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Experiments on quantum materials have uncovered many interesting quantum phases ranging from superconductivity to a variety of topological quantum matter including the recently observed fractional quantum anomalous Hall insulators. The findings have come in parallel with the development of approaches to probe the rich excitations inherent in such systems. In contrast to observing electrically charged excitations, the detection of charge-neutral electronic excitations in condensed matter remains difficult, though they are essential to understanding a large class of strongly correlated phases. Low-energy neutral excitations are especially important in characterizing unconventional phases featuring electron fractionalization, such as quantum spin liquids, spin ices, and insulators with neutral Fermi surfaces. In this perspective, we discuss searches for neutral fermionic, bosonic, or anyonic excitations in unconventional insulators, highlighting theoretical and experimental progress in probing excitonic insulators, new quantum spin liquid candidates and emergent correlated insulators based on two-dimensional layered crystals and moir\'e materials. We outline the promises and challenges in probing and utilizing quantum insulators, and discuss exciting new opportunities for future advancements offered by ideas rooted in next-generation quantum materials, devices, and experimental schemes., Comment: 23 pages, 4 figures; A Perspective/Review Article published in Nature

Motivated by the recent appearance of the trillium lattice in the search for materials hosting spin liquids, we study the ground state of the classical Heisenberg model on its linegraph, the trilline lattice. We find that this network realises the recently proposed notion of a fragile spin liquid in three dimensions. Additionally, we analyze the Ising case and argue for a possible $\mathbb{Z}_2$ quantum spin liquid phase in the corresponding quantum dimer model. Like the well-known $U(1)$ spin liquids, the classical phase hosts moment fractionalisation evidenced in the diluted lattice, but unlike these, it exhibits exponential decay both in spin correlations and interactions between fractionalised moments. This provides the first instance of a purely short-range correlated classical Heisenberg spin liquid in three dimensions.

The ergodicity postulate, a foundational pillar of Gibbsian statistical mechanics predicts that a periodically driven (Floquet) system in the absence of any conservation law heats to a featureless `infinite temperature' state. Here, we find--for a clean and interacting generic spin chain subject to a {\it strong} driving field--that this can be prevented by the emergence of {\it approximate but stable} conservation-laws not present in the undriven system. We identify their origin: they do not necessarily owe their stability to familiar protections by symmetry, topology, disorder, or even high energy costs. We show numerically, {\it in the thermodynamic limit,} that when required by these emergent conservation-laws, the entanglement-entropy density of an infinite subsystem remains zero over our entire simulation time of several decades in natural units. We further provide a recipe for designing such conservation laws with high accuracy. Finally, we present an ensemble description, which we call the strongly driven ensemble incorporating these constraints. This provides a way to control many-body chaos through stable Floquet-engineering. Strong signatures of these conservation-laws should be experimentally accessible since they manifest in all length and time scales. Variants of the spin model we have used, have already been realized using Rydberg-dressed atoms.

Disorder-free localization (DFL) is a phenomenon as striking as it appears to be simple: a translationally invariant state evolving under a disorder-free Hamiltonian failing to thermalize. It is predicted to occur in a number of quantum systems exhibiting emergent or native \emph{local} symmetries. These include models of lattice gauge theories and, perhaps most simply, some two-component spin chains. Though well-established analytically for special soluble examples, numerical studies of generic systems have proven difficult. Moreover, the required local symmetries are a challenge for any experimental realization. Here, we show how a canonical model of DFL can be efficiently implemented on gate-based quantum computers, which relies on our efficient encoding of three-qubit gates. We show that the simultaneous observation of the absence of correlation spreading and tunable entanglement growth to a volume law provides an ideal testbed for benchmarking the capabilities of quantum computers. In particular, the availability of a soluble limit allows for a rigorous prediction of emergent localization length scales and tunable time scales for the volume law entanglement growth, which are ideal for testing capabilities of scalable quantum computers., Comment: $10$ pages, $3$ figures

One of the most challenging problems in the computational study of localization in quantum manybody systems is to capture the effects of rare events, which requires sampling over exponentially many disorder realizations. We implement an efficient procedure on a quantum processor, leveraging quantum parallelism, to efficiently sample over all disorder realizations. We observe localization without disorder in quantum many-body dynamics in one and two dimensions: perturbations do not diffuse even though both the generator of evolution and the initial states are fully translationally invariant. The disorder strength as well as its density can be readily tuned using the initial state. Furthermore, we demonstrate the versatility of our platform by measuring Renyi entropies. Our method could also be extended to higher moments of the physical observables and disorder learning.

Motivated by the rapid experimental progress in twisted van der Waals materials, we study the triangular trimer model as a representative framework for extended Wannier orbitals in twisted bilayer graphene at 1/3-filling. This deceptively simple model exhibits a rich suite of complex phases, including unusual excitations exhibiting the physics of fractionalization and fractons. For our investigations, we carry out extensive Monte Carlo simulations using an efficient cluster algorithm. The so-obtained finite-temperature phase diagram reveals a novel polar fluid and an ordered brick-wall phase characterized by fractionally charged $e/3$ excitations with subdimensional lineonic dynamics. Notably, we identify a critical trimer liquid phase for the particularly simple model of hard trimers. For this, we derive a new field theory which takes the form of a U(1)$\times$U(1) gauge theory. Its $e/3$ monomers are fractionalized bionic excitations: they carry a {\it pair} of emergent gauge charges, as evidenced by algebraic correlations with two distinct exponents. These field theoretical predictions offer theoretical grounds for numerical observations of critical exponents. Our study highlights the triangular trimer model as a new key platform for investigating fractionalization and fractons, where trimer liquid bionic monomers can transform into lineons or fractons in proximate phases, and calls for experimental investigations of this physics in twisted van der Waals materials and a broader class of systems with intermediate-range interactions., Comment: 8+13 pages

