New approach in the theory of low-dimensional Heisenberg magnets

Antsygina, T. N.; Slyusarev, V. A.

doi:10.1007/bf01015896

Cited by 9 publications

(7 citation statements)

References 10 publications

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“…A thorough discussion of the relevance of the Sinai model in condensed matter physics and a review of the early results obtained may be found in [11,12]. In this paper we analyse the continuum version of the Sinai model, using techniques similar to those used in the theory of disordered quantum systems [14] and that have already been used in the study of the Sinai model [3,10]. We rederive the results already known in a very compact fashion which in addition sheds more light on the nature of these solutions.…”

Section: Introductionmentioning

confidence: 99%

Exact results on Sinai's diffusion

Comtet¹,

Dean²

1998

J. Phys. A: Math. Gen.

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Abstract:We study the continuum version of Sinai's problem of a random walker in a random force field in one dimension. A method of stochastic representations is used to represent various probability distributions in this problem (mean probability density function and first passage time distributions). This method reproduces already known rigorous results and also confirms directly some recent results derived using approximation schemes. We demonstrate clearly, in the Sinai scaling regime, that the disorder dominates the problem and that the thermal distributions tend to zero-one laws.

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Section: Introductionmentioning

confidence: 99%

Exact results on Sinai's diffusion

Comtet¹,

Dean²

1998

J. Phys. A: Math. Gen.

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“…It therefore captures all the statistical properties of τ (k) once the distribution of ξ(x) is known. Denoting by x c the correlation length of V (x) and assuming that x c and k −1 are the smallest length scales of the system, then it was proven in [1] that the variable ξ(x) is a Brownian motion of the form:…”

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confidence: 99%

Universality of the Wigner Time Delay Distribution for One-Dimensional Random Potentials

1999

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We show that the distribution of the time delay for one-dimensional random potentials is universal in the high energy or weak disorder limit. Our analytical results are in excellent agreement with extensive numerical simulations carried out on samples whose sizes are large compared to the localisation length (localised regime). The case of small samples is also discussed (ballistic regime). We provide a physical argument which explains in a quantitative way the origin of the exponential divergence of the moments. The occurence of a log-normal tail for finite size systems is analysed. Finally, we present exact results in the low energy limit which clearly show a departure from the universal behaviour.The problem of quantum scattering by chaotic or disordered systems is encountered in many fields ranging from atomic or molecular physics as well as in the scattering of electromagnetic microwaves. Some properties of the scattering process are well captured through the concept of time delay. This quantity, which goes back to Eisenbud and Wigner [7], is related to the time spent in the interaction region by a wave packet of energy peaked at E. It can be expressed in terms of the derivative of the S matrix with respect to the energy. In the context of chaotic scattering the approach based on random matrix theory (RMT) provides a statistical description of the time delays. This problem was first studied by a supersymmetric approach [11] and in [13] by using a statistical analysis. This latter work provides a derivation for the one channel case for the different universality classes. Recently it served as a starting point for [3] where the N channel distribution is shown to be given by the Laguerre ensemble of RMT. In spite of its success, such a description by RMT is not entirely satisfactory, in particular it does not apply to strictly one-dimensional systems [10] for which strong localisation effects occur. Furthermore it does not shed much light on the physical mechanisms which are responsible for the universal distribution. In this work we explore another approach by considering the scattering by a one-dimensional random potential. In this case the existence of universal distributions was first conjectured in [6] on the basis of a comparative study of two different models. This was further supported by [16] where the random potential is still of a different kind.The purpose of this letter is to present a new derivation that accounts for the universality and also to provide a physical picture that explains the origin of the algebraic tail of the distribution in terms of resonances. Further details will be given elsewhere [22]. To begin with, let us briefly recall the model. We consider the Schrödinger equation on the half line x 0:

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“…3, for a bipartite two-dimensional antiferromagnet at the temperatures t = 0.2 and t = 0.4, the ratio P int is plotted as a function of magnetic (m H ) and staggered (m) field strength. 6 One observes that the interaction in stronger fields is quite large, amounting up to about fifteen percent compared to the noninteracting magnon gas contribution. In the entire parameter space we consider, the genuine spin-wave interaction in the pressure is attractive and tends to zero when the magnetic field is turned off.…”

Section: Two-dimensional Antiferromagnetsmentioning

confidence: 99%

“…The impact of external magnetic fields on antiferromagnetic systems -both in two and three spatial dimensions -has been studied by various authors employing different techniques: (modified) spin-wave theory [1][2][3][4][5][6][7][8][9][10][11], Green's functions [12][13][14][15][16], series expansions [17][18][19][20], Monte Carlo simulations [21][22][23], exact diagonalization [24,25], and yet other methods [26][27][28][29][30][31][32][33][34][35][36][37].…”

Section: Introductionmentioning

confidence: 99%

Antiferromagnetic Dispersion Relations and Nature of Magnon Pressure

Hofmann¹

2020

Preprint

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We derive higher-order corrections in the magnon dispersion relations for two-and three-dimensional antiferromagnets exposed to magnetic and staggered fields that are mutually aligned. "Dressing" the magnons is the prerequisite to separate the low-temperature representation of the pressure into a piece due to noninteracting magnons and a piece that corresponds to the magnon-magnon interaction. Both in two and three spatial dimensions, the interaction in the pressure turns out to be attractive. While concrete figures refer to the spin-1 2 square-lattice and the spin-1 2 simple cubic lattice antiferromagnet, our results are valid for arbitrary bipartite geometry.

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New approach in the theory of low-dimensional Heisenberg magnets

Cited by 9 publications

References 10 publications

Exact results on Sinai's diffusion

Exact results on Sinai's diffusion

Universality of the Wigner Time Delay Distribution for One-Dimensional Random Potentials

Antiferromagnetic Dispersion Relations and Nature of Magnon Pressure

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