section: 2mentioning

confidence: 86%

“…A number of authors have developed and utilized the following 'path integral representation' of the quantum Ising model, see for example [7,8,15,16,27,33] and the recent surveys to be found in [24,28]. Let S = S β be the circle of circumference β, which we think of as the interval [0, β] with its two endpoints identified.…”

Section: 2mentioning

confidence: 99%

“…2.2]. As pointed out in [17], the appendix of [16] contains a type of switching argument for the mean-field model. A principal difference between that argument and those of [17,28] and the current work is that it uses the classical switching lemma developed in [1], applied to a discretized version of the mean-field system.…”

Section: Remark 13mentioning

confidence: 99%

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The Phase Transition of the Quantum Ising Model is Sharp

Björnberg

Grimmett

2009

J Stat Phys

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Abstract. An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d + 1 dimensions. A so-called 'random-parity' representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.

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Section: 2mentioning

confidence: 86%

Section: 2mentioning

confidence: 99%

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The Phase Transition of the Quantum Ising Model is Sharp

Björnberg

Grimmett

2009

J Stat Phys

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“…in its stochastic representation via Feynman-Kac, [1], [3] and [7]. We are indebted to D. Ioffe for pointing out the connection and for useful comments.…”

Section: Introductionmentioning

confidence: 99%

Layered Systems at the Mean Field Critical Temperature

et al. 2015

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We consider the Ising model on Z × Z where on each horizontal line {(x, i), x ∈ Z}, the interaction is given by a ferromagnetic Kac potential with coupling strength J γ (x, y) ∼ γJ(γ(x − y)) at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength ǫ and prove that for every ǫ > 0 the systems exhibits phase transition provided γ > 0 is small enough.

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“…Stochastic representations/path integral approach frequently provides a useful intuition and insight into the structure of quantum spin states. Numerous examples include [2,3,8,10,12,17,18,22,26]. In this work we rely on a path integral approach and related large deviations techniques, and derive global logarithmic asymptotics of ground states for a class of quantum mean field models in transverse field.…”

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confidence: 99%

Ground States for Mean Field Models with a Transverse Component

Ioffe

Levit

2013

J Stat Phys

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Abstract. We investigate global logarithmic asymptotics of ground states for a family of quantum mean field models. Our approach is based on a stochastic representation and a combination of large deviation and weak KAM techniques. The spin-1 2 case is worked out in more detail. 1.The model and the result 1.1. Introduction. Stochastic representations/path integral approach frequently provides a useful intuition and insight into the structure of quantum spin states. Numerous examples include [2,3,8,10,12,17,18,22,26]. In this work we rely on a path integral approach and related large deviations techniques, and derive global logarithmic asymptotics of ground states for a class of quantum mean field models in transverse field. These asymtotics limits are identified as weak-KAM [16] type solutions of certain Hamilton-Jacobi equations. In principle, such solutions are not unique, and an additional refined analysis along the lines of [14,19,20] is needed for recovering the correct asymptotic ground state. This issue is addressed in more detail for the spin 1 2 -case. In particular, our results imply logarithmic asymptotics of ground states for models with p-body interactions [5]. In the case of Laplacian with periodic potential a weak KAM approach to semiclassical asymptotics was already employed in [1].Our stochastic representation gives rise to a family of continuous time Markov chains on a simplex ∆The transition rates are enhanced by a factor of N , and the chain moves in a potential of the type N F . Ground states are Perron-Frobenius eigenfunctions of the corresponding generators. On the concluding stages of this work we have learned about the series of papers [23][24][25]. The models we consider here essentially fall into a much more general framework studied in these works. The authors of [23][24][25] extend an analysis of Schrödinger operators [14,19,20] on R d to lattice operators on Z d , and they develop powerful techniques, which go well beyond the scope of our work, and which enable a complete asymptotic expansion of low lying eigenvalues and eigenfunctions in neighbourhoods of potential wells. The paper is organized as follows: The class of models is described in Subsection 1.2, and the results are formulated in Subsection 1.4. Main steps of our approach are explained in Section 2, whereas some of the proofs are relegated to Section 3. The

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The Phase Diagram of the Quantum Curie-Weiss Model

Cited by 36 publications

References 21 publications

The Phase Transition of the Quantum Ising Model is Sharp

The Phase Transition of the Quantum Ising Model is Sharp

Layered Systems at the Mean Field Critical Temperature

Ground States for Mean Field Models with a Transverse Component

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