A crucial ingredient for scalable fault-tolerant quantum computing is the construction of logical qubits with low error rates and intrinsic noise protection. We propose a cross-platform construction for such hardware-level noise-protection in which the qubits are protected from depolarizing (relaxation) and dephasing errors induced by local noise. These logical qubits arise from the entanglement between two internal degrees of freedom, hence - entanglemons. Our construction is based on the emergence of collective degrees of freedom from a generalized coherent state construction, similar in spirit to spin coherent states, of a set of such internally entangled units. These degrees of freedom, for a finite number of units, parametrize the quantized version of complex projective space $\mathbb{C}$P(3). The noise protection of the entanglemon qubit is then a consequence of a weakly coupled emergent degree of freedom arising due to the non-linear geometry of complex projective space. We present two simple models for entanglemons which are platform agnostic, provide varying levels of protection and in which the qubit basis states are the two lowest energy states with a higher energy gap to other states. We end by commenting on how entanglemons could be realized in platforms ranging from superconducting circuits and trapped ion platforms to possibly also quantum Hall skyrmions in graphene and quantum dots in semiconductors. The inherent noise protection in our models combined with the platform agnosticism highlights the potential of encoding information in additional weakly coupled emergent degrees of freedom arising in non-linear geometrical spaces and curved phase spaces, thereby proposing a different route to achieve scalable fault-tolerance., Comment: 15+6 pages, 6 figures; comments welcome

We present a general framework to implement massive Nambu-Goldstone quasi-particles in driven many-body systems. The underlying mechanism leverages an explicit Lie group structure imprinted into an effective Hamiltonian that governs the dynamics of slow degrees of freedom; the resulting emergent continuous symmetry is weakly explicitly broken, giving rise to a massive Nambu-Goldstone mode, with a spectral mass gap scaling linearly with the drive period. We discuss explicit and experimentally implementable realizations, such as Heisenberg-like spin models that support gapped spin-wave excitations. We provide a protocol to certify the existence of the massive Nambu-Goldstone mode from the dynamics of specific observables, and analyse the dispersion spectrum and their lifetime in the presence of weak explicit symmetry breaking.

The Chern-Hopf insulator is an unconventional three-dimensional topological insulator with a bulk gap and gapless boundary states without protection from global discrete symmetries. This study investigates its fate in the presence of disorder. We find it stable up to moderate disorder by analyzing the surface states and the zero energy bulk density of states using large-scale numerical simulation and the self-consistent Born approximation. The disordered Chern-Hopf insulator shows reentrant behavior: the disorder initially enhances the topological phase before driving it across an insulator-diffusive metal transition. We examine the associated critical exponents via finite-size scaling of the bulk density of states, participation entropy, and two-terminal conductance. We estimate the correlation length exponent $\nu\simeq 1.0(1)$, consistent with the clean two-dimensional Chern universality and distinct from the integer quantum Hall exponent., Comment: 9 pages, 8 figures

Motivated by recent experiments on Google's sycamore NISQ platform on the spin transport resulting from a non-unitary periodic boundary drive of an XXZ chain, we study a classical variant thereof by a combination of analytical and numerical means. We find the classical model reproduces the quantum results in remarkable detail, and provides an analytical handle on the nature and shape of the spin transport's three distinct regimes: ballistic (easy-plane), subdiffusive (isotropic) and insulating (easy-axis). Further, we show that this phenomenology is remarkably robust to the inclusion of bond disorder -- albeit that the transient dynamics approaching the steady states differs qualitatively between the clean and disordered cases -- providing an accessible instance of ballistic transport in a disordered setting., Comment: 14 pages, 10 figures

We build Floquet-driven capactive circuit networks to realize topological states of matter in the frequency domain. We find the Floquet circuit network equations of motion to reveal a potential barrier which effectively acts as a boundary in frequency space. By implementing a Su-Shrieffer-Heeger Floquet lattice model and measuring the associated circuit Laplacian and characteristic resonances, we demonstrate how topological edge modes can nucleate at such a frequency boundary., Comment: 4 pages, 3 figures, supplement

Condensed matter systems can host emergent `vacua' with particles, fields and dimension different from that of the universe we inhabit. Motivated by the appearance of emergent gauge fields with both electric and magnetic charges, we consider the fate of electromagnetism in two dimensions in such a setting. We find that generically electromagnetic fields are damped due to resistive effects. However, we can still identify a magnetohydrodynamic regime. It exhibits magnetosonic waves which, unlike in $d=3$, are isotropic, to which there is a contribution from the odd viscosity coefficient present in the system. Further, we find a dynamo action, which unlike in $d=3$ requires compressibility of the flow., Comment: 10 pages, 1 figure

We investigate a simple variant of spin ice whose degenerate ground states are densely packed monopole configurations. An applied field drives this model through a Z2 confinement transition, in absence of any local order parameter. Instead, this hidden order turns out to be diagnosed by a string order invisible to any local probe. We describe the transition in terms of a bosonic field theory with a pairing term, as well as a Kramers-Wannier duality into a 3D Ising model, which establishes its Z2 nature. This topological transition can be thought of as a variant of the celebrated U(1) Kasteleyn transition; but instead of the traditional '3/2'-order kink, the system shows critical scaling expected near a 3D Ising transition. Remarkably, however, the magnetic response scales with the critical exponent not of the susceptibility, but of the specific heat.

