The history of spin ice

Skjærvø S.H., Marrows C.H., Stamps R.L., Heyderman L.J.

Artificial spin ices consist of nanomagnets arranged on the sites of various periodic and aperiodic lattices. They have enabled the experimental investigation of a variety of fascinating phenomena such as frustration, emergent magnetic monopoles and phase transitions that have previously been the domain of bulk spin crystals and theory, as we discuss in this Review. Artificial spin ices also show promise as reprogrammable magnonic crystals and, with this in mind, we give an overview of the measurements of fast dynamics in these magnetic metamaterials. We survey the variety of geometries that have been implemented, in terms of both the form of the nanomagnets and the lattices on which they are placed, including quasicrystalline systems and artificial spin systems in 3D. Different magnetic materials can also be incorporated to modify anisotropies and blocking temperatures, for example. With this large variety of systems, the way is open to discover new phenomena, and we complete this Review with possible directions for the future. Artificial spin ices are metamaterials displaying fascinating phenomena arising from the collective behaviour of nanoscale magnets. We review recent developments in terms of emergent magnetic monopoles, phase transitions, dynamics and geometries, and discuss future directions for research and potential applications.

Edberg R., Sandberg L.Ø., Bakke I.M., Haubro M.L., Folkers L.C., Mangin-Thro L., Wildes A., Zaharko O., Guthrie M., Holmes A.T., Sørby M.H., Lefmann K., Deen P.P., Henelius P.

The magnetically frustrated spin ice family of materials is host to numerous exotic phenomena such as magnetic monopole excitations and macroscopic residual entropy extending to low temperature. A finite-temperature ordering transition in the absence of applied fields has not been experimentally observed in the classical spin ice materials Dy2Ti2O7 and Ho2Ti2O7. Such a transition could be induced by the application of pressure, and in this work we consider the effects of uniaxial pressure on classical spin ice systems. Theoretically we find that the pressure induced ordering transition in Dy2Ti2O7 is strongly affected by the dipolar interaction. We also report measurements on the neutron structure factor of Ho2Ti2O7 under pressure, and compare the experimental results to the predictions of our theoretical model.

Barry K., Zhang B., Anand N., Xin Y., Vailionis A., Neu J., Heikes C., Cochran C., Zhou H., Qiu Y., Ratcliff W., Siegrist T., Beekman C.

We present an extensive study on the effect of substrate orientation, strain, stoichiometry and defects on spin ice physics in Ho$_2$Ti$_2$O$_7$ thin films grown onto yttria-stabilized-zirconia substrates. We find that growth in different orientations produces different strain states in the films. All films exhibit similar c-axis lattice parameters for their relaxed portions, which are consistently larger than the bulk value of 10.10 \AA. Transmission electron microscopy reveals anti-site disorder and growth defects to be present in the films, but stuffing is not observed. The amount of disorder depends on the growth orientation, with the (110) film showing the least. Magnetization measurements at 1.8 K show the expected magnetic anisotropy and saturation magnetization values associated with a spin ice for all orientations; shape anisotropy is apparent when comparing in and out-of-plane directions. Significantly, only the (110) oriented films display the hallmark spin ice plateau state in magnetization, albeit less well-defined compared to the plateau observed in a single crystal. Neutron scattering maps on the more disordered (111) oriented films show the Q=0 phase previously observed in bulk materials, but the Q=X phase giving the plateau state remains elusive. We conclude that the spin ice physics in thin films is modified by defects and strain, leading to a reduction in the temperature at which correlations drive the system into the spin ice state.

Dusad R., Kirschner F.K., Hoke J.C., Roberts B.R., Eyal A., Flicker F., Luke G.M., Blundell S.J., Davis J.C.

