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Physical review. B, Condensed matter

Crossover renormalization-group approach to semi-infinite inhomogeneous critical behavior

A. M. Nemirovsky, Zhen‐Gang Wang, Karl F. Freed

Published Sep 1, 1987• 3 citations
Publisher

Abstract

The inhomogeneous nature of critical phenomena in semi-infinite systems is studied by calculating the full crossover for both the two-point correlation function and the surface susceptibility ${\ensuremath{\chi}}_{11}$(z) as a function of the distance z from the surface at z=0 for a semi-infinite system near the special transition. We introduce z-dependent counterterms to derive the renormalization-group (RG) equation describing this full crossover. The solution to the RG equation in the one-loop approximation is obtained explicitly and in the long-distance regime is shown to approach bulk behavior exponentially fast. In contrast to the usual approach, our RG equation also contains all information about the short-distance (near-surface) behavior of the theory and its crossover to the bulk. Applications of our method to the descriptions of crossovers for other problems of critical phenomena with several competing lengths are also discussed.

DOI
10.1103/physrevb.36.3755

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The inhomogeneous nature of critical phenomena in semi-infinite systems is studied by calculating the full crossover for both the two-point correlation function and the surface susceptibility \ensuremathχ11{\ensuremath{\chi}}_{11}\ensuremathχ11​(z) as a function of the distance z from the surface at z=0 for a semi-infinite system near the special transition. We introduce z-dependent counterterms to derive the renormalization-group (RG) equation describing this full crossover. The solution to the RG equation in the one-loop approximation is obtained explicitly and in the long-distance regime is shown to approach bulk behavior exponentially fast. In contrast to the usual approach, our RG equation also contains all information about the short-distance (near-surface) behavior of the theory and its crossover to the bulk. Applications of our method to the descriptions of crossovers for other problems of critical phenomena with several competing lengths are also discussed.
We study the surface critical behavior of semi-infinite systems belonging to the bulk universality class of the Ising model. Special attention is paid to the local behavior of experimentally relevant quantities such as the order parameter and the correlation function in the crossover regimes between different surface universality classes, where the surface field h1 and the temperature enhancement c can induce additional macroscopic length scales. Starting from the field-theoretical φ4 model and employing renormalization-group improved perturbation theory (ϵ-expansion), explicit results for the local behavior of the correlation and structure function (zero-loop) and the order parameter (one-loop) are derived. Supplementing earlier studies that focused on the special transition, we here pay particular attention to the situation where a large c suppresses the tendency to order in the surface (ordinary transition) but a surface field h1 generates a small surface magnetization m1. Our results are in good agreement with recent phenomenological considerations and Monte Carlo studies devoted to similar questions and, combined with the latter, provide a much more detailed understanding of the local properties of systems with weak surface fields.
The addition of a spatial dependence of the form $\ensuremath{\Delta}J={\mathrm{Az}}^{\ensuremath{-}y}$ to the short-range couplings of a semi-infinite spin system with surface at z=0z=0z=0 and bulk exponent $\ensuremath{\nu}$ is predicted to modify the surface critical behavior for $y<{\ensuremath{\nu}}^{\ensuremath{-}1}$ but not for $y>{\ensuremath{\nu}}^{\ensuremath{-}1}$. This information is implied by the scaling relation ${\ensuremath{\varphi}}_{y}=1\ensuremath{-}\ensuremath{\nu}y$ proposed for the crossover exponent. Crossover effects associated with long-range couplings of the surface spins are also discussed.
The surface critical behavior of semi-infinite systems belonging to the Ising universality class with short-range interactions is investigated for supercritical surface enhancement -c>0 and vanishing surface field h1{\mathit{h}}_{1}h1​. Renormalization-group improved perturbation theory is applied to the standard semi-infinite scalar ${\mathrm{\ensuremath{\varphi}}}^{4}$ model in d=4-\ensuremath{\epsilon} dimensions to compute the order-parameter profile to one-loop order both for temperatures T with \ensuremath{\tau}\ensuremath{\equiv}(T-Tcb{\mathit{T}}_{\mathit{c}\mathit{b}}Tcb​)/Tcb{\mathit{T}}_{\mathit{c}\mathit{b}}Tcb​\ensuremath{\gtrsim}0 and \ensuremath{\tau}\ensuremath{\lesssim}0. The associated scaling functions are found to cross smoothly over from their short-distance behavior for distances z\ensuremath{\ll}${\ensuremath{\xi}}_{\mathit{b}}$ (=bulk correlation length) to their long-distance behavior for z\ensuremath{\gg}${\ensuremath{\xi}}_{\mathit{b}}$ without showing the peculiar nonmonotonic behavior asserted by Peliti and Leibler [J. Phys. C 16, 2635 (1983)]. Furthermore, the short-distance behavior of the profiles is shown to be fully consistent with a \ensuremath{\Vert}\ensuremath{\tau}${\mathrm{\ensuremath{\Vert}}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ singularity of the surface magnetization m1{\mathit{m}}_{1}m1​ plus a regular background term; that is, in contrast to results published recently by other authors, the amplitudes A+{\mathit{A}}_{+}A+​ and ${\mathit{A}}_{\mathrm{\ensuremath{-}}}$ of the contributions ${\mathit{A}}_{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\tau} to m1{\mathit{m}}_{1}m1​ linear in \ensuremath{\tau}>0 or \ensuremath{\tau}0 agree to one-loop order. Finally, we confirm that the universal profiles for the critical adsorption of fluids (governed by the critical-adsorption fixed point at c=+\ensuremath{\infty} and h1{\mathit{h}}_{1}h1​=\ensuremath{\infty}) agree with the previous ones pertaining to the negative-c-transition fixed point at c=-\ensuremath{\infty} and h1{\mathit{h}}_{1}h1​=0.
A semi-microscopic scaling-field theory is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotic critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field tricritical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint.
An introduction into the theory of boundary critical phenomena and the application of the field-theoretical renormalization group method to these is given. The emphasis is on a discussion of surface critical behavior at bulk critical points of magnets, binary alloys, and fluids. Yet a multitude of related phenomena are mentioned. The most important distinct surface universality classes that may occur for a given universality class of bulk critical behavior are described, and the respective boundary conditions of the associated field theories are discussed. The short-distance singularities of the order-parameter profile in the diverse asymptotic regimes are surveyed.
An impurity in a Tomonaga-Luttinger liquid leads to a crossover between the short- and long-distance regime which describes many physical phenomena. However, a calculation of the entire crossover of correlation functions over different length scales has been difficult. We develop a powerful numerical method based on the infinite density matrix renormalization group. By utilizing infinite boundary conditions we can obtain correlation functions within a finite-size window that contains the impurity. For the S=1/2S=1/2S=1/2 chain, we demonstrate that a full crossover can be precisely obtained, and that their limiting behaviors show a good agreement with field-theory predictions.