The discovery of the quantum Hall effect founded the field of topological condensed matter physics. Its amazingly accurate quantisation of the Hall conductance, now enshrined in quantum metrology, is topologically protected: it is stable against any reasonable perturbation. Conversely, topological protection thus implies a form of censorship, as it completely hides any local information from the observer. The spatial distribution of the current in the sample is such a piece of information, which however has now become accessible thanks to spectacular experimental advances. It is an old question whether an original, and intuitively compelling, picture of the current flowing in a narrow channel along the sample edge is the physically correct one. Motivated by recent experiments $\textit{locally}$ imaging the quantized current flow in a Chern insulating (Bi, Sb)$_2$Te$_3$ heterostructure, [Rosen et al., PRL 129, 246602 (2022); Ferguson et.al, Nat. Mater. 22, 1100-1105 (2023)], we theoretically demonstrate the possibility of a broad `edge state' meandering away from the sample boundary deep into the sample bulk. Further, we show that varying experimental parameters permits continuously tuning between narrow edge states and meandering channels all the way to incompressible bulk transport. This accounts for various features observed in, and differing between, experiments. Overall, this underscores the robustness of topological condensed matter physics, but it also unveils a phenomenological richness hidden by topological censorship -- much of which we believe remains to be discovered., Comment: 10 pages, 8 figures

Digital quantum simulation of many-body dynamics relies on Trotterization to decompose the target time evolution into elementary quantum gates operating at a fixed equidistant time discretization. Recent advances have outlined protocols enabling more efficient adaptive Trotter protocols, which have been shown to exhibit a controlled error in the dynamics of local observables and correlation functions. However, it has remained open to which extent the errors on the actual generator of the dynamics, i.e., the target many-body Hamiltonian, remain controlled. Here, we propose to use quantum Hamiltonian learning to numerically obtain the effective Hamiltonian and apply it on the recently introduced ADA-Trotter algorithm as a concrete demonstration. Our key observation is that deviations from the target generator remain bounded on all simulation times. This result suggests that the ADA-Trotter not only generates reliable digital quantum simulation of local dynamics, but also controllably approximates the global quantum state of the target system. Our proposal is sufficiently general and readily applicable to other adaptive time-evolution algorithms., Comment: 5+5 pages, 4+4 figures

We study how sharp signatures of fractionalization emerge in nonlinear spectroscopy experiments on spin liquids with separated energy scales. Our model is that of dipolar-octupolar rare earth pyrochlore materials, prime candidates for realising quantum spin ice. This family of three dimensional quantum spin liquids exhibits fractionalization of spin degrees of freedom into spinons charged under an emergent $U(1)$ gauge field. We show that the technique of two dimensional coherent spectroscopy (2DCS) can identify clear signatures of fractionalised spinon dynamics in dipolar-octupolar quantum spin ices. However, at intermediate temperatures, spinon dynamics are heavily constrained in the presence of an incoherent spin background, leading to a broad 2DCS response. At lower temperatures, a sharp signal emerges as the system enters a coherent spin liquid state. This lower temperature signal can in turn distinguish between zero-flux and $\pi$-flux forms of quantum spin ice., Comment: 6 pages, 4 figures

The relaxation behaviour of isolated quantum systems taken out of equilibrium is among the most intriguing questions in many-body physics. Quantum systems out of equilibrium typically relax to thermal equilibrium states by scrambling local information and building up entanglement entropy. However, kinetic constraints in the Hamiltonian can lead to a breakdown of this fundamental paradigm due to a fragmentation of the underlying Hilbert space into dynamically decoupled subsectors in which thermalisation can be strongly suppressed. Here, we experimentally observe Hilbert space fragmentation (HSF) in a two-dimensional tilted Bose-Hubbard model. Using quantum gas microscopy, we engineer a wide variety of initial states and find a rich set of manifestations of HSF involving bulk states, interfaces and defects, i.e., d = 2, 1 and 0 dimensional objects. Specifically, uniform initial states with equal particle number and energy differ strikingly in their relaxation dynamics. Inserting controlled defects on top of a global, non-thermalising chequerboard state, we observe highly anisotropic, sub-dimensional dynamics, an immediate signature of their fractonic nature. An interface between localized and thermalising states in turn displays dynamics depending on its orientation. Our results mark the first observation of HSF beyond one dimension, as well as the concomitant direct observation of fractons, and pave the way for in-depth studies of microscopic transport phenomena in constrained systems, Comment: 14 pages, 12 figures

Our understanding of phases of matter relies on symmetry breaking, one example being water ice whose crystalline structure breaks the continuous translation symmetry of space. Recently, breaking of time translation symmetry was observed in systems not in thermal equilibrium. The associated notion of time crystallinity has led to a surge of interest, raising the question about the extent to which highly controllable quantum simulators can generate rich and tunable temporal orders, beyond the conventional classification of order in static systems. Here, we investigate different kinds of partial temporal orders, stabilized by non-periodic yet structured drives, which we call rondeau order. Using a $^{13}$C-nuclear-spin diamond quantum simulator, we report the first experimental observation of a -- tunable degree of -- short-time disorder in a system exhibiting long-time stroboscopic order. This is based on a novel spin control architecture that allows us to implement a family of drives ranging from structureless via structured random to quasiperiodic and periodic drives. Leveraging a high throughput read-out scheme, we continuously observe the spin polarization over 105 pulses to probe rondeau order, with controllable lifetimes exceeding 4 seconds. Using the freedom in the short-time temporal disorder of rondeau order, we show the capacity to encode information in the response of observables. Our work broadens the landscape of observed nonequilibrium temporal order, paving the way for new applications harnessing driven quantum matter., Comment: 6+8 pages

