An anyon-chain-like lattice model with symmetry described by the Ising fusion category is studied. Combining numerical and analytical studies, we uncover a rich phase diagram that contains three phases: a symmetric critical phase and two categorical symmetry breaking phases. The symmetric phase lies in the same universality class as the usual critical Ising model. The first symmetry-breaking phase, dubbed the categorical ferromagnetic phase, has the Ising fusion category fully broken and exhibits a threefold ground-state degeneracy, as expected from the generalized Landau paradigm. The other symmetry-breaking phase is analogous to a conventional antiferromagnet: it breaks lattice translation and part of the Ising fusion category, and therefore is termed the categorical antiferromagnetic phase. Unlike ordinary antiferromagnetic states associated with finite invertible symmetry breaking, this phase itself is critical, being described by a fourfold degenerate Ising conformal field theory. We argue more generally that antiferromagnetic states associated with broken non-invertible symmetries have a large low-energy manifold that grows exponentially in system size, due to the greater-than-one quantum dimension of domain walls. We also numerically study the transitions between the three phases. The transition between the symmetric and categorical ferromagnetic phase is described by the c=7/10 tricritical Ising CFT, while the transition between the symmetric and categorical antiferromagnetic phases is less understood. Our numerical data suggest that the latter transition is continuous and described by a conformal field theory with central charge c=3/2.

Odd-parity magnetoresistance (OMR), characterized by its asymmetric response to magnetic field reversal, has remained limited in practical use due to its typically small magnitude and limited control in conventional magnetic metals. Here, we report a gate-tunable OMR effect in high-mobility AlGaN/GaN two-dimensional electron gas (2DEG) heterostructures, exhibiting a giant OMR coefficient exceeding 60%. Dual-gate control enables independent tuning of carrier density, achieving a highly linear and reversible OMR response, which is further enhanced by device geometry optimization. The effect arises from spatially engineered gradients of the Hall coefficient and magnetic-field-induced symmetry breaking. These results establish AlGaN/GaN 2DEGs as a robust and tunable platform for high-performance OMR devices, with potential applications in nonreciprocal electronics and magnetic sensing.

Symmetry can be intrinsically broken in topological phases due to inherent incompatibilities, a phenomenon known as enforced symmetry breaking (ESB) in the framework of topological order. In our previous work, we developed a systematic framework to understand ESB within 2D invertible topological order. Meanwhile, the origin of parity violation in the Standard Model remains one of the most profound mysteries in physics, with no clear explanation to date. In this study, we explore the ESB of parity symmetry by three-dimensional fermionic topological order (fTO), offering potential insights into the origins of parity violation. As the simplest example, here we consider an fTO related to the intrinsic interacting fermionic SPT phase protected by Z_2^f× Z_2× Z_8 symmetry in three dimensions. We show that time-reversal symmetry (TRS) with T^2=1 on physical fermions is incompatible with such fTO; then, through the so-called crystalline equivalence principle, we show that the parity symmetry is also incompatible with it. In comparison, conventional TRS with T^2=P_f remains compatible to this fTO. We also discuss a general framework to study the ESB phenomenon for 3D fTO.

Recently, many studies are focused on generalized global symmetry, a mixture of both invertible and non-invertible symmetries in various space-time dimensions. The complete structure of generalized global symmetry is described by higher fusion category theory. In this paper, We first review the construction of fusion 2-category symmetry $\Sigma \cal B$ where $\cal B$ is a a braided fusion category. In particular, we elaborate on the monoidal structure of $\Sigma \cal B$ which determines fusion rules and controls the dynamics of topological operators/defects. We then take $\Sigma \mathrm{sVec}$ as an example to demonstrate how we calculate fusion rule, quantum dimension and 10j-symbol of the fusion 2-category. With our algorithm, all these data can be efficiently encoded and computed in computer program. The complete program will be uploaded to github soon. Our work can be thought as explicitly computing the representation theory of $\cal B$, in analogy to, for example the representation theory of $SU(2)$. The choice of basis bimodule maps are in analogy to the Clebsch-Gordon coefficients and the 10j-symbol are in analogy to the 6j-symbol.

