Short‐range ±J interaction Ising spin glass in a transverse field on a Bethe lattice: a quantum‐spherical approach

Abstract

We consider the short-range interaction disordered quantum Ising model with symmetric binary ±J bond distribution on the Bethe lattice (with coordination number z). The system exhibits quantum phase transition separating the spin glass and disordered phases where the quantum effect are regulated by a parameter ∆ describing the transverse field. By introducing a mapping of the quantum Hamiltonian of the model onto a soft-spin action we consider it truncated version in a form of the solvable quantized spherical model. Quantum dynamics is examined via various correlation functions on the infinite tree which are evaluated in a closed form.

The Hubbard model on the Bethe lattice via variational uniform tree states: metal-insulator transition and a Fermi liquid

“…Solutions on the finite coordination number Bethe lattice provide a better approximation to thermodynamic quantities than the mean-field approximation (corresponding to the infinite dimensional limit) [26][27][28]. Models studied on the Bethe lattice include classical and quantum spin models [29][30][31][32][33][34][35], spin glass systems [36][37][38][39], the Bose Hubbard model [40], and models of Anderson localization [41][42][43][44][45]. The fermionic Hubbard model on the finite version of the z = 3 Bethe lattice (known as a Cayley tree) has also been studied previously using a variant of the density matrix renormalization group (DMRG) algorithm [46], but only the case of half filling was studied (which is a charge insulator) and only local ground state quantities given (energy, staggered magnetization and its fluctuations, and neighboring spin correlations).…”

Absence of Goldstone bosons on the Bethe lattice

“…In addition, we note variational work on the ground state of the quantum Ising model 7 and on a spherical model of a spin glass. 8 In this paper we study the quantum unfrustrated ͑ferro-magnetic͒ problem on the Bethe lattice analytically. For technical reasons, we do this nominally in the form of the spherical model for a scalar field but it is essentially also the large N f limit of the nearest neighbor O͑N f ͒ quantum rotor model on the Bethe lattice.…”

“…4,5,6 This has led to computational results on the quantum Ising model, the quantum Ising spin glass and on the Bose-Hubbard model. In addition, we note variational work on the ground state of the quantum Ising model 7 and on a spherical model of a spin glass 8 .…”