Physical Review A

Critical behavior of an Ising model of classical spins in a transverse field

Theodore W. Burkhardt, J. D. Gunton

Published Jun 1, 1974• 6 citations

Abstract

The critical behavior of an Ising model of classical spins in a transverse field $H$ is derived for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$ using an $\ensuremath{\epsilon}$ expansion based on the Wilson renormalization group. To first order in $\ensuremath{\epsilon}$ the model exhibits a line of critical points ${T}_{c}(H)$ with the same fixed point and critical exponents as in previous studies of the Wilson effective Hamiltonian.

DOI: 10.1103/physreva.9.2802

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Effective-field renormalization-group study for the transverse Ising model in a quantum-spin system

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In this paper we suggest a method that combines the effective-field renormalization group with the discretized path-integral method to study the critical behavior of the transverse Ising model in a quantum-spin system. We find the results on a critical transverse field and critical exponent at zero temperature to be closely comparable to other results. This demonstrates the utility of this method in studying critical properties of quantum-spin systems.

Physical review. B, Condensed matterDec 1, 198921 citations

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Quantum effects in the renormalization group approach to phase transitions

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Physical Review A

Critical behavior of an Ising model of classical spins in a transverse field

Theodore W. Burkhardt, J. D. Gunton

Published Jun 1, 1974• 6 citations

Publisher

Abstract

The critical behavior of an Ising model of classical spins in a transverse field

H

is derived for dimension

d=4\ensuremath{-}\ensuremath{\epsilon}

using an

\ensuremath{\epsilon}

expansion based on the Wilson renormalization group. To first order in

\ensuremath{\epsilon}

the model exhibits a line of critical points

{T}_{c}(H)

with the same fixed point and critical exponents as in previous studies of the Wilson effective Hamiltonian.

DOI: 10.1103/physreva.9.2802

Related papers

Effective-field renormalization-group study for the transverse Ising model in a quantum-spin system

Qing Jiang, Z. Y. Li

Physical review. B, Condensed matterDec 1, 198921 citations

Ising model with a transverse field in two dimensions: Phase diagram and critical properties from a real-space renormalization group

Zvi Friedman

Physical review. B, Condensed matterFeb 1, 197824 citations

Zero-temperature renormalization-group method for quantum systems. III. Ising model in a transverse field in two dimensions

K. A. Penson, Rémi Jullien, Pierre M. Pfeuty

A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.

Physical review. B, Condensed matterMay 1, 197961 citations

Transverse spin correlations of the random transverse-field Ising model

Ferenc Iglói, István A. Kovács

The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in $d=1,2$ , and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In $d=1$ the results are related to dimer-dimer correlations in the random antiferromagnetic $XX$ chain and have been tested by numerical calculations using free-fermionic techniques.The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in $d=1,2$, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In $d=1$ the results are related to dimer-dimer correlations in the random antiferromagnetic $XX$ chain and have been tested by numerical calculations using free-fermionic techniques.

arXiv:1712.07467Physical review. B./Physical review. BMar 21, 20184 citations

Renormalization group study of the two-dimensional random transverse-field Ising model

István A. Kovács, Ferenc Iglói

The infinite-disorder fixed point of the random transverse-field Ising model is expected to control the critical behavior of a large class of random quantum and stochastic systems having an order parameter with discrete symmetry. Here we study the model on the square lattice with a very efficient numerical implementation of the strong disorder renormalization group method, which makes us possible to treat finite samples of linear size up to $L=2048$ . We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates: $x=0.982(15)$ and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.The infinite-disorder fixed point of the random transverse-field Ising model is expected to control the critical behavior of a large class of random quantum and stochastic systems having an order parameter with discrete symmetry. Here we study the model on the square lattice with a very efficient numerical implementation of the strong disorder renormalization group method, which makes us possible to treat finite samples of linear size up to $L=2048$. We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates: $x=0.982(15)$ and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.

arXiv:1005.4740Physical Review BAug 31, 2010105 citations

Quantum effects in the renormalization group approach to phase transitions

A. P. Young

The renormalization group approach to critical phenomena is developed for quantum mechanical problems with non-commuting operators. Applying the theory to the spin 1 / 2 Ising model with a transverse field Gamma it is found that the critical exponents are those of the Ising model with Gamma =0 if the transition occurs at T>0. However, for transitions at T=0 the critical behaviour of the d-dimensional transverse system corresponds to that of the (d+1)-dimensional Ising model with Gamma =0, in agreement with series expansion predictions. At T=0 the dynamic, as well as static, critical behaviour is given by mean field theory for d>3.The renormalization group approach to critical phenomena is developed for quantum mechanical problems with non-commuting operators. Applying the theory to the spin 1 / 2 Ising model with a transverse field Gamma it is found that the critical exponents are those of the Ising model with Gamma =0 if the transition occurs at T>0. However, for transitions at T=0 the critical behaviour of the d-dimensional transverse system corresponds to that of the (d+1)-dimensional Ising model with Gamma =0, in agreement with series expansion predictions. At T=0 the dynamic, as well as static, critical behaviour is given by mean field theory for d>3.

Journal of Physics C Solid State PhysicsAug 7, 1975108 citations

The critical behavior of an Ising model of classical spins in a transverse field

H

is derived for dimension

d=4\ensuremath{-}\ensuremath{\epsilon}

using an

\ensuremath{\epsilon}

expansion based on the Wilson renormalization group. To first order in

\ensuremath{\epsilon}

the model exhibits a line of critical points

{T}_{c}(H)

with the same fixed point and critical exponents as in previous studies of the Wilson effective Hamiltonian.

d=1,2

, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In

d=1

the results are related to dimer-dimer correlations in the random antiferromagnetic

XX

chain and have been tested by numerical calculations using free-fermionic techniques.

L=2048

. We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates:

x=0.982(15)

and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.