Dynamics and transport of the Z2 spin liquid: application to kappa-(ET)2Cu2(CN)3

Signatures of Fractional Statistics in Nonlinear Pump-Probe Spectroscopy

We show that the presence of anyons in the excitation spectrum of a two-dimensional system can be inferred from nonlinear spectroscopic quantities. In particular, we consider pump-probe spectroscopy, where a sample is irradiated by two light pulses with an adjustable time delay between them. The relevant response coefficient exhibits a universal form that originates from the statistical phase acquired when anyons created by the first pulse braid around those created by the second. This behavior is shown to be qualitatively unchanged by nonuniversal physics including nonstatistical interactions and small nonzero temperatures. In magnetic systems, the signal of interest can be measured using currently available terahertz-domain probes, highlighting the potential usefulness of nonlinear spectroscopic techniques in the search for quantum spin liquids.

Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface

The Kitaev model with an applied magnetic field in the [Formula: see text] direction shows two transitions: from a nonabelian gapped quantum spin liquid (QSL) to a gapless QSL at [Formula: see text] and a second transition at a higher field [Formula: see text] to a gapped partially polarized phase, where K is the strength of the Kitaev exchange interaction. We identify the intermediate phase to be a gapless U(1) QSL and determine the spin structure function [Formula: see text] and the Fermi surface [Formula: see text] of the gapless spinons using the density matrix renormalization group (DMRG) method for large honeycomb clusters. Further calculations of static spin-spin correlations, magnetization, spin susceptibility, and finite temperature-specific heat and entropy corroborate the gapped and gapless nature of the different field-dependent phases. In the intermediate phase, the spin-spin correlations decay as a power law with distance, indicative of a gapless phase.

Critical spin liquid versus valence-bond glass in a triangular-lattice organic antiferromagnet

In the quest for materials with unconventional quantum phases, the organic triangular-lattice antiferromagnet κ-(ET)2Cu2(CN)3 has been extensively discussed as a quantum spin liquid (QSL) candidate. The description of its low temperature properties has become, however, a particularly challenging task. Recently, an intriguing quantum critical behaviour was suggested from low-temperature magnetic torque experiments. Here we highlight significant deviations of the experimental observations from a quantum critical scenario by performing a microscopic analysis of all anisotropic contributions, including Dzyaloshinskii-Moriya and multi-spin scalar chiral interactions. Instead, we show that disorder-induced spin defects provide a comprehensive explanation of the low-temperature properties. These spins are attributed to valence bond defects that emerge spontaneously as the QSL enters a valence-bond glass phase at low temperature. This theoretical treatment is applicable to a general class of frustrated magnetic systems and has important implications for the interpretation of magnetic torque, nuclear magnetic resonance, thermal transport and thermodynamic experiments.

Low-Energy Excitations in Quantum Spin Liquids Identified by Optical Spectroscopy

The electrodynamic response of organic spin liquids with highly frustrated triangular lattices has been measured in a wide energy range. While the overall optical spectra of these Mott insulators are governed by transitions between the Hubbard bands, distinct in-gap excitations can be identified at low temperatures and frequencies, which we attribute to the quantum-spin-liquid state. For the strongly correlated β^{'}-EtMe_{3}Sb[Pd(dmit)_{2}]_{2}, we discover enhanced conductivity below 175  cm^{-1}, comparable to the energy of the magnetic coupling J≈250  K. For ω→0, these low-frequency excitations vanish faster than the charge-carrier response subject to Mott-Hubbard correlations, resulting in a dome-shaped band peaked at 100  cm^{-1}. Possible relations to spinons, magnons, and disorder are discussed.

Electrodynamics of quantum spin liquids

Quantum spin liquids attract great interest due to their exceptional magnetic properties characterized by the absence of long-range order down to low temperatures despite the strong magnetic interaction. Commonly, these compounds are strongly correlated electron systems, and their electrodynamic response is governed by the Mott gap in the excitation spectrum. Here we summarize and discuss the optical properties of several two-dimensional quantum spin liquid candidates. First we consider the inorganic material herbertsmithite ZnCu₃(OH)₆Cl₂ and related compounds, which crystallize in a kagome lattice. Then we turn to the organic compounds [Formula: see text]-EtMe₃Sb[Pd(dmit)₂]₂, κ-(BEDT-TTF)₂Ag₂(CN)₃ and κ-(BEDT-TTF)₂Cu₂(CN)₃, where the spins are arranged in an almost perfect triangular lattice, leading to strong frustration. Due to differences in bandwidth, the effective correlation strength varies over a wide range, leading to a rather distinct behavior as far as the electrodynamic properties are concerned. We discuss the spinon contributions to the optical conductivity in comparison to metallic quantum fluctuations in the vicinity of the Mott transition.

