arXiv subjects
Publications and source records attributed to Xiao-Gang Wen.
Given a premodular category $\mathcal{C}$, we show that its $R$-symbol can be recovered from its $T$-matrice, fusion coefficients and some 2nd generalized Frobenius-Schur indicators. In particular, if $\mathcal{C}$ is modular, its $R$-symbols for a certain gauge choice are completely determined by its modular data.
We study Wigner crystals (WCs) induced by a strong Coulomb interaction from the ring-like Fermi surface of a Mexican-hat dispersion $ε_k= c_2k^2+c_4k^4$. We design orbital shape in order to minimize the energy of the WC, and find that a low ground-state energy requires an orbital shape with a depletion of electrons near $k=0$. To capture the Coulomb-induced correlations, we include a Jastrow factor as well as a factor describing the correlation between electrons and doped vacancies. Using variational Monte Carlo calculations, we calibrate the effective band parameters $c_2$ and $c_4$ to reproduce the two transitions observed experimentally as the electron density is lowered: from a spin-valley-polarized Fermi liquid with a disk-like Fermi surface, to one with a ring-like Fermi surface, and finally to a WC. We find that, even with an optimized orbital shape that depletes electrons near $k=0$, a WC with hole self-doping near $k=0$ can still be energetically favorable near the WC transition, provided that the electron-vacancy correlation is included. We also estimate the dispersion of the doped hole.
We perform the variational Monte Carlo calculation for recently proposed chiral superconducting states driven by strong Coulomb interactions. We compare the resulting energetics of these electronic phases for the electron dispersion relation $E_k = c_2 k^2+c_4 k^4$. Motivated by the recent discovery of chiral superconductivity in rhombohedral graphene systems, we apply our analysis to relevant parameter regimes. We demonstrate that topological chiral superconducting phases (including a spin-unpolarized state) can be energetically favored over the spin-valley polarized Fermi liquid above the density of Wigner crystal phase. Our results show that the preference for chiral superconductivity is strongest when $c_2$ lies between zero and a negative value corresponding to a Fermi sea on the verge of forming a hole pocket around $k=0$. This finding suggests that superconductivity can arise from pure repulsive Coulomb interactions in systems with an almost flat band bottom, without relying on the pairing instability of a Fermi surface. This mechanism opens a new pathway to superconductivity beyond the conventional BCS mechanism.
We develop a general framework for the statistics of mixed-dimensional excitations subject to intertwined conservation laws, extending the familiar Fermi statistics with conserved particle number. We define statistics microscopically through a \emph{hopping-operator algebra}: a local operator subalgebra (LOsA) generated by operators that locally move or deform excitations while preserving the conservation law. Nontrivial statistics arise when this subalgebra is nontrivial. We first focus on LOsAs that encode \emph{pointed} conservation laws. These give rise to invertible excitations, whose fusion rules are exactly those of the symmetry defects of a higher group $\cG$. For such $\cG$-conserved excitations in $d$-dimensional space, we show that the corresponding LOsA -- and hence the statistics it defines -- is classified by a cohomology class $[ω] \in H^{d+2}(B\cG;\R/\Z)$, where changing $[ω]$ by a coboundary corresponds merely to a rephasing of the local operators. We further provide a holographic realization: excitations with this prescribed conservation law and statistics live on the boundary of a $\cG$ higher-group gauge theory in $(d+1)$-dimensional space, twisted by $[ω]$. More generally, non-pointed conservation laws and the associated statistics of non-invertible excitations are defined by a pair: a LOsA together with its excitation-complex representation. This is equivalent to the pair consisting of a LOsA and its Hilbert-space representation, which is the data defining a generalized symmetry. Consequently, non-pointed conservation laws and their statistics in $d$-dimensional space are classified by fusion $d$-categories, just as generalized symmetries are. The higher-group results above are the fully-pointed special cases of this more general classification.
