We have studied the field induced magnetic structures in the spin liquid Tb$_2$Ti$_2$O$_7$, in a wide temperature (0.3$<$$T$$<$270 K) and field (0$<$$H$$<$7 T) range, by single crystal neutron diffraction with $\bf{H}$ // [110] axis. We combined unpolarized neutron data with polarized ones, analyzed within the local susceptibility model. A ferromagnetic-like structure with $\bf{k}$ = 0 propagation vector is induced, whose local order at low field and low temperature is akin to spin ice. The four Tb ions separate in $α$ and $β$ chains having different values of the magnetic moments, which is quantitatively explained by taking the crystal field anisotropy into account. Above 2 T and below 2 K, an antiferromagnetic-like structure with $\bf{k}$ = (0,0,1) is induced besides the $\bf{k}$ = 0 structure. It shows a reentrant behavior and extends over a finite length scale. It occurs together with a broadening of the nuclear peaks, which suggests a field induced distortion and magnetostriction effect.
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 diagonalisation, numerical linked-cluster and series expansions, we find that the previously-studied U(1) quantum spin liquid, in its pi-flux phase, transforms into a nematic quantum spin liquid at a high-symmetry, SU(2…
Recent experiments on the spin-ice material Dy2Ti2O7 suggest that the Pauling "ice entropy", characteristic of its classical Coulombic spin-liquid state, may be lost at low temperatures [D. Pomaranski et al., Nature Phys. 9, 353 (2013)]. However, despite nearly two decades of intensive study, the nature of the equilibrium ground state of spin ice remains uncertain. Here we explore how long-range dipolar interactions D, short-range exchange interactions, and quantum fluctuations combine to determine the ground state of dipolar spin ice. We identify a new organisational principle, namely that or…
Frustration in magnetic interactions can give rise to disordered ground states with subtle and beautiful properties. The spin ices Ho2Ti2O7 and Dy2Ti2O7 exemplify this phenomenon, displaying a classical spin liquid state, with fractionalized magnetic--monopole excitations. Recently there has been great interest in closely-related "quantum spin ice" materials, following the realization that anisotropic exchange interactions could convert spin ice into a massively-entangled, quantum, spin liquid, where magnetic monopoles become the charges of an emergent quantum electrodynamics. Here we show tha…
Quantum spin liquids are highly entangled ground states of quantum systems with emergent gauge structure, fractionalized spinon excitations, and other unusual properties. While these features clearly distinguish quantum spin liquids from conventional, mean-field-like states at zero temperature (T), their status at T>0 is less clear. Strictly speaking, it is known that most quantum spin liquids lose their identity at non-zero temperature, being in that case adiabatically transformable into a trivial paramagnet. This is the case for the U(1) quantum spin liquid states recently proposed to occur …
We have studied the Tb$_2$Sn$_2$O$_7$ frustrated magnet by neutron diffraction under isotropic pressure of 4.6 GPa, combined with uniaxial pressure of 0.3 GPa, in the temperature range 0.06 K$<$T$<$100 K. Magnetic order persists under pressure but the ordered spin ice structure stabilized at ambient pressure below 1.3 K partly transforms into an antiferromagnetic one. The long range ordered moment at 0.06 K is reduced under pressure, which is interpreted by a pressure induced enhancement of the spin liquid fluctuations. Above the ordering transition, short range spin correlations are affected …
Inelastic neutron scattering reveals a broad continuum of excitations in Pr$_2$Zr$_2$O$_7$, the temperature and magnetic field dependence of which indicate a continuous distribution of quenched transverse fields ($Δ$) acting on the non-Kramers Pr$^{3+}$ crystal field ground state doublets. Spin-ice correlations are apparent within 0.2 meV of the Zeeman energy. A random phase approximation provides an excellent account of the data with a transverse field distribution $ρ(Δ)\propto (Δ^2+Γ^2)^{-1}$ where $ Γ=0.27(1)$ meV. Established during high temperature synthesis due to an underlying structura…
