The Ising spin glass in a transverse field revisited. Results of two fermionic models (original) (raw)

2002, Physica a-Statistical Mechanics and Its Applications

In the calculation of the quantum mechanical partition function special tools are needed to deal with the non-commuting operators forming the Hamiltonian. The method more currently used in the study of short-range and infinite range spin glasses in a transverse field is the Trotter-Suzuki formula , that maps a system of quantum spins in d-dimensions to a classical system of spins in (d + 1)-dimensions, and it is suited to perform numerical studies. Another way of dealing with the non-commutativity of quantum mechanical operators is to use Feynman's path integral formulations and to introduce time-ordening by means of an imaginary time 0 ≤ τ ≤ β, where β is the inverse temperature. Both methods use unitarity and closure to express the quantum mechanical partition function as a trace over M intermediate time steps τ ≈ O(1 M) , when M → ∞ , and they must lead to the same exact solution. We may ask, however, if the same holds true as regards to approximations. In the present work we investigate this problem by representing the spin operators as bilinear combinations of fermions fields in the Hamiltoniam for a long range Ising spin glass in a transverse field. This path integral formulation has a natural application in problems in condensed matter theory, where the fermions operators represent electrons that also participate in other physical processes. A possible criticism, however, may be that the spin eigenstates at each site do not belong to one irreducible representation S Z = ± 1 2 , but they are labeled instead by the fermionic occupation numbers n σ = 0 or 1 , giving two more unwanted states with S z = 0. We call this the "four states" (4S) model, and despite the presence of these two unwanted states the 4S-Ising spin glass model describes a spin glass transition with the same characteristics as the Sherrington-Kirkpatrick (SK) model in a replica symmetric theory. A way to get rid of the unwanted states consists in fixing the occupation number n i↑ + n i↓ by means of an integral constraint at every site. We refer this as the "two states" 2S-Ising spin glass model. We analyse the 4S-Ising and 2S-Ising spin glass models in a transverse field, within the static approximation in a replica symmetric theory. The static ansatz neglects time fluctuations and may be considered an approximation similar to mean field theory. When Γ = 0 the static approximation reproduces the exact results obtained by other methods, in particular for the 2S-Ising spin glass model we recover SK equations. The results in both models are very similar; they both exhibit a critical spin glass temperature T c (Γ) that decreases when the strength Γ of the transverse field increases, until it reaches a quantum critical point(QCP) at Γ c , T c (Γ c) = 0. The value of Γ c is the same for both models and the 4S-Ising and 2S-Ising models are identical close to the QCP.