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Path-integral Fujikawa's approach to anomalous virial theorems and equations of state for complex 2D bosonic fields with contact interactions, and for other systems with SO(2,1) symmetry

CARLOS R. ORDÓÑEZ

Published Mar 4, 2015• 0 citations
OpenAlex

Abstract

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

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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
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.
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.
A system of two-species, one-dimensional fermions, with an attractive two-body interaction of the derivative-delta type, features a scale anomaly. In contrast to the well-known two-dimensional case with contact interactions, and its one-dimensional cousin with three-body interactions (studied recently by some of us and others), the present case displays dimensional transmutation featuring a power-law rather than a logarithmic behavior. We use both the Schrödinger equation and quantum field theory to study bound and scattering states, showing consistency between both approaches. We show that the expressions for the reflection (R) and the transmission (T) coefficients of the renormalized, anomalous derivative-delta potential are identical to those of the regular delta potential. The second-order virial coefficient is calculated analytically using the Beth–Uhlenbeck formula, and we make comments about the proper ϵB→0 (where ϵB is the bound-state energy) limit. We show the impact of the quantum anomaly (which appears as the binding energy of the two-body problem, or equivalently as Tan's contact) on the equation of state and on other universal relations. Our emphasis throughout is on the conceptual and structural aspects of this problem.
The relationship between two-dimensional SO(2,1)\text{SO}(2,1)SO(2,1) conformal anomalies in nonrelativistic systems and the virial expansion is explored using recently developed path-integral methods. In the process, the Beth-Uhlenbeck formula for the shift of the second virial coefficient $\ensuremath{\delta}{b}_{2}$ is obtained, as well as a virial expansion for the Tan contact. A possible extension of these techniques for higher orders in the virial expansion is discussed.
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.