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. Remarkably, the spin ices provide one of very few experimentally realised examples of fractionalization because their elementary excitations can be regarded as magnetic monopoles and, over some temperature range, the 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 class…
Pyrochlore iridates and pyrochlore ices are two families of materials where novel quantum phenomena are intertwined with strong spin-orbit coupling, substantial electron correlation and geometrical frustration. Motivated by the puzzling experiments on two pyrochlore systems Pr2Ir2O7 and Yb2Ti2O7, we study the proximate Ising orders and the quantum phase transition out of quantum spin ice U(1) quantum spin liquid (QSL). We apply the electromagnetic duality of the compact quantum electrodynamics to analyze the "magnetic monopoles" condensation for U(1) QSL. The monopole condensation transition r…
An exact solution is obtained for a model of itinerant electrons coupled to ice-rule variables on the tetrahedron Husimi cactus, an analogue of the Bethe lattice of corner-sharing tetrahedra. It reveals a quantum critical point with the emergence of non-Fermi-liquid behavior in melting of the "charge ice" insulator. The electronic structure is compared with the numerical results for the pyrochlore-lattice model to elucidate the physics of electron systems interacting with the tetrahedron ice rule.
The Coulombic quantum spin liquid in quantum spin ice is an exotic quantum phase of matter that emerges on the pyrochlore lattice and is currently actively searched for. Motivated by recent experiments on the Yb-based breathing pyrochlore material Ba$_3$Yb$_2$Zn$_5$O$_{11}$, we theoretically study the phase diagram and magnetic properties of the relevant spin model. The latter takes the form of a quantum spin ice Hamiltonian on a breathing pyrochlore lattice, and we analyze the stability of the quantum spin liquid phase in the absence of the inversion symmetry which the lattice breaks explicit…
Quantum spin liquids (QSL) have generated considerable excitement as phases of matter with emergent gauge structures and fractionalized excitations. In this context, phase transitions out of QSLs have been widely discussed as Higgs transitions from deconfined to confined phases of a lattice gauge theory. However the possibility of a wider range of novel phases, occuring between these two limits, has yet to be systematically explored. In this Letter, we develop a formalism which allows for interactions between fractionalised quasiparticles coming from the constraint on the physical Hilbert spac…
It is a salient experimental fact that a large fraction of candidate spin liquid materials freeze as the temperature is lowered. The question naturally arises whether such freezing is intrinsic to the spin liquid ("disorder-free glassiness") or extrinsic, in the sense that a topological phase simply coexists with standard freezing of impurities. Here, we demonstrate a surprising third alternative, namely that freezing and topological liquidity are inseparably linked. The topological phase reacts to the introduction of disorder by generating degrees of freedom of a new type (along with interact…
Competing interactions in the so-called spin-ice compounds stabilize a frustrated ground-state with finite zero-point entropy and, interestingly, emergent magnetic monopole excitations. The properties of these monopoles are at the focus of recent research with particular emphasis on their quantum dynamics. It is predicted that each monopole also possesses an electric dipole moment, which allows to investigate their dynamics via the dielectric function ε(ν). Here, we report on broadband spectroscopic measurements of ε(ν) in Dy2Ti2O7 down to temperatures of 200mK with a specific focus on the cri…
