Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics C Solid State Physics

Critical behaviour of the spherical model with enhanced surface exchange

Michael N. Barber, David Jasnow, Surjit Singh, R. A. Weiner

Published Oct 7, 1974• 20 citations
Publisher

Abstract

The local surface susceptibility chi 1,1(T) is calculated and analysed in detail for a ferromagnetic spherical model on a d-dimensional hypercubic half-space bounded by a free surface in which the exchange coupling, J(1+ Delta ) with Delta >0, is enhanced over the bulk value J. For Delta > Delta c=(2d-2)-1, chi 1,1(T) is found to diverge at a temperature Tcx( Delta ) above the bulk critical temperature, while for Delta < Delta c, chi 1,1(T) behaves as for a surface without enhancement ( Delta =0). The nature of the crossover phenomena is explored; it is shown that for Delta near Delta c and T near Tcx( Delta ), chi 1,1(T) can be represented in a scaled form in accord with the general theory of crossover phenomena.

DOI
10.1088/0022-3719/7/19/009

Related papers

Surface critical exponents for a three-dimensional modified spherical model

Daniel M. Danchev, J.G. Brankov, Magdy E. Amin

Abstract. A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some ρ> 0, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility χ1,1 has been evaluated exactly. For ρ = 1 we find that χ1,1 is finite at the bulk critical temperature Tc, in contrast with the recently derived value γ1,1 = 1 in the case of just one global spherical constraint. The result γ1,1 = 1 is recovered only if ρ = ρc = 2 − (12Kc) −1, where Kc is the dimensionless critical coupling. When ρ> ρc, χ1,1 diverges exponentially as T → T + c. An effective hamiltonian which leads to an exactly solvable model with γ1,1 = 2, the value for the n → ∞ limit of the corresponding O(n) model, is proposed too.

•Nov 19, 2012•6 citations

Surface effects on magnetic phase transitions

Kurt Binder, Pierre C. Hohenberg

The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange ($0<{J}_{s}<J$), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below ${T}_{c}$, thus providing a possible explanation for previous experiments. For sufficiently strong negative ${J}_{s}$, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.

Physical review. B, Solid state•Mar 1, 1974•566 citations

Surface critical exponents for a three-dimensional modified spherical model

D M Danchev, J G Brankov, M E Amin

A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann - Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some , and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals 1). The surface susceptibility has been evaluated exactly. For we find that is finite at the bulk critical temperature , in contrast to the recently derived value of in the case of just one global spherical constraint. The result is only recovered if , where is the dimensionless critical coupling. When , diverges exponentially as . An effective Hamiltonian is also proposed which leads to an exactly solvable model with , the value for the limit of the corresponding model.

arXiv:cond-mat/9706159•Journal of Physics A Mathematical and General•Aug 21, 1997•7 citations

Quantum crossover in the transverse Ising model

Ian D. Lawrie

The critical behaviour of d dimensional Ising models in a transverse magnetic field, Gamma , is investigated using a quantal generalisation of the Landau-Ginzburg-Wilson free-energy functional introduced by Hertz and Young (1976). A variational approximation of the Hartree type allows a detailed analysis: behaviour near the critical line, Gamma = Gammac(T) is governed by the exponents of the d dimensional spherical model (which are mean-field-like for d>or=4). However, the point ( Gamma ,T)=( Gammac(0),0) is multicritical in nature, and characterised by the exponents of the (d+1) dimensional spherical model. Crossover scaling functions, valid near the multicritical point, are found for the longitudinal susceptibility and free energy. They involve the crossover exponent phiT; for 2<d<or=3 this is equal to the (d+1) dimensional correlation length exponent, nud+1=1/(d-1); for 3<d<or=4, we find phiT=1/2(4-d), while for d>4, no crossover occurs. These results are shown to be consistent, for all d>2, with the finite-size scaling analogy suggested by Suzuki (1976) and others.

