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Physical Review B

Quantum spherical spin model on hypercubic lattices

M. H. Oliveira, Ernesto P. Raposo, Mauricio D. Coutinho-Filho

Published Nov 1, 2006• 18 citations
Publisher

Abstract

We present an alternative treatment to the quantum spherical spin model on $(d\ensuremath{\ge}2)$-dimensional hypercubic lattices, focusing on the effects of quantum $(g)$ and thermal $(T)$ 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 $d\ensuremath{\ge}2$ quantum ($g={g}_{c}$, $T=0$) and $d\ensuremath{\ge}3$ thermal $[T={T}_{c}(g)]$ 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, $g>{g}_{c}$. Conversely, the null gap for $T\ensuremath{\le}{T}_{c}(g\ensuremath{\ne}0)$ becomes finite as $T\ensuremath{\rightarrow}{T}_{c}(g\ensuremath{\ne}0{)}^{+}$; thus, quantum fluctuations suppress the critical prefactors of observables near ${T}_{c}(g\ensuremath{\ne}0)$, though they are irrelevant to the universality class shared with the gapless classical spherical model. The results on the entropy and specific heat in $g\ensuremath{\ne}0$ circumvent the drawback in classical spherical models concerning the third law of thermodynamics, as $T\ensuremath{\rightarrow}0$.