Recently there have been a number of experiments on Ce$_2$Zr$_2$O$_7$ and Ce$_2$Sn$_2$O$_7$, suggesting that these materials host a three-dimensional quantum spin liquid with emergent photons and fractionalized spinon excitations. However, the interpretation of the data to determine the precise nature of the quantum spin liquids is still under debate. The Kramers doublet in Ce$^{3+}$ local moment offers unusual pseudo-spin degrees of freedom as the $x$ and $z$ components transform as a dipole and $y$ component as an octupole. This leads to a variety of possible quantum spin liquid (or quantum …
We carry out an analytical study of quantum spin ice, a U$(1)$ quantum spin liquid close to the classical spin ice solution for an effective spin $1/2$ model with anisotropic exchange couplings $J_{zz}$, $J_{\pm}$ and $J_{z\pm}$ on the pyrochlore lattice. Starting from the quantum rotor model introduced by Savary and Balents in Phys. Rev. Lett. 108, 037202 (2012), we retain the dynamics of both the spinons and gauge field sectors in our treatment. The spinons are described by a bosonic representation of quantum XY rotors while the dynamics of the gauge field is captured by a phenomenological H…
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 …
We have studied the field-dependent ac magnetic susceptibility of single crystals of Dy2Ti2O7 spin ice along the [111] direction in the temperature range 1.8 K - 7 K. Our data reflect the onset of local spin ice order in the appearance of different field regimes. In particular, we observe a prominent feature at approximately 1.0 T that is a precursor of the low-temperature metamagnetic transition out of field-induced kagome ice, below which the kinetic constraints imposed by the ice rules manifest themselves in a substantial frequency-dependence of the susceptibility. Despite the relatively hi…
We present a theory of a hybrid quantum liquid state, $\textit{quantum spin-dipole liquid}$ (QSDL), in a hydrogen-bonded electron system, by combining a quantum proton ice and Anderson's resonating valence bond spin liquid theory, motivated by the recent experimental discovery of a quantum spin liquid with proton fluctuations in $κ$-H$_3$(Cat-EDT-TTF)$_2$ (a.k.a. H-Cat). In our theory, an electron spin liquid and a proton dipole liquid are realized simultaneously in the ground state called $\textit{quantum valence bond ice}$. In this state, neither of them can be established independently of t…
We present the full phase diagram of the dumbbell model of spin ice as a function of temperature, chemical potential and staggered chemical potential which breaks the translational lattice symmetry in favour of charge crystal ordering. We observe a double winged structure with five possible phases, monopole fluid (spin ice), fragmented single monopole crystal phases and double monopole crystal, the zinc blend structure. Our model provides a skeleton for liquid-liquid phase transitions and for the winged structures observed for itinerant magnets under pressure and external field. We relate our …
The mathematics of gauge theories lies behind many of the most profound advances in physics in the last 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 physic…
We combine two aspects of magnetic frustration, multiferroicity and emergent quasi-particles in spin liquids, by studying magneto-electric monopoles. Spin ice offers to couple these emergent topological defects to external fields, and to each other, in unusual ways, making possible to lift the degeneracy underpinning the spin liquid and to potentially stabilize novel forms of charge crystals, opening the path to a "magnetic crystallography". In developing the general phase diagram including nearest-neighbour coupling, Zeeman energy, electric and magnetic dipolar interactions, we uncover the em…
Recent work has highlighted remarkable effects of classical thermal fluctuations in the dipolar spin ice compounds, such as "artificial magnetostatics", manifesting as Coulombic power-law spin correlations and particles behaving as diffusive "magnetic monopoles". In this paper, we address quantum spin ice, giving a unifying framework for the study of magnetism of a large class of magnetic compounds with the pyrochlore structure, and in particular discuss Yb2Ti2O7 and extract its full set of Hamiltonian parameters from high field inelastic neutron scattering experiments. We show that fluctuatio…
