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arXiv:1503.02585 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Path-integral derivation of the nonrelativistic scale anomaly

Chris L. Lin, CARLOS R. ORDÓÑEZ

Published Apr 16, 2015• 8 citations
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Abstract

In this paper we calculate the scale anomaly for a quantum field theoretic 2D-nonrelativistic Bose gas with contact interactions using Fujikawa's method, both in vacuum and in many-body systems. The use of path integrals for these problems is novel and motivated by a recently developed path-integral framework for addressing questions about scaling in these systems. A natural class of regulators is found that produces the correct value of the anomaly traditionally calculated via other methods, e.g., diagrammatically via the $\ensuremath{\beta}$ function.

DOI
10.1103/physrevd.91.085023

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In this paper we calculate the scale anomaly for a quantum field theoretic 2D-nonrelativistic Bose gas with contact interactions using Fujikawa's method, both in vacuum and in many-body systems. The use of path integrals for these problems is novel and motivated by a recently developed path-integral framework for addressing questions about scaling in these systems. A natural class of regulators is found that produces the correct value of the anomaly traditionally calculated via other methods, e.g., diagrammatically via the \ensuremathβ\ensuremath{\beta}\ensuremathβ function.
We derive anomalous equations of state for non-relativistic 2D complex bosonic felds with contact interactions, using Fujikawa's path-integral approach to anomalies and scaling arguments. In the process, we derive an anomalous virial theorem for such systems. The methods used are easily generalizable for other 2D systems, including fermionic ones, and of different spatial dimensionality, all of which share a classical SO(2,1) Schrodinger symmetry. The discussion is of a more formal nature and is intended mainly to shed light on the structure of anomalies in 2D many-body systems. The practicality of these ideas rests upon being able to compute in detail the Fujikawa Jacobian that contains the anomaly. This and other technical and conceptual issues are discussed at the end of the paper
We regulate in Euclidean space the Jacobian under scale transformations for two-dimensional nonrelativistic fermions and bosons interacting via contact interactions and compare the resulting scaling anomalies. For fermions, Grassmannian integration inverts the Jacobian; however, this effect is canceled by the regularization procedure and a result similar to that of bosons is attained. We show the independence of the result with respect to the regulating function, and show the robustness of our methods by comparing the procedure with an effective potential method using both cutoff and $\ensuremath{\zeta}$-function regularization.
The virial theorem for nonrelativistic complex fields in D spatial dimensions and with arbitrary many-body potential is derived, using path-integral methods and scaling arguments recently developed to analyze quantum anomalies in low-dimensional systems. The potential appearance of a Jacobian J due to a change of variables in the path-integral expression for the partition function of the system is pointed out, although in order to make contact with the literature most of the analysis deals with the J=1 case. The virial theorem is recast into a form that displays the effect of microscopic scales on the thermodynamics of the system. From the point of view of this paper the case usually considered, J=1 , is not natural, and the generalization to the case J≠1 is briefly presented.
We show that a system of three species of one-dimensional fermions, with an attractive three-body contact interaction, features a scale anomaly directly related to the anomaly of two-dimensional fermions with two-body contact forces. We show, furthermore, that those two cases (and their multispecies generalizations) are the only nonrelativistic systems with contact interactions that display a scale anomaly. While the two-dimensional case is well known and has been under study both experimentally and theoretically for years, the one-dimensional case presented here has remained unexplored. For the latter, we calculate the impact of the anomaly on the equation of state, which appears through the generalization of Tan's contact for three-body forces, and determine the pressure at finite temperature. In addition, we show that the third-order virial coefficient is proportional to the second-order coefficient of the two-dimensional two-body case.
We develop an approach of calculating the many-body path integral based on the linked cluster expansion method. First, we derive a linked cluster expansion and we give the diagrammatic rules for calculating the free energy and the pair distribution function g(r)g(r)g(r) as a systematic power-series expansion in the particle density. We also present a structured Pad\'e approximation scheme in the momentum space to determine g(r)g(r)g(r). The calculated g(r)g(r)g(r) for distinguishable particles interacting with Lennard-Jones and hard-sphere potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path-integral Monte Carlo simulation. Our method is applicable to a wide range of problems of current general interest and may be extended to the case of identical particles and, in particular, to the case of the many-fermion problem.
We analyze, from a canonical quantum field theory (QFT) perspective, the problem of one-dimensional particles with three-body attractive interactions, which was recently shown to exhibit a scale anomaly identical to that observed in two-dimensional (2D) systems with two-body interactions. We study in detail the properties of the scattering amplitude including both bound and scattering states, using cutoff and dimensional regularization, and clarify the connection between the scale anomaly derived from thermodynamics to the nonvanishing non-relativistic trace of the energy–momentum tensor.