Spin-lattice decoupling in a triangular-lattice quantum spin liquid

A quantum spin liquid (QSL) is an exotic state of matter in condensed-matter systems, where the electron spins are strongly correlated, but conventional magnetic orders are suppressed down to zero temperature because of strong quantum fluctuations. One of the most prominent features of a QSL is the presence of fractionalized spin excitations, called spinons. Despite extensive studies, the nature of the spinons is still highly controversial. Here we report magnetocaloric-effect measurements on an organic spin-1/2 triangular-lattice antiferromagnet, showing that electron spins are decoupled from a lattice in a QSL state. The decoupling phenomena support the gapless nature of spin excitations. We further find that as a magnetic field is applied away from a quantum critical point, the number of spin states that interact with lattice vibrations is strongly reduced, leading to weak spin–lattice coupling. The results are compared with a model of a strongly correlated QSL near a quantum critical point.

A number of materials have been proposed as realizations of exotic quantum spin liquids but many important properties are difficult to establish. Isono et al. show evidence for spin-lattice decoupling in an organic material, which may help resolve conflicting results about the existence of a spin-excitation gap.

Unusual Phonon Heat Transport in α-RuCl_{3}: Strong Spin-Phonon Scattering and Field-Induced Spin Gap

The honeycomb Kitaev-Heisenberg model is a source of a quantum spin liquid with Majorana fermions and gauge flux excitations as fractional quasiparticles. Here we unveil the highly unusual low-temperature heat conductivity κ of α-RuCl_{3}, a prime candidate for realizing such physics: beyond a magnetic field of B_{c}≈7.5  T, κ increases by about one order of magnitude, both for in-plane as well as out-of-plane transport. This clarifies the unusual magnetic field dependence unambiguously to be the result of severe scattering of phonons off putative Kitaev-Heisenberg excitations in combination with a drastic field-induced change of the magnetic excitation spectrum. In particular, an unexpected, large energy gap arises, which increases linearly with the magnetic field, reaching remarkable ℏω_{0}/k_{B}≈50  K at 18 T.

Randomness-induced quantum spin liquid on honeycomb lattice

Quantum entanglement in magnetic materials is expected to yield a quantum spin liquid (QSL), in which strong quantum fluctuations prevent magnetic ordering even at zero temperature. This topic has been one of the primary focuses of condensed-matter science since Anderson first proposed the resonating valence bond state in a certain spin-1/2 frustrated magnet in 1973. Since then, several candidate materials featuring frustration, such as triangular and kagome lattices, have been reported to exhibit liquid-like behavior. However, the mechanisms that stabilize the liquid-like states have remained elusive. Here, we present a QSL state in a spin-1/2 honeycomb lattice with randomness in the exchange interaction. That is, we successfully introduce randomness into the organic radial-based complex and realize a random-singlet (RS) state (or valence bond glass). All magnetic and thermodynamic experimental results indicate the liquid-like behaviors, which are consistent with those expected in the RS state. Our results suggest that the randomness or inhomogeneity in the actual systems stabilize the RS state and yield liquid-like behavior.

Anomalous metallic behaviour in the doped spin liquid candidate κ-(ET)4Hg2.89Br8

Quantum spin liquids are exotic Mott insulators that carry extraordinary spin excitations. Therefore, when doped, they are expected to afford metallic states with unconventional magnetic excitations. Here, we report experimental results which are suggestive of a doped spin liquid with anomalous metallicity in a triangular-lattice organic conductor. The spin susceptibility is nearly perfectly scaled to that of a non-doped spin liquid insulator in spite of the metallic state. Furthermore, the charge transport that is confined in the layer at high temperatures becomes sharply deconfined on cooling, coinciding with the rapid growth of spin correlations or coherence as signified by a steep decrease in spin susceptibility. The present results substantiate the desired doped spin liquid and suggest a strange metal, in which the coherence of the underlying spin liquid promotes the deconfinement of charge from the layers while preserving the non-Fermi-liquid nature.

It is expected that introducing charge carriers into an exotic quantum spin liquid state may lead to an unconventional metal but there are no clear realizations of a metallic spin liquid. Here, the authors present a spin liquid candidate that also shows evidence of strange metal behavior.

Dynamical Reduction of the Dimensionality of Exchange Interactions and the "Spin-Liquid" Phase of κ-(BEDT-TTF)_{2}X

We show that the anisotropy of the effective spin model for the dimer Mott insulator phase of κ-(BEDT-TTF)_{2}X salts is dramatically different from that of the underlying tight-binding model. Intradimer quantum interference results in a model of coupled spin chains, where frustrated interchain interactions suppress long-range magnetic order. Thus, we argue, the "spin liquid" phase observed in some of these materials is a remnant of the Tomonaga-Luttinger physics of a single chain. This is consistent with previous experiments and resolves some outstanding puzzles.

