Non-Hermitian Holography

Quantum theory can be formulated with certain non-Hermitian Hamiltonians. An anti-linear involution, denoted by PT, is a symmetry of such Hamiltonians. In the PT-symmetric regime the non-Hermitian Hamiltonian is related to a Hermitian one by a Hermitian similarity transformation. We extend the concept of non-Hermitian quantum theory to gauge-gravity duality. Non-Hermiticity is introduced via boundary conditions in asymptotically AdS spacetimes. At zero temperature the PT phase transition is identified as the point at which the solutions cease to be real. Surprisingly for solutions containing black holes real solutions can be found well outside the quasi-Hermitian regime. These backgrounds are however unstable to fluctuations which establishes the persistence of the holographic dual of the PT phase transition at finite temperature.

Introduction One of the basic axioms of quantum mechanics is that the dynamics of a quantum system is generated by a Hermitian Hamiltonian. It comes then as a surprise that meaningful quantum mechanics can be formulated for certain non-Hermitian Hamiltonians, the so-called PT-symmetric quantum mechanics [1,2]. We quickly review the salient features of this PT-symmetric quantum mechanics using a simple example [2]. It will serve as a guideline to construct a non-Hermitian holographic model. Consider the Hamiltonian of a two state system State A is unstable and decays with decay rate 2Γ whereas state B is also unstable but suffers exponential growth with the same (inverse) rate. Both states can also transform into each other with amplitude g. The interpretation of such Hamiltonians is that the physical system under consideration is subject to exactly balanced gain and loss terms with external sources and sinks. Since gain and loss is balanced one expects that it is possible for the system to reach a time independent steady state. Indeed the eigenvalues of the Hamiltonian (1) are real as long as the interaction is stronger than the gain/loss terms, |g| > Γ. The gain/loss terms are exchanged by time-reversal T , which in quantum mechanics is just complex conjugation. They are also exchanged by the permutation of the subsystems A and B represented by the matrix P = 0 1 1 0 . The combined action PT leaves the Hamiltonian invariant. The so called PT symmetric regime is the one in which the eigenvalues are real. For |g| < Γ the eigenvalues come in complex conjugate pairs; this is the PT broken regime, and the transition between the two is known as PT phase transition [2].
Let us now discuss a slightly different aspect of the Hamiltonian (1). As pointed out in [3][4][5] a Hamiltonian in the PT symmetric regime is related to a Hermitian one by a similarity transformation. In our case we can start from the fact that every Hermitian Hamiltonian acting on a two-state system can be written as (3) Every two Hamiltonians of this form can be transformed into each other by an SU (2) transformation D( α) = exp(i α 2 σ) via H 2 = D † H 2 D. For example we start with a Hamiltonian with g = (g , 0, 0). An SU (2) transformation generated by σ 2 /2 brings the Hamiltonian into the form If we now analytically continue to imaginary values of the parameter α = iα we find This Hamiltonian is indeed of the form of (1) with g = g cosh(α) and Γ = g sinh(α). The restriction g 2 > Γ 2 is automatically fulfilled. The unitary matrix D(α) −1 = D(α) † becomes the Hermitian one η(α) = η(α) † upon the analytic continuation, and η −1 (α) = η(−α). In the regime of real eigenvalues the Hamiltonian (1) is quasi-Hermitian H 2,nh = η(α) −1 H 2 η(α) [3]. Notice that H 2 is invariant under conjugation with D(α) and a compensating rotation of the couplings g → R(α) g. Hence we can generate the non-Hermitian Hamiltonian from the Hermitian one by transforming the couplings g = (g , 0, 0) withR The case |g| = Γ is special. The Hamiltonian is no longer quasi-Hermitian but it can be reached by taking the limitα → ∞, g → 0 while keeping the product fixed. These special values of the couplings are generically known as "exceptional points".
The guiding principle for constructing the holographic model will be to select an operator that transforms under a simple Lie group and implement the transformation to the non-Hermitian theory as a transformation on the couplings. A typical example in field theory would be to select a Dirac mass termΨΨ and do a complexified axial transformation to obtainΨγ 5 Ψ [6][7][8][9]. Once the quasi-Hermitian theory is obtained it can be extended to the exceptional point and beyond.
