Quantum ice: a quantum Monte Carlo study

Signatures of Spinon Dynamics and Phase Structure of Dipolar-Octupolar Quantum Spin Ices in Two-Dimensional Coherent Spectroscopy

We study how sharp signatures of fractionalization emerge in nonlinear spectroscopy experiments on spin liquids with separated energy scales. Our model is that of dipolar-octupolar rare earth pyrochlore materials, prime candidates for realizing quantum spin ice. This family of three-dimensional quantum spin liquids exhibits fractionalization of spin degrees of freedom into spinons charged under an emergent U(1) gauge field. We show that the technique of two-dimensional coherent spectroscopy can identify clear signatures of fractionalized spinon dynamics in dipolar-octupolar quantum spin ices. However, at intermediate temperatures, spinon dynamics are heavily constrained in the presence of an incoherent spin background, leading to a broad two-dimensional coherent spectroscopy response. At lower temperatures, a sharp signal emerges as the system enters a coherent spin liquid state. This lower temperature signal can in turn distinguish between zero-flux and π-flux forms of 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.

Prethermal Time-Crystalline Spin Ice and Monopole Confinement in a Driven Magnet

Studies of systems far from equilibrium open up new avenues for investigating exotic phases of matter. A driven-dissipative frustrated spin system is examined in this study, and we suggest an out-of-equilibrium nonmagnetic phase where the spins do not order but adhere to the ice rule in space and establish a long-range crystalline order in time. In contrast to the conventional spin ice, the dynamics of monopoles is confined due to the nonequilibrium feature of our model. Possible experimental realizations of our model are discussed.

Exact diagonalization for a 16-site spin-1/2 pyrochlore cluster

We find exact solutions to the Hamiltonian of a 16-site spin-1/2 pyrochlore cluster with nearest neighbour exchange interactions. The methods of group theory (symmetry) are used to completely block-diagonalize the Hamiltonian, yielding precise details about symmetry of the eigenstates, in particular those components which arespin icestates, in order to evaluate the spin ice density at finite temperature. At low enough temperatures, a 'perturbed' spin ice phase, where the '2-in-2-out' ice rule is largely obeyed, is clearly outlined within the four parameter space of the general model of exchange interactions. The quantum spin ice phase is expected to exist within these boundaries.

Anomalous magnetoresistance by breaking ice rule in Bi2Ir2O7/Dy2Ti2O7 heterostructure

While geometrically frustrated quantum magnets host rich exotic spin states with potentials for revolutionary quantum technologies, most of them are necessarily good insulators which are difficult to be integrated with modern electrical circuit. The grand challenge is to electrically detect the emergent fluctuations and excitations by introducing charge carriers that interact with the localized spins without destroying their collective spin states. Here, we show that, by designing a Bi2Ir2O7/Dy2Ti2O7 heterostructure, the breaking of the spin-ice rule in insulating Dy2Ti2O7 leads to a charge response in the conducting Bi2Ir2O7 measured as anomalous magnetoresistance during the field-induced Kagome ice-to-saturated ice transition. The magnetoresistive anomaly also captures the characteristic angular and temperature dependence of this ice-rule-breaking transition, which has been understood as magnetic monopole condensation. These results demonstrate a novel heteroepitaxial approach for electronically probing the transition between exotic insulating spin states, laying out a blueprint for the metallization of frustrated quantum magnets.

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.

Uncovering Footprints of Dipolar-Octupolar Quantum Spin Ice from Neutron Scattering Signatures

Recent experiments on Ce_{2}Zr_{2}O_{7} suggest that this material may host a novel form of quantum spin ice, a three-dimensional quantum spin liquid with an emergent photon. The Ce^{3+} local moments on the pyrochlore lattice are described by pseudospin-1/2 degrees of freedom, whose components transform as dipolar and octupolar moments under symmetry operations. In principle, there exist four possible quantum spin ice regimes, depending on whether the Ising component is in the dipolar or octupolar channel, and two possible flux configurations of the emergent gauge field. In this Letter, using exact diagonalization and molecular dynamics, we investigate the equal-time and dynamical spin structure factors in all four quantum spin ice regimes using quantum and classical calculations. Contrasting the distinct signatures of quantum and classical results for the four possible quantum spin ice regimes and elucidating the role of quantum fluctuations, we show that the quantum structure factor computed for the π-flux octupolar quantum spin ice regime is most compatible with the neutron scattering results on Ce_{2}Zr_{2}O_{7}.

