Shannon and Rényi mutual information in quantum critical spin chains

We study the Shannon mutual information in one-dimensional critical spin chains, following a recent conjecture [Alcaraz and Rajabpour, Phys. Rev. Lett. 111, 017201 (2013)], as well as Rényi generalizations of it. We combine conformal field theory (CFT) arguments with numerical computations in lattice discretizations with central charge $c=1$ and $c=1/2$. For a periodic system of length $L$ cut into two parts of length $\ell$ and $L-\ell$, all our results agree with the general shape dependence $I_n(\ell ,L)=(b_n/4)ln(\frac{L}{\pi }sin\frac{\pi \ell }{L})$, where $b_n$ is a universal coefficient. For the free boson CFT we show from general arguments that $b_n=c=1$. At $c=1/2$ we conjecture a result for $n>1$. We perform extensive numerical computations in Ising chains to confirm this, and also find $b_1\simeq 0.4801629\left(2\right)$, a nontrivial number which we do not understand analytically. Open chains at $c=1/2$ and $n=1$ are even more intriguing, with a shape-dependent logarithmic divergence of the Shannon mutual information.

Figure 1

Space-time geometry. (a) Cut cylinder, corresponding to a periodic chain. (b) Cut strip, corresponding to an open chain. By convention the position of the slit is at $\tau =0$ and $x\in [0;\ell ].$

Figure 2

Numerical checks for the periodic $XXZ$ chain. We plot the $c$ estimate (43) for two different values of the anisotropy $(\Delta =\pm 0.4)$ and 3 values of the Rényi index $(n=0.5,1,1.5)$. The data show excellent agreement with the conformal scaling and a central charge $c=1$. Inset: Example of convergence of the $c$ estimate in the finite-size data.

Figure 3

Second Rényi mutual information $I_2(\ell ,L)-I_2(L/2,L)$ in the periodic Ising chain, and comparison with CFT (black curve). Inset: Central charge extraction. Due to the huge computational costs involved, not all data are shown for the biggest system size $L=56$ in the $z$ basis.

Figure 4

Numerical extraction of $b_n$ in the XX chain. Black dots: Extrapolation from the entropy in the infinite limit, up to subsystem sizes $\ell =38$. Blue stars: Discrete derivative method, up to total system size $L=72$. The data are compared to the CFT prediciton of Eq. (50) (thick red curve).

Figure 5

Numerical extraction of $b_n$ in the periodic ICTF. Black dots: Extrapolation from the entropy in the infinite limit, up to subsystem sizes $\ell =40$, in the $z$ basis. Gray triangles: Discrete derivative method, up to total system size $L=68$, $z$ basis. Blue stars: Derivative method, up to $L=36$, $x$ basis. Red curve is the CFT result, assuming the phase transition argument.

Figure 6

Fourth RMI $I_4(\ell ,L)$ in the periodic XX chain and “conformal” curve with $b_4$ given by (51). We show the data for $L=24,40,56$. Not all data points are shown for $L=56$ due to their huge computational costs. Inset: Extraction of $b_4$, and comparison with (51) including error bars.

Figure 7

Shannon mutual information in the ICTF. The data for $L=20,36$ are shown in the $x$ basis, and $L=24,40,56$ in the $z$ basis. The conformal scaling is obeyed, with a nontrivial value of $b_1$, very close to $0.48$.

Figure 8

Numerical extraction of the shape-dependent logarithmic prefactor in the XY chain on the Ising critical line. We use system sizes up to $\ell =36$ for the fits. $\gamma =1$ (ICTF) as well as $\gamma =0.5$ and $\gamma =1.5$ are shown. The results clearly support the universality of this shape function.

Figure 9

Conformal mapping from the infinite cylinder with a slit to the upper half plane.

Figure 10

Conformal mapping from the infinite strip with a slit to the upper half plane. In our convention the slit is at $0\le \mathrm{Re}\left(w\right)\le \ell$ and $\mathrm{Im}\left(w\right)=0$.

Figure 11

Extraction of the ${\alpha }_n$ exponent of Eq. (B4), for $n=0.5,1,1.5,2$. The data for $n=0.5$ and $n=1$ agree very well with the known exponents ${\alpha }_{1/2}=1$ and ${\alpha }_1=1/4$. The data for $n=1.5,2$ are consistent with an exponent zero, and therefore an ordering of the spins, at the binding of the book.

Figure 12

Spin-spin correlation exponent in the “Rényified” ground state ${|\psi \rangle }_{\left(n\right)}$, as a function of $n$. We show the finite-size exponent extracted from the two methods exposed in the text, and for system sizes $L=24$ and $L=44$.

Figure 13

Graphical representation of the Dyson-Gaudin gas corresponding to different boundary conditions. The spin configuration $|\uparrow \downarrow \uparrow \uparrow \downarrow \uparrow \downarrow \downarrow \rangle$ with $L=8$ is shown. (a) Periodic case; the particles (up spins) are in blue. (b) Open case: each blue particle in the upper part interacts with the others and their mirror images (in lighter blue). (c) Open Neuman case: the particles are the original ones, in blue, or the mirror images of the holes, in green. All of them interact with each other.