In 1991, it was proposed that fourfold-degenerate Landau levels formed by a single species of electrons could host a non-Abelian fractional quantum Hall (FQH) state with Fibonacci anyons at filling fraction $ν= \frac{2}{3}$. In this work, we investigate how such degenerate Landau levels can be realized in rhombohedral-stacked tetralayer graphene. We identify the following key conditions which may stabilize the Fibonacci state: (1) A magnetic field of around 20 Tesla is required if surface and interior carbons have the same energy level. If substrate hybridization raises the surface carbon energy level by $Δ_2 = 30$\,meV relative to interior carbon, the required field will have a larger range: 15 -- 20 Tesla. For $Δ_2 = 45$\,meV, the range reaches a maximum: 7 -- 20 Tesla. (2) The displacement field must be tuned to achieve Landau level degeneracy. $ν= \frac{3}{5}$ Fibonacci FQH states may also be realized in pentalayer rhombohedral graphene with a magnetic field of 12 Tesla, and $ν= \frac12$ states with Ising anyons may occur in trilayer graphene for magnetic fields of 12 -- 20 Tesla at $Δ_2 = 0$ or 5 -- 20 Tesla at $Δ_2 = 45$\,meV. We also study a simple interaction model to explore spin/valley polarization effects, and we see that the Fibonacci statemay occur at $ν= 2/3 + $ integer filling fractions, where the integer is 0 and 4 for sufficiently weak interaction, or can shift to 2 and 5 under a stronger interaction. The case $Δ_2 = 45$\,meV also produces states at negative filling fraction, e.g. $-\frac23$, $-4\frac23$. Here $ν$ is defined with respect to the Hall conductance, $σ_{xy} = ν\frac{e^2}{h}$.
Scientific knowledge is increasingly dispersed across vast and heterogeneous scientific literature, where important claims are often implicit, evolving, and internally debated. While large language models (LLMs) have shown impressive performance in information extraction and summarization, their ability to recover latent scientific consensus remains unclear. Here, we investigate this problem in the context of high-temperature superconductivity (HTS), a long-standing and highly debated topic in condensed matter physics, as a challenging testbed. Using near 18,000 highly-cited publications over the past seven decades, we construct a structured knowledge graph linking competing superconducting mechanisms, material families, evidential modalities, and citation relations. We find that LLM-extracted representations recover coherent and physically interpretable structures, including family-dependent mechanism profiles, evidence-specific correlations, and citation-mediated temporal evolution of scientific beliefs. Ablation studies on LLM further show that the global structure remains robust across prompting, decoding, and model variations. Our results suggest that LLMs can indeed serve as scalable tools for deciphering scientific knowledge in domains characterized by competing interpretations and evolving knowledge.
Low-energy emanant and emergent symmetries can be anomalous, higher-group, or non-invertible. A way to systematically capture the properties of such symmetries is through the topological orders in one-higher dimension, known as symmetry topological orders (symTOs). Consequently, identifying the emergent or emanant symmetry of a system is not simply a matter of determining its group structure, but rather of computing the corresponding symTO. In this work, we develop a method to compute the symTO of 1+1D systems by analyzing their low-energy spectra under closed boundary conditions with all possible symmetry twists. Following this approach, we show that the gapless antiferromagnetic (AF) spin-$1/2$ Heisenberg model possesses an exact emanant symTO corresponding to the $D_8$ quantum double, when the global symmetry is restricted to the $\mathbb{Z}_2^x \times \mathbb{Z}_2^z$ subgroup of the $SO(3)$ spin-rotation symmetry and lattice translations. Moreover, this model exhibits an emergent $SO(4)$ symmetry, whose exact components are described jointly by automorphisms of the $D_8$ quantum double and the $SO(3)$ spin-rotations. Using the condensable algebras of the emanant symTO, we further identify several other phases that may be accessible by modifying interactions among low-energy excitations: (1) a gapped dimer phase, connected to the AF phase via an $SO(4)$ rotation, (2) a commensurate collinear ferromagnetic phase that breaks translation by one site with a $ω\sim k^2$ mode, (3) an incommensurate, translation-symmetric ferromagnetic phase featuring both $ω\sim k^2$ and $ω\sim k$ modes, (4) and an incommensurate ferromagnetic phase that breaks translation by one site with both $ω\sim k^2$ and $ω\sim k$ modes.