Magnetic monopoles1–3 are hypothetical elementary particles with quantized magnetic charge. In principle, a magnetic monopole can be detected by the quantized jump in magnetic flux that it generates upon passage through a superconducting quantum interference device (SQUID)4. Following the theoretical prediction that emergent magnetic monopoles should exist in several lanthanide pyrochlore magnetic insulators5,6, including Dy2Ti2O7, the SQUID technique has been proposed for their direct detection6. However, this approach has been hindered by the high number density and the generation–recombination fluctuations expected of such thermally generated monopoles. Recently, theoretical advances have enabled the prediction of the spectral density of magnetic-flux noise from monopole generation–recombination fluctuations in these materials7,8. Here we report the development of a SQUID-based flux noise spectrometer and measurements of the frequency and temperature dependence of magnetic-flux noise generated by Dy2Ti2O7 crystals. We detect almost all of the features of magnetic-flux noise predicted for magnetic monopole plasmas7,8, including the existence of intense magnetization noise and its characteristic frequency and temperature dependence. Moreover, comparisons of simulated and measured correlation functions of the magnetic-flux noise indicate that the motions of magnetic charges are strongly correlated. Intriguingly, because the generation–recombination time constant for Dy2Ti2O7 is in the millisecond range, magnetic monopole flux noise amplified by SQUID is audible to humans. Magnetic-flux noise measurements with a SQUID-based spectrometer demonstrate the presence of a magnetic monopole plasma in Dy2Ti2O7.

Raban V., Suen C.T., Berthier L., Holdsworth P.C.

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 results to recent experiments on Ho$_2$Ir$_2$O$_7$ and propose a wide ranging set of new experiments that exploit the phase diagram, including high pressure protocols, dynamical scaling of Kibble-Zurek form and universal violations of the fluctuation-dissipation theorem.

Gaudet J., Smith E. ., Dudemaine J., Beare J., Buhariwalla C. ., Butch N. ., Stone M. ., Kolesnikov A. ., Xu G., Yahne D. ., Ross K. ., Marjerrison C. ., Garrett J. ., Luke G. ., Bianchi A. ., et. al.

Neutron scattering measurements on the pyrochlore magnet Ce$_2$Zr$_2$O$_7$ reveal an unusual crystal field splitting of its lowest $J$=5/2 multiplet, such that its ground state doublet is composed of m$_J$=$\pm$3/2, giving these doublets a dipole - octupole (DO) character with local Ising anisotropy. Its magnetic susceptibility shows weak antiferromagnetic correlations with $\theta_{CW}$=-0.4(2)K, leading to a naive expectation of an All-In, All-Out ordered state at low temperatures. Instead our low energy inelastic neutron scattering measurements show a dynamic quantum spin ice state, with suppressed scattering near |$\textbf{Q}$|=0, and no long range order at low temperatures. This is consistent with recent theory predicting symmetry enriched U(1) quantum spin liquids for such DO doublets decorating the pyrochlore lattice. Finally, we show that disorder, especially oxidation of powder samples, is important in Ce$_2$Zr$_2$O$_7$ and could play an important role in the low temperature behaviour of this material.

Paulsen C., Giblin S.R., Lhotel E., Prabhakaran D., Matsuhira K., Balakrishnan G., Bramwell S.T.

Extensive work on single molecule magnets has identified a fundamental mode of relaxation arising from the nuclear-spin assisted quantum tunnelling of nearly independent and quasi-classical magnetic dipoles. Here we show that nuclear-spin assisted quantum tunnelling can also control the dynamics of purely emergent excitations: magnetic monopoles in spin ice. Our low temperature experiments were conducted on canonical spin ice materials with a broad range of nuclear spin values. By measuring the magnetic relaxation, or monopole current, we demonstrate strong evidence that dynamical coupling with the hyperfine fields bring the electronic spins associated with magnetic monopoles to resonance, allowing the monopoles to hop and transport magnetic charge. Our result shows how the coupling of electronic spins with nuclear spins may be used to control the monopole current. It broadens the relevance of the assisted quantum tunnelling mechanism from single molecular spins to emergent excitations in a strongly correlated system. Spin ice compounds have localised excitations that behave as magnetic monopoles which move by hopping from site to site, creating a chain of spins. Here the authors show that the hyperfine coupling between electron and nuclear spins is an important part of the mechanism underlying monopole motion.

Bovo L., Rouleau C.M., Prabhakaran D., Bramwell S.T.