Quantum criticality in an organic spin-liquid insulator κ-(BEDT-TTF)2Cu2(CN)3

A quantum spin-liquid state, an exotic state of matter, appears when strong quantum fluctuations enhanced by competing exchange interactions suppress a magnetically ordered state. Generally, when an ordered state is continuously suppressed to 0 K by an external parameter, a quantum phase transition occurs. It exhibits critical scaling behaviour, characterized only by a few basic properties such as dimensions and symmetry. Here we report the low-temperature magnetic torque measurements in an organic triangular-lattice antiferromagnet, κ-(BEDT-TTF)₂Cu₂(CN)₃, where BEDT-TTF stands for bis(ethylenedithio)tetrathiafulvalene. It is found that the magnetic susceptibilities derived from the torque data exhibit a universal critical scaling, indicating the quantum critical point at zero magnetic field, and the critical exponents, γ=0.83(6) and νz=1.0(1). These exponents greatly constrain the theoretical models for the quantum spin liquid, and at present, there is no theory to explain the values, to the best of our knowledge.

Quantum spin liquids emerge when quantum fluctuations suppress a magnetically ordered state. Here the authors measure magnetic torque in κ-(BEDT-TTF)₂Cu₂(CN)₃, showing universal critical scaling in the magnetic susceptibilities, with critical exponents incompatible with known models for quantum spin liquids.

Pressure-Tuned Exchange Coupling of a Quantum Spin Liquid in the Molecular Triangular Lattice κ-(ET)_{2}Ag_{2}(CN)_{3}

The effects of pressure on a quantum spin liquid are investigated in an organic Mott insulator κ-(ET)_{2}Ag_{2}(CN)_{3} with a spin-1/2 triangular lattice. The application of negative chemical pressure to κ-(ET)_{2}Cu_{2}(CN)_{3}, which is a well-known sister Mott insulator, allows for extensive tuning of antiferromagnetic exchange coupling, with J/k_{B}=175-310  K, under hydrostatic pressure. Based on ^{13}C nuclear magnetic resonance measurements under pressure, we uncover universal scaling in the static and dynamic spin susceptibilities down to low temperatures ∼0.1k_{B}T/J. The persistent fluctuations and residual specific heat coefficient are consistent with the presence of gapless low-lying excitations. Our results thus demonstrate the fundamental finite-temperature properties of a quantum spin liquid in a wide parameter range.

Probing the Mott physics in κ-(BEDT-TTF)₂X salts via thermal expansion

In the field of interacting electron systems the Mott metal-to-insulator (MI) transition represents one of the pivotal issues. The role played by lattice degrees of freedom for the Mott MI transition and the Mott criticality in a variety of materials are current topics under debate. In this context, molecular conductors of the κ-(BEDT-TTF)2X type constitute a class of materials for unraveling several aspects of the Mott physics. In this review, we present a synopsis of literature results with focus on recent expansivity measurements probing the Mott MI transition in this class of materials. Progress in the description of the Mott critical behavior is also addressed.

Destroying a topological quantum bit by condensing Ising vortices

The imminent realization of topologically protected qubits in fabricated systems will provide not only an elementary implementation of fault-tolerant quantum computing architecture, but also an experimental vehicle for the general study of topological order. The simplest topological qubit harbours what is known as a Z2 liquid phase, which encodes information via a degeneracy depending on the system's topology. Elementary excitations of the phase are fractionally charged objects called spinons, or Ising flux vortices called visons. At zero temperature, a Z2 liquid is stable under deformations of the Hamiltonian until spinon or vison condensation induces a quantum-phase transition destroying the topological order. Here we use quantum Monte Carlo to study a vison-induced transition from a Z2 liquid to a valence-bond solid in a quantum dimer model on the kagome lattice. Our results indicate that this critical point is beyond the description of the standard Landau paradigm.

A quantum fidelity study of the anisotropic next-nearest-neighbour triangular lattice Heisenberg model

