The T = 0 random-field Ising model on a Bethe lattice with large coordination number: hysteresis and metastable states

Martin L. Rosinberg, Gilles Tarjus, Francisco J. Perez-Reche

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Abstract

In order to elucidate the relationship between rate-independent hysteresis and metastability in disordered systems driven by an external field, we study the Gaussian RFIM at T = 0 on regular random graphs (Bethe lattice) of finite connectivity z and compute to O(1/z) (i.e. beyond mean field) the quenched complexity associated with the one-spin-flip stable states with magnetization m as a function of the magnetic field H. When the saturation hysteresis loop is smooth in the thermodynamic limit, we find that it coincides with the envelope of the typical metastable states (the quenched complexity vanishes exactly along the loop and is strictly positive everywhere inside). On the other hand, the occurrence of a jump discontinuity in the loop (associated with an infinite avalanche) can be traced back to the existence of a gap in the magnetization of the metastable states for a range of applied fields, and the envelope of the typical metastable states is then reentrant. These findings confirm and complete earlier analytical and numerical studies.
Original languageEnglish
Article numberP03003
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2009
Issue numberMarch 2009
DOIs
Publication statusPublished - Mar 2009
Externally publishedYes

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Bethe Lattice
Metastable States
Hysteresis
coordination number
metastable state
Random Field
Ising model
Ising Model
hysteresis
Magnetization
Envelope
Metastability
Hysteresis Loop
Disordered Systems
envelopes
Field Study
Avalanche
Strictly positive
Thermodynamic Limit
Flip

Cite this

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abstract = "In order to elucidate the relationship between rate-independent hysteresis and metastability in disordered systems driven by an external field, we study the Gaussian RFIM at T = 0 on regular random graphs (Bethe lattice) of finite connectivity z and compute to O(1/z) (i.e. beyond mean field) the quenched complexity associated with the one-spin-flip stable states with magnetization m as a function of the magnetic field H. When the saturation hysteresis loop is smooth in the thermodynamic limit, we find that it coincides with the envelope of the typical metastable states (the quenched complexity vanishes exactly along the loop and is strictly positive everywhere inside). On the other hand, the occurrence of a jump discontinuity in the loop (associated with an infinite avalanche) can be traced back to the existence of a gap in the magnetization of the metastable states for a range of applied fields, and the envelope of the typical metastable states is then reentrant. These findings confirm and complete earlier analytical and numerical studies.",
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The T = 0 random-field Ising model on a Bethe lattice with large coordination number : hysteresis and metastable states. / Rosinberg, Martin L.; Tarjus, Gilles; Perez-Reche, Francisco J.

In: Journal of Statistical Mechanics: Theory and Experiment, Vol. 2009, No. March 2009, P03003, 03.2009.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The T = 0 random-field Ising model on a Bethe lattice with large coordination number

T2 - hysteresis and metastable states

AU - Rosinberg, Martin L.

AU - Tarjus, Gilles

AU - Perez-Reche, Francisco J.

PY - 2009/3

Y1 - 2009/3

N2 - In order to elucidate the relationship between rate-independent hysteresis and metastability in disordered systems driven by an external field, we study the Gaussian RFIM at T = 0 on regular random graphs (Bethe lattice) of finite connectivity z and compute to O(1/z) (i.e. beyond mean field) the quenched complexity associated with the one-spin-flip stable states with magnetization m as a function of the magnetic field H. When the saturation hysteresis loop is smooth in the thermodynamic limit, we find that it coincides with the envelope of the typical metastable states (the quenched complexity vanishes exactly along the loop and is strictly positive everywhere inside). On the other hand, the occurrence of a jump discontinuity in the loop (associated with an infinite avalanche) can be traced back to the existence of a gap in the magnetization of the metastable states for a range of applied fields, and the envelope of the typical metastable states is then reentrant. These findings confirm and complete earlier analytical and numerical studies.

AB - In order to elucidate the relationship between rate-independent hysteresis and metastability in disordered systems driven by an external field, we study the Gaussian RFIM at T = 0 on regular random graphs (Bethe lattice) of finite connectivity z and compute to O(1/z) (i.e. beyond mean field) the quenched complexity associated with the one-spin-flip stable states with magnetization m as a function of the magnetic field H. When the saturation hysteresis loop is smooth in the thermodynamic limit, we find that it coincides with the envelope of the typical metastable states (the quenched complexity vanishes exactly along the loop and is strictly positive everywhere inside). On the other hand, the occurrence of a jump discontinuity in the loop (associated with an infinite avalanche) can be traced back to the existence of a gap in the magnetization of the metastable states for a range of applied fields, and the envelope of the typical metastable states is then reentrant. These findings confirm and complete earlier analytical and numerical studies.

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