The origin of the lowest-temperature anomaly reported several years ago using a polycrystalline sample of the spin ice compound Dy2Ti2O7 had remained unresolved. Here we finally clarify its origin by susceptibility measurements down to 65 mK using single crystals under accurate control of the magnetic fields in two independent directions. We demonstrate that the transition is induced under a subtle field combination that precisely cancel the nearest-neighbor spin interactions acting on the spins on the triangular lattice within the pyrochlore structure. Contrary to the other two fieldinduced t…
We derive an effective $Z_2$ gauge theory to describe the quantum kagome ice (QKI) state that has been observed by Carrasquilla $\textit{et. al.}$ in Monte Carlo studies of the $S = 1/2$ kagome XYZ model in a Zeeman field. The numerical results on QKI are consistent with, but do not confirm or rule out, the hypothesis that it is a $Z_2$ spin liquid. Our effective theory allows us to explore this hypothesis and make a striking prediction for future numerical studies, namely that symmetry-protected vison zero modes arise at lattice disclination defects, leading to a Curie defect term in the spin…
We study the U(1) quantum spin liquid on the pyrochlore spin ice systems. For the non-Kramers doublets such as Pr$^{3+}$ and Tb$^{3+}$, we point out that the inelastic neutron scattering not only detects the low-energy gauge photon, but also contains the continuum of the "magnetic monopole" excitations. Unlike the spinons, these "magnetic monopoles" are purely of quantum origin and have no classical analogue. We further point out that the "magnetic monopole" experiences a background dual "$π$" flux due to the spin-1/2 nature of the local moment when the "monopole" hops on the dual diamond latt…
Ho2Ti2O7 and Dy2Ti2O7 spin ices exhibit elementary excitations akin to magnetic monopoles. Here we focus on Tb2Ti2O7 spin liquid, where correlated magnetic moments keep fluctuating down to very low temperatures. Using a monopole picture, we have re-analyzed the field-induced magnetic structure previously determined by neutron diffraction in Tb2Ti2O7. We show that under a high field applied along a [110] direction, Tb2Ti2O7 orders as a three dimensional arrangement of monopole and antimonopole double layers. In contrast, Ho2Ti2O7 spin ice in the same conditions behaves as a monopole-free state.…
Actively shought since the turn of the century, two-dimensional quantum spin liquids (QSLs) are exotic phases of matter where magnetic moments remain disordered even at extremely low temperatures. Despite ongoing searches, QSLs remain elusive, due to a lack of concrete knowledge of the microscopic mechanisms that inhibit magnetic order in real materials. Here, we study a theoretical model for a broad class of frustrated magnetic rare-earth pyrochlore materials called "quantum spin ices". When subject to an external magnetic field along the [111] crystallographic direction, the resulting spin i…
We study the effects of lattice deformations on the Kagome spin ice, with Ising spins coupled by nearest neighbor exchange and long range dipolar interactions, in the presence of in-plane magnetic fields. We describe the lattice energy according to the Einstein model, where each site distortion is treated independently. Upon integration of lattice degrees of freedom, effective quadratic spin interactions arise. Classical MonteCarlo simulations are performed on the resulting model, retaining up to third neighbor interactions, under different directions of the magnetic field. We find that, as th…
Quantum spin ice in pyrochlore lattice exemplifies three dimensional frustrated spin systems.In existing studies, Bose-Einstein condensation of bosonic spinons gives rise to magnetically ordered ground state.A truly liquid quantum spin superfluid state that manifests a deconfined Higgs condensate of spinons is demonstrated.This state is shown to occur in the fully antiferromagnetic case of the non-Kramers quantum spin ice with a fluctuations-induced first-order quantum phase transition from the $U(1)$ spin liquid to the spin superfluid.The spin superfluid density jump is obtained analytically.
