We study the Shannon and R\'enyi mutual information (MI) in the ground state (GS) of different critical quantum spin chains. Despite the apparent basis dependence of these quantities we show the existence of some particular basis (we will call them conformal basis) whose finite-size scaling function is related to the central charge
c of the underlying conformal field theory of the model. In particular, we verified that for large index
n, the MI of a subsystem of size $\ensuremath{\ell}$ in a periodic chain with
L sites behaves as $\frac{c}{4}\frac{n}{n\ensuremath{-}1}ln[\frac{L}{\ensuremath{\pi}}sin(\frac{\ensuremath{\pi}\ensuremath{\ell}}{L})]$, when the ground-state wave function is expressed in these special conformal basis. This is in agreement with recent predictions. For generic local basis, we will show that, although in some cases ${b}_{n}ln[\frac{L}{\ensuremath{\pi}}sin(\frac{\ensuremath{\pi}\ensuremath{\ell}}{L})]$ is a good fit to our numerical data, in general, there is no direct relation between
bn and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model,
Q=3,4 quantum Potts chain, quantum Ashkin-Teller chain, and the XXZ quantum chain. We will also present some additional results of the Shannon mutual information (
n=1), for the parafermionic
ZQ quantum chains with
Q=5,6,7, and 8.
We study the mutual information between two lattice blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems. Quantum q-state Potts model and transverse-field spin-1/2 XY model are considered numerically by using the infinite matrix product state approach. As a system parameter varies, block-block mutual information exhibit singular behaviors that enable us to identify the critical points for the quantum phase transitions. As happens with von Neumann entanglement entropy of single block, at critical points, block-block mutual information for two adjacent blocks show a logarithmic leading behavior with increasing the size of the blocks, which yields the central charge c of the underlying conformal field theory, as it should be. It seems that two disjoint blocks show a similar logarithmic growth of the mutual information as a characteristic property of critical systems but the proportional coefficients of the logarithmic term are very different from the central charges. As the separation between the two lattice blocks increases, the mutual information reveals a consistent power-law decaying behavior for various truncation dimensions and lattice-block sizes. The critical exponent of block-block mutual information in the thermodynamic limit is estimated by extrapolating the exponents of power-law decaying regions for finite truncation dimensions. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. Whereas the critical exponents have different values to a degree of distinction for the different universality classes. As the lattice-block size becomes bigger, the critical exponent becomes smaller. We discuss a relation between the exponents of block-block mutual information and correlation with the Shatten one-norm of block-block correlation.
A practical use of the entanglement entropy in a 1D quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3)lnℓ for an interval of length ℓ in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2D conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and three-state Potts models.
We study the generalized mutual information ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{I}}_{n}$ of the ground state of different critical quantum chains. The generalized mutual information definition that we use is based on the well established concept of the R\'enyi divergence. We calculate this quantity numerically for several distinct quantum chains having either discrete
Z(Q) symmetries (
Q-state Potts model with
Q=2,3,4 and
Z(Q) parafermionic models with
Q=5,6,7,8 and also Ashkin-Teller model with different anisotropies) or the
U(1) continuous symmetries (Klein-Gordon field theory,
XXZ and spin-1 Fateev-Zamolodchikov quantum chains with different anisotropies). For the spin chains these calculations were done by expressing the ground-state wave functions in two special bases. Our results indicate some general behavior for particular ranges of values of the parameter
n that defines ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{I}}_{n}$. For a system, with total size
L and subsystem sizes $\ensuremath{\ell}$ and $L\ensuremath{-}\ensuremath{\ell}$, the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{I}}_{n}$ has a logarithmic leading behavior given by $\frac{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{n}}{4}log[\frac{L}{\ensuremath{\pi}}sin(\frac{\ensuremath{\pi}\ensuremath{\ell}}{L})]$ where the coefficient ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{n}$ is linearly dependent on the central charge
c of the underlying conformal field theory describing the system's critical properties.
An existing conjecture states that the Shannon mutual information contained in the ground-state wave function of conformally invariant quantum chains, on periodic lattices, has a leading finite-size scaling behavior that, similarly as the von Neumann entanglement entropy, depends on the value of the central charge of the underlying conformal field theory describing the physical properties. This conjecture applies whenever the ground-state wave function is expressed in some special basis (conformal basis). Its formulation comes mainly from numerical evidences on exactly integrable quantum chains. In this paper, the above conjecture was tested for several general nonintegrable quantum chains. We introduce new families of self-dual
Z(Q) symmetric quantum chains
(Q=2,3,...). These quantum chains contain nearest-neighbor as well next-nearest-neighbor interactions (coupling constant
p). In the cases
Q=2 and
Q=3, they are extensions of the standard quantum Ising and three-state Potts chains, respectively. For
Q=4 and $Q\ensuremath{\ge}5$, they are extensions of the Ashkin-Teller and
Z(Q) parafermionic quantum chains. Our studies indicate that these models are interesting on their own. They are critical, conformally invariant, and share the same universality class in a continuous critical line. Moreover, our numerical analysis for
Q=2−−8 indicate that the Shannon mutual information exhibits the conjectured behavior irrespective if the conformally invariant quantum chain is exactly integrable or not. For completeness we also calculated, for these new families of quantum chains, the two existing generalizations of the Shannon mutual information, which are based on the R\'enyi entropy and on the R\'enyi divergence.
We find a connection between logarithmic formation probabilities of two disjoint intervals of quantum critical spin chains and the Casimir energy of two aligned needles in two dimensional classical critical systems. Using this connection we provide a formula for the logarithmic formation probability of two disjoint intervals in generic
1+1 dimensional critical systems. The quantity is depenedent on the full structure of the underlying conformal field theory and so useful to find the universality class of the critical system. The connection that we find also provides a very efficient numerical method to calculate the Casimir energy between needles using quantum critical chains. The agreement between numerical results performed on critical transverse field Ising model and XX chain with our exact results is very good. We also comment on the mutual Rényi information of two disjoint intervals.