Quantum Spin Ice Research Articles

Fluctuation-induced first-order quantum phase transition of theU(1)spin liquid in a pyrochlore quantum spin ice

We predict, using a free-energy calculation, that the quantum phase transition between the $U(1)$ quantum spin liquid (QSL) and antiferromagnet (AFM) phases in certain pyrochlore quantum spin ice is first order rather than second order. This change in order from second to first order is induced by gauge fluctuations, which are explicitly taken into account at gauge theory level in our effective low-energy theory. We therefore have identified a fluctuation-induced first-order quantum phase transition in pyrochlore quantum spin ice. We explicitly derive the quantum free-energy description of this QSL-to-AFM phase transition and predict that it is a weakly first-order phase transition. We also briefly discuss the experimental relevance of this result.

Unconventional magnetic ground state in Yb2Ti2O7

We report low temperature specific heat and muon spin relaxation/rotation ($\mu$SR) measurements on both polycrystalline and single crystal samples of the pyrochlore magnet Yb$_2$Ti$_2$O$_7$. This system is believed to possess a spin Hamiltonian supporting a Quantum Spin Ice (QSI) ground state and to display sample variation in its low temperature heat capacity. Our two samples exhibit extremes of this sample variation, yet our $\mu$SR measurements indicate a similar disordered low temperature state down to 16 mK in both. We report little temperature dependence to the spin relaxation and no evidence for ferromagnetic order, in contrast to recent reports by Chang \emph{et al.} (Nat. Comm. {\bf 3}, 992 (2012)). Transverse field (TF) $\mu$SR measurements show changes in the temperature dependence of the muon Knight shift which coincide with heat capacity anomalies. We are therefore led to propose that Yb$_2$Ti$_2$O$_7$ enters a hidden order ground state below $T_c\sim265$ mK where the nature of the ordered state is unknown but distinct from simple long range order.

Crystal-field states of Pr3+in the candidate quantum spin ice Pr2Sn2O7

Neutron time-of-flight spectroscopy has been employed to study the crystal-field splitting of Pr$^{3+}$ in the pyrochlore stannate Pr$_{2}$Sn$_{2}$O$_{7}$. The crystal field has been parameterized from a profile fit to the observed neutron spectrum. The single-ion ground state is a well isolated non-Kramers doublet of $\Gamma_3^+$ symmetry with a large Ising-like anisotropy, $\chi_{zz} /\chi_\perp \approx 60$ at 10 K, but with a significant admixture of terms $| M_J \neq \pm J >$ which can give rise to quantum zero-point fluctuations. This magnetic state satisfies the requirements for quantum spin ice behavior.

Spin liquid regimes at nonzero temperature in quantum spin ice

Quantum spin liquids are highly entangled ground states of quantum systems with an 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 nonzero temperature, being then adiabatically transformable into a trivial paramagnet. This is the case for the $U(1)$ quantum spin liquid states recently proposed to occur in the quantum spin ice pyrochlores. Here we propose, however, that in practical terms, the latter quantum spin liquids can be regarded as phases distinct from the high-temperature paramagnet. Through a combination of gauge mean-field theory calculations and physical reasoning, we argue that these systems sustain both quantum spin liquid and thermal spin liquid phases, dominated by quantum fluctuations and entropy, respectively. These phases are separated by a first-order ``thermal confinement'' transition such that, for temperatures below the transition, spinons and emergent photons are coherently propagating excitations, and above it the dynamics is classical. Even for parameters for which the ground state is magnetically ordered and not a quantum spin liquid, this strong first-order transition occurs, preempting conventional Landau-type criticality. We argue that this picture explains the anomalously low-temperature phase transition observed in the quantum spin ice material Yb${}_{2}$Ti${}_{2}$O${}_{7}$.

