Quantum spherical spin model on hypercubic lattices

Abstract

We present an alternative treatment to the quantum spherical spin model on ͑d Ն 2͒-dimensional hypercubic lattices, focusing on the effects of quantum ͑g͒ and thermal ͑T͒ fluctuations, under a uniform magnetic field h, on the correlation function, correlation length, entropy, specific heat, and energy gap in the excitation spectrum. Explicit expressions for such quantities are provided close to the d Ն 2 quantum ͑g = g c , T =0͒ and d Ն 3 thermal ͓T = T c ͑g͔͒ phase transitions in h = 0, including the low-T quantum regimes near the quantum critical point. In particular, the calculation of the correlation function and correlation length generalizes the results on the g = 0 classical spherical model. At T = 0, the zero-field system is gapless at and below g c ; however, a gap opens in the quantum-disordered ground state, g Ͼ g c . Conversely, the null gap for T Յ T c ͑g 0͒ becomes finite as T → T c ͑g 0͒ + ; thus, quantum fluctuations suppress the critical prefactors of observables near T c ͑g 0͒, though they are irrelevant to the universality class shared with the gapless classical spherical model. The results on the entropy and specific heat in g 0 circumvent the drawback in classical spherical models concerning the third law of thermodynamics, as T → 0.

Supersymmetric extension of the quantum spherical model

“…The extension to the quantum domains has been considered by various authors and has raised much attention in the context of quantum phase transitions [21][22][23]. In particular, we mention studies in some quantum versions of the spherical model [20,[24][25][26][27][28][29], and also including some ingredients of the statistical mechanics, such as the influence of random fields [30], spin glasses [31][32][33][34][35], frustration [36], competing interactions [37], and the quantum Lifshitz point [38]. In this work we extend these studies by considering a supersymmetric version of the spherical model with special attention to the existence of critical points and the determination of critical dimensions.…”

Axiomatic construction of quantum Langevin equations

“…For short-ranged interactions, and dimensions 2 < d < 4, its second-order phase transition is in an universality class distinct from mean-field theory. The quantum spherical model is defined by the hamiltonian [51,40,66,52,67]…”

Stochastic quantization of the spherical model and supersymmetry

“…In this appendix, we wish to discuss the stochastic quantization of the mean spherical model whose constraint is imposed as a thermal average r S 2 r = N (not strictly as in (2)) in contrast with other approaches as the canonical and the path integral quantization methods applied to this specific model [7,9,[18][19][20]. Let us consider the classical Hamiltonian…”