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Highly Frustrated Magnetism (wHFM21)

\noindent The kagome Heisenberg antiferromagnet decorated with quantum spins is a fascinating model to look for exotic quantum states including the elusive spin liquids. Despite the apparent simplicity of this model, there is still no consensus on the exact nature of the ground state and the excitation spectrum. Nevertheless, the current availability of quantum kagome materials, each with their own deviation to the pure Heisenberg model, allows to confront a fair number of experimental results to theoretical predictions and study the effect of perturbations inherent to real materials. I will focus on herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$. This is the emblematic compound in this family of materials because it is the closest realization to the pure Heisenberg model so far ($J\sim180$~K), with a nonmagnetic dynamical ground state.\\ \noindent I will present recent $^{17}$O NMR measurements on a high-quality single crystal of the local static spin susceptibility and spin dynamics (T$_1$ relaxation) at several field values [Ref. 1]. At variance with a previous study [Ref. 2], we used contrast methods to single out the contributions of oxygen sites close and far from interlayer copper defects. This allowed us to show unambiguously that the excitation spectrum is gapless, restoring some convergence with a Dirac cone model now advocated for in most of numerical works [Ref. 3 and Ref. 4]. While herbertsmithite displays a gapless spin liquid-like behavior in zero field, we confirmed the presence of a field-induced instability toward a spin-solid phase with largely suppressed moments $\lesssim0.1$~$\mu_\mathrm{B}$ and gapped excitations at sub-kelvin temperatures, which had already been observed on a polycrystalline sample [Ref. 5].\\ \noindent References: \begin{itemize} \item [Ref. 1] Khuntia et al., \emph{Nat. Phys.} 16, 469-474 (2020) \item [Ref. 2] Fu et al., \emph{Science} 350, 655-658 (2015) \item [Ref. 3] Ran et al., \emph{Phys. Rev. Lett.} 98, 117205 (2007) \item [Ref. 4] He et al., \emph{Phys. Rev. X} 7, 031020 (2017) \item [Ref. 5] Jeong et al., \emph{Phys. Rev. Lett.} 107, 237201 (2011) \end{itemize}

Beyond the absence of long-range magnetic orders, the most prominent feature of the elusive quantum spin liquid (QSL) state is the existence of fractionalized spin excitations. To study the spin dynamics, we perform Raman scattering on single crystals of the kagome QSL candidate Cu3Zn(OH)6FBr and the control compound EuCu3(OH)6Cl3 with an antiferromagnetic order. In Cu3Zn(OH)6FBr, the ideal kagome structure is confirmed down to low temperatures without any lattice distortion, and we observe a remarkable $E_{2g}$ Raman continuum in good agreement with the theoretical prediction for the kagome QSL. We identify a unique one spinon-antispinon pair continuum in Cu3Zn(OH)6FBr, in contrast to a sharp magnon peak in EuCu3(OH)6Cl3. The magnon peak emerges from the one-pair continuum and can be regarded as the spinon-antispinon bound state. The comparative studies demonstrate that the magnetic Raman continuum, in particular, the one-pair component is strong evidence for fractional spinon excitations in Cu3Zn(OH)6FBr.

Kagome lattices can host frustrated magnetism and topological insulator phases. Two-dimensional (2D) metal-organic frameworks (MOFs) allow for the bottom-up synthesis of such lattices and fine-tuning their properties. We report the synthesis of a 2D MOF with copper atoms and dicyanoanthracene molecules on a silver Ag(111) surface. We observe the Kondo effect using scanning tunnelling spectroscopy, providing evidence of local magnetic moments in this system. From density functional theory and a mean-field Hubbard model, we find that electron-electron interactions drive magnetic ordering in this system rather than any intrinsic magnetic moments, providing evidence of strong electronic correlations. We also model the effects of other substrates on the magnetic and electronic properties of the MOF.

We numerically study the effects of non-magnetic impurities (vacancies) in the spin-S Heisenberg antiferromagnet on the kagomé lattice. For a range of low but nonzero temperatures, and spin values that extend down to S=2, we find that the magnetization response to an external magnetic field is consistent with the response of emergent "half-orphan" degrees of freedom that are expected to dominate the response of the corresponding classical magnet in a similar temperature range whenever there are two vacancies on the same triangle. Specifically, for all spin values we have considered (from S=1/2 to S=4), there is a large enhancement of the local susceptibility of the lone spin on such a triangle with two vacancies; in the presence of a uniform magnetic field h, this lone-spin behaves effectively as an almost free spin S in an effective field h/2. Quite remarkably, in the zero temperature limit, the ground state in the presence of a half-orphan has a non-zero total spin value SGS that shows a trend similar to S/2 when S≥2. These qualitative aspects of the response differ strikingly from the more conventional response of diluted samples without such half-orphan degrees of freedom. We discuss how these findings could be checked experimentally. arXiv:2009.01711

