These lecture notes provide an introduction to quantum spin glasses, with a particular emphasis on their application to open problems in cosmology and holography. Focusing on the spherical $p$ -spin model, we develop a complete and self-contained analysis of its thermodynamic and dynamical properties. We introduce the replica method, discuss both replica-symmetric and one-step replica symmetry-broken solutions, and characterise the resulting glassy phase. The framework is then extended to the quantum model, where we study the emergence of slow dynamics and the role of quantum correlations through Euclidean and real-time correlation functions. Finally, we explore how concepts such as replica symmetry breaking and disorder naturally arise in cosmological and holographic systems related to black hole fragmentation, highlighting spin glass models as useful theoretical laboratories for these phenomena. These notes are based on a series of lectures held during the La Ricotta Summer School: The Disordered Universe 2025 from 21 July to 1 August 2025.These lecture notes provide an introduction to quantum spin glasses, with a particular emphasis on their application to open problems in cosmology and holography. Focusing on the spherical $p$-spin model, we develop a complete and self-contained analysis of its thermodynamic and dynamical properties. We introduce the replica method, discuss both replica-symmetric and one-step replica symmetry-broken solutions, and characterise the resulting glassy phase. The framework is then extended to the quantum model, where we study the emergence of slow dynamics and the role of quantum correlations through Euclidean and real-time correlation functions. Finally, we explore how concepts such as replica symmetry breaking and disorder naturally arise in cosmological and holographic systems related to black hole fragmentation, highlighting spin glass models as useful theoretical laboratories for these phenomena. These notes are based on a series of lectures held during the La Ricotta Summer School: The Disordered Universe 2025 from 21 July to 1 August 2025.

arXiv:2610.36997Oct 27, 20260 citations

Replicon modes and stability of critical behaviour of disordered systems with respect to the continuous replica symmetry breaking

Andrei A. Fedorenko

A field-theory approach is used to investigate the 'spin-glass effects' on the critical behaviour of systems with weak temperature-like quenched disorder. The renormalization group (RG) analysis of the effective Hamiltonian of a model with replica symmetry breaking (RSB) potentials of a general type is carried out in the two-loop approximation. The fixed point (FP) stability, recently found within the one-step RSB RG treatment, is further explored in terms of replicon eigenvalues. We find that the traditional FPs, which are usually considered to describe the disorder-induced universal critical behaviour, remain stable when the continuous RSB modes are taken into account.A field-theory approach is used to investigate the 'spin-glass effects' on the critical behaviour of systems with weak temperature-like quenched disorder. The renormalization group (RG) analysis of the effective Hamiltonian of a model with replica symmetry breaking (RSB) potentials of a general type is carried out in the two-loop approximation. The fixed point (FP) stability, recently found within the one-step RSB RG treatment, is further explored in terms of replicon eigenvalues. We find that the traditional FPs, which are usually considered to describe the disorder-induced universal critical behaviour, remain stable when the continuous RSB modes are taken into account.

arXiv:cond-mat/0301433Journal of Physics A Mathematical and GeneralJan 22, 200313 citations

Spontaneous symmetry breaking in quantum systems

Franco Strocchi

ScholarpediaJan 1, 201211 citations

Site-disordered spin systems in the Gaussian variational approximation

David S. Dean, David Lancaster

We define a replica field theory describing finite dimensional site-disordered spin systems by introducing the notion of grand canonical disorder, where the number of spins in the system is random but quenched. A general analysis of this field theory is made using the Gaussian variational or Hartree - Fock method, and illustrated with several specific examples. Irrespective of the form of interaction between the spins this approximation predicts a spin-glass phase. We discuss the replica symmetric phase at length, explicitly identifying the correlator that diverges at the spin-glass transition. We also discuss the form of continuous replica symmetry breaking found just below the transition. Finally we show how an analysis of ferromagnetic ordering indicates a breakdown of the approximation.We define a replica field theory describing finite dimensional site-disordered spin systems by introducing the notion of grand canonical disorder, where the number of spins in the system is random but quenched. A general analysis of this field theory is made using the Gaussian variational or Hartree - Fock method, and illustrated with several specific examples. Irrespective of the form of interaction between the spins this approximation predicts a spin-glass phase. We discuss the replica symmetric phase at length, explicitly identifying the correlator that diverges at the spin-glass transition. We also discuss the form of continuous replica symmetry breaking found just below the transition. Finally we show how an analysis of ferromagnetic ordering indicates a breakdown of the approximation.

arXiv:cond-mat/9606033Journal of Physics A Mathematical and GeneralJan 7, 19973 citations

Emergent symmetries in disordered quantum spin systems

Victor L. Quito, Pedro L. S. Lopes, José A. Hoyos +1 more

Bulletin of the American Physical SocietyMar 15, 20170 citations

These lecture notes provide an introduction to quantum spin glasses, with a particular emphasis on their application to open problems in cosmology and holography. Focusing on the spherical

p

-spin model, we develop a complete and self-contained analysis of its thermodynamic and dynamical properties. We introduce the replica method, discuss both replica-symmetric and one-step replica symmetry-broken solutions, and characterise the resulting glassy phase. The framework is then extended to the quantum model, where we study the emergence of slow dynamics and the role of quantum correlations through Euclidean and real-time correlation functions. Finally, we explore how concepts such as replica symmetry breaking and disorder naturally arise in cosmological and holographic systems related to black hole fragmentation, highlighting spin glass models as useful theoretical laboratories for these phenomena. These notes are based on a series of lectures held during the La Ricotta Summer School: The Disordered Universe 2025 from 21 July to 1 August 2025.