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.

Low-temperature high-frequency dynamic magnetic susceptibility of classical spin-ice Dy2Ti2O7

Radio-frequency (14.6 MHz) AC magnetic susceptibility,χAC', of Dy₂Ti₂O₇was measured using self-oscillating tunnel-diode resonator. Measurements were made with the excitation AC field parallel to the superimposed DC magnetic field up to 5 T in a wide temperature range from 50 mK to 100 K. At 14.6 MHz, a known broad peak ofχAC'(T)from kHz-range audio-frequency measurements around 15 K for both [111] and [110] directions shifts to 45 K, continuing the Arrhenius activated behavior with the same activation energy barrier ofE_a≈ 230 K. Magnetic field dependence ofχAC'along [111] reproduces previously reported low-temperature two-in-two-out to three-in-one-out spin configuration transition at about 1 T, and an intermediate phase between 1 and 1.5 T. The boundaries of the intermediate phase show reasonable overlap with the literature data and connect at a critical endpoint of the first order transition line, suggesting that these features are frequency independent. An unusual upturn of the magnetic susceptibility atT→ 0 was observed in magnetic fields between 1.5 T and 2 T for both magnetic field directions, before fully polarized configuration sets in above 2 T.

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.

Dipole-dipole interaction induced phases in hydrogen-bonded squaric acid crystal

We study the finite-temperature phase diagram of proton ordering of a quasi-two dimensional hydrogen-bonded system, namely the squaric acid crystal (H₂C₄O₄) using quantum Monte Carlo. We take into account the four-spin plaquette interaction at the zeroth order followed by next nearest neighbor Ising interaction within a plaquette, dipole-dipole interaction and an external transverse magnetic field respectively. Using an improvised loop algorithm within the stochastic series expansion (SSE) quantum Monte Carlo method, we find two distinct phases as we increase the temperature and magnetic-field. One of the phase is the Π_f, the phase with long range ferroelectric order and the other being an intermediate state with strong local correlations, i.e, a quantum liquid-like state Π_ql. The transition to Π_f shows a very small anomalous peak in the specific heat with strong dependence of critical temperature on the strength of dipole-dipole interaction. The presence of the small peak is attributed to the absence of macroscopic degeneracy in the presence of dipole-dipole interaction and re-entrance of such degeneracy to some extent at small temperature. The work also discusses an intricate connection of quantum fluctuation and thermal fluctuation in the presence of competing interaction with entropic effects.

Quantum Versus Classical Spin Fragmentation in Dipolar Kagome Ice Ho3Mg2Sb3O14

A promising route to realize entangled magnetic states combines geometrical frustration with quantum-tunneling effects. Spin-ice materials are canonical examples of frustration, and Ising spins in a transverse magnetic field are the simplest many-body model of quantum tunneling. Here, we show that the tripod-kagome lattice material Ho3Mg2Sb3O14 unites an icelike magnetic degeneracy with quantum-tunneling terms generated by an intrinsic splitting of the Ho3+ ground-state doublet, which is further coupled to a nuclear spin bath. Using neutron scattering and thermodynamic experiments, we observe a symmetry-breaking transition at T * ≈ 0.32 K to a remarkable state with three peculiarities: a concurrent recovery of magnetic entropy associated with the strongly coupled electronic and nuclear degrees of freedom; a fragmentation of the spin into periodic and icelike components; and persistent inelastic magnetic excitations down to T ≈ 0.12 K . These observations deviate from expectations of classical spin fragmentation on a kagome lattice, but can be understood within a model of dipolar kagome ice under a homogeneous transverse magnetic field, which we survey with exact diagonalization on small clusters and mean-field calculations. In Ho3Mg2Sb3O14, hyperfine interactions dramatically alter the single-ion and collective properties, and suppress possible quantum correlations, rendering the fragmentation with predominantly single-ion quantum fluctuations. Our results highlight the crucial role played by hyperfine interactions in frustrated quantum magnets and motivate further investigations of the role of quantum fluctuations on partially ordered magnetic states.