DOI
10.1103/physrevb.74.184101

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We present an alternative treatment to the quantum spherical spin model on (d\ensuremath≥2)(d\ensuremath{\ge}2)(d\ensuremath≥2)-dimensional hypercubic lattices, focusing on the effects of quantum (g)(g)(g) and thermal (T)(T)(T) fluctuations, under a uniform magnetic field hhh, 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\ensuremath≥2d\ensuremath{\ge}2d\ensuremath≥2 quantum (g=gcg={g}_{c}g=gc​, T=0T=0T=0) and d\ensuremath≥3d\ensuremath{\ge}3d\ensuremath≥3 thermal [T=Tc(g)][T={T}_{c}(g)][T=Tc​(g)] phase transitions in h=0h=0h=0, including the low-TTT quantum regimes near the quantum critical point. In particular, the calculation of the correlation function and correlation length generalizes the results on the g=0g=0g=0 classical spherical model. At T=0T=0T=0, the zero-field system is gapless at and below gc{g}_{c}gc​; however, a gap opens in the quantum-disordered ground state, g>gcg>{g}_{c}g>gc​. Conversely, the null gap for T\ensuremath≤Tc(g\ensuremath≠0)T\ensuremath{\le}{T}_{c}(g\ensuremath{\ne}0)T\ensuremath≤Tc​(g\ensuremath=0) becomes finite as T\ensuremath→Tc(g\ensuremath≠0)+T\ensuremath{\rightarrow}{T}_{c}(g\ensuremath{\ne}0{)}^{+}T\ensuremath→Tc​(g\ensuremath=0)+; thus, quantum fluctuations suppress the critical prefactors of observables near Tc(g\ensuremath≠0){T}_{c}(g\ensuremath{\ne}0)Tc​(g\ensuremath=0), 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\ensuremath≠0g\ensuremath{\ne}0g\ensuremath=0 circumvent the drawback in classical spherical models concerning the third law of thermodynamics, as T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0.
The quantum version of the random-field spherical XY\mathrm{XY}XY model on a ddd-dimensional hypercubic lattice is presented in the boson space, and solved exactly. We recover the Imry-Ma result concerning the lower critical dimension dc=4,{d}_{c}=4,dc​=4, and calculate the critical exponents near the critical temperature. In particular, we obtain a general phase diagram of the model with arbitrary dimension ddd when d>dc.d>{d}_{c}.d>dc​. The ferromagnetic ordering is reduced by quantum fluctuations, and destroyed completely by random-field fluctuations with sufficiently large values of the random-field variance \ensuremath{\sigma}. The entropy and the specific heat vanish as Td/2{T}^{d/2}Td/2 at low temperatures. Since the model is equivalent to a Bose system, we show a superfluid-Mott insulator transition at a critical chemical potential ${\ensuremath{\mu}}_{c}.$
Abstract The entropy, the internal energy and the specific heat of a d-dimensional quantum spherical model with a long-range interaction, the decreasing of which is controlled by a parameter σ (0 < σ ≤ 2), are studied close to its quantum critical point in the context of the finite-size scaling (FSS) theory for d = σ. The obtained specific heat critical exponent σ does not depend on σ and satisfies the quantum hyperscaling relation. Based on the derived FSS forms and asymptotes of the universal scaling functions of the considered quantities, the leading temperature dependencies of the entropy, the internal energy and the specific heat are obtained in both the renormalized classical and the quantum disordered regions of the phase diagram. It has been shown that when the temperature T → +0, the entropy and the specific heat in the renormalized classical region tend to zero as powers of T, while in the quantum disordered region they tend to zero exponentially.
The thermodynamic properties of a ddd-dimensional classical vector-spin system in a random magnetic field are calculated explicitly in the spherical-model limit. The system is characterized by significant spin-glass order of the Edwards-Anderson type even when no transition to a state with spontaneous magnetization occurs. This spin-glass order should result in observable effects in the magnetic susceptibility and specific heat. The properties of the random-field spherical model are compared with those reported for the corresponding Ising model. The behavior of the coherence length is identical in both models at their respective lower critical dimensionalities.
Explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of a spherical model of spins on a d-dimensional hypercubical lattice, of size L1{L}_{1}L1​\ifmmode\times\else\texttimes\fi{}L2{L}_{2}L2​\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}Ld{L}_{d}Ld​, under antiperiodic boundary conditions. The relevant scaling functions that govern the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis [Phys. Rev. B 30, 322 (1984)] on the hyperuniversality of finite systems are verified. Approach towards standard critical behavior, both for TTc{T}_{c}Tc​(\ensuremath{\infty}) and T>Tc{T}_{c}Tc​(\ensuremath{\infty}), is examined. In the former case, the approach generally takes place through a power law; only in some special situations does one obtain an exponential law instead. In the latter case, the approach is generally exponential, except for the susceptibility of the system which (somewhat surprisingly) displays a finite-size effect dominantly determined by the surface-to-volume ratio of the lattice.
We study a quantum extension of the spherical p-spin-glass model using the imaginary-time replica formalism. We solve the model numerically and we discuss two analytical approximation schemes that capture most of the features of the solution. The phase diagram and the physical properties of the system are determined in thermodynamic equilibrium and under conditions of marginal stability. In both cases, the phase diagram consists of two qualitatively different regions. If the transition temperature is higher than a certain value T∗,{T}^{*},T∗, quantum effects are qualitatively irrelevant and the phase transition is second order, as in the classical case. When quantum fluctuations depress the transition temperature below T∗,{T}^{*},T∗, the transition becomes first order. The susceptibility is discontinuous and shows hysteresis across the first-order line. This behavior is reminiscent of that observed in the dipolar Ising spin-glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ in an external transverse magnetic field. We discuss the thermodynamics and the stationary dynamics of both states. The spectrum of magnetic excitations of the equilibrium spin-glass state is gapped, leading to an exponentially small specific heat at low temperatures. That of the marginally stable state is gapless and its specific heat varies linearly with temperature, as generally observed in glasses at low temperature. We show that the properties of the marginally stable state are closely related to those obtained in studies of the real-time dynamics of the system weakly coupled to a quantum thermal bath. Finally, we discuss a possible application of our results to the problem of polymers in random media.
A quantum spherical spin model with an antiferromagnetic coupling JJJ on the AB2A{B}_{2}AB2​ chain is studied, whose topology is of interest in the context of ferrimagnetic polymers and oxocuprates. A ferrimagnetic long-range order is found at the only critical point g=T=h=0g=T=h=0g=T=h=0, where TTT denotes the temperature, hhh the magnetic field, and ggg the quantum coupling constant in energy units. The approach to the critical point, with diverging correlation length $\ensuremath{\xi}$ and cell susceptibility ${\ensuremath{\chi}}_{\text{cell}}$, is characterized through several paths in the g,T,h{g,T,h}g,T,h parameter space: for $(T∕J)\ensuremath{\rightarrow}0$ and g=h=0g=h=0g=h=0, ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕{T}^{2}$, as also found in several classical and quantum spherical and Heisenberg models; for $(h∕J)\ensuremath{\rightarrow}0$ and g=T=0g=T=0g=T=0, ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕h$; and for $(g∕J)\ensuremath{\rightarrow}0$ and T=h=0T=h=0T=h=0, $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕\sqrt{g}$, thus evidencing an essential singularity due to quantum fluctuations. In any path chosen the relation ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}{\ensuremath{\xi}}^{2}$ is satisfied. For finite ggg and TTT a field-induced short-range ferrimagnetism occurs to some extent in the g,T,h{g,T,h}g,T,h space, as confirmed by the analysis of the local spin averages, cell magnetization with a rapid increase for very low fields, and spin-spin correlation functions. The asymptotic limits of the correlation functions are also provided with respect to ggg, TTT, hhh, and spin distance. The analysis of the entropy and specific heat reveals that the quantum fluctuations fix the well-known drawback of classical spherical models concerning the third law of thermodynamics.