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Physica A Statistical Mechanics and its Applications

Quantum vector spin-glass in a field: results for general spin

Yadin Y. Goldschmidt, Pik-Yin Lai

Published Sep 1, 1991• 4 citations
Publisher
DOI
10.1016/0378-4371(91)90199-m

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We consider the quantum Sherrington–Kirkpatrick (SK) spin-glass model with a transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures β > 0. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with Rd-valued spins. This enables us to use the results of Panchenko who generalized [Ann. Probab. 46(2), 829–864 (2018); ibid, 46(2), 865–896 (2018)] the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynamic limit of the free energy of the quantum SK model as the d → ∞ limit of the free energies of the d-dimensional approximations of the model.
Several publications dealing with the infinite-range quantum XY and Heisenberg spin glasses in an external magnetic field K in the z direction share a common supposition: the correlation between the x and y components of the spins is zero. Using the Matsubara imaginary-time formalism we show that this assumption holds only for those models where K=0. With the help of the Trotter-Suzuki formulation we investigate the phase diagram of the XY model for spin S=1/2, and find that it is different from those previously reported because of the nonzero xy correlation. It is also argued that for a general spin S in the XY model at zero temperature the spin-glass phase persists up to Kc{\mathit{K}}_{\mathit{c}}Kc​=2S.