Geometrical frustration and a high magnetic field are keys to explore unconventional quantum states. In the spin-1/2 kagome antiferromagnet, several magnetization plateaus are predicted to occur. Interestingly, these plateaus can be described as crystals of "emergent magnons" i.e. magnons localized inside a hexagon. To observe crystallization of the emergent magnon, we performed Faraday rotation measurement of two kagome antiferromagnets up to 190 T. First in a newly synthesized Cd-kapellasite, we found an unprecedented series of magnetization plateaus, which can be interepreted as either close or loose packing of the emergent magnon. Second in Herbertsmithite we observed a robust 1/3 plateau despite presence of significant chemical disorder. These results hint at stabillization of the emergent magnon in real kagome antiferromagnets.

The apparent fragmentation of magnetic moments in spin ice systems into coexisting longitudinal, transverse and harmonic parts is a direct consequence of emergent electromagnetism in these systems. At the microscopic level, moments of fixed length act as elements of this emergent lattice field from which topological defects - magnetic monopoles, can be excited. By construction, the leftover is the sum of the divergence-free (transverse) and harmonic parts. In quantum spin ice systems the emergence is even more striking, as small quantum fluctuations give to the transverse part the dynamics of a compact, U (1) gauge theory reminiscent of quantum electromagnetism. The low energy emergent photon excitation is a signature of a U (1) spin liquid. In this talk we concentrate on a two dimensional analogue, the partially ordered topological phase of kagome spin ice, stabilised either through long range interactions or by placing an external field along the [111] direction in pyrochlore spin ice. In the classical case the transverse part remains disordered, corresponding to a Coulomb phase with ferromagnetic, dipolar correlations and maps directly to a problem of close packed, hard-core dimers. Using degenerate perturbation theory, the transverse fragment of the spins of an XXZ hamiltonian project onto a quantum dimer model on the dual, honeycomb lattice. We discuss the stability of this phase in the presence of small quantum fluctuations and the consequences for magnetic correlations. We show that the onset of the predicted long-range dimer order is explicitly reflected in the transverse part of the emergent field, accompanied by enhanced quantum fluctuations. As a consequence, the dimer ordering can be directly observed by neutron scattering and we compare the structure factor of this order to existing QMC simulations of quantum kagome ice.

I will discuss spontaneous dimerization and emergent criticality in a spin-3/2 chain with antiferromagnetic nearest-neighbor J1, next-nearest-neighbor J2 and three-site J3 interactions. In the absence of three-site interaction J3, I will provide evidence that the model undergoes a remarkable sequence of three phase transitions as a function of J2/J1, going successively through a critical commesurate phase, a partially dimerized gapped phase, a critical floating phase with quasi-long-range incommensurate order, to end up in a fully dimerized phase at very large J2/J1. In the field theory language, this implies that the coupling constant of the marginal operator responsible for dimerization changes sign three times. For large enough J3, the fully dimerized phase is stabilized for all J2, and the phase transitions between the critical and floating phases and the dimerized phase are both Wess-Zumino-Witten (WZW) SU(2)_3 along part of the boundary and turn first order at some point due to the presence of a marginal operator in the WZW SU(2)_3 model. By contrast, the transition between the two dimerized phase is always first order, and the phase transitions between the partially dimerized phase and the critical phases are Kosterlitz-Thouless. Finally, I will discuss the intriguing spin-1/2 edge states that emerge in the partially dimerized phase for even chains. Unlike their counterparts in the spin-1 chain, they are not confined and disappear upon increasing J2 in favour of a reorganization of the dimerization pattern.

The quasi-1D magnetic insulator CoNb2O6 hosts a paradigmatic example of a quantum phase transition in the Ising universality class. In addition, inelastic neutron scattering investigations of the spin-dynamics at low temperature uncovered a rich phenomenology both in the ferromagnetic and in the paramagnetic phases. However, the understanding of experimental data in these regimes is limited by the knowledge of the spin-exchange Hamiltonian of the material. Analysing the spatial symmetry of the material, I will present a microscopic Hamiltonian which reproduces the entirety of the experimental phenomenology observed to date. The symmetry analysis recalls that a chain in the material is buckled, with two sites in each unit cell related by a glide symmetry. In particular, the two sites-unit cell allows for the presence of staggered couplings, which are fundamental in reproducing the experimental data. Furthermore, the glide symmetry plays also a major role in the phase transition itself. In fact, the material lacks an on-site Ising symmetry and the Ising phase transition can be described as a spontaneous symmetry breaking of this symmetry. In relation to this I will discuss how the glide-symmetry breaking can be observed in the inelastic neutron scattering data. Finally, I will explain how kinematical considerations related to the glide symmetry can be used to interpret the phenomenon of quasi-particle breakdown in the paramagnetic phase. [1] Fava, Coldea, Parameswaran, Proc. Nat. Acad. Sci. USA 117, 25219 (2020)

