--- abstract: 'We study the thermodynamics of a family of deformed CFTs, using free CFT as the UV fixed point. We discuss free CFTs with double trace deformations and deformed Lifshitz fixed points, and construct their RG flows. We evaluate partition functions and central charges in terms of the beta functions, which are governed by the corresponding effective potentials. We present analytical expressions for the partition functions and central charges of free fermions and bosons with the deformed potential of the Lifshitz fixed points and evaluate them in terms of certain special functions, the Meijer G-functions, at arbitrary values of the deformation parameter. We establish the general behavior of these partition functions. We investigate their limits for small and large values of the deformation parameter. We also study deformed free scalar field theory on $S^1 \times \mathbb{R}$ space with Lifshitz scaling. In this case we explicitly evaluate the partition function and determine the two point function of the theory for arbitrary values of the deformation parameter. We derive a duality relation between small and large values of the deformation parameter. The partition function and the energy-momentum tensor can be used to study the properties of the theory. Using these tools we investigate some of the physical properties of the theory.' author: - Siavash Aslanbeigi - Hossein Bakhram - 'Ali M. Ghezelbash' - Pouria Pedram title: 'Thermodynamics of deformed conformal field theories: A model of non-relativistic quantum mechanics on $\mathbb{R} \times S^1$' --- Introduction {#section_introduction} ============ CFT has served as a very effective tool for studying various problems in physics in recent years. Particularly, the AdS/CFT correspondence [@Maldacena:1997re] has found a lot of applications in theoretical physics, mostly in the context of condensed matter systems [@Hartnoll:2009sz; @Ammon:2011hz; @Sachdev:2010um]. The CFT description of the gravitational systems has been studied in many papers (for a review see e.g. [@Hartnoll:2009sz; @McGreevy:2009xe; @Maldacena:2016upp; @Hubeny:2010ry; @Klebanov:2011gs; @Sachdev:2010um; @Nishioka:2009un; @Hartman:2014oaa; @Heemskerk:2010hk]). Although the construction of a bulk gravity dual is often very hard, the CFT description is usually simpler than the bulk theory, and this motivates us to have an effective technique for treating CFTs. Some time ago Polchinski discovered that there is a special class of deformed CFTs which have special thermodynamic properties and are equivalent to free theories [@Polchinski:1991uq]. In this work, we generalize this kind of deformed CFTs and discuss some of their properties. In the first part of this paper, we study the thermodynamics of deformed CFTs using the example of free fermions and bosons at large $N$ in the limit of large temperature. We show that this limit leads to a new class of Lifshitz-type scaling theories with non-trivial deformed IR fixed points. We examine their thermodynamics in terms of the free energy in a similar way as for Lifshitz fixed points. In the second part of this paper, we consider a family of double-trace deformations. This kind of deformations was first proposed in [@Dubovsky:2010ye; @Dubovsky:2011tu] and considered in a variety of works [@Ghezelbash:2012qn; @Baggio:2012rr; @Keeler:2014bra; @Kim:2013nva; @Donnelly:2015hta; @Ghezelbash:2013dda; @Das:2014jna; @Ghezelbash:2015pda]. The IR behavior of these theories is governed by the IR fixed points, which can be Lifshitz or hyperscaling ones. A related problem is the emergence of an IR Lifshitz-type fixed point in the quantum critical region of a 2D superconductor [@Gubser:2012gy]. We generalize these Lifshitz-type deformations and study their thermodynamics in the large temperature limit. We study the critical point of the transition from this new kind of CFT to the Gaussian CFT. We also find a similar phase transition for a family of deformed UV fixed points in the holographic context, and study their thermodynamics. In the third part of this paper, we consider a scalar quantum field theory in 1+1 dimensions with a potential that has the Lifshitz type scaling at large $N$. We use the free boson and free fermion CFTs as UV fixed points, and the Lifshitz type scalar field theory to model the UV deformed UV CFTs. We use the partition function to calculate the vacuum energy and central charge, and we find them as functions of the deformation parameter. We study the limits of small and large values of the deformation parameter. It is worth mentioning that there has been an interesting duality between the large values of the deformation parameter and the free scalar theory in the IR. We study this duality by studying the two point functions of the theory and comparing them for small and large values of the deformation parameter. We find that the central charge and free energy, in addition to their $N$ dependence, depend on the parameters of the potential in terms of some specific functions such as the Lambert function or the generalized hypergeometric functions. We also calculate their large temperature limits, which are functions of $\epsilon$ with $\epsilon$ representing the Lifshitz deformation parameter. The structure of this paper is as follows: We discuss the thermodynamics of deformed free CFT in section \[sec\_free\], the thermodynamics of the deformed potential theories in section \[sec\_potential\], and the thermodynamics of the deformed scalar field theory in section \[sec\_scalar\]. In section \[sec\_therm\] we investigate the limits of small and large values of the parameters. We summarize our results in section \[sec\_conclusion\] and appendices \[appendix\_sliced\] and \[appendix\_free\_energy\]. We discuss the generalized hypergeometric functions used in this paper in appendix \[appendix\_hyper\]. Thermodynamics of free CFTs {#sec_free} =========================== In this section we consider a family of deformed free field theories. We first consider one family of deformed free Dirac fermions in the large $N$ limit of the large $k$ limit. We calculate the partition function and free energy in terms of the deformation parameter $k$. We also obtain the central charge in terms of the Lambert function of the deformation parameter, $k$, where the partition function is used to extract the free energy. We next consider another family of deformed free Dirac fermions. We use this family to study the thermodynamics of non-relativistic fermions. We use the two point function to determine the form of the spectral function and obtain the dispersion relation for a free fermionic field theory on $\mathbb{R}^3$. We find that the spectral function can have both Dirac and Schrödinger types of boundary conditions and we propose that there is a duality between the two spectral functions. We briefly discuss the non-relativistic free scalar field theory on $\mathbb{R}$ to consider some general properties of these theories. We study their thermodynamics and find their free energy as a function of the deformation parameter. It is worth mentioning that in the large $N$ limit, these families of deformed free Dirac fermions in $d=1+1$ with the large $k$ limit can be modeled by (free) massive theories on $\mathbb{R}^{1+1}$ with general potential as argued in [@Gubser:2009qt] (and references therein), but the fermions can have different choices of boundary conditions. There is a connection between these theories and the theories with Lifshitz scaling [@Kachru:2008yh]. We briefly discuss the behavior of the free energy and free energy density of massive fermions with $2 < \Delta < 4$ at large $N$. One family of deformed Dirac fermions: Free Majorana fermions {#sec_free_fermions} ------------------------------------------------------------- We begin our study with the partition function of the one family of free Majorana fermions: $$\label{eq_partition_function_free_fermion} Z=e^{F}\int d\Psi d\bar{\Psi} \exp{-\int \frac{d^3p}{(2\pi)^3}\Big{(} \bar{\Psi}_p \Psi_p - M \bar{\Psi}_p \Psi_{-p}\Big{)}},$$ where $\Psi_p$ and $\bar{\Psi}_p$ are the Dirac spinors. We express $Z$ as a path integral. The large $k$ limit is defined as $k\to\infty$ and $N\to\infty$ with $k^2/N\to 0$. To perform the large $k$ limit, we expand the exponentials in the integrand in powers of $k$, use the integral $$\label{eq_integral_representation} e^{\alpha} = \lim_{\nu\to0}\frac{1}{\nu\Gamma(-\nu)}\int_0^\infty dx \, x^{\nu-1} e^{-\alpha x},$$ and make a change of variable from $p$ to $x$, so that the partition function becomes $$\label{eq