Vertex models are an important class of statistical mechanical system that admit exact solutions and exotic physics. Applications include water ice, ferro- and antiferro-electrics, spin ice and artificial spin ice. Here we show that it is possible to engineer spin ice films with atomic-layer precision down to the monolayer limit. Specific heat measurements show that these films, which have a fundamentally different symmetry to bulk spin ice, realise systems close to the two-dimensional F-model, with exotic phase transitions on topologically-constrained configurational manifolds. Our results show how spin ice thin films can release the celebrated Pauling entropy of spin ice without an anomaly in the specific heat. They also significantly expand the class of vertex models available to experiment. Magnetic spin ice compounds are described by vertex models, which have been intensively studied for their exotic properties. Bovo et al. show thin films of Dy2Ti2O7 have structures distinct from bulk crystals and come close to realising the two-dimensional F-model, which has an unusual ordering transition in the Berezinskii–Kosterlitz–Thouless class.

Kaiser V., Bloxsom J., Bovo L., Bramwell S.T., Holdsworth P.C., Moessner R.

The low-temperature picture of dipolar spin ice in terms of the Coulomb fluid of its fractionalised magnetic monopole excitations has allowed analytic and conceptual progress far beyond its original microscopic spin description. Here we develop its thermodynamic treatment as a `magnetolyte', a fluid of singly and doubly charged monopoles, an analogue of the electrochemical system ${\rm 2 H_2O = H_3O^+ +OH^- = H_4O^{2+} + O^{2-}}$, but with perfect symmetry between oppositely charged ions. For this lattice magnetolyte, we present an analysis based on Debye-H\"uckel theory, which is accurate at all temperatures and incorporates `Dirac strings' imposed by the microscopic ice rule constraints at the level of Pauling's approximation. Our results are in close agreement with the specific heat from numerical simulations as well as new experimental measurements with an improved lattice correction, which we present here, on the spin ice materials $\mathrm{Ho_2Ti_2O_7}$ and $\mathrm{Dy_2Ti_2O_7}$. Our study of the magnetolyte shows how electrochemistry can emerge in non-electrical systems. We also provide new experimental tests of Debye-H\"uckel theory and its extensions. The application of our results also yields insights into the electrochemical behaviour of water ice and liquid water, which are closely related to the spin ice magnetolyte.

Giblin S. ., Twengström M., Bovo L., Ruminy M., Bartkowiak M., Manuel P., Andresen J. ., Prabhakaran D., Balakrishnan G., Pomjakushina E., Paulsen C., Lhotel E., Keller L., Frontzek M., Capelli S. ., et. al.

Determining the fate of the Pauling entropy in the classical spin ice material Dy$_2$Ti$_2$O$_7$ with respect to the third law of thermodynamics has become an important test case for understanding the existence and stability of ice-rule states in general. The standard model of spin ice - the dipolar spin ice model - predicts an ordering transition at $T\approx 0.15$ K, but recent experiments by Pomaranski $et\ al.$ suggest an entropy recovery over long time scales at temperatures as high as $0.5$ K, much too high to be compatible with theory. Using neutron scattering and specific heat measurements at low temperatures and with long time scales ($0.35$ K$/10^6$ s and $0.5$ K$/10^5$ s respectively) on several isotopically enriched samples we find no evidence of a reduction of ice-rule correlations or spin entropy. High-resolution simulations of the neutron structure factor show that the spin correlations remain well described by the dipolar spin ice model at all temperatures. Further, by careful consideration of hyperfine contributions, we conclude that the original entropy measurements of Ramirez $et\ al.$ are, after all, essentially correct: the short-time relaxation method used in that study gives a reasonably accurate estimate of the equilibrium spin ice entropy due to a cancellation of contributions.

Sibille R., Gauthier N., Yan H., Ciomaga Hatnean M., Ollivier J., Winn B., Filges U., Balakrishnan G., Kenzelmann M., Shannon N., Fennell T.