The van-der-Waals compounds TMPX3 (e.g. FePS3) have proven to be ideal examples of two-dimensional antiferromagnets on a honeycomb lattice. At ambient pressure, the FePS3 features zigzag ferromagnetic chains which are coupled antiferromagnetically with its neighbours. For a long time, the frustrated magnetic configuration was understood in terms of a spin Hamiltonian with equal exchange parameters between equivalent neighbours. Recent study [1], however, introduced a biquadratic exchange term to resolve the discrepancies between experimental neutron diffraction data and theoretical modelling. In addition, the TMPX3 compounds are Mott or charge-transfer insulators. Our recent studies [2-4] have reported pressure-induced insulator-to-metal transitions, together with novel magnetic and crystalline phases. There are also reports of superconductivity in a related member of this family of compounds [5]. To further understand the magnetic frustration at ambient pressure, we performed first-principles calculations and DFT+U studies to elucidate the magnetoelastic coupling between lattice and magnetic frustration. A random structure search is also performed to understand the structure and physical property evolution with pressure. Our computational explorations are expected to guide the discovery of structural and magnetic phases in the vdW TMPX3 compounds. [1] A. R. Wildes, et al., J. Appl. Phys. 127, 223903 (2020). [2] C. R. S. Haines, et al., Phys. Rev. Lett. 121, 266801 (2018). [3] M. J. Coak, et al., J. Phys. Condens. Matter 32, 124003 (2020). [4] M.J. Coak, et al., Phys. Rev. X, Oct (2020) Accepted [5] Y. Wang, et al., Nat. Commun. 9, 1914 (2018).

The physical realization of $Z_2$ topological order as encountered in the paradigmatic toric code has proven to be an elusive goal. In this talk, I will show that this phase of matter can be created in a two-dimensional array of strongly interacting Rydberg atoms on a ruby lattice. We will first consider a Rydberg blockade model: this effectively realizes a monomer-dimer model on the kagome lattice with a single-site kinetic term, for which we observe a Z_2 spin liquid using the numerical density matrix renormalization group method. Next, this phase is shown to persist upon including realistic, algebraically-decaying van der Waals interactions. Moreover, one can directly access the topological loop operators of this model, which can be measured experimentally using a dynamic protocol, providing a ``smoking gun'' experimental signature of the topological phase. Time permitting, I will show how to trap an emergent anyon and realize different topological boundary conditions, and more broadly discuss the implications for exploring fault-tolerant quantum memories. This talk is based on work with Mikhail Lukin and Ashvin Vishwanath (arxiv:2011.12310).

We build and probe a $\mathbb{Z}_2$ spin liquid in a programmable quantum device, the D-Wave DW-2000Q. To realize this state of matter, we design a Hamiltonian with combinatorial gauge symmetry using only pairwise-qubit interactions and a transverse field, i.e., interactions which are accessible in this quantum device. The combinatorial gauge symmetry remains exact along the full quantum annealing path, landing the system onto the classical 8-vertex model at the endpoint of the path. The output configurations from the device allows us to directly observe the loop structure of the model. Moreover, we deform the Hamiltonian so as to vary the weights of the 8 vertices and show that we can selectively attain the 6-vertex (ice model), or drive the system into a ferromagnetic state. We present studies of the phase diagram of the system as function of the 8-vertex deformations and effective temperature, which we control by varying the relative strengths of the programmable couplings, and we show that the experimental results are consistent with theoretical analysis. Finally, we identify additional capabilities that, if added to these quantum devices, would allow us to realize $\mathbb{Z}_2$ quantum spin liquids on which to build topological qubits.

In the context of quantum magnetism, there are a broad range of materials which are accurately described by models of weakly coupled spin dimers. Such models are well-understood theoretically, and exhibit magnetic field-induced phase transitions which are naturally described in terms of equivalent Bose-Einstein condensation (BEC) problems. Recently, experimental data on a system of weakly coupled dimers, Yb$_2$Si$_2$O$_7$, revealed an asymmetric BEC dome. To study this behavior, we examine modifications to the Heisenberg model on a breathing honeycomb lattice, showing that this physics can be explained by competing anisotropic perturbations. We employ a gamut of analytical and numerical techniques to show that the anisotropy yields a field driven phase transition from a state with broken Ising symmetry to a phase which breaks no symmetries and crosses over to the polarized limit. From the BEC perspective, this yields insights into the behavior of bosonized spin models in which an external magnetic field couples to a non-conserved quantity.

