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arXiv:2209.14253 · Physical review. B./Physical review. B

Universality in the tripartite information after global quenches

Vanja Marić, Maurizio Fagotti

Published Oct 31, 2023• 16 citations
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Abstract

We consider macroscopically large 3-partitions $(A,B,C)$ of connected subsystems $A\ensuremath{\cup}B\ensuremath{\cup}C$ in infinite quantum spin chains and study the R\'enyi-$\ensuremath{\alpha}$ tripartite information ${I}_{3}^{(\ensuremath{\alpha})}(A,B,C)$. At equilibrium in clean 1D systems with local Hamiltonians it generally vanishes. A notable exception is the ground state of conformal critical systems, in which ${I}_{3}^{(\ensuremath{\alpha})}(A,B,C)$ is known to be a universal function of the cross ratio $x=|A||C|/[(|A|+|B|)(|C|+|B|)]$, where $|A|$ denotes $A$'s length. We identify different classes of states that, under time evolution with translationally invariant Hamiltonians, locally relax to states with a nonzero (R\'enyi) tripartite information, which furthermore exhibits a universal dependency on $x$. We report a numerical study of ${I}_{3}^{(\ensuremath{\alpha})}$ in systems that are dual to free fermions, propose a field-theory description, and work out their asymptotic behavior for $\ensuremath{\alpha}=2$ in general and for generic $\ensuremath{\alpha}$ in a subclass of systems. This allows us to infer the value of ${I}_{3}^{(\ensuremath{\alpha})}$ in the scaling limit $x\ensuremath{\rightarrow}{1}^{\ensuremath{-}}$, which we call ``residual tripartite information''. If nonzero, our analysis points to a universal residual value $\ensuremath{-}log2$ independently of the R\'enyi index $\ensuremath{\alpha}$, and hence applies also to the genuine (von Neumann) tripartite information.

DOI
10.1103/physrevb.108.l161116

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Abstract We study stationary states emerging after global quenches in which the time evolution is under local Hamiltonians that possess semilocal conserved operators. In particular, we study a model that is dual to quantum XY chain. We show that a localized perturbation in the initial state can turn an exponential decay of spatial correlations in the stationary state into an algebraic decay. We investigate the consequences on the behavior of the (Rényi-α) entanglement entropies, focusing on the tripartite information of three adjacent subsystems. In the limit of large subsystems, we show that in the stationary state with the algebraic decay of correlations the tripartite information exhibits a non-zero value with a universal dependency on the cross ratio, while it vanishes in the stationary state with the exponential decay of correlations.
A bstract We take the paradigm of interacting spin chains, the Heisenberg spin- 12 \frac{1}{2} 21​ 1 2 XXZ model, as a reference system and consider interacting models that are related to it by Jordan-Wigner transformations and restrictions to sub-chains. An example is the fermionic analogue of the gapless XXZ Hamiltonian, which, in a continuum scaling limit, is described by the massless Thirring model. We work out the Rényi- α entropies of disjoint blocks in the ground state and extract the universal scaling functions describing the Rényi- α tripartite information in the limit of infinite lengths. We consider also the von Neumann entropy, but only in the limit of large distance. We show how to use the entropies of spin blocks to unveil the spin structures of the underlying massless Thirring model. Finally, we speculate about the tripartite information after global quenches and conjecture its asymptotic behaviour in the limit of infinite time and small quench. The resulting conjecture for the “residual tripartite information”, which corresponds to the limit in which the intervals’ lengths are infinitely larger than their (large) distance, supports the claim of universality recently made studying noninteracting spin chains. Our mild assumptions imply that the residual tripartite information after a small quench of the anisotropy in the gapless phase of XXZ is equal to − log 2.
We study the tripartite information I_3 of free fermions on two-dimensional lattices partitioned into three adjacent strips of width w. Translation invariance yields the exact decomposition I_3 = \sum_{k_y} g(k_F(k_y) w), where g(z) is a universal function of the scaling variable z = k_F w, determined by the spectrum of the sine-kernel (Slepian) integral operator. We find that g(z) has a unique zero at z* = 1.329 +/- 0.001: modes with k_F w < z* violate monogamy of mutual information (g > 0), while modes with k_F w > z* satisfy it (g < 0). The central analytical result is g(z) = cz + O(z^3 ln z) with c = 3 ln(4/3)/pi ~ 0.2747, derived from the rank-1 limit of the sine kernel. Two exact cancellations -- of the z ln z area-law terms and of the z^2 terms -- are intrinsic to the I_3 combination. The coefficient c generalizes to n-partite information: c_n = (n/pi) ln R_n with R_n a rational number from binomial combinatorics. For Renyi entropy of index alpha, we show that g_alpha(z) ~ z^alpha for alpha < 2 and g_2(z) = -(8/pi^3)z^3: von Neumann entropy (alpha = 1) uniquely gives linear sensitivity to Lifshitz transitions, while Renyi-2 gives only cubic sensitivity. We verify all predictions on square, triangular, and cubic lattices. DMRG calculations on the interacting t-V model provide evidence that the linear coefficient is independent of the Luttinger parameter K.
We show that the Rényi tripartite information I3(α)I_3^{(α)}I3(α)​ of free fermions exhibits a qualitatively ααα-dependent scaling at small Fermi momentum, in sharp contrast to bipartite entropy where only the prefactor changes. In the rank-1 regime (z=kFw≪1z = k_F w \ll 1z=kF​w≪1), I3(α)I_3^{(α)}I3(α)​ receives contributions from two competing channels -- a fractional-moment channel ∼zα\sim z^α∼zα (active for non-integer ααα) and a polynomial channel ∼zm\sim z^m∼zm from the first nonvanishing inclusion-exclusion moment σmσ_mσm​ -- yielding the scaling exponent βm(α)=min⁡(α,m)β_m(α) = \min(α, m)βm​(α)=min(α,m) for mmm-partite information of mmm adjacent strips. Integer Rényi indices α=2,3,…α= 2, 3, \ldotsα=2,3,… are anomalous: the fractional channel closes and the exponent jumps to mmm or higher. A direct consequence is a replica obstruction: Im(n)/Im(1)∼zm−1→0I_m^{(n)}/I_m^{(1)} \sim z^{m-1} \to 0Im(n)​/Im(1)​∼zm−1→0 for all integer n≥2n \geq 2n≥2, so the leading von Neumann signal cannot be reconstructed from integer Rényi data at the level of leading scaling -- a situation with no bipartite analog. Conversely, negativity-based measures (α=1/2α= 1/2α=1/2) give a 20×20\times20× enhanced signal compared to von Neumann. We derive the underlying product formula for the coefficient c(wA,wB,wD)c(w_A, w_B, w_D)c(wA​,wB​,wD​), prove an mmm-partite generating function for the inclusion-exclusion moments, and verify all results numerically to high precision.
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 ccc of the underlying conformal field theory of the model. In particular, we verified that for large index nnn, the MI of a subsystem of size $\ensuremath{\ell}$ in a periodic chain with LLL 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{b}_{n}bn​ and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model, Q=3,4Q=3,4Q=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=1n=1n=1), for the parafermionic ZQ{Z}_{Q}ZQ​ quantum chains with Q=5,6,7Q=5,6,7Q=5,6,7, and 8.