In a quantum spin liquid, the magnetic moments of the constituent electron spins evade classical long-range order to form an exotic state that is quantum entangled and coherent over macroscopic length scales1,2. Such phases offer promising perspectives for device applications in quantum information technologies, and their study can reveal new physics in quantum matter. Quantum spin ice is an appealing proposal of one such state, in which the fundamental ground state properties and excitations are described by an emergent U(1) lattice gauge theory3–7. This quantum-coherent regime has quasiparticles that are predicted to behave like magnetic and electric monopoles, along with a gauge boson playing the role of an artificial photon. However, this emergent lattice quantum electrodynamics has proved elusive in experiments. Here we report neutron scattering measurements of the rare-earth pyrochlore magnet Pr2Hf2O7 that provide evidence for a quantum spin ice ground state. We find a quasi-elastic structure factor with pinch points—a signature of a classical spin ice—that are partially suppressed, as expected in the quantum-coherent regime of the lattice field theory at finite temperature. Our result allows an estimate for the speed of light associated with magnetic photon excitations. We also reveal a continuum of inelastic spin excitations, which resemble predictions for the fractionalized, topological excitations of a quantum spin ice. Taken together, these two signatures suggest that the low-energy physics of Pr2Hf2O7 can be described by emergent quantum electrodynamics. If confirmed, the observation of a quantum spin ice ground state would constitute a concrete example of a three-dimensional quantum spin liquid—a topical state of matter that has so far mostly been explored in lower dimensionalities. A detailed and systematic neutron scattering study of rare-earth pyrochlore magnet Pr2Hf2O7 provides evidence for a quantum spin ice state, and emergent lattice quantum electrodynamics consistent with theoretical predictions.

Östman E., Stopfel H., Chioar I., Arnalds U.B., Stein A., Kapaklis V., Hjörvarsson B.

The modification of geometry and interactions in two-dimensional magnetic nanosystems has enabled a range of studies addressing the magnetic order1–6, collective low-energy dynamics7,8 and emergent magnetic properties5, 9,10 in, for example, artificial spin-ice structures. The common denominator of all these investigations is the use of Ising-like mesospins as building blocks, in the form of elongated magnetic islands. Here, we introduce a new approach: single interaction modifiers, using slave mesospins in the form of discs, within which the mesospin is free to rotate in the disc plane 11 . We show that by placing these on the vertices of square artificial spin-ice arrays and varying their diameter, it is possible to tailor the strength and the ratio of the interaction energies. We demonstrate the existence of degenerate ice-rule-obeying states in square artificial spin-ice structures, enabling the exploration of thermal dynamics in a spin-liquid manifold. Furthermore, we even observe the emergence of flux lattices on larger length scales, when the energy landscape of the vertices is reversed. The work highlights the potential of a design strategy for two-dimensional magnetic nano-architectures, through which mixed dimensionality of mesospins can be used to promote thermally emergent mesoscale magnetic states. Coupling strengths differ between neighbours in square artificial spin ices, resulting in the loss of degeneracy. Introducing mesospins on vertices of the array alleviates this problem, by tuning the strength and ratio of the interaction energies.

Bramwell S.T.

Many liquid or liquid-like states remain stable down to temperatures well below the interaction energy scale, where mean-field theory predicts an ordering transition. In magnetism, correlated states such as spin ice and the spin liquid have been described as Coulomb phases, governed by an emergent gauge principle. In the physical chemistry of polar liquids, systems that evade mean field order have, in contrast, been described by Onsager’s theory of the reaction field. Here we observe that in the low-temperature limit, Onsager’s theory may be cast as a prototypical theory of the Coulomb phase. However at finite temperature, it describes a distinct geometrical state, characterised by harmonic functions. This state, labelled here the ‘harmonic phase’, is shown to occur experimentally in spin ice, a dipolar lattice system. It is suggested to be relevant to more general dipolar liquids. Spin ice materials can be described using idealised models of frustrated magnetism and have motivated a revisiting of the theory of interacting dipolar systems. Bramwell shows that Onsager’s theory of polar liquids describes the Coulomb phase and predicts a distinct ‘harmonic phase’ at finite temperature.

