Motivated by the search for unconventional orders in frustrated quantum magnets, we present a multi-method investigation into the nature of the quantum phase diagram of the spin-1/2 Heisenberg model on the maple-leaf lattice with three symmetry-inequivalent nearest-neighbor interactions. It has been argued that the parameter regime with antiferromagnetic couplings on hexagons J h and ferromagnetic couplings on triangles J t and dimer J d bonds is potentially host to a cornucopia of emergent phases with unconventional orders. Our analysis indeed identifies an extended region where any conventional dipolar magnetic order is absent. A hexagonal singlet state is found in the region around J d = J t = 0, while a dimerized hexagonal singlet order of a lattice nematic character appears proximate to the phase boundary with the c120 degrees antiferromagnetic order. Interestingly, upon traversing the bulk of the paramagnetic (PM) region, we find a variety of distinct correlation profiles, which are qualitatively different from those of the hexagonal singlet and dimerized hexagonal singlet orders but feature no appreciable spin-nematic response, while the boundary with the ferromagnetic phase shows evidence of spin-nematic order. This PM region is thus likely host to an ensemble of nonmagnetic phases, which could putatively include quantum spin liquids. Our phase diagram is built from a complementary application of state-of-the-art implementations of the cluster mean-field and pseudo-fermion functional renormalization group approaches, together with an unconstrained Luttinger-Tisza treatment of the model providing insights from the semi-classical limit.

Conditional probability distributions describe the effect of learning an initially unknown classical state through Bayesian inference. Here we demonstrate the existence of a learning transition, having signatures in the long distance behavior of conditional correlation functions, in the two-dimensional classical Ising model. This transition, which arises when learning local energy densities, extends all the way from the infinite-temperature paramagnetic state down to the thermal critical state. The intersection of the line of learning transitions and the thermal Ising transition is a new tricritical point. Our model for learning also exactly describes the effects of weak measurements on ground states of frustration-free quantum Hamiltonians, which interpolate between the toric code and a paramagnet. Notably, the location of the above tricritical point implies that the quantum memory defined by the degenerate ground states in the topological phase is robust to weak measurement, even when the initial state is arbitrarily close to the quantum phase transition separating topological and trivial phases. Our analysis uses a replica field theory combined with the renormalization group, and we chart out the phase diagram using a combination of tensor network and Monte Carlo techniques. Our methods can be extended to study the more general effects of learning on both classical and quantum states. The learning induced critical states can be realized in classical or quantum devices.

Abstract With recent advances in terahertz (THz) sources and detection, two-dimensional coherent spectroscopy (2DCS), which allows one to probe nonlinear responses, now reaches the meV regime relevant for quasiparticle excitations in magnetic materials. This opens a promising route to reveal many-body phenomena that evade linear-response probes. To date, most experimental applications have focused on classical magnets, and a solid demonstration in a quantum magnet has yet to be established. Here we present a theoretical study of 2DCS in CoNb 2 O 6 , a quasi-one-dimensional Ising magnet that is believed to host fractionalized spinons which at low temperatures are confined by weak interchain coupling. Our analysis, which builds on an experimentally constrained effective S = 1/2 Hamiltonian, is found to reveal unambiguous 2DCS signatures of spinon deconfinement above the low-temperature ordered phase. Using a four-spinon approximation, we track these 2DCS signatures by sequentially building a faithful microscopic model for CoNb 2 O 6 , starting from the exactly solvable one-dimensional transverse-field Ising model (1 d TFIM) and successively adding additional interactions. In particular, adding a bond-dependent staggered YZ interaction to the 1 d -TFIM already reproduces many key spectral features of the full material Hamiltonian. Within this TFIM+YZ model, we find a series of bound states, including a four-spinon bound state that is distinct from the familiar two-spinon bound states. We further find that introducing a confinement potential suppresses sharp spinon-echo features, which are thought to reflect an underlying continuum of fractionalized excitations. Our results provide concrete predictions for future THz 2DCS experiments on CoNb 2 O 6 and related quasi-one-dimensional quantum magnets.

