Dilational symmetry-breaking in thermodynamics

Abstract

Using thermodynamic relations and dimensional analysis we derive a general formula for the thermodynamical trace 2E − DP for nonrelativistic systems and E − DP for relativistic systems, where D is the number of spatial dimensions, in terms of the microscopic scales of the system within the grand canonical ensemble. We demonstrate the formula for several cases, including anomalous systems which develop scales through dimensional transmutation. Using this relation, we make explicit the connection between dimensional analysis and the virial theorem. This paper is focused mainly on the non-relativistic aspects of this relation.

Virial expansion for the Tan contact and Beth-Uhlenbeck formula from two-dimensional SO(2,1) anomalies

“…where E = energy density = H A , A = 2D volume, P is the pressure, and D the dimensionality of space (D=2 in this paper). The {E k } are a set of energy parameters that may include bound state energies as well as those formed from dimensionful couplings constants in S E [16]. In our case, there is no dimensionful coupling constant (c is dimensionless) and there is only one bound state energy −E b , E b > 0 (we will use E b in Eq.…”

“…where f (z i , βE k ) is a dimensionless function of dimensionless variables and z i = e βµi is the fugacity corresponding to µ i . It is straightforward to show that [16]…”

“…In this paper, we show that δb 2 is indeed produced entirely by the anomaly. We use a path-integral approach inspired by the work of [10][11][12][13][14][15][16]. In the process, we will describe the virial expansion of the Tan contact (which in 2D is interpreted as the anomaly [17]), as well as a procedure to compute δb n , n ≥ 2, using the Hubbard-Stratonovich (HS) representation of the partition function.…”

Virial theorem, boundary conditions, and pressure for massless Dirac electrons

“…The equation (B5) can be interpreted as a counterpart of (14) for the many-body wave function subject to momentum cutoff. The scaling analysis of a system of massless Dirac electrons can be also found in [29].…”

“…For three-dimensional Dirac and Weyl semimetals, various boundary conditions are proposed [27,28]. Other possible anomalies in scaling properties of a system of massless Dirac electrons can also give rise to additional terms in the virial theorem [29].…”