In frustrated magnets, most of the focus has been devoted to the physics near zero temperature. However, even in the presence of very strong frustration, the majority of frustrated magnetic materials ultimately develop long-range order or display spin-glass freezing at a nonzero critical temperature $T_c$. It therefore seems natural to ask what behavior near $T_c$ may inform on the zero-temperature ground state physics. We consider such a situation in Er$_2$Sn$_2$O$_7$, a pyrochlore antiferromagnet known to be strongly frustrated and residing at the boundary of two classical phases. We specifically study a recurrent aspect of frustrated magnetic systems observed at nonzero temperature: reentrance. Reentrance refers to when a system, after developing an ordered phase, returns to (reenters) its original, less ordered, phase as some external parameter is continuously tuned. Reentrance has been found in a variety of systems, from spin glasses to black hole thermodynamics, but its presence is often unexpected, and the microscopic mechanisms behind it have not been studied in detail. We use a combination of experimental and theoretical techniques to study a single crystal of the frustrated pyrochlore magnet Er$_2$Sn$_2$O$_7$, taking advantage of the recent advance in synthesis which enables the production of rare-earth stannate single crystals. We show Er$_2$Sn$_2$O$_7$ has multiple instances of reentrance in its field vs temperature phase diagram for fields along the three high symmetry directions ([100], [110], and [111]). Through classical Monte Carlo simulations, mean field theory, and classical linear spin-wave expansion, we propose that the origins of reentrance are linked to $T=0$ multi-phase competition induced by either the applied field or the inherent proximity to a competing zero field phase. The ground state phase competition enhances thermal fluctuations that entropically stabilize the ordered phase, leading to increased transition temperatures for certain field values. Our work represents a detailed look into the mechanisms responsible for reentrance in frustrated magnets, which may serve as a guide for other reentrant phenomena in physics.

We characterize the ground state magnetism of pyrochlore Yb$_2$Ti$_2$O$_7$ in a [111] magnetic field using inelastic neutron scattering from the CNCS spectrometer at SNS. Our measurements reveal broadened but coherent zero-field excitations and sharp high-field spin waves modes. Comparison to linear spin wave theory allows us to distinguish between different proposed Hamiltonians for Yb$_2$Ti$_2$O$_7$ and reveals features of antiferromagnetism in the low-field spectrum. High-field [111] spin wave fits show that Yb$_2$Ti$_2$O$_7$ is extremely close to an antiferromagnetic phase boundary. This proximity allows for stable regions of antiferromagnetism, which may be responsible for the unusual magnetic behavior of this compound.

Magnetic pyrochlore oxides have attracted significant interest due to their geometrically frustrated lattice, which acts to suppress long range magnetic order and leads to a variety of unusual magnetic ground states and exotic excitations. Some of the most studied pyrochlores are the spin ices Ho2Ti2O7 and Dy2Ti2O7, where the ferromagnetically frustrated rare earth ions result in magnetic monopole quasiparticles. Despite extensive work on spin ices, a reliable experimental indicator of the density of magnetic monopoles in spin-ice systems is yet to be found. In this talk, we present magnetisation and resistivity measurements on new single crystals of the pyrochlore iridate Ho2Ir2O7, where the low-temperature holmium spin ice is coupled to antiferromagnetically ordered itinerant iridium moments. Our experiments, in combination with dipolar Monte Carlo simulations of the rare earth moments using an effective representation of the Ir moments as local fields [1], show that the magnetoresistance is highly sensitive to the density of monopoles in a way that holds promise to quantitatively measure it in experiments. We argue that this is due to iridium conduction electrons scattering off the Coulombic magnetic and dipolar electric [2] field of the monopoles. Our result provides a powerful and versatile new tool for the study of spin-ice systems. We further show that, for certain orientations, an applied magnetic field couples strongly to the antiferromagnetically ordered iridium ions via the large holmium moments. This allows us to manipulate the antiferromagnetic domains via an external field, which is evidenced by the appearance of a strong and correlated hysteresis in both the magnetisation and magnetoresistance. The precise control of antiferromagnetic domain walls is a key goal for the design of next-generation spintronic devices [3]. Our results provide a way forward to how this target may be achieved using externally applied magnetic fields and a peculiar interplay between frustrated localised moments and itinerant antiferromagnetism. [1] E. Lefrançois, V. Cathelin, E. Lhotel, J. Robert, P. Lejay, C. V. Colin, B. Canals, F. Damay, J. Ollivier, B. Fåk, L. C. Chapon, R. Ballou, and V. Simonet, Fragmentation in spin ice from magnetic charge injection. Nat. Commun. 8, 209 (2017). [2] D. I. Khomskii, Electric dipoles on magnetic monopoles in spin ice. Nat. Commun. 3, 904 (2012). [3] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Antiferromagnetic spintronics. Rev. Mod. Phys. 90, 015005 (2018).

