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arXiv:1905.02281 · Physical review. B./Physical review. B

Specific heat maximum as a signature of Mott physics in the two-dimensional Hubbard model

G. Sordi, C. Walsh, P. Sémon, A.–M. S. Tremblay

Published Sep 12, 2019• 16 citations
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Abstract

Recent experiments on cuprates show that as a function of doping, the normal-state specific heat sharply peaks at the doping ${\ensuremath{\delta}}^{*}$, where the pseudogap ends at low temperature. This finding is taken as the thermodynamic signature of a quantum critical point, whose nature has not yet been identified. Here we present calculations for the two-dimensional Hubbard model in the doped Mott insulator regime, which indicate that the specific heat anomaly can arise from the finite-temperature critical end point of a first-order transition between a pseudogap phase with dominant singlet correlations and a metal. As a function of doping at the temperature of the end point, the specific heat diverges. Upon increasing temperature, the peak becomes broader. The diverging correlation length is associated with uniform density fluctuations. No broken symmetries are needed. These anomalies also occur at half-filling as a function of interaction strength, and are relevant for organic superconductors and ultracold atoms.

DOI
10.1103/physrevb.100.121105

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Recent experiments on cuprates show that as a function of doping, the normal-state specific heat sharply peaks at the doping \ensuremathδ∗{\ensuremath{\delta}}^{*}\ensuremathδ∗, where the pseudogap ends at low temperature. This finding is taken as the thermodynamic signature of a quantum critical point, whose nature has not yet been identified. Here we present calculations for the two-dimensional Hubbard model in the doped Mott insulator regime, which indicate that the specific heat anomaly can arise from the finite-temperature critical end point of a first-order transition between a pseudogap phase with dominant singlet correlations and a metal. As a function of doping at the temperature of the end point, the specific heat diverges. Upon increasing temperature, the peak becomes broader. The diverging correlation length is associated with uniform density fluctuations. No broken symmetries are needed. These anomalies also occur at half-filling as a function of interaction strength, and are relevant for organic superconductors and ultracold atoms.
For doped two-dimensional Mott insulators in their normal state, the challenge is to understand the evolution from a conventional metal at high doping to a strongly correlated metal near the Mott insulator at zero doping. To this end, we solve the cellular dynamical mean-field equations for the two-dimensional Hubbard model using a plaquette as the reference quantum impurity model and continuous-time quantum Monte Carlo method as impurity solver. The normal-state phase diagram as a function of interaction strength UUU, temperature TTT, and filling nnn shows that, upon increasing nnn toward the Mott insulator, there is a surface of first-order transition between two metals at nonzero doping. That surface ends at a finite temperature critical line originating at the half-filled Mott critical point. Associated with this transition, there is a maximum in scattering rate as well as thermodynamic signatures. These findings suggest a new scenario for the normal-state phase diagram of the high temperature superconductors. The criticality surmised in these systems can originate not from a T === 0 quantum critical point, nor from the proximity of a long-range ordered phase, but from a low temperature transition between two types of metals at finite doping. The influence of Mott physics therefore extends well beyond half-filling.
Experiments on layered materials call for a study of the influence of short-range spin correlations on the Mott transition. To this end, we solve the cellular dynamical mean-field equations for the Hubbard model on a plaquette with continuous-time quantum Monte Carlo calculations. The normal-state phase diagram as a function of temperature T, interaction strength U, and filling n reveals that upon increasing n towards the insulator, there is a surface of first-order transition between two metals at nonzero doping. For T above the critical end line there is a maximum in scattering rate.
