Infinite-range Ising Spin Glass Research Articles

Stability of the quantum Sherrington-Kirkpatrick spin glass model.

I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find 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 (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).

Code-division multiple-access multiuser demodulator by using quantum fluctuations.

We examine the average-case performance of a code-division multiple-access (CDMA) multiuser demodulator in which quantum fluctuations are utilized to demodulate the original message within the context of Bayesian inference. The quantum fluctuations are built into the system as a transverse field in the infinite-range Ising spin glass model. We evaluate the performance measurements by using statistical mechanics. We confirm that the CDMA multiuser modulator using quantum fluctuations achieve roughly the same performance as the conventional CDMA multiuser modulator through thermal fluctuations on average. We also find that the relationship between the quality of the original information retrieval and the amplitude of the transverse field is somehow a "universal feature" in typical probabilistic information processing, viz., in image restoration, error-correcting codes, and CDMA multiuser demodulation.

Two-sublattice fermionic Ising spin glass model in a transverse field: Asymmetric replica symmetry breaking solution

We investigate a two-sublattice fermionic infinite-range Ising spin glass (SG) model in a transverse field Γ by one-step replica symmetry breaking (1S-RSB) theory. In this model, a parallel magnetic field H breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor a non-ergodic mixed phase ( SG + AF ) characterizing an asymmetric RSB region. The Γ field introduces a quantum spin flip mechanism that suppresses the magnetic orders leading them to quantum critical points. The competition between disorder and quantum fluctuations is analyzed by 1S-RSB solution in the asymmetric RSB region.

Replica symmetry breaking in a quantum spin glass-antiferromagnetic Kondo lattice

The competition between the Kondo effect and the glassy magnetic order has been studied in a theoretical model of a Kondo lattice with an intrasite Kondo interaction. The spin glass (SG) and the antiferromagnetic (AF) orderings are described by two Kondo sublattices with infinite-range Ising SG interactions among localized spins and the disordered interactions can occur with spins of same sublattices and between spins of distinct sublattices. A transverse field Γ is introduced in the effective model as a quantum mechanism to produce spin flipping. The problem is formulated in a Grassmann path integral formalism. The disorder is treated within the replica trick in one-step replica symmetry breaking (1S-RSB). The static ansatz is adopted to get a mean-field expression for the free energy and order parameters. Results show a transition from the AF order to an RSB region with a finite staggered magnetization (mixed phase) when temperature T decreases for low values of the Kondo interaction. The SG phase is not observed below the mixed phase for 1S-RSB solution, in contrast with previous replica symmetry (RS) results. The Γ field suppresses the Neel temperature leading it to a quantum critical point.

One-step replica symmetry breaking solution for a highly asymmetric two-sublattice fermionic Ising spin-glass model in a transverse field

The one-step replica symmetry breaking (RSB) is used to study a two-sublattice fermionic infinite-range Ising spin glass (SG) model in a transverse field $\Gamma$. The problem is formulated in a Grassmann path integral formalism within the static approximation. In this model, a parallel magnetic field $H$ breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF) characterizing an asymmetric RSB region. In this region, intra-sublattice disordered interactions $V$ increase the difference between the RSB solutions of each sublattice. The freezing temperature shows a higher increase with $H$ when $V$ enhances. A discontinue phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intra-sublattice ferromagnetic average coupling. The $\Gamma$ field introduces a quantum spin flip mechanism that suppresses the magnetic orders leading them to quantum critical points. Results suggest that the quantum effects are not able to restore the RS solution. However, in the asymmetric RSB region, $\Gamma$ can produce a stable RS solution at any finite temperature for a particular sublattice while the other sublattice still presents RSB solution for the special case in which only the intra-sublattice spins couple with disordered interactions.