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.

Quantum Criticality of Two-Dimensional Quantum Magnets with Long-Range Interactions

We study the critical breakdown of two-dimensional quantum magnets in the presence of algebraically decaying long-range interactions by investigating the transverse-field Ising model on the square and triangular lattice. This is achieved technically by combining perturbative continuous unitary transformations with classical Monte Carlo simulations to extract high-order series for the one-particle excitations in the high-field quantum paramagnet. We find that the unfrustrated systems change from mean-field to nearest-neighbor universality with continuously varying critical exponents. In the frustrated case on the square lattice the system remains in the universality class of the nearest-neighbor model independent of the long-range nature of the interaction, while we argue that the quantum criticality for the triangular lattice is terminated by a first-order phase transition line.

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.

Instabilities of a U(1) Quantum Spin Liquid in Disordered Non-Kramers Pyrochlores

Quantum spin liquids (QSLs) are exotic phases of matter exhibiting long-range entanglement and supporting emergent gauge fields. A vigorous search for experimental realizations of these states has identified several materials with properties hinting at QSL physics. A key issue in understanding these QSL candidates is often the interplay of weak disorder of the crystal structure with the spin liquid state. It has recently been pointed out that in at least one important class of candidate QSLs-pyrochlore magnets based on non-Kramers ions such as Pr^{3+} or Tb^{3+}-structural disorder can actually promote a U(1) QSL ground state. Here we set this proposal on a quantitative footing by analyzing the stability of the QSL state in the minimal model for these systems: a random transverse field Ising model. We consider two kinds of instability, which are relevant in different limits of the phase diagram: condensation of spinons and confinement of the U(1) gauge fields. Having obtained stability bounds on the QSL state, we apply our results directly to the disordered candidate QSL Pr_{2}Zr_{2}O_{7}. We find that the available data for currently studied samples of Pr_{2}Zr_{2}O_{7} are most consistent with it a ground state outside the spin liquid regime, in a paramagnetic phase with quadrupole moments near saturation due to the influence of structural disorder.

Dynamics of Topological Excitations in a Model Quantum Spin Ice

We study the quantum spin dynamics of a frustrated XXZ model on a pyrochlore lattice by using large-scale quantum Monte Carlo simulation and stochastic analytic continuation. In the low-temperature quantum spin ice regime, we observe signatures of coherent photon and spinon excitations in the dynamic spin structure factor. As the temperature rises to the classical spin ice regime, the photon disappears from the dynamic spin structure factor, whereas the dynamics of the spinon remain coherent in a broad temperature window. Our results provide experimentally relevant, quantitative information for the ongoing pursuit of quantum spin ice materials.

Molecular Dynamics of Hexamethylbenzene at Low Temperatures: Evidence of Unconventional Magnetism Based on Rotational Motion of Protons

The types of magnetism known to date are all mainly based on contributions from electron motion. We show how rotational motion of protons (H⁺ ) within the methyl groups in hexamethylbenzene (C₆ (CH₃ )₆ ) also contribute significantly to the magnetic susceptibility. Starting from below 118 K, as the rotational motion of the methyl groups set in, an associated magnetic moment positive in nature due to charge of the protons renders the susceptibility to become anomalously dependent on temperature. Starting from 20 K, the susceptibility diverges with decreasing temperature indicative of spin-spin interactions between methyl groups aligned in a previously unclassified type of anti-ferromagnetic configuration. Complementary dielectric constant measurements also show the existence of magneto-dielectric coupling. Our findings allow for the study of strongly correlated systems that are based on a species that possesses much slower dynamics.

