Physical review. B, Condensed matter

Random-field effect on the quantum ferromagnetic XY model

Yu‐qiang Ma

Published Jul 1, 1998• 1 citation

Abstract

The quantum version of the random-field spherical $\mathrm{XY}$ model on a $d$-dimensional hypercubic lattice is presented in the boson space, and solved exactly. We recover the Imry-Ma result concerning the lower critical dimension ${d}_{c}=4,$ and calculate the critical exponents near the critical temperature. In particular, we obtain a general phase diagram of the model with arbitrary dimension $d$ when $d>{d}_{c}.$ The ferromagnetic ordering is reduced by quantum fluctuations, and destroyed completely by random-field fluctuations with sufficiently large values of the random-field variance \ensuremath{\sigma}. The entropy and the specific heat vanish as ${T}^{d/2}$ at low temperatures. Since the model is equivalent to a Bose system, we show a superfluid-Mott insulator transition at a critical chemical potential ${\ensuremath{\mu}}_{c}.$

DOI: 10.1103/physrevb.58.282

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Physical review. B, Condensed matter

Random-field effect on the quantum ferromagnetic XY model

Yu‐qiang Ma

Published Jul 1, 1998• 1 citation

Publisher

Abstract

The quantum version of the random-field spherical

\mathrm{XY}

model on a

d

-dimensional hypercubic lattice is presented in the boson space, and solved exactly. We recover the Imry-Ma result concerning the lower critical dimension

{d}_{c}=4,

and calculate the critical exponents near the critical temperature. In particular, we obtain a general phase diagram of the model with arbitrary dimension

d

when

d>{d}_{c}.

The ferromagnetic ordering is reduced by quantum fluctuations, and destroyed completely by random-field fluctuations with sufficiently large values of the random-field variance \ensuremath{\sigma}. The entropy and the specific heat vanish as

{T}^{d/2}

at low temperatures. Since the model is equivalent to a Bose system, we show a superfluid-Mott insulator transition at a critical chemical potential

{\ensuremath{\mu}}_{c}.

DOI: 10.1103/physrevb.58.282

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We propose a soluble quantum spherical XY ferromagnet with a random field in the boson space. We obtain a general expression of the critical temperature below which the ordered ferromagnet phase appears. The Imry–Ma result concerning the lower critical dimension is recovered, and the critical exponents near the critical temperature are calculated. We show that the random-field fluctuations rather than the quantum fluctuations dominate the phase transition and critical behavior of the system. The entropy vanishes as at low temperatures, contrary to the classical spherical model.We propose a soluble quantum spherical XY ferromagnet with a random field in the boson space. We obtain a general expression of the critical temperature below which the ordered ferromagnet phase appears. The Imry–Ma result concerning the lower critical dimension is recovered, and the critical exponents near the critical temperature are calculated. We show that the random-field fluctuations rather than the quantum fluctuations dominate the phase transition and critical behavior of the system. The entropy vanishes as at low temperatures, contrary to the classical spherical model.

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Quantum spherical spin model on hypercubic lattices

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Lower critical dimension of the random-field XY model and the zero-temperature critical line

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arXiv:2203.11153Physical review. EJul 29, 202211 citations

Phase Transitions in Gaussian Random Fields. II: Ferromagnets in d = 4 + Formula Space Dimensions

Fumihiko Tanaka

Critical behaviors of a ferromagnetic spin system under quenched random magnetic fields are investigated near above four space dimensions. Critical temperature Tc(m, µ, ε) as a function of the number of spin components m(≥2) and strength µ of the random field is obtained in a power series of ε ≡ d-4 (>0) by means of the Migdal-Polyakov renormalization group theory. Critical exponents ν and η are also calculated in powers of ε. It is shown that they are in agreement with those of (2 + ε)-dimensional pure system for sufficiently close to Tc. In the limit m→∞, present results reduce to d = 4 + ε case of our previous results for the spherical model.Critical behaviors of a ferromagnetic spin system under quenched random magnetic fields are investigated near above four space dimensions. Critical temperature Tc(m, µ, ε) as a function of the number of spin components m(≥2) and strength µ of the random field is obtained in a power series of ε ≡ d-4 (>0) by means of the Migdal-Polyakov renormalization group theory. Critical exponents ν and η are also calculated in powers of ε. It is shown that they are in agreement with those of (2 + ε)-dimensional pure system for sufficiently close to Tc. In the limit m→∞, present results reduce to d = 4 + ε case of our previous results for the spherical model.