Replica symmetry breaking in the transverse-field Ising spin-glass model: Two fermionic representations

We analyze the infinite range Ising spin glass in a transverse-field $\ensuremath{\Gamma}$ below the critical temperature by a one step replica symmetry breaking theory. The set of $n$ replicas is divided in $r$ blocks of $m$ replicas each. We present results for different values of the block-size parameter $m$. The spin operators are represented by bilinear combinations of fermionic fields and we compare the results of two models: In the four-states (4-S) model the diagonal ${S}_{i}^{z}$ operator has two unphysical vanishing eigenvalues, that are suppressed by a restraint in the two-states (2-S) model. In the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature ${T}_{c}(\ensuremath{\Gamma})$ that decreases when $\ensuremath{\Gamma}$ increases, until it reaches a quantum critical point at the same value of ${\ensuremath{\Gamma}}_{c}$ and they are both unstable under replica symmetry breaking in the whole spin glass phase. Below the critical temperature we present results for the order parameters and free energy.

The Nature of the Specific Heat of Ising Spin Glass

The specific heat of finite dimensional (two- and three-dimentional) ±J Ising models and infinite-range Ising spin glass (Sherrington-Kirkpatrick (SK) model) has been investigated by Monte Carlo simulations from a new point of view. The region where a part of the specific heat takes a negative value has been investigated, which is characteristic of frustrated systems and reflects the non-trivial degeneracy of the ground state. For each model, the region mentioned above is found to be fairly large in the corresponding phase diagram, which includes the Nishimori line.

Fermionic Ising glasses with BCS pairing interaction in the presence of a transverse field

In the present work we have analyzed a fermionic infinite-ranged Ising spin glass with a local BCS coupling in the presence of transverse field. This model has been obtained by tracing out the conduction electrons degrees of freedom in a superconducting alloy. The transverse field is applied in the resulting effective model. The problem is formulated in the path integral formalism where the spins operators are represented by bilinear combination of Grassmann fields. The problem can be solved by combining previous approaches used to study a fermionic Heisenberg spin glass and a Ising spin glass in a transverse field. The results are shown in a phase diagram T/ J versus Γ/ J ( J is the standard deviation of the random coupling J ij ) for several values of g (the strength of the pairing interaction). For small g, the line transition T c( Γ) between the normal-paramagnetic phase and the spin glass phase decreases when Γ increases, until it reaches a quantum critical point. For increasing g, a PAIR phase (where there is formation of local pairs) has been found which disappears when it is close to Γ c showing that the transverse field tends to inhibit the PAIR phase.

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

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 imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of ${C}_{H}/T$ near the glass transition temperature ${T}_{g},$ associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.

The Ising spin glass in a transverse field revisited. Results of two fermionic models

We analyze the infinite range Ising spin glass in a transverse field Γ by using Grassmann variables in a field theory where the spin operators are represented by bilinear combinations of fermionic fields. We compare the results of two fermionic models. In the four state (4S)-model the diagonal S i z operator has two vanishing eigenvalues, that are suppressed by a restraint in the two states (2S)-model. Within a replica symmetric theory and in the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature T c ( Γ) that decreases when Γ increases, until it reaches a quantum critical point at the same value of Γ c and they are both unstable under replica symmetry breaking in the whole spin glass phase. Our results are in agreement with recent renormalization group calculations in three-dimensional systems.

Fractal geometry of spin-glass models

Stability and diversity are two key properties that living entities share with spin glasses, where they are manifested through the breaking of the phase space into many valleys or local minima connected by saddle points. The topology of the phase space can be conveniently condensed into a tree structure, akin to the biological phylogenetic trees, whose tips are the local minima and internal nodes are the lowest-energy saddles connecting those minima. For the infinite-range Ising spin glass with p-spin interactions, we show that the average size-frequency distribution of saddles obeys a power law $<\psi(w) > \sim w^{-D}$, where w=w(s) is the number of minima that can be connected through saddle s, and D is the fractal dimension of the phase space.

Da Costa-Nobre-Yokoi Model with Spin S

An infinite-ranged Ising spin glass model with spin S, including S=∞, with uniform biquadratic interactions is investigated using replica symmetry (RS) solutions, de Almeida and Thouless (AT) stability analyses, and one-step replica symmetry breaking (RSB) solutions. We find that the model with integral spin S and the model with half-integral spin S exhibit different thermodynamic behavior. Namely, we show that for the model with integral spin S, certain physical quantities display two-sublattice structure, but that for the model with half-integral spin S, including S=∞, this is not the case.

