This is an introductory review of the physics of quantum spin liquid states. Quantum magnetism is a rapidly evolving field, and recent developments reveal that the ground states and low-energy physics of frustrated spin systems may develop many exotic behaviors once we leave the regime of semiclassical approaches. The purpose of this article is to introduce these developments. The article begins by explaining how semiclassical approaches fail once quantum mechanics become important and then describe the alternative approaches for addressing the problem. Mainly spin-1/2 systems are discussed, and most of the time is spent in this article on one particular set of plausible spin liquid states in which spins are represented by fermions. These states are spin-singlet states and may be viewed as an extension of Fermi liquid states to Mott insulators, and they are usually classified in the category of so-called SU(2), U(1), or Z2 spin liquid states. A review is given of the basic theory regarding these states and the extensions of these states to include the effect of spin-orbit coupling and to higher spin (S>1/2) systems. Two other important approaches with strong influences on the understanding of spin liquid states are also introduced: (i) matrix product states and projected entangled pair states and (ii) the Kitaev honeycomb model. Experimental progress concerning spin liquid states in realistic materials, including anisotropic triangular-lattice systems [κ-(ET)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2], kagome-lattice system [ZnCu3(OH)6Cl2], and hyperkagome lattice system (Na4Ir3O8), is reviewed and compared against the corresponding theories.

Quantum spin liquid states

We review studies of the electromagnetic response of various classes of correlated electron materials including transition metal oxides, organic and molecular conductors, intermetallic compounds with $d$- and $f$-electrons as well as magnetic semiconductors. Optical inquiry into correlations in all these diverse systems is enabled by experimental access to the fundamental characteristics of an ensemble of electrons including their self-energy and kinetic energy. Steady-state spectroscopy carried out over a broad range of frequencies from microwaves to UV light and fast optics time-resolved techniques provide complimentary prospectives on correlations. Because the theoretical understanding of strong correlations is still evolving, the review is focused on the analysis of the universal trends that are emerging out of a large body of experimental data augmented where possible with insights from numerical studies.

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Electrodynamics of correlated electron materials

Transport of electrons in a single molecule junction is the simplest problem in the general subject area of molecular electronics. In the past few years, this area has been extended to probe beyond the simple tunnelling associated with large energy gaps between electrode Fermi level and molecular levels, to deal with smaller gaps, with near-resonance tunnelling and, particularly, with effects due to interaction of electronic and vibrational degrees of freedom. This overview is devoted to the theoretical and computational approaches that have been taken to understanding transport in molecular junctions when these vibronic interactions are involved. After a short experimental overview, and discussion of different test beds and measurements, we define a particular microscopic model Hamiltonian. That overall Hamiltonian can be used to discuss all of the phenomena dealt with subsequently. These include transition from coherent to incoherent transport as electron/vibration interaction increases in strength, inelastic electron tunnelling spectroscopy and its interpretation and measurement, affects of interelectronic repulsion treated at the Hubbard level, noise in molecular transport junctions, non-linear conductance phenomena, heating and heat conduction in molecular transport junctions and current-induced chemical reactions. In each of these areas, we use the same simple model Hamiltonian to analyse energetics and dynamics. While this overview does not attempt survey the literature exhaustively, it does provide appropriate references to the current literature (both experimental and theoretical). We also attempt to point out directions in which further research is required to answer cardinal questions concerning the behaviour and understanding of vibrational effects in molecular transport junctions. (Some figures in this article are in colour only in the electronic version)

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Molecular transport junctions: vibrational effects

We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The ``algorithmic discontinuities'' between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic $S=1/2$ Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the $\mathrm{XY}$ model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For $L\ifmmode\times\else\texttimes\fi{}L$ lattices with L up to $64,$ we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: ${\ensuremath{\chi}}_{\ensuremath{\perp}}=0.0659\ifmmode\pm\else\textpm\fi{}0.0002.$

Quantum Monte Carlo with directed loops.

This article reviews the static and dynamic properties of spontaneous superstructures formed by electrons. Representations of such electronic crystals are charge density waves (CDW) and spin density waves in inorganic as well as organic low-dimensional materials. A special attention is paid to the collective effects in pinning and sliding of these superstructures, and the glassy properties at low temperature. Charge order and charge disproportionation which occur in organic materials resulting from correlation effects are analysed. Experiments under magnetic field, and more specifically field-induced CDWs are discussed. Properties of meso-and nanostructures of CDWs are also reviewed.

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Electronic crystals: an experimental overview

We present a study of the effects of simultaneous charge and spin frustration on the two-dimensional strongly correlated quarter-filled band on an anisotropic triangular lattice. Our conclusions are based on exact diagonalization studies that include electron-electron interactions as well as adiabatic electron-phonon coupling terms treated self-consistently. The broken-symmetry states that dominate in the weakly frustrated region near the rectangular lattice limit are the well known antiferromagnetic state with in-phase lattice dimerization along one direction and the Wigner crystal state with the checkerboard charge order. For moderate to strong frustration, however, the dominant phase is a novel spin-singlet paired-electron crystal (PEC), consisting of pairs of charge-rich sites separated by pairs of charge-poor sites. The PEC, with coexisting charge order and spin gap in two dimensions, is the quarter-filled band equivalent of the valence bond solid (VBS) that can appear in the frustrated half-filled band within antiferromagnetic spin Hamiltonians. We discuss the phase diagram as a function of onsite and intersite Coulomb interactions as well as electron-phonon coupling strength. We speculate that the spin-bonded pairs of the PEC can become mobile for even stronger frustration, giving rise to a paired-electron liquid. We discuss the implications of the PEC concept for understanding several classes of quarter-filled band materials that display unconventional superconductivity, focusing, in particular, on organic charge transfer solids. Our work points out the need to go beyond quantum spin liquid concepts for highly frustrated organic charge-transfer solids such as $\ensuremath{\kappa}$-(BEDT-TTF)${}_{2}$Cu${}_{2}$(CN)${}_{3}$ and EtMe${}_{3}$Sb[Pd(dmit)${}_{2}$]${}_{2}$, which we believe show frustration-induced charge disproportionation at low temperatures. We discuss possible application to layered cobaltates and quarter-filled band spinels.