Antiferromagnetic spin ice correlations at (12,12,12) in the ground state of the pyrochlore magnet Tb2Ti2O7

We present high-resolution single crystal time-of-flight neutron scattering measurements on the candidate quantum spin liquid pyrochlore Tb2Ti2O7 at low temperature and in a magnetic field. At ~70 mK and in zero field, Tb2Ti2O7 reveals diffuse magnetic elastic scattering at (1/2,1/2,1/2) positions in reciprocal space, consistent with short-range correlated regions based on a two-in, two-out spin ice configuration on a doubled conventional unit cell. This elastic scattering is separated from very low-energy magnetic inelastic scattering by an energy gap of ~0.06-0.08 meV. The elastic signal disappears under the application of small magnetic fields and upon elevating temperature. Pinch-point-like elastic diffuse scattering is observed near (1,1,1) and (0,0,2) in zero field at ~70 mK, in agreement with Fennell et al. (Ref. 1), supporting the quantum spin ice interpretation of Tb2Ti2O7.

Generic quantum spin ice

We consider possible exotic ground states of quantum spin ice as realized in rare earth pyrochlores. Prior work [Savary and Balents, Phys. Rev. Lett. 108, 037202 (2012).] introduced a gauge mean-field theory (gMFT) to treat spin or pseudospin Hamiltonians for such systems, reformulated as a problem of bosonic spinons coupled to a $U(1)$ gauge field. We extend gMFT to treat the most general nearest-neighbor exchange Hamiltonian, which contains a further exchange interaction. This term leads to interactions between spinons and requires a significant extension of gMFT, which we provide. As an application, we focus especially on the non-Kramers materials Pr${}_{2}T{M}_{2}$O${}_{7}$ ($TM=$ Sn, Zr, Hf, and Ir), for which the additional term is especially important, but for which an Ising-planar exchange coupling discussed previously is forbidden by time-reversal symmetry. In this case, when the planar $\mathit{XY}$ exchange is unfrustrated, we perform a full analysis and find three quantum ground states: a $U(1)$ quantum spin liquid (QSL), an antiferroquadrupolar ordered state and a noncoplanar ferroquadrupolar ordered one. We also consider the case of frustrated $\mathit{XY}$ exchange, and find that it favors a $\ensuremath{\pi}$-flux QSL, with an emergent line degeneracy of low-energy spinon excitations. This feature greatly enhances the stability of the QSL with respect to classical ordering.

Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice

The "spin ice" state found in the rare earth pyrochlore magnets Ho2Ti2O7 and Dy2Ti2O7 offers a beautiful realisation of classical magnetostatics, complete with magnetic monopole excitations. It has been suggested that in "quantum spin ice" materials, quantum-mechanical tunnelling between different ice configurations could convert the magnetostatics of spin ice into a quantum spin liquid which realises a fully dynamical, lattice-analogue of quantum electromagnetism. Here we explore how such a state might manifest itself in experiment, within the minimal microscopic model of a such a quantum spin ice. We develop a lattice field theory for this model, and use this to make explicit predictions for the dynamical structure factor which would be observed in neutron scattering experiments on a quantum spin ice. We find that "pinch points", seen in quasi-elastic scattering, which are the signal feature of a classical spin ice, fade away as a quantum ice is cooled to its zero-temperature ground state. We also make explicit predictions for the ghostly, linearly dispersing magnetic excitations which are the "photons" of this emergent electromagnetism. The predictions of this field theory are shown to be in quantitative agreement with Quantum Monte Carlo simulations at zero temperature.

Artificial Light in a Quantum Spin Ice

Artificial Light in a Quantum Spin Ice

Vindication ofYb2Ti2O7as 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.

Low-temperature magnetization in geometrically frustrated Tb2Ti2O7

The nature of the low temperature ground state of the pyrochlore compound Tb2Ti2O7 remains a puzzling issue. Dynamic fluctuations and short-range correlations persist down to 50 mK, as evidenced by microscopic probes. In parallel, magnetization measurements show irreversibilities and glassy behavior below 200 mK. We have performed magnetization and AC susceptibility measurements on four single crystals down to 57 mK. We did not observe a clear plateau in the magnetization as a function of field along the [111] direction, as suggested by the quantum spin ice model. In addition to a freezing around 200 mK, slow dynamics are observed in the AC susceptibility up to 4 K. The overall frequency dependence cannot be described by a canonical spin-glass behavior.

