Bridging two quantum quench problems – local joining quantum quench and Möbius quench – and their holographic dual descriptions

Spatial inhomogeneity is ubiquitous in quantum many-body problems and can lead to a rich variety of physics. On the one hand, it can take the form of randomness or disorder. An important phenomenon caused by disorder is the Anderson localization [1], which plays an important role in the integer quantum hall effect for example. On the other hand, it can also be introduced in a more controlled manner, such as a harmonic trap of cold atomic gas [2]. For example, quantum field theory can be studied on a curved spacetime [3], a setting that appears in many contexts of physics [4].

In this paper, we are interested in a particular kind of spatial inhomogeneity that is introduced to many-body quantum systems in one spatial dimension. It can be obtained from the regular, homogeneous Hamiltonian $H_{0}$ – given as a spatial integral of the Hamiltonian density $h(x)$ as $H_{0}=\int dx\,h(x)$ – by deforming it by introducing an envelop function $f(x)$, $H=\int dx\,f(x)\,h(x)$. In particular, we will be interested in the so-called Möbius deformation and sine-square deformation (SSD). (See Sec. IV for the choice of the envelope function and more details.) One of the initial motivations for these deformations was to study many-body systems with open boundaries numerically while suppressing the boundary effects [5]. Amazingly, at a conformal quantum critical point, an SSD Hamiltonian has exactly the same ground state as that of the regular Hamiltonian with periodic boundary condition [6, 7, 8]. More recently, the Möbius deformation and SSD have been used to study non-equilibrium dynamics [9, 10, 11, 12]. Inhomogeneities in CFT are also studied in [13, 3, 14, 15, 16, 17] for example.

The spatial deformation of the above kind appears in various contexts. Another example is the modular Hamiltonian (also known as the entanglement Hamiltonian), defined for a reduced density matrix for a subregion, is given in some cases by a spatial deformation of the regular Hamiltonian (in the above sense). The modular Hamiltonian for the ground state (vacuum) of a relativistic invariant theory, when half of the total space is traced out, is nothing but the Rindler Hamiltonian, and the evolution by a spatially deformed Hamiltonian appears in that context [18, 19]. In high-energy physics, understanding the evolution by modular Hamiltonians is important to study the structure of spacetime through the AdS/CFT correspondence [20, 21, 22, 23, 24, 25, 26].

Although the action of these spatially deformed Hamiltonians on special states is understood through the relation to the undeformed Hamiltonians, the properties of general excited states are yet to be understood. To develop a deeper understanding, in this paper, we will consider two seemingly different quantum quench problems in the context of (1+1)-dimensional conformal field theory (CFT). First, we consider a quantum quench process, which we call the Möbius quench, where the system is initially prepared for the ground state of CFT (with the regular Hamiltonian) on a finite interval. At $t=0$, the system’s Hamiltonian is changed from the regular Hamiltonian to the Möbius Hamiltonian. This quench problem was studied in Ref. [9]. Quench problems with more general spatial deformations are studied in [27]. In the second quench problem, we initially consider two decoupled systems (ground states of CFT), each defined on a finite interval of equal length. The two systems are joined or “glued” at $t=0$ and then time-evolved by the uniform CFT Hamiltonian of the coupled intervals [28]. We call this the local quantum quench. Quantum quenches in CFT, including local quantum quench, were studied in various context [29, 30, 31, 32, 33, 34, 35, 36].

One of the main results of the paper is to establish the equivalence between these two problems. Not only does the equivalence allow us to relate the time-dependent physical observables, but also to gain a deeper understanding of aspects of these quantum quench problems. Here, we note that in (1+1)d CFT many non-equilibrium problems are related to each other by conformal mappings. In particular, all quantum quench problems for which the relevant spacetime geometry can be mapped to the upper half-plane are related to each other. These include, e.g., inhomogeneous global quenches, finite-size global quenches, splitting local quenches, double local quenches, Floquet CFT, etc. They only differ by the space-dependent Weyl transformation and coordinate transformation. Especially the Weyl transformation doesn’t affect the time evolution. For example, physical observables in these quench problems exhibit eternal oscillations, albeit the CFT in question can be a fast quantum information scrambler (in the limit of large central charge). The oscillations can be attributed to the underlying ${\it SL}(2,\mathbb{R})$ structure of the Möbius Hamiltonian [9]. The non-trivial mapping between the two problems also allows us to construct their holographic dual (AdS/CFT) descriptions easily. We find that in the holographic dual descriptions, the so-called end-of-the-world (EOW) brane is involved in the bulk [37, 38], the dynamics of which describes the time-dependence of physical observables (entanglement entropy, energy density). Here, we note that the EOW brane is a key ingredient of holographic duality for boundary CFT (BCFT). We will also speculate that similar equivalence relations can be established for a wider class of quantum quench problems.