We use the computer algebra system GAP to classify modular data up to rank 12. This extends the previously obtained classification of modular data up to rank 6. Our classification includes all the modular data from modular tensor categories up to rank 12, with a few possible exceptions at rank 12 and levels $5,7$ and $14$. Those exceptions are eliminated up to a certain bound by an extensive finite search in place of required infinite search. Our list contains a few potential unitary modular data which are not known to correspond to any unitary modular tensor categories (such as those from Kac-Moody algebra, twisted quantum doubles of finite group, as well as their Abelian anyon condensations). It remains to be shown if those potential modular data can be realized by modular tensor categories or not. We provide some evidence that all may be constructed from centers of near-group categories or gauging group symmetries of known modular tensor categories, with the exception of a total of five cases at rank 11 (with $D^2 =1964.590$) and 12 (with $D^2 =3926.660$). The classification of modular data corresponds to a classification of modular tensor categories (up to modular isotopes which are not expected to be present at low ranks). The classification of modular tensor categories leads to a classification of gapped quantum phases of matter in 2-dimensional space for bosonic lattice systems with no symmetry, as well as a classification of generalized symmetries in 1-dimensional space.
We explore 1+1 dimensional (1+1D) gapped phases in systems with non-invertible symmetries, focusing on symmetry-protected topological (SPT) phases (defined as gapped phases with non-degenerate ground states), as well as SPT orders (defined as the differences between gapped/gapless phases with identical bulk excitations spectrum). For group-like symmetries, distinct SPT phases share identical bulk excitations and always differ by SPT orders. However, for certain non-invertible symmetries, we discover novel SPT phases that have different bulk excitations and thus do not differ by SPT orders. Additionally, we also study the spontaneous symmetry-breaking (SSB) phases of non-invertible symmetries. Unlike group-like symmetries, non-invertible symmetries lack the concept of subgroups, which complicates the definition of SSB phases as well as their identification. This challenge can be addressed by employing the symmetry-topological-order (symTO) framework for the symmetry. The Lagrangian condensable algebras and automorphisms of the symTO facilitate the classification of gapped phases in systems with such symmetries, enabling the analysis of both SPT and SSB phases (including those that differ by SPT orders). Finally, we apply this methodology to investigate gapless phases in symmetric systems and to study gapless phases differing by SPT orders.
For a given parent fractional quantum Hall (FQH) state at filling fraction $ν$, the hierarchy construction produces FQH states at nearby filling fractions $\{ν_n\}$ by condensing minimally charged quasiholes or quasiparticles of the parent state into their own FQH states. The hierarchy construction has been useful for relating families of FQH states and for the experimental identification of the topological order of parent states via the presence of daughter states. We reinterpret the hierarchy construction as a two-step procedure: stacking with a second FQH state and condensing a condensable algebra of bosons. This two-step procedure can be applied to both abelian and non-abelian FQH states, and it does not require calculations with a wavefunction. We show this construction reproduces the hierarchies for the Laughlin and Pfaffian states, and can be applied further to propose hierarchies for various non-abelian FQH states.
Two quantum theories which look different but are secretly describing the same low-energy physics are said to be dual to each other. When realized in the Topological Holography formalism, duality corresponds to changing the gapped boundary condition on the top boundary of a topological field theory, which determines the symmetry of the system, while not affecting the bottom boundary where all the dynamics take place. In this paper, we show that duality in the Topological Holography formalism can be realized with a Sequential Quantum Circuit applied to the top boundary. As a consequence, the Hamiltonians before and after the duality mapping have exactly the same spectrum in the corresponding symmetry sectors, and the entanglement in the corresponding low-energy eigenstates differs by at most an area law term.