The spin-1/2 Heisenberg model on the pyrochlore lattice is an iconic frustrated three-dimensional spin system with a rich phase diagram. Besides hosting several ordered phases, the case with only nearest-neighbor antiferromagnetic interactions is debated for potentially realizing a spin-liquid ground state. Here, we contest this hypothesis with an extensive numerical investigation using both exact diagonalization and several complementary variational techniques. Specifically, we employ a Pfaffian-type many-variable Monte Carlo ansatz and convolutional neural network quantum states for calculations with up to $3\times 4^3$ and $3 \times 3^3$ spins, respectively. We demonstrate that these techniques yield consistent results, allowing for reliable extrapolations to the thermodynamic limit. Our main results are (1) a determination of the phase transition between the putative spin liquid phase and the neighboring magnetically ordered phase and (2) a careful characterization of the ground state in terms of symmetry breaking tendencies. We find clear indications for spontaneously broken inversion and rotational symmetry, calling the quantum spin-liquid scenario into question. Our work showcases how many-variable variational techniques can be used to make progress in answering challenging questions about three-dimensional frustrated quantum magnets.

We use a combination of three computational methods to investigate the notoriously difficult frustrated three dimensional pyrochlore $S=1/2$ quantum antiferromagnet, at finite temperature, $T$: canonical typicality for a finite cluster of $2\times 2 \times 2$ unit cells (i.e. $32$ sites), a finite-$T$ matrix product state method on a larger cluster with $48$ sites, and the numerical linked cluster expansion (NLCE) using clusters up to $25$ lattice sites, including non-trivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at $T = 0.57 J$, which is stable across finite size clusters and converged in the series expansion. At $T\approx 0.25J$ (the limit of convergence of our method), the residual entropy per spin is $0.47 k_B \ln2$, which is relatively large compared to other frustrated models at this temperature. We also observe a non-monotonic dependence on $T$ of the magnetization at low magnetic fields, reflecting the dominantly non-magnetic character of the low-energy states. A detailed comparison of our results to measurements for the $S=1$ material NaCaNi$_2$F$_7$ yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material Dy$_2$Ti$_2$O$_7$.

We address the ground state properties of the long-standing and much-studied three dimensional quantum spin liquid candidate, the $S=\frac 1 2$ pyrochlore Heisenberg antiferromagnet. By using $SU(2)$ DMRG, we are able to access cluster sizes of up to 108 spins. Our most striking finding is a robust spontaneous inversion symmetry breaking, reflected in an energy density difference between the two sublattices of tetrahedra, familiar as a starting point of earlier perturbative treatments. We also determine the ground state energy, $E_0/N_\text{sites} = -0.488(12) J$, by combining extrapolations of DMRG with those of a numerical linked cluster expansion. These findings suggest a scenario in which a finite-temperature spin liquid regime gives way to a symmetry-broken state at low temperatures.

Quantum spin liquids are phases of matter with fractionalized excitations and emergent gauge fields. However, distinct and unambiguous signatures of spin liquids are hard to come by, especially in the experimentally relevant scenario of non-zero temperatures. We show that the dynamics of spinon production in Coulomb spin liquids, as measured by the dynamic structure factor, contain such distinct signatures due to dramatic interaction effects and unusual energy scales. These include Sommerfeld enhancement of the pair production cross section due to the emergent Coulomb interaction and Cerenkov radiation due to the presence of slow photons. Varying the temperature provides characteristic changes in these effects due to the bath of thermal photons and magnetic monopoles. Our results are consistent with recent numerics in lattice models and point to the important role of interactions in understanding dynamics of spin liquids.

Condensed matter systems act as mini-universes with emergent low-energy properties drastically different from those of the standard model. One prominent example is quantum spin ice which hosts an emergent quantum electrodynamics (QED), one whose magnetic monopoles set it apart from the familiar QED of the world we live in. In addition to new exotic excitations, the emergent QED's parameters such as the speed of light and fine-structure constant can be vastly different from those in vacuum QED. Here, I discuss recent work where we calculate the emergent fine structure constant in the canonical quantum spin ice model using exact diagonalization techniques. We find that is more than an order of magnitude greater than that in vacuum QED. Furthermore, by engineering the microscopic Hamiltonian, we find that the fine structure constant, the emergent speed of light, and all other parameters of the emergent QED are tunable. In particular, the fine structure constant can be tuned all the way from zero up to what is believed to be the strongest possible coupling beyond which QED confines. This points to quantum spin ice as an ideal candidate to explore new regimes of QED in magnetic materials and synthetic realizations.

