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Physical review. B, Condensed matter

Spherical model for superfluidity in a restricted geometry

Shmuel Fishman, Timothy A. L. Ziman

Published Aug 1, 1982• 12 citations
Publisher

Abstract

The spherical model is solved on a hypercubic lattice in $d$ dimensions, each bond of which is decorated with $l$ spins. The thermodynamic functions and the helicity modulus, analogous to a superfluid density, are calculated. We find that at least two spherical fields are required for the model to exhibit low-temperature properties that can approximate reasonably those of $O(n)$ models. The heuristic prediction that the critical temperature behaves as ${T}_{c}(l)\ensuremath{\sim}{(l+1)}^{\ensuremath{-}1}$ is checked for the model and found to hold quite accurately even for small $l(\ensuremath{\gtrsim}2)$. The helicity modulus and magnetization of the two-constraint spherical model are found to scale approximately with the critical temperature, but the relation between them is more complex than in the undecorated model. This relation is used to check heuristic arguments concerning the helicity modulus at low temperatures. We comment on the relevance to physical systems, in particular, the problem of boson condensation in a restricted geometry.

DOI
10.1103/physrevb.26.1258

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The critical temperatures Kc and the nearest-neighbour correlation functions Uc have been calculated for the d-dimensional hypercubic Ising ferromagnet for d=4 to d=30, by generalizing a recent theory reported for d=3. The results are of relevance for Monte Carlo Renormalization Group work involving critical properties in random fields, which calls for a knowledge of Kc in the limit of zero field. As d becomes large, Kc and Uc are found to approach the spherical model values. Polynomial expressions in 1/d are fitted to the calculated values, using the analogous spherical model expressions as guide. Comparison is made with the corresponding expressions of Fisher and Gaunt. Uc/(1−Kc), independent of d for the spherical model, is investigated; it is seen to be approximately independent of lattice structure for d=2 and for d=3.

