Quantum spherical spin models

Abstract

A recently introduced class of quantum spherical spin models is considered in detail. Since the spherical constraint already contains a kinetic part, the Hamiltonian need not have kinetic term. As a consequence, situations with or without momenta in the Hamiltonian can be described, which may lead to different symmetry classes. Two models that show this difference are analyzed. Both models are exactly solvable and their phase diagram is analyzed. A transversal external field leads to a phase transition line that ends in a quantum critical point. The two considered symmetries of the Hamiltonian considered give different critical phenomena in the quantum critical region. The model with momenta is argued to be analog to the large-N limit of an SU(N ) Heisenberg ferromagnet, and the model without momenta shares the critical phenomena of an SU(N ) Heisenberg antiferromagnet.

Supersymmetric quantum spherical spins

“…It is interesting to observe the following point in the constraint structure in (23). We have the same three constraints as in the o-shell formulation, implemented by the Lagrange multipliers µ, ξ, and ξ .…”

“…We have the same three constraints as in the o-shell formulation, implemented by the Lagrange multipliers µ, ξ, and ξ . On the other hand, the Lagrange multiplier γ, which in the o-shell formulation implemented the last constraint of (6), in the on-shell expression (23) it can be thought as implementing a constraint as an average with a Gaussian distribution instead of a delta due to the term proportional to γ 2 . Its equation of motion is…”

“…Quantized versions of the spherical model go back to [19,20] and in more recent years to [21][22][23]. In [21] the introduction of quantum fluctuations was proposed as natural mechanism to fix the anomalous low-temperature behavior of the classical counterpart (the entropy diverges for T → 0).…”

Supersymmetric extension of the quantum spherical model

“…We need to construct an action in the superspace such that the corresponding Lagrangian reproduces (23) after integration over θ andθ , and further take into account the constraints (27) and (28). Initially, let us consider the kinetic term.…”

“…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.…”