Journal of Physics C Solid State Physics•Mar 28, 1978•38 citations

Finite-size effects in the spherical model of ferromagnetism: Zero-field susceptibility under antiperiodic boundary conditions

Surjit Singh, Raj K. Pathria, Michael Ellis Fisher

The overall zero-field susceptibility \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} of a finite-sized spherical model of spins under various antiperiodic boundary conditions is reexamined with a view to explaining the finite-size effects of an algebraic nature found recently by Singh and Pathria. The cause of this ``unexpected'' behavior at temperatures above the bulk critical temperature ${T}_{c}$(\ensuremath{\infty}) is seen to lie in the spatial variation of the local susceptibility which, on averaging over the system, leads precisely to the effects found previously. Below ${T}_{c}$(\ensuremath{\infty}), the influence of antiperiodic conditions is even more severe, in that not only are the finite-size amplitudes for \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} modified but, for the local susceptibility, new exponents also appear.

Physical review. B, Condensed matter•May 1, 1986•10 citations

Magnetic-Phase Transitions of Ising Surfaces with Modified Surface-Bulk Coupling: a Monte Carlo Study

Michele Vendruscolo, Mauro Rovere, Annalisa Fasolino

We present the Monte Carlo phase diagram of Ising surfaces with intrasurface ( J s ) and surface-bulk ( J 1 ) interactions modified with respect to the bulk exchange interaction J . The new parameter J 1 strongly affects the position of the Binder-Landau multi critical point and the crossover from three-dimensional to two-dimensional behaviour. The surface critical exponents β 1 and ϕ are determined by crossover scaling analysis. A detailed comparison with the results of the mean-field equations is also presented. The Monte Carlo phase diagram shows that stronger modifications of the surface couplings, with respect to those obtained by mean field, are needed to produce an ordered surface phase on a paramagnetic bulk. This is particularly relevant at low values of J s ; in the Monte Carlo simulation the surface is disordered for all values of J 1 when J s < (0.4 ÷ 0.5) J , while the mean field predicts that the ordered phase is present even in the limit J s → 0.