H Yan, O Benton, LDC Jaubert, N Shannon On a lattice without Lorentz symmetry of spacetime, many exotic versions of gauge theory other than the conventional electrodynamics can emerge. Among them, the symmetric rank-2 U(1) gauge theories are particularly interesting [1]. With electric and gauge fields being symmetric tensors, these theories enforce not only charge but also dipole or other multipole conservation laws. As a consequence, they may host charge excitations, dubbed “fractons”, that are entirely immobile in the system. Fracton physics has sparked tremendous interest across quantum information, beyond-topological-order quantum phases of matter, and high energy theory [2]. But relatively little is known about how to realise them in experiment, due to the complexity of the prototype fracton models. Here we report a surprisingly simple frustrated magnetic system that realizes a classical symmetric tensor gauge theory [3]. The model is a breathing pyrochlore lattice with dominant Heisenberg interactions between nearest spins, and Dzyaloshinskii–Moriya interactions on half the tetrahedra. We discover a classical spin liquid phase, with its low-energy effective theory described by a symmetric rank-2 electric field constrained by the Gauss’s laws of a vector charge. Its vector charge excitations are fractons. We also make explicit experimental predictions of such a spin liquid. More specifically, the four-fold pinch point will be observed in neutron scattering. Given the common ingredients needed in the model, we hope it will guide future material synthesis in search of fracton matter. In particular, Yb-based breathing pyrochlores are identified to be promising candidates. [1] M. Pretko, Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids, Phys. Rev. B 95, 115139 (2017) [2] M. Pretko, X. Chen, Y. You, Fracton phases of matter, Int. J. of Mod. Phys. A 35, 06, 2030003 (2020) [3] H. Yan, O. Benton, L. D. C. Jaubert, and Nic Shannon, Rank-2 U(1) Spin Liquid on the Breathing Pyrochlore Lattice, Phys. Rev. Lett. 124, 127203 (2020)

In this talk, I will show that the defects of the valence plaquette solid(VPS) order parameter, in addition to possessing non-trivial quantum numbers, have fracton mobility constraints in the VPS phase. The spinon inside a single vortex cannot move freely in any direction, while a dipolar pair of vortices with spinon pairs can only move perpendicular to its dipole moment. These mobility constraints, while they persist near QCP, can potentially inhibit the condensation of spinons and preclude a continuous transition from the VPS to the Néel antiferromagnet. Instead, the VPS melting transition can be driven by the proliferation of spinon dipoles. In particular, we argue that a 2d VPS can melt into a stable gapless phase in the form of an algebraic bond liquid with algebraic correlations and long-range entanglement. Such a bond liquid phase yields a concrete example of the elusive 2d Bose metal with symmetry fractionalization and UV-IR mixing.

Classical spin liquids (CSLs) are high-entropy, low-temperature phases of matter formed in classical spin systems subject to a local constraint. They show up in some of the most celebrated problems in frustrated magnetism, most famously spin ice. CSLs can be considered as classical analogues of quantum spin liquids (QSLs), and are in some sense simpler than their quantum counterparts, being without quantum coherence. Strangely, however, while the classification of QSLs has a well-developed machinery via projective symmetry group analysis, CSLs have no such systematic classification scheme. Here, we propose an approach to classifying classical spin liquids using topological invariants. By this means we are able to distinguish both algebraic and short-range correlated CSLs from trivial paramagnetic states and to establish the possibility of topological phase transitions between classical spin liquids. Examples will be given of models in two and three dimensions which exhibit a series of such phase transitions. We also see that the concept of ``symmetry protected topological phases'' has relevance to CSLs, just as it does to quantum systems. This work thus establishes a new perspective on an important class of frustrated magnets.

Quantum magnets provide the simplest example of strongly interacting quantum matter, yet they continue to resist a comprehensive understanding above one spatial dimension. We explore the Dirac spin liquid (DSL) in 2D lattices, a version of Quantum Electrodynamics (QED3) with four flavors of Dirac fermions coupled to photons. Importantly, its excitations include magnetic monopoles that drive confinement, and the symmetry actions on monopoles contain crucial information about the DSL states. The underlying band topology of spinon insulators, e.g., wannier insulator protected by rotation etc, determines the elusive Berry phase of monopole. The stability of the DSL is enhanced on triangular lattices compared to square(bipartite) lattices. We obtain the universal signatures of the DSL on triangular and kagome lattices, including those of monopole excitations, as a guide to numerics and experiments on existing materials. Even when unstable, the DSL helps unify and organize the plethora of competing orders in correlated two-dimensional materials. Time permitting I will describe recent results on chiral spin liquid states on the triangular lattice and the superconducting/metallic phases that emerge on lightly doping them