We construct a family of one-dimensional (1D) quantum lattice models based on $G$-graded unitary fusion category $\mathcal{C}_G$. This family realize an interpolation between the anyon-chain models and edge models of 2D symmetry-protected topological states, and can be thought of as edge models of 2D symmetry-enriched topological states. The models display a set of unconventional global symmetries that are characterized by the input category $\mathcal{C}_G$. While spontaneous symmetry breaking is also possible, our numerical evidence shows that the category symmetry constrains the the models to the extent that the low-energy physics has a large likelihood to be gapless.

Owing to the chirality of Weyl nodes characterized by the first Chern number, a Weyl system supports one-way chiral zero modes under a magnetic field, which underlies the celebrated chiral anomaly. As a generalization of Weyl nodes from three-dimensional to five-dimensional physical systems, Yang monopoles are topological singularities carrying nonzero second-order Chern numbers c_{2}=±1. Here, we couple a Yang monopole with an external gauge field using an inhomogeneous Yang monopole metamaterial and experimentally demonstrate the existence of a gapless chiral zero mode, where the judiciously designed metallic helical structures and the corresponding effective antisymmetric bianisotropic terms provide the means for controlling gauge fields in a synthetic five-dimensional space. This zeroth mode is found to originate from the coupling between the second Chern singularity and a generalized 4-form gauge field-the wedge product of the magnetic field with itself. This generalization reveals intrinsic connections between physical systems of different dimensions, while a higher-dimensional system exhibits much richer supersymmetric structures in Landau level degeneracy due to the internal degrees of freedom. Our study offers the possibility of controlling electromagnetic waves by leveraging the concept of higher-order and higher-dimensional topological phenomena.

It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking by topological orders. In this work, we systematically study enforced breaking of a general finite group $G_f$ by a class of topological orders, namely 0D, 1D and 2D fermionic invertible topological orders. Mathematically, the symmetry group $G_f$ is a central extension of a bosonic group $G$ by the fermion parity group $Z_2^f$, characterized by a 2-cocycle $\lambda$ $\in H^2(G,Z_2)$. With some minor assumptions and for given $G$ and $\lambda$, we are able to obtain a series of criteria on the existence or non-existence of enforced symmetry breaking by the fermionic invertible topological orders. Using these criteria, we discover many examples that are not known previously. For 2D systems, we define the physical quantities to describe symmetry-enriched invertible topological orders and derive some obstruction functions using both fermionic and bosonic languages. In the latter case which is done via gauging the fermion parity, we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetry-enriched topological states, with the conditions inherited from the original fermionic system. We also study enforced breaking of the continuous group $SU_f(N)$ by 2D invertible topological orders through a different argument.

We discover three-dimensional intertwined Weyl phases, by developing a theory to create topological phases. The theory is based on intertwining existing topological gapped and gapless phases protected by the same crystalline symmetry. The intertwined Weyl phases feature both unconventional Weyl semimetallic (monopole charge>1) and higher-order topological phases, and more importantly, their exotic intertwining. While the two phases are independently stabilized by the same symmetry, their intertwining results in the specific distribution of them in the bulk. The construction mechanism allows us to combine different kinds of unconventional Weyl semimetallic and higher-order topological phases to generate distinct phases. Remarkably, on 2D surfaces, the intertwining causes the Fermi-arc topology to change in a periodic pattern against surface orientation. This feature provides a characteristic and feasible signature to probe the intertwining Weyl phases. Moreover, we provide guidelines for searching candidate materials, and elaborate on emulating the intertwined double-Weyl phase in cold-atom experiments.

Studying strong electron correlations has been an essential driving force for pushing the frontiers of condensed matter physics. In particular, in the vicinity of correlation-driven quantum phase transitions (QPTs), quantum critical fluctuations of multiple degrees of freedom facilitate exotic many-body states and quantum critical behaviours beyond Landau’s framework 1 . Recently, moiré heterostructures of van der Waals materials have been demonstrated as highly tunable quantum platforms for exploring fascinating, strongly correlated quantum physics 2 – 22 . Here we report the observation of tunable quantum criticalities in an experimental simulator of the extended Hubbard model with spin–valley isospins arising in chiral-stacked twisted double bilayer graphene (cTDBG). Scaling analysis shows a quantum two-stage criticality manifesting two distinct quantum critical points as the generalized Wigner crystal transits to a Fermi liquid by varying the displacement field, suggesting the emergence of a critical intermediate phase. The quantum two-stage criticality evolves into a quantum pseudo criticality as a high parallel magnetic field is applied. In such a pseudo criticality, we find that the quantum critical scaling is only valid above a critical temperature, indicating a weak first-order QPT therein. Our results demonstrate a highly tunable solid-state simulator with intricate interplay of multiple degrees of freedom for exploring exotic quantum critical states and behaviours.

