Quantum spin liquids.

Spin liquids are quantum phases of matter with a variety of unusual features arising from their topological character, including "fractionalization"-elementary excitations that behave as fractions of an electron. Although there is not yet universally accepted experimental evidence that establishes that any single material has a spin liquid ground state, in the past few years a number of materials have been shown to exhibit distinctive properties that are expected of a quantum spin liquid. Here, we review theoretical and experimental progress in this area.

Observation of two types of charge-density-wave orders in superconducting La2-xSrxCuO4.

The discovery of charge- and spin-density-wave (CDW/SDW) orders in superconducting cuprates has altered our perspective on the nature of high-temperature superconductivity (SC). However, it has proven difficult to fully elucidate the relationship between the density wave orders and SC. Here, using resonant soft X-ray scattering, we study the archetypal cuprate La2-xSrxCuO4 (LSCO) over a broad doping range. We reveal the existence of two types of CDW orders in LSCO, namely CDW stripe order and CDW short-range order (SRO). While the CDW-SRO is suppressed by SC, it is partially transformed into the CDW stripe order with developing SDW stripe order near the superconducting Tc. These findings indicate that the stripe orders and SC are inhomogeneously distributed in the superconducting CuO2 planes of LSCO. This further suggests a new perspective on the putative pair-density-wave order that coexists with SC, SDW, and CDW orders.

Holon Wigner Crystal in a Lightly Doped Kagome Quantum Spin Liquid.

We address the problem of a lightly doped spin liquid through a large-scale density-matrix renormalization group study of the t-J model on a kagome lattice with a small but nonzero concentration δ of doped holes. It is now widely accepted that the undoped (δ=0) spin-1/2 Heisenberg antiferromagnet has a spin-liquid ground state. Theoretical arguments have been presented that light doping of such a spin liquid could give rise to a high temperature superconductor or an exotic topological Fermi liquid metal. Instead, we infer that the doped holes form an insulating charge-density wave state with one doped hole per unit cell, i.e., a Wigner crystal. Spin correlations remain short ranged, as in the spin-liquid parent state, from which we infer that the state is a crystal of spinless holons, rather than of holes. Our results may be relevant to kagome lattice herbertsmithite upon doping.

Ideal charge-density-wave order in the high-field state of superconducting YBCO.

The existence of charge-density-wave (CDW) correlations in cuprate superconductors has now been established. However, the nature of the CDW ground state has remained uncertain because disorder and the presence of superconductivity typically limit the CDW correlation lengths to only a dozen unit cells or less. Here we explore the field-induced 3D CDW correlations in extremely pure detwinned crystals of YBa2Cu3O2 (YBCO) ortho-II and ortho-VIII at magnetic fields in excess of the resistive upper critical field ([Formula: see text]) where superconductivity is heavily suppressed. We observe that the 3D CDW is unidirectional and possesses a long in-plane correlation length as well as significant correlations between neighboring CuO2 planes. It is significant that we observe only a single sharply defined transition at a critical field proportional to [Formula: see text], given that the field range used in this investigation overlaps with other high-field experiments including quantum oscillation measurements. The correlation volume is at least two to three orders of magnitude larger than that of the zero-field CDW. This is by far the largest CDW correlation volume observed in any cuprate crystal and so is presumably representative of the high-field ground state of an "ideal" disorder-free cuprate.

Three-dimensional charge density wave order in YBa2Cu3O6.67 at high magnetic fields.

Charge density wave (CDW) correlations have been shown to universally exist in cuprate superconductors. However, their nature at high fields inferred from nuclear magnetic resonance is distinct from that measured with x-ray scattering at zero and low fields. We combined a pulsed magnet with an x-ray free-electron laser to characterize the CDW in YBa2Cu3O6.67 via x-ray scattering in fields of up to 28 tesla. While the zero-field CDW order, which develops at temperatures below ~150 kelvin, is essentially two dimensional, at lower temperature and beyond 15 tesla, another three-dimensionally ordered CDW emerges. The field-induced CDW appears around the zero-field superconducting transition temperature; in contrast, the incommensurate in-plane ordering vector is field-independent. This implies that the two forms of CDW and high-temperature superconductivity are intimately linked.

One Hole in the Two-Leg t-J Ladder and Adiabatic Continuity to the Noninteracting Limit.