Holography Gravitational theories with a negative cosmological constant and asymptotically anti-de Sitter boundary conditions allow for a dual interpretation in terms of strongly coupled quantum systems [10]. This can be used to construct gravity models that are dual to interesting quantum many body phenomena [11,12]. We will now construct the holographic dual to a non-Hermitian quantum field theory along the same lines as outlined before. The key is that in the holographic duality the asymptotic values of the fields encode the couplings of the dual field theory.
In gauge-gravity duality every global symmetry of the dual field theory is promoted to a gauge symmetry in the bulk. To copy our construction for non-Hermitian theories we therefore need at least a U (1) gauge symmetry. In order to introduce couplings that transform under this symmetry we also need a charged bulk field. We simply choose a complex scalar field in the bulk that is charged with charge q under the U (1) symmetry. These are the minimal ingredients to construct our non-Hermitian holographic model. Its action is that of the holographic superconductor [13]. The quartic potential is needed for the model to have domain wall solutions interpolating between two AdS geometries.
For concreteness we will from now on choose d = 3 corresponding to the space-time dimensions of the dual field theory. Furthermore we set Λ = −d(d − 1)/(2L 2 ).
The equations of motion are The unperturbed theory is defined by choosing the asymptotics of the metric. We assume coordinates in which the metric takes the form and demand that for small values of z so that it becomes AdS 4 at z ∼ 0, and accordingly the conformal boundary is z → 0.
To implement the non-Hermiticity we proceed in the following manner. First we choose general boundary . Henceforth we set the bulk scalar mass to be m 2 = − 2 L 2 such that ∆ = 1 and set L = 1. For the numerical solutions we choose v = 3/2 and q = 1. Notice that, unlike in the holographic superconductor [14], we explicitly break the U (1) symmetry by the boundary conditions and we do not introduce a chemical potential. Next we promote the theory to a non-Hermitian one by analytically continuing α → iα. We also set eα = 1+x 1−x and thus obtain the non-Hermitian boundary conditions The U (1) symmetry is now crucial. We note that the action (7) is invariant under global complexified U (1) transformations, φ → eαφ andφ → e −αφ . This means that automatically any bulk geometry with non-Hermitian boundary conditions will be the same as an Hermitian one with with boundary conditions Equivalence of the non-Hermitian and Hermitian theories in the exactly PT-symmetric regime has also been argued for in quantum theory in [15].
Nevertheless it is interesting to see explicitly what happens at the border of the quasi-Hermitian regime in holography. To do so we look for solutions with non-Hermitian boundary values. We take the ansatz (13) so that the asymptotic behavior for this new fields reads ψ ≈ M z + O z 2 , where O corresponds to the vev of the dual operator. To find the background we take ψ(z) to be real. Notice that the gauge symmetry in the bulk gives rise to the constraint φφ − φ φ = 0 which is solved by our ansatz. Finally, the equations of motion (8) T = 0 solutions: We will now look for zero temperature solutions that correspond to domain wall geometries. We integrate numerically the equations (14) from a regular solution in the deep IR at large z [13] u(z) = 1 + 1 6v + . . . , which asymptotes to AdS 4 with radius 6v/(1 + 6v) realizing a conformal IR fixed point in the dual theory. χ 0 and ψ 1 are two free parameters we use to shoot towards the desired boundary conditions in the UV. The resulting solutions are domain wall geometries interpolating between two AdS 4 spaces.
The IR boundary conditions (16) make clear that real solutions can only exist for |x| ≤ 1. For |x| > 1 the ground state spontaneously breaks PT and, as we will see, the dual bulk geometry becomes complex.