Twengström M., Bovo L., Gingras M.J., Bramwell S.T., Henelius P.

The demagnetizing factor N is of both conceptual interest and practical importance. Considering localized magnetic moments on a lattice, we show that for non-ellipsoidal samples, N depends on the spin dimensionality (Ising, XY, or Heisenberg) and orientation, as well as the sample shape and susceptibility. The generality of this result is demonstrated by means of a recursive analytic calculation as well as detailed Monte Carlo simulations of realistic model spin Hamiltonians. As an important check and application, we also make an accurate experimental determination of N for a representative collective paramagnet (i.e. the Dy2Ti2O7 spin ice compound) and show that the temperature dependence of the experimentally determined N agrees closely with our theoretical calculations. Our conclusion is that the well established practice of approximating the true sample shape with "corresponding ellipsoids" for systems with long-range interactions will in many cases overlook important effects stemming from the microscopic aspects of the system under consideration.

Perrin Y., Canals B., Rougemaille N.

All of the characteristics of the square-ice model are observed in an artificial square-ice system in which the two sublattices of nanomagnets are slightly vertically separated. Geometric frustration in atomic lattices, such as those in water ice and magnetic materials called spin ice, leads to rich physical behaviour. So-called artificial spin ice, consisting of two-dimensional lattices of nanomagnets, was developed as a means of modelling these systems. Magnetic ordering, where neighbouring magnets try to align such that one is spin-up and the other spin-down, cannot be optimized owing to geometric frustration in these systems, which can be directly imaged. Until now it has not been possible to realize a fundamental 'square ice' model in these artificial systems that is of direct relevance to real materials. Nicolas Rougemaille and colleagues have solved this problem by designing a square lattice in which one of the two sublattices of nanomagnets is slightly vertically displaced. This enables them to directly observe the predicted spin-liquid state. Artificial spin-ice systems are lithographically patterned arrangements of interacting magnetic nanostructures that were introduced as way of investigating the effects of geometric frustration in a controlled manner1,2,3,4. This approach has enabled unconventional states of matter to be visualized directly in real space5,6,7,8,9,10,11,12,13,14,15,16,17,18, and has triggered research at the frontier between nanomagnetism, statistical thermodynamics and condensed matter physics. Despite efforts to create an artificial realization of the square-ice model—a two-dimensional geometrically frustrated spin-ice system defined on a square lattice—no simple geometry based on arrays of nanomagnets has successfully captured the macroscopically degenerate ground-state manifold of the model19. Instead, square lattices of nanomagnets are characterized by a magnetically ordered ground state that consists of local loop configurations with alternating chirality1,20,21,22,23,24,25,26. Here we show that all of the characteristics of the square-ice model are observed in an artificial square-ice system that consists of two sublattices of nanomagnets that are vertically separated by a small distance. The spin configurations we image after demagnetizing our arrays reveal unambiguous signatures of a Coulomb phase and algebraic spin-spin correlations, which are characterized by the presence of ‘pinch’ points in the associated magnetic structure factor. Local excitations—the classical analogues of magnetic monopoles27—are free to evolve in an extensively degenerate, divergence-free vacuum. We thus provide a protocol that could be used to investigate collective magnetic phenomena, including Coulomb phases28 and the physics of ice-like materials.

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Sengodan N., Dzubinska A., Arun K., Vimaljith A.R., Nagalakshmi R., Čurlik I., Iľkovič S., Reiffers M.

Anonymous

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Magnetization noise measurements on the spin ice Dy2Ti2O7 have revealed a remarkable “pink noise” power spectrum S(f,T) below 4 K, including evidence of magnetic monopole excitations diffusing in a fractal landscape. However, at higher temperatures, the reported values of the anomalous exponent b(T) describing the high-frequency tail of S(f,T) are not easy to reconcile with other results in the literature, which generally suggest significantly smaller deviations from the Brownian motion value of b=2, that become negligible above T=20 K. We accurately estimate b(T) at temperatures between 2 and 20 K, using ac susceptibility measurements that, crucially, stretch up to the relatively high frequency of f=106 Hz. We show that previous noise measurements underestimate b(T) and we suggest reasons for this. Our results establish deviations in b(T) from b=2 up to about 20 K. However, we confirm that b(T) is sample dependent: The details of this dependence agree in part, though not completely, with previous studies of the effect of crystal defects on monopole population and diffusion. Our results establish the form of b(T) which characterizes the subtle, and evolving, nature of monopole diffusion in the dense Coulomb fluid, a highly correlated state, where several dynamical processes combine. Published by the American Physical Society 2025

