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arXiv:0905.2106 · Physical Review B

Thermal expansion and Grüneisen parameter in quantum Griffiths phases

Thomas Vojta

Published Jul 1, 2009• 12 citations
arXivHTMLPDF

Abstract

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 $\text{ln}({T}_{0}/T)$ with vanishing temperature $T$ 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 ${\text{CePd}}_{1\ensuremath{-}x}{\text{Rh}}_{x}$.

DOI
10.1103/physrevb.80.041101

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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 [ln(T0/T)]1+1/(\ensuremathν\ensuremathψ){[\text{ln}({T}_{0}/T)]}^{1+1/(\ensuremath{\nu}\ensuremath{\psi})}[ln(T0​/T)]1+1/(\ensuremathν\ensuremathψ) where \ensuremathν\ensuremath{\nu}\ensuremathν and \ensuremathψ\ensuremath{\psi}\ensuremathψ 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 CePd1\ensuremath−xRhx{\text{CePd}}_{1\ensuremath{-}x}{\text{Rh}}_{x}CePd1\ensuremath−x​Rhx​.
Abstract The magnetic analog of the Grüneisen parameter, that is, 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 low‐temperature behavior at the exotic infinite‐randomness quantum critical point and in the associated quantum Griffiths phases. We present extensive numerical simulations showing that the magnetic Grüneisen parameter diverges logarithmically with decreasing temperature in the quantum Griffiths phase. It changes sign right at criticality. These results confirm a recent strong‐disorder renormalization group theory. We also compare our findings to the behavior of the clean transverse‐field Ising chain.
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
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 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.
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 TTT 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.
We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the nonthermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"uneisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.