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arXiv:2506.10396 · Physical Review Research

Multientropy and the dihedral measures at quantum critical points

Jonathan Harper, Ali Mollabashi, Tadashi Takayanagi, Kenya Tasuki

Published Nov 20, 2025• 4 citations
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Abstract

The multientropy and dihedral measures are a class of tractable measures for multipartite entanglement, which are labeled by the Rényi index (or replica number) n as in the Rényi entanglement entropy. The purpose of this article is to demonstrate that these quantities are useful probes of quantum critical points by examining concrete examples. In particular, we compute the multientropy and dihedral measures in the ( 1 + 1 )-dimensional massless free scalar field theory on a lattice and in the transverse-field Ising model. For n = 2 , we find that the numerical results in both lattice theories quantitatively agree with those from conformal field theoretic calculations. For n = 3 and n = 4 , we provide predictions of these measures for the massless scalar field theory.

DOI
10.1103/67k4-vc74

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The multientropy and dihedral measures are a class of tractable measures for multipartite entanglement, which are labeled by the Rényi index (or replica number) n as in the Rényi entanglement entropy. The purpose of this article is to demonstrate that these quantities are useful probes of quantum critical points by examining concrete examples. In particular, we compute the multientropy and dihedral measures in the ( 1 + 1 )-dimensional massless free scalar field theory on a lattice and in the transverse-field Ising model. For n = 2 , we find that the numerical results in both lattice theories quantitatively agree with those from conformal field theoretic calculations. For n = 3 and n = 4 , we provide predictions of these measures for the massless scalar field theory.
Multi-invariants are local-unitary invariants of state replicas introduced as potential probes of multipartite entanglement and correlations in quantum many-body systems. In this paper, we investigate two multi-invariants for tripartite pure states, namely multientropy and dihedral invariant. We compute the (genuine) multientropy for Lifshitz ground states and obtain its analytical continuation to noninteger values of Rényi index. We show that the genuine multientropy can be expressed in terms of mutual information and logarithmic negativity, a relation that also holds for stabilizer states. For general tripartite pure states, we demonstrate that dihedral invariants are related to Rényi reflected entropies. In particular, we show that the dihedral permutations of replicas are equivalent to the reflected construction, or alternatively to the realignment of density matrices.
At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multisheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+11+11+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2+12+12+1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling.
Although entanglement is a well studied property in the context of quantum systems, the ability to measure it in Monte Carlo methods is relatively new. \nThrough measures of the Renyi entanglement entropy and mutual information one is able to examine and characterize criticality, pinpoint phase transitions, and probe universality. \nWe describe the most basic algorithms for calculating these quantities in straightforward Monte Carlo methods and state of the art techniques used in high performance computing. \nThis description emphasizes the core principal of these measurements and allows one to both build an intuition for these quantities and how they are useful in numerical studies. \n \nUsing the Renyi entanglement entropy we demonstrate the ability to detect thermal phase transitions in the Ising model and XY model without use of an order parameter. \nThe scaling near the critical point also shows signatures identifying the universality class of the model. \nImproved methods are explored using extended ensemble techniques that can increase calculation efficiency, and show good agreement with the standard approach. \nWe explore the "ratio trick" at finite temperature and use it to explore the quantum critical fan of the one dimensional transverse field Ising model, showing agreement with finite temperature and finite size scaling from field theory. \nThis same technique is used at zero temperature to explore the geometric dependence of the entanglement entropy and examine the universal scaling functions in the two dimensional transverse field Ising model. \nAll of this shows the multitude of ways in which the study of the Renyi entanglement entropy can be efficiently and practically used in conventional and exotic condensed matter systems, and should serve as a reference for those wishing to use it as a tool.
Universal logarithmic terms in the entanglement entropy appear at quantum critical points (QCPs) in one dimension (1D) and have been predicted in 2D at QCPs described by 2D conformal field theories. The entanglement entropy in a strip geometry at such QCPs can be obtained via the "Shannon entropy" of a 1D spin chain with open boundary conditions. The Shannon entropy of the XXZ chain is found to have a logarithmic term that implies, for the QCP of the square-lattice quantum dimer model, a logarithm with universal coefficient ±0.25. However, the logarithm in the Shannon entropy of the transverse-field Ising model, which corresponds to entanglement in the 2D Ising conformal QCP, is found to have a singular dependence on the replica or Rényi index resulting from flows to different boundary conditions at the entanglement cut.
Although the leading-order scaling of entanglement entropy is non-universal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in non-interacting field theories, however it typically requires numerical calculation to access them in interacting theories. In this paper, we use large-scale T = 0 quantum Monte Carlo simulations to examine in detail the second Rényi entropy of entangled regions at the QCP in the transverse-field Ising model in 2 + 1 space–time dimensions—a fixed point for which there is no exact result for the scaling of entanglement entropy. We calculate a universal coefficient of a vertex-induced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shape-dependence of the Rényi entropy for finite-size toroidal lattices divided into two entangled cylinders by smooth boundaries. Remarkably, we find that the dependence on cylinder length follows a shape-dependent function calculated previously by Stephan et al (2013 New J. Phys. 15 015004) at the QCP corresponding to the 2 + 1 dimensional quantum Lifshitz free scalar field theory. The quality of the fit of our data to this scaling function, as well as the apparent cutoff-independent coefficient that results, presents tantalizing evidence that this function may reflect universal behaviour across these and other very disparate QCPs in 2 + 1 dimensional systems.
Rényi entropies Sq are useful measures of quantum entanglement; they can be calculated from traces of the reduced density matrix raised to power q, with q ≥ 0. For (d + 1)-dimensional conformal field theories, the Rényi entropies across Sd−1 may be extracted from the thermal partition functions of these theories on either (d+1)-dimensional de Sitter space or R×Hd, where Hd is the d-dimensional hyperbolic space. These thermal partition functions can in turn be expressed as path integrals on branched coverings of the (d+ 1)-dimensional sphere and S1 × Hd, respectively. We calculate the Rényi entropies of free massless scalars and fermions in d = 2, and show how using zeta-function regularization one finds agreement between the calculations on the branched coverings of S3 and on S1 × H2. Analogous calculations for massive free fields provide monotonic interpolating functions between the Rényi entropies at the Gaussian and the trivial fixed points. Finally, we discuss similar Rényi entropy calculations in d> 2.