Transmon qubits arise from the quantization of nonlinear resonators, systems that are prone to the buildup of strong, possibly chaotic, fluctuations. Such instabilities will likely affect fast gate operations which involve the transient population of higher excited states outside the computational subspace. Here we show that a statistical analysis of the instantaneous eigenphases of the time evolution operator, in particular of their curvatures, allows for identifying the subspace affected by chaotic fluctuations. Our analysis shows that fast entangling gates, operating at speeds close to the so-called quantum speed limit, contain transient regimes where the dynamics indeed becomes partially chaotic for just two transmons.

Emergent gauge fields and Coulomb liquids have long been central to the physics of frustrated pyrochlore magnets, yet their realization beyond conventional, i.e., rank-one U(1), spin ice and into fully quantum higher-rank regimes has remained elusive. Here we provide strong evidence for this physics in the spin-1/2 quantum Heisenberg antiferromagnet on the breathing pyrochlore lattice with symmetry-allowed Dzyaloshinskii-Moriya interactions, using the pseudofermion functional renormalization group. We find a robust quantum analog of the rank-one U(1) Coulomb liquid together with compelling signatures of a putative quantum analog of the rank-two U(1) Coulomb liquid, distinguished by their characteristic multifold pinch-point morphologies in momentum space. This Letter therefore establishes a minimal three-dimensional setting in which signatures of gauge theories of different rank emerge within a single microscopic spin Hamiltonian. In addition, quantum fluctuations qualitatively reshape the classical nearest-neighbor atlas of phases, causing an incommensurate spiral instability and an extended nondipolar quantum paramagnetic regime, both absent in the classical model. Our results establish the breathing pyrochlore as a promising and experimentally relevant platform where higher-rank gauge constraints, conventional magnetic order, and fluctuation-driven quantum phases compete on equal footing, opening a direct route to diagnosing emergent gauge structure in three-dimensional quantum magnets.

Floquet quantum error-correcting codes provide an operationally economical route to fault tolerance by dynamically generating stabilizer structures using only two-body Pauli measurements. But while it is well established that stabilizer codes in higher spatial dimensions gain additional levels of intrinsic robustness, higher-dimensional Floquet codes have hitherto been explored only in limited scope. Here we introduce a 3d generalization of a Floquet code whose instantaneous stabilizer group realizes a 3d fermionic toric code, while crucially preserving all three logical qubits throughout the entire measurement sequence. One central ingredient is the identification of a 3d lattice geometry that generalizes the features of the Kekulé lattice underlying the 2d Hastings-Haah code - specifically, a structure where deleting any one edge color yields a two-color subgraph that decomposes into short, closed loops rather than homologically nontrivial chains. This loop property avoids the collapse of logical information that plagues naive sequential two-color measurement schedules on many 3d lattices. Although, for our lattice geometry, a simple 3-round cycle that sequentially measures the three types of parity checks does not expose the full error syndrome set, we show that one can append a measurement sequence to extract the missing syndromes without disturbing the logical subspace. Beyond code design, 3d tricoordinated lattice geometries define a family of 3d monitored Kitaev models, in which random measurements of the non-commuting parity checks give rise to dynamically created entangled phases with nontrivial topology. In discussing the general structure of their underlying phase diagrams and, in particular, the existence of certain quantum critical points, we again make a connection to the general preservation of logical information in time-ordered Floquet protocols.

Spin-orbit coupling locks spin direction and spatial orientation and generates, in semi-classical magnets, a local spin easy-axis and associated ordering. Quantum spin-1/2’s defy this fate: rather than spins becoming locally anisotropic, the spin-spin interactions do. Consequently interactions become dependent on the spatial orientation of bonds between spins, prime theoretical examples of which are Kitaev magnets. Bond-directional interactions imply the existence of bond-directional magnetic modes, spin excitations that render crystallographically equivalent bonds magnetically inequivalent, which yet have remained elusive experimentally. Here we show that resonant inelastic X-ray scattering allows us to explicitly probe the bond-directional character of magnetic excitations. To do so, we use a scattering plane spanned by one bond and the corresponding spin component and scan a range of momentum transfer that encompasses multiple Brillouin zones. Applying this approach to Na2IrO3 we establish the different bond-directional characters of magnetic excitations at ~10 meV and ~45 meV. The bond-directional nearest-neighbor excitations are a fingerprint of the dominant Kitaev interactions and even prevail in the magnetically ordered state.

