Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0105129 · Physical Review Letters

Absence of Spontaneous Magnetic Order at Nonzero Temperature in One- and Two-Dimensional Heisenberg and XY Systems with Long-Range Interactions

P. Bruno

Published Sep 7, 2001• 174 citations
arXivHTMLPDF

Abstract

The Mermin-Wagner theorem is strengthened so as to rule out magnetic long-range order at $T>0$ in one- or two-dimensional Heisenberg and XY systems with long-range interactions decreasing as ${R}^{\ensuremath{-}\ensuremath{\alpha}}$ with a sufficiently large exponent $\ensuremath{\alpha}$. For oscillatory interactions, ferromagnetic long-range order at $T>0$ is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}1(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)$ or $\ensuremath{\alpha}>5/2(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$. For systems with monotonically decreasing interactions, ferro- or antiferromagnetic long-range order at $T>0$ is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}2D$.

DOI
10.1103/physrevlett.87.137203

Related papers

Finite-temperature critical behaviors in 2D long-range quantum Heisenberg model

Jiarui Zhao, Menghan Song, Yang Qi +2 more

Abstract The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/ r α ), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2 D . However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of S U (2) symmetry in the ferromagnetic Heisenberg model with 1/ r α -form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.

arXiv:2306.01044•npj Quantum Materials•Oct 14, 2023•20 citations

Absence of Ordering in Certain Isotropic Systems

Michael F. Thorpe

The method of Mermin and Wagner [Phys. Rev. Lett. 17, 1133 (1966)] is used to show that one- and two-dimensional spin systems interacting with a general isotropic interaction H=12 ∑ ijn Iij(n) (Si·Sj)n,where the exchange interactionsIij(n) are of finite range, cannot order in the sense that 〈Oi〉=0for all traceless operators Oi defined at a single site i. Mermin and Wagner have proved the above for the case n = 1 with Oi = Si, i.e., for the Heisenberg Hamiltonian. The proof allows us to rule out the possibility that a small isotropic biquadratic exchange (Si·Sj)2 could induce ferromagnetism or antiferromagnetism in a two-dimensional Heisenberg system.

Journal of Applied Physics•Mar 15, 1971•30 citations

Classical Heisenberg ferromagnetic chain with long-range interactions: A spectral density approach

A. Cavallo, F. Cosenza, L. De Cesare

The thermodynamic properties of a classical spin-$S$ Heisenberg ferromagnetic chain with long-range interactions in the presence of an external magnetic field are investigated by means of the spectral density method within the framework of the two-time Green's functions in classical statistical mechanics. For exchange interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}},$ it is shown that long-range order exists at finite temperature for $1<\ensuremath{\alpha}<2$ and that no transition to a ferromagnetic phase at any finite temperature occurs for $\ensuremath{\alpha}>~2,$ consistently with a recent extension of the Mermin-Wagner theorem to spin models with long-range interactions. Besides, by means of a suitable modification of the lowest-order moment equations for the spectral density, the critical temperature and the main critical exponents for $1<\ensuremath{\alpha}<2$ and the low-temperature behavior of the paramagnetic susceptibility for $\ensuremath{\alpha}>~2$ are also obtained. Our results appear in a reasonable agreement with the available exact ones and Monte Carlo calculations.

Physical review. B, Condensed matter•Nov 27, 2002•14 citations

Absence of Spontaneous Spin Symmetry Breaking in 1D and 2D Quantum Ferromagnetic Systems with Bilinear and Biquadratic Exchange Interactions

Roberto Zivieri

Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.

Symmetry•Dec 11, 2020•6 citations

Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models

N. David Mermin, H. Wagner

It is rigorously proved that at any nonzero temperature, a one- or two-dimensional isotropic spin-$S$ Heisenberg model with finite-range exchange interaction can be neither ferromagnetic nor antiferromagnetic. The method of proof is capable of excluding a variety of types of ordering in one and two dimensions.

Physical Review Letters•Nov 28, 1966•8,362 citations

Néel order in the ground state of Heisenberg antiferromagnetic chains with long-range interactions

J. Rodrigo Parreira, Oscar Bolina, J Fernando Perez

We consider the ground state of one-dimensional antiferromagnets with long-range interactions with Hamiltonian given by where , with J>0. We prove Néel order for all if the spin s is sufficiently large. We also prove the absence of long-range order when for any spin value.

Journal of Physics A Mathematical and General•Feb 21, 1997•11 citations
The Mermin-Wagner theorem is strengthened so as to rule out magnetic long-range order at T>0T>0T>0 in one- or two-dimensional Heisenberg and XY systems with long-range interactions decreasing as R\ensuremath−\ensuremathα{R}^{\ensuremath{-}\ensuremath{\alpha}}R\ensuremath−\ensuremathα with a sufficiently large exponent \ensuremathα\ensuremath{\alpha}\ensuremathα. For oscillatory interactions, ferromagnetic long-range order at T>0T>0T>0 is ruled out if \ensuremathα\ensuremath≥1(D=1)\ensuremath{\alpha}\ensuremath{\ge}1(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)\ensuremathα\ensuremath≥1(D=1) or \ensuremathα>5/2(D=2)\ensuremath{\alpha}>5/2(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)\ensuremathα>5/2(D=2). For systems with monotonically decreasing interactions, ferro- or antiferromagnetic long-range order at T>0T>0T>0 is ruled out if \ensuremathα\ensuremath≥2D\ensuremath{\alpha}\ensuremath{\ge}2D\ensuremathα\ensuremath≥2D.
Abstract The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/ r α ), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2 D . However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of S U (2) symmetry in the ferromagnetic Heisenberg model with 1/ r α -form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.
The method of Mermin and Wagner [Phys. Rev. Lett. 17, 1133 (1966)] is used to show that one- and two-dimensional spin systems interacting with a general isotropic interaction H=12 ∑ ijn Iij(n) (Si·Sj)n,where the exchange interactionsIij(n) are of finite range, cannot order in the sense that 〈Oi〉=0for all traceless operators Oi defined at a single site i. Mermin and Wagner have proved the above for the case n = 1 with Oi = Si, i.e., for the Heisenberg Hamiltonian. The proof allows us to rule out the possibility that a small isotropic biquadratic exchange (Si·Sj)2 could induce ferromagnetism or antiferromagnetism in a two-dimensional Heisenberg system.
The thermodynamic properties of a classical spin-SSS Heisenberg ferromagnetic chain with long-range interactions in the presence of an external magnetic field are investigated by means of the spectral density method within the framework of the two-time Green's functions in classical statistical mechanics. For exchange interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}},$ it is shown that long-range order exists at finite temperature for $1<\ensuremath{\alpha}<2$ and that no transition to a ferromagnetic phase at any finite temperature occurs for $\ensuremath{\alpha}>~2,$ consistently with a recent extension of the Mermin-Wagner theorem to spin models with long-range interactions. Besides, by means of a suitable modification of the lowest-order moment equations for the spectral density, the critical temperature and the main critical exponents for $1<\ensuremath{\alpha}<2$ and the low-temperature behavior of the paramagnetic susceptibility for $\ensuremath{\alpha}>~2$ are also obtained. Our results appear in a reasonable agreement with the available exact ones and Monte Carlo calculations.
Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.
It is rigorously proved that at any nonzero temperature, a one- or two-dimensional isotropic spin-SSS Heisenberg model with finite-range exchange interaction can be neither ferromagnetic nor antiferromagnetic. The method of proof is capable of excluding a variety of types of ordering in one and two dimensions.