How a closed system thermalizes, especially in the absence of global conservation laws but in the presence of disorder and interactions, is one of the central questions in non-equilibrium statistical mechanics. We explore this for a disordered, periodically driven Ising chain. Our numerical results reveal inhomogeneous thermalization leading to a distribution of thermalization timescales within a single disordered sample, which we encode via a distribution of effective local temperatures. Using this, we find an excellent collapse $\textit{without}$ $\textit{any}$ $\textit{fitting}$ $\textit{parameters}$ of the local relaxation dynamics for the entire range of disorder values in the ergodic regime when adapting the disorder-averaged diagonal entanglement entropy as internal `time' of the system. This approach evidences a remarkably uniform parametrization of the dynamical many-body evolution of local temperature within the otherwise highly heterogeneous ergodic regime, independent of the strength of the disorder., Comment: 7 pages, 6 figures

A recent advance in the study of emergent magnetic monopoles was the discovery that monopole motion is restricted to dynamical fractal trajectories (J. Hall\'en et al, Science 378, 1218 (2022)) thus explaining the characteristics of magnetic monopole noise spectra (Dusad, R. et al. Nature 571, 234 (2019); Samarakoon, A. M. et al. Proc. Natl. Acad. Sci. 119, e2117453119 (2022)). Here we apply this new theory to explore the dynamics of field-driven monopole currents, finding them comprised of two quite distinct transport processes: initially swift fractal rearrangements of local monopole configurations followed by conventional monopole diffusion. This theory also predicts a characteristic frequency dependence of the dissipative loss-angle for AC-field-driven currents. To explore these novel perspectives on monopole transport, we introduce simultaneous monopole current control and measurement techniques using SQUID-based monopole current sensors. For the canonical material Dy2Ti2O7, we measure ${\Phi}(t)$, the time-dependence of magnetic flux threading the sample when a net monopole current $J(t) = \dot{\Phi}(t)/\mu_0$ is generated by applying an external magnetic field $B_0(t)$. These experiments find a sharp dichotomy of monopole currents, separated by their distinct relaxation time-constants before and after $t \approx 600 {\mu}s$ from monopole current initiation. Application of sinusoidal magnetic fields $B_0(t) = Bcos({\omega}t)$ generates oscillating monopole currents whose loss angle ${\theta}(f)$ exhibits a characteristic transition at frequency $f \approx 1.8$ kHz over the same temperature range. Finally, the magnetic noise power is also dichotomic, diminishing sharply after $t \approx 600 {\mu}s$. This complex phenomenology represents a new form of heterogeneous dynamics generated by the interplay of fractionalization and local spin configurational symmetry.

Superdiffusion is surprisingly easily observed even in systems without the integrability underpinning this phenomenon. Indeed, the classical Heisenberg chain -- one of the simplest many-body systems, and firmly believed to be non-integrable -- evinces a long-lived regime of anomalous, superdiffusive spin dynamics at finite temperature. Similarly, superdiffusion persists for long timescales, even at high temperature, for small perturbations around a related integrable model. Eventually, however, ordinary diffusion is believed to be asymptotically restored. We examine the timescales governing the lifetime of the superdiffusive regime, and argue that it diverges algebraically fast -- both in deviation from the integrable limit, and at low temperature, where we find $t^* \sim T^{-\zeta}$ with an exponent possibly as large as $\zeta = 8$. This can render the crossover to ordinary diffusion practically inaccessible., Comment: 4 pages

We present a general driving protocol for many-body systems to generate a sequence of prethermal regimes, each exhibiting a lower symmetry than the preceding one. We provide an explicit construction of effective Hamiltonians exhibiting these symmetries. This imprints emergent quasi-conservation laws hierarchically, enabling us to engineer the respective symmetries and concomitant orders in nonequilibrium matter. We provide explicit examples, including spatiotemporal and topological phenomena, as well as a spin chain realizing the symmetry ladder $\text{SU(2)}{\rightarrow}\text{U(1)} {\rightarrow} \mathbb{Z}_2{\rightarrow} E$.

Inelastic neutron scattering (INS) measurements of powder D$_3(^{7}$Li)($^{193}$Ir)$_2$O$_6$ reveal low energy magnetic excitations with a scattering cross section that is broad in $|Q|$ and consistent with a Kitaev spin-liquid (KSL) state. The magnetic nature of the excitation spectrum is demonstrated by longitudinally polarized neutron studies. The total magnetic moment of 1.7(2)$\mu_B$/Ir inferred from the total magnetic scattering cross section is consistent with the effective moment inferred from magnetic susceptibility data and expectations for the $J_{\rm eff}=1/2$ single ion state. The rise in the dynamic correlation function ${\cal S}(Q,\omega)$ for $\hbar\omega<5~$meV can be described by a nearest-neighbor Kitaev model with interaction strength $K\approx-13(5)$~meV. Exchange disorder associated with the mixed D-Li site could play an important role in stabilizing the low $T$ quantum fluctuating state., Comment: 5 pages, 4 figures

Classical spin liquids (CSLs) have proved to be a fruitful setting for the emergence of exotic gauge theories. Vacancy clusters in CSLs can introduce gauge charges into the system, and the resulting behavior in turn reveals the nature of the underlying theory. We study these effects for a series of CSLs on the honeycomb lattice. We find that dilution leads to the emergence of effective free spins with tuneable, and generally irrational, size. For a specific higher-rank CSL, described by a symmetric tensor gauge fields, dilution produces non-decaying spin textures with a characteristic quadrupolar angular structure, and infinite-ranged interactions between dilution clusters.