The rest of this paper is organized as follows. In Sec. III, we review the local quench on a finite strip and study the entanglement and energy-density dynamics. In Sec. IV, we study the Möbius quench and discover its relation to the local quench problem. Using this relation, we also study the energy-momentum tensor dynamics in the Möbius quench. In Sec. V, we construct the holographic dual of the Möbius quench using the relation between the two quench problems. In particular, we study the end-of-the-world brane dynamics and compare it with the entanglement dynamics in CFT analysis. We conclude in Sec. VI, and provide some future discussions.

First, we consider global quenches on an infinite line $(-\infty,\infty)$. We imagine that first we have a gapped deformation of conformal field theory and then suddenly turn off the deformation term. The initial gapped ground state is evolved by the homogeneous CFT Hamiltonian. To approximate the gapped ground state, we use the smeared boundary state [39, 40, 32]

The evolution of entanglement entropy on a half line $[0,\infty)$ becomes ¹¹1Here we simply omit a non-universal term, which is denoted as $\tilde{c}_{1}^{\prime}$ in [40].

Here $z_{\epsilon}$ is a UV cutoff. Note that entanglement entropy for even an infinite line is well-defined reflecting the fact that the initial state only has short-range entanglement. Entanglement entropy for an infinite line grows linearly in time forever. When we consider a finite interval instead, entanglement entropy saturates when the time reaches the half of the length of the interval divided by the speed of sounds.

Next, we consider the Rindler quenches. In this problem, we start from the ground state $\ket{G}$ of the homogeneous Hamiltonian $H_{0}=\int_{0}^{\infty}h(x)dx$ on a half line $[0,\infty)$. Then, we change the Hamiltonian to the Rindler Hamiltonian $H_{1}=a\int_{0}^{\infty}xh(x)dx$ and evolve the original state $\ket{G}$ by the Rindler Hamiltonian. Here $a$ is a parameter of the dimension of the inverse of the length. This is equivalent to putting CFT on a curved spacetime

Then, the evolution of entanglement entropy for an infinite line $[x,\infty)$ is given by

Two quench problems show a similar evolution of entanglement entropy. Actually, after identifying the parameter $a=\frac{2\pi}{\beta}$ and changing the cutoff $z_{\epsilon}=\frac{\epsilon}{ax}$, the entropy for a global quench becomes

and we can obtain exactly the same evolution of the entropy for Rindler quenches.

Actually, these two problems are related in a more direct manner. First, the Euclidean version of the metric (II.3) is

where we used the coordinate transformation

The coordinate transformation suggests that the ground state of the homogeneous Hamiltonian is equivalent to the boundary state with the finite amount of Euclidean evolution by the Rindler Hamiltonian:

Next, changing the spacial coordinate $x=e^{a\rho}/a$, we obtain

which means that after Weyl transformation $ds^{2}\to e^{-2a\rho}ds^{2}$, the Euclidean path integral is equivalent to that of Calabrese-Cardy state preparation for global quenches [39, 40, 32]. These two show that the correlation functions after Rindler quench is Weyl equivalent to those after global quenches:

In particular, we can apply this relation to twist operators to study entanglement entropy and we can deduce the relation (II.5). In this manner, we can explain the relation between Rindler quench and global quench following their path integral representations and coordinate transformations among them.

In Ref. [28], the authors studied a local quantum quench process in the context of (1+1)d CFT. In this process, the system is initially “cut” into two independent subsystems. To be specific, we consider two intervals of equal length ($=L_{{\it eff}}/2)$, $[-L_{{\it eff}}/2,0]$, and $[0,L_{{\it eff}}/2]$. (Here, denoting the total system size by $L_{{\it eff}}$ may look bizarre. The motivation for this will become clear when we later make contact with the Möbius quench.) The system is initially prepared as the tensor product of the ground states of the two intervals. These two intervals are then glued together at time $t=0$. Namely, for $t>0$, the system time-evolves by the Hamiltonian for the single interval of length $L_{{\it eff}}$.

The quench process can be analyzed by using the Euclidean path integral on a “pants” geometry, which is represented in Fig. 2. We use $w=y+i\tau$ to coordinatize this geometry where $\tau$ and $y$ represent Euclidean temporal and spatial coordinates, respectively. We regularize this excited state by the Euclidean path integral for Euclidean time $\alpha$.