The Ising spin glass in a transverse field and the Kondo model

We establish the existence of an asymptotic equivalence between the infinite-range Ising spin glass in a transverse magnetic field at its quantum critical point, and the single-impurity Kondo model. The analogy between these two problems provides a useful tool for the study of the physics of diluted Kondo alloys.

Replica analysis of thep-spin interaction Ising spin-glass model

The thermodynamics of the infinite-range Ising spin glass with p-spin interactions in the presence of an external magnetic field h is investigated analytically using the replica method. We give emphasis to the analysis of the transition between the replica symmetric and the one-step replica symmetry breaking regimes. In particular, we derive analytical conditions for the onset of the continuous transition, as well as for the location of the tricritical point at which the transition between those two regimes becomes discontinuous.

Dynamics of the Infinite-Range Ising Spin-Glass Model in a Transverse Field

We use quantum Monte Carlo methods and various analytic approximations to study the infinite-range Ising spin-glass model in a transverse field in the disordered phase. We focus on the behavior of the frequency dependent susceptibility of the system above and below the critical field. We establish that, at the quantum critical point, there exists an equivalence between the long-time behavior of this model and that of the single-impurity Kondo model. Our predictions for the long-time dynamics of the model are in good agreement with experimental results on ${\mathrm{LiHo}}_{0.167}{\mathrm{Y}}_{0.833}{\mathrm{F}}_{4}$.

Landscape statistics of thep-spin Ising model

The statistical properties of the local optima (metastable states) of the infinite range Ising spin glass with p-spin interactions in the presence of an external magnetic field h are investigated analytically. The average number of optima as well as the typical overlap between pairs of identical optima are calculated for general p. Similarly to the thermodynamic order parameter, for p>2 and small h the typical overlap q_t is a discontinuous function of the energy. The size of the jump in q_t increases with p and decreases with h, vanishing at finite values of the magnetic field.

Mean-field theory of the transverse-field Ising spin glass in the classical limit

An infinite-range transverse-field Ising spin glass is studied in the classical limit. At the $T=0$ phase transition the nonlinear susceptibility diverges as a function of the field with an exponent of $1/2$ which is equal to the quantum estimates but without a multiplicative logarithmic correction. The replica-symmetric solution fails at $T=0$ below the critical value of the transverse field. Equations that break the replica symmetry are constructed and are shown to be harder to solve than in the absence of the transverse field. In the absence of the transverse field their solution agrees with that of Sommers and Dupont [J. Phys. C 17, 5785 (1984)].

Solving the Schrödinger equation for the Sherrington - Kirkpatrick model in a transverse field

By numerically solving the Schr\"oedinger equation for small sizes we investigate the quantum critical point of the infinite-range Ising spin glass in a transverse field at zero temperature. Despite its simplicity the method yields accurate information on the value of the critical field and critical exponents. We obtain $\Gamma_c=1.47\pm 0.01$ and check that exponents are in agreement with analytical approaches.

Quantum Monte Carlo study of the infinite-range Ising spin glass in a transverse field

We study the zero-temperature behavior of the infinite-ranged Ising spin glass in a transverse field. Using spin summation techniques and Monte Carlo methods we characterize the zero-temperature quantum transition. Our results are well compatible with a value $\nu=\frac{1}{4}$ for the correlation length exponent, $z=4$ for the dynamical exponent and an algebraic decay $t^{-1}$ for the imaginary-time correlation function. The zero-temperature relaxation of the energy in the presence of the transverse field shows that the system monotonically reaches the ground state energy due to tunneling processes and displays strong glassy effects.

Effects of a Gaussian random field in the Sherrington-Kirkpatrick spin glass.

The infinite-range Ising spin glass in the presence of a Gaussian random field is investigated by means of the replica method. The replica-symmetric solution and its instability are discussed; it is shown that the Almeida-Thouless instability regions are always reduced by the random field. The crossover exponent \ensuremath{\varphi} associated with the irreversibility line is shown to be nonuniversal in the presence of a random field, varying in a range from 1 to 3. Some of our results are compared with recent experimental measurements performed in the diluted antiferromagnet ${\mathrm{Fe}}_{\mathit{x}}$${\mathrm{Zn}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{F}}_{2}$.