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arXiv:1305.1239 · Physical Review Letters

Universal Behavior of the Shannon Mutual Information of Critical Quantum Chains

Francisco Castilho Alcaraz, M. A. Rajabpour

Published Jul 2, 2013• 70 citations
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Abstract

We consider the Shannon mutual information of subsystems of critical quantum chains in their ground states. Our results indicate a universal leading behavior for large subsystem sizes. Moreover, as happens with the entanglement entropy, its finite-size behavior yields the conformal anomaly c of the underlying conformal field theory governing the long-distance physics of the quantum chain. We study analytically a chain of coupled harmonic oscillators and numerically the Q-state Potts models (Q=2, 3, and 4), the XXZ quantum chain, and the spin-1 Fateev-Zamolodchikov model. The Shannon mutual information is a quantity easily computed, and our results indicate that for relatively small lattice sizes, its finite-size behavior already detects the universality class of quantum critical behavior.

DOI
10.1103/physrevlett.111.017201

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We consider the Shannon mutual information of subsystems of critical quantum chains in their ground states. Our results indicate a universal leading behavior for large subsystem sizes. Moreover, as happens with the entanglement entropy, its finite-size behavior yields the conformal anomaly c of the underlying conformal field theory governing the long-distance physics of the quantum chain. We study analytically a chain of coupled harmonic oscillators and numerically the Q-state Potts models (Q=2, 3, and 4), the XXZ quantum chain, and the spin-1 Fateev-Zamolodchikov model. The Shannon mutual information is a quantity easily computed, and our results indicate that for relatively small lattice sizes, its finite-size behavior already detects the universality class of quantum critical behavior.
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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)Z(Q)Z(Q) symmetric quantum chains (Q=2,3,...)(Q=2,3,...)(Q=2,3,...). These quantum chains contain nearest-neighbor as well next-nearest-neighbor interactions (coupling constant p)p)p). In the cases Q=2Q=2Q=2 and Q=3Q=3Q=3, they are extensions of the standard quantum Ising and three-state Potts chains, respectively. For Q=4Q=4Q=4 and $Q\ensuremath{\ge}5$, they are extensions of the Ashkin-Teller and Z(Q)Z(Q)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−−8Q=2--8Q=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 study the R\'enyi entanglement entropy and the Shannon mutual information for a class of spin-1/2 quantum critical XXZ chains with random coupling constants which are partially correlated. In the XX case, distinctly from the usual uncorrelated random case where the system is governed by an infinite-disorder fixed point, the correlated-disorder chain is governed by finite-disorder fixed points. Surprisingly, we verify that, although the system is not conformally invariant, the leading behavior of the R\'enyi entanglement entropies are similar to those of the clean (no randomness) conformally invariant system. In addition, we compute the Shannon mutual information among subsystems of our correlated-disorder quantum chain and verify the same leading behavior as the n=2n=2n=2 R\'enyi entanglement entropy. This result extends a recent conjecture concerning the same universal behavior of these quantities for conformally invariant quantum chains. For the generic spin-1/2 quantum critical XXZ case, the true asymptotic regime is identical to that in the uncorrelated disorder case. However, these finite-disorder fixed points govern the low-energy physics up to a very long crossover length scale, and the same results as in the XX case apply. Our results are based on exact numerical calculations and on a numerical strong-disorder renormalization group.
Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\ensuremath{\alpha}$-R\'enyi entropy of a single interval for several critical quantum chains. We considered models with U(1)U(1)U(1) symmetry such as the spin-1/2 XXZXXZXXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the 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)Z(Q)Z(Q) symmetries (QQQ-state Potts model with Q=2,3,4Q=2,3,4Q=2,3,4 and Z(Q)Z(Q)Z(Q) parafermionic models with Q=5,6,7,8Q=5,6,7,8Q=5,6,7,8 and also Ashkin-Teller model with different anisotropies) or the U(1)U(1)U(1) continuous symmetries (Klein-Gordon field theory, XXZXXZXXZ 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 nnn that defines ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{I}}_{n}$. For a system, with total size LLL 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 ccc of the underlying conformal field theory describing the system's critical properties.
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.