Progress of Theoretical PhysicsMay 1, 19782 citations

Variational study of the random-field XY model

Thomas Garel, Giulia Iori, Henri Orland

A disorder-dependent Gaussian variational approach is applied to the $d$ -dimensional ferromagnetic $\mathrm{XY}$ model in a random field. The randomness yields a nonextensive contribution to the variational free energy, implying a random mass term in correlation functions. The Imry-Ma low-temperature result, concerning the existence ( $d>4$ ) or absence ( $d<4$ ) of long-range order is obtained in a transparent way. The physical picture which emerges below $d=4$ is that of a marginally stable mixture of domains. We also calculate within this variational scheme, disorder-dependent correlation functions, as well as the probability distribution of the Imry-Ma domain size.A disorder-dependent Gaussian variational approach is applied to the $d$-dimensional ferromagnetic $\mathrm{XY}$ model in a random field. The randomness yields a nonextensive contribution to the variational free energy, implying a random mass term in correlation functions. The Imry-Ma low-temperature result, concerning the existence ($d>4$) or absence ($d<4$) of long-range order is obtained in a transparent way. The physical picture which emerges below $d=4$ is that of a marginally stable mixture of domains. We also calculate within this variational scheme, disorder-dependent correlation functions, as well as the probability distribution of the Imry-Ma domain size.

arXiv:cond-mat/9512057Physical review. B, Condensed matterFeb 1, 199621 citations

The quantum version of the random-field spherical

\mathrm{XY}

model on a

d

-dimensional hypercubic lattice is presented in the boson space, and solved exactly. We recover the Imry-Ma result concerning the lower critical dimension

{d}_{c}=4,

and calculate the critical exponents near the critical temperature. In particular, we obtain a general phase diagram of the model with arbitrary dimension

d

when

d>{d}_{c}.

{T}^{d/2}

at low temperatures. Since the model is equivalent to a Bose system, we show a superfluid-Mott insulator transition at a critical chemical potential

{\ensuremath{\mu}}_{c}.

{\mathrm{d}}_{\mathrm{c}}

=2, where

{\mathrm{d}}_{\mathrm{c}}

is the lower critical dimension. It is shown that the critical temperature

{\mathrm{T}}_{\mathrm{c}}

below which a ferromagnetic phase appears increases with increasing the anisotropy parameter, and remains finite even for anisotropy parameter D\ensuremath{\gg}:1.

We present an alternative treatment to the quantum spherical spin model on $(d\ensuremath{\ge}2)$-dimensional hypercubic lattices, focusing on the effects of quantum

(g)

and thermal

(T)

fluctuations, under a uniform magnetic field

h

, on the correlation function, correlation length, entropy, specific heat, and energy gap in the excitation spectrum. Explicit expressions for such quantities are provided close to the $d\ensuremath{\ge}2$ quantum (

g={g}_{c}

T=0

) and $d\ensuremath{\ge}3$ thermal

[T={T}_{c}(g)]

phase transitions in

h=0

, including the low-

T

quantum regimes near the quantum critical point. In particular, the calculation of the correlation function and correlation length generalizes the results on the

g=0

classical spherical model. At

T=0

, the zero-field system is gapless at and below

{g}_{c}

; however, a gap opens in the quantum-disordered ground state,

g>{g}_{c}

. Conversely, the null gap for $T\ensuremath{\le}{T}_{c}(g\ensuremath{\ne}0)$ becomes finite as $T\ensuremath{\rightarrow}{T}_{c}(g\ensuremath{\ne}0{)}^{+}$; thus, quantum fluctuations suppress the critical prefactors of observables near ${T}_{c}(g\ensuremath{\ne}0)$, though they are irrelevant to the universality class shared with the gapless classical spherical model. The results on the entropy and specific heat in $g\ensuremath{\ne}0$ circumvent the drawback in classical spherical models concerning the third law of thermodynamics, as $T\ensuremath{\rightarrow}0$.

A disorder-dependent Gaussian variational approach is applied to the

d

-dimensional ferromagnetic

\mathrm{XY}

model in a random field. The randomness yields a nonextensive contribution to the variational free energy, implying a random mass term in correlation functions. The Imry-Ma low-temperature result, concerning the existence (

d>4

) or absence (

d<4

) of long-range order is obtained in a transparent way. The physical picture which emerges below

d=4

is that of a marginally stable mixture of domains. We also calculate within this variational scheme, disorder-dependent correlation functions, as well as the probability distribution of the Imry-Ma domain size.