We introduce 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 the two-in/two-out ice states to include branching structures built from three-i…
We develop a theory of the dynamical response of a minimal model of quantum spin ice (QSI) by means of inelastic light scattering. In particular, we are interested in the Raman response of the fractionalized U(1) spin liquid realized in the XXZ QSI. We show that the low-energy Raman intensity is dominated by spinon and gauge fluctuations. We find that the Raman response in the QSI state of that model appears only in the $T_{2g}$ polarization channel. We show that the Raman intensity profile displays a broad continuum from the spinons and coupled spinon and gauge fluctuations, and a low-energy …
We consider a classical model of charges $\pm q$ on a pyrochlore lattice in the presence of long range Coulomb interactions. This model first appeared in the early literature on charge order in magnetite. In the limit where the interactions become short-ranged, the model has a ground state with an extensive entropy and dipolar charge-charge correlations. When long range interactions are introduced, the exact degeneracy is broken. We study the thermodynamics of the model and show the presence of a correlated charge liquid within a temperature window in which the physics is well described as a l…
Quantum effect is expected to dictate the behaviour of physical systems at low temperature. For quantum magnets with geometrical frustration, quantum fluctuation usually lifts the macroscopic classical degeneracy, and exotic quantum states emerge. However, how different types of quantum processes entangle wave functions in a constrained Hilbert space is not well understood. Here, we study the topological entanglement entropy (TEE) and the thermal entropy of a quantum ice model on a geometrically frustrated kagome lattice. We find that the system does not show a $Z_2$ topological order down to …
Unbiased quantum Monte-Carlo simulations are performed on the nearest-neighbor spin-$\frac{1}{2}$ pyrochlore XXZ model with an antiferromagnetic longitudinal and a weak ferromagnetic transverse exchange couplings, $J$ and $J_\perp$. The specific heat exhibits a broad peak at $T_{\mathrm{CSI}}\sim0.2J$ 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 $|J_\perp|\ll J$, as in classical spin ice. On further cooling, the entropy restarts decaying for $J_\perp>J_{\perp c}\…
Arrested Relaxation in a Disorder-Free Coulomb Spin Liquid
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究院）；School of Physics, Georgia Institute of Technology（佐治亚理工学院物理学院）；Max-Planck-Institut für Physik komplexer Systeme（马克斯·普朗克复杂系统物理研究所）
Classical symmetry enriched topological orders and distinct monopole charges for dipole-octupole spin ices
International Center for Quantum Materials, Peking University（北京大学量子材料中心）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；The University of Hong Kong, Shenzhen Institute of Research and Innovation（香港大学深圳研究院）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学量子器件北京市重点实验室）
Spin ice is a frustrated magnetic system that at low temperatures exhibits a Coulomb phase, a classical spin liquid with topological order and deconfined excitations. This work establishes the presence of a Coulomb phase with coexisting ferromagnetic order in a microscopic model of classical spin ice subject to uniaxial lattice distortion. General theoretical arguments are presented for the presence of such a phase, and its existence is confirmed using Monte Carlo results. This example is used to illustrate generic properties of spin liquids with magnetic order, including deconfinement of mono…
We study the coupling between conventional (Maxwell) and emergent electrodynamics in quantum spin ice, a 3+1-dimensional $U(1)$ quantum spin liquid. We find that a uniform electric field can be used to tune the properties of both the ground state and excitations of the spin liquid. In particular, it induces emergent birefringence, rendering the speed of the emergent light anisotropic and polarization-dependent. A sufficiently strong electric field triggers a quantum phase transition into new $U(1)$ quantum spin liquid phases which trap emergent electric $π$-fluxes. The flux patterns of these n…
Motivated by the rapid experimental progress of quantum spin ice materials, we study the dynamical properties of pyrochlore spin ice in the U(1) spin liquid phases. In particular, we focus on the spinon excitations that appear at high energies and show up as an excitation continuum in the dynamic spin structure factor. The keen connection between the crystal symmetry fractionalization of the spinons and the spectral periodicity of the spinon continuum is emphasized and explicitly demonstrated. When the spinon experiences a background $π$ flux and the spinon continuum exhibits an enhanced spect…
