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arXiv:2404.08389 · Physical Review A

Rényi entropy of the permutationally invariant part of the ground state across a quantum phase transition

Yuki Miyazaki, Giacomo Marmorini, Nobuo Furukawa, Daisuke Yamamoto

Published Nov 18, 2024• 0 citations
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Abstract

We investigate the role of the permutationally invariant part of the density matrix (PIDM) in capturing the properties of the ground state of the system during a quantum phase transition. In the context of quantum state tomography, PIDM is known to be obtainable with only a low number of measurement settings, namely $O({L}^{2})$, where $L$ is the system size. Considering the transverse-field Ising chain and the $XXZ$ chain as examples, we compute the second-order R\'enyi entropy of PIDM for the ground state by using the density-matrix renormalization-group algorithm. In the transverse-field Ising chain with ferromagnetic interaction, the ground state is permutationally invariant both in the limits of zero and infinite field, leading to a vanishing R\'enyi entropy of PIDM. The latter exhibits a broad peak as a function of the transverse field around the quantum critical point, which gets more pronounced for larger system size. In the antiferromagnetic case, the peak structure disappears and the R\'enyi entropy diverges like $O(L)$ in the whole field range of the ordered phase. In the $XXZ$ model with ferromagnetic interaction, the ground state is permutationally invariant in the full parameter regime of easy-axis anisotropy, causing the R\'enyi entropy of PIDM to vanish. In the easy-plane region, the R\'enyi entropy of PIDM grows almost linearly as a function of the system size. In the antiferromagnetic case, the R\'enyi entropy of PIDM takes finite values at any point in parameter space; importantly, its minimum appears at the phase-transition point. We discuss the cause of these behaviors of the R\'enyi entropy of PIDM, examining the possible application of this experimentally tractable quantity to the analysis of phase-transition phenomena.