Domain walls between different topological phases are one of the most interesting phenomena that reveal the non-trivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$. As simple examples, we numerically study several finite groups(including both Abelian and non-Abelian finite group such as $S_3$) in $2$D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.

We study the classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the Z(2) fermion parity. On the other hand, we uncover new classes of intrinsically fermionic SPT phases that are only enabled by interactions, both in 2D and 3D. Several 3D Majorana-type fermionic SPT phases are identified. Moreover, several new instances of Lieb-Schultz-Mattis-type theorems for Majorana-type fermionic SPT phases are obtained and we discuss their interpretations from the perspective of bulk-boundary correspondence.

We study non-Hermitian spatial symmetries-a class of symmetries that have no counterparts in Hermitian systems-and study how normal and exceptional semimetals can be stabilized by these symmetries. Different from internal ones, spatial symmetries act nonlocally in momentum space and enforce global constraints on both band degeneracies and topological quantities at different locations. In deriving general constraints on band degeneracies and topological invariants, we demonstrate that non-Hermitian spatial symmetries are on an equal footing with, but are essentially different from Hermitian ones. First, we discover the nonlocal Hermitian conjugate pair of exceptional or normal band degeneracies that are enforced by non-Hermitian spatial symmetries. Remarkably, we find that these pairs lead to the symmetry-enforced violation of the Fermion doubling theorem in the long-time limit. Second, with the topological constraints, we unravel that a certain exceptional manifold is only compatible with and stabilized by non-Hermitian spatial symmetries but is intrinsically incompatible with Hermitian spatial symmetries. We illustrate these findings using two three-dimensional models of a non-Hermitian Weyl semimetal and an exceptional unconventional Weyl semimetal. Experimental cold-atom realizations of both models are also proposed.

Abelian Chern-Simons theory, characterized by the so-called K matrix, has been quite successful in characterizing and classifying Abelian fractional quantum Hall effect as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT (fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry G(f). Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry G(f) have three new features: (1) they may support gapless Majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if G(f) is a nontrivial extension of bosonic symmetry Gb by Z(2)(f), and (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge n - 1 for type-III bosonic SPT phases protected by (Z(n))(3) symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used in these examples could be generalized to derive the edge theories of fSPT phases with arbitrary unitary Abelian total symmetry G(f).

A 40-year-old puzzle in transition metal pentatellurides ZrTe_{5} and HfTe_{5} is the anomalous peak in the temperature dependence of the longitudinal resistivity, which is accompanied by sign reverses of the Hall and Seebeck coefficients. We give a plausible explanation for these phenomena without assuming any phase transition or strong interaction effect. We show that, due to intrinsic thermodynamics and diluteness of the conducting electrons in these materials, the chemical potential displays a strong dependence on the temperature and magnetic field. With that, we compute resistivity, Hall and Seebeck coefficients in zero field, and magnetoresistivity and Hall resistivity in finite magnetic fields, in all of which we reproduce the main features that are observed in experiments.

W. B. Rui,1, ∗ Zhen Zheng,1 Moritz M. Hirschmann,2 Song-Bo Zhang,3 Chenjie Wang,1, † and Z. D. Wang1, ‡ Department of Physics and HKU-UCAS Joint Institute for Theoretical and Computational Physics at Hong Kong, The University of Hong Kong, Pokfulam Road, Hong Kong, China Max-Planck-Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany (Dated: May 19, 2021)

