Criticality in correlated quantum matter

At quantum critical points (QCP) \cite{Pfeuty:1971,Young:1975,Hertz:1976,Chakravarty:1989,Millis:1993,Chubukov:1 994,Coleman:2005} there are quantum fluctuations on all length scales, from microscopic to macroscopic lengths, which, remarkably, can be observed at finite temperatures, the regime to which all experiments are necessarily confined. A fundamental question is how high in temperature can the effects of quantum criticality persist? That is, can physical observables be described in terms of universal scaling functions originating from the QCPs? Here we answer these questions by examining exact solutions of models of correlated systems and find that the temperature can be surprisingly high. As a powerful illustration of quantum criticality, we predict that the zero temperature superfluid density, $\rho_{s}(0)$, and the transition temperature, $T_{c}$, of the cuprates are related by $T_{c}\propto\rho_{s}(0)^y$, where the exponent $y$ is different at the two edges of the superconducting dome, signifying the respective QCPs. This relationship can be tested in high quality crystals.Comment: Final accepted version not including minor stylistic correction

Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid

The Landau paradigm of classifying phases by broken symmetries was demonstrated to be incomplete when it was realized that different quantum Hall states could only be distinguished by more subtle, topological properties. Today, the role of topology as an underlying description of order has branched out to include topological band insulators, and certain featureless gapped Mott insulators with a topological degeneracy in the groundstate wavefunction. Despite intense focus, very few candidates for these topologically ordered "spin liquids" exist. The main difficulty in finding systems that harbour spin liquid states is the very fact that they violate the Landau paradigm, making conventional order parameters non-existent. Here, we uncover a spin liquid phase in a Bose-Hubbard model on the kagome lattice, and measure its topological order directly via the topological entanglement entropy. This is the first smoking-gun demonstration of a non-trivial spin liquid, identified through its entanglement entropy as a gapped groundstate with emergent Z2 gauge symmetry.Comment: 4+ pages, 3 figure

Dynamics and transport near quantum-critical points

The physics of non-zero temperature dynamics and transport near quantum-critical points is discussed by a detailed study of the O(N)-symmetric, relativistic, quantum field theory of a N-component scalar field in $d$ spatial dimensions. A great deal of insight is gained from a simple, exact solution of the long-time dynamics for the N=1 d=1 case: this model describes the critical point of the Ising chain in a transverse field, and the dynamics in all the distinct, limiting, physical regions of its finite temperature phase diagram is obtained. The N=3, d=1 model describes insulating, gapped, spin chain compounds: the exact, low temperature value of the spin diffusivity is computed, and compared with NMR experiments. The N=3, d=2,3 models describe Heisenberg antiferromagnets with collinear N\'{e}el correlations, and experimental realizations of quantum-critical behavior in these systems are discussed. Finally, the N=2, d=2 model describes the superfluid-insulator transition in lattice boson systems: the frequency and temperature dependence of the the conductivity at the quantum-critical coupling is described and implications for experiments in two-dimensional thin films and inversion layers are noted.Comment: Lectures presented at the NATO Advanced Study Institute on "Dynamical properties of unconventional magnetic systems", Geilo, Norway, April 2-12, 1997, edited by A. Skjeltorp and D. Sherrington, Kluwer Academic, to be published. 46 page

Algebraic charge liquids

High temperature superconductivity emerges in the cuprate compounds upon changing the electron density of an insulator in which the electron spins are antiferromagnetically ordered. A key characteristic of the superconductor is that electrons can be extracted from them at zero energy only if their momenta take one of four specific values (the `nodal points'). A central enigma has been the evolution of the zero energy electrons in the metallic state between the antiferromagnet and the superconductor, and recent experiments yield apparently contradictory results. The oscillation of the resistance in this metal as a function of magnetic field indicate that the zero energy electrons carry momenta which lie on elliptical `Fermi pockets', while ejection of electrons by high intensity light indicates that the zero energy electrons have momenta only along arc-like regions. We present a theory of new states of matter, which we call `algebraic charge liquids', which arise naturally between the antiferromagnet and the superconductor, and reconcile these observations. Our theory also explains a puzzling dependence of the density of superconducting electrons on the total electron density, and makes a number of unique predictions for future experiments.Comment: 6+8 pages, 2 figures; (v2) Rewritten for broader accessibility; (v3) corrected numerical error in Eq. (5

On the break in the single-particle energy dispersions and the `universal' nodal Fermi velocity in the high-temperature copper-oxide superconductors

Recent data from angle-resolved photoemission experiments published by Zhou et al. [Nature, Vol. 423, 398 (2003)] concerning a number of hole-doped copper-oxide-based high-temperature superconductors reveal that in the nodal directions of the underlying square Brillouin zones (i.e. the directions along which the d-wave superconducting gap is vanishing) the Fermi velocities for some finite range of k inside the Fermi sea and away from the nodal Fermi wavevector k_F are to within an experimental uncertainty of approximately 20% the same both in all the compounds investigated and over a wide range of doping concentrations and that, in line with earlier experimental observations, at some characteristic wavevector k_* away from k_F the Fermi velocities undergo a sudden change, with this change (roughly speaking, a finite discontinuity) being the greatest (smallest) in the case of underdoped (overdoped) compounds. In this paper we present a rigorous analysis concerning the implications of these observations. [Short abstract]Comment: 29 pages, 4 postscript figures. Brought into conformity with the published versio

