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Traditional magnetic sub-Kelvin cooling relies on the nearly free local moments in hydrate paramagnetic salts, whose utility is hampered by the dilute magnetic ions and low thermal conductivity. Here we propose to use instead fractional excitations inherent to quantum spin liquids (QSLs) as an alternative, which are sensitive to external fields and can induce a very distinctive magnetocaloric effect. With state-of-the-art tensor-network approach, we compute low-temperature properties of Kitaev honeycomb model. For the ferromagnetic case, strong demagnetization cooling effect is observed due to the nearly free $Z_2$ vortices via spin fractionalization, described by a paramagnetic equation of state with a renormalized Curie constant. For the antiferromagnetic Kitaev case, we uncover an intermediate-field gapless QSL phase with very large spin entropy, possibly due to the emergence of spinon Fermi surface. Potential realization of topological excitation cooling in Kitaev materials is also discussed, which may offer a promising pathway to circumvent existing limitations in the paramagnetic hydrates., Comment: 10 pages, 4 figures; supplementary materials; to appear in Nat. Commun. (2024)

Two-dimensional mirror symmetry-enriched topological orders can be studied using the folding approach: it can be folded along the mirror axis and turned into a bilayer system on which the mirror symmetry acts as a $\mathbb Z_2$ layer-exchange symmetry. How mirror-symmetry enriches the topological order is then encoded at the mirror axis, which is a gapped boundary of the folded bilayer system. Based on anyon-condensation theory, we classify possible $\mathbb Z_2$-symmetric gapped boundaries of the folded system. In particular, we derive an $H^2$ obstruction function, which corresponds to an $H^3$ obstruction for topological orders enriched by the time-reversal symmetry instead of mirror symmetry. We demonstrate that states with a nontrivial $H^2$ obstruction function can be constructed on the surface of a three-dimensional mirror SET order., Comment: 13 pages, 5 figures

The energy-efficient and brain-like information processing abilities of Spiking Neural Networks (SNNs) have attracted considerable attention, establishing them as a crucial element of brain-inspired computing. One prevalent challenge encountered by SNNs is the trade-off between inference speed and accuracy, which requires sufficient time to achieve the desired level of performance. Drawing inspiration from animal behavior experiments that demonstrate a connection between decision-making reaction times, task complexity, and confidence levels, this study seeks to apply these insights to SNNs. The focus is on understanding how SNNs make inferences, with a particular emphasis on untangling the interplay between signal and noise in decision-making processes. The proposed theoretical framework introduces a new optimization objective for SNN training, highlighting the importance of not only the accuracy of decisions but also the development of predictive confidence through learning from past experiences. Experimental results demonstrate that SNNs trained according to this framework exhibit improved confidence expression, leading to better decision-making outcomes. In addition, a strategy is introduced for efficient decision-making during inference, which allows SNNs to complete tasks more quickly and can use stopping times as indicators of decision confidence. By integrating neuroscience insights with neuromorphic computing, this study opens up new possibilities to explore the capabilities of SNNs and advance their application in complex decision-making scenarios., Comment: 27 pages, 6 figures, 3 tables

The pyrochlore iridates have become ideal platforms to unravel fascinating correlated and topolog?ical phenomena that stem from the intricate interplay among strong spin-orbit coupling, electronic correlations, lattice with geometric frustration, and itinerancy of the 5d electrons. The all-in-all?out antiferromagnetic state, commonly considered as the magnetic ground state, can be dramatically altered in reduced dimensionality, leading to exotic or hidden quantum states inaccessible in bulk. Here, by means of magnetotransport, resonant elastic and inelastic x-ray scattering experiments, we discover an emergent quantum disordered state in (111) Y2Ir2O7 thin films (thickness less than 30 nm) per?sisting down to 5 K, characterized by dispersionless magnetic excitations. The anomalous Hall effect observed below an onset temperature near 135 K corroborates the presence of chiral short-range spin configurations expressed in non-zero scalar spin chirality, breaking the macroscopic time-reversal symmetry. The origin of this chiral state is ascribed to the restoration of magnetic frustration on the pyrochlore lattice in lower dimensionality, where the competing exchange interactions together with enhanced quantum fluctuations suppress any long-range order and trigger spin-liquid-like behavior with degenerate ground-state manifold.

