$T\overline{T}$ and $J\overline{T}$ Deformations in Quantum Mechanics

Let us consider a $d=2$ Lorentz invariant $QFT$ on a cylinder $\mathbb{R}\times S^{1}$ of radius $R$ with a stress tensor $T_{\mu\nu}$ which is conserved. Conservation implies

Then, the $T\overline{T}$ operator of such a system is defined as

Let the Lagrangian density of such a theory (before deformation) be denoted by $\mathcal{L}_{0}$. Note that the $T\overline{T}$ operator is a composite operator and must be defined by point splitting. That the coincident limit is finite up to total derivative terms has been proved in Zamolodchikov:2004ce .

Next, let us consider an RG flow in the space of theories parametrized by the affine parameter $\lambda$ such that at each point on the trajectory in the space of theories, the flow is triggered by the operator $T\overline{T}(\lambda)$ at that point on the trajectory. Setting $\lambda=0$ gives back the seed theory one started with.

Let the Lagrangian of the deformed theory at an arbitrary point on the RG trajectory be given by $\mathcal{L}_{\lambda}$. Then the deformation of a $d=2$ Lorentz invariant $QFT$ by the operator $T\overline{T}$ is given by the flow equation

Although this deformation is irrelevant, it turns out to be solvable. The deformed spectrum, $E(\lambda;E_{0},P)$, can be expressed as a function of the spectrum $E_{0}$ and momentum $P$ of the undeformed theory as

Note that for $\lambda>0$, the deformed spectrum is unitary as opposed to for $\lambda<0$, in which case the spectrum becomes complex above some threshold.

To define a $J\overline{T}$ deformation of a CFT, let us start with a $d=2$ CFT on a cylinder of radius $R$ that contains a conserved left moving $U(1)$ current $J$. Then the flow equation of $J\overline{T}$ deformation is given by

where $\alpha$ is the coupling, $J(\alpha)$ is the deformed left-moving $U(1)$ current and $\overline{T}(\alpha)$ is the deformed right-moving component of the stress tensor. Similar to the $T\overline{T}$ deformation, the $J\overline{T}$ deformation is irrelevant, but unlike the previous case, $J\overline{T}$ deformation breaks Lorentz invariance.

It has been argued in Chakraborty:2018vja that all along the RG flow,

1. (i)

   the left-moving component of the stress tensor, $T(\alpha)$, remains holomorphic i.e.  $\overline{\partial}T(\alpha)=0$,

2. (ii)

   the left moving component of the global $U(1)$ current remains holomorphic i.e. $\overline{\partial}J(\alpha)=0$,

3. (iii)

   the quantity $T(\alpha)-{1\over 2}J(\alpha)^{2}$ is independent of the coupling $\alpha$.

From the above three conditions one can construct the deformed spectrum that takes the following form Guica:2017lia ; Chakraborty:2018vja :

where, $Q$ is the undeformed left-moving $U(1)$ charge. Note that in this case the spectrum above some threshold, becomes complex for both signs of the coupling.

Next, one can consider a deformation by a general linear combination of $T\overline{T}$, $J\overline{T}$ and $T\overline{J}$. The flow equations are given by

where, $\alpha_{\pm}$ are respectively the $J\overline{T}$ and $T\overline{J}$ coupling. Defining the theory is a bit involved, so we will refer to the references LeFloch:2019rut ; Chakraborty:2019mdf for more details. The deformed spectrum takes the following form

with,

where, $Q_{L,R}$ are the left and right-moving $U(1)$ charges. For $A\geq 0$ the deformed spectrum is unitary, else there is a threshold above which the spectrum becomes complex.

In the following section, we are going to draw motivations from the expressions of the deformed spectrum in $d=2$ to define an analogue of $T\overline{T}$ and $J\overline{T}$ deformations in one-dimensional quantum mechanical system.