We derive a series of quantitative bulk-boundary correspondences for 3D bosonic and fermionic symmetry-protected topological (SPT) phases under the assumption that the surface is gapped, symmetric and topologically ordered, i.e., a symmetry-enriched topological (SET) state. We consider those SPT phases that are protected by the mirror symmetry and continuous symmetries that form a group of U(1), SU(2) or SO(3). In particular, the fermionic cases correspond to a crystalline version of 3D topological insulators and topological superconductors in the famous ten-fold-way classification, with the time-reversal symmetry replaced by the mirror symmetry and with strong interaction taken into account. For surface SETs, the most general interplay between symmetries and anyon excitations is considered. Based on the previously proposed dimension reduction and folding approaches, we re-derive the classification of bulk SPT phases and define a complete set of bulk topological invariants for every symmetry group under consideration, and then derive explicit expressions of the bulk invariants in terms of surface topological properties (such as topological spin, quantum dimension) and symmetry properties (such as mirror fractionalization, fractional charge or spin). These expressions are our quantitative bulk-boundary correspondences. Meanwhile, the bulk topological invariants can be interpreted as anomaly indicators for the surface SETs which carry ’t Hooft anomalies of the associated symmetries whenever the bulk is topologically non-trivial. Hence, the quantitative bulk-boundary correspondences provide an easy way to compute the ’t Hooft anomalies of the surface SETs. Moreover, our anomaly indicators are complete. Our derivations of the bulk-boundary correspondences and anomaly indicators are explicit and physically transparent. The anomaly indicators obtained in this work can be straightforwardly translated to their timereversal counterparts that apply to the usual topological insulators and topological superconductors, due to a known correspondence between mirror and time-reversal topological phases.

We construct a lattice model of topological order (kagome quantum spin liquids) and solve it with unbiased quantum Monte Carlo simulations. A three-stage anyon condensation with two transitions from a Z(2) boxed times Z(2) topological order to a Z(2) topological order and eventually to a trivial symmetric phase is revealed. These results provide concrete examples of phase transitions between topological orders in quantum magnets. The designed quantum spin liquid model and its numerical solution offer a playground for further investigations on vestigial anyon condensation.

Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for 3D interacting fermion systems. Most surprisingly, we discover new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with emergent fermionic particles). On the other hand, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also gives rise to an alternative way to classify FSPT phases. We further compare the classification results for FSPT phases with arbitrary Abelian unitary total symmetry Gf and find systematical agreement with previous studies.

Without the constraint imposed by Hermiticity, non-Hermitian systems enjoy greater freedom than Hermitian ones. While the non-Hermitian ramification of non-spatial (internal) symmetries has been revealed, spatial symmetries remain to be explored. Here, we identify intrinsically non-Hermitian spatial symmetries using the same reasoning as non-spatial symmetry ramification. The symmetry endows exceptional topological semimetals with global topological structures, by preserving exceptional points but altering their topological invariants nonlocally. Furthermore, the global band configuration in the bulk is strongly constrained by non-Hermitian spatial symmetry, due to the intertwining of left and right eigen-systems at symmetry-related locations in momentum space. We illustrate our theory using two novel topological phases: exceptional unconventional Weyl semimetals and exceptional triple-point semimetals, in which the global structures of exceptional points, exceptional lines, and higher-order exceptional points are stabilized by one or more non-Hermitian spatial symmetries. We propose a cold-atom experiment to realize the exceptional unconventional Weyl semimetals.

Studies in recent years show that the delivery process, the choice of delivery mode, and the delivery outcome of primiparas are affected by their mental state. With the transformation of the medical model from the single biomedical model to the biopsychosocial medical model, the influence of social psychological factors on maternal psychological state has aroused heated discussion among clinical scholars. In this study, 480 cases of normal primiparas who had regular prenatal examination and delivered in hospital were selected as the research object. The Hospital Anxiety and Depression Scale (HAD) was used to record the depressive anxiety state of all study subjects in the third trimester, and we analyzed the effects of age, education background of pregnant women and their spouses, the nature of jobs, family income, prenatal and pregnancy health status, medical insurance status, attend pregnant women’s school, and accompanying status of family member on their depressive anxiety state. The influence of depression and anxiety on delivery mode was analyzed. The result showed that the age, health status during prenatal and pregnancy, accompanying status of family members, and attend pregnant women’s school are the independent risk factors affecting depression and anxiety status of primiparas during late pregnancy. Antenatal depression and anxiety have certain influence on the choice of the delivery mode of pregnant women. Targeted psychological intervention for primiparas with high-risk factors is helpful to improve their psychological state and reduce the rate of cesarean section.