Harmonic generation provides an efficient tool for the study of ultrafast nonlinear dynamics. We report on time-resolved optical-pump terahertz-harmonic-generation spectroscopic investigation of ultrafast nonlinearity in a prototypical three-dimensional Dirac semimetal Cd$_3$As$_2$. A transient population inversion characterized by excessive nonthermal Dirac electrons and holes is found to be very sensitive and responsive to a periodic terahertz drive, leading to very efficient terahertz third-harmonic generation. Based on the Boltzmann transport theory, we analyze the terahertz field-driven kinetics of the transient Dirac fermions that is responsible for the observed strong terahertz nonlinearity., Comment: 6 pages, 4 figures

Spontaneous symmetry breaking and more recently entanglement are two cornerstones of quantum matter. We introduce the notion of anisotropic entanglement ordered phases, where the spatial profile of spin-pseudospin entanglement spontaneously lowers the four-fold rotational symmetry of the underlying crystal to a two-fold one, while the charge density retains the full symmetry. The resulting phases, which we term $\textit{entanglement smectic}$ and $\textit{entanglement stripe}$, exhibit a rich Goldstone mode spectrum and a set of phase transitions as a function of underlying anisotropies. We discuss experimental consequences of such anisotropic entanglement phases distinguishing them from more conventional charge or spin stripes. Our discussion of this interplay between entanglement and spontaneous symmetry breaking focuses on multicomponent quantum Hall systems realizing textured Wigner crystals, as may occur in graphene or possibly also in moir\'e systems, highlighting the rich landscape and properties of possible entanglement ordered phases., Comment: 4.5+6.5 pages;Published version

We propose the strongly tilted Bose-Hubbard model as a natural platform to explore Hilbert-space fragmentation (HSF) and fracton dynamics in two-dimensions, in a setup and regime readily accessible in optical lattice experiments. Using a perturbative ansatz, we find HSF when the model is tuned to the resonant limit of on-site interaction and tilted potential. First, we investigate the quench dynamics of this system and observe numerically that the relaxation dynamics strongly depends on the chosen initial state -- one of the key signature of HSF. Second, we identify fractonic excitations with restricted mobility leading to anomalous transport properties. Specifically, we find excitations that show one dimensional diffusion (z = 1/2) as well as excitations that show subdiffusive behaviour in two dimensions (z = 3/4). Using a cellular automaton, we analyze their dynamics and compare to an effective hydrodynamic description., Comment: 4+4 pages, 7 figures

Two dimensional coherent spectroscopy (2DCS) probes the nonlinear optical response of correlated systems. An interesting application is the study of fractionalized excitations, which are challenging to distinguish unambiguously in linear response. Here we demonstrate how the sensitivity of optical matrix elements to variations in the photon polarization allows one to probe different aspects of low lying excitations in models of the candidate fractionalized materials Ce$_2$Zr$_2$O$_7$ and Nd$_2$Zr$_2$O$_7$, which host effective one-dimensional spin chains when subjected to a [110] magnetic field. We show how both fractionalized spinon excitations or conventional magnons can be picked out in the 2DCS response, and how the response from polarized spin chains can be used to probe the dipolar-octupolar mixing angle $\theta$ through the relative intensity of one- and two-magnon signals. Further, we find that a $[001]$ polarization of the probe field is particularly sensitive to lower band edge of the spinon continuum and can be used as a measure of the proximity of a quantum critical point in Ce$_2$Zr$_2$O$_7$. 2DCS can thus be employed to provide invaluable and detailed information both on the constituent degrees of freedom of a quantum material and on their collective behaviour., Comment: 23 pages, 17 figures

The Ising antiferromagnets on the triangular and on the pyrochlore lattices are two of the most iconic examples of magnetic frustration, paradigmatically illustrating many exotic properties such as emergent gauge fields, fractionalisation, and topological order. In this work, we show that the two instances of frustration can, remarkably, be combined in a single system, where they coexist without inducing conventional long range ordering. We show that the system undergoes a first order phase transition upon lowering the temperature, into a yet different frustrated phase that we characterise to exhibit nematic order. We argue that an extensive degeneracy survives down to zero temperature, at odds with a customary Pauling estimate. Dynamically, we find evidence of anomalous noise in the power spectral density, arising from an effectively anisotropic fractal motion of monopoles at low temperature., Comment: 19 pages, 18 figures

We investigate many-body dynamics where the evolution is governed by unitary circuits through the lens of `Krylov complexity', a recently proposed measure of complexity and quantum chaos. We extend the formalism of Krylov complexity to unitary circuit dynamics and focus on Floquet circuits arising as the Trotter decomposition of Hamiltonian dynamics. For short Trotter steps the results from Hamiltonian dynamics are recovered, whereas a large Trotter step results in different universal behavior characterized by the existence of local maximally ergodic operators: operators with vanishing autocorrelation functions, as exemplified in dual-unitary circuits. These operators exhibit maximal complexity growth, act as a memoryless bath for the dynamics, and can be directly probed in current quantum computing setups. These two regimes are separated by a crossover in chaotic systems. Conversely, we find that free integrable systems exhibit a nonanalytic transition between these different regimes, where maximally ergodic operators appear at a critical Trotter step., Comment: 22 pages, 16 figures