In quantum error correction, the error threshold provides essential quantitative guidance for the ability to bring about fault-tolerance through decoding the effects of incoherent noise, weak measurement or inference. However, the numerical value of an error threshold is typically only accessible through large-scale numerical simulations of the underlying noise model. Here we introduce an analytical estimate of error thresholds falling into the Nishimori universality class via a Fourier–Walsh projection scheme that maps the critical point of the underlying disorder-free statistical-mechanics model to the Born-disordered Nishimori critical point. Using a minimal replica theory approach, this closed-form estimate is obtained from a projection of the exact replicated single-bond weight which we find to reproduce (within a percentage point) the known numerical thresholds of random-bond and random-plaquette Ising models / ℤ_2 stabilizer codes in spatial dimensions d=2-5, and extends to Potts variables with q≤4. The main application of our projection scheme is to ℤ_q surface codes, whose decoding problem maps to the disordered q-state clock model. For q≥5 the clean clock model has two Berezinskii–Kosterlitz–Thouless transitions, which the projection maps to two Nishimori temperatures that bound an intermediate information-critical phase. The resulting threshold values not only accurately agree with recent decohered-ℤ_q-toric-code numerics, but are found to satisfy the Gilbert–Varshamov self-dual entropy relation ln q ≃ H_q(T_1^∗)+H_q(T_2^∗), although no duality condition is imposed in the construction. Our approach thereby points to a deeper connection between the clean and Born-disordered models, while allowing for instant analytical estimates of error thresholds for a variety of stabilizer codes.

Motivated by a previous Ising study, we identify a higher Nishimori line in the learning phase diagram of the 2D q-state Potts model (2 < q≤ 4) under bond-energy measurements. This higher Nishimori line meets the critical temperature line of the Potts model, in a higher Nishimori critical point – a tricritical point at finite inference strength that separates a paramagnetic, a ferromagnetic and a 'spin-glass' phase. With analytical tools, we discuss the general structure of the rich phase diagram, which contains two unstable and three stable fixed points, and obtain a number of exact results for universal quantities, including the decay exponent of the Edwards-Anderson correlator, using a Gaussian measurement protocol which allows for exact calculations. Using extensive numerical tools, we confirm these statements for a generic, discrete q-state measurement protocol and determine precise numerical estimates for the location of higher and ordinary Nishimori critical points as well as RG flows between the various fixed points. We also discuss the Casimir effective central charges of the critical points in the learning phase diagram, and their monotonic decrease along measurement-induced RG flows, as established non-perturbatively by the c-effective theorem and its extensions, and contrast it to the monotonic increase along the corresponding RG flows in the random-bond Potts model. Finally, we discuss a general argument based on Elitzur's theorem that establishes stability of the ordinary Nishimori critical points in their respective learning phase diagrams. Equivalently, our results describe a monitored deformed ℤ_q toric code where the tricritical higher Nishimori point is an 'information' critical point that separates stable quantum, classical, and no memory phases.

We revisit a learning-induced tricritical point, at which three phases with strong, weak, and broken Z_2 symmetry meet, in the phase diagram of a deformed toric code wavefunction subjected to weak measurements. This setting is exactly dual to a classical Bayesian inference phase diagram of the 2D classical Ising model. Here we demonstrate that this tricritical point lies on a distinct higher Nishimori line, which has an emergent gauge-invariant formulation, just like the ordinary Nishimori line but with a higher replica symmetry as a replica stat-mech model in the replica number R→2 limit, where disorder is averaged according to the Born rule. As such, the learning tricritical point is in fact a higher Nishimori critical point. Using this identification, we obtain a number of exact results at this higher Nishimori critical point; e.g., we show that the power-law exponent of the Edwards-Anderson correlation function is exactly equal to that of the spin correlation function at the unmeasured Ising critical point and verify this in numerical simulations. Using the tools of the proof of a c-effective theorem [arXiv:2507.07959], we show that the Casimir effective central charge c_eff decreases under renormalization group (RG) flow from the higher Nishimori critical point to the unmeasured 2D Ising critical point, and is thus greater than 1/2. This is corroborated by extensive numerical simulations finding c_eff = 0.522(1). The analytical result also explains, with a physically motivated assumption, the numerically observed increase of the Casimir effective central charge under the RG flow from the ordinary Nishimori critical point to the clean Ising critical point in the random-bond Ising model. We also discuss higher Nishimori criticality in general dimensions D>1.