Europhysics Letters (EPL)•Nov 15, 1992•14 citations
The local surface susceptibility chi 1,1(T) is calculated and analysed in detail for a ferromagnetic spherical model on a d-dimensional hypercubic half-space bounded by a free surface in which the exchange coupling, J(1+ Delta ) with Delta >0, is enhanced over the bulk value J. For Delta > Delta c=(2d-2)-1, chi 1,1(T) is found to diverge at a temperature Tcx( Delta ) above the bulk critical temperature, while for Delta < Delta c, chi 1,1(T) behaves as for a surface without enhancement ( Delta =0). The nature of the crossover phenomena is explored; it is shown that for Delta near Delta c and T near Tcx( Delta ), chi 1,1(T) can be represented in a scaled form in accord with the general theory of crossover phenomena.
Abstract. A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some ρ> 0, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility χ1,1 has been evaluated exactly. For ρ = 1 we find that χ1,1 is finite at the bulk critical temperature Tc, in contrast with the recently derived value γ1,1 = 1 in the case of just one global spherical constraint. The result γ1,1 = 1 is recovered only if ρ = ρc = 2 − (12Kc) −1, where Kc is the dimensionless critical coupling. When ρ> ρc, χ1,1 diverges exponentially as T → T + c. An effective hamiltonian which leads to an exactly solvable model with γ1,1 = 2, the value for the n → ∞ limit of the corresponding O(n) model, is proposed too.
The surface scaling theory previously presented by the authors is developed further, and derived heuristically from a cluster model. Monte Carlo calculations are carried out to obtain the spatial and temperature dependence of the magnetization in Ising and Heisenberg systems with free surfaces. The exponent ${\ensuremath{\beta}}_{1}$ of the (surface) layer magnetization is shown to agree with the scaling value (${\ensuremath{\beta}}_{1}\ensuremath{\approx}\frac{2}{3}$) previously derived. In the Heisenberg system, the results at low temperature agree with a spin-wave calculation by Mills and Maradudin. Ising models with modified exchange ${J}_{s}=J(1+\ensuremath{\Delta})\ensuremath{\ne}J$ on the surface are considered, both in mean-field theory and by means of high-temperature-series expansions. The critical value ${\ensuremath{\Delta}}_{c}$ for surface ordering is found from the series to be 0.6, compared to the mean-field value of 0.25. For $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ there is a temperature region in which the surface behaves like a bulk two-dimensional Ising model near its phase transition. The critical exponents experience a crossover at $\ensuremath{\Delta}={\ensuremath{\Delta}}_{c}$, which is reflected in poorly behaved series, and effective exponents differing from the true ones for $\ensuremath{\Delta}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{c}$. In the case of weakened surface exchange (0<Js<J0<{J}_{s}<J0<Js​<J), the layer magnetization is shown to fit a linear temperature dependence over a large temperature range below Tc{T}_{c}Tc​, thus providing a possible explanation for previous experiments. For sufficiently strong negative Js{J}_{s}Js​, mean-field theory predicts that the surface will order antiferromagnetically while the bulk is ferromagnetic.
A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann - Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some , and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals 1). The surface susceptibility has been evaluated exactly. For we find that is finite at the bulk critical temperature , in contrast to the recently derived value of in the case of just one global spherical constraint. The result is only recovered if , where is the dimensionless critical coupling. When , diverges exponentially as . An effective Hamiltonian is also proposed which leads to an exactly solvable model with , the value for the limit of the corresponding model.
The critical behaviour of d dimensional Ising models in a transverse magnetic field, Gamma , is investigated using a quantal generalisation of the Landau-Ginzburg-Wilson free-energy functional introduced by Hertz and Young (1976). A variational approximation of the Hartree type allows a detailed analysis: behaviour near the critical line, Gamma = Gammac(T) is governed by the exponents of the d dimensional spherical model (which are mean-field-like for d>or=4). However, the point ( Gamma ,T)=( Gammac(0),0) is multicritical in nature, and characterised by the exponents of the (d+1) dimensional spherical model. Crossover scaling functions, valid near the multicritical point, are found for the longitudinal susceptibility and free energy. They involve the crossover exponent phiT; for 2<d<or=3 this is equal to the (d+1) dimensional correlation length exponent, nud+1=1/(d-1); for 3<d<or=4, we find phiT=1/2(4-d), while for d>4, no crossover occurs. These results are shown to be consistent, for all d>2, with the finite-size scaling analogy suggested by Suzuki (1976) and others.
The overall zero-field susceptibility \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} of a finite-sized spherical model of spins under various antiperiodic boundary conditions is reexamined with a view to explaining the finite-size effects of an algebraic nature found recently by Singh and Pathria. The cause of this ``unexpected'' behavior at temperatures above the bulk critical temperature Tc{T}_{c}Tc​(\ensuremath{\infty}) is seen to lie in the spatial variation of the local susceptibility which, on averaging over the system, leads precisely to the effects found previously. Below Tc{T}_{c}Tc​(\ensuremath{\infty}), the influence of antiperiodic conditions is even more severe, in that not only are the finite-size amplitudes for \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{} modified but, for the local susceptibility, new exponents also appear.
We present the Monte Carlo phase diagram of Ising surfaces with intrasurface ( J s ) and surface-bulk ( J 1 ) interactions modified with respect to the bulk exchange interaction J . The new parameter J 1 strongly affects the position of the Binder-Landau multi critical point and the crossover from three-dimensional to two-dimensional behaviour. The surface critical exponents β 1 and ϕ are determined by crossover scaling analysis. A detailed comparison with the results of the mean-field equations is also presented. The Monte Carlo phase diagram shows that stronger modifications of the surface couplings, with respect to those obtained by mean field, are needed to produce an ordered surface phase on a paramagnetic bulk. This is particularly relevant at low values of J s ; in the Monte Carlo simulation the surface is disordered for all values of J 1 when J s < (0.4 ÷ 0.5) J , while the mean field predicts that the ordered phase is present even in the limit J s → 0.