Flocks of animals represent a fascinating archetype of collective behavior in the macroscopic classical world, where the constituents, such as birds, concertedly perform motions and actions as if being one single entity. Here, we address the outstanding question of whether flocks can also form in the microscopic world at the quantum level. For that purpose, we introduce the concept of active quantum matter by formulating a class of models of active quantum particles on a one-dimensional lattice. We provide both analytical and large-scale numerical evidence that these systems can give rise to quantum flocks. A key finding is that these flocks, unlike classical ones, exhibit distinct quantum properties by developing strong quantum coherence over long distances. We propose that quantum flocks could be experimentally observed in Rydberg atom arrays. Our work paves the way towards realizing the intriguing collective behaviors of biological active particles in quantum matter systems. We expect that this opens up a path towards a yet totally unexplored class of nonequilibrium quantum many-body systems with unique properties., Comment: 4+ pages and 8 figures

Bosonic collective modes are ubiquitous in metals, but over a wide range of energy and momenta suffer from Landau damping, decaying into the continuum of particle-hole excitations. Here we point out that interactions can suppress this decay, protecting a finite fraction of the total spectral weight associated with the collective mode, e.g. a plasmon. The underlying mechanism is level repulsion between a discrete mode and the continuum. We demonstrate the effect using a number of simplified models of strongly correlated Fermi-liquid metals, including a ``solvable" random flavor model in the large$-N$ limit. We discuss in detail the possibility of observing such an avoided decay for plasmons in (moir\'e) graphene-like systems., Comment: 13 pages, 6 figures

Digital quantum simulation relies on Trotterization to discretize time evolution into elementary quantum gates. On current quantum processors with notable gate imperfections, there is a critical tradeoff between improved accuracy for finer timesteps, and increased error rate on account of the larger circuit depth. We present an adaptive Trotterization algorithm to cope with time-dependent Hamiltonians, where we propose a concept of piecewise "conserved" quantities to estimate errors in the time evolution between two (nearby) points in time; these allow us to bound the errors accumulated over the full simulation period. They reduce to standard conservation laws in the case of time-independent Hamiltonians, for which we first developed an adaptive Trotterization scheme [PRX Quantum 4, 030319]. We validate the algorithm for a time-dependent quantum spin chain, demonstrating that it can outperform the conventional Trotter algorithm with a fixed step size at a controlled error., Comment: 7+5pages, 5+2 figures. Accepted in PRL

Driven many-body systems typically experience heating due to the lack of energy conservation. Heating may be suppressed for time-periodic drives, but little is known for less regular drive protocols. In this work, we investigate the heating dynamics in aperiodically kicked systems, specifically those driven by quasi-periodic Thue-Morse or a family of random sequences with $n$-multipolar temporal correlations. We demonstrate that multiple heating channels can be eliminated even away from the high-frequency regime. The number of eliminated channels increases with multipolar order $n$. We illustrate this in a classical kicked rotor chain where we find a long-lived prethermal regime. When the static Hamiltonian only involves the kinetic energy, the prethermal lifetime $t^*$ can strongly depend on the temporal correlations of the drive, with a power-law dependence on the kick strength $t^*\sim K^{-2n}$, for which we can account using a simple linearization argument.

The non-equilibrium dynamics of domain wall initial states in a classical anisotropic Heisenberg chain exhibits a striking coexistence of apparently linear and non-linear behaviours: the propagation and spreading of the domain wall can be captured quantitatively by \textit{linear}, i.e. non-interacting, spin wave theory absent its usual justifications; while, simultaneously, for a wide range of easy-plane anisotropies, emission can take place of stable topological solitons -- a process and objects intrinsically associated with interactions and non-linearities. The easy-axis domain wall only has transient dynamics, the isotropic one broadens diffusively, while the easy-plane one yields a pair of ballistically counter-propagating domain walls which, unusually, broaden \textit{subdiffusively}, their width scaling as $t^{1/3}$., Comment: 4 + 4 pages

We study a classical Ising model on the honeycomb lattice with local two-body interactions and present strong evidence that at low temperature it realizes a higher-rank Coulomb liquid with fracton excitations. We show that the excitations are (type-I) fractons, appearing at the corners of membranes of spin flips. Because of the three-fold rotational symmetry of the honeycomb lattice, these membranes can be locally combined such that no excitations are created, giving rise to a set of ground states described as a liquid of membranes. We devise a cluster Monte-Carlo algorithm purposefully designed for this problem that moves pairs of defects, and use it to study the finite-temperature behavior of the model. We show evidence for a first order transition from a high-temperature paramagnet to a low-temperature phase whose correlations precisely match those predicted for a higher-rank Coulomb phase., Comment: 12 pages in total (6 pages main text + 6 pages supplementary material), 4 figures

The hallmark of highly frustrated systems is the presence of many states close in energy to the ground state. Fluctuations between these states can preclude the emergence of any form of order and lead to the appearance of spin liquids. Even on the classical level, spin liquids are not all alike: they may have algebraic or exponential correlation decay, and various forms of long wavelength description, including vector or tensor gauge theories. Here, we introduce a classification scheme, allowing us to fit the diversity of classical spin liquids (CSLs) into a general framework as well as predict and construct new kinds. CSLs with either algebraic or exponential correlation-decay can be classified via the properties of the bottom flat band(s) in their soft-spin Hamiltonians. The classification of the former is based on the algebraic structures of gapless points in the spectra, which relate directly to the emergent generalized Gauss's laws that control the low temperature physics. The second category of CSLs, meanwhile, are classified by the fragile topology of the gapped bottom band(s). Utilizing the classification scheme we construct new models realizing exotic CSLs, including one with anisotropic generalized Gauss's laws and charges with subdimensional mobility, one with a network of pinch-line singularities in its correlation functions, and a series of fragile topological CSLs connected by zero-temperature transitions., Comment: 38 pages, 23 figures. Extended version of the short summary paper arXiv:2305.00155

