Aspects of pseudoentropy in field theories

In this paper, we explore properties of pseudoentropy in quantum field theories and spin systems from several approaches. pseudoentropy is a generalization of entanglement entropy such that it depends on both an initial and final state and has a clear gravity dual via the AdS/CFT. We numerically analyze a class of free scalar field theories and the XY spin model. This reveals basic properties of pseudoentropy in quantum many-body systems, namely, the area-law behavior, the saturation behavior, and the nonpositivity of difference between the pseudoentropy and averaged entanglement entropy in the same quantum phase. In addition, our numerical analysis finds an example where the strong subadditivity of pseudoentropy gets violated. Interestingly, we find that the nonpositivity of the difference can be violated only if the initial and final state belong to different quantum phases. We also present analytical arguments, which support these properties by both conformal field theoretic and holographic calculations. When the initial and final state belong to different topological phases, we expect a gapless mode localized along an interface, which enhances the pseudoentropy, leading to the violation of the nonpositivity of the difference. Moreover, we also compute the time evolution of pseudoentropy after a global quantum quench, where we observe that the imaginary part of pseudoentropy shows an interesting characteristic behavior.

Figure 1

$S\left({\tau }_A^{1|2}\right)$ as a function of the size of the subsystem $\ell$. We set $L=100$ and $z_1=z_2=1$. The solid curves are $(1/3)log\left[\right(L/\pi )sin(\pi \ell /L\left)\right]+\text{const.}$, where the constant term is defined in (3.7). We chose $m_1={10}^{-6}$. The value of $m_2$ is $2\times {10}^{-6},4\times {10}^{-6},6\times {10}^{-6},8\times {10}^{-6}$ as indicated.

Figure 2

(a) $\mathrm{\Delta }{S}_{12}$ for free scalar with $z_1=z_2=1$. Here we set $m_1={10}^{-5}$ and $m_2={10}^{-4}$. We have small $\ell$ and $L$-dependence but it is negligible up to 3 or 4 digit. Therefore, the second term of (3.7) essentially explains this negative value. We set $L=100,150,200$ from the top graph to the bottom one. (b) The $z$ dependence of $\mathrm{\Delta }{S}_{12}$ for $z>1$. Here we set $L=2000,m_1={10}^{-7},m_2={10}^{-8}$, and $z_1=z_2\equiv z$. It can be perfectly explained by the equation (3.10). Note that we can also produce the same results from smaller total systems and see the perfect agreement in the whole subregions. We set $z=2,3,4,5,6,7$ as indicated.

Figure 3

(a) The pseudoentropy as a function of the subsystem size $\ell$ when we chose $m_1={10}^{-3}$ and $m_2={10}^{-5}$ for various values of $z_1=z_2$. We set $z_1=z_2=1,2,3,4$ as indicated. (b) The pseudoentropy when we set $z_1=3$ and $m_1=m_2={10}^{-5}$. We chose the total system $L=100$. We chose $z_2=1,2,3,4,5$ as indicated.

Figure 4

Dynamical exponent dependence of the pseudoentropy. We set $m_1=m_2={10}^{-5}$ and $\ell =30$ on an infinite lattice. The pseudoentropy saturates at the value of the larger $z$. We set $z_1=1,2,3,4$ as indicated.

Figure 5

(a) The pseudoentropy for fixed $m_1={10}^{-3}$ with various mass $m_2$. We chose $m_2={10}^{-5},{10}^{-4},2.5\times {10}^{-4}$ as indicated. (b) The regularized pseudoentropy for $m_1={10}^{-3}$ and $m_2=2.5\times {10}^{-4}$. As a reference, we also plot the entanglement entropy for vacuum with $c=1$ (orange or upper curve). Note that this formula is valid only in the small subsystem such that $m_i\ell \ll 1$. Out of this regime, as we can see from the figure (b), there is a small deviation.

Figure 6

Mass and dynamical exponent dependence of $\mathrm{\Delta }{S}_{12}$. Here we set $\ell =30$ on an infinite lattice. (a) $m_2$ dependence with $m_1={10}^{-5}$ and fixed $z_1=z_2$. We chose $z_1=z_2=1,2,3,4$ as indicated. (b) $z_2$ dependence with $m_1=m_2={10}^{-5}$ and fixed $z_1$. These figures ensure the quadratic behavior of $\mathrm{\Delta }{S}_{12}$ when the parameter is small. We chose $z_1=1,2,3,4$ as indicated.