Disorder-Induced Quantum Spin Liquid in Spin Ice Pyrochlores

We propose that in a certain class of magnetic materials, known as non-Kramers "spin ice," disorder induces quantum entanglement. Instead of driving glassy behavior, disorder provokes quantum superpositions of spins throughout the system and engenders an associated emergent gauge structure and set of fractional excitations. More precisely, disorder transforms a classical phase governed by a large entropy, classical spin ice, into a quantum spin liquid governed by entanglement. As the degree of disorder is increased, the system transitions between (i) a "regular" Coulombic spin liquid, (ii) a phase known as "Mott glass," which contains rare gapless regions in real space, but whose behavior on long length scales is only modified quantitatively, and (iii) a true glassy phase for random distributions with large width or large mean amplitude.

Quantum spin liquids: a review

Quantum spin liquids may be considered 'quantum disordered' ground states of spin systems, in which zero-point fluctuations are so strong that they prevent conventional magnetic long-range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, which is of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons, which are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments in relation to study quantum spin liquids, and to the diverse probes used therein.

A novel type of splayed ferromagnetic order observed in Yb2Ti2O7

The pyrochlore insulator Yb2Ti2O7 has attracted the attention of experimentalists and theoreticians alike for about 15 years. Conflicting neutron diffraction data on the possible existence of magnetic Bragg reflections at low temperature have been published. Here we report the observation of magnetic Bragg reflections by neutron powder diffraction at 60 mK. The magnetic diffraction pattern is analyzed using representation theory. We find Yb2Ti2O7 to be a splayed ferromagnet as reported for Yb2Sn2O7, a sibling compound with also dominating ferromagnetic interactions as inferred from the positive Curie-Weiss temperature. However, the configuration of the magnetic moment components perpendicular to the easy axis is of the all-in-all-out type in Yb2Ti2O7 while it is two-in-two-out in the Yb2Sn2O7. An overall experimental picture of the magnetic properties emerges.

Spin slush in an extended spin ice model

We present a new classical spin liquid on the pyrochlore lattice by extending spin ice with further neighbour interactions. We find that this disorder-free spin model exhibits a form of dynamical heterogeneity with extremely slow relaxation for some spins, while others fluctuate quickly down to zero temperature. We thus call this state spin slush, in analogy to the heterogeneous mixture of solid and liquid water. This behaviour is driven by the structure of the ground-state manifold which extends the celebrated two-in/two-out ice states to include branching structures built from three-in/one-out, three-out/one-in and all-in/all-out tetrahedra defects. Distinctive liquid-like patterns in the magnetic correlations serve as a signature of this intermediate range order. Possible applications to materials as well the effects of quantum tunnelling are discussed.

A spin-liquid with pinch-line singularities on the pyrochlore lattice

The mathematics of gauge theories lies behind many of the most profound advances in physics in the past 200 years, from Maxwell's theory of electromagnetism to Einstein's theory of general relativity. More recently it has become clear that gauge theories also emerge in condensed matter, a prime example being the spin-ice materials which host an emergent electromagnetic gauge field. In spin-ice, the underlying gauge structure is revealed by the presence of pinch-point singularities in neutron-scattering measurements. Here we report the discovery of a spin-liquid where the low-temperature physics is naturally described by the fluctuations of a tensor field with a continuous gauge freedom. This gauge structure underpins an unusual form of spin correlations, giving rise to pinch-line singularities: line-like analogues of the pinch points observed in spin-ice. Remarkably, these features may already have been observed in the pyrochlore material Tb₂Ti₂O₇.

Neutron scattering measurements of spin-ice materials contain signature pinch-point singularities, demonstrating the existence of an emergent electromagnetic gauge field. Here, the authors propose a system in which correlations manifest in pinch lines, which may have already been observed experimentally.