For the quantum phase transition in the transverse-field Ising chain, Kramers-Wannier duality not only protects its critical properties but also pinpoints the location of the phase transition. Its role in out-of-equilibrium, monitored dynamics, however, remains largely unexplored beyond time-periodic Floquet protocols where self-duality turns into a statistical average symmetry. Here we explore the emergence of dynamical self-duality in the absence of time-translation symmetry by investigating the monitored dynamics of one-dimensional Ising/Majorana chains where measurements are arranged in a quasiperiodic Fibonacci sequence. We find that the dynamical extension of this non-invertible symmetry to an out-of-equilibrium setting allows one to organize the dynamical phase diagram of entangled phases, both predicting the transition locations and protecting universal critical behavior. Analytically and numerically, we identify two distinct critical lines, both related to the golden ratio, for Born-rule weak measurements and for random Clifford projective measurements. The latter coincides with the transition of a pure imaginary-time evolution, which can be viewed as a post-selected trajectory. The universality classes of the long-time critical steady states at Fibonacci times are determined, while the transient dynamics between Fibonacci times is deformed by measurements, realizing dynamical measurement-altered quantum criticality in real time.

Quantum circuits offer a versatile platform for simulating digital quantum dynamics and uncovering novel states of nonequilibrium quantum matter. One principal example are measurement-induced phase transitions arising from nonunitary dynamics in monitored circuits, which employ midcircuit measurements as an essential building block next to standard unitary gates. Although a comprehensive understanding of the dynamics in generic circuits is still evolving, we contend that monitored quantum circuits give rise to robust phases of dynamic matter, which-akin to Hamiltonian ground-state phases-yield emergent universal behavior, which can be categorized based on circuit symmetries and spatial dimensionality. To illustrate this concept, we focus on measurement-only quantum circuits within symmetry classes BDI and D, which are measurement-only circuit adaptations of the paradigmatic Kitaev and Yao-Kivelson models, embodying particle-hole-symmetric Majorana fermions with or without time reversal. We establish a general framework (Majorana loop models) for both symmetry classes (in arbitrary spatial dimensions) to provide access to the phenomenology of the entanglement dynamics in these circuits, displaying both an area-law phase of localized Majorana loops and a delocalized, highly entangled Majorana liquid phase. The two phases are separated by a continuous transition displaying quantum Lifshitz scaling, albeit with critical exponents of two distinct, non-Hamiltonian universality classes. The loop model framework provides not only analytical understanding of these universality classes in terms of nonlinear sigma models but also allows for highly efficient numerical techniques capable of simulating excessively large circuits with up to 108 qubits. We utilize this framework to accurately determine universal probes that distinguish both the entangled phases and the critical points of the two symmetry classes. Our work thereby further solidifies the concept of emergent circuit phases and their phase transitions.

Frustrated magnets can elude the paradigm of conventional symmetry breaking and instead exhibit signatures of emergent symmetries at low temperatures. Such symmetries arise from "accidental" degeneracies within the ground-state manifold and have been explored in a number of disparate models, in both two and three dimensions. Here we report the systematic construction of a family of classical spin models that, for a wide variety of lattice geometries with triangular motifs in one, two, and three spatial dimensions, such as the kagome or hyperkagome lattices, exhibit an emergent, continuous U(1) symmetry. This is particularly surprising because the underlying Hamiltonian actually has very little symmetry-a bond-directional, off-diagonal exchange model inspired by the microscopics of spin-orbit entangled materials (the Gamma ' model). The construction thus allows for a systematic study of the interplay between the emergent continuous U(1) symmetry and the underlying discrete Hamiltonian symmetries in different lattices across different spatial dimensions. We discuss the impact of thermal and quantum fluctuations in lifting the accidental ground-state degeneracy via the thermal and quantum order-by-disorder mechanisms, and how spatial dimensionality and lattice symmetries play a crucial role in shaping the physics of the model. Complementary Monte Carlo simulations, for representative one-, two-, and three-dimensional lattice geometries, provide a complete account of the thermodynamics and confirm our analytical expectations.

