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Journal of Applied Physics

Scaling functions for quantum crossover

Ian D. Lawrie, Michael Ellis Fisher

Published Mar 1, 1978• 6 citations
Publisher

Abstract

The critical properties of d-dimensional Ising models in a transverse magnetc field, Hx=Γ, are considered. A quantal generalization of the Landau-Ginzburg-Wilson free-energy functional, introduced by Hertz and Young, has been treated by a variational approximation of the Hartree type. Behavior near the critcal line, Γ=Γc(T), is governed by the exponents of the d-dimensional spherical model. However, quantal crossover occurs on approach to the T=0 multicritical point at Γ=Γc(0) and this is characterized by the exponents of the (d+1) -dimensional spherical model (which are mean-field-like for d+1≳4). Crossover scaling functions, valid near the multicritical point, are reported for the susceptibility. The quantal crossover exponent φT, is equal to νd+1=1/(d−1), the (d+1) -dimensional correlation length exponent for 2⩽d⩽3, as suggested by ideas of Suzuki, but for 3<d⩽4 we find instead φT= (1/2)(4−d); for d≳4 no crossover occurs.

DOI
10.1063/1.324987

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arXiv:1802.05627•Physical review. B./Physical review. B•Jun 18, 2018•32 citations
The critical properties of d-dimensional Ising models in a transverse magnetc field, Hx=Γ, are considered. A quantal generalization of the Landau-Ginzburg-Wilson free-energy functional, introduced by Hertz and Young, has been treated by a variational approximation of the Hartree type. Behavior near the critcal line, Γ=Γc(T), is governed by the exponents of the d-dimensional spherical model. However, quantal crossover occurs on approach to the T=0 multicritical point at Γ=Γc(0) and this is characterized by the exponents of the (d+1) -dimensional spherical model (which are mean-field-like for d+1≳4). Crossover scaling functions, valid near the multicritical point, are reported for the susceptibility. The quantal crossover exponent φT, is equal to νd+1=1/(d−1), the (d+1) -dimensional correlation length exponent for 2⩽d⩽3, as suggested by ideas of Suzuki, but for 3<d⩽4 we find instead φT= (1/2)(4−d); for d≳4 no crossover occurs.
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 d-dimensional, quantum-mechanical Ising model in a transverse field Gamma displays a critical line, Gamma = Gamma c (T), which terminates in a multicritical point, ( Gamma ,T)=( Gamma c (0),0), with the exponents of the (d+1)-dimensional, nonquantal Ising model. Analogous behaviour is found for a nonquantal, (d+1)-dimensional n-component model of finite thickness, L in the last dimension, with the replacement Gamma to T,T to 1/L. A field theoretic method is used to study crossover behaviour in the vicinity of the multicritical point, for 3<d<4, where the multicritical exponents are mean-field-like. Multicritical behaviour is found to be governed by a Gaussian-like fixed point of the renormalisation group, yielding the crossover exponent, phi T = 1 / 2 (4-d). Crossover scaling functions are constructed to first order in epsilon =4-d for the longitudinal susceptibility and specific heat.
The susceptibility of a quantum X-Y model in a transverse field Gamma is studied in the vicinity of the multicritical point ( Gamma c ,T=0). For dimensions d between two and four we obtain a Hartree type approximation an equation for the corresponding crossover function which is solvable analytically in three dimensions. The critical line Gamma c (T)- Gamma c (0) approximately T psi is asymptotically described by the shift exponent psi =d/2 in accord with self-consistent spin wave theory. For the variable of the crossover scaling function T/( Gamma - Gamma c (T)) phi one obtains the crossover exponent phi =(4-d)/2 which is characteristic for the Gaussian multicritical point. A dimensional-analysis argument suggests that this is valid beyond the approximation.
The equation of state of the transverse Ising model in a longitudinal field B is investigated in the vicinity of the zero-temperature multicritical point by using the renormalisation group method recently proposed by the author and K. Walasek (1983). For the critical value of transverse field the scaling crossover form of the relation between the longitudinal magnetic field and magnetisation M is obtained as B approximately M delta q Psi (z) with z approximately TM - phi beta q/, where phi , delta q and beta q denote the crossover and multicritical exponents. The crossover scaling function Psi (z) is calculated for the spatial dimensions d<or=3 and 3<d<4 to the first order in epsilon '=3-d and epsilon =4-d, respectively.
The quantum-classical crossover critical behaviour of the Ising model in a transverse field Gamma is first studied in one dimension where exact results are obtained. The correspondence between the quantum-classical crossover behaviour of this d-dimensional quantum model and the finite-size crossover behaviour of an equivalent (d+1)-dimensional classical model exact for d=1 is extended to higher dimensions. The crossover depends strongly on the relative magnitude of the crossover exponent phi and of the critical line-shift exponent Psi . In particular, if Pi > Psi approaches the critical line along a path of constant transverse field less than the critical field Gamma c and a low temperature, only classical behaviour is seen, quantum effects do not show up. There is some controversy about the value of Psi . Experiments are suggested to give an answer.
We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the nonthermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"uneisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.