Dynamics of Topological Excitations in a Model Quantum Spin Ice

Spectroscopic Signatures of Fractionalization in Octupolar Quantum Spin Ice

Recent investigations on the dipolar-octupolar compounds Ce_{2}Zr_{2}O_{7} and Ce_{2}Sn_{2}O_{7} suggest that they may stabilize so-called π-flux octupolar quantum spin ice (π-O-QSI), a novel three-dimensional quantum spin liquid hosting emergent photons. Confirmation of such an exotic phase would require the prediction of a distinctive signature and its subsequent experimental observation. So far, however, theoretical predictions for any such sharp smoking-gun signatures are lacking. In this Letter, we thoroughly investigate O-QSI using an extension of gauge mean-field theory. This framework produces a phase diagram consistent with previous work and an energy-integrated neutron scattering signal with intensity-modulated rod motifs, as reported in experiments and numerical studies. We predict that the dynamical spin structure factor of π-O-QSI is characterized by a broad continuum with three distinctive peaks as a consequence of the two mostly flat spinon bands. These three peaks should be measurable by high-resolution inelastic neutron scattering. Such spectroscopic signatures would be clear evidence for the realization of π-flux quantum spin ice.

Unlocking the general relationship between energy and entanglement spectra via the wormhole effect

Based on the path integral formulation of the reduced density matrix, we develop a scheme to overcome the exponential growth of computational complexity in reliably extracting low-lying entanglement spectrum from quantum Monte Carlo simulations. We test the method on the Heisenberg spin ladder with long entangled boundary between two chains and the results support the Li and Haldane's conjecture on entanglement spectrum of topological phase. We then explain the conjecture via the wormhole effect in the path integral and show that it can be further generalized for systems beyond gapped topological phases. Our further simulation results on the bilayer antiferromagnetic Heisenberg model with 2D entangled boundary across the (2 + 1)D O(3) quantum phase transition clearly demonstrate the correctness of the wormhole picture. Finally, we state that since the wormhole effect amplifies the bulk energy gap by a factor of β, the relative strength of that with respect to the edge energy gap will determine the behavior of low-lying entanglement spectrum of the system.

Dynamical Axions in U(1) Quantum Spin Liquids

Since their proposal nearly half a century ago, physicists have sought axions in both high energy and condensed matter settings. Despite intense and growing efforts, to date, experimental success has been limited, with the most prominent results arising in the context of topological insulators. Here, we propose a novel mechanism whereby axions can be realized in quantum spin liquids. We discuss the necessary symmetry requirements and identify possible experimental realizations in candidate pyrochlore materials. In this context, the axions couple to both the external and the emergent electromagnetic fields. We show that the interaction between the axion and the emergent photon leads to a characteristic dynamical response, which can be measured experimentally in inelastic neutron scattering. This Letter sets the stage for studying axion electrodynamics in the highly tunable setting of frustrated magnets.

Triangular lattice quantum dimer model with variable dimer density

Quantum dimer models are known to host topological quantum spin liquid phases, and it has recently become possible to simulate such models with Rydberg atoms trapped in arrays of optical tweezers. Here, we present large-scale quantum Monte Carlo simulation results on an extension of the triangular lattice quantum dimer model with terms in the Hamiltonian annihilating and creating single dimers. We find distinct odd and even [Formula: see text] spin liquids, along with several phases with no topological order: a staggered crystal, a nematic phase, and a trivial symmetric phase with no obvious broken symmetry. We also present dynamic spectra of the phases, and note implications for experiments on Rydberg atoms.

Emergent Fine Structure Constant of Quantum Spin Ice Is Large

Condensed-matter systems provide alternative "vacua" exhibiting emergent low-energy properties drastically different from those of the standard model. A case in point is the emergent quantum electrodynamics (QED) in the fractionalized topological magnet known as quantum spin ice, whose magnetic monopoles set it apart from the familiar QED of the world we live in. Here, we show that the two greatly differ in their fine structure constant α, which parametrizes how strongly matter couples to light: α_{QSI} is more than an order of magnitude greater than α_{QED}≈1/137. Furthermore, α_{QSI}, the emergent speed of light, and all other parameters of the emergent QED, are tunable by engineering the microscopic Hamiltonian. We find that α_{QSI} can be tuned all the way from zero up to what is believed to be the strongest possible coupling beyond which QED confines. In view of the small size of its constrained Hilbert space, this marks out quantum spin ice as an ideal platform for studying exotic quantum field theories and a target for quantum simulation. The large α_{QSI} implies that experiments probing candidate condensed-matter realizations of quantum spin ice should expect to observe phenomena arising due to strong interactions.