DOI
10.1103/physreva.110.052422

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We investigate the role of the permutationally invariant part of the density matrix (PIDM) in capturing the properties of the ground state of the system during a quantum phase transition. In the context of quantum state tomography, PIDM is known to be obtainable with only a low number of measurement settings, namely O(L2)O({L}^{2})O(L2), where LLL is the system size. Considering the transverse-field Ising chain and the XXZXXZXXZ chain as examples, we compute the second-order R'enyi entropy of PIDM for the ground state by using the density-matrix renormalization-group algorithm. In the transverse-field Ising chain with ferromagnetic interaction, the ground state is permutationally invariant both in the limits of zero and infinite field, leading to a vanishing R'enyi entropy of PIDM. The latter exhibits a broad peak as a function of the transverse field around the quantum critical point, which gets more pronounced for larger system size. In the antiferromagnetic case, the peak structure disappears and the R'enyi entropy diverges like O(L)O(L)O(L) in the whole field range of the ordered phase. In the XXZXXZXXZ model with ferromagnetic interaction, the ground state is permutationally invariant in the full parameter regime of easy-axis anisotropy, causing the R'enyi entropy of PIDM to vanish. In the easy-plane region, the R'enyi entropy of PIDM grows almost linearly as a function of the system size. In the antiferromagnetic case, the R'enyi entropy of PIDM takes finite values at any point in parameter space; importantly, its minimum appears at the phase-transition point. We discuss the cause of these behaviors of the R'enyi entropy of PIDM, examining the possible application of this experimentally tractable quantity to the analysis of phase-transition phenomena.
We present an analytical study of the quantum phase transition between the topologically ordered toric-code-model ground state and the disordered spin-polarized state. The phase transition is induced by applying an external magnetic field, and the variation in topological order is detected via two nonlocal quantities: the Wilson loop and the topological R\'enyi entropy of order 2. By exploiting an equivalence with the transverse-field Ising model and considering two different variants of the problem, we investigate the field dependence of these quantities by means of an exact treatment in the exactly solvable variant and complementary perturbation theories around the limits of zero and infinite fields in both variants. We find strong evidence that the phase transition point between topological order and disorder is marked by a discontinuity in the topological R\'enyi entropy and that the two phases around the phase transition point are characterized by its different constant values. Our results therefore indicate that the topological R\'enyi entropy is a proper topological invariant: its allowed values are discrete and can be used to distinguish between different phases of matter.
We compute the Renyi entropy in a one-dimensional transverse-field quantum Ising model by employing a swapping operator acting on the states which are prepared from the neural network methods. In the static ground state, Renyi entropy can uncover the critical point of the quantum phase transition from paramagnetic to ferromagnetic. At the critical point, the relation between the Renyi entropy and the subsystem size satisfies the predictions from conformal field theory. In the dynamical case, we find coherent oscillations of the Renyi entropy after the end of the linear quench. These oscillations have universal frequencies which may come from the superpositions of excited states. The asymptotic form of the Renyi entropy implies a new length scale away from the critical point. This length scale is also verified by the overlap of the reduced Renyi entropy against the dimensionless subsystem size.
We investigate the scaling of the entanglement spectrum and of the R\'enyi block entropies and determine its universal aspects in the ground state of critical and noncritical one-dimensional quantum spin models. In all cases, the scaling exhibits an oscillatory behavior that terminates at the factorization point and whose frequency is universal. Parity effects in the scaling of the R\'enyi entropies for gapless models at zero field are thus shown to be a particular case of such universal behavior. Likewise, the absence of oscillations for the Ising chain in transverse field is due to the vanishing value of the factorizing field for this particular model. In general, the transition occurring at the factorizing field between two different scaling regimes of the entanglement spectrum corresponds to a quantum transition to the formation of finite-range, ordered structures of quasidimers, quasitrimers, and quasipolymers. This entanglement-driven transition is superimposed to and independent of the long-range magnetic order in the broken symmetry phase. Therefore, it conforms to recent generalizations that identify and classify the quantum phases of matter according to the structure of ground-state entanglement patterns. We characterize this form of quantum order by a global order parameter of entanglement defined as the integral, over blocks of all lengths, of the R\'enyi entropy of infinite order. Equivalently, it can be defined as the integral of the bipartite single-copy or geometric entanglement. The global entanglement order parameter remains always finite at fields below the factorization point and vanishes identically above it.
In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin 1/2 Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number k fixing the polarization in the subsystem conservation of Sz and with respect to the irreducible representations of the Sn group. Analytical expression for the RDM elements and for the RDM spectrum are derived for states of arbitrary permutational symmetry and for arbitrary polarizations. The temperature dependence and scaling of the VNE across a finite temperature phase transition is discussed and the RDM moments and the Rényi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.
We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power α of the density matrix of the block. We calculate the asymptotic for L → ∞ analytically in terms of Klein's elliptic λ-function. We study the limiting entropy as a function of its parameter α. We show that up to the trivial addition terms and multiplicative factors, and after a proper rescaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map α → α −1 and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when α is an integer power of 2; this includes the purity tr ρ 2 . We also analyze the behavior of the entropy as α → 0 and ∞ and at the critical magnetic field and in the isotropic limit (XX model).
The interplay of quantum and classical fluctuations in the vicinity of a quantum critical point (QCP) gives rise to various regimes or phases with distinct quantum character. In this work, we show that the R\'enyi entropy is a precious tool to characterize the phase diagram of critical systems not only around the QCP but also away from it, thanks to its capability to detect the emergence of local moments at finite temperature. For an efficient evaluation of the R\'enyi entropy, we introduce an algorithm based on a path-integral Langevin dynamics combined with a previously proposed thermodynamic integration method built on regularized paths. We apply this framework to study the critical behavior of a linear chain of anharmonic oscillators, a particular realization of the ${\ensuremath{\phi}}^{4}$ model. We fully resolved its phase diagram, as a function of both temperature and interaction strength. At finite temperature, we find a sequence of three regimes---para, disordered, and antiferro---met as the interaction is increased. The R\'enyi entropy divergence coincides with the crossover between the para and disordered regimes, which shows no temperature dependence. The occurrence of the antiferro regime, on the other hand, is temperature dependent. The two crossover lines merge in proximity of the QCP, at zero temperature, where the R\'enyi entropy is sharply peaked. Via its subsystem-size scaling, we confirm that the transition belongs to the two-dimensional Ising universality class. This phenomenology is expected to happen in all ${\ensuremath{\phi}}^{4}$-like systems, as well as in the elusive water-ice transition across phases VII, VIII, and X.