The motivation for studying $T\overline{T}$ deformation in quantum mechanics is two-fold. Firstly, to understand $JT$ gravity with a finite radial cutoff in the bulk $AdS_{2}$ Gross:2019ach ; Gross:2019uxi ; Iliesiu:2020zld and secondly to study a solvable deformation of quantum mechanics that would give rise to a deformed spectrum with a square root branch cut singularity like the one that appears in two dimensions (4). The $T\overline{T}$ flow equation as defined in Gross:2019ach ; Gross:2019uxi takes the following form:

where, $S_{E}$ is the euclidean action of the deformed theory, $\tau$ is the Euclidean time, and $T=T^{\tau}_{\tau}=H$ is the stress tensor of the system. This leads to the differential equation for the deformed spectrum

which solves as

The spectrum agrees precisely with the energy of a two dimensional black hole in $AdS_{2}$ with a finite radial cutoff Gross:2019ach for $\lambda<0$. Note that, the limit $\lambda\to 0$ leads to the undeformed spectrum. As in the case of two dimensions, the deformed spectrum is unitary for $\lambda>0$ whereas for $\lambda<0$, there is a threshold above which the spectrum becomes complex. One can thus read off the deformed Hamiltonian operator as

An obvious question that one may raise at this point is: what would be an analogue of $J\overline{T}$ deformation in quantum mechanics? In the absence of any nice holographic interpretation, we would like to draw motivation from the spectrum (6) of a $CFT_{2}$ deformed by the $J\overline{T}$ operator. The deformed spectrum in two dimensions has the properties that it has a square root branch cut much like the case of $T\overline{T}$ and that for both signs of the $J\overline{T}$ coupling, the spectrum is non-unitary. Inspired from (6), let us define the deformed spectrum of a one-dimensional quantum mechanical system with a global $U(1)$ charge $Q$ (which in quantum mechanics is also the conserved current $J$) as follows:⁷⁷7Note that in quantum mechanics, there are no space directions. Thus in defining the deformed spectrum in one dimension, one sets the momentum $P=0$.

where $\alpha$ is the coupling of the deforming operator, and $\ell$ is some length scale associated with the field space of undeformed theory (e.g. for a free compact scalar $y\sim y+2\pi\sqrt{\ell}$, $\sqrt{\ell}$ is the radius of the compact boson in the configuration space.).⁸⁸8In the canonical formalism, the charge $Q$ is dimensionless in all dimensions. The coupling $\alpha$ has length dimension $1/2$. Thus to make the quantity $\alpha/\sqrt{\ell}$ dimensionless, $\ell$ must have dimension of length.

The flow equation that would give rise to such a deformed spectrum is given by

This leads to the following differential equation of the deformed spectrum

which solves as (24).⁹⁹9Note that, $Q=0$ and $\alpha^{2}\to-2\lambda$ give rise to the flow equation and deformed spectrum for the $T\overline{T}$ case.¹⁰¹⁰10Recall that we assumed $Q$ remains undeformed all along the flow.

In this subsection, we are going to turn on a deformation that is a general linear combination of $T\overline{T}$ and $J\overline{T}$. As stated in subsection 3.2, the $J\overline{T}$ deformation alone makes the theory non-unitary. In the discussion that follows, we are going to argue that in the presence of both these deformations, there exist a region in the parameter space (i.e. the space spanned by $\lambda$ and $\alpha$) where the deformed theory is unitary. To define such a deformation, we are, as before, going to draw motivation from the deformed spectrum (10), (11). Thus let us define the spectrum of a one-dimensional quantum mechanical system with a global $U(1)$ charge $Q$ to be of the form¹¹¹¹11Note that, one can further generalize the deformed spectrum by keeping both $\alpha_{\pm}$ and $Q_{L/R}$ from (10) and defining resulting spectrum as the spectrum obtained by deforming a quantum mechanical system by a general linear combination of three operators namely $T\overline{T}$, $J\overline{T}$ and $T\overline{J}$. The charges $Q_{L/R}$ can be thought of as charges coming from two global $U(1)$’s.

where

and as before $\lambda$ and $\alpha$ are respectively the $T\overline{T}$ and $J\overline{T}$ coupling. It’s obvious from the deformed spectrum that the theory is unitary in the regime in the parameter space where $h>0$. This corresponds to the region above the red curve (i.e. the locus $\lambda=\alpha^{2}/2$) in figure 1. For $h<0$, which corresponds to the region below the red curve in figure 1, the deformed spectrum above some threshold becomes complex. Approaching the red curve from the above (i.e. $h\to+\infty$) gives an interesting intermediate regime where the theory is unitary and exhibits many exotic behaviors, that we will discuss later.

