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arXiv:1004.4898 · Physical Review Letters

Scaling Theory of the Mott Transition and Breakdown of the Grüneisen Scaling Near a Finite-Temperature Critical End Point

Lorenz Bartosch, M De Souza, Michael Lang

Published Jun 14, 2010• 47 citations
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Abstract

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.

DOI
10.1103/physrevlett.104.245701

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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.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We explain, in a consistent manner, the set of seemingly conflicting experiments on the finite temperature Mott critical point, and demonstrate that the Mott transition is in the Ising universality class. We show that, even though the thermodynamic behavior of the system near such critical point is described by an Ising order parameter, the global conductivity can depend on other singular observables and, in particular, on the energy density. Finally, we show that in the presence of weak disorder the dimensionality of the system has crucial effects on the size of the critical region that is probed experimentally.
Phenomenological theory of the Mott transition is presented. When the critical temperature of the Mott transition is much higher than the quantum degeneracy temperature, the transition is essentially described by the Ising universality class. Below the critical temperature, phase separation or first-order transition occurs. However, if the critical point is involved in the Fermi degeneracy region, a marginal quantum critical point appears at zero temperature. The originally single Mott critical point generates subsequent many unstable fixed points through various Fermi surface instabilities induced by the Mott criticality characterized by the diverging charge susceptibility or doublon susceptibility. This occurs in marginal quantum-critical region. Charge, magnetic and superconducting instabilitites compete severely under these critical charge fluctuations. The quantum Mott transition triggers multi-furcating criticality, which goes beyond the conventional concept of multicriticality in quantum phase transitions. Near the quantum Mott transition, the criticality generically drives growth of inhomogeneous structure in the momentum space with singular points of flat dispersion on the Fermi surface. The singular points determine the quantum dynamics of the Mott transition by the dynamical exponent z=4z=4z=4. We argue that many of filling-control Mott transitions are classified to this category. Recent numerical results as well as experimental results on strongly correlated systems including transition metal oxides, organic materials and 3^33He layer adsorbed on a substrate are consistently analyzed especially in two-dimensional systems.
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.
Chapter 25 established the scaling behaviour of correlation functions at criticality, T = Tc . This chapter studies the critical domain which is defined by the property that the correlation length is large with respect to the microscopic scale, but finite. Using results proven in Chapter 10, it first demonstrates strong scaling above Tc : in the critical domain above Tc , all correlation functions, after rescaling, can be expressed in terms of universal correlation functions, in which the scale of distance is provided by the correlation length. The first part of the chapter is restricted to Ising-like systems. It then generalizes the results to N-component order parameters in Section 26.6. In Section 26.7, shows how to expand the universal two-point function when T approaches Tc , using the short distance expansion. Finally, the appendix discusses the energy correlation function when the specific heat exponent vanishes.