https://link.aps.org/accepted/10.1103/PhysRevB.83.245106

Paired electron crystal: Order from frustration in the quarter-filled band

We present a detailed numerical study of the Hubbard-Holstein model in one dimension at half filling, including full finite-frequency quantum phonons. At half filling, the effects of the electronphonon and electron-electron interactions compete, with the Holstein phonon coupling acting as an effective negative Hubbard onsite interaction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or U > 0 Mott insulator phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We present results confirming the intermediate metallic phase, and show that the Luttinger liquid correlation exponent K� > 1 in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of onsite Hubbard U, phonon coupling constant, and phonon frequency. Electron-phonon (e-ph) interactions can give rise to a number of interesting effects in low-dimensional materials, including superconductivity as well as charge-density wave and insulating phenomena. Frequently these materials feature strong electron-electron (e-e) interactions as well, leading to very rich phase diagrams that combine lattice, charge, and spin (magnetic) orderings. We will focus specifically on materials where the electrons are coupled to localized vibrational modes, which may be of relatively high frequency. This type of e-ph interaction is most studied in molecular crystal materials, including the quasi-one- and quasi-two-dimensional organic superconductors[1] and fullerene superconductors [2]. In all of these materials, a fundamental question is whether the effects of e-e and e-ph interactions compete or cooperate with each other. In this Letter, we examine this issue within one of the most basic models. We find that despite the two interactions each separately favoring insulating states, together they can mediate an unexpected metallic phase with superconducting (SC) pair correlations. The model we consider is the one-dimensional (1D) Hubbard-Holstein model (HHM), with Hamiltonian H = −t X j,� (c †+1,� cj,� + h.c.) + U X j nj,"nj,# (1)

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Intermediate phase of the one dimensional half-filled Hubbard-Holstein model.

We examine two recently proposed models of charge ordering (CO) in the nominally $\frac{1}{4}$-filled, quasi-one-dimensional (1D) organic charge-transfer solids (CTS). The two models are characterized by site charge density ``cartoons'' $\dots{}1010\dots{}$ and $\dots{}1100\dots{},$ respectively. We use the Peierls-extended Hubbard model to incorporate both electron-electron $(e\ensuremath{-}e)$ and electron-phonon $(e\ensuremath{-}ph)$ interactions. We first compare the results, for the purely electronic Hamiltonian, of exact many-body calculations with those of Hartree-Fock (HF) mean-field theory. We find that HF gives qualitatively and quantitatively incorrect values for the critical nearest-neighbor Coulomb repulsion ${(V}_{c})$ necessary for $\dots{}1010\dots{}$ order to become the ground state. Second, we establish that spin-Peierls order can occur in either the $\dots{}1100\dots{}$ and $\dots{}1010\dots{}$ states and calculate the phase diagram including both on-site and intrasite $e\ensuremath{-}\mathrm{ph}$ interactions. Third, we discuss the expected temperature dependence of the CO and metal-insulator transitions for both $\dots{}1010\dots{}$ and $\dots{}1100\dots{}$ CO states. Finally, we show that experimental observations clearly indicate the $\dots{}1100\dots{}$ CO in the 1:2 anionic CTS and the $(\mathrm{TMTSF}{)}_{2}X$ materials, while the results for $(\mathrm{TMTTF}{)}_{2}X$ with narrower one-electron bandwidths are more ambiguous, likely because the nearest-neighbor Coulomb interaction in these materials is near ${V}_{c}.$

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Pattern of charge ordering in quasi-one-dimensional organic charge-transfer solids

We report exact calculations of magnetic and superconducting pair-pair correlations for the half-filled band Hubbard model on an anisotropic triangular lattice. Our results for the magnetic phases are similar to those obtained with other techniques. The superconducting pair-pair correlations at distances beyond nearest neighbor decrease monotonically with increasing Hubbard interaction U for all anisotropy, indicating the absence of frustration-driven superconductivity within the model.

Absence of superconductivity in the half-filled band Hubbard model on the anisotropic triangular lattice.

We investigate theoretically charge ordered states on the anisotropic triangular lattice characteristic of the theta-(BEDT-TTF)_2 X materials. Using exact diagonalization studies, we establish that the charge order (CO) pattern corresponds to a ``horizontal'' stripe structure, with ...1100... CO along the two directions with larger electron hopping (p-directions), and ...1010... CO along the third direction (c-direction). The CO is accompanied by co-operative bond dimerizations along all three directions in the highest spin state. In the lowest spin state bonds along the p-directions are tetramerized. Our theory explains the occurence of a charge-induced high temperature transition as well as a spin gap transition at lower temperature.

R. T. Clay

Papers

Paired electron crystal: Order from frustration in the quarter-filled band

Intermediate phase of the one dimensional half-filled Hubbard-Holstein model.

Pattern of charge ordering in quasi-one-dimensional organic charge-transfer solids

Absence of superconductivity in the half-filled band Hubbard model on the anisotropic triangular lattice.

Charge ordering in theta-(BEDT-TTF)_2 X materials