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arXiv:cond-mat/9912424

Schrodinger equation approach to non-linear $σ$-models in the large N-limit

B. N. Shalaev

Published Dec 23, 1999• 0 citations
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Abstract

Non-linear d-dimensional vector $σ$-models are studied in the large N-limit. It is found that a two-point correlation function obeys a standard Schrodinger equation for a free quantum particle moving in the $δ$-function quantum well. The threshold problem for bound states in this equation is shown to be equivalent to a critical behavior of these models above and below the Curie point. The SU(N)- symmetric Ginzburg-Landau (GL) $σ$-model subject to a uniform magnetic field H is considered in the large-N limit within the Schrodinger equation approach. A upper critical magnetic field line $H_{c2}(T)$ of type-II superconductors for an arbitrary external H is obtained without exploiting the lowest Landau level (LLL) approximation. Both low-H perturbation expansion terms and exponentially small corrections to the LLL approximation are calculated. Correspondences between the one-particle quantum mechanics and critical phenomena as well as some applications of the above method to other models of statistical mechanics are also discussed.

DOI
10.48550/arxiv.cond-mat/9912424

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Schrodinger equation approach to non-linear σσσ-models in the large N-limit
Non-linear d-dimensional vector σσσ-models are studied in the large N-limit. It is found that a two-point correlation function obeys a standard Schrodinger equation for a free quantum particle moving in the δδδ-function quantum well. The threshold problem for bound states in this equation is shown to be equivalent to a critical behavior of these models above and below the Curie point. The SU(N)- symmetric Ginzburg-Landau (GL) σσσ-model subject to a uniform magnetic field H is considered in the large-N limit within the Schrodinger equation approach. A upper critical magnetic field line Hc2(T)H_{c2}(T)Hc2​(T) of type-II superconductors for an arbitrary external H is obtained without exploiting the lowest Landau level (LLL) approximation. Both low-H perturbation expansion terms and exponentially small corrections to the LLL approximation are calculated. Correspondences between the one-particle quantum mechanics and critical phenomena as well as some applications of the above method to other models of statistical mechanics are also discussed.
Nonlinear d -dimensional vector σ models, such as O ( N ), SU ( N ), and CP N , are considered in the limit of an infinite number of components N . It is shown that the equation for the two-point correlation function in these models is similar to the Schrödinger equation for a quantum particle moving in a δ-function potential well (− T )δ( x ), where T is the temperature. This equation adequately describes the systems under study both above and below the Curie point. Within this approach, the critical behavior of the SU ( N )-invariant Ginzburg-Landau model in an external uniform magnetic field is determined in the vicinity of the upper critical magnetic field. The critical indices in this case are the same as in the spherical model in a random magnetic field. An exact equation describing the H c 2 ( T ) curve of continuous phase transitions is derived, which allows one to determine the asymptotes of this curve in strong and weak fields. The relation between the one-particle Schrödinger equation and critical phenomena is analyzed, and applications of this method to various models in solid state physics and statistical mechanics are discussed.
In three-dimensional O(N)O(N)O(N) models, we investigate the low-momentum behavior of the two-point Green's function G(x)G(x)G(x) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotationally invariant fixed point. Several approaches are exploited, such as strong-coupling expansion of lattice NNN-vector model, and 1/N1/N1/N expansion, field-theoretical methods within the ${\ensuremath{\varphi}}^{4}$ continuum formulation. Non-Gaussian corrections to the universal low-momentum behavior of G(x)G(x)G(x) are evaluated, and found to be very small. In nonrotationally invariant physical systems with O(N)O(N)O(N)-invariant interactions, the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent $\ensuremath{\rho}$, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At $N=\ensuremath{\infty}$ one finds $\ensuremath{\rho}=2$. We show that, for all values of N> 0N>~0N> 0, $\ensuremath{\rho}\ensuremath{\simeq}2$.
