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arXiv:1503.05843 · Advances in High Energy Physics

Virial Theorem for Nonrelativistic Quantum Fields in D Spatial Dimensions

Chris L. Lin, Carlos R. Ordóñez

Published Jan 1, 2015• 8 citations
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Abstract

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.

DOI
10.1155/2015/796275

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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 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
It is shown by dimensional analysis that Pauli's ground-state-energy theorem, which expresses the ground-state energy in the form of a coupling-constant integration of the potential energy, and the virial theorem are equivalent for a many-particle system interacting with a homogeneous interparticle potential ($\ensuremath{\propto}{r}^{l}$) in arbitrary (ddd) dimensions. The finite-temperature case is also discussed.
The field theory of nonrelativistic fermions interacting via contact interactions can be used to calculate the properties of few-body systems of cold atoms confined in harmonic traps. The state-operator correspondence of nonrelativistic conformal field theory (NRCFT) shows that the energy eigenvalues (in oscillator units) of NNN harmonically trapped fermions can be calculated from the scaling dimensions of NNN-fermion operators in the NRCFT. They are also in one-to-one correspondence with zero energy, scale-invariant solutions to the NNN-body problem in free space. We show that these two mappings of the trapped fermion problem to free space problems are related by an automorphism of the SL(2,R)SL(2,R)SL(2,R) algebra of the conformal symmetry of fermions at the unitary limit. This automorphism exchanges the internal Hamiltonian of the gas with the trapping potential and hence provides a method for deriving virial theorems for trapped Fermi gases at the unitary limit. We also show that the state-operator correspondence can be applied directly in three spatial dimensions by calculating the scaling dimensions of two- and three-fermion operators and finding agreement with known exact results for energy levels of two and three trapped fermions at the unitary limit.
Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, we have derived an equation using the time-independent Schr\"odinger equation, which can be treated as a sufficient condition for a given stationary quantum state. A change of boundary condition does not affect these virial equations. In the hard confining condition, the perturbing (confining potential) does not affect the expression; it merely shifts the boundary from infinity to a finite region. In the soft case, on the contrary, the final expression includes contributions from the perturbing term. These are demonstrated numerically for several representative enclosed systems like harmonic oscillators (one-dimensional and three-dimensional) and hydrogen atoms. Its applicability in various other confinements (including angular) has been discussed. In essence, a virial equation has been proposed for free and confined quantum systems, from simple arguments.
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.
For the system of noninteracting fermions in a one-body potential , the local virial theorems (LVT) are relations, at a given point in space, between this potential, kinetic energy and particle densities. It was recently shown (Brack et al 2010 J. Phys. A: Math. Theor. 43 255204) that for d-dimensional linear and also for isotropic harmonic oscillator potentials these LVTs are exactly satisfied. We present alternative and simple proofs of these theorems, by consideration of the canonical or Bloch density matrix and its relation to the kinetic energy density. The explicit analytical forms of the Bloch density matrix are used for the above-mentioned potentials to achieve the proofs. For the case of linear potential, we obtain a more general result for the so-called semilocal virial theorem, and for the harmonic oscillator potential case we derive a new relationship between the diagonal part of the canonical bloch density and the kinetic energy density.