Journal of Applied Physics•Nov 1, 1982•8 citations
The spherical model is solved on a hypercubic lattice in ddd dimensions, each bond of which is decorated with lll spins. The thermodynamic functions and the helicity modulus, analogous to a superfluid density, are calculated. We find that at least two spherical fields are required for the model to exhibit low-temperature properties that can approximate reasonably those of O(n)O(n)O(n) models. The heuristic prediction that the critical temperature behaves as Tc(l)\ensuremath∼(l+1)\ensuremath−1{T}_{c}(l)\ensuremath{\sim}{(l+1)}^{\ensuremath{-}1}Tc​(l)\ensuremath∼(l+1)\ensuremath−1 is checked for the model and found to hold quite accurately even for small l(\ensuremath≳2)l(\ensuremath{\gtrsim}2)l(\ensuremath≳2). The helicity modulus and magnetization of the two-constraint spherical model are found to scale approximately with the critical temperature, but the relation between them is more complex than in the undecorated model. This relation is used to check heuristic arguments concerning the helicity modulus at low temperatures. We comment on the relevance to physical systems, in particular, the problem of boson condensation in a restricted geometry.
The field-theoretical renormalization-group techniques previously developed for the description of the mestastable phases of isotropic, homogeneous, ddd-dimensional systems, which (close to their critical temperature) can be described in terms of the Landau-Ginzburg-Wilson functional integral with a two-component order parameter, and applied to obtain the scaling form of the helicity modulus, defined as a measure of the free-energy increment associated with a metastable, helical twisting of the order parameter. The general scaling form of this quantity is determined even in the presence of an external helical field, for arbitrary helix pitch and temperature. In the case of vanishing external field, this quantity represents the superfluid density and is a function of the temperature and the superfluid velocity (which corresponds to the inverse of the helix pitch). In the limit of small velocities this expression exhibits the well-known Josephson asymptotic form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{\nu}(d\ensuremath{-}2)}={({T}_{c}\ensuremath{-}T)}^{2\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}\ensuremath{\eta}}$. An explicit $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion for the helicity modulus is presented in the case of arbitrary external field to the one-loop approximation.
The ordered state of a ddd-dimensional isotropic system with an nnn-vector ($n\ensuremath{\ge}2$) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus $\ensuremath{\Upsilon}(T)$ which measures the free-energy increment associated with "twisting" the direction of the order parameter. For a Bose system the superfluid density is seen to be ${\ensuremath{\rho}}_{s}(T)={(\frac{m}{\ensuremath{\hbar}})}^{2}\ensuremath{\Upsilon}(T)$. A critical exponent vvv is defined by $\ensuremath{\Upsilon}(T)\ensuremath{\sim}{|T\ensuremath{-}{T}_{c}|}^{v}$ as $T\ensuremath{\rightarrow}{T}_{c}$; for an ideal Bose gas and spherical model ($n\ensuremath{\rightarrow}\ensuremath{\infty}$), v=1v=1v=1 is an exact result for all d>2d>2d>2. The difficulties of defining a correlation length in the ordered phase are discussed. A full scaling theory of the correlations avoids these problems and may be linked to a phenomenological hydrodynamic approach, to clarify and rederive Josephson's relation $v=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$. This reduces to $v=(d\ensuremath{-}2)\ensuremath{\nu}$ (used by some authors with d=3d=3d=3), only if one accepts ddd-dependent, "hyperscaling" relations such as $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$; however, both these latter relations fail for the ideal Bose gas when d>4d>4d>4. An alternative derivation of the formula $v=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$ is based on the scaling theory for systems with a large but finite dimension.
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)(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 $d\ensuremath{\ge}2$ quantum (g=gcg={g}_{c}g=gc​, T=0T=0T=0) and $d\ensuremath{\ge}3$ 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 $T\ensuremath{\le}{T}_{c}(g\ensuremath{\ne}0)$ 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 $T\ensuremath{\rightarrow}0$.
The helicity modulus, which is the stiffness associated with a twisted order parameter, for the two-dimensional Hubbard model is calculated for the equivalent cases of (i) attractive on-site interaction (negative U) with arbitrary strength, arbitrary electron density, and zero magnetic field and (ii) repulsive on-site interaction (positive U) with arbitrary strength, at half-filling and in an arbitrary magnetic field. An explicit formula for the helicity modulus is derived using the Bogoliubov-Hartree-Fock approximation. An improved value for the helicity modulus is obtained by performing variational Monte Carlo calculations using a Gutzwiller projected trial wave function. To within a small correction term the helicity modulus is found to be given by -1/8 the average kinetic energy. The variational Monte Carlo calculation is found to increase the value of the helicity modulus by a small amount (about 5% for intermediate values of the interaction strength) compared to the results from the Bogoliubov-Hartree-Fock approximation. In the case of attractive interaction, from a comparison with the Kosterlitz-Thouless relation between critical temperature and helicity modulus, the critical temperature for a Kosterlitz-Thouless transition is calculated and a phase diagram is obtained. An optimal critical temperature is found for an intermediate value of U. We discuss connections of our results with results in the literature on the Hubbard model using the random-phase approximation and quantum Monte Carlo calculations.
The critical temperatures Kc and the nearest-neighbour correlation functions Uc have been calculated for the d-dimensional hypercubic Ising ferromagnet for d=4 to d=30, by generalizing a recent theory reported for d=3. The results are of relevance for Monte Carlo Renormalization Group work involving critical properties in random fields, which calls for a knowledge of Kc in the limit of zero field. As d becomes large, Kc and Uc are found to approach the spherical model values. Polynomial expressions in 1/d are fitted to the calculated values, using the analogous spherical model expressions as guide. Comparison is made with the corresponding expressions of Fisher and Gaunt. Uc/(1−Kc), independent of d for the spherical model, is investigated; it is seen to be approximately independent of lattice structure for d=2 and for d=3.