# The Free Energy of a Quantum Sherrington–Kirkpatrick Spin-Glass Model for Weak Disorder

@article{Leschke2019TheFE, title={The Free Energy of a Quantum Sherrington–Kirkpatrick Spin-Glass Model for Weak Disorder}, author={Hajo Leschke and Sebastian Rothlauf and Rainer Ruder and Wolfgang L. Spitzer}, journal={Journal of Statistical Physics}, year={2019} }

<jats:p>We extend two rigorous results of <jats:sc>Aizenman</jats:sc>, <jats:sc>Lebowitz</jats:sc>, and <jats:sc>Ruelle</jats:sc> in their pioneering paper of 1987 on the <jats:sc>Sherrington</jats:sc>–<jats:sc>Kirkpatrick</jats:sc> spin-glass model without external magnetic field to the quantum case with a “transverse field” of strength <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathsf {b}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML…

## 8 Citations

Free energy of the quantum Sherrington–Kirkpatrick spin-glass model with transverse field

- Physics
- 2019

We consider the quantum Sherrington-Kirkpatrick (SK) spin-glass model with transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures…

The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

- PhysicsJournal of Statistical Physics
- 2021

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…

Generalized Random Energy Models in a Transversal Magnetic Field: Free Energy and Phase Diagrams

- Physics
- 2020

We determine explicit variational expressions for the free energy of mean-field spin glasses in a transversal magnetic field, whose glass interaction is given by a hierarchical Gaussian potential as…

Spectral Analysis of the Quantum Random Energy Model

- Physics
- 2022

The Quantum Random Energy Model (QREM) is a random matrix of Anderson-type which describes effects of a transversal magnetic field on Derrida’s spin glass. The model exhibits a glass phase as well as…

Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model

- Physics, Mathematics
- 2022

We study the ﬂuctuations of time-additive random observables in the stochastic dynamics of a system of N non-interacting Ising spins. We mainly consider the case of all-to-all dynamics where…

Universality of replica-symmetry breaking in the transverse field Sherrington-Kirkpatrick model

- Physics
- 2022

The existence theorem for replica-symmetry breaking (RSB) in the transverse field SherringtonKirkpatrick (SK) model is extended to the model with a general random exchange interactions. The relation…

Existence of Replica-Symmetry Breaking in Quantum Glasses.

- PhysicsPhysical review letters
- 2021

By controlling quantum fluctuations via the Falk-Bruch inequality we give the first rigorous argument for the existence of a spin-glass phase in the quantum Sherrington-Kirkpatrick model with a…

Phase Diagram of the Quantum Random Energy Model

- PhysicsJournal of Statistical Physics
- 2020

We prove Goldschmidt’s formula (Goldschmidt in Phys Rev B 47:4858–4861, 1990) for the free energy of the quantum random energy model. In particular, we verify the location of the first order and the…

## References

SHOWING 1-10 OF 118 REFERENCES

Free energy of the quantum Sherrington–Kirkpatrick spin-glass model with transverse field

- Physics
- 2019

We consider the quantum Sherrington-Kirkpatrick (SK) spin-glass model with transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures…

Stability of the quantum Sherrington-Kirkpatrick spin glass model.

- PhysicsPhysical review. E
- 2017

It is found that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0.

Possible ergodic-nonergodic regions in the quantum Sherrington-Kirkpatrick spin glass model and quantum annealing.

- PhysicsPhysical review. E
- 2018

We explore the behavior of the order parameter distribution of the quantum Sherrington-Kirkpatrick model in the spin glass phase using Monte Carlo technique for the effective Suzuki-Trotter…

Quantum fluctuations in the transverse Ising spin glass model: A field theory of random quantum spin systems

- Physics
- 2007

We develop a mean-field theory for random quantum spin systems using the spin coherent state path integral representation. After the model is reduced to the mean-field one-body Hamiltonian, the…

The Thermodynamic Limit in Mean Field Spin Glass Models

- Mathematics
- 2002

Abstract: We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as…

Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model

- Mathematics
- 1991

AbstractWe prove that ifĤNis the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity
$$\bar q_N = N^{ - 1} \sum \left\langle {S_l } \right\rangle _H^2 $$
converges in the variance to a…

A dynamic theory of transverse freezing in the Sherrington-Kirkpatrick Ising model

- Physics
- 1988

The effect of a transverse field Gamma on spin-glass freezing in the Sherrington-Kirkpatrick version of the quantum Ising model is studied by using the thermo-field-dynamic (TFD) method (the…

Infinite-range Ising spin glass with a transverse field under the static approximation

- Physics
- 2002

In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the…

The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case

- Mathematics
- 1995

We study the fluctuations of free energy, energy and entropy in the high temperature regime for the Sherrington-Kirkpatrick model of spin glasses. We introduce here a new dynamical method with the…

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

- Mathematics
- 2003

Abstract: By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick…