Bipartite fidelity and Loschmidt echo of the bosonic conformal interface

We study the quantum quench problem for a class of bosonic conformal interfaces by computing the Loschmidt echo and the bipartite fidelity. The quench can be viewed as a sudden change of boundary conditions parametrized by $\theta$ when connecting two one-dimensional critical systems. They are classified by $S\left(\theta \right)$ matrices associated with the current scattering processes on the interface. The resulting Loschmidt echo of the quench has long time algebraic decay $t^{-\alpha }$, whose exponent also appears in the finite size bipartite fidelity as $L^{-\frac{\alpha }{2}}$. We perform analytic and numerical calculations of the exponent $\alpha$, and find that it has a quadratic dependence on the change of $\theta$ if the prior and post-quench boundary conditions are of the same type of $S$, while remaining $\frac{1}{4}$ otherwise. Possible physical realizations of these interfaces include, for instance, connecting different quantum wires (Luttinger liquids), quench of the topological phase edge states, etc., and the exponent can be detected in an x-ray edge singularity-type experiment.

Figure 1

Cut-and-join quench protocol. (Left) Prepare the ground states of the two separated chains and join them at $t=0$, then time evolve with the whole chain Hamiltonian. (Right) Space-time diagram of the cut-and-join protocol. The solid line represents the boundaries of the two disconnected chains. It is totally reflective for the incident particles on both sides. The dashed line is the world line of the junction, which we will call interface. It could either be totally transparent or partially permeable, depending on the types of theories A and B.

Figure 2

Folding picture for the penetrable interface. (Left) World line of the penetrable interface. ${\partial }_{\pm }{\varphi }^{1,2}$ denote the left- and right-going modes in their respective domains. (Right) Folding operation that sends ${\varphi }^2\left(x\right)$ to ${\varphi }^2(-x)$. The dashed line represents the impenetrable boundary for the resulting tensor theory. The arrow represents the incoming and outgoing particles scattered by the interface.

Figure 3

Fidelity of connecting two CFTs. The horizontal axis is the imaginary time. Evolution along the two semi-infinite stripes produces the ground states of the disconnected and connected chain Hamiltonians. The right diagram is the result of folding the lower part of the diagram up, so that all the boundaries are now boundary states. The solid dot represents the boundary condition changing (bcc) operator. Here D (Dirichlet), N (Neumman), and $\lambda$ (permeable interface parametrized by $\lambda )$ are possible choices of boundary conditions.

Figure 4

Loschmidt echo of connecting two CFTs. Evolution along the two infinitely extended sides produces the ground state of the disconnected chain Hamiltonians. They sandwich the evolution of the connected chains. In the folding picture on the right, $a,b,c$ represent the most general boundary conditions of the chains (for example, $a$ and $c$ are DN according to the left figure).

Figure 5

Mapping from a strip to the upper half plane $\xi =exp\left(\frac{\pi z}{L}\right)$. The two black dots represent possible locations of the boundary condition changing (bcc) operators. The dot inside the blue semicircle has coordinate $\xi =1$, which is the image of the point connecting $a$ and $b$ boundaries. The other dot $\xi =0$ corresponds to the connection between $a$ and $c$ boundaries at $-\infty$. To evaluate the diagram, we add the outer blue semicircle centered at $\xi =1$ with radius $R_{\xi }$ to be the IR cutoff and map it to the cylinder with $w=ln(\xi -1)$.

Figure 6

The dashed (solid) lines are gluing (completely reflective) boundary conditions. Red arrows are the directions of Hamiltonian flow that propagates the dashed line boundary state to the solid line boundary state. (Left) Diagram of the Loschmidt echo that reduces to a partition function with imaginary time in the horizontal direction. The blue semicircles of radius $\epsilon$ are the UV regulators and they are identified as periodic boundaries in the direction perpendicular to the red arrow (equal time slice). (Middle) Image of the map $\xi =\frac{z}{\tau -z}$. The two semicircles have radii ${(\tau /\epsilon )}^{\pm 1}$, respectively. (Right) Image of $w=ln\xi$. It is a cylinder by identifying the blue lines and the standard radial quantization procedure can be applied.

Figure 7

The Loschmidt echo decay exponent of the process in $\text{DD}\to \lambda$, with gluing condition $S_1\left(\theta \right)$. We work with the total system size $N=30000$ sites, and parameters $m={10}^{-8},k=1$. The lattice constant is set to unity. The blue dots representing the numerical results lie on the red analytic line. As predicted, the echo exponents are all equal for different values of $\theta$. (Inset) An example of Loschmidt echo with $\theta =0.02\pi$ shown in the log-log scale. The dashed line denotes the expected power law of $t^{-0.25}$. Finite size effect does not emerge before $t={10}^3$, which sets the right boundary of the range we fit. See main text for the curve fitting method.

Figure 8

The slope of the free energy for (a) the Loschmidt echo and (b) the bipartite fidelity of the process $\text{DN}\to \lambda$. The total system size is $N=35000$ sites with the same parameters as in Fig. 7. The numerical value of exponents follow a quadratic relation as predicted. There is still visible deviation from the analytic results in (a) due to the subtlety of the zero mode; see the discussion in the main text. (a) (Inset) From the top to bottom, we show the power law decay of the Loschmidt echo with $\theta =0.02\pi ,0.12\pi$ and $0.24\pi$. Finite size effect does not emerge before $t={10}^4$. We use the same curve fitting method as described in Fig. 7.

Figure 9

The decay exponent of (a) the Loschmidt echo and (b) the bipartite fidelity of the process $\text{P}\to \lambda$. The parameters are the same as those in Fig. 8. The plot is symmetric with respect to $\theta =0.25\pi$ as predicted. The deviation around $\frac{\pi }{4}$ in (a) is small but visible; see the discussion in the main text. (a) (Inset) From top to bottom, we show the power law decay of the Loschmidt echo with $\theta =0.04\pi ,0.08\pi$, and $0.18\pi$. Finite size effect does not emerge before $t={10}^4$. We use the same curve fitting method as described for Fig. 7.

Figure 10

The decay exponent of the Loschmidt echo for the process in Eq. (43). The boundary condition $c$, which is now different from $a$, changes the scaling completely from the analytic quadratic curve. The deficit to the analytic value is roughly $\frac{1}{8}$ for $\theta >\frac{\pi }{4}$, It becomes smaller and approaches zero for small $\theta$.

Figure 11

Partition function between the two boundary states of $S_i\left({\theta }_1\right)$ and that of $S_j\left({\theta }_2\right)$.

Figure 12

Partition function of Hamiltonian with DN and $\lambda$ boundary conditions. We unfold the cylinder and the new stripe has N and D boundary conditions on the top and bottom plus a $\lambda$ junction in the middle. The length $L$ here is the height of the unfolded cylinder $2\pi$.