Whether or not anomalies in the thermal conductivity in insulating cuprates can be attributed to antiferromagnetic order and magnons in a 2D Mott insulator remains an intriguing open question. To shed light on this issue, we investigate the thermal conductivity $\ensuremath{\kappa}$ and its relationship with the specific heat cv{c}_{v}cv​ in the half-filled 2D single-band Hubbard model, using the numerically exact determinant quantum Monte Carlo algorithm and maximum entropy analytic continuation. At low temperatures where the charge degrees of freedom are gapped-out and cv{c}_{v}cv​ exhibits a clear magnon peak, we observe that thermal conductivity $\ensuremath{\kappa}$ also tends to form a peak at similar temperatures. Reducing temperature further produces a sharp upturn in $\ensuremath{\kappa}$, associated with an increasing mean-free path. We identify this as the high-temperature side of the anomalous peak in insulating cuprates, where the mean-free path eventually is cut off by other scattering effects, including phonons, disorder, and physical size. Different scattering effects in our model are identified and analyzed in the thermal diffusivity.
Qualitative changes in thermodynamic and single-particle properties characterize the transition between the pseudogapped electronic liquid and the Fermi liquid. Recent cold-atom experiments on a simulator of the Hubbard model with nearest-neighbor hoppings [L. H. Kendrick , .] showed that the isothermal compressibility κ ( δ ) has a maximum as a function of doping δ . Here, we use the two-particle self-consistent plus approach to explain these experiments and connect the maximum in κ ( δ ) to the transformation of the single-particle spectrum from the pseudogapped to the metallic regime. This elucidates the nature of the pseudogap. Specifically, the maximum in κ ( δ ) practically coincides with the doping at which the precursor of the lower ( π , π ) spin-density-wave (SDW) band at the antinodal point crosses the zero-frequency ω = 0 . The Knight shift χ sp ( 0 , 0 ) ( δ ) , as a function of doping, should also have a maximum. In addition, we demonstrate that two-particle self-consistent plus correctly predicts a maximum in the temperature dependence of the Knight shift χ sp ( 0 , 0 ) ( T ) , consistent with recent ultracold-atom experiments [T. Chalopin , .]. The maxima in both quantities should exist, at sufficiently low temperatures ( T ), in both the intermediate U ≈ U Mott and weak U < U Mott interaction limits. In both limits, the mechanism is critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at δ max ( T ) can be small, controlled not by static but by dynamic critical thermal fluctuations. We also find that the SDW fluctuations are incommensurate at δ = δ max . We predict that, at low T , the multiple peaks in the spin susceptibility in the incommensurate case lead to more than two SDW precursor peaks in the spectral function and density of states. By allowing access to parameter regimes relevant to cuprates—including further-neighbor hopping ( t ′ , t ′′ ) and low temperatures, our work provides a high-impact tool for further studies by the broader community.
The transition between a metallic and a pseudogap phase in high-Tc{T}_{c}Tc​ cuprate superconductors, which is the subject of recent experimental investigations, is studied theoretically within the Hubbard model. We apply the cluster dynamical impurity approximation to the anisotropic Hubbard model on the square lattice at zero temperature and finite doping. This approach can detect a first-order transition between two metallic states: a pseudogap state at low doping with a depleted density of states at the Fermi level and a correlated metal at higher doping. This transition was seen first in the cluster dynamical mean-field theory at finite temperatures by Sordi et al. [Phys. Rev. Lett. 108, 216401 (2012)] on the isotropic Hubbard model. Here we investigate this transition at zero temperature and as a function of on-site interaction UUU, anisotropy ty/tx{t}_{y}/{t}_{x}ty​/tx​, and doping. We find a first-order transition line which ends at a critical point, which occurs around ${t}_{y}/{t}_{x}\ensuremath{\sim}0.5$.
The thermodynamic properties of specific heat, entropy, chemical potential, spin susceptibility ${\ensuremath{\chi}}_{s},$ and charge susceptibility ${\ensuremath{\chi}}_{c}$ are studied as functions of temperature and doping within the two-dimensional Hubbard model with various U/t=4−−12.U/t=4--12.U/t=4−−12. Quantities are calculated using the finite-temperature Lanczos method with additional phase averaging for a system of $4\ifmmode\times\else\texttimes\fi{}4$ sites. Results show that the entropy at low T reaches a maximum near half-filling at the electron density $n\ensuremath{\sim}1\ifmmode\pm\else\textpm\fi{}0.15$ in the whole regime of studied U/t.U/t.U/t. The pseudogap in ${\ensuremath{\chi}}_{s}(T)$ becomes clearly pronounced for U/t> 8U/t>~8U/t> 8 while ${\ensuremath{\chi}}_{c}$ shows a maximum close to half-filling. The relation of results to those within the $t\ensuremath{-}J$ model and to experiments is discussed.