The Free Energy of the GREM with Random Magnetic Field

Arguin, Louis-Pierre; Persechino, Roberto

doi:10.1007/978-3-030-29077-1_3

Cited by 3 publications

(14 citation statements)

References 7 publications

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“…3, is a generalization of Theorem 1.4 in [22], which addresses the case without a longitudinal field, h = 0. In the classical case without transversal magnetic field, b = 0, it generalizes the results of [7], which covers the case that h is constant, and of [4,5], which treats the special case of a REM or two-level GREM in a random magnetic field.…”

Section: The Quantum Grem With a Random Longitudinal Fieldmentioning

confidence: 59%

“…therein). Unlike for the SK-model, implementing the longitudinal field naively in GREM models causes the frozen phase to expand [4,5,7]. Derrida and Gardner [14] therefore suggested a hierarchical implementation of the longitudinal magnetic field, which then leads again to a destabilization of the frozen phase.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The main aim of this subsection to prove the following Theorem 3, which extends the discussion of the two-level GREM in [5] to the general n-level GREM. To this end, we will need to introduce some notation.…”

Section: The Grem With a Random Magnetic Fieldmentioning

confidence: 87%

“…Based on the already established results and methods in [4,5,20,22], the proof of Theorem 1 is straightforward but quite lengthy. Before we move on to the details, we outline our proof strategy which consists of three main steps:…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…First, we need to generalize the results in [4,5] on the REM and two-level GREM with a random magnetic field to the general n-level GREM (see Theorem 3). Following [4,5] closely, the argument is based on a large-deviation principle for the entropy which transforms the computation of the limit to a linear optimization problem with non-linear constraints. 2.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

Manai

Warzel

2021

J Stat Phys

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We determine explicitly and discuss in detail the effects of the joint presence of a longitudinal and a transversal (random) magnetic field on the phases of the Random Energy Model and its hierarchical generalization, the GREM. Our results extent known results both in the classical case of vanishing transversal field and in the quantum case for vanishing longitudinal field. Following Derrida and Gardner, we argue that the longitudinal field has to be implemented hierarchically also in the Quantum GREM. We show that this ensures the shrinking of the spin glass phase in the presence of the magnetic fields as is also expected for the Quantum Sherrington–Kirkpatrick model.

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Section: The Quantum Grem With a Random Longitudinal Fieldmentioning

confidence: 59%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: The Grem With a Random Magnetic Fieldmentioning

confidence: 87%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

See 3 more Smart Citations

The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

Manai

Warzel

2021

J Stat Phys

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Generalized random energy models in a transversal magnetic field: Free energy and phase diagrams

Manai

Warzel

2022

Prob. Math. Phys.

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We determine explicit variational expressions for the quantum free energy of mean-field spin glasses in a transversal magnetic field, whose glass interaction is given by a hierarchical Gaussian potential as in Derrida's generalized random energy model (GREM), its continuous version (CREM), or the nonhierarchical GREM. The corresponding phase diagrams, which generally include glass transitions as well as transitions to quantum paramagnetic and mixed quantum-classical phases, are discussed. In the glass phase, the free energy is generally determined by both the parameters of the classical model and the transversal field. MSC2020: 82B44, 82D30.

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The de Almeida-Thouless Line in Hierarchical Quantum Spin Glasses

Manai,

Warzel

2021

Preprint

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We determine explicitly and discuss in detail the effects of the joint presence of a longitudinal and a transversal (random) magnetic field on the phases of the Random Energy Model (REM) and its hierarchical generalization, the GREM. Our results extent known results both in the classical case of vanishing transversal field and in the quantum case for vanishing longitudinal field. Following Derrida and Gardner, we argue that the longitudinal field has to be implemented hierarchically also in the Quantum GREM. We show that this ensures the shrinking of the spin glass phase in the presence of the magnetic fields as also expected for the Quantum Sherrington-Kirkpatrick model.

show abstract

The Free Energy of the GREM with Random Magnetic Field

Cited by 3 publications

References 7 publications

The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

Generalized random energy models in a transversal magnetic field: Free energy and phase diagrams

The de Almeida-Thouless Line in Hierarchical Quantum Spin Glasses

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