Figure 7

Plot for PMI as a function of distance $d$ between two subsystems $A$ and $B$. (a) Almost massless regime. Here we set $L=300,{\ell }_A={\ell }_B=40,m_1={10}^{-6},m_2={10}^{-5}$ and $z_1=z_2\equiv z$. (b) Massive regime. Here we set $m_1={10}^{-3},m_2={10}^{-2}$, and other parameters are the same as the almost massless ones. In both pictures (a) and (b) we chose $z_1=z_2=1,2,3,4$ as indicated.

Figure 8

The $z$ dependence of $\mathrm{\Delta }{S}_{12}$ for two-interval cases. Here we set $L=300,{\ell }_A={\ell }_B=40,m_1={10}^{-6},m_2={10}^{-5}$, and $z_1=z_2\equiv z$. We can see perfect agreement with analytic results. We chose $z_1=z_2=1,2,3,4$ as indicated.

Figure 9

Plot for MI and PMI as a function of distance $d$ between two subsystems $A$ and $B$. For clarity, we plot reguralized version, $I_{\text{reg.}}=I\left(d\right)-I\left(0\right)$ and $I_{\text{reg.}}^{1|2}=I^{1|2}\left(d\right)-I^{1|2}\left(0\right)$. Here we set $L=300,{\ell }_A={\ell }_B=40,m_1={10}^{-6},m_2={10}^{-5}$, and $z_1=z_2\equiv z$. This plot suggests that $I^{1|2}$ and $I$ are the same up to the constant term as described in (3.17). We chose $z_1=z_2=1,2,3,4$ as indicated.

Figure 10

The $z$ dependence of the difference between pseudoentropy and averaged entanglement entropies, $\mathrm{\Delta }{S}_{12}$ for two-interval cases. Here we set $L=300,{\ell }_A={\ell }_B=40,m_1={10}^{-3},m_2={10}^{-2}$, and $z_1=z_2\equiv z$. For $z>1$ case, the minimum value attaches the one for almost massless regime (lower graph) (3.22).

Figure 11

The breaking of the monotonicity of PMI or strong subadditivity of PE. Here we set ${\ell }_A=80,{\ell }_B={\ell }_C=40$, and $m_1=m_2={10}^{-4}$ and take infinite size limit. We can see the similar saturation behavior as the pseudoentropy. For the sake of resolution, we did not plot the values at $d=0$, which take positive values. We set $z_1=1$. We also chose $z_2=2,3,4,5$ as indicated.

Figure 12

The recovery of breaking from the low-energy dispersion relation (3.25). Here we again set ${\ell }_A=80,{\ell }_B={\ell }_C=40$, $z_1=1$, and $m_1=m_2={10}^{-4}$ and take infinite size limit. We chose $z_2=2,3,4,5$ as indicated.

Figure 13

This figure suggests the breaking of monotonicity indeed persists even in the continuum limit. Here we set $m_1=m_2={10}^{-4}$ and $z_1=1,z_2=5$. The minimum value becomes smaller as we enlarge the number of lattice in the subsystem. We set $l_B=l_C=30,40,50$ as indicated.

Figure 14

The phase structure of the quantum XY model. The bold blue line ($\gamma =0$ and $0\le \lambda <1$) and the bold green line ($\lambda =1$ and $\gamma >0$) are the two critical lines. The red dashed line ($\gamma =1$) corresponds to the transverse Ising model. The ground state becomes degenerate at the black-dashed line (${\lambda }^2+{\gamma }^2=1$).

Figure 15

Crossing the XY critical line. (b) and (c) correspond to correlator method for subregion ${\ell }_x=50$ on an infinite lattice, (d) and (e) correspond to direct calculation with ${\ell }_x=7$ on a periodic lattice with 14 sites. In the plots (b)–(d), the one with a sharp peak around ${\lambda }_2=1$ is the average $(S_1+S_2)/2$, while the other is the pseudoentropy $S\left({\tau }^{1|2}\right)$.

Figure 16

Crossing the XY critical line along the Ising line. (b) and (c) correspond to the correlator method for subregion ${\ell }_x=50$ on an infinite lattice, (d) and (e) correspond to direct diagonalization with ${\ell }_x=7$ on a periodic lattice with 14 sites. In the plots (b)–(d), the one with a sharp peak around ${\lambda }_2=1$ is the average $(S_1+S_2)/2$, while the other is the pseudoentropy $S\left({\tau }^{1|2}\right)$.