Motivated by the magnetism of pyrochlore oxides, we consider the effect of quantum fluctuations in the most general symmetry-allowed nearest-neighbor Kramers exchange Hamiltonian on the pyrochlore lattice. At the classical level, this Hamiltonian exhibits a rich landscape of classical spin liquids and a variety of nonconventional magnetic phases. In contrast, much remains unclear for the quantum model, where quantum fluctuations have the potential to alter the classical landscape and stabilize novel magnetic phases. Employing state-of-the-art pseudofermion functional renormalization group calculations for the spin-1/2 model, we determine the quantum phase diagram at relevant cross-sections, where the classical model hosts an algebraic nodal rank-2 spin liquid and a spin nematic order. We find large regions in parameter space where dipolar magnetic order is absent, and, based on known fingerprints in the correlation functions, we suggest that this nonconventional region is composed of an ensemble of distinct phases stabilized by quantum fluctuations. Our results hint at the existence of a spin nematic phase, and we identify the quantum analog of the classical rank-2 spin liquid. Furthermore, we highlight the importance of assessing the subtle interplay of quantum and thermal fluctuations in reconciling the experimental findings on the nature of magnetic order in Yb2Ti2O7.

Frustrated magnets can elude the paradigm of conventional symmetry breaking and instead exhibit signatures of emergent symmetries at low temperatures. Such symmetries arise from "accidental" degeneracies within the ground state manifold and have been explored in a number of disparate models, in both two and three dimensions. Here we report the systematic construction of a family of classical spin models that, for a wide variety of lattice geometries with triangular motifs in one, two and three spatial dimensions, such as the kagome or hyperkagome lattices, exhibit an emergent, continuous U(1) symmetry. This is particularly surprising because the underlying Hamiltonian actually has very little symmetry - a bond-directional, off-diagonal exchange model inspired by the microscopics of spin-orbit entangled materials (the Γ^'-model). The construction thus allows for a systematic study of the interplay between the emergent continuous U(1) symmetry and the underlying discrete Hamiltonian symmetries in different lattices across different spatial dimensions. We discuss the impact of thermal and quantum fluctuations in lifting the accidental ground state degeneracy via the thermal and quantum order-by-disorder mechanisms, and how spatial dimensionality and lattice symmetries play a crucial role in shaping the physics of the model. Complementary Monte Carlo simulations, for representative one-, two-, and three-dimensional lattice geometries, provide a complete account of the thermodynamics and confirm our analytical expectations.

For the surface code, topological quantum order allows one to encode logical quantum information in a robust, long-range entangled many-body quantum state. However, if an observer probes this quantum state by performing measurements on the underlying qubits, thereby collecting an ensemble of highly correlated classical snapshots, two closely related questions arise: (i) do measurements decohere the topological order of the quantum state; and (ii) how much of the logical information can one learn from the snapshots? Here we address these questions for measurements in a uniform basis on all qubits. We find that for generic measurement angles, sufficiently far away from the Clifford X, Y, and Z directions (such as the X+Y+Z basis) the logical information is never lost in one of the following two ways: (i) for weak measurement, the topological order is absolutely robust; (ii) for projective measurement, the quantum state inevitably collapses, but the logical quantum information is faithfully transferred from the quantum system to the observer in the form of a tomographically complete classical shadow. At these generic measurement angles and in the projective-measurement limit, the measurement ensemble enforced by Born probabilities can be represented by a 2D tensor network that can be fermionized into a disordered, free-fermion network model in symmetry class DIII, which gives rise to a Majorana "metal" phase. When the measurement angle is biased towards the X or Z limits, a critical angle indicates the threshold of a learning transition beyond which the classical shadow no longer reveals full tomographic information (but reduces to a measurement of the logical X or Z state). This learning transition can be described in the language of the network model as a "metal to insulator" transition...

