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arXiv:cond-mat/0403345 · Physical Review B

Thermodynamics and quantum criticality in cuprate superconductors

Jan Zaanen, Baharak Hosseinkhani

Published Aug 31, 2004• 16 citations
arXivHTMLPDF

Abstract

We will present elementary scaling arguments focused on the thermodynamics in the proximity of the quantum critical point in the cuprate superconductors. Extending the analysis centered on the Gr\"uneisen parameter by Rosch, Si, and co-workers to the cuprates, we demonstrate that a combination of specific-heat and chemical potential measurements can reveal the nature of the zero temperature singularity. From the known specific-heat data it follows that the effective number of time dimensions has to equal the number of space dimensions, while we find a total of six scaling laws governing the temperature and density dependence of the chemical potential, revealing directly the coupling constant scaling dimension.

DOI
10.1103/physrevb.70.060509

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We will present elementary scaling arguments focused on the thermodynamics in the proximity of the quantum critical point in the cuprate superconductors. Extending the analysis centered on the Gr"uneisen parameter by Rosch, Si, and co-workers to the cuprates, we demonstrate that a combination of specific-heat and chemical potential measurements can reveal the nature of the zero temperature singularity. From the known specific-heat data it follows that the effective number of time dimensions has to equal the number of space dimensions, while we find a total of six scaling laws governing the temperature and density dependence of the chemical potential, revealing directly the coupling constant scaling dimension.
Using the scaling theory of quantum critical phenomena we explore the occurrence of universal critical behavior at the insulator-to-superconductor and superconductor-to-normal state transitions at zero temperature.Experimentally, these phase transitions are driven by doping and correspond to critical end points of the phase transition łine in the temperature-hole concentration plane.Provided that the order parameter is a complex scalar in two dimensions, and that the London relation between superfluid number density and magnetic penetration depth holds, the scaling theory predicts universal behavior close to the insulator-to-superconductor transition.In particular, transition temperature and zero temperature penetration depth are universally related and the sheet resistance adopts a universal value.These predictions agree remarkably well with available experimental data and provide useful constraints for a microscopic theory.
For strongly correlated electrons the relation between total number of charge carriers ne{n}_{e}ne​ and the chemical potential $\ensuremath{\mu}$ reveals for large Coulomb energy the apparently paradoxical pinning of $\ensuremath{\mu}$ within the Mott gap, as observed in high-Tc{T}_{c}Tc​ cuprates. By unraveling consequences of the nontrivial topology of the charge gauge U(1) group and the associated ground state degeneracy we found a close kinship between the pinning of $\ensuremath{\mu}$ and the zero-temperature divergence of the charge compressibility $\ensuremath{\kappa}\ensuremath{\sim}\ensuremath{\partial}{n}_{e}∕\ensuremath{\partial}\ensuremath{\mu}$, which marks a novel quantum criticality governed by topological charges rather than Landau principle of the symmetry breaking.
Superfluids and superconductors can be either fully gapped or gapless with nodal structures in momentum space. Both the gapped and nodal phases can be topologically protected and possess nontrivial topological invariants. Topological quantum phase transitions exist at zero temperature between different gapped phases or between gapped and nodal phases. These phase transitions can be driven by chemical potential and/or spin exchange fields. The two phases separated by the phase transition can have the same local order but differ in global topology. Thus, these topological quantum phase transitions cannot be described by the Landau paradigm of symmetry breaking. Although some aspects of these transitions, such as the change of topological invariants and gapless boundary states, have been discussed before, a complete theory of these transitions has yet to be developed. In this dissertation, we construct effective field theories to study the universality and thermodynamic signatures of these transitions. We find four different universality classes in superfluids and superconductors. Certain thermodynamic quantities, such as compressibility or spin susceptibility, change non-analytically across the transitions. For certain time-reversal symmetry breaking fields that lead to bulk phase transitions, there also exist topological phase transitions on the surface. All the topological phase transitions studied in this dissertation only exist at zero temperature. At finite temperature, different states are connected by smooth crossovers. There exists a quantum critical region at finite temperature near the quantum critical point (QCP). In this quantum critical region, thermodynamic quantities have universal scaling dependence on temperature dictated by the universality class of the QCP. We argue that these scaling properties can be used to probe and differentiate these QCPs. These bulk and surface topological quantum phase transitions are discussed in various concrete models, such as chiral and time-reversal invariant p-wave superfluids, topological superconductors of emergent Dirac fermions, and topological superconducting model of CuₓBi₂Se₃.
If the zero-field transition in high-temperature superconductors such as ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ is a critical point in the universality class of the three-dimensional XY\mathrm{XY}XY model, then the general theory of critical phenomena predicts the existence of a critical region in which thermodynamic functions have a characteristic scaling form. We report an attempt to calculate the universal scaling function associated with the specific heat, for which experimental data have become available in recent years. Scaling behavior is extracted from a renormalization-group analysis, and the 1/N1/N1/N expansion is adopted as a means of approximation. The estimated scaling function is qualitatively similar to that observed experimentally, and also to the lowest-Landau-level scaling function used by some authors to provide an alternative interpretation of the same data. Unfortunately, the 1/N1/N1/N expansion is not sufficiently reliable at small values of N for a quantitative fit to be feasible.
Abstract Materials tuned to a quantum critical point display universal scaling properties as a function of temperature T and frequency ω. A long-standing puzzle regarding cuprate superconductors has been the observed power-law dependence of optical conductivity with an exponent smaller than one, in contrast to T-linear dependence of the resistivity and ω-linear dependence of the optical scattering rate. Here, we present and analyze resistivity and optical conductivity of La2−xSrxCuO4 with x = 0.24. We demonstrate ℏω/kBT scaling of the optical data over a wide range of frequency and temperature, T-linear resistivity, and optical effective mass proportional to ∼ln⁡T\sim \ln T∼lnT ~ ln T corroborating previous specific heat experiments. We show that a T, ω-linear scaling Ansatz for the inelastic scattering rate leads to a unified theoretical description of the experimental data, including the power-law of the optical conductivity. This theoretical framework provides new opportunities for describing the unique properties of quantum critical matter.