One-dimensional spin chains are investigated by constructing a mapping of the anisotropic Heisenberg model in the limit Delta to 1 onto the non-linear Schrodinger model (a gas of bosons with delta function interactions). Three applications of this mapping are given. First it is shown that the Bethe ansatz solutions of the anisotropic spin-1/2 Heisenberg model go over to the continuous Bethe ansatz solutions of the non-linear Schrodinger model. Then bound-state energies for arbitrary spin are calculated and as a third application a nearly identical mapping of the anisotropic spin-1/2 Heisenberg model in the limit Delta to 0 onto a non-linear Schrodinger model for fermion fields is given. Finally the authors construct an anisotropic SU(N) model. The continuum limit of this model is shown to be the (N-1)-component non-linear Schrodinger model, which can be solved by means of the nested Bethe ansatz.
The goal of this thesis is to provide a practical method to calculate, in scalar field theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric and broken phases to perform high precision calculations of the Green's functions at zero momentum. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting β reach a critical value βc. We show that the round-off errors on the magnetic susceptibility grow like (βc − β) −1 near criticality. We show that the systematic errors (finite truncation and volume) can be controlled with an exponential precision and reduced to a level lower than numerical errors. We probe the numerical errors made in Renormalization Group (RG) calculations by varying slightly the resealing factor of the fields and resealing back in order to get the same (if there were no round-off errors) magnetic susceptibility. We find a significant departure from Gaussian behavior. In addition, the average value differs from the exact answer by a quantity which is of the same order as the standard deviation. We calculate the critical exponents γ and the subleading exponent Δ of Dyson's model by direct fits of the zero-momentum two-point function calculated with an Ising and a Landau-Ginzburg measure and by linearization about the non-trivial fixed point for O(N) model up to N = 5. We extend the method to the low temperature phase and show that accurate values of the connected one-, two- and three- point functions can be obtained at large volume and in a limited range of constant external field coupled linearly to the field variable. We show that the leading exponents obey the scaling and hyperscaling relations. We discuss the concept of universality in terms of critical exponents, correction amplitudes and the universal coupling constant for different Landau-Ginzburg measures. We compare our accurate numerical results with those obtained from perturbative methods.
We present a perturbative approach which yields, for the first time, an infrared-stable fixed point in the Ginzburg-Landau (GL) model at the oneloop level. The calculations are done in d = 3 dimensions and below Tc, where the renormalization group functions can be expressed directly as functions of the Ginzburg parameter κ which is the ratio between the two fundamental scales of the problem, the penetration depth λ and the correlation length ξ. We find a charged fixed point for κ> 1 / √ 2, that is, in the type II regime, where ∆κ ≡ κ − 1 / √ 2 is shown to be a natural expansion parameter. This parameter controls a momentum space instability in the two-point correlation function of the order field. This instability appears at a nonzero wave-vector p0 whose magnitude scales like ∼ ∆κ ¯ β, with a critical exponent ¯ β = 1/2 in the one-loop approximation, a behavior known from magnetic systems with a Lifshitz point in the phase diagram. This momentum space instability is argued to be the origin of the negative η-exponent of the order field.
The effect of the long-range correlated random magnetic fields which behave like 〈h(x)h(y)〉〈h(x)h(y)〉〈h(x)h(y)〉 $\ensuremath{\sim}\frac{1}{{|x\ensuremath{-}y|}^{d\ensuremath{-}\ensuremath{\sigma}}}$ on the critical phenomena is discussed with the use of the nonlinear $\ensuremath{\sigma}$ model. The crossover between the two critical behaviors dominated by the long-range and short-range disorder fixed points is shown to occur at $(m\ensuremath{-}2)\ensuremath{\sigma}\ensuremath{-}d+4=0$ (mmm is the spin dimensionality). The critical exponents calculated at the long-range disorder fixed point around the lower critical dimensionality $4+\ensuremath{\sigma}$ are not the same as those in $d\ensuremath{-}2$ expansions in the pure system.