Vibrating-Coil Magnetometry of the Spin Liquid Properties ofTb2Ti2O7

We have explored the spin liquid state in Tb(2)Ti(2)O(7) 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 Ho(2)Ti(2)O(7) and Dy(2)TiTi(2)O(7). The magnetic phase diagram inferred from the magnetization is essentially isotropic, without evidence of magnetization plateaus as anticipated for so-called quantum spin ice, predicted theoretically for [111] when quantum fluctuations renormalize the interactions. Instead, the magnetization for T≪T* agrees semiquantitatively with the predictions of "all-in-all-out" (AIAO) antiferromagnetism. Taken together, this suggests that the spin liquid state in Tb(2)Ti(2)O(7) is akin to an incipient AIAO antiferromagnet.

Quantum Strings in Quantum Spin Ice

We study quantum spin ice in an external magnetic field applied along a <100> direction. When quantum spin fluctuations are weak, elementary excitations are quantum strings with monopoles at their ends manifested as multiple spin wave branches in the dynamical structure factor. Strong quantum fluctuations make the string tension negative and give rise to the deconfinement of monopoles. We discuss our results in the light of recent neutron scattering experiments in Yb2Ti2O7.

Quantum Ice: A Quantum Monte Carlo Study

Ice states, in which frustrated interactions lead to a macroscopic ground-state degeneracy, occur in water ice, in problems of frustrated charge order on the pyrochlore lattice, and in the family of rare-earth magnets collectively known as spin ice. Of particular interest at the moment are "quantum spin-ice" materials, where large quantum fluctuations may permit tunnelling between a macroscopic number of different classical ground states. Here we use zero-temperature quantum Monte Carlo simulations to show how such tunnelling can lift the degeneracy of a spin or charge ice, stabilizing a unique "quantum-ice" ground state-a quantum liquid with excitations described by the Maxwell action of (3+1)-dimensional quantum electrodynamics. We further identify a competing ordered squiggle state, and show how both squiggle and quantum-ice states might be distinguished in neutron scattering experiments on a spin-ice material.

Quantum Excitations in Quantum Spin Ice

A form of quantum electrodynamics emerges from interacting spins at low temperatures in the spin ice $Y\phantom{\rule{0}{0ex}}{b}_{2}\phantom{\rule{0}{0ex}}T\phantom{\rule{0}{0ex}}{i}_{2}\phantom{\rule{0}{0ex}}{O}_{7}$.

Quantum fluctuations in the effective pseudospin-12model for magnetic pyrochlore oxides

The effective quantum pseudospin-1/2 model for interacting rare-earth magnetic moments, which are locally described with atomic doublets, is studied theoretically for magnetic pyrochlore oxides. It is derived microscopically for localized Pr^{3+} 4f moments in Pr_2TM_2O_7 (TM = Zr, Sn, Hf, and Ir) by starting from the atomic non-Kramers magnetic doublets and performing the strong-coupling perturbation expansion of the virtual electron transfer between the Pr 4f and O 2p electrons. The most generic form of the nearest-neighbor anisotropic superexchange pseudospin-1/2 Hamiltonian is also constructed from the symmetry properties, which is applicable to Kramers ions Nd^{3+}, Sm^{3+}, and Yb^{3+} potentially showing large quantum effects. The effective model is then studied by means of a classical mean-field theory and the exact diagonalization on a single tetrahedron and on a 16-site cluster. These calculations reveal appreciable quantum fluctuations leading to quantum phase transitions to a quadrupolar state as a melting of spin ice for the Pr^{3+} case. The model also shows a formation of cooperative quadrupole moment and pseudospin chirality on tetrahedrons. A sign of a singlet quantum spin ice is also found in a finite region in the space of coupling constants. The relevance to the experiments is discussed.

Dynamically Induced Frustration as a Route to a Quantum Spin Ice State inTb2Ti2O7via Virtual Crystal Field Excitations and Quantum Many-Body Effects

The Tb2Ti2O7 pyrochlore magnetic material is attracting much attention for its spin liquid state, failing to develop long-range order down to 50 mK despite a Curie-Weiss temperature thetaCW approximately -14 K. In this Letter we reinvestigate the theoretical description of this material by considering a quantum model of independent tetrahedra to describe its low-temperature properties. The naturally tuned proximity of this system near a Néel to spin ice phase boundary allows for a resurgence of quantum fluctuation effects that lead to an important renormalization of its effective low-energy spin Hamiltonian. As a result, Tb2Ti2O7 is argued to be a quantum spin ice. We put forward an experimental test of this proposal using neutron scattering on a single crystal.