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arXiv:2306.00566 · Physical review. B./Physical review. B

Grüneisen parameter as an entanglement compass and the breakdown of the Hellmann-Feynman theorem

Lucas Squillante, L. S. Ricco, Aniekan Magnus Ukpong, Roberto E. Lagos, Antonio Carlos Ferreira Seridonio, M De Souza

Published Oct 6, 2023• 11 citations
arXivHTMLPDF

Abstract

The Gr\"uneisen ratio $\mathrm{\ensuremath{\Gamma}}$, i.e., the singular part of the ratio of thermal expansion to the specific heat, has been broadly employed to explore both finite $T$ and quantum critical points (QCPs). For a genuine quantum phase transition (QPT), thermal fluctuations are absent and thus the thermodynamic $\mathrm{\ensuremath{\Gamma}}$ cannot be employed. We propose a quantum analog to $\mathrm{\ensuremath{\Gamma}}$ that computes entanglement as a function of a tuning parameter $\ensuremath{\lambda}$ and show that QPTs take place only for systems in which the ground-state energy depends on $\ensuremath{\lambda}$ nonlinearly. Furthermore, we demonstrate the breakdown of the Hellmann-Feynman theorem in the thermodynamic limit at any QCP. We showcase our approach using the quantum one-dimensional Ising model with a transverse field and Kane's quantum computer. The slowing down of the dynamics and thus the ``creation of mass'' close to any QCP/QPT is also discussed.

DOI
10.1103/physrevb.108.l140403

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The Gr"uneisen ratio \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ, i.e., the singular part of the ratio of thermal expansion to the specific heat, has been broadly employed to explore both finite TTT and quantum critical points (QCPs). For a genuine quantum phase transition (QPT), thermal fluctuations are absent and thus the thermodynamic \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ cannot be employed. We propose a quantum analog to \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ that computes entanglement as a function of a tuning parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ and show that QPTs take place only for systems in which the ground-state energy depends on \ensuremathλ\ensuremath{\lambda}\ensuremathλ nonlinearly. Furthermore, we demonstrate the breakdown of the Hellmann-Feynman theorem in the thermodynamic limit at any QCP. We showcase our approach using the quantum one-dimensional Ising model with a transverse field and Kane's quantum computer. The slowing down of the dynamics and thus the ``creation of mass'' close to any QCP/QPT is also discussed.
We consider the behavior of the Gr\"uneisen parameter, the ratio between thermal expansion and specific heat, at pressure-tuned infinite-randomness quantum-critical points and in the associated quantum Griffiths phases. We find that the Gr\"uneisen parameter diverges as ln(T0/T)\text{ln}({T}_{0}/T)ln(T0​/T) with vanishing temperature TTT in the quantum Griffiths phases. At the infinite-randomness critical point itself, the Gr\"uneisen parameter behaves as ${[\text{ln}({T}_{0}/T)]}^{1+1/(\ensuremath{\nu}\ensuremath{\psi})}$ where $\ensuremath{\nu}$ and $\ensuremath{\psi}$ are the correlation length and tunneling exponents. Analogous results hold for the magnetocaloric effect at magnetic-field-tuned transitions. We contrast clean and dirty systems, we discuss subtle differences between Ising and Heisenberg symmetries, and we relate our findings to recent experiments on ${\text{CePd}}_{1\ensuremath{-}x}{\text{Rh}}_{x}$.
The magnetic analog of the Grüneisen parameter, i.e., the magnetocaloric effect, is a valuable tool for studying field-tuned quantum phase transitions. We determine the magnetic Grüneisen parameter of the one-dimensional random transverse-field Ising model, focusing on its lowtemperature behavior at the exotic infinite-randomness quantum critical point and in the associated quantum Griffiths phases. 1 Introduction Zero-temperature quantum phase transitions are characterized by a competition between different quantum ground states. The resulting quantum fluctuations often cause unconventional behavior of thermodynamic and transport properties in the vicinity of the
The Grüneisen ratio, defined as Γg≡(1/T)(∂T/∂g)SΓ_g \equiv (1/T) (\partial T/\partial g)_SΓg​≡(1/T)(∂T/∂g)S​, serves as a highly sensitive probe for detecting quantum critical points (QCPs) driven by an external feild ggg and for characterizing the magnetocaloric effect (MCE). Near a QCP, the Grüneisen ratio displays a universal divergence which is governed by a universality-class-dependent scaling function stemming from the scale invariance. In this work, we systematically investigate the universal scaling functions of Grüneisen ratio in both one-dimensional (1D) and two-dimensional (2D) quantum spin systems, including the transverse-field Ising model, the spin-1/2 Heisenberg model, the quantum qqq-state Potts model (q=3,4q=3,4q=3,4) and the J1J_1J1​-J2J_2J2​ columnar dimer model. Our approach employs the thermal tensor-network method for infinite-size 1D systems and the stochastic series expansion quantum Monte Carlo (SSE QMC) simulations for 2D systems, enabling precise calculations of the Grüneisen ratio near QCPs. Through data collapse analysis, we extract the corresponding scaling functions, which establish quantitative frameworks to interpret magnetocaloric experiments and guide the development of ultralow-temperature refrigeration.
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.
Studies of the Gr\"uneisen ratio, i.e., the ratio between thermal expansion and specific heat, have become a powerful tool in the context of quantum criticality since it was shown theoretically that the Gr\"uneisen ratio displays characteristic power-law divergencies upon approaching the transition point of a continuous quantum phase transition. Here we show that the Gr\"uneisen ratio also diverges at a symmetry-enhanced first-order quantum phase transition, albeit with mean-field exponents, as the enhanced symmetry implies the vanishing of a mode gap which is finite away from the transition. We provide explicit results for simple pseudospin models, both with and without Goldstone modes in the stable phases, and discuss implications.
The magnetic Grüneisen ratio, defined as Γ g ≡ (1/ T )(∂ T /∂ g ) S , characterizes the magnetocaloric effect (MCE) driven by an external field g and serves as a highly sensitive probe for field-induced quantum critical points (QCPs). Near a QCP, Γ g displays diverging behaviors described by a universal scaling function depending on the universality class. Here, we systematically investigate the universal scaling functions of Grüneisen ratio Γ g in typical one-dimensional (1D) and two-dimensional (2D) quantum spin systems, including the transverse-field Ising model, XY model, q -state quantum Potts model ( q = 3, 4), and the J 1 - J 2 columnar dimer model. Our approach employs the linearized tensor renormalization group (LTRG) method for infinite-size 1D systems and the stochastic series expansion quantum Monte Carlo (SSE QMC) simulations for 2D systems, enabling precise calculations of the Γ g near QCPs. We extract universal scaling functions via data collapse analysis. This universal magnetocaloric framework offers a quantitative tool for analyzing measurements in quantum materials and for guiding the search for quantum critical cooling applications.