Spin Dynamics and Unconventional Coulomb Phase in Nd_{2}Zr_{2}O_{7}

We investigate the temperature dependence of the spin dynamics in the pyrochlore magnet Nd_{2}Zr_{2}O_{7} by neutron scattering experiments. At low temperature, this material undergoes a transition towards an "all-in-all-out" antiferromagnetic phase and the spin dynamics encompass a dispersionless mode, characterized by a dynamical spin ice structure factor. Unexpectedly, this mode is found to survive above T_{N}≈300  mK. Concomitantly, elastic correlations of the spin ice type develop. These are the signatures of a peculiar correlated paramagnetic phase which can be considered as a new example of Coulomb phase. Our observations near T_{N} do not reproduce the signatures expected for a Higgs transition, but show reminiscent features of the "all-in-all-out" order superimposed on a Coulomb phase.

Amplitude Mode in Quantum Magnets via Dimensional Crossover

We investigate the amplitude (Higgs) mode associated with longitudinal fluctuations of the order parameter at the continuous spontaneous symmetry breaking phase transition. In quantum magnets, due to the fast decay of the amplitude mode into low-energy Goldstone excitations, direct observation of this mode represents a challenging task. By focusing on a quasi-one-dimensional geometry, we circumvent the difficulty and investigate the amplitude mode in a system of weakly coupled spin chains with the help of quantum Monte Carlo simulations, stochastic analytic continuation, and a chain-mean field approach combined with a mapping to the field-theoretic sine-Gordon model. The amplitude mode is observed to emerge in the longitudinal spin susceptibility in the presence of a weak symmetry-breaking staggered field. A conventional measure of the amplitude mode in higher dimensions, the singlet bond mode, is found to appear at a lower than the amplitude mode frequency. We identify these two excitations with the second (first) breather of the sine-Gordon theory, correspondingly. In contrast to higher-dimensional systems, the amplitude and bond order fluctuations are found to carry significant spectral weight in the quasi-1D limit.

Theoretical scheme for finite-temperature dynamics of Kitaev's spin liquids

In this article, we review the theoretical formulation of finite temperature dynamics of Kitaev's spin liquid. We present the exact analytical solution of the dynamical spin correlation function at the integrable limit of Kitaev's model, on the basis of (2018Phys. Rev. B98220404). By combining the analytical solution with the equilibrium classical Monte-Carlo scheme, we construct a formulation to access the finite temperature dynamics of Kitaev's spin liquid exactly, with a reasonable amount of computational cost. This formulation is based on the real-time representation, which enables us to directly access the experimental observables defined in real frequency, without analytical continuation. The real-time scheme is essential to capturing the resonant features of the spectrum accurately, which occurs e.g. in the chiral spin liquid phase with isolated Majorana zero modes. Accordingly, this scheme provides an effective approach to address the nature of fractional excitations in Kitaev's spin liquid. As an application, we address the detection of zero mode around the site vacancy through the local resonant spectrum and discuss how the character of Kitaev's spin liquid emerges in its dynamical signature.

Giant isotropic magneto-thermal conductivity of metallic spin liquid candidate Pr2Ir2O7 with quantum criticality

Spin liquids are exotic states with no spontaneous symmetry breaking down to zero-temperature because of the highly entangled and fluctuating spins in frustrated systems. Exotic excitations like magnetic monopoles, visons, and photons may emerge from quantum spin ice states, a special kind of spin liquids in pyrochlore lattices. These materials usually are insulators, with an exception of the pyrochlore iridate Pr2Ir2O7, which was proposed as a metallic spin liquid located at a zero-field quantum critical point. Here we report the ultralow-temperature thermal conductivity measurements on Pr2Ir2O7. The Wiedemann-Franz law is verified at high fields and inferred at zero field, suggesting no breakdown of Landau quasiparticles at the quantum critical point, and the absence of mobile fermionic excitations. This result puts strong constraints on the description of the quantum criticality in Pr2Ir2O7. Unexpectedly, although the specific heats are anisotropic with respect to magnetic field directions, the thermal conductivities display the giant but isotropic response. This indicates that quadrupolar interactions and quantum fluctuations are important, which will help determine the true ground state of this material.