We present the results of first-principle calculations using the Vienna Ab-initio Simulation Package (VASP) for a new class of organometallics labeled TM3C6O6 (TM =Sc, Ti, V, Cr, Fe, Co, Ni and Cu) in the form of planar, two-dimensional, periodic free-standing layers. These materials, which can be produced by on-surface coordination on metallic surfaces, have a kagome lattice of TM ions. Calculating the structural properties, we show that all considered materials have local magnetic moments in the ground state, but four of them (with Fe, Co, Ni and Cu) show spin-crossover behavior by changing …
The search for two dimensional quantum spin liquids, exotic magnetic states with an entangled ground state remaining disordered down to zero temperature, has been a great challenge in frustrated magnetism during the last decades. Recently, fractionalized excitations, called spinons, emerging from these states, have been evidenced in kagome and triangular antiferromagnets. In contrast, quantum ferromagnetic spin liquids in two dimensions, namely quantum kagome ices, have been less investigated, yet their classical counterparts exhibit amazing properties, magnetic monopole crystals as well as ma…
Understanding the nature and behaviour of excitations in quantum spin liquids, and in topological phases of matter in general, is of fundamental importance and has proven crucial for experimental detection and characterisation of candidate materials. Current theoretical and numerical techniques, however, have limited capabilities, especially when it comes to studying gapped excitations. Here, we propose a semiclassical numerical method to study systems whose spin liquid behaviour is underpinned by perturbative ring-exchange Hamiltonians. Our method can readily access both thermodynamic and spe…
Geometrically frustrated materials, such as spin ice or kagome lattice, are known to exhibit exotic Hall effect phenomena due to spin chirality. We explore Hall effect mechanism in an artificial honeycomb spin ice of Nd--Sn element using Hall probe and polarized neutron reflectivity measurements. In an interesting observation, a strong enhancement in Hall signal at relatively higher temperature of $T$ $\sim$ 20 K is detected. The effect is attributed to the planar Hall effect due to magnetic moment configuration in spin ice state in low field application. In the antiferromagnetic state of neod…
We have explored the spin liquid state in Tb2Ti2O7 with vibrating coil magnetometry down to 0.04 K under magnetic fields up to 5 T. We observe magnetic history dependence below $T < 0.2$K reminiscent of the classical spin ice systems Ho2Ti2O7 and Dy2Ti2O7. The magnetic phase diagram inferred from the magnetization is essentially isotropic, without evidence of magnetization plateaux as anticipated for so-called quantum spin ice, predicted theoretically for [111] when quantum fluctuations renormalize the interactions. Instead, the magnetization for $T \ll T*$ agrees semi-quantitatively with the …
We present numerical studies of dipolar spin ice in the presence of a magnetic field slightly tilted away from the [111] axis. We find a first-order transition from a kagome ice to a $\mathbf{q}=$X state when the external field is tilted toward the $[11\bar{2}]$ direction. This is consistent with the anomalous critical scattering previously observed in the neutron scattering experiment on the spin ice material $\textrm{Ho}_{2}\textrm{Ti}_{2}\textrm{O}_{7}$ in a tilted field [Nat. Phys. 3, 566 (2007)]. We show that this ordering originates from the antiferromagnetic alignment of spin chains on …
If spin liquids have been famously defined by what they are not, i.e. ordered, the past years have seen the frontier between order and spin liquid starting to fade, with a growing number of materials whose low-temperature physics cannot be explained without co-existence of (partial) magnetic order and spin fluctuations. Here we study an example of such co-existence in the presence of magnetic dipolar interactions, related to spin ice, where the order is long range and the fluctuations support a Coulomb gauge field. Topological defects are effectively coupled via energetic and entropic Coulomb …
We investigate a detail of a dodecamer cluster ordering in a double-exchange spin ice model on a kagomé lattice. In frustrated systems, ordinary spin orderings are suppressed and macroscopic degeneracy remains down to low temperatures. In some frustrated systems, the degeneracy is lifted due to residual interactions and cluster orderings are stabilized. In the present model, the spin ice state is first formed at intermediate temperatures, and further entropies are released at lower temperatures as the dodecamer phase emerges. Since the spin symmetry is not broken in the dodecamer phase, there …