Employing large-scale quantum Monte Carlo simulations, we find in magnetized interacting Dirac fermion model, there emerges a new and universal collective Larmor-Silin spin wave mode in the transverse dynamical spin susceptibility. Such mode purely originates from the interaction among Dirac fermions and distinguishes itself from the usual particle-hole continuum with finite lifetime and clear dispersion, both at small and large momenta in a large portion of the Brillouin zone. Our unbiased numerical results offer the dynamic signature of this new collective excitations in interacting Dirac fermion systems, and provide experimental guidance for inelastic neutron scattering, electron spin resonance and other spectroscopic approaches in the investigation of such universal collective modes in quantum Moire materials, topological insulators and quantum spin liquid materials under magnetic field, with quintessential interaction nature beyond the commonly assumed noninteracting Dirac fermion or spinon approximations., Comment: 12 pages, 6 figures

We propose a new method to efficiently compute the entanglement entropy (EE) of quantum many-body systems. Our approach, called the incremental SWAP operator method , combines the simplicity of the SWAP operator used in projector quantum Monte Carlo simulations with recent advances in precisely computing exponential observables using incremental algorithms. We apply this technique to obtain accurate EE data at reduced computational cost for 1d and 2d antiferromagnetic Heisenberg models with different bipartition schemes. Using the computed EE data, we extract the area law coefficient, universal logarithmic corrections from Goldstone modes, and the geometric constant, finding quantitative agreement with analytical predictions. Moreover, for the first time in an unbiased numerical simulation of 2d antiferromagnetic Heisenberg model, we successfully obtain reliable universal logarithmic corrections from sharp corners that match expected theoretical values. The consistency between our numerical results and theoretical calculations demonstrates the power of our approach for accessing challenging universal entanglement properties. The extensions of our method to other quantum spin/boson models and the interacting fermion models, are outlined.

Sensory perception originates from the responses of sensory neurons, which react to a collection of sensory signals linked to various physical attributes of a singular perceptual object. Unraveling how the brain extracts perceptual information from these neuronal responses is a pivotal challenge in both computational neuroscience and machine learning. Here we introduce a statistical mechanical theory, where perceptual information is first encoded in the correlated variability of sensory neurons and then reformatted into the firing rates of downstream neurons. Applying this theory, we illustrate the encoding of motion direction using neural covariance and demonstrate high-fidelity direction recovery by spiking neural networks. Networks trained under this theory also show enhanced performance in classifying natural images, achieving higher accuracy and faster inference speed. Our results challenge the traditional view of neural covariance as a secondary factor in neural coding, highlighting its potential influence on brain function., Comment: 18 pages, 5 figures

Strange correlators are useful tools for diagnosing symmetry-protected topological states from their bulk wave functions. We study strange correlators for one-dimensional fermionic symmetry-protected topological states using fixed-point wave functions, and show that a combination of strange correlators constructed using fermion annihilation operators and bosonic order parameters can fully diagnose the classification from such wave functions. By converting the strange correlator to a correlation in a one-dimensional statistical problem described by transfer matrices, we show that when the wave function is topologically nontrivial, the corresponding strange correlator exhibits a long-range-order behavior, which can be analyzed from the symmetry properties of the transfer matrices constructed from the fixed-point wave function. Our general argument is demonstrated by several concrete examples., Comment: 12 pages

