Quantum Criticality in Condensed Matter

The following sections are included:

• PREFACE
• CONTENTS

The last few years have seen significant experimental progress in characterizing the copper-based hole-doped high temperature superconductors in the regime of low hole density, p. Quantum oscillations, NMR, X-ray, and STM experiments have shed much light on the nature of the ordering at low temperatures. We review evidence that the order parameter in the non-Lanthanum-based cuprates is a d-form factor density-wave. This novel order acts as an unexpected window into the electronic structure of the pseudogap phase at higher temperatures in zero field: we argue in favor of a ‘fractionalized Fermi liquid’ (FL*) with 4 pockets of spin S = 1/2, charge +e fermions enclosing an area specified by p.

The heavy quasiparticle bands in Kondo materials which originate in the hybridization of f- and conduction electrons exhibit numerous, sometimes coexisting, broken symmetry phases. Most notable are unconventional superconductivity, itinerant small moment antiferromagnetism and hidden order of higher order multipoles of f-electrons which all lead to a gapping of the heavy bands. In rare cases the chemical potential lies within the hybridization gap and the ground state is a Kondo semiconductor without ordering. The dynamical magnetic response of such gapped f-electron systems has been investigated with inelastic neutron scattering. It was found that collective spin exciton modes which are due to residual quasiparticle interactions appear below the threshold of superconducting or hidden order gap or directly the hybridzation gap . The spin exciton resonance is commonly located around a zone boundary vector Q with nesting properties in the normal state. In the superconducting case its appearance gives a strong criterion for the gap symmetry requesting a sign change Δ_k+Q = −Δ_k due to the coherence factors. Therefore this many body effect with fundamental importance may also be used as a tool to discriminate between proposed gap models. While the spin resonance has been observed for many compounds we restrict our discussion here exclusively to the small group of f-electron superconductors CeCoIn₅, CeCu₂Si₂ and UPd₂Al₃, hidden order Kondo compounds CeB₆ and URu₂Si₂ as well as the Kondo semiconductor YbB₁₂.

The evolution with magnetic fields of the heavy f-derived quasiparticles in YbRh₂Si₂ and the temperature dependence of the 4f spectral function in ambient field are calcualted by means of the Renormalized Band method. The Zeeman splitting of the Kondo resonance induces a series of Lifshitz transitions. The theoretically predicted critical magnetic fields agree quantitatively with the positions of anomalies in transport properties. To calculate the 4f spectral function of the periodic solid we use the single-impurity many-body t-matrix for the f-channels. We find that the dispersive heavy quasiparticle structures persist up to rather high temperatures in agreement with recent ARPES experiments. Implications for the interpretation of the observed “anomaly line” are discussed.

A common feature of magnetic impurity systems with strongly correlated electrons is the emergence of a unique low energy scale, the Kondo temperature T_K, such that the low energy thermodyamics and the response functions are universal functions of the temperature T and frequency ω in terms of T_K. We show that this type of universality occurs on the approach to the critical points in a two-impurity model with the emergence of a single energy scale T*, such that T* → 0 at the critical points. We also find this convergence of energy scales in infinite dimensional lattice models, such as the Hubbard and Hubbard–Holstein models on the approach to the metal to insulator transitions. This form of universality is consistent with the ω/T scaling as T* → 0 as observed in dynamic response functions in some non-Fermi–liquid heavy fermion systems near the quantum critical point.

I am concerned in these lectures with the breakdown of the particle concept in strongly correlated electron matter. I first show that the standard procedure for counting particles, namely Luttinger's theorem, breaks down anytime polelike excitations are replaced by ones that have a divergent self-energy. Such a breakdown obtains in electronic systems whose pole-like excitations do not extend to the edge of the Brillouin zone, as in Fermi arcs in the cuprates. Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, unparticles are the natural candidate to explain the presence of charged degrees of freedom that have no particle content. The continuous mass formulation of unparticles is recast as an action in anti de Sitter space. Such an action serves as the generating functional for the propagator. This mapping fixes the scaling dimension of the unparticle to be and ensures that the corresponding propagator has zeros with d the spacetime dimension of the unparticle field. The general dynamical mechanism by which bulk operators, such as the Pauli term, couple to the scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energy is reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicates that the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.

Fidelity approach to quantum phase transitions uses the overlap between ground states of the system to gain some information about its quantum phases. Such an overlap is called fidelity. We illustrate how this approach works in the one dimensional quantum Ising model in the transverse field. Several closedform analytical expressions for fidelity are discussed. An example of what insights fidelity provides into the dynamics of quantum phase transitions is carefully described. The role of fidelity in central spin systems is pointed out.

Ideally, in an adiabatic quantum computation or quantum state preparation, a simple ground state of an initial Hamiltonian is evolved adiabatically to an interesting ground state of a final Hamiltonian. Unfortunately, the simple and the interesting are often different enough to be separated by a quantum phase transition. Due to a vanishing energy gap between the ground state and excited states at the critical point, near the transition the adiabaticity is bound to fail. This failure is quantified by a quantum version of the Kibble-Zurek mechanism (KZM). In these notes I introduce KZM in its general textbook form, based on adiabatic/impulse approximation, and then support the picture by an exact solution of the integrable transverse field quantum Ising chain and an approximate one of the non-integrable Bose-Hubbard model. The last model illustrates typical problems with adiabatic quantum state preparation that are encoutered in atomic quantum simulators.

When a closed quantum system is driven out of equilibrium, expectation values of general observables start oscillating in time. This article reviews the general theory of such temporal fluctuations. We first survey some results on the strength of such temporal fluctuations. For example, temporal fluctuations are exponentially small in the system's volume for generic systems whereas they fall-off algebraically in integrable systems. We then concentrate on the so-called quench scenario where the system is driven out-of-equilibrium under the application of a sudden perturbation. For sufficiently small perturbations, temporal fluctuations of physical observables can be characterized in full generality and can be used as an effective tool to probe quantum criticality of the underlying model. In the off-critical region the distribution becomes Gaussian. Close to criticality the distribution becomes a universal function uniquely characterized by a single critical exponent, that we compute explicitly. This contrasts standard equilibrium quantum fluctuations for which the critical distribution depends on a numerable set of critical coefficients and is known only for limited examples.