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arXiv:cond-mat/0606246 · Physical Review B

One-step replica symmetry breaking solution for a fermionic Ising spin glass in a transverse field

Fábio M. Zimmer, S. G. Magalhães

Published Jul 14, 2006• 9 citations
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Abstract

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 $m$ (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 $m$ 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.

DOI
10.1103/physrevb.74.012202

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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→0T\ensuremath{\rightarrow}0T\ensuremath→0, in which the system is driven by quantum fluctuations.
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.
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.
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 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 ${T}_{c}(\ensuremath{\Gamma})$ 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 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.
We investigate the inverse freezing in the fermionic Ising spin-glass model in a transverse field $\ensuremath{\Gamma}$. The grand canonical potential is calculated in the static approximation, replica symmetry, and one-step replica symmetry breaking parisi scheme. It is argued that the average occupation per site nnn is strongly affected by $\ensuremath{\Gamma}$. As a consequence, the boundary phase is modified and, therefore, the reentrance associated with the inverse freezing is modified too.
Parisi's replica-symmetry-breaking scheme is applied to the quantum Sherrington-Kirkpatrick Ising spin glass in a transverse magnetic field. The order-parameter function q(x) is calculated near Tf{\mathit{T}}_{\mathit{f}}Tf​, while for all temperatures the first stage of replica-symmetry breaking is calculated. Internal energy, entropy, and magnetic susceptibility are calculated in order to investigate the quality of both the replica-symmetric and the replica-symmetry-broken theories.