© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Progress of Theoretical Physics

Phenomenological Theory of Spin-Glasses and Some Rigorous Results

Masuo Suzuki

Published Oct 1, 1977• 359 citations

Abstract

A general phenomenological theory of spin-glasses is presented, which predicts the weak singularities (like cusps) of susceptibilities and specific heat near the transition point TSG and also predicts generally the divergence of the second derivative of the nonlinear susceptibility with respect to a magnetic field in zero field at TSG. A similar second derivative of specific heat is also shown to diverge at TSG except for the molecular field theory. Scaling equations of state for magnetization m and spin-glass order-parameter q are derived, which yield new scaling relations of critical exponents for m(T), q(T), χ0(T), C0(T) and the corresponding nonlinear quantities. These general results explain qualitatively very well the magnetic field dependence of magnetization and specific heat observed experimentally. A non-uniform phenomenological Hamiltonian is also proposed for the renormalization group approach, which yields the critical dimensionality dc = 6, as usual. Intermediate random statistics and real replica method are proposed. There is also presented an exactly soluble model showing a spin-glass like phase transition.

DOI: 10.1143/ptp.58.1151

Related papers

Spin-glass transition in a magnetic field: A renormalization group study

Iveta R. Pimentel, T. Temesvári, C. De Dominicis

We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.

arXiv:cond-mat/0204615•Physical review. B, Condensed matter•Jun 6, 2002•50 citations

Scaling theory for the quantum spin-glass transition

M. A. Contínentino, B. Boechat, Raimundo R. dos Santos

We present a scaling theory for the low-temperature phase transition of the Ising spin glass in a transverse magnetic field (\ensuremath{\Gamma}). The theory provides relations and bounds on some critical exponents, which are supported by results from actual calculations. A renormalization-group analysis of the zero-temperature transition in the presence of a longitudinal symmetry-breaking field enables us to estimate the critical exponents \ensuremath{\beta} and \ensuremath{\gamma}, associated with the Edwards-Anderson order parameter and the nonlinear susceptibility, respectively. We have found no indications of the transition being of first order.

Physical review. B, Condensed matter•Nov 1, 1994•10 citations

Spin Glass in a Field: A New Zero-Temperature Fixed Point in Finite Dimensions

Maria Chiara Angelini, Giulio Biroli

By using real-space renormalization group (RG) methods, we show that spin glasses in a field display a new kind of transition in high dimensions. The corresponding critical properties and the spin-glass phase are governed by two nonperturbative zero-temperature fixed points of the RG flow. We compute the critical exponents and discuss the RG flow and its relevance for three-dimensional systems. The new spin-glass phase we discovered has unusual properties, which are intermediate between the ones conjectured by droplet and full replica symmetry-breaking theories. These results provide a new perspective on the long-standing debate about the behavior of spin glasses in a field.

arXiv:1409.1011•Physical Review Letters•Mar 4, 2015•49 citations

Divergence of the Transverse Nonlinear Susceptibility along the GT Line in Heisenberg Spin Glasses

Masuo Suzuki

It it shown phenomenologically that the transverse nonlinear susceptibility x2 along the Gabay-Toulouse line in m-component spin glasses, behaves as x2∝(T-Tsg(Hz))-γ near the transition point Tsg(Hz), and that Tsg(Hz)-Tsg(0) ∝Hz4 β/(β+ γ) with a magnetic field Hz and the critical exponent β of the spin glass order parameter.

Progress of Theoretical Physics•Mar 1, 1985•3 citations

Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses

Juan Jesus Ruiz-Lorenzo

Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.

arXiv:2006.12930•WORLD SCIENTIFIC eBooks•Jun 24, 2020•1 citation

Evidence of a second-order phase transition in the six-dimensional Ising spin glass in a field

Miguel Aguilar-Janita, Víctor Martín-Mayor, J. Moreno-Gordo +1 more

The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.

arXiv:2306.00569•Physical review. E•May 10, 2024•8 citations

A general phenomenological theory of spin-glasses is presented, which predicts the weak singularities (like cusps) of susceptibilities and specific heat near the transition point TSG and also predicts generally the divergence of the second derivative of the nonlinear susceptibility with respect to a magnetic field in zero field at TSG. A similar second derivative of specific heat is also shown to diverge at TSG except for the molecular field theory. Scaling equations of state for magnetization m and spin-glass order-parameter q are derived, which yield new scaling relations of critical exponents for m(T), q(T), χ0(T), C0(T) and the corresponding nonlinear quantities. These general results explain qualitatively very well the magnetic field dependence of magnetization and specific heat observed experimentally. A non-uniform phenomenological Hamiltonian is also proposed for the renormalization group approach, which yields the critical dimensionality dc = 6, as usual. Intermediate random statistics and real replica method are proposed. There is also presented an exactly soluble model showing a spin-glass like phase transition.

We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.

We present a scaling theory for the low-temperature phase transition of the Ising spin glass in a transverse magnetic field (\ensuremath{\Gamma}). The theory provides relations and bounds on some critical exponents, which are supported by results from actual calculations. A renormalization-group analysis of the zero-temperature transition in the presence of a longitudinal symmetry-breaking field enables us to estimate the critical exponents \ensuremath{\beta} and \ensuremath{\gamma}, associated with the Edwards-Anderson order parameter and the nonlinear susceptibility, respectively. We have found no indications of the transition being of first order.

By using real-space renormalization group (RG) methods, we show that spin glasses in a field display a new kind of transition in high dimensions. The corresponding critical properties and the spin-glass phase are governed by two nonperturbative zero-temperature fixed points of the RG flow. We compute the critical exponents and discuss the RG flow and its relevance for three-dimensional systems. The new spin-glass phase we discovered has unusual properties, which are intermediate between the ones conjectured by droplet and full replica symmetry-breaking theories. These results provide a new perspective on the long-standing debate about the behavior of spin glasses in a field.

Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.

The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.