We carry out density-matrix-renormalization group (DMRG) calculations for the problem of one doped hole in a two-leg t-J ladder. Recent studies have concluded that exotic "Mott" physics-arising from the projection onto the space of no double-occupied sites-is manifest in this model system, leading to charge localization and a new mechanism for charge modulation. In contrast, we show that there is no localization and that the charge-density modulation arises when the minimum in the quasiparticle dispersion moves away from π. Although singular changes in the quasiparticle dispersion do occur as a function of model parameters, all of the DMRG results can be qualitatively understood from a noninteracting "band-structure" perspective.

Enhancement of superconductivity near a nematic quantum critical point.

We consider a low T_{c} metallic superconductor weakly coupled to the soft fluctuations associated with proximity to a nematic quantum critical point (NQCP). We show that (1) a BCS-Eliashberg treatment remains valid outside of a parametrically narrow interval about the NQCP, (2) the symmetry of the superconducting state (d wave, s wave, p wave) is typically determined by the noncritical interactions, but T_{c} is enhanced by the nematic fluctuations in all channels, and (3) in 2D, this enhancement grows upon approach to criticality up to the point at which the weak coupling approach breaks down, but in 3D, the enhancement is much weaker.

From quantum matter to high-temperature superconductivity in copper oxides.

The discovery of high-temperature superconductivity in the copper oxides in 1986 triggered a huge amount of innovative scientific inquiry. In the almost three decades since, much has been learned about the novel forms of quantum matter that are exhibited in these strongly correlated electron systems. A qualitative understanding of the nature of the superconducting state itself has been achieved. However, unresolved issues include the astonishing complexity of the phase diagram, the unprecedented prominence of various forms of collective fluctuations, and the simplicity and insensitivity to material details of the 'normal' state at elevated temperatures.

Microscopic model of quasiparticle wave packets in superfluids, superconductors, and paired Hall states.

We study the structure of Bogoliubov quasiparticles, bogolons, the fermionic excitations of paired superfluids that arise from fermion (BCS) pairing, including neutral superfluids, superconductors, and paired quantum Hall states. The naive construction of a stationary quasiparticle in which the deformation of the pair field is neglected leads to a contradiction: it carries a net electrical current even though it does not move. However, treating the pair field self-consistently resolves this problem: in a neutral superfluid, a dipolar current pattern is associated with the quasiparticle for which the total current vanishes. When Maxwell electrodynamics is included, as appropriate to a superconductor, this pattern is confined over a penetration depth. For paired quantum Hall states of composite fermions, the Maxwell term is replaced by a Chern-Simons term, which leads to a dipolar charge distribution and consequently to a dipolar current pattern.

Weakly coupled Pfaffian as a type I quantum Hall liquid.

The Pfaffian phase in the proximity of a half-filled Landau level is understood to be a p+ip superconductor of composite fermions. We consider the properties of this paired quantum Hall phase when the pairing energy is small, i.e., in the weak-coupling, BCS limit, where the coherence length is much larger than the charge screening length. We find that, as in a type I superconductor, vortices attract so that, upon varying the magnetic field from its magic value at ν=5/2, the system exhibits Coulomb frustrated phase separation. We propose that the weakly and strongly coupled Pfaffians exemplify a general dichotomy between type I and type II quantum Hall fluids.

From a single-band metal to a high-temperature superconductor via two thermal phase transitions.

The nature of the pseudogap phase of cuprate high-temperature superconductors is a major unsolved problem in condensed matter physics. We studied the commencement of the pseudogap state at temperature T* using three different techniques (angle-resolved photoemission spectroscopy, polar Kerr effect, and time-resolved reflectivity) on the same optimally doped Bi2201 crystals. We observed the coincident, abrupt onset at T* of a particle-hole asymmetric antinodal gap in the electronic spectrum, a Kerr rotation in the reflected light polarization, and a change in the ultrafast relaxational dynamics, consistent with a phase transition. Upon further cooling, spectroscopic signatures of superconductivity begin to grow close to the superconducting transition temperature (T(c)), entangled in an energy-momentum-dependent manner with the preexisting pseudogap features, ushering in a ground state with coexisting orders.

Hidden quasi-one-dimensional superconductivity in Sr2RuO4.