A quantum liquid of magnetic octupoles on the pyrochlore lattice

Spin liquids are highly correlated yet disordered states formed by the entanglement of magnetic dipoles1. Theories define such states using gauge fields and deconfined quasiparticle excitations that emerge from a local constraint governing the ground state of a frustrated magnet. For example, the '2-in-2-out' ice rule for dipole moments on a tetrahedron can lead to a quantum spin ice2-4 in rare-earth pyrochlores. However, f-electron ions often carry multipole degrees of freedom of higher rank than dipoles, leading to intriguing behaviours and 'hidden' orders5-6. Here we show that the correlated ground state of a Ce3+-based pyrochlore, Ce2Sn2O7, is a quantum liquid of magnetic octupoles. Our neutron scattering results are consistent with a fluid-like state where degrees of freedom have a more complex magnetization density than that of magnetic dipoles. The nature and strength of the octupole-octupole couplings, together with the existence of a continuum of excitations attributed to spinons, provides further evidence for a quantum ice of octupoles governed by a '2-plus-2-minus' rule7-8. Our work identifies Ce2Sn2O7 as a unique example of frustrated multipoles forming a 'hidden' topological order, thus generalizing observations on quantum spin liquids to multipolar phases that can support novel types of emergent fields and excitations.

Rank-2 U(1) Spin Liquid on the Breathing Pyrochlore Lattice

Higher-rank generalizations of electrodynamics have recently attracted considerable attention because of their ability to host "fracton" excitations, with connections to both fracton topological order and gravity. However, the search for higher-rank gauge theories in experiment has been greatly hindered by the lack of materially relevant microscopic models. Here we show how a spin liquid described by rank-2 U(1) gauge theory can arise in a magnet on the breathing pyrochlore lattice. We identify Yb-based breathing pyrochlores as candidate systems, and make explicit predictions for how the rank-2 U(1) spin liquid would manifest itself in experiment.

Spectroscopy of Spinons in Coulomb Quantum Spin Liquids

We calculate the effect of the emergent photon on threshold production of spinons in U(1) Coulomb spin liquids such as quantum spin ice. The emergent Coulomb interaction modifies the threshold production cross-section dramatically, changing the weak turn-on expected from the density of states to an abrupt onset reflecting the basic coupling parameters. The slow photon typical in existing lattice models and materials suppresses the intensity at finite momentum and allows profuse Cerenkov radiation beyond a critical momentum. These features are broadly consistent with recent numerical and experimental results.

Order out of a Coulomb Phase and Higgs Transition: Frustrated Transverse Interactions of Nd_{2}Zr_{2}O_{7}

The pyrochlore material Nd_{2}Zr_{2}O_{7} with an "all-in-all-out" (AIAO) magnetic order shows novel quantum moment fragmentation with gapped flat dynamical spin ice modes. The parametrized spin Hamiltonian with a dominant frustrated ferromagnetic transverse term reveals a proximity to a U(1) spin liquid. Here we study the magnetic excitations of Nd_{2}Zr_{2}O_{7} above the ordering temperature (T_{N}) using high-energy-resolution inelastic neutron scattering. We find strong spin ice correlations at zero energy with the disappearance of gapped magnon excitations of the AIAO order. It seems that the gap to the dynamical spin ice closes above T_{N} and the system enters a quantum spin ice state competing with and suppressing the AIAO order. Classical Monte Carlo simulations, molecular dynamics, and quantum boson calculations support the existence of a Coulombic phase above T_{N}. Our findings relate the magnetic ordering of Nd_{2}Zr_{2}O_{7} with the Higgs mechanism and provide explanations for several previously reported experimental features.

Kosterlitz-Thouless melting of magnetic order in the triangular quantum Ising material TmMgGaO4

Frustrated magnets hold the promise of material realizations of exotic phases of quantum matter, but direct comparisons of unbiased model calculations with experimental measurements remain very challenging. Here we design and implement a protocol of employing many-body computation methodologies for accurate model calculations—of both equilibrium and dynamical properties—for a frustrated rare-earth magnet TmMgGaO₄ (TMGO), which explains the corresponding experimental findings. Our results confirm TMGO is an ideal realization of triangular-lattice Ising model with an intrinsic transverse field. The magnetic order of TMGO is predicted to melt through two successive Kosterlitz–Thouless (KT) phase transitions, with a floating KT phase in between. The dynamical spectra calculated suggest remnant images of a vanishing magnetic stripe order that represent vortex–antivortex pairs, resembling rotons in a superfluid helium film. TMGO therefore constitutes a rare quantum magnet for realizing KT physics, and we further propose experimental detection of its intriguing properties.

TmMgGaO₄ is one of a number of recently-synthesized quantum magnets that are proposed to realize important theoretical models. Here the authors demonstrate the agreement between detailed experimental measurements and state-of-the-art predictions based on the 2D transverse-field triangular lattice Ising model.