Quantum measurements performed on a subsystem of a quantum many-body state can generate entanglement for its remaining constituents. The whole system including the measurement record is described by a hybrid mixed state, which can exhibit exotic phase transitions and critical phenomena. We demonstrate that generic measurement-induced phase transitions (MIPTs) can be cast as decoherence-induced critical mixed states in one higher dimension, by constructing a projected entangled pair state (PEPS) prior to decoherence or measurement. In this context, a deeper conceptual understanding of such mixed-state criticality is called for, particularly with regard to algebraic symmetry as an advanced organizing principle for such entangled states of matter. Integrating these connections we investigate the role of self-dual symmetry – a fundamental notion in theoretical physics – in mixed states, showing that the decoherence of electric (e) and magnetic (m) vortices from the 2D bulk of the toric code, or equivalently, a 2D cluster state with symmetry-protected topological order, can leave a (1+1)D quantum critical mixed state protected by a weak Kramers-Wannier self-dual symmetry. The corresponding self-dual critical bulk is described by the N->1 limit of the 2D Non-linear Sigma Model in symmetry class D with target space SO(2N)/U(N) at Θ-angle π, and represents a "measurement-version" of the Cho-Fisher network model subjected to Born-rule randomness...

Motivated by the recent introduction of a U(1)-symmetric toric code (TC) model, we investigate symmetry-based deformations of topological order by systematically deconstructing the Gauss-law-enforcing star terms of the TC Hamiltonian. This "term-dropping" protocol introduces global symmetries that go beyond the alternative framework of "ungauging" topological order in symmetry-deformed models and gives rise to models such as the U(1)TC or XY TC. These models inherit (emergent) subsystem symmetries (from the original 1-form symmetry of the TC) that can give rise to (subextensive) ground-state degeneracies, which can still be organized by the eigenvalues of Wilson loop operators. However, we demonstrate that these models do not support topological or fracton order (as has been conjectured in the literature) due to the loss of (emergent) gauge symmetry. An extreme deformation of the TC is the quantum dimer model (QDM), which we discuss along the family of symmetry-deformed models from the perspective of subsystem symmetries, sublattice modulation, and quantum order-by-disorder mechanisms resulting in rich phase diagrams. For the QDM, this allows us to identify an emergent SO(2) symmetry for what appears to be a gapless ground state (by numerical standards) that is unstable to the formation of a plaquette valence bond solid upon sublattice modulation.

Local moments with a spin S>1/2 can exhibit a rich variety of elementary quasiparticle excitations, such as quadrupolar excitations, that go beyond the dipolar magnons of conventional spin-1/2 systems. However, the experimental observation of such quadrupolar excitations is often challenging due to the dipolar selection rules of many linear response probes, rendering them invisible. Here we show that non-linear spectroscopy, in the form of two-dimensional coherent spectroscopy (2DCS), can be used to reveal quadrupolar excitations. Considering a family of spin-1 Heisenberg ferromagnets with single-ion easy-axis anisotropy as an example, we explicitly calculate their 2DCS signature by combining exact diagonalization and generalized spin wave theory. We further demonstrate that 2DCS can provide access to the quadrupolar weight of an excitation, analogous to how linear response provides access to the dipolar weight. Our work highlights the potential of non-linear spectroscopy as a powerful tool to diagnose multipolar excitations in quantum magnets.

In circuit-based quantum state preparation, qubit loss and coherent errors are circuit imperfections that imperil the formation of long-range entanglement beyond a certain threshold. The critical theory at the threshold is a continuous entanglement transition known to be described by a (2+0)-dimensional non-unitary conformal field theory which, for the two types of imperfections of certain circuits, is described by either percolation or Nishimori criticality, respectively. Here we study the threshold behavior when the two types of errors simultaneously occur and show that, when moving away from the Clifford-regime of projective stabilizer measurements, the percolation critical point becomes unstable and the critical theory flows to Nishimori universality. We track this critical renormalization group (RG) crossover flow by mapping out the entanglement phase diagrams, parametrized by the probability and strength of random weak measurements, of two dual protocols preparing surface code or GHZ-class cat states from a parent cluster state via constant-depth circuits. Extensive numerical simulations, using hybrid Gaussian fermion and tensor network / Monte Carlo sampling techniques on systems with more than a million qubits, demonstrate that an infinitesimal deviation from the Clifford regime leads to a sudden, strongly non-monotonic entanglement growth at the incipient non-unitary RG flow. We argue that spectra of scaling dimensions of both the percolation and Nishimori fixed points exhibit multifractality. For percolation, we provide the exact (non-quadratic) multifractal spectrum of exponents, while for the Nishimori fixed point we show high-precision numerical results for five leading exponents characterizing multifractality.