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Journal of Physics C Solid State Physics

Quantum effects in the critical properties of X-Y model in a transverse magnetic field

Paul R. Gerber, H Beck

Published Oct 28, 1977• 46 citations
Publisher

Abstract

The critical properties of a quantum-mechanical X-Y model in a transverse magnetic field are calculated. At zero temperature near the critical field Gamma c a Gaussian type behaviour is observed for dimensions higher than two. The exponent which describes the rescaling of frequencies is here z=2. For d<2 a renormalisation group treatment produces a non-trivial fixed point. However, the exponents remain unchanged to first order in epsilon =2-d.

DOI
10.1088/0022-3719/10/20/009

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Journal of Physics A Mathematical and General•Dec 1, 1980•23 citations
The critical properties of a quantum-mechanical X-Y model in a transverse magnetic field are calculated. At zero temperature near the critical field Gamma c a Gaussian type behaviour is observed for dimensions higher than two. The exponent which describes the rescaling of frequencies is here z=2. For d<2 a renormalisation group treatment produces a non-trivial fixed point. However, the exponents remain unchanged to first order in epsilon =2-d.
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 field theoretical renormalisation group method, which correctly describes the critical as well as the multicritical (at zero temperature) behaviours of the quantum mechanical d-dimensional Ising model in a transverse magnetic field, is proposed. Due to the introduction of the additional renormalisation the singularities associated with zero temperature have been removed. The crossover scaling form of the inverse correlation length and longitudinal susceptibility are considered for the following spatial dimensions d:2<d<3, d=3 and 3<d<4. For 2<d<3, d=3 and 3<d<4 the calculations are performed to the first order in 3-d and 4-d, respectively.
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 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 critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.
The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.