Spinon Walk in Quantum Spin Ice

We study a minimal model for the dynamics of spinons in quantum spin ice. The model captures the essential strong coupling between the spinon and the disordered background spins. We demonstrate that the spinon motion can be mapped to a random walk with an entropy-induced memory in imaginary time. Our numerical simulation of the spinon walk indicates that the spinon propagates as a massive quasiparticle at low energy despite its strong coupling to the spin background at the microscopic energy scale. We discuss the experimental implications of our findings.

Possible observation of highly itinerant quantum magnetic monopoles in the frustrated pyrochlore Yb2Ti2O7

The low-energy elementary excitations in frustrated quantum magnets have fascinated researchers for decades. In frustrated Ising magnets on a pyrochlore lattice possessing macroscopically degenerate spin-ice ground states, the excitations have been discussed in terms of classical magnetic monopoles, which do not contain quantum fluctuations. Here we report unusual behaviours of magneto-thermal conductivity in the disordered spin-liquid regime of pyrochlore Yb₂Ti₂O₇, which hosts frustrated spin-ice correlations with large quantum fluctuations owing to pseudospin-1/2 of Yb ions. The analysis of the temperature and magnetic field dependencies shows the presence of gapped elementary excitations. We find that the gap energy is largely suppressed from that expected in classical monopoles. Moreover, these excitations propagate a long distance without being scattered, in contrast to the diffusive nature of classical monopoles. These results suggests the emergence of highly itinerant quantum magnetic monopole, which is a heavy quasiparticle that propagates coherently in three-dimensional spin liquids.

Spin excitations above the ground state of spin ices, geometrically frustrated pyrochlore magnets, are equivalent to pairs of oppositely charged deconfined magnetic monopoles which may propagate through the system. Here, the authors evidence highly itinerant quantum magnetic monopole excitations in Yb₂Ti₂O₇.

Numerical Evidence of Quantum Melting of Spin Ice: Quantum-to-Classical Crossover

Unbiased quantum Monte Carlo simulations are performed on the nearest-neighbor spin-1/2 pyrochlore XXZ model with an antiferromagnetic longitudinal and the weak ferromagnetic transverse exchange couplings, J and J_{⊥}. The specific heat exhibits a broad peak at T_{CSI}~0.2J associated with a crossover to a classical Coulomb liquid regime showing a suppressed spin-ice monopole density, a broadened pinch-point singularity, and the Pauling entropy for |J_{⊥}|≪J, as in classical spin ice. On further cooling, the entropy restarts decaying for J_{⊥}>J_{⊥c}∼-0.104J, producing another broad specific heat peak for a crossover to a bosonic quantum Coulomb liquid, where the spin correlation contains both photon and quantum spin-ice monopole contributions. With negatively increasing J_{⊥} across J_{⊥c}, a first-order thermal phase transition occurs from the quantum Coulomb liquid to an XY ferromagnet. Relevance to magnetic rare-earth pyrochlore oxides is discussed.

Coulomb Liquid Phases of Bosonic Cluster Mott Insulators on a Pyrochlore Lattice

Employing large-scale quantum Monte Carlo simulations, we reveal the full phase diagram of the extended Hubbard model of hard-core bosons on the pyrochlore lattice with partial fillings. When the intersite repulsion is dominant, the system is in a cluster Mott insulator phase with an integer number of bosons localized inside the tetrahedral units of the pyrochlore lattice. We show that the full phase diagram contains three cluster Mott insulator phases with 1/4, 1/2, and 3/4 boson fillings, respectively. We further demonstrate that all three cluster Mott insulators are Coulomb liquid phases and its low-energy property is described by the emergent compact U(1) quantum electrodynamics. In addition to measuring the specific heat and entropy of the cluster Mott insulators, we investigate the correlation function of the emergent electric field and verify it is consistent with the compact U(1) quantum electrodynamics description. Our result sheds light on the magnetic properties of various pyrochlore systems, as well as the charge physics of the cluster magnets.