The spin-orbit entangled (SOE) Jeff-state has been a fertile ground to study novel quantum phenomena. Contrary to the conventional weakly correlated Jeff=1/2 state of 4d and 5d transition metal compounds, the ground state of CuAl2O4 hosts a Jeff=1/2 state with a strong correlation of Coulomb U. Here, we report that surprisingly Cu2+ ions of CuAl2O4 overcome the otherwise usually strong Jahn-Teller distortion and instead stabilize the SOE state, although the cuprate has relatively small spin-orbit coupling. From the x-ray absorption spectroscopy and high-pressure x-ray diffraction studies, we obtained definite evidence of the Jeff=1/2 state with a cubic lattice at ambient pressure. We also found the pressure-induced structural transition to a compressed tetragonal lattice consisting of the spin-only S=1/2 state for pressure higher than Pc=8 GPa. This phase transition from the Mott insulating Jeff=1/2 to the S=1/2 states is a unique phenomenon and has not been reported before. Our study offers a rare example of the SOE Jeff-state under strong electron correlation and its pressure-induced transition to the S=1/2 state.

Band topology in electronic systems is known to have profound consequences on various observable properties in semi-metals and certain insulators. Insights from this field have reached into many areas of physics and in this talk we describe certain universal signatures of band topology in propagating bosonic quasi-particles. We show that the Dirac magnon material CoTiO3 provides an experimental demonstration of a universal winding of the inelastic neutron scattering intensity around linear touching points of magnetic excitations - magnons and spin-orbit excitons - that originates from the isospin texture of the quasiparticle wavefunction in momentum space. https://arxiv.org/abs/2007.04199

The appearance of nontrivial phases in Kitaev materials exposed to an external magnetic field has recently been a subject of intensive studies. Here, we elucidate the relation between the field-induced ground states of the classical and quantum spin models proposed for such materials, by using infinite density matrix renormalization group and linear spin wave theory. We consider an extended Kitaev model with additional symmetric off-diagonal spin exchanges, $\Gamma$ and $\Gamma'$. Focusing on the magnetic field along the $[111]$ direction, we explain the origin of a nematic paramagnet, which breaks the lattice-rotational symmetry and exists in an extended window of magnetic field. We show, that this phenomenon can be understood as the effect of quantum order-by-disorder in the frustrated ferromagnet phase with a continuous manifold of degenerate ground states discovered in the corresponding classical model. We present dynamical spin structure factors to enable comparison with inelastic neutron scattering experiments on Kitaev materials in an external magnetic field along the $[111]$ direction. In particular, the nematic paramagnet exhibits a characteristic pseudo-Goldstone mode which results from the lifting of a continuous degeneracy via quantum fluctuations.

cently it was realized that the zigzag magnetic order in Kitaev materials can be stabilized by small negative off-diagonal interactions called the $\Gamma'$ terms. To fully understand the effect of the $\Gamma'$ interactions, we investigate the quantum $K$-$\Gamma$-$\Gamma'$ model on the honeycomb lattice using the variational Monte Carlo method. Two multinode Z$_2$ quantum spin liquids (QSLs) are found at $\Gamma'>0$, one of which is the previously found proximate Kitaev spin liquid called the PKSL14 state which shares the same projective symmetry group (PSG) with the Kitaev spin liquid. A remarkable result is that a $\pi$-flux state with a distinct PSG appears at larger $\Gamma'$. The $\pi$-flux state is characterized by an enhanced periodic structure in the spinon dispersion in the original Brillouin zone (BZ), which is experimentally observable. Interestingly, two PKSL8 states are competing with the $\pi$-flux state and one of them can be stabilized by six-spin ring-exchange interactions. The physical properties of these nodal QSLs are studied by applying magnetic fields and the results depend on the number of cones. Our study infers that there exist a family of zero-flux QSLs that contain $6n+2, n\in\mathbb Z$ Majorana cones and a family of $\pi$-flux QSLs containing $4(6n+2)$ cones in the original BZ. It provides guidelines for experimental realization of non-Kitaev QSLs in relevant materials.

Mechanical strain can generate a pseudo-magnetic field, and hence Landau levels (LL), for low energy excitations of quantum matter in two dimensions. We study the collective state of the fractionalised Majorana fermions arising from residual generic spin interactions in the central LL, where the projected Hamiltonian reflects the spin symmetries in intricate ways: emergent U (1) and particle-hole symmetries forbid any bilinear couplings, leading to an intrinsically strongly interacting system; also, they allow the definition of a filling fraction, which is fixed at 1/2. We argue that the resulting many-body state is gapless within our numerical accuracy, implying ultra-short-ranged spin correlations, while chirality correlators decay algebraically. This amounts to a Kitaevnon-Fermi'spin liquid, and shows that interacting Majorana Fermions can exhibit intricate behaviour akin to fractional quantum Hall physics in an insulating magnet.