In the past decade, there has been a systematic investigation of symmetry-protected topological (SPT) phases in interacting fermion systems. Specifically, by utilizing the concept of equivalence classes of finite-depth fermionic symmetric local unitary (FSLU) transformations and the fluctuating decorated symmetry domain wall picture, a large class of fixed-point wave functions have been constructed for fermionic SPT (FSPT) phases. Remarkably, this construction coincides with the Atiyah-Hirzebruch spectral sequence, enabling a complete classification of FSPT phases. However, unlike bosonic SPT phases, the stacking group structure in fermion systems proves to be much more intricate. The construction of fixed-point wave functions does not explicitly provide this information. In this paper, we employ FSLU transformations to investigate the stacking group structure of FSPT phases. Specifically, we demonstrate how to compute stacking FSPT data from the input FSPT data in each layer, considering both unitary and anti-unitary symmetry, up to 2+1 dimensions. As concrete examples, we explicitly compute the stacking group structure for crystalline FSPT phases in all 17 wallpaper groups and the mixture of wallpaper groups with onsite time-reversal symmetry using the fermionic crystalline equivalence principle. Importantly, our approach can be readily extended to higher dimensions, offering a versatile method for exploring the stacking group structure of FSPT phases., Comment: Correct some errors in 1+1D case. Add more examples for 2+1D crystalline FSPT protected by symmetries being the mixture of wallpaper groups with onsite time-reversal symmetry

We investigate the role of superconducting phase fluctuations in generic 2D superconductors. For nodal d-wave superconductors, it is found that the thermal (static) phase fluctuations in the normal state significantly broaden the d-wave nodes, and lead to the pseudogap and accompanying Fermi arcs in a quite general manner. The formation of Fermi arcs can be depicted by the intertwinement between d-wave superconductivity and the scattering of Cooper pairs. To support our theoretical findings, we numerically report the observation of Fermi arcs in a concrete lattice model, proposed originally by X. Y. Xu and T. Grover in Phys. Rev. Lett. 126, 217002 (2021), using the determinant quantum Monte Carlo (DQMC) method without the infamous sign problem. As far as we noticed, it is the first time in a highly-correlated model that the Fermi arcs are identified with unbiased DQMC simulations. Despite the conciseness of our theory, our simulation results confirm the theoretical predictions qualitatively., Comment: 6 pages, 5 figures for the main text; 8 pages, 5 figures for the supplementary material

We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products and conjecture that it achieves its minimal value at Frobenius inner product. We prove our conjecture in the case of matrices. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry., Comment: 31 pages, 2 tables

Quantum loop and dimer models are archetypal examples of correlated systems with local constraints. Obtaining generic solutions for these models is difficult due to the lack of controlled methods to solve them in the thermodynamic limit. Nevertheless, these solutions are of immediate relevance to both statistical and quantum field theories, as well as the rapidly growing experiments in Rydberg atom arrays and quantum moir\'e materials, where the interplay between correlation and local constraints gives rise to a plethora of novel phenomena. In a recent work [X. Ran, Z. Yan, Y.-C. Wang, et al, arXiv:2205.04472 (2022)], it was found through sweeping cluster quantum Monte Carlo (QMC) simulations and field theory analysis that the triangular lattice quantum loop model (QLM) hosts a rich ground state phase diagram with lattice nematic, vison plaquette (VP) crystals, and the $\mathbb{Z}_2$ quantum spin liquid (QSL) close to the Rokhsar-Kivelson point. Here, we focus on the continuous quantum critical point separating the VP and QSL phases and demonstrate via both static and dynamic probes in QMC simulations that this transition is of the (2+1)D cubic* universality. In this transition, the fractionalized visons in QSL condense to give rise to the crystalline VP phase, while leaving their trace in the anomalously large anomalous dimension exponent and pronounced continua in the dimer and vison spectra compared with those at the conventional cubic or O(3) quantum critical points., Comment: 7+4 pages, 4+3 figures

In this article we exhibit new explicit families of congruences for the overpartition function, making effective the existence results given previously by Treneer. We give infinite families of congruences modulo $m$ for $m = 5, 7, 11$, and finite families for $m = 13, 17, 19$., Comment: Added an author, improved results

The primary focus of this paper is overpartitions, a type of partition that plays a significant role in $q$-series theory. In 2006, Treneer discovered an explicit infinite family of congruences of overpartitions modulo $5$. In our research, we have identified explicit infinite families of congruences of overpartitions modulo $3,7,11$. This work reveals the connection between overpartitions and half-integral modular forms.

