Computational access to lattice and long-wavelength physics in quantum mutual information

TL;DR

The relevance of lattice effects on the family of RMI mutual informations for ground states of models with conformal field theory descriptions is analyzed and distinct regions in the $\alpha$-$z$ plane are identified, where RMI corrections due to the lattice are relevant or irrelevant.

Abstract

Quantum mutual information is an important tool for characterizing correlations in quantum many-body systems, but its numerical evaluation is often prohibitively expensive. While some variants of Rényi Mutual Information (RMI) are computationally more tractable, it is not clear whether they correctly capture the long-wavelength physics or are dominated by UV effects, which is of key importance in lattice simulations. We analyze the relevance of lattice effects on the family of $α$-$z$ Rényi mutual informations for ground states of models with conformal field theory descriptions. On the example of massless free fermions we identify distinct regions in the $α$-$z$ plane, where RMI corrections due to the lattice are relevant or irrelevant. We further support these findings with MPS calculations on the transverse field Ising model (TFIM). Our results, accompanied by the open-source Julia package QMICalc$.$jl, provide guidance to using RMI in quantum-many body physics numerical computations.

Figures (12)

• Figure 1: Partition of a chain (black circles) into two disjoint subsystems: $A$ (yellow disks) and $B$ (blue disks), with lengths $\ell_A$ and $\ell_B$, respectively, separated by a distance $d$, all measured in units of the lattice constant. The mutual information $I(A:B)$ encodes all correlations between the two subsystems, as illustrated by the red lines and shading connecting them.
• Figure 2: (a) Schematic "phase" diagram of the relevance of lattice effects in the $\alpha$-$z$ Rényi mutual information (RMI) for the ground state of models whose low-energy physics is described by a conformal field theory (CFT). Each point in the $\alpha$-$z$ plane corresponds to a distinct RMI. Variants from region I (blue) predominantly encode long-wavelength information, with lattice corrections that decay with the subsystem size $\ell$. In region III (red), on the other hand, lattice effects dominate, causing the mutual information to diverge with $\ell$ and rendering long-wavelength information inaccessible. The transition region II (light green) lies between these two, and each variant within it is associated with a finite length scale $\ell^*$ governing the accessibility of long-wavelength information. (b) Regions in the space of $\alpha$-$z$ RMIs, where the data processing inequality (DPI) [\ref{['eq:DPI']}] is proven or conjectured to be satisfied (shaded in yellow) Audenaert2015, along with well-known special cases (blue points and black arrows indicating points outside the displayed range) and one-parameter subfamilies (blue lines). (c) Two examples of $\alpha$-$z$ RMI $I_{\alpha,z}$: $(\alpha,z)=(2,2)$ (left) from region I, and $(\alpha,z)=(2,1)$ (right) from region III. Each data point corresponds to $I_{\alpha,z}(A:B)$ for the partition with $\ell_A=\ell_B=\ell$ and separation $d$, plotted as a function of the cross ratio $x$ [\ref{['eq:cross-ratio']}]. The data is computed for the ground state of the free-fermion model given in \ref{['eq:Hamiltonian-free-fermions']}. In the left plot (region I) data points for different subsystem sizes collapse on a single curve determined by $x$ with only small corrections, while in the right one (region II) there is a strong dependence on the geometric quantities.
• Figure 3: Proxy $\widetilde{\Delta I}_{\alpha,z}$ [\ref{['eq:lattice-contribution-proxy']}] for the lattice contribution $\Delta I_{\alpha,z}$ to the $\alpha$-$z$ Rényi mutual information with $(\alpha,z)=(1.5,1)$ as a function of $\ell=\ell_A=\ell_B$ in a double-logarithmic plot. Data points for odd and even $\ell$ are shown as yellow disks and blue squares, respectively. The red line represents a linear least-squares fit to the logarithmically rescaled data points with $\ell \in [31,100]$ (gray vertical lines), with slope $\delta = -2.03681(23)$. Inset: Convergence of raw data $\Delta I_{\alpha,z}(x,\ell)$ to $I_{\alpha,z}^{\mathrm{CFT}(x)}$ for $\ell\to\infty$ (gray horizontal line).
• Figure 4: The $\alpha$-$z$ phase diagrams of the relevance of lattice effects in the $\alpha$-$z$ Rényi mutual information of a partition with $x=1/4$ and $\ell_A=\ell_B=\ell$ for the ground state of massless free fermions. The relevance is measured by the scaling exponent $\delta$ (see legend to the right) of the lattice contribution $\Delta I_{\alpha,z}$, as defined by the proxy $\widetilde{\Delta I}_{\alpha,z}\sim\ell^\delta$ [\ref{['eq:lattice-contribution-proxy']}], extracted from fits to the range (a) $31\leq\ell\leq 50$ and (b) $2\leq\ell\leq 11$. In (a) solid black lines indicate how the position of the transition from negative (irrelevant) to positive (relevant) scaling exponent $\delta$ changes for choices of $x$ different from $x=\frac{1}{4}$. In both panels the black dashed lines represent the boundaries of the regions I, II and III illustrated in \ref{['fig:azRMI-phase-diagram_summary']}. Note that $\delta=0$, as extracted from the proxy, actually indicates a logarithmic scaling with $\ell$: $\Delta I_{\alpha,z}\sim\log\ell$. In (a), a neighborhood of $(1.2,0.6)$ is excluded due to $\widetilde{\Delta I}_{\alpha,z}$ changing sign as a function of $\ell$ near the fitting range before becoming a power law again, preventing an estimation of $\delta$ for this range. Estimating it for smaller $\ell$, as in (b), is possible but we observe that the exponent becomes anomalously small. See \ref{['fig:App:DeltaI-vs-l:exclusion']} for a discussion and explanation.
• Figure 5: Scaling exponent $\delta$ of the lattice contribution extracted form the proxy $\widetilde{\Delta I}_{\alpha,z}\sim\ell^\delta$ [\ref{['eq:lattice-contribution-proxy']}] to the $\alpha$-$z$ Rényi mutual information near its transition from negative to positive value. Exemplary data is shown for $z=1$, $x=1/4$, and $\ell_A=\ell_B=\ell$, but the qualitative behavior is unchanged for other parameter choices. The different curves correspond to exponents $\delta$ extracted from different ranges of $\ell$ of size $\Delta\ell = 20$ with the maximal value indicated by the color according to the legend on the right. Inset: Proxy as a function of $\ell$ close but to the left of the transition ($\alpha$ marked by a vertical gray line in the main figure). As is characteristic of the transition region, the proxy increases at small $\ell$ but then reverses behavior at $\ell=\ell^*$ and decreases with $\ell$.