Anonymous

We present a dynamic scaling theory to describe relaxation dynamics following a magnetic-field quench near an unconventional phase transition in the magnetic material spin ice. Starting from a microscopic model, we derive an effective description for the critical dynamics in terms of the seeding and growth of string excitations, and use this to find scaling forms in terms of time, reduced temperature, and monopole fugacity. We confirm the predictions of scaling theory using Monte Carlo simulations, which also show good quantitative agreement with analytical expressions valid in the limit of low monopole density. As well as being relevant for experiments in the spin ice materials, our results open the way for the study of dynamic critical properties in a family of unconventional classical phase transitions. Published by the American Physical Society 2025

Gao B., Desrochers F., Tam D.W., Kirschbaum D.M., Steffens P., Hiess A., Nguyen D.H., Su Y., Cheong S., Paschen S., Kim Y.B., Dai P.

Wu T., Chang Y., Wu A., Terilli M., Wen F., Kareev M., Choi E.S., Graf D., Zhang Q., Gu L., Wang Z., Pixley J.H., Chakhalian J.

In magnetic pyrochlore materials, the interplay of spin-orbit coupling, electronic correlations, and geometrical frustration gives rise to exotic quantum phases, including topological semimetals and spin ice. While these phases have been observed in isolation, the interface-driven phenomena emerging from their interaction have never been realized previously. Here, we report on the discovery of interfacial electronic anisotropy and rotational symmetry breaking at a heterostructure consisting of the Weyl semimetal Eu 2 Ir 2 O 7 and spin ice Dy 2 Ti 2 O 7 . Subjected to magnetic fields, we unveil a sixfold anisotropic transport response that is theoretically accounted by a Kondo-coupled heterointerface, where the spin ice’s field-tuned magnetism induces electron scattering in the Weyl semimetal’s topological Fermi-arc states. Furthermore, at elevated magnetic fields, we reveal a twofold anisotropic response indicative of the emergence of a symmetry-broken many-body state. This discovery showcases the potential of pyrochlore frustrated magnet/topological semimetal heterostructures in search of emergent interfacial phenomena.

Tokura Y., Motome Y., Ueda K.

Abstract Pyrochlore oxides with chemical formula of A 2 B 2 O7 exhibit a diverse range of electronic properties as a representative family of quantum materials. These properties mostly stem from strong electron correlations at the transition metal B site and typical geometrical frustration effects on the pyrochlore lattice. Furthermore, the coupling between the magnetic moments of the rare-earth A site and the conduction electrons at the B site, along with the relativistic spin–orbit coupling particularly affecting the 4d/5d electrons at the B site, gives rise to the topological characteristics of the correlated electrons. This review paper focuses on the metal–insulator transitions in pyrochlore oxides as evidence of the strong electron correlation, which is highlighted as a rich source of intriguing charge dynamics coupled with frustrated spin-orbital entangled magnetism.

Gorsd M., Grigera S.A., Borzi R.A.

Smith E.M., Lhotel E., Petit S., Gaulin B.D.