The Euclidean geometry is mapped to the upper half plane by the conformal transformation [28]

Physical observables can then be computed from the corresponding correlations on the upper half-plane. For example, the one -oint function of a primary operator $O(w)$ with conformal dimension $\Delta$ $\braket{O(w,\bar{w})}=A_{O}|({1}/{2\text{Im}\,\xi})({d\xi}/{dw})|^{\Delta}$ where $O(w,\bar{w})$ is a primary operator, $\Delta=2h$ is the scaling dimension of $O$ with the conformal weight $h$ and $A_{O}$ is the one point function. Introducing the mapping to the strip $\xi=e^{i\frac{\pi}{L}\zeta}$, which will use later, we can also write the map as

The entanglement entropy can be obtained from the correlation function of the twist-anti-twist operators [41]. The Euclidean time $\tau$ is analytically continued to Lorentzian time $t$, $\tau\to it$. For general $t$, and for the subsystem of an interval $y\in[-L_{{\it eff}}/2,+l/2]$, we obtain the following expression for the entanglement entropy:

where $c$ is the central charge, $z_{\epsilon}$ is a UV cutoff, and

In particular, at $t=0$, the entanglement entropy just after joining is

Here $z_{\epsilon}$ is a UV cutoff. The profile of the dynamical entanglement entropy is plotted in Fig. 3. Here, we consider the difference between (III.3) and the ground state entanglement entropy, $S_{{\it diff}}=S_{A}(t,l)-S_{A}^{{\it ground}}(l),$ where $S_{A}^{{\it ground}}(l)=\frac{c}{6}\log(\frac{2L_{{\it eff}}}{\pi z_{\epsilon}}\cos\frac{\pi l}{L_{{\it eff}}})$ is the entanglement entropy of the ground state on the same strip.

The conformal map (III.1) to the upper half-plane also allows us to compute the time-dependence of the energy-momentum tensor by

Explicitly, the holomorphic and anti-holomorphic components of the energy-momentum tensor are given by

In Fig. 4, we plot the right-moving part of the stress tensor. Right at the moment of the quench, the stress tensor is sharply peaked at $y=0$, and then propagates to the right. Once the peaks hit the boundaries at $y=\pm L_{{\it eff}}/2$, they get reflected back. After that the energy-momentum tensor profile exhibits an eternal oscillation. Note that at $t=L_{{\it eff}}/2$ the peak looks like to jump from one boundary to the other. This jump actually captures the reflection correctly since the right moving excitation is reflected to the left moving excitation at the boundaries.

Note that the eternal oscillations are universal in any two-dimensional CFTs. In particular, we will encounter the oscillation even in chaotic CFTs like holographic CFTs. Similar non-thermalizing behaviors are also found in global quenches with boundaries [42, 43].

In this section, we consider another quantum quench problem, the Möbius quench [9], which is seemingly different from the local quantum quench considered in the previous section. In the Möbius quench, we start from the ground state $|{\it GS}\rangle$ of (1+1)d CFT on a finite interval of length $L$, $H_{0}|\Psi_{0}\rangle=E_{{\it GS}}|\Psi_{0}\rangle$. Here, $H_{0}$ is the (regular) Hamiltonian of CFT on a finite interval, and given in terms of the energy density operator as $H_{0}=\int^{L}_{0}dx\,h(x)$. At $t=0$, we suddenly change the Hamiltonian from $H_{0}$ to the Möbius Hamiltonian $H_{\text{M\"{o}bius}}=\int_{0}^{L}dxf_{\gamma}(x)h(x)$ with

Here, $\gamma$ is a real positive parameter. As we send $\gamma\to 0$ and $\gamma\to+\infty$, the Möbius Hamiltonian reduces to the regular Hamiltonian $H_{0}$ and the sine-square deformed (SSD) Hamiltonian, respectively [5, 44, 6, 45, 46, 47, 7, 48, 49, 8, 50, 51, 52, 53, 11, 54, 55, 56, 57, 58, 59]. In [10], the Möbius quench starting from a thermal initial state was studied. In holographic theories, the Möbius quench induces a non-trivial dynamics (time-dependent deformation) of the black hole horizon. As we will demonstrate later, the current Möbius quench induces a non-trivial dynamics of the EOW brane.

The Möbius Hamiltonian effectively changes the total system size from $L$ to $L_{{\it eff}}$ where $L$ and $L_{{\it eff}}$ are related by [52, 53]

Specifically, there is a conformal transformation that maps the spacetime (cylinder of circumference $L$) with $H_{\text{M\"{o}bius}}$ as the Hamiltonian, to another spacetime (cylinder of circumference $L_{{\it eff}}$) with $H_{0}$ as the Hamiltonian. In the limit $\gamma\to+\infty$ (the SSD limit), $L_{{\it eff}}\to+\infty$.