We exhibit an exactly solvable example of a SU(2) symmetric Majorana spin liquid phase, in which quenched disorder leads to random-singlet phenomenology. More precisely, we argue that a strong-disorder fixed point controls the low temperature susceptibility $\chi(T)$ of an exactly solvable $S=1/2$ model on the decorated honeycomb lattice with quenched bond disorder and/or vacancies, leading to $\chi(T) = {\mathcal C}/T+ {\mathcal D} T^{\alpha(T) - 1}$ where $\alpha(T) \rightarrow 0$ as $T \rightarrow 0$. The first term is a Curie tail that represents the emergent response of vacancy-induced spin textures spread over many unit cells: it is an intrinsic feature of the site-diluted system, rather than an extraneous effect arising from isolated free spins. The second term, common to both vacancy and bond disorder (with different $\alpha(T)$ in the two cases) is the response of a random singlet phase, familiar from random antiferromagnetic spin chains and the analogous regime in phosphorus-doped silicon (Si:P).

Since 2006, the Kitaev honeycomb model has attracted significant attention due to the exactly solvable spin-liquid ground state with fractionalized Majorana excitations [1] and the possible materialization in magnetic Mott insulators with strong spin-orbit couplings [2]. Recently, the 5d-electron compound H$_{3}$LiIr$_{2}$O$_{6}$ has shown to be a strong candidate of Kitaev physics considering the absence of long-range ordered magnetic state [3]. In this work [4], we demonstrate that a finite density of random vacancies gives rise to a remarkable pile up of low-energy states and is possibly related to the observed low-temperature upturn in the specific heat measurement of H$_{3}$LiIr$_{2}$O$_{6}$. We study both the free-flux and the vacancy-induced bound-flux background and their responses to additional time-reversal symmetry-breaking term, which imitates the magnetic field in real experiments. [1] A. Kitaev, Ann. Phys. 321, 2 (2006). [2] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). [3] K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R. Takano,Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli, and H. Takagi, Nature 554, 341 (2018). [4] W.-H. Kao, J. Knolle, G. B. Halász, R. Moessner, N. B. Perkins, arXiv:2007.11637 (2020).

Kitaev magnets have gained a huge amount of interest in recent years due to the possibility of such materials hosting a highly-entangled quantum spin liquid (QSL) ground state. Such materials are characterized by a honeycomb lattice structure and can be analytically solved by mapping the localized spins into itinerant and localized Majorana excitations. Interestingly, recent experiments on Ag3LiIr2O6 have shown that spiral incommensurate order coexists with a metal-like phase characterized by a finite Sommerfeld coefficient. In this work, we describe this exotic phase by considering the effects of interaction on the statistical energy of Landau-Fermi quasiparticles near the Majorana Fermi surface. The resulting specific heat is found to be dominated by a quadratic-temperature dependence with a strong non-analytic contribution, in agreement with present experiments on the silver-lithium iridate at zero and finite external magnetic field. Co-Authors: Faranak Bahrami, Roman Movshovich, Xiao Chen, Kevin Bedell, & Fazel Tafti. Work at BC was supported by the John H. Rourke Endowment Fund and the NSF award number DMR–1708929. Work at Los Alamos was conducted under the auspices of the U.S. Department of Energy.

Kitaev materials are prime examples where the orbital and spin degrees of freedom can not be understood separately, and instead are formulated jointly through so-called "pseudospins". In contrast to conventional spin-lattice coupling, the spin-orbital nature of the pseudospins foreshadows a much more delicate coupling to the lattice. Using large-scale first-principles simulations we obtain a magnetoelastic Hamiltonian of $\alpha$-RuCl$_3$, that reveals a highly nontrivial interplay of different magnetic interactions with the lattice [1]. We reproduce and explain recently measured magnetostriction [2], using exact diagonalization on our magnetoelastic model, disentangling contributions related to different anisotropic interactions and $g$ factors. Uniaxial strain perpendicular to the honeycomb planes is predicted to reorganize the relative coupling strengths, strongly enhancing the Kitaev interaction while simultaneously weakening the other anisotropic exchanges under compression. Uniaxial strain may therefore pose a fruitful route to experimentally tune $\alpha$-RuCl$_3$ nearer to the Kitaev limit. [1] D.A.S Kaib, S. Biswas, K. Riedl, S.M. Winter, R. Valenti, arXiv: 2008.08616 (2020). [2] S. Gass et al., PRB 101, 245158 (2020).