The main result of the paper is the Fibonacci-like property of the partition function. The partition function $p(n)$ has a property: $p(n) \leq p(n-1) + p(n-2)$. Our result shows that if we impose certain restrictions on the partition, then the inequality becomes an equality. Furthermore, we extend this result to cases with a greater number of summands., Comment: First draft

The main result of the paper is the existence of an infinitely many families of Ramanujan-type congruences for $b_4(n)$ and $b_6(n)$ modulo primes $m \geq 2$ and $m \geq 5$, respectively. We provide new examples of congruences for $b_4(n)$ and $b_6(n)$. Moreover, we find two infinite explicit infinite families of congruences for $b_4(n)$ modulo $3$., Comment: 13 pages. v2: Append results for b_4(n) modulo 2, b_6(n) modulo 5. v3: We find two explicit infinite families of b_4(n) modulo 3. See Theorem 1.2

The study of quantum criticality and entanglement in systems with long-range (LR) interactions is still in its early stages, with many open questions remaining. In this work, we investigate critical exponents and scaling of entanglement entropies (EE) in the LR bilayer Heisenberg model using large-scale quantum Monte Carlo (QMC) simulations and the recently developed nonequilibrium increment algorithm for measuring EE. By applying modified (standard) finite-size scaling (FSS) above (below) the upper critical dimension and field theory analysis, we obtain precise critical exponents in three regimes: the LR Gaussian regime with a Gaussian fixed point, the short-range (SR) regime with Wilson-Fisher (WF) exponents, and a LR non-Gaussian regime where the critical exponents vary continuously from LR Gaussian to SR values. We compute the R\'enyi EE both along the critical line and in the N\'eel phase and observe that as the LR interaction is enhanced, the area-law contribution in EE gradually vanishes both at quantum critical points (QCPs) and in the N\'eel phase. The log-correction in EE arising from sharp corners at the QCPs also decays to zero as LR interaction grows, whereas the log-correction for N\'eel states, caused by the interplay of Goldstone modes and restoration of the symmetry in a finite system, is enhanced as LR interaction becomes stronger. We also discuss relevant experimental settings to detect these nontrivial properties in critical behavior and entanglement information for quantum many-body systems with LR interactions., Comment: 5pages, 4 figures

The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension $D\le 2$. For long-range interactions with a power-law form ($1/r^{\alpha}$), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when $\alpha\ge 2D$. However, the situation for $\alpha \in (2,4)$ at $D=2$ is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of $SU(2)$ symmetry in the ferromagnetic Heisenberg model with $1/r^{\alpha}$-form long-range interactions at $D=2$. We determine critical exponents through finite-size analysis for $\alpha<3$ (above the upper critical dimension with Gaussian fixed point) and $3\le\alpha<4$ (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem's scope.

Building robust, interpretable, and secure AI system requires quantifying and representing uncertainty under a probabilistic perspective to mimic human cognitive abilities. However, probabilistic computation presents significant challenges for most conventional artificial neural network, as they are essentially implemented in a deterministic manner. In this paper, we develop an efficient probabilistic computation framework by truncating the probabilistic representation of neural activation up to its mean and covariance and construct a moment neural network that encapsulates the nonlinear coupling between the mean and covariance of the underlying stochastic network. We reveal that when only the mean but not the covariance is supervised during gradient-based learning, the unsupervised covariance spontaneously emerges from its nonlinear coupling with the mean and faithfully captures the uncertainty associated with model predictions. Our findings highlight the inherent simplicity of probabilistic computation by seamlessly incorporating uncertainty into model prediction, paving the way for integrating it into large-scale AI systems., Comment: Code is available in https://github.com/AwakerMhy/probabilistic-computing-mnn

