Physical Review Letters

Exact Scaling of Spin-Wave Correlations in the 2D XY Ferromagnet with Dipolar Forces

А. Кашуба

Published Oct 17, 1994• 15 citations

Abstract

Long-range spin-wave correlation functions in the ordered phase of a 2D $\mathrm{XY}$ ferromagnet with 2D dipolar forces are governed by the 2D smectic $A$ Hamiltonian and possess exact scaling. The latter confirms from a different perspective the exact results obtained recently for the 2D smectic $A$. Exact scaling holds in the ordered phase of the 2D $\mathrm{XY}$ ferromagnet with 3D dipolar forces, with the dimension of the spatial anisotropy being ${\ensuremath{\Delta}}_{x}=\frac{4}{3}$ and the anomalous dimension of spin being ${\ensuremath{\eta}}_{\ensuremath{\varphi}}=\frac{1}{6}$.

DOI: 10.1103/physrevlett.73.2264

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Renormalization Theory for Ordered Phase of 2D XY Magnet with Dipolar Interaction

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In the ordered phase of 2D XY ferromagnet the dipolar interaction between spins induces a relevant interaction between spin-waves. Although the density of spin-waves is small at low temperature, this long-range interaction correlates two spatially separated spin-waves. Anomalous scaling of spin-wave correlation functions is found in both statics and dynamics cases.

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Physical Review Letters

Exact Scaling of Spin-Wave Correlations in the 2D XY Ferromagnet with Dipolar Forces

А. Кашуба

Published Oct 17, 1994• 15 citations

Publisher

Abstract

Long-range spin-wave correlation functions in the ordered phase of a 2D

\mathrm{XY}

ferromagnet with 2D dipolar forces are governed by the 2D smectic

A

Hamiltonian and possess exact scaling. The latter confirms from a different perspective the exact results obtained recently for the 2D smectic

A

. Exact scaling holds in the ordered phase of the 2D

\mathrm{XY}

ferromagnet with 3D dipolar forces, with the dimension of the spatial anisotropy being

{\ensuremath{\Delta}}_{x}=\frac{4}{3}

and the anomalous dimension of spin being

{\ensuremath{\eta}}_{\ensuremath{\varphi}}=\frac{1}{6}

DOI: 10.1103/physrevlett.73.2264

Related papers

Renormalization Theory for Ordered Phase of 2D XY Magnet with Dipolar Interaction

А. Кашуба

International Journal of Modern Physics BMay 30, 19980 citations

Spin-spin correlation function of the two-dimensional XY model with weak site or bond dilution

O. Kapikranian

The spin-spin correlation function of the two-dimensional (2D) $XY$ model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature-dependent exponent $\ensuremath{\eta}=\ensuremath{\eta}(T/J)$ ( $J$ is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D $XY$ model with site or bond dilution this exponent depends on the concentration $p$ of removed sites or bonds as well. Knowing the slope $\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p$ at point $p=0$ , one can predict the value of the exponent for small dilution concentrations: ${\ensuremath{\eta}(p)\ensuremath{\simeq}\ensuremath{\eta}(0)+p(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}$. As shown in this paper, the spin-wave Hamiltonian allows one to obtain exact results for this slope: ${(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}=T/(2J)+O({(T/J)}^{2})$ and $T/(\ensuremath{\pi}J)+O({(T/J)}^{2})$ for site and for bond dilution, respectively.The spin-spin correlation function of the two-dimensional (2D) $XY$ model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature-dependent exponent $\ensuremath{\eta}=\ensuremath{\eta}(T/J)$ ($J$ is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D $XY$ model with site or bond dilution this exponent depends on the concentration $p$ of removed sites or bonds as well. Knowing the slope $\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p$ at point $p=0$, one can predict the value of the exponent for small dilution concentrations: ${\ensuremath{\eta}(p)\ensuremath{\simeq}\ensuremath{\eta}(0)+p(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}$. As shown in this paper, the spin-wave Hamiltonian allows one to obtain exact results for this slope: ${(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}=T/(2J)+O({(T/J)}^{2})$ and $T/(\ensuremath{\pi}J)+O({(T/J)}^{2})$ for site and for bond dilution, respectively.