Revealing fermionic quantum criticality from new Monte Carlo techniques

This review summarizes recent developments in the study of fermionic quantum criticality, focusing on new progress in numerical methodologies, especially quantum Monte Carlo methods, and insights that emerged from recently large-scale numerical simulations. Quantum critical phenomena in fermionic systems have attracted decades of extensive research efforts, partially lured by their exotic properties and potential technology applications, and partially awakened by the profound and universal fundamental principles that govern these quantum critical systems. Due to the complex and non-perturbative nature, these systems face the most difficult and challenging problems in the study of modern condensed matter physics, and many important fundamental problems remain open. Recently, new developments in model design and algorithm improvements enabled unbiased large-scale numerical solutions to be achieved in the close vicinity of these quantum critical points, which paves a new pathway towards achieving controlled conclusions through combined efforts of theoretical and numerical studies, as well as possible theoretical guidance for experiments in heavy-fermion compounds, Cu-based and Fe-based superconductors, ultra-cold fermionic atomic gas, twisted graphene layers, etc, where signatures of fermionic quantum criticality exist.

Spectrum of Itinerant Fractional Excitations in Quantum Spin Ice

We study the quantum dynamics of fractional excitations in quantum spin ice. We focus on the density of states in the two-monopole sector, ρ(ω), as this can be connected to the wave-vector-integrated dynamical structure factor accessible in neutron scattering experiments. We find that ρ(ω) exhibits a strikingly characteristic singular and asymmetric structure that provides a useful fingerprint for comparison to experiment. ρ(ω) obtained from the exact diagonalization of a finite cluster agrees well with that, from the analytical solution of a hopping problem on a Husimi cactus representing configuration space, but not with the corresponding result on a face-centered cubic lattice, on which the monopoles move in real space. The main difference between the latter two lies in the inclusion of the emergent gauge field degrees of freedom, under which the monopoles are charged. This underlines the importance of treating both sets of degrees of freedom together, and it presents a novel instance of dimensional transmutation.

Diagnosing Fractionalization from the Spin Dynamics of Z_{2} Spin Liquids on the Kagome Lattice by Quantum Monte Carlo Simulations

Based on large-scale quantum Monte Carlo simulations, we examine the dynamical spin structure factor of the Balents-Fisher-Girvin kagome lattice spin-1/2 model, which is known to harbor an extended Z_{2} quantum spin liquid phase. We use a correlation-matrix sampling scheme combined with a stochastic analytic continuation method to resolve the spectral functions of this anisotropic quantum spin model with a three-site unit cell. Based on this approach, we monitor the spin dynamics throughout the phase diagram of this model, from the XY-ferromagnetic region to the Z_{2} quantum spin liquid regime. In the latter phase, we identify a gapped two-spinon continuum in the transverse scattering channel, which is faithfully modeled by an effective spinon tight-binding model. Within the longitudinal channel, we identify gapped vison excitations and exhibit indications for the translational symmetry fractionalization of the visons via an enhanced spectral periodicity.

Dynamical Signature of Symmetry Fractionalization in Frustrated Magnets

The nontriviality of quantum spin liquids (QSLs) typically manifests in the nonlocal observables that signify their existence; however, this fact actually casts a shadow on detecting QSLs with experimentally accessible probes. Here, we provide a solution by unbiasedly demonstrating a dynamical signature of anyonic excitations and symmetry fractionalization in QSLs. Employing large-scale quantum Monte Carlo simulation and stochastic analytic continuation, we investigate the extended XXZ model on the kagome lattice, and find out that, across the phase transitions from Z_{2} QSLs to different symmetry breaking phases, spin spectral functions can reveal the presence and condensation of emergent anyonic spinon and vison excitations, in particular, the translational symmetry fractionalization of the latter, which can be served as the dynamical signature of the seemingly ephemeral QSLs in spectroscopic techniques such as inelastic neutron or resonance (inelastic) x-ray scatterings.

Quantum Spin Ice with Frustrated Transverse Exchange: From a π-Flux Phase to a Nematic Quantum Spin Liquid

Quantum spin ice materials, pyrochlore magnets with competing Ising and transverse exchange interactions, have been widely discussed as candidates for a quantum spin-liquid ground state. Here, motivated by quantum chemical calculations for Pr pyrochlores, we present the results of a study for frustrated transverse exchange. Using a combination of variational calculations, exact diagonalization, numerical linked-cluster and series expansions, we find that the previously studied U(1) quantum spin liquid, in its π-flux phase, transforms into a nematic quantum spin liquid at a high-symmetry, SU(2) point.