In figure 1 we show numerical solutions for several values |x| < 1. Since M is the only dimension-full scale, all solutions at fixed x with M = 0 are equivalent. Then we can explore the space of solutions by simply fixing M = 1 and searching for domain walls at different values of 0 ≤ x ≤ 1. We find that the domain wall shifts towards the IR as x is increased, and in the limit x → 1 it moves all the way to infinity. Indeed, as is clear from (14), at x = 1 the scalar decouples from the metric which becomes AdS 4 , while ψ = M z+ O z 2 is now an exact solution corresponding to a scalar with m 2 = −2 in AdS 4 . Finally, for |x| > 1 we find solutions that are complex along the bulk while still meeting the real UV boundary condition ψ(0) = M . In particular, for each value of x we obtain a pair of solutions complex conjugate to each other and featuring a purely imaginary vev O . We leave the investigation of these complex solutions for future study but note that similar complex solutions have been discussed recently in a different context [16]. T > 0 solutions: To determine what happens as we heat up the system we now study solutions with an horizon at z = z h where the blackening factor u(z h ) = 0 and with the horizon temperature.
Integrating from the horizon and imposing the same UV boundary conditions we now expect a family of solutions characterized by two dimensionless parameters M/T and x. Interestingly, we find that at fixed M/T we are able to obtain real solutions for 0 ≤ x ≤ x c , with x c > 1 and monotonically increasing with M/T . In figure 2 we plot the vev O /M 2 as a function of x for different values of M/T . Notice that for 1 < x < x c two different branches of solutions exist. Finally, beyond x c we only find complex solutions (with real values of M/T ).
How is it that we are finding a seemingly valid background of the theory in the PT broken regime 1 < x ≤ x c ? As we will show next, these solutions have a tachyon in their spectrum and are therefore unstable In order to asses the stability of our finite temperature solutions we now study the quasinormal modes (QNM) of the system. More precisely we look for spatially homogeneous solutions to the time dependent linearized equations of motion with ingoing boundary conditions at the black hole horizon. These fluctuations can be organized in several decoupled sectors. We focus on the one containing the temporal component of the gauge field e −iωt a t (z) and a particular combination of the fluctua- tions of the scalar fields defined through the constraint This constraint results from requiring that the Einstein's equation of motion are satisfied without turning on any new metric degree of freedom. Solving (18) for δφ when x > 0 (analogously one solves for δφ when x < 0), the equations of motion for δφ and δφ become equivalent, and we are left with the following two coupled differential equations We integrate these equations numerically, imposing ingoing boundary conditions at the horizon and to obtain the frequencies of the QNMs we use the determinant method [17].
In figure 3 we plot the purely imaginary QNM that becomes the pseudo-diffusive one at x = 0. This is the would-be hydrodynamic mode corresponding to charge diffusion, which becomes gapped once M is switched on. As x is increased the purely imaginary gap decreases, vanishing at exactly x = 1. Recall that at x = 1 the scalar decouples from the geometry and we recover the hydro diffusive mode. Crucially, for x > 1 the mode crosses into the upper half plane, thus becoming tachyonic and signaling the instability of those finite temperature solutions beyond the PT regime. One could ask next if this instability leads to a new background for x > 1.
For the system at hand the only possibility would be that a background with a spontaneous nonzero charge density (A t (z) = −ρ z + . . . ) exists for x > 1. Yet a thorough numerical search has failed to produce such a background (even after relaxing the requirement that the fields be real). We thus believe that there is no endpoint for this instability indicating that the system does not have a true ground state in the PT broken regime.

Conclusion and Outlook
We have successfully constructed a model of a strongly coupled quantum systems with non-Hermitian couplings via the holographic duality. The PT phase transition takes an interesting form at finite temperature: real solutions exist even for a region of values |x| > 1, but they turn out to be unstable to small fluctuations. While our model falls into the bottom-up class it is easily generalized to models directly derived from string theory such as the ones in [18][19][20]. We expect our findings to hold also in these models. There are many possible generalizations of our work. Spontaneous symmetry breaking and Goldstone modes in PT field theories have recently been discussed in [21][22][23][24][25][26]. This could be generalized to holographic systems using the methods of [27][28][29]. It would also be interesting to understand if a similar picture holds for the PT phase transition at finite temperature in weakly coupled perturbative field theory.