Fractionalized charge excitations in a spin liquid on partially filled pyrochlore lattices

We study the Mott transition from a metal to cluster Mott insulators in the 1/4- and 1/8-filled pyrochlore lattice systems [corrected]. It is shown that such Mott transitions can arise due to charge localization in clusters or in tetrahedron units, driven by the nearest-neighbor repulsive interaction. The resulting cluster Mott insulator is a quantum spin liquid with a spinon Fermi surface, but at the same time a novel fractionalized charge liquid with charge excitations carrying half the electron charge. There exist two emergent U(1) gauge fields or "photons" that mediate interactions between spinons and charge excitations, and between fractionalized charge excitations themselves, respectively. In particular, it is suggested that the emergent photons associated with the fractionalized charge excitations can be measured in x-ray scattering experiments. Various other experimental signatures of the exotic cluster Mott insulator are discussed in light of candidate materials with partially filled bands on the pyrochlore lattice.

Order-by-disorder and quantum Coulomb phase in quantum square ice

We show that quantum square ice-namely, the two-dimensional version of proton or spin ice with tunable quantum tunneling of the electric or magnetic dipole moment-exhibits a quantum spin-liquid phase supporting fractionalized spinons. This phase corresponds to a thermally induced, deconfined quantum Coulomb phase of a two-dimensional lattice gauge theory. It emerges at finite, yet exceedingly low temperatures from the melting of two distinct order-by-disorder phases appearing in the ground state: a plaquette valence-bond solid for low tunneling; and a canted Néel state for stronger tunneling. The latter phases appear via the highly nonlinear effect of quantum fluctuations within the degenerate manifold of ice-rule states, and they can be identified as the two competing ground states of a discrete lattice gauge theory (quantum link model) emerging as the effective Hamiltonian of the system within degenerate perturbation theory.

Gauge field and the confinement-deconfinement transition in hydrogen-bonded ferroelectrics

Quantum melting of a ferroelectric moment in the frustrated hydrogen-bonded system with the "ice rule" is studied theoretically by using quantum Monte Carlo simulation. The large number of nearly degenerate configurations are described as the gauge degrees of freedom; i.e., the model is mapped to a lattice gauge theory, which shows the confinement-deconfinment transition (CDT). The dipole-dipole interaction J(2), on the other hand, explicitly breaks the gauge symmetry leading to the ferroelectric transition at finite temperature T. It is found that the crossover from the FT to CDT manifests itself in the reduced correlation length of the polarization ξ(FT)∼Δ(K-K(c))(-ν), with Δ∝√J(2) while K(c) and ν remains finite in the limit J(2)→0. In contrast, the Curie-Weiss-like law for the susceptibility χ and the spontaneous polarization behaves smoothly and the length scale ξ(CDT), related to the molecular symmetry and volume for CDT, does not reduce in this limit.

Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets

The spin ice materials, including Ho2Ti2O7 and Dy2Ti2O7, are rare-earth pyrochlore magnets which, at low temperatures, enter a constrained paramagnetic state with an emergent gauge freedom. Spin ices provide one of very few experimentally realized examples of fractionalization because their elementary excitations can be regarded as magnetic monopoles and, over some temperature range, spin ice materials are best described as liquids of these emergent charges. In the presence of quantum fluctuations, one can obtain, in principle, a quantum spin liquid descended from the classical spin ice state characterized by emergent photon-like excitations. Whereas in classical spin ices the excitations are akin to electrostatic charges with a mutual Coulomb interaction, in the quantum spin liquid these charges interact through a dynamic and emergent electromagnetic field. In this review, we describe the latest developments in the study of such a quantum spin ice, focusing on the spin liquid phenomenology and the kinds of materials where such a phase might be found.

Antiferromagnetic metal and Mott transition on Shastry-Sutherland lattice

The Shastry-Sutherland lattice, one of the simplest systems with geometrical frustration, which has an exact eigenstate by putting singlets on diagonal bonds, can be realized in a group of layered compounds and raises both theoretical and experimental interest. Most of the previous studies on the Shastry-Sutherland lattice are focusing on the Heisenberg model. Here we opt for the Hubbard model to calculate phase diagrams over a wide range of interaction parameters, and show the competing effects of interaction, frustration and temperature. At low temperature, frustration is shown to favor a paramagnetic metallic ground state, while interaction drives the system to an antiferromagnetic insulator phase. Between these two phases, there are an antiferromagnetic metal phase and a paramagnetic insulator phase (which should consist of a small plaquette phase and a dimer phase) resulting from the competition of the frustration and the interaction. Our results may shed light on more exhaustive studies about quantum phase transitions in geometrically frustrated systems.