Classical spin liquids (CSL) lack long-range magnetic order and are characterized by an extensive ground state degeneracy. We propose a classification scheme of CSLs based on the structure of the flat bands of their Hamiltonians. Depending on absence or presence of the gap from the flat band, the CSL are classified as algebraic or fragile topological, respectively. Each category is further classified: the algebraic case by the nature of the emergent Gauss's law at the gap-closing point(s), and the fragile topological case by the homotopy of the eigenvector winding around the Brillouin zone. Previously identified instances of CSLs fit snugly into our scheme, which finds a landscape where algebraic CSLs are located at transitions between \fragile topological ones. It also allows us to present a new, simple family of models illustrating that landscape, which hosts both fragile topological and algebraic CSLs, as well as transitions between them., Comment: 6 pages, 3 figures. The extended discussion is in our long paper arXiv:2305.19189

We introduce a model to study magnon scattering in skyrmion crystals, sandwiched between ferromagnets which act as the source of magnons. Skyrmions are topological objects while skyrmion crystals break internal and translational symmetries, thus our setup allows us to study the interplay of topology and symmetry breaking. Starting from a basis of holomorphic theta functions, we construct an analytical ansatz for such a junction with finite spatially modulating topological charge density in the central region and vanishing in the leads. We then construct a suitably defined energy functional for the junction and derive the resulting equations of motion, which resemble a Bogoliubov-de Gennes-like equation. Using analytical techniques, field theory, heuristic models and microscopic recursive transfer-matrix numerics, we calculate the spectra and magnon transmission properties of the skyrmion crystal. We find that magnon transmission can be understood via a combination of low-energy Goldstone modes and effective emergent Landau levels at higher energies. The former manifests in discrete low-energy peaks in the transmission spectrum which reflect the nature of the Goldstone modes arising from symmetry breaking. The latter, which reflect the topology, lead to band-like transmission features, from the structure of which further details of the excitation spectrum of the skyrmion crystal can be inferred. Such characteristic transmission features are absent in competing phases of the quantum Hall phase diagram, and hence provide direct signatures of skyrmion crystal phases and their spectra. Our results directly apply to quantum Hall heterojunction experiments in monolayer graphene with the central region doped slightly away from unit filling, a $\nu = 1:1 \pm \delta \nu: 1$ junction and are also relevant to junctions formed by metallic magnets or in junctions with artificial gauge fields., Comment: 20+6 pages, 9+2 figures, comments welcome

We study the transport and equilibration properties of a classical Heisenberg chain, whose couplings are random variables drawn from a one-parameter family of power-law distributions. The absence of a scale in the couplings makes the system deviate substantially from the usual paradigm of diffusive spin hydrodynamics, and exhibit a regime of subdiffusive transport with an exponent changing continuously with the parameter of the distribution. We propose a solvable phenomenological model that correctly yields the subdiffusive exponent, thereby linking local fluctuations in the coupling strengths to the long-time, large-distance behaviour. It also yields the finite-time corrections to the asymptotic scaling, which can be important in fitting the numerical data. We show how such exponents undergo transitions as the distribution of the coupling gets wider, marking the passage from diffusion to a regime of slow diffusion, and finally to subdiffusion., Comment: 10+3 pages, 5+4 figures

We study the phase diagram of a bilayer quantum spin liquid model with Kitaev-type interactions on a square lattice. We show that the low energy limit is described by a $\pi$-flux Hubbard model with an enhanced SO(4) symmetry. The antiferromagnetic Mott transition of the Hubbard model signals a magnetic fragmentation transition for the spin and orbital degrees of freedom of the bilayer. The fragmented "N\'eel order" features a non-local string order parameter for an in-plane N\'eel component, in addition to an anisotropic local order parameter. The associated quantum order is characterized by an emergent $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ gauge field when the N\'eel vector is along the $\hat{z}$ direction, and a $\mathbb{Z}_2$ gauge field otherwise. We underpin these results with a perturbative calculation, which is consistent with the field theory analysis. We conclude with a discussion on the low energy collective excitations of these phases and show that the Goldstone boson of the $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ phase is fractionalized and non-local., Comment: 26 pages, 7 figures

Topological magnets host two sets of gauge fields: that of native Maxwell electromagnetism, thanks to the magnetic dipole moment of its constituent microscopic moments; and that of the emergent gauge theory describing the topological phase. Here, we show that in quantum spin ice, the emergent magnetic charges of the latter carry native electric charge of the former. We both provide a general symmetry-based analysis underpinning this result, and discuss a microscopic mechanism which binds a native electric charge to the emergent magnetic one. This has important ramifications. First and foremost, an applied electric field gives rise to an emergent magnetic field. This in turn exerts an `inverted' Lorentz force on moving emergent electric/native magnetic charges. This can be probed via what we term a magnetic Nernst effect: applying an electric field perpendicular to a temperature gradient yields a magnetisation perpendicular to both. Finally, and importantly as a further potential experimental signature, a thermal gas of emergent magnetic charges will make an activated contribution to the optical conductivity at low temperatures., Comment: 11 pages, 2 figures

