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Physical Review Letters

Renormalization Group and Exact Solubility in One Dimension: Generalized Sutherland Model

Eugene B. Kolomeisky

Published Sep 19, 1994• 5 citations
Publisher

Abstract

We introduce a generalized Sutherland model: a system of one-dimensional fermions or bosons interacting by a two-body potential which has a short-range interaction of finite range and amplitude, and an inverse-square tail. Using renromalization-group arguments we analyze the ground-state and correlation properties of this family of models. We reproduce known exact results as special cases of the general model and point out a correspondence between the exactly solvable problems and special initial or final conditions of the renormalization transformation, which we believe to be a general criterion. This enables us to predict a series of new presumably exact results.

DOI
10.1103/physrevlett.73.1648

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We introduce a generalized Sutherland model: a system of one-dimensional fermions or bosons interacting by a two-body potential which has a short-range interaction of finite range and amplitude, and an inverse-square tail. Using renromalization-group arguments we analyze the ground-state and correlation properties of this family of models. We reproduce known exact results as special cases of the general model and point out a correspondence between the exactly solvable problems and special initial or final conditions of the renormalization transformation, which we believe to be a general criterion. This enables us to predict a series of new presumably exact results.
We consider a low-density system of Bose particles in general dimensionality interacting through a potential having a short-range part and a long-range tail that falls off like 1/rs{\mathit{r}}^{\mathit{s}}rs, focusing mainly on the special case s=2. We use a renormalization-group approach for the computation of the ground-state properties, which we argue is asymptotically exact in the dilute limit. The strength of the long-range part determines fixed-point values for the amplitude of the short-range part; several phases can occur, depending on the relative strengths of the short-range part and the fixed-point amplitudes. For the case in which the physics is determined by the flow away from the unstable fixed point (or when there are no fixed points at all), there results a collapsed state. When the flow is towards the stable fixed point, the outcome depends on whether the fixed-point value is negative or positive: In the former case we predict a kind of bound state, which is characterized by nontrivial correlations, even though the particles are not pairwise bound; in the latter case, a density expansion for the ground-state properties is given. The leading term of this expansion gives an exact result (a generalization of the one-dimensional result previously given by Sutherland) whenever the renormalization flow ends at the stable fixed point (including in particular the case that the flow starts at the fixed point). For the case that the renormalization flow stops before reaching the fixed point, the expansion parameter is proportional to the density raised to a nonuniversal power. The case s>2 can be reduced to the treatment of purely short-range interactions, whereas the repulsive case of s2 cannot be handled without additionally introducing a background of opposite charge. The case of attraction for s2 always leads to a collapse.
We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order. Calogero-Sutherland type models are a class of exactly solvable models. Starting from the inception [1], till today, these models continue to be of interest, especially due to their exact solvability [2]. Moreover, these models are related to (1 + 1)-dimensional conformal field theory, random matrices and a host of other things [3]. Some variants of the models have attracted attention in very recent past [3–5]. The original model of Calogero [1] described N particles in one dimension, interacting through an inverse square law and a two-body harmonic interaction. The latter interaction was added to the effect of complete discretization of the spectrum. This model gave in to an analytic treatment yielding an exact solution. Sutherland [6] considered a variant of this problem where the harmonic interaction was replaced by a harmonic well containing the N particles. It was shown that one can exactly solve for the ground state of the system. The norm of the ground state wave-function was shown to be of a form that coincides with the joint probability density function of the eigenvalues of random matrices from an orthogonal, unitary or symplectic Gaussian ensemble, provided, the
A system of fermions or bosons interacting in one dimension via a two-body potential of the form V(r)=gr2V(r)=\frac{g}{{r}^{2}}V(r)=r2g​ is known to have a ground state of Jastrow type. The boson system is explored in this paper within the hypernetted-chain scheme to study the various ground-state properties, and the results are compared with the exact results available for this system. The results are expected to have important bearing on spin correlations in the Gutzwiller state.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
We investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.