Four-coloring model and frustrated superfluidity in the diamond lattice

We propose a novel four-coloring model which describes "frustrated superfluidity" of p-band bosons in the diamond optical lattice. The superfluid phases of the condensate wave functions on the diamond-lattice bonds are mapped to four distinct colors at low temperatures. The fact that a macroscopic number of states satisfy the constraints that four differently colored bonds meet at the same site leads to an extensive degeneracy in the superfluid ground state at the classical level. We demonstrate that the phase of the superfluid wave function as well as the orbital angular momentum correlations exhibit a power-law decay in the degenerate manifold that is described by an emergent magnetostatic theory with three independent flux fields. Our results thus provide a novel example of critical superfluid phase with algebraic order in three dimensions. We further show that quantum fluctuations favor a Néel ordering of orbital angular moments with broken sublattice symmetry through the order-by-disorder mechanism.

Quantum fluctuations in spin-ice-like Pr2Zr2O7

Spin ice is a magnetic analog of H2O ice that harbors dense static disorder. Dipolar interactions between classical spins yield a frozen frustrated state with residual configurational Pauling entropy and emergent magnetic monopolar quasiparticles. Introducing quantum fluctuations is of great interest as this could melt spin ice and allow coherent propagation of monopoles. Here, we report experimental evidence for quantum dynamics of magnetic monopolar quasiparticles in a new class of spin ice based on exchange interactions, Pr2Zr2O7. Narrow pinch point features in otherwise diffuse elastic neutron scattering reflects adherence to a divergence-free constraint for disordered spins on long time scales. Magnetic susceptibility and specific heat data correspondingly show exponentially activated behaviors. In sharp contrast to conventional ice, however, >90% of the neutron scattering is inelastic and devoid of pinch points furnishing evidence for magnetic monopolar quantum fluctuations.

Crystal shape-dependent magnetic susceptibility and Curie law crossover in the spin ices Dy2Ti2O7 and Ho2Ti2O7

We present an experimental determination of the isothermal magnetic susceptibility of the spin ice materials Dy2Ti2O7 and Ho2Ti2O7 in the temperature range 1.8-300 K. The use of spherical crystals has allowed accurate correction for demagnetizing fields and allowed the true bulk isothermal susceptibility χT(T) to be estimated. This has been compared against a theoretical expression based on a Husimi tree approximation to the spin ice model. Agreement between experiment and theory is excellent at T > 10 K, but systematic deviations occur below that temperature. Our results largely resolve an apparent disagreement between neutron scattering and bulk measurements that has been previously noted. They also show that the use of non-spherical crystals in magnetization studies of spin ice may introduce very significant systematic errors, although we note some interesting--and possibly new--systematics concerning the demagnetizing factor in cuboidal samples. Finally, our results show how experimental susceptibility measurements on spin ices may be used to extract the characteristic energy scale of the system and the corresponding chemical potential for emergent magnetic monopoles.

Dynamical splayed ferromagnetic ground state in the quantum spin ice Yb(2)Sn(2)O(7)

From magnetic, specific heat, (170)Yb Mössbauer effect, neutron diffraction, and muon spin relaxation measurements on polycrystalline Yb(2)Sn(2)O(7), we show that below the first order transition at 0.15 K all of the Yb(3+) ions are long-range magnetically ordered and each has a moment of 1.1 μ(B) which lies at ≃ 10° to a common fourfold cubic axis. The four sublattice moments have four different directions away from this axis and are therefore noncoplanar. We term this arrangement splayed ferromagnetism. This ground state has a dynamical component with a fluctuation rate in the megahertz range. The net ferromagnetic exchange interaction has an anisotropy that favors the local threefold axis. We discuss our results in terms of the phase diagram proposed by Savary and Balents [Phys. Rev. Lett. 108, 037202 (2012)] for a pyrochlore lattice of Kramers 1/2 effective spins.