We show that the interplay between spin and charge fluctuations in Sr2RuO4 leads unequivocally to triplet pairing which has a hidden quasi-one-dimensional character. The resulting superconducting state spontaneously breaks time-reversal symmetry and is of the form Δ ~(p(x)+ip(y))z(^) with sharp gap minima and a d vector that is only weakly pinned. The superconductor lacks robust chiral Majorana fermion modes along the boundary. The absence of topologically protected edge modes could explain the surprising absence of experimentally detectable edge currents in this system.

Polar Kerr-effect measurements of the high-temperature YBa2Cu3O6+x superconductor: evidence for broken symmetry near the pseudogap temperature.

The polar Kerr effect in the high-T_(c) superconductor YBa2Cu3O6+x was measured at zero magnetic field with high precision using a cyogenic Sagnac fiber interferometer. We observed nonzero Kerr rotations of order approximately 1 microrad appearing near the pseudogap temperature T(*) and marking what appears to be a true phase transition. Anomalous magnetic behavior in magnetic-field training of the effect suggests that time reversal symmetry is already broken above room temperature.

Stability of nodal quasiparticles in superconductors with coexisting orders.

We establish a condition for the perturbative stability of zero energy nodal points in the quasiparticle spectrum of superconductors in the presence of coexisting commensurate orders. The nodes are found to be stable if the Hamiltonian is invariant under time reversal followed by a lattice translation. The principle is demonstrated with a few examples. Some experimental implications of various types of assumed order are discussed in the context of the cuprate superconductors.

Dynamical layer decoupling in a stripe-ordered high-T(c) superconductor.

In the stripe-ordered state of a strongly correlated two-dimensional electronic system, under a set of special circumstances, the superconducting condensate, like the magnetic order, can occur at a nonzero wave vector corresponding to a spatial period double that of the charge order. In this case, the Josephson coupling between near neighbor planes, especially in a crystal with the special structure of La(2-x)Ba(x)CuO(4), vanishes identically. We propose that this is the underlying cause of the dynamical decoupling of the layers recently observed in transport measurements at x = 1/8.

Hysteresis and noise from electronic nematicity in high-temperature superconductors.

An electron nematic is a translationally invariant state which spontaneously breaks the discrete rotational symmetry of a host crystal. In a clean square lattice, the electron nematic has two preferred orientations, while dopant disorder favors one or the other orientations locally. In this way, the electron nematic in a host crystal maps to the random field Ising model. Since the electron nematic has anisotropic conductivity, we associate each Ising configuration with a resistor network and use what is known about the random field Ising model to predict new ways to test for local electronic nematic order (nematicity) using noise and hysteresis. In particular, we have uncovered a remarkably robust linear relation between the orientational order and the resistance anisotropy which holds over a wide range of circumstances.

Gap-inhomogeneity-induced electronic states in superconducting Bi2Sr2CaCu2O(8+delta).

In this Letter, we analyze, using scanning tunneling spectroscopy, the density of electronic states in nearly optimally doped Bi2Sr2CaCu2O(8+delta) in zero magnetic field. Focusing on the superconducting gap, we find patches of what appear to be two different phases in a background of some average gap, one with a relatively small gap and sharp large coherence peaks and one characterized by a large gap with broad weak coherence peaks. We compare these spectra with calculations of the local density of states for a simple phenomenological model in which a 2xi0 x 2xi0 patch with an enhanced or suppressed d-wave gap amplitude is embedded in a region with a uniform average d-wave gap.

Nodal quasiparticles in stripe ordered superconductors.

We study the properties of a quasi-one-dimensional superconductor which consists of an alternating array of two inequivalent chains. This model is a simple caricature of a striped high temperature superconductor, and is more generally a theoretically controllable system in which the superconducting state emerges from a non-Fermi-liquid normal state. Even in this limit, " d-wave-like" order parameter symmetry is natural, but the superconducting state can either have a complete gap in the quasiparticle spectrum, or gapless "nodal" quasiparticles. We also find circumstances in which antiferromagnetic order (typically incommensurate) coexists with superconductivity.

Thermodynamics of the interplay between magnetism and high-temperature superconductivity.

Copper-oxide-based high-temperature superconductors have complex phase diagrams with multiple ordered phases. It even appears that the highest superconducting transition temperatures for certain cuprates are found in samples that display simultaneous onset of magnetism and superconductivity. We show here how the thermodynamics of fluid mixtures-a touchstone for chemistry as well as hard and soft condensed matter physics-accounts for this startling observation, as well as many other properties of the cuprates in the vicinity of the instability toward "striped" magnetism.