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Novel phase transition in two-dimensional xy-models with long-range interaction

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The purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction.We investigate a two-dimensional xy-model extended by the dipolar interaction.Describing our system by a nonlinear σ-model and using renormalization group methods we predict a phase transition to an ordered state.This transition is due to the long-range dipolar interaction.The ferromagnetic phase is governed by a low temperature fixed-point with infinite dipolar coupling.In the critical regime we find exponential behavior for the correlation length and the order parameter in contrast to the usual power laws.The nature of the transition shows a striking similarity to the Kosterlitz-Thouless transition.We show that there is a whole class of longrange xy-models leading to such non-standard behavior.Parameterizing the divergencies in terms of the correlation length we are able to calculate the critical exponents.These exponents are correct in any loop order.The purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction.We investigate a two-dimensional xy-model extended by the dipolar interaction.Describing our system by a nonlinear σ-model and using renormalization group methods we predict a phase transition to an ordered state.This transition is due to the long-range dipolar interaction.The ferromagnetic phase is governed by a low temperature fixed-point with infinite dipolar coupling.In the critical regime we find exponential behavior for the correlation length and the order parameter in contrast to the usual power laws.The nature of the transition shows a striking similarity to the Kosterlitz-Thouless transition.We show that there is a whole class of longrange xy-models leading to such non-standard behavior.Parameterizing the divergencies in terms of the correlation length we are able to calculate the critical exponents.These exponents are correct in any loop order.

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Walter Hofstetter, Malte Henkel

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Equation of state and scaling relations for isotropic ferromagnets with dipolar interactions

Amnon Aharony A, Alastair D. Bruce

The scaling equation of state for ferromagnets with both isotropic exchange and dipolar interactions near the critical point is derived to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where $d$ is the dimensionality of space and is also equal to the number of spin components. The resulting scaling function is found to be rather close numerically to its pure short-range counterpart at $\ensuremath{\epsilon}=1$, although the functional forms are quite different. Several universal quantities are derived; in particular, the ratio of the amplitudes of the zero-field specific heats above and below ${T}_{c}$ is found to be $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=\frac{6}{5}+O(\ensuremath{\epsilon})$, compared with $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=1+O(\ensuremath{\epsilon})$ for pure short-range interactions.The scaling equation of state for ferromagnets with both isotropic exchange and dipolar interactions near the critical point is derived to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where $d$ is the dimensionality of space and is also equal to the number of spin components. The resulting scaling function is found to be rather close numerically to its pure short-range counterpart at $\ensuremath{\epsilon}=1$, although the functional forms are quite different. Several universal quantities are derived; in particular, the ratio of the amplitudes of the zero-field specific heats above and below ${T}_{c}$ is found to be $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=\frac{6}{5}+O(\ensuremath{\epsilon})$, compared with $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=1+O(\ensuremath{\epsilon})$ for pure short-range interactions.

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Magnetism in the two-dimensional dipolar XY model