Brownian motion and quantum dynamics of magnetic monopoles in spin ice

Spin ice illustrates many unusual magnetic properties, including zero point entropy, emergent monopoles and a quasi liquid–gas transition. To reveal the quantum spin dynamics that underpin these phenomena is an experimental challenge. Here we show how crucial information is contained in the frequency dependence of the magnetic susceptibility and in its high frequency or adiabatic limit. The typical response of Dy₂Ti₂O₇ spin ice indicates that monopole diffusion is Brownian but is underpinned by spin tunnelling and is influenced by collective monopole interactions. The adiabatic response reveals evidence of driven monopole plasma oscillations in weak applied field, and unconventional critical behaviour in strong applied field. Our results clarify the origin of the relatively high frequency response in spin ice. They disclose unexpected physics and establish adiabatic susceptibility as a revealing characteristic of exotic spin systems.

Isolated magnetic monopoles are usually forbidden, but can arise as quasiparticles in magnetically frustrated spin-ice materials. Bovo et al. explore the classical and quantum natures of these excitations through their influence on the high frequency magnetic susceptibility of dysprosium titanate.

Generalized longitudinal susceptibility for magnetic monopoles in spin ice

The generalized longitudinal susceptibility χ(q,ω) affords a sensitive measure of the spatial and temporal correlations of magnetic monopoles in spin ice. Starting with the monopole model, a mean field expression for χ(q,ω) is derived as well as expressions for the mean square longitudinal field and induction at a point. Monopole motion is shown to be strongly correlated, and both spatial and temporal correlations are controlled by the dimensionless monopole density x which defines the ratio of the magnetization relaxation rate and the monopole hop rate. Thermal effects and spin-lattice relaxation are also considered. The derived equations are applicable in the temperature range where the Wien effect for magnetic monopoles is negligible. They are discussed in the context of existing theories of spin ice and the following experimental techniques: DC and AC magnetization, neutron scattering, neutron spin echo and longitudinal and transverse field μSR. The monopole theory is found to unify diverse experimental results, but several discrepancies between theory and experiment are identified. One of these, concerning the neutron scattering line shape, is explained by means of a phenomenological modification to the theory.

Vindication of Yb2Ti2O7 as a model exchange quantum spin ice

We use numerical linked-cluster expansions to compute the specific heat C(T) and entropy S(T) of a quantum spin ice Hamiltonian for Yb2Ti2O7 using anisotropic exchange interactions, recently determined from inelastic neutron scattering measurements, and find good agreement with experimental calorimetric data. This vindicates Yb2Ti2O7 as a model quantum spin ice. We find that in the perturbative weak quantum regime, such a system has a ferrimagnetic ordered ground state, with two peaks in C(T): a Schottky anomaly signaling the paramagnetic to spin ice crossover, followed at a lower temperature by a sharp peak accompanying a first-order phase transition to the ordered state. We suggest that the two C(T) features observed in Yb2Ti2O7 are associated with the same physics. Spin excitations in this regime consist of weakly confined spinon-antispinon pairs. We anticipate that the conventional ground state with exotic quantum dynamics will prove a prevalent characteristic of many real quantum spin ice materials.

Power-law spin correlations in the pyrochlore antiferromagnet Tb(2)Ti(2)O(7)

We investigate the low-temperature state of the rare-earth pyrochlore Tb(2)Ti(2)O(7) using polarized neutron scattering. Tb(2)Ti(2)O(7) is often described as an antiferromagnetic spin liquid with spin correlations extending over lengths comparable to individual tetrahedra of the pyrochlore lattice. We confirm this picture at 20 K but find that at 0.05 K the data contain evidence of pinch-point scattering, suggesting that the low temperature state of Tb(2)Ti(2)O(7) has power-law spin correlations.