The highly irregular spiking activity of cortical neurons and behavioral variability suggest that the brain could operate in a fundamentally probabilistic way. Mimicking how the brain implements and learns probabilistic computation could be a key to developing machine intelligence that can think more like humans. In this work, we propose a theory of stochastic neural computing (SNC) in which streams of noisy inputs are transformed and processed through populations of nonlinearly coupled spiking neurons. To account for the propagation of correlated neural variability, we derive from first principles a moment embedding for spiking neural network (SNN). This leads to a new class of deep learning model called the moment neural network (MNN) which naturally generalizes rate-based neural networks to second order. As the MNN faithfully captures the stationary statistics of spiking neural activity, it can serve as a powerful proxy for training SNN with zero free parameters. Through joint manipulation of mean firing rate and noise correlations in a task-driven way, the model is able to learn inference tasks while simultaneously minimizing prediction uncertainty, resulting in enhanced inference speed. We further demonstrate the application of our method to Intel's Loihi neuromorphic hardware. The proposed theory of SNC may open up new opportunities for developing machine intelligence capable of computing uncertainty and for designing unconventional computing architectures.

There is no doubt that the information hidden in entanglement entropy (EE), for example, the $n$-th order R\'enyi EE, i.e., $S^{A}_n=\frac{1}{1-n}\ln \Tr (\rho_A^n)$ where $\rho_A=\mathrm{Tr}_{\overline{A}}\rho$ is the reduced density matrix, can be used to infer the organizing principle of 2D interacting fermion systems, ranging from spontaneous symmetry breaking phases, quantum critical points to topologically ordered states. It is far from clear, however, whether the EE can actually be obtained with the precision required to observe these fundamental features -- usually in the form of universal finite size scaling behavior. Even for the prototypical 2D interacting fermion model -- the Hubbard model, to all existing numerical algorithms, the computation of the EE has not been succeeded with reliable data that the universal scaling regime can be accessed. Here we explain the reason for these unsuccessful attempts in EE computations in quantum Monte Carlo simulations in the past decades and more importantly, show how to overcome the conceptual and computational barrier with the incremental algorithm, such that the stable computation of the EE in 2D interacting fermion systems can be achieved and universal scaling information can be extracted. Relevance towards the experimental 2D interacting fermion systems is discussed., Comment: 6+4 pages, 4+4 figures

The deconfined quantum critical point (DQCP) -- the enigmatic incarnation of the quantum phase transition beyond the Landau-Ginzburg-Wilson paradigm of symmetries and their spontaneous breaking -- has been proposed and actively pursued for more than two decades. Various 2D quantum many-body lattice models, both in spin/boson and fermion representations have been tested with the state-of-the-art numerical techniques and field-theoretical analyses, and yet, the conclusion is still controversial. Experimental realizations of DQCP in the quantum magnet SrCu$_2$(BO$_3$)$_2$ and superconducting quantum criticality in 2D material have either shown first order transition or intermediate phase. The tension between the lattice scale details and the requirement from continuum limit, manifested in the form of the inconsistent critical scaling behavior and violations of generic conformal bootstrap bound, has not been resolved. Here we solve these decades-long controversies from the new and fundamental perspective of the quantum entanglement. We develop the incremental algorithm to compute the entanglement entropy at a fermionic DQCP with unprecedentedly accurate data and reveal the universal coefficient of the logarithmic correction therein is negative and at odds with positivity requirement of the conformal field theory. Together with results in other 2D DQCP lattice models (both in fermion and spin systems), our discoveries clearly demonstrate the 2D SU(2) antiferromagnet to valence bond solid DQCPs are not conformal fixed point and naturally explain the experimental difficulties in finding them. This marks the end of the beginning of unambiguous finding of the quantum phase transitions truely beyond the Landau-Ginzburg-Wilson paradigm, since its suggestion two decades ago., Comment: 9+5 pages, 3+1 figures