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arXiv:2102.01699 · Physical review. B./Physical review. B

Divergence of the Grüneisen ratio at symmetry-enhanced first-order quantum phase transitions

Charlotte Beneke, Matthias Vojta C

Published May 18, 2021• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.103.174420

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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 Gr\"uneisen ratio (GR) has emerged as a superb tool for the diagnosis of quantum phase transitions; it diverges algebraically upon approaching critical points of continuous phase transitions. However, this paradigm has been challenged recently by observations of a finite GR for self-dual criticality and divergent GR at symmetry-enhanced first-order transitions. To unveil the fascinating GR further, we exemplify the idea by studying an exactly solvable quantum Ising model with Dzyaloshinskii-Moriya interaction, which harbors a ferromagnetic phase, a paramagnetic phase, and a chiral Luttinger liquid. Although the self-dual criticality of the ferromagnetic-paramagnetic transition is undermined by the Dzyaloshinskii-Moriya interaction, we find that the GR at the transition is still finite albeit with an increasing value, signifying a proximate self-dual relation. By contrast, the GR at the transition between the gapped ferromagnetic phase and the gapless Luttinger liquid diverges and changes its sign when crossing the first-order transition. This implies that the GR could also probe the first-order transition between the gapped and gapless phases.
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 ${\text{CePd}}_{1\ensuremath{-}x}{\text{Rh}}_{x}$.
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.
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 discuss a scaling theory of the lattice response in the vicinity of a finite-temperature critical end point. The thermal expansivity is shown to be more singular than the specific heat such that the Grüneisen ratio diverges as the critical point is approached, except for its immediate vicinity. More generally, we express the thermal expansivity in terms of a scaling function which we explicitly evaluate for the two-dimensional Ising universality class. Recent thermal expansivity measurements on the layered organic conductor κ-(BEDT-TTF)2X close to the Mott transition are well described by our theory.
directions. By tuning coupling strengths along distinct directions, the system exhibits different symmetries, ranging from a discrete Z2 to a continuous U(1) symmetry. The anisotropic coupled-top model displays a discrete Z2 symmetry, and the symmetry breaking induced by strong coupling drives a quantum phase transition from a disordered paramagnetic phase to an ordered ferromagnetic or antiferromagnetic phase. In particular, the isotropic coupled-top model possesses a continuous U(1) symmetry, whose breaking gives rise to the Goldstone mode. The phase boundary can be well captured by the mean-field approach, characterized by the distinct behaviors of the order parameter. Higher-order quantum effects beyond the mean-field contribution can be achieved by mapping the large spins to bosonic operators via the Holstein-Primakoff transformation. For the anisotropic coupled-top model with Z2 symmetry, the energy gap closes, and both quantum fluctuations and entanglement entropy diverge near the critical point, signaling the onset of second-order quantum phase transitions. Strikingly, when U(1) symmetry is broken, the energy gap vanishes beyond the critical point, yielding a novel critical exponent of 1, rather than 1/2 for Z2 symmetry breaking. The rich symmetry structure of the coupled-top model underpins its role as a paradigmatic model for studying quantum phase transitions and exploring associated physical phenomena.