Ground- and excited-state quantum fidelities in combination with generalized quantum fidelity susceptibilites, obtained from exact diagonalizations, are used to explore the phase diagram of the anisotropic next-nearest-neighbour triangular Heisenberg model. Specifically, the J'-J2 plane of this model, which connects the J1-J2 chain and the anisotropic triangular lattice Heisenberg model, is explored using these quantities. Through the use of a quantum fidelity associated with the first excited-state, in addition to the conventional ground-state fidelity, the BKT-type transition and Majumdar-Ghosh point of the J1-J2 chain (J'=0) are found to extend into the J'-J2 plane and connect with points on the J2=0 axis thereby forming bounded regions in the phase diagram. These bounded regions are then explored through the generalized quantum fidelity susceptibilities χρ, χ₁₂₀°, χD and χCAF which are associated with the spin stiffness, 120° spiral order parameter, dimer order parameter and collinear antiferromagnetic order parameter respectively. These quantities are believed to be extremely sensitive to the underlying phase and are thus well suited for finite-size studies. Analysis of the fidelity susceptibilities suggests that the J', J2≪J phase of the anisotropic triangular model is either a collinear antiferromagnet or possibly a gapless disordered phase that is directly connected to the Luttinger phase of the J1-J2 chain. Furthermore, the outer region is dominated by incommensurate spiral physics as well as dimer order.

Gapless quantum spin liquid in an organic spin-1/2 triangular-lattice κ-H3(Cat-EDT-TTF)2

We report the results of SQUID and torque magnetometry of an organic spin-1/2 triangular-lattice κ-H(3)(Cat-EDT-TTF)(2). Despite antiferromagnetic exchange coupling at 80-100 K, we observed no sign of antiferromagnetic order down to 50 mK owing to spin frustration on the triangular lattice. In addition, we found nearly temperature-independent susceptibility below 3 K associated with Pauli paramagnetism. These observations suggest the development of gapless quantum spin liquid as the ground state. On the basis of a comparative discussion, we point out that the gapless quantum spin liquid states in organic systems share a possible mechanism, namely the formation of a band with a Fermi surface possibly attributed to spinons.

Theory of a competitive spin liquid state for weak Mott insulators on the triangular lattice

We propose a novel quantum spin liquid state that can explain many of the intriguing experimental properties of the low-temperature phase of the organic spin liquid candidate materials κ-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2. This state of paired fermionic spinons preserves all symmetries of the system, and it has a gapless excitation spectrum with quadratic bands that touch at momentum k[over →]=0. This quadratic band touching is protected by symmetries. Using variational Monte Carlo techniques, we show that this state has highly competitive energy in the triangular lattice Heisenberg model supplemented with a realistically large ring-exchange term.

Novel Pauli-paramagnetic quantum phase in a Mott insulator

In Mott insulators, the strong electron-electron Coulomb repulsion localizes electrons. In dimensions greater than one, their spins are usually ordered antiferromagnetically at low temperatures. Geometrical frustrations can destroy this long-range order, leading to exotic quantum spin liquid states. However, their magnetic ground states have been a long-standing mystery. Here we show that a quantum spin liquid state in the organic Mott insulator EtMe(3)Sb[Pd(dmit)(2)](2) (where Et is C(2)H(5)-, Me is CH(3)-, and dmit is 1,3-dithiole-2-thione-4,5-dithiolate) with two-dimensional triangular lattice has Pauli-paramagnetic-like low-energy excitations, which are a hallmark of itinerant fermions. Our torque magnetometry down to low temperatures (30 mK) up to high fields (32 T) reveals distinct residual paramagnetic susceptibility comparable to that in a half-filled two-dimensional metal, demonstrating the magnetically gapless nature of the ground state. Moreover, our results are robust against deuteration, pointing toward the emergence of an extended 'quantum critical phase', in which low-energy spin excitations behave as in paramagnetic metals with Fermi surface, despite the frozen charge degree of freedom.

Successive phase transitions and extended spin-excitation continuum in the S=1/2 triangular-lattice antiferromagnet Ba3CoSb2O9

Using magnetic, thermal, and neutron measurements on single-crystal samples, we show that Ba3CoSb2O9 is a spin-1/2 triangular-lattice antiferromagnet with the c axis as the magnetic easy axis and two magnetic phase transitions bracketing an intermediate up-up-down phase in magnetic field applied along the c axis. A pronounced extensive neutron-scattering continuum above spin-wave excitations, observed below T(N), implies that the system is in close proximity to one of two spin-liquid states that have been predicted for a 2D triangular lattice.

Highly mobile gapless excitations in a two-dimensional candidate quantum spin liquid

The nature of quantum spin liquids, a novel state of matter where strong quantum fluctuations destroy the long-range magnetic order even at zero temperature, is a long-standing issue in physics. We measured the low-temperature thermal conductivity of the recently discovered quantum spin liquid candidate, the organic insulator EtMe3Sb[Pd(dmit)2]2. A sizable linear temperature dependence term is clearly resolved in the zero-temperature limit, indicating the presence of gapless excitations with an extremely long mean free path, analogous to excitations near the Fermi surface in pure metals. Its magnetic field dependence suggests a concomitant appearance of spin-gap-like excitations at low temperatures. These findings expose a highly unusual dichotomy that characterizes the low-energy physics of this quantum system.