Thermally fluctuating superconductors in two dimensions

We describe the different regimes of finite temperature dynamics in the vicinity of a zero temperature superconductor to insulator quantum phase transition in two dimensions. New results are obtained for a low temperature phase-only hydrodynamics, and for the intermediate temperature quantum-critical region. In the latter case, we obtain a universal relationship between the frequency-dependence of the conductivity and the value of the d.c. resistance.Comment: Presentation completely revised; 4 pages, 2 figure

Hour-glass magnetic spectrum in an insulating, hole-doped antiferromagnet

Superconductivity in layered copper-oxide compounds emerges when charge carriers are added to antiferromagnetically-ordered CuO2 layers. The carriers destroy the antiferromagnetic order, but strong spin fluctuations persist throughout the superconducting phase and are intimately linked to super-conductivity. Neutron scattering measurements of spin fluctuations in hole-doped copper oxides have revealed an unusual `hour-glass' feature in the momentum-resolved magnetic spectrum, present in a wide range of superconducting and non-superconducting materials. There is no widely-accepted explanation for this feature. One possibility is that it derives from a pattern of alternating spin and charge stripes, an idea supported by measurements on stripe-ordered La1.875Ba0.125CuO4. However, many copper oxides without stripe order also exhibit an hour-glass spectrum$. Here we report the observation of an hour-glass magnetic spectrum in a hole-doped antiferromagnet from outside the family of superconducting copper oxides. Our system has stripe correlations and is an insulator, which means its magnetic dynamics can conclusively be ascribed to stripes. The results provide compelling evidence that the hour-glass spectrum in the copper-oxide superconductors arises from fluctuating stripes.Comment: 13 pages, 4 figures, to appear in Natur

Quantum magnetism and criticality

Magnetic insulators have proved to be fertile ground for studying new types of quantum many body states, and I survey recent experimental and theoretical examples. The insights and methods transfer also to novel superconducting and metallic states. Of particular interest are critical quantum states, sometimes found at quantum phase transitions, which have gapless excitations with no particle- or wave-like interpretation, and control a significant portion of the finite temperature phase diagram. Remarkably, their theory is connected to holographic descriptions of Hawking radiation from black holes.Comment: 39 pages, 10 figures, review article for non-specialists; (v2) added clarifications and references; (v3) minor corrections; (v4) added footnote on hydrodynamic long-time tail

Disentangling Cooper-pair formation above Tc from the pseudogap state in the cuprates

The discovery of the pseudogap in the cuprates created significant excitement amongst physicists as it was believed to be a signature of pairing, in some cases well above the room temperature. In this "pre-formed pairs" scenario, the formation of pairs without quantum phase rigidity occurs below T*. These pairs condense and develop phase coherence only below Tc. In contrast, several recent experiments reported that the pseudogap and superconducting states are characterized by two different energy scales, pointing to a scenario, where the two compete. However a number of transport, magnetic, thermodynamic and tunneling spectroscopy experiments consistently detect a signature of phase-fluctuating superconductivity above leaving open the question of whether the pseudogap is caused by pair formation or not. Here we report the discovery of a spectroscopic signature of pair formation and demonstrate that in a region of the phase diagram commonly referred to as the "pseudogap", two distinct states coexist: one that persists to an intermediate temperature Tpair and a second that extends up to T*. The first state is characterized by a doping independent scaling behavior and is due to pairing above Tc, but significantly below T*. The second state is the "proper" pseudogap - characterized by a "checker board" pattern in STM images, the absence of pair formation, and is likely linked to Mott physics of pristine CuO2 planes. Tpair has a universal value around 130-150K even for materials with very different Tc, likely setting limit on highest, attainable Tc in cuprates. The observed universal scaling behavior with respect to Tpair indicates a breakdown of the classical picture of phase fluctuations in the cuprates.Comment: 9 pages, 4 figure

Anisotropic Structure of the Order Parameter in FeSe0.45Te0.55 Revealed by Angle Resolved Specific Heat

The symmetry and structure of the superconducting gap in the Fe-based superconductors are the central issue for understanding these novel materials. So far the experimental data and theoretical models have been highly controversial. Some experiments favor two or more constant or nearly-constant gaps, others indicate strong anisotropy and yet others suggest gap zeros ("nodes"). Theoretical models also vary, suggesting that the absence or presence of the nodes depends quantitatively on the model parameters. An opinion that has gained substantial currency is that the gap structure, unlike all other known superconductors, including cuprates, may be different in different compounds within the same family. A unique method for addressing this issue, one of the very few methods that are bulk and angle-resolved, calls for measuring the electronic specific heat in a rotating magnetic field, as a function of field orientation with respect to the crystallographic axes. In this Communication we present the first such measurement for an Fe-based high-Tc superconductor (FeBSC). We observed a fourfold oscillation of the specific heat as a function of the in-plane magnetic field direction, which allowed us to identify the locations of the gap minima (or nodes) on the Fermi surface. Our results are consistent with the expectations of an extended s-wave model with a significant gap anisotropy on the electron pockets and the gap minima along the \Gamma M (or Fe-Fe bond) direction.Comment: 32 pages, 7 figure