Note that in the limit $\alpha\to 0$, (33) gives rise to the $T\overline{T}$ deformed spectrum (16) and in the limit $\lambda\to 0$, it gives the $J\overline{T}$ deformed spectrum (24). As specified earlier there is an intermediate regime $h\to\infty$ where the deformed spectrum takes the form

The flow equations that would give rise to such a deformed spectrum are given by

This leads to the following differential equations in the deformed spectrum:

which solves as (33).

The discussion here is in the same spirit as appears in Gross:2019uxi . Let us consider a set of $n$ operators $\mathcal{O}_{i}(\tau_{i})$ $\forall i=\{1,2,\cdots,n\}$ that doesn’t change upon deformation inserted at $\tau_{i}$. For simplicity lets also assume that $\tau_{1}>\tau_{2}>\cdots>\tau_{n-1}>0$.Then the n-point thermal correlator of the deformed theory is given by

where $\beta_{i}=\tau_{i}-\tau_{i+1}$ for $i=0,1,\cdots,n-1$, $\tau_{0}=\beta$ and $\beta_{n-1}=\tau_{n-1}$. Using the kernel formula (40) and (41), one can express the deformed n-point thermal correlation function in terms of the undeformed correlation function $G_{0}^{(n)}(\beta^{\prime},\{\tau^{\prime}_{i}\})$ as

where $\beta^{\prime}_{i}=\tau^{\prime}_{i}-\tau^{\prime}_{i+1}$ for $i=0,1,\cdots,n-1$, $\tau^{\prime}_{0}=\beta^{\prime}$ and $\beta^{\prime}_{n-1}=\tau^{\prime}_{n-1}$.

As a simple example let us consider the deformation of the 2-point functions of conformal quantum mechanics at zero chemical potential. The 2-point functions of operators of dimension $\Delta$ in conformal quantum mechanics is given by DeAlfaro:1977kq ; Chamon:2011xk

Thus, using (43) the 2-point function of a conformal quantum mechanics deformed by a general linear combination of $T\overline{T}$ and $J\overline{T}$ at zero chemical potential (i.e. $\mu=0$) is given by ¹²¹²12In (45), $K_{n}(x)$ is the modified Bessel function of second kind, not to be confused with the kernel $K$.

Note that the $\lambda$ and $\alpha$ dependences in (45) come in the combination $h$. Thus, in the limit $h=\infty$ (i.e. $\lambda=\alpha^{2}/2$), the two point function of the deformed theory takes the same form as that of the undeformed theory. This implies that in the absence of chemical potential, there exists a regime in the parameter space (i.e. for all points on the red curve in figure 1) where the theory “looks undeformed”. This is reminiscent of the fact that the deformed energy spectrum in the limit $h\to\infty$, at some fixed charge remains unchanged up to some multiplicative factor.

Starting from a conformal quantum mechanics, the $T\overline{T}$ and the $J\overline{T}$ deformation break certain symmetries (e.g. scale invarience) of the seed theory. It’s likely that in the regime in the parameter space where $h=\infty$, there is a restoration of some amount of broken symmetries. This is reflected in the fact that the 2-point function “looks undeformed” in this limit. It’s as if that the effects of the $T\overline{T}$ and $J\overline{T}$ deformations cancel each other in this regime in the parameter space. In the grand canonical ensemble, where the system is coupled to a finite chemical potential, there we do expect to see some changes.

For $h<0$, the deformed Euclidean 2-point function (45) becomes complex. This is related to the non-unitarity of the theory for $h<0$.

In this subsection we are going to present another application of the kernel formula (40) and (41) in analyzing the thermodynamics of a certain class of quantum systems. This is going to reveal the non-local nature of the theory in the UV.

Let us consider systems with linear specific heat in the presence of a finite chemical potential ¹³¹³13In $d=1$ quantum mechanics, systems like Schwarzian models exhibit linear specific heat Stanford:2017thb . The kind of systems we are interested in (46) could be thought of as a charged versions of the usual Schwarzian models. This could also be relevant in understanding the holographic dual of $T\overline{T}+J\overline{T}$ deformed charged SYK model.. The partition function of such a thermodynamic systems schematically takes the following form

where $c_{1,2}$ are system dependent positive constants proportional to the number of degrees of freedom of the system. Using the kernel formula, one can derive the deformed partition function and analyze the thermodynamics of the deformed system. Let’s do it in two steps: first lets analyze the $T\overline{T}$ case and then analyze the more general case namely $T\overline{T}+J\overline{T}$ at zero chemical potential of the deformed theory.