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Physics Letters A

Dependence of critical exponents upon symmetry, dimensionality, potential-range and strength of interaction

Masuo Suzuki

Published Jan 1, 1972• 13 citations
Publisher
DOI
10.1016/0375-9601(72)90958-9

Related papers

Critical Exponents for Long-Range Interactions. I

Masuo Suzuki

The critical exponents r and a are derived in powers of 1/n to first order for the n- vector model (or n-component system) with long-range interactions.The results show that the exponents have discontinuity with respect to the potential-range " at <T=2 for d>2 (d: dimensionality).It is also shown that they depend upon d and "• rather independently.The connection of the present results with the ones derived in E-expansion is briefly discussed.The 1/n-expansion of other critical exponents such as {J, iJ, "' and 71 are also discussed.§ 1. Introduction and summary Critical exponents characteristic of critical phenomena depend upon the dimensionality d, symmetry (or the degrees of freedom, n) and potential-range s (or effective potential-range rJ).Among these three parameters, the d-and n- dependences of critical exponents have been investigated 1 l-4 l mainly by using an expansion method combined with renormalization group techniques. 5> However, the potential-range s (or rJ) is also a very important parameter for critical phenomena, in particular for the purpose of discussing critical behavior in systems of restricted dimensionality, in which there would occur no phase transition otherwise.6>One of our main purposes in this paper is to investigate how the critical exponents depend upon the three parameters d, n and (J, particularly to clarify the mutual relationship between dimensionality d and potential-range s (or rJ).For this purpose, the critical exponents for general d and s (or rJ) are calculated, (I) by applying 1/n-expansion 7 > to n-vector model with potential-range s, and (II) by extending s-expansion 1 >• 2 > to a generalized effective Hamiltonian of Wilson-type 8 >• 6 l (essentially equivalent to the n-vector model with long-range interactions).Following Wilson 2 l and Fisher,ll for any real value d(>1), we make use of the formulae (2n)-a S d"-kf(k~) =Ka f"ka-~(k~)dk,Downloaded from https://academic.oup.com/ptp/article/49/2/424/1856924 by guest on 12 August 2023Critical Exponents for Long-Range Interactions.I with Ka = 2-<"-t>n-"12 [T(d/2) J-t.The effective potential-range (J is defined by rJ=s for O<s<2 and rJ=2 for s>2 through the interaction

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Renormalization Group and Critical Exponents for Classical Heisenberg Ferromagnet with Dipole–Dipole Interaction

S. Fakioǧlu

Abstract The renormalization group technique is applied to the three‐dimensional classical Heisenberg spin system with modified dipole–dipole interaction. For this system, the values and dependence of several critical exponents are investigated. These exponents are: α, the specific heat exponent; β and δ, the magnetization exponents; γ, the susceptibility exponent; η, the correlation function exponent; and v, the correlation length exponent. All the critical exponents except β depend upon the potential range σ. When the potential range σ = 3/2, the modified dipole–dipole results agree with mean‐field values of the critical exponents.

physica status solidi (b)•Mar 1, 1980•8 citations

Scaling theory of the dependence of critical behaviour on dimensionality

Y. Imry

Physics Letters A•Oct 1, 1972•7 citations

Testing the universality of the many-body metal-insulator transition by time evolution of a disordered one-dimensional ultracold fermionic gas

Masaki Tezuka, Antonio M. Garcı́a-Garcı́a

It is now possible to study experimentally the combined effect of disorder and interactions in cold atom physics. Motivated by these developments we investigate the dynamics around the metal-insulator transition (MIT) in a one-dimensional Fermi gas with short-range interactions in a quasiperiodic potential by the time-dependent density-matrix renormalization group technique. By tuning disorder and interactions we study the MIT from the weakly to the strongly interacting limit. The MIT is not universal as time evolution, well described by a process of anomalous diffusion, depends qualitatively on the interaction strength. By using scaling ideas we relate the parameter that controls the diffusion process with the critical exponent that describes the divergence of the localization length. In the limit of strong interactions theoretical arguments suggest that the motion at the MIT tends to ballistic and critical exponents approach mean-field predictions.

arXiv:1109.4037•Physical Review A•Mar 13, 2012•11 citations
The critical exponents r and a are derived in powers of 1/n to first order for the n- vector model (or n-component system) with long-range interactions.The results show that the exponents have discontinuity with respect to the potential-range " at <T=2 for d>2 (d: dimensionality).It is also shown that they depend upon d and "• rather independently.The connection of the present results with the ones derived in E-expansion is briefly discussed.The 1/n-expansion of other critical exponents such as {J, iJ, "' and 71 are also discussed.§ 1. Introduction and summary Critical exponents characteristic of critical phenomena depend upon the dimensionality d, symmetry (or the degrees of freedom, n) and potential-range s (or effective potential-range rJ).Among these three parameters, the d-and n- dependences of critical exponents have been investigated 1 l-4 l mainly by using an expansion method combined with renormalization group techniques. 5> However, the potential-range s (or rJ) is also a very important parameter for critical phenomena, in particular for the purpose of discussing critical behavior in systems of restricted dimensionality, in which there would occur no phase transition otherwise.6>One of our main purposes in this paper is to investigate how the critical exponents depend upon the three parameters d, n and (J, particularly to clarify the mutual relationship between dimensionality d and potential-range s (or rJ).For this purpose, the critical exponents for general d and s (or rJ) are calculated, (I) by applying 1/n-expansion 7 > to n-vector model with potential-range s, and (II) by extending s-expansion 1 >• 2 > to a generalized effective Hamiltonian of Wilson-type 8 >• 6 l (essentially equivalent to the n-vector model with long-range interactions).Following Wilson 2 l and Fisher,ll for any real value d(>1), we make use of the formulae (2n)-a S d"-kf(k~) =Ka f"ka-~(k~)dk,Downloaded from https://academic.oup.com/ptp/article/49/2/424/1856924 by guest on 12 August 2023Critical Exponents for Long-Range Interactions.I with Ka = 2-<"-t>n-"12 [T(d/2) J-t.The effective potential-range (J is defined by rJ=s for O<s<2 and rJ=2 for s>2 through the interaction
Abstract The renormalization group technique is applied to the three‐dimensional classical Heisenberg spin system with modified dipole–dipole interaction. For this system, the values and dependence of several critical exponents are investigated. These exponents are: α, the specific heat exponent; β and δ, the magnetization exponents; γ, the susceptibility exponent; η, the correlation function exponent; and v, the correlation length exponent. All the critical exponents except β depend upon the potential range σ. When the potential range σ = 3/2, the modified dipole–dipole results agree with mean‐field values of the critical exponents.
It is now possible to study experimentally the combined effect of disorder and interactions in cold atom physics. Motivated by these developments we investigate the dynamics around the metal-insulator transition (MIT) in a one-dimensional Fermi gas with short-range interactions in a quasiperiodic potential by the time-dependent density-matrix renormalization group technique. By tuning disorder and interactions we study the MIT from the weakly to the strongly interacting limit. The MIT is not universal as time evolution, well described by a process of anomalous diffusion, depends qualitatively on the interaction strength. By using scaling ideas we relate the parameter that controls the diffusion process with the critical exponent that describes the divergence of the localization length. In the limit of strong interactions theoretical arguments suggest that the motion at the MIT tends to ballistic and critical exponents approach mean-field predictions.