We review a key subset of the experimental studies that have recently focused on cubic pyrochlore magnets whose pseudospin-1 $/$ 2 degrees of freedom have mixed dipolar and octupolar character. We discuss how this comes about and how the character of the pseudospin-1 $/$ 2 can be experimentally determined. The minimal spin Hamiltonian for such magnetic insulators is known to give rise to a rich phase diagram with both disordered U(1) quantum spin ice (QSI) states and all-in–all-out (AIAO) noncollinear ordered states, each with dipolar and octupolar character. We focus primarily on experimental studies on two such single crystal systems, the $\TimesFont{J}$ = 5 $/$ 2 Ce2Zr2O7 and the $\TimesFont{J}$ = 9 $/$ 2 Nd2Zr2O7. We make the case that Ce2Zr2O7 is an excellent QSI ground-state candidate material, close to the border between QSIs with dipolar and octupolar symmetry. Nd2Zr2O7 exhibits an AIAO ordered phase, featuring an order parameter consisting of dipolar and octupolar magnetic moments. It is found to reside close to a QSI phase boundary and features dynamic fragmentation in its excitation spectrum.

Wu S., Zhao L., Song W., Tan M., Jin F., Ying T., Yin J., Zhang Q.

Oliveira S., Santos J.P., Sá Barreto F.C.

Abstract In this work, we presented a new theoretical approach based on Mean-Field Theory, employing a hybrid Hamiltonian (spin/charge) in the spin-1/2 antiferromagnetic Ising model on lattices exhibiting geometric frustration. The study was conducted using the mean-field theory derived from Bogoliubov’s inequality to obtain a generic expression for the free energy in any frustrated lattice. To validate this theoretical approach, we applied the model to both the pyrochlore and kagome lattices occupied by spin-1/2 described by antiferromagnetic Ising model. The results revealed key features of geometric frustration in the studied structures, consistent with previous results reported in the literature, such as residual entropy, the characteristic behavior of the specific heat, and the emergence of plateaus in the magnetization curves.

Gutiérrez-Llorente A.

Topological quantum materials that show strongly correlated electrons as well as topological order, for which spin–orbit coupling is a key ingredient, exhibit novel states of matter. One such example is the family of pyrochlore iridates, featuring strong spin–orbital coupling, strong electron interactions, as well as geometric frustration, making them an ideal platform to study novel topological phases. High-quality epitaxial pyrochlore iridate films, although challenging to produce, provide a pathway to explore unconventional behaviors and unravel the intrinsic properties of these largely unexplored materials. Additionally, designing interfaces with specific properties is crucial to creating multilayered devices that can achieve significant technological breakthroughs using topological states of these materials. This article reviews experimental research on epitaxial pyrochlore iridate thin films, discussing evidence of topological phases found in them. Future research directions are outlined, which include exploring the rich tunability offered by chemical doping, especially when combined with the design of epitaxial heterostructures.

Chung K.T., Gingras M.J.

We introduce a geometrically frustrated classical Ising model, dubbed the “spin-vorticity model,” whose ground-state manifold is a classical spin liquid: a 2-form Coulomb phase. We study the thermodynamics of this model both analytically and numerically, exposing the presence of algebraically decaying correlations and an extensive ground-state entropy, and give a comprehensive account of its ground-state properties and excitations. Each classical ground state may be decomposed into collections of closed two-dimensional membranes, supporting fractionalized string excitations attached to the edges of open membranes. At finite temperature, the model can then be described as a gas of closed strings in a background of fluctuating membranes. The emergent gauge structure of this spin liquid is naturally placed in the language of 2-form electrodynamics, which describes one-dimensional charged strings coupled to a rank-2 antisymmetric gauge field. After establishing the classical spin-vorticity model, we consider perturbing it with quantum exchange interactions, deriving an effective “membrane exchange” model of the quantum dynamics, analogous to ring exchange in quantum spin ice. We demonstrate the existence of a Rokhsar-Kivelson point where the quantum ground state is an equal-weight superposition of all classical ground-state configurations, i.e., a quantum spin liquid. The quantum aspects of this spin liquid are exposed by mapping the membrane exchange model to a strongly coupled frustrated 2-form U(1) lattice gauge theory. We further demonstrate how to quantize the string excitations by coupling a 1-form string field to the 2-form U(1) gauge field, thus mapping a quantum spin model to a 2-form U(1) gauge-Higgs model. We discuss the stability of the gapless deconfined phase of this gauge theory and the possibility of realizing a class of quantum phases of matter: 2-form U(1) quantum spin liquids. Published by the American Physical Society 2025