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arXiv:cond-mat/0505522 · Physical Review B

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

Eduardo Muccillo Monteiro Santos, Alba Theumann

Published Jul 19, 2005• 4 citations
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Abstract

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.

DOI
10.1103/physrevb.72.014447

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We analyze the infinite range Ising spin glass in a transverse-field \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ below the critical temperature by a one step replica symmetry breaking theory. The set of nnn replicas is divided in rrr blocks of mmm replicas each. We present results for different values of the block-size parameter mmm. 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 Siz{S}_{i}^{z}Siz​ 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 Tc(\ensuremathΓ){T}_{c}(\ensuremath{\Gamma})Tc​(\ensuremathΓ) that decreases when \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ increases, until it reaches a quantum critical point at the same value of \ensuremathΓc{\ensuremath{\Gamma}}_{c}\ensuremathΓc​ 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 replica-symmetry-breaking (RSB) solution of the infinite-range Ising spin glass in the presence of a transverse field is obtained. The quenched free energy and the phase boundary of the glass transition temperature versus the transverse field are calculated at first-step RSB without using the static approximation. We demonstrate that replica symmetry (RS) has to be broken in the spin-glass phase by comparing the free energies of the RSB and RS solutions. No evidence is found to support an intermediate spin-glass phase with replica symmetry.
In the present work, a fermionic infinite-range Ising spin glass (SG) model with a transverse field Γ is studied. The model is analyzed in a one-lattice structure (quantum SG) and in a two-sublattice structure (SG/antiferromagnetism (AF) competition). In these two cases, the exchange interactions are random variables that follow Gaussian probability distributions (SK interactions). For the two-sublattice model, spins of distinct sublattices can couple with an SK-type interaction with an AF average. For a general case of this model, the disordered interactions can also occur with spins of same sublattices. The problem is formulated in a Grassmann path integral formalism within the static ansatz. The replica method is applied to treat the disorder, in which the replica symmetry (RS) and one-step replica symmetry breaking (1P-RSB) approximations are used. The one-lattice model results provide evidences that the 1S-RSB can be used to improve the physical description of the ordered quantum SG phase treated by the replica method. In the two-sublattice 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 (Tf ) shows a higher increase with H when V enhances. A discontinuous 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 Γ 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. Therefore, the presence of both fields, Γ and H, produces effects that are in competition on Tf for instance. However, in the asymmetric RSB region, Γ 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.
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 $\ensuremath{\Gamma}$. The problem is formulated in a Grassmann path integral formalism within the static approximation. In this model, a parallel magnetic field HHH breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF)(\mathrm{SG}+\mathrm{AF})(SG+AF) characterizing an asymmetric RSB region. In this region, intrasublattice disordered interactions VVV increase the difference between the RSB solutions of each sublattice. The freezing temperature shows a higher increase with HHH when VVV enhances. A discontinuous phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intrasublattice ferromagnetic average coupling. The $\ensuremath{\Gamma}$ 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, $\ensuremath{\Gamma}$ 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 intrasublattice spins couple with disordered interactions.
The fermionic Ising spin glass models in a transverse field are investigated in a Grassmann path integral formalism. The Parisi's scheme of one-step replica symmetry breaking (RSB) is used within the static ansatz. This formalism has already been applied in a theory in which mmm (Parisi's block-size parameter) is taken as a constant [Santos and Theumann Phys. Rev. B 72, 014447 (2005)]. Now, it is extended to consider mmm as a variational parameter. In this case, the results show that RSB is present when $T\ensuremath{\rightarrow}0$, in which the system is driven by quantum fluctuations.
The present work analyzes the fermionic Ising spin glass model in a transverse magnetic field. The problem is written in the Grand Canonical ensemble with the spin operators represented by bilinear combinations of fermionic fields. The transverse field is a spin "flip" mechanism and inserts quantum fluctuations, which can lead the transition temperature Tj to quantum critical point (QCP). In this case, the phase transition from spin glass to paramagnetic phases occurs in T = o. On the other hand, charge fluctuations controlled by chemical potentiall-L show the existence of a tricritical point. The replica method with replica symmetry and static approximation are used to solve this problem. However, the replica symmetry solution shows some problems, like negative entropy in low temperatures. Thus, the stability conditions are derived to test the generated solution of this IDodel. The main purpose of this work is to study the competition between spin glass and paramagnetic phases with both quantum and charge fluctuations. After the stability conditions are derived, a detailed sight into the generated solution is shown under the form of diagrams phase. The stability conditions, however, show another applications in the present problem, once they can be used, for instance, in the tricritical point derivation.
We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field ΓΓΓ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of ΓΓΓ and for system sizes N≤16N \le 16N≤16. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of ΓΓΓ. The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at Γc≃1.5Γ_c \simeq 1.5Γc​≃1.5. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get z=2.1±0.1z=2.1 \pm 0.1z=2.1±0.1 which is in contradiction with a recent estimate (z=4z=4z=4). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for Γ<ΓcΓ< Γ_cΓ<Γc​.