Note on the Mermin‐Wagner Theorem

Krzemiński, S.

doi:10.1002/pssb.2220740249

Cited by 4 publications

(6 citation statements)

References 16 publications

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“…Thorpe [28] considers the case of ferromagnetism in phenomenological models with double and higher-order exchange terms; similar results were obtained in the multi-sublattice case by Krzemiński. [29] These methods differ from others in that the Hamiltonian is written as a series expansion in terms of spin spherical harmonics, thus offering a more systematic way of evaluating the Bogoliubov inequality using the defining properties of spin spherical harmonics. A closely related proof, using spherical tensor operators, has been put forward for the problem of ordering in quadrupolar systems of restricted dimensionality.…”

Section: Spin Systemsmentioning

confidence: 99%

The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results

Gelfert¹,

Nolting²

2001

J. Phys.: Condens. Matter

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After a brief discussion of the Bogoliubov inequality and possible generalizations thereof, we present a complete review of results concerning the Mermin-Wagner theorem for various many-body systems, geometries and order parameters. We extend the method to cover magnetic phase transitions in the Periodic Anderson Model as well as certain superconducting pairing mechanisms for Hubbard films. The relevance of the Mermin-Wagner theorem to approximations in manybody physics is discussed on a conceptual level.

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Section: Spin Systemsmentioning

confidence: 99%

The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results

Gelfert¹,

Nolting²

2001

J. Phys.: Condens. Matter

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“…This proof was obtained by using the Bogoliubov's inequality and developing a mathematical approach based on writing terms of the spin equivalent of the spherical harmonics. A few years later, Krzemiński used the same mathematical method and came to similar conclusions demonstrating the absence of long-range order also in multi-sublattice systems [25]. The absence of off-diagonal longrange order and of long-range order was demonstrated in 2D ferromagnetic lattices described via the Heisenberg model by Suzuki [26] who also showed that, in systems with higher spins leading to a different definition of order parameter, off-diagonal long-range order can exist and the order parameter can remain finite.…”

Section: Introductionmentioning

confidence: 75%

“…Finally, the absence of spontaneous breaking of continuous and internal symmetries, of crystalline ordering in 2D systems and uniqueness of equilibrium states was proved by Frölich and Pfister [27] according to a unified approach based on Araki's relative entropy concept. However, for the systems studied in [24][25][26][27], the analysis was mainly mathematical and the physical effects of the biquadratic exchange interaction were not investigated.…”

Section: Introductionmentioning

confidence: 99%

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

Zivieri

2020

Symmetry

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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.

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“…Since then various more precise and more general versions have been considered (see Refs. 300,[306][307][308][309][310][311] ). These considerations of symmetry broken systems are important in order to establish whether or not long-range order is possible in various concrete situation.…”

Section: Bogoliubov's Inequality and The Mermin-wagner Theoremmentioning

confidence: 99%

Bogoliubov's Vision: Quasiaverages and Broken Symmetry to Quantum Protectorate and Emergence

Kuzemsky

2010

Int. J. Mod. Phys. B

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In the present interdisciplinary review we focus on the applications of the symmetry principles to quantum and statistical physics in connection with some other branches of science. The profound and innovative idea of quasiaverages formulated by N. N. Bogoliubov, gives the so-called macro-objectivation of the degeneracy in domain of quantum statistical mechanics, quantum field theory and in the quantum physics in general. We discuss the complementary unifying ideas of modern physics, namely: spontaneous symmetry breaking, quantum protectorate and emergence. The interrelation of the concepts of symmetry breaking, quasiaverages and quantum protectorate was analyzed in the context of quantum theory and statistical physics. The chief purposes of this paper were to demonstrate the connection and interrelation of these conceptual advances of the many-body physics and to try to show explicitly that those concepts, though different in details, have a certain common features. Several problems in the field of statistical physics of complex materials and systems (e.g. the chirality of molecules) and the foundations of the microscopic theory of magnetism and superconductivity were discussed in relation to these ideas.

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Note on the Mermin‐Wagner Theorem

Cited by 4 publications

References 16 publications

The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results

The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results

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

Bogoliubov's Vision: Quasiaverages and Broken Symmetry to Quantum Protectorate and Emergence

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