Figure 17

Crossing the XX critical line. (b) is given by the correlator method for subregion ${\ell }_x=100$ on an infinite lattice and (c) is given by direct diagonalization with ${\ell }_x=7$ on a periodic lattice with 14 sites. In the plots (b)–(d), the one with a sharp peak around ${\lambda }_2=0$ is the average $(S_1+S_2)/2$, while the other is the pseudoentropy $S\left({\tau }^{1|2}\right)$.

Figure 18

Crossing the Ising line. (b) and (c) correspond to the correlator method for subregion ${\ell }_x=50$ on an infinite lattice, (d) and (e) correspond to direct diagonalization with ${\ell }_x=7$ on a periodic lattice with 14 sites. In the plots (b)–(d), the upper graph is the average $(S_1+S_2)/2$, while the lower one is the pseudoentropy $S\left({\tau }^{1|2}\right)$.

Figure 19

Crossing the degenerate semi circle. Panel (b) is given by the correlator method for subregion ${\ell }_x=100$ on an infinite lattice and panel (c) is given by direct diagonalization with ${\ell }_x=7$ on a periodic lattice with 14 sites. Panel (d) shows to the imaginary part of the pseudoentropy. In the plots (b)–(d), the upper, middle and lower graph at ${\gamma }_2=0.8$, are $\mathrm{Re}\left[S\left({\tau }^{1|2}\right)\right]$, $S_2$ and $(S_1+S_2)/2$, respectively.

Figure 20

The Euclidean path integral corresponding to ${\tau }^{1|2}\left(t\right)$ (upper panel, parameterized by $z=x+i\tau$) and the conformal transformation mapping it to a right-half plane (lower panel, parameterized by $\xi$).

Figure 21

Panel (a) we show the time evolution of the real part of pseudoentropy and the entanglement entropies of both input states for scalar theory with $z=z_1=z_2=1$, $m_1=1$, $m_2=2$ and $m={10}^{-5}$ for subregion of length $L_A=100$ on an infinite lattice. In panel (b) we show the time evolution of the imaginary part of pseudoentropy compared with the time derivative of the entanglement entropy of the input states. In panel (c) we show the time evolution of $\mathrm{\Delta }S$. Panels (d), (e), and (f) are similar to (a), (b), and (c) with $z=z_1=z_2=2$. Panels (g), (h), and (i) are similar plots for free fermion analytic results where we set $\epsilon ={10}^{-5}$, ${\alpha }_1=2$ and ${\alpha }_2=3/4$. In the plot (a), (d), and (g), the upper, middle and lower graph describe $S\left({\rho }_1\right)$, $\mathrm{Re}S\left({\tau }^{1|2}\right)$, and $S\left({\rho }_2\right)$, respectively. In the plot (b), (e), and (h), the upper, middle, and lower graph describe $\mathrm{Im}S\left({\tau }^{1|2}\right)$, $dS\left({\rho }_2\right)/dt$, and $dS\left({\rho }_1\right)/dt$, respectively.

Figure 22

A sketch of the AdS/BCFT correspondence. The left half shows a 2D manifold $\mathrm{\Sigma }$ (shown in grey) on which the BCFT is defined. Its gravity dual $\mathcal{M}$ is shown in the right half. $\mathcal{M}$ is a portion of a 3D asymptotically AdS spacetime with $\partial \mathcal{M}=\mathrm{\Sigma }\cup Q$. $Q$ (shown in blue) is an end-of-the-world brane with Neumann boundary condition (5.28) imposed on it, and $\partial Q=\partial \mathrm{\Sigma }$.

Figure 23

The AdS/CFT correspondence for a Euclidean strip. The asymptotic boundary $\mathrm{\Sigma }$ and the end-of-the-world brane $Q$ are shown in grey and blue respectively.

Figure 24

A $x=\mathrm{const}.$ slice of the right panel of Fig. 23. The solid grey (blue) line shows the asymptotic boundary $\mathrm{\Sigma }$ (end-of-the-world brane $Q$), and the dotted grey line shows the asymptotic boundary of a regular Euclidean BTZ black hole. For positive (negative) tension $T$, the gravity dual looks like the left (right) panel where the larger (smaller) portion is taken.

Figure 25

The conformal map from a n-replicated upper complex planes ${\mathrm{\Sigma }}_n^{+}$ to a Pizza slice geometry $P_n^{+}$. We wrote $n=3$ explicitly in the above pictures. The colored regions in the left and right picture consist of $n=3$ disconnected regions, which are ${\mathrm{\Sigma }}_n^{+}$ and $P_n^{+}$, respectively. The $n$-sheeted surface with both colored and white region in the left represents ${\mathrm{\Sigma }}_n$.