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arXiv:2303.04768 · Physical review. B./Physical review. B

Rényi entropy of a quantum anharmonic chain at nonzero temperature

Miha Srdinšek, Michele Casula, Rodolphe Vuilleumier

Published Dec 7, 2023• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.108.245121

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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 \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 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 \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4-like systems, as well as in the elusive water-ice transition across phases VII, VIII, and X.
Quantum phase transitions and quantum entanglement can be studied by quantum Rényi entropy, a generalization of Von Neumann entropy. However, Rényi entropy calculations have been plagued by their computational complexity. In this thesis we present a new numerical technique based on thermodynamic integration via regularized integration paths. The optimal path is constructed by minimizing the variance of the Rényi entropy estimator leading to an efficient evaluation of its quantity for complex quantum systems of distinguishable particles with unprecedented entanglement and size. We implement the method in path integral Monte Carlo (PIMC) and path integral molecular dynamics (PIMD) schemes. We demonstrate in the case of the one-dimensional Ising model in transverse magnetic field, that the method outperforms some of the existing state-of-the-art numerical methods. We then study the complex of formic acid dimer and discover that its two shared protons are entangled even above room temperature. By the means of hybrid PIMC-PIMD scheme we then fully resolve the phase diagram of the discrete ϕ4 model, and show that the peak of the Rényi entropy at finite temperature exactly coincides with the quantum critical point (QCP) and the freezing of the field in the imaginary time, interpreted as the local moments formation. We then use the Rényi entropy to determine the phase diagram of high-pressure water ice phases VIII, VII and X. The quantum Rényi entropy of protons sharply peaks at the pressure of 43 GPa, which is a signature of a QCP. The peak precisely coincides with the freezing of protons in the imaginary time, where phase X is lost, and the formation of a macroscopic polarization at lower temperature, characteristic of phase VIII. In this way we are able to scan the full phase diagram of high pressure water ice by looking at the changes in Rényi entropy behavior as a function of pressure and temperature.
Stabilizer Rényi entropy has emerged as a probe of nonstabilizerness in quantum many-body systems, but its scaling structure beyond critical points remains poorly understood compared with entanglement entropy. Recent field-theory approaches indicate that stabilizer entropy contains universal critical data and boundary-sensitive terms, raising the question of how these structures extend into massive and crossover regimes. We address this problem for a broad class of finite-range spin chains at Rényi index one-half. We derive exact finite-size formulas for both full periodic chains and finite intervals of the infinite chain, making the universal crossover from critical to noncritical behavior analytically accessible. In periodic geometry, the entropy obeys a volume law away from criticality and exhibits a universal finite-size crossover controlled by the competition between system size and correlation length. We also show that the large-scale SRE density develops a cusp across the field-tuned critical line, while the XX endpoint is governed by a distinct scaling regime associated with the saturation point. In the subsystem geometry, the interval entropy separates bulk critical behavior from boundary contributions generated by the way the finite region cuts the infinite chain. The crossover from critical to massive behavior is then encoded in boundary constants and universal functions controlled by the correlation length. Through exact stabilizer-entropy correspondences, the scaling theory extends to internal XY reductions, Finite-range spin chains, and Cluster--Ising representatives. Our results provide an exact lattice benchmark for the emerging QFT description of stabilizer entropy beyond isolated conformal points.
Up to now, probing the quantum phase transition (QPT) and quantum critical (QC) phenomena at finite temperatures in one-dimensional (1D) spin systems still lacks an in-depth understanding. Herein, we study the QPT and thermodynamics of 1D spin-1/2 anisotropic Heisenberg antiferromagnetic chains by Green’s function theory. The quantum phase diagram is renormalized by the anisotropy (∆), which manifests a quantum critical point (QCP) hc = 1 + ∆ signaling the transition from gapless Tomonaga–Luttinger liquid (TLL) to gapped ferromagnetic (FM) state, demonstrated by the magnetic entropy and thermal Drude weight. At low temperatures, it is shown that two crossover temperatures fan out a QC regime and capture the QCP via the linear extrapolation to zero temperature. In addition, around QCP, the QC scaling is performed by analyzing the entropy and thermal Drude weight to extract the critical exponents (α, δ, and β) that fulfill the Essamm–Fisher scaling law, which provides a novel thermodynamic means to detect QPT for experiment. Furthermore, scaling hypothesis equations with two rescaled manners are proposed to testify the scaling analysis, for which all the data points fall on a universal curve or two independent branches for the plot against rescaled field or temperature, implying the self-consistency and reliability of the obtained critical exponents.
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action, we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an ϵϵϵ-expansion with ϵ=1ϵ=1ϵ=1 by Sachdev [Phys. Rev. B 55, 142 (1997)].
The information-geometric origin of fidelity susceptibility and its utility as a universal probe of quantum criticality in many-body settings have been widely discussed. Here we explore the metric response of quantum relative entropy (QRE), by tracing out all but nnn adjacent sites from the ground state of spin chains of finite length NNN, as a parameter of the corresponding Hamiltonian is varied. The diagonal component of this metric defines a susceptibility of the QRE that diverges at quantum critical points (QCPs) in the thermodynamic limit. We study two spin-1/21/21/2 models as examples, namely the integrable transverse field Ising model (TFIM) and a non-integrable Ising chain with three-spin interactions. We demonstrate distinct scaling behaviors for the peak of the QRE susceptibility as a function of NNN: namely a square logarithmic divergence in TFIM and a power-law divergence in the non-integrable chain. This susceptibility encodes uncertainty of entanglement Hamiltonian gradients and is also directly connected to other information measures such as Petz-Rényi entropies. We further show that this susceptibility diverges even at finite NNN if the subsystem size, nnn, exceeds a certain value when the Hamiltonian is tuned to its classical limits due to the rank of the RDMs being finite; unlike the divergence associated with the QCPs which require N→∞N \rightarrow \inftyN→∞.
We analyze several thermodynamic properties of the two-dimensional Kondo necklace using finite-temperature stochastic series expansion. In agreement with previous zero-temperature findings the model is shown to exhibit a quantum critical point (QCP), separating an antiferromagnetic from a paramagnetic dimerized state at a critical Kondo exchange-coupling strength ${J}_{c}\ensuremath{\approx}1.4$. We evaluate the temperature dependent uniform and staggered structure factors as well as the uniform and staggered susceptibilities and the local ``impurity'' susceptibility close to the QCP as well as in the ordered and quantum disordered phase. The crossover between the classical, renormalized classical, and quantum critical regime is analyzed as a function of temperature and Kondo coupling.