Björn Sbierski, Marcus Bintz, Shubhayu Chatterjee +3 more

Motivated by a recent experiment on a square-lattice Rydberg atom array realizing a long-range dipolar XY model [C. Chen et al., Nature (London) 616, 691 (2023)], we numerically study the model's equilibrium properties. We obtain the phase diagram, critical properties, entropies, variance of the magnetization, and site-resolved correlation functions. We consider both ferromagnetic and antiferromagnetic interactions and apply quantum Monte Carlo and pseudo-Majorana functional renormalization group techniques, generalizing the latter to a $U(1)$ symmetric setting. Our simulations perform extensive thermometry in dipolar Rydberg atom arrays and establish conditions for adiabaticity and thermodynamic equilibrium. On the ferromagnetic side of the experiment, we determine the entropy per particle $S/N\ensuremath{\approx}0.5$, close to the one at the critical temperature, ${S}_{c}/N=0.585(15)$ . The simulations suggest the presence of an out-of-equilibrium plateau at large distances in the correlation function, thus motivating future studies on the nonequilibrium dynamics of the system.Motivated by a recent experiment on a square-lattice Rydberg atom array realizing a long-range dipolar XY model [C. Chen et al., Nature (London) 616, 691 (2023)], we numerically study the model's equilibrium properties. We obtain the phase diagram, critical properties, entropies, variance of the magnetization, and site-resolved correlation functions. We consider both ferromagnetic and antiferromagnetic interactions and apply quantum Monte Carlo and pseudo-Majorana functional renormalization group techniques, generalizing the latter to a $U(1)$ symmetric setting. Our simulations perform extensive thermometry in dipolar Rydberg atom arrays and establish conditions for adiabaticity and thermodynamic equilibrium. On the ferromagnetic side of the experiment, we determine the entropy per particle $S/N\ensuremath{\approx}0.5$, close to the one at the critical temperature, ${S}_{c}/N=0.585(15)$. The simulations suggest the presence of an out-of-equilibrium plateau at large distances in the correlation function, thus motivating future studies on the nonequilibrium dynamics of the system.

arXiv:2305.03673Physical review. B./Physical review. BApr 12, 202422 citations

Long-range spin-wave correlation functions in the ordered phase of a 2D

\mathrm{XY}

ferromagnet with 2D dipolar forces are governed by the 2D smectic

A

Hamiltonian and possess exact scaling. The latter confirms from a different perspective the exact results obtained recently for the 2D smectic

A

. Exact scaling holds in the ordered phase of the 2D

\mathrm{XY}

ferromagnet with 3D dipolar forces, with the dimension of the spatial anisotropy being

{\ensuremath{\Delta}}_{x}=\frac{4}{3}

and the anomalous dimension of spin being

{\ensuremath{\eta}}_{\ensuremath{\varphi}}=\frac{1}{6}

The spin-spin correlation function of the two-dimensional (2D)

XY

model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature-dependent exponent $\ensuremath{\eta}=\ensuremath{\eta}(T/J)$ (

J

is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D

XY

model with site or bond dilution this exponent depends on the concentration

p

of removed sites or bonds as well. Knowing the slope $\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p$ at point

p=0

, one can predict the value of the exponent for small dilution concentrations: ${\ensuremath{\eta}(p)\ensuremath{\simeq}\ensuremath{\eta}(0)+p(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}$. As shown in this paper, the spin-wave Hamiltonian allows one to obtain exact results for this slope: ${(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}=T/(2J)+O({(T/J)}^{2})$ and $T/(\ensuremath{\pi}J)+O({(T/J)}^{2})$ for site and for bond dilution, respectively.

d

is the dimensionality of space and is also equal to the number of spin components. The resulting scaling function is found to be rather close numerically to its pure short-range counterpart at $\ensuremath{\epsilon}=1$, although the functional forms are quite different. Several universal quantities are derived; in particular, the ratio of the amplitudes of the zero-field specific heats above and below

{T}_{c}

is found to be $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=\frac{6}{5}+O(\ensuremath{\epsilon})$, compared with $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=1+O(\ensuremath{\epsilon})$ for pure short-range interactions.

U(1)

symmetric setting. Our simulations perform extensive thermometry in dipolar Rydberg atom arrays and establish conditions for adiabaticity and thermodynamic equilibrium. On the ferromagnetic side of the experiment, we determine the entropy per particle $S/N\ensuremath{\approx}0.5$, close to the one at the critical temperature,

{S}_{c}/N=0.585(15)

. The simulations suggest the presence of an out-of-equilibrium plateau at large distances in the correlation function, thus motivating future studies on the nonequilibrium dynamics of the system.