We report finite-size topology in the quintessential time-reversal (TR) invariant systems, the quantum spin Hall insulator (QSHI) and the three-dimensional, strong topological insulator (STI): previously-identified helical or Dirac cone boundary states of these phases hybridize in wire or slab geometries with one open boundary condition for finite system size, and additional, topologically-protected, lower-dimensional boundary modes appear for open boundary conditions in two or more directions. For the quasi-one-dimensional (q(2-1)D) QSHI, we find topologically-protected, quasi-zero-dimensional (q(2-2)D) boundary states within the hybridization gap of the helical edge states, determined from q(2-1)D bulk topology characterized by topologically non-trivial Wilson loop spectra. We show this finite-size topology furthermore occurs in 1T'-WTe2 in ribbon geometries with sawtooth edges, based on analysis of a tight-binding model derived from density-functional theory calculations, motivating experimental investigation of our results. In addition, we find quasi-two-dimensional (q(3-1)D) finite-size topological phases occur for the STI, yielding helical boundary modes distinguished from those of the QSHI by a non-trivial magneto-electric polarizability linked to the original 3D bulk STI. Finite-size topological phases therefore exhibit signatures associated with the non-trivial topological invariant of a higher-dimensional bulk. Finally, we find the q(3-2)D STI also exhibits finite-size topological phases, finding the first signs of topologically-protected boundary modes of codimension greater than 1 due to finite-size topology. Finite-size topology of four or higher-dimensional systems is therefore possible in experimental settings without recourse to thermodynamically large synthetic dimensions., Comment: References updated

We show how one can generate domain walls that separate high- and low-density regions with opposite momenta in the ground state of a harmonically trapped Bose-Einstein condensate using a density-dependent gauge potential. Within a Gross-Pitaevskii framework, we elucidate the distinct roles of vector and scalar potentials and how they lead to synthetic electromagnetic fields that are localized at the domain wall. In particular, the kinetic energy cost of a steep density gradient is compensated by an electrostatic field that pushes particles away from a special value of density. We show numerically in one dimension that such a domain wall is more prominent for repulsive contact interactions, and becomes metastable at strong electric fields through a first-order phase transition that ends at a critical point as the field is reduced. Our findings build on recent experimental developments and may be realized with cold atoms in a shaken optical lattice, providing insights into collective phenomena arising from dynamical gauge fields., Comment: 7 pages, 5 figures

Time-dependent driving holds the promise of realizing dynamical phenomenon absent in static systems. Here, we introduce a correlated random driving protocol to realize a spatiotemporal order that cannot be achieved even by periodic driving, thereby extending the discussion of time translation symmetry breaking to randomly driven systems. We find a combination of temporally disordered micro-motion with prethermal stroboscopic spatiotemporal long-range order. This spatiotemporal order remains robust against generic perturbations, with an algebraically long prethermal lifetime where the scaling exponent strongly depends on the symmetry of the perturbation, which we account for analytically., Comment: 9 pages, 8 figures; References are updated. We also add discussions about the hyperuniform driving

Linear spin wave theory (LSWT) is the standard technique to compute the spectra of magnetic excitations in quantum materials. In this paper, we show that LSWT, even under ordinary circumstances, may fail to implement the symmetries of the underlying ordered magnetic Hamiltonian leading to spurious degeneracies. In common with pseudo-Goldstone modes in cases of quantum order-by-disorder these degeneracies tend to be lifted by magnon-magnon interactions. We show how, instead, the correct symmetries may be restored at the level of LSWT. In the process we give examples, supported by nonperturbative matrix product based time evolution calculations, where symmetries dictate that there should be a topological magnon gap but where LSWT fails to open up this gap. We also comment on possible spin split magnons in MnF$_2$ and similar rutiles by analogy to recently proposed altermagnets.

We show that certain lattice gauge theories exhibiting disorder-free localization have a characteristic response in spatially averaged spectral functions: a few sharp peaks combined with vanishing response in the zero frequency limit. This reflects the discrete spectra of small clusters of kinetically active regions formed in such gauge theories when they fragment into spatially finite clusters in the localized phase due to the presence of static charges. We obtain the transverse component of the dynamic structure factor, which is probed by neutron scattering experiments, deep in this phase from a combination of analytical estimates and a numerical cluster expansion. We also show that local spectral functions of large finite clusters host discrete peaks whose positions agree with our analytical estimates. Further, information spreading, diagnosed by an unequal time commutator, halts due to real space fragmentation. Our results can be used to distinguish the disorder-free localized phase from conventional paramagnetic counterparts in those frustrated magnets which might realize such an emergent gauge theory., Comment: 5 pages, 3 figures

Fractals -- objects with non-integer dimensions -- occur in manifold settings and length scales in nature, ranging from snowflakes and lightning strikes to natural coastlines. Much effort has been expended to generate fractals for use in many-body physics. Here, we identify an emergent dynamical fractal in a disorder-free, stoichiometric three-dimensional magnetic crystal in thermodynamic equilibrium. The phenomenon is born from constraints on the dynamics of the magnetic monopole excitations in spin ice, which restrict them to move on the fractal. This observation explains the anomalous exponent found in magnetic noise experiments in the spin ice compound Dy$_2$Ti$_2$O$_7$, and it resolves a long standing puzzle about its rapidly diverging relaxation time. The capacity of spin ice to exhibit such striking phenomena holds promise of further surprising discoveries in the cooperative dynamics of even simple topological many-body systems., Comment: 13 pages, 11 figures

Dual-unitary circuits are paradigmatic examples of exactly solvable yet chaotic quantum many-body systems, but solvability naturally goes along with a degree of non-generic behaviour. By investigating the effect of weakly broken dual-unitarity on the spreading of local operators we study whether, and how, small deviations from dual-unitarity recover fully generic many-body dynamics. We present a discrete path-integral formula for the out-of-time-order correlator and use it to recover a butterfly velocity smaller than the light-cone velocity, $v_B < v_{LC}$ , and a diffusively broadening operator front, two generic features of ergodic quantum spin chains absent in dual-unitary circuit dynamics. We find that the butterfly velocity and diffusion constant are determined by a small set of microscopic quantities and that the operator entanglement of the gates plays a crucial role., Comment: (6+15) pages, 5 figures; Accepted version; corrected some typos in the supplementary material