We investigate a mechanism to produce superconductivity by strong purely repulsive interactions for flat dispersion $\varepsilon \sim k^4$, without using pairing instability in Fermi-liquid. The resulting superconductors break both time-reversal and reflection symmetries in the orbital motion of electrons, and exhibit non-trivial topological order. Our findings suggest that this topological chiral superconductivity is more likely to emerge near or between fully spin-valley polarized metallic phase and Wigner crystal phase. These topological chiral superconductors can be fully or partially spin-valley polarized. For partial spin-valley polarization, the ratios of electron densities associated with different spin-valley quantum numbers are quantized as simple rational numbers. Furthermore, many of these topological chiral superconductors exhibit charge-4 or higher condensation, neutral quasiparticles with fractional statistics, and/or gapless chiral edge states. Two of the topological chiral superconductors are in the same phases as the ``spin''-triplet or spinless $p+ \textrm{i} p$ BCS superconductor, while others are in different phases than any BCS superconductors. The same mechanism is also used to produce anyon superconductivity between fractional anomalous quantum Hall states in the presence of a periodic potential.
We study the nonlinear $σ$-model in ${(d+1)}$-dimensional spacetime with connected target space $K$ and show that, at energy scales below singular field configurations (such as vortices), it has an emergent non-invertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $d$-representations of a discrete $d$-group $\mathbb{G}^{(d)}$ (i.e. the emergent symmetry is the dual of the invertible $d$-group $\mathbb{G}^{(d)}$ symmetry). The $d$-group $\mathbb{G}^{(d)}$ is determined such that its classifying space $B\mathbb{G}^{(d)}$ is given by the $d$-th Postnikov stage of $K$. In $(2+1)$D and for finite $\mathbb{G}^{(2)}$, this symmetry is always holo-equivalent to an invertible ${0}$-form (ordinary) symmetry with potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $σ$-model spontaneously breaks this symmetry, and when $\mathbb{G}^{(d)}$ is finite, it is described by the deconfined phase of $\mathbb{G}^{(d)}$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free $S^2$ nonlinear $σ$-model in ${(3+1)}$D and show that it has an emergent non-invertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this non-perturbative result is different from the results obtained using the $S^N$ and $\mathbb{C}P^{N-1}$ nonlinear $σ$-models in the large-$N$ limit.
A characteristic property of a gapless liquid state is its emergent symmetry and dual symmetry, associated with the conservation laws of symmetry charges and symmetry defects respectively. These conservation laws, considered on an equal footing, can't be described simply by the representation theory of a group (or a higher group). They are best described in terms of a topological order (TO) with gappable boundary in one higher dimension; we call this the symTO of the gapless state. The symTO can thus be considered a fingerprint of the gapless state. We propose that a largely complete characterization of a gapless state, up to local-low-energy equivalence, can be obtained in terms of its maximal emergent symTO. In this paper, we review the symmetry/topological-order (Symm/TO) correspondence and propose a precise definition of maximal symTO. We discuss various examples to illustrate these ideas. We find that the 1+1D Ising critical point has a maximal symTO described by the 2+1D double-Ising topological order. We provide a derivation of this result using symmetry twists in an exactly solvable model of the Ising critical point. The critical point in the 3-state Potts model has a maximal symTO of double (6,5)-minimal-model topological order. As an example of a noninvertible symmetry in 1+1D, we study the possible gapless states of a Fibonacci anyon chain with emergent double-Fibonacci symTO. We find the Fibonacci-anyon chain without translation symmetry has a critical point with unbroken double-Fibonacci symTO. In fact, such a critical theory has a maximal symTO of double (5,4)-minimal-model topological order. We argue that, in the presence of translation symmetry, the above critical point becomes a stable gapless phase with no symmetric relevant operator.
Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) $S^{\,}_3$ symmetry and the other with non-invertible $\mathsf{Rep}(S^{\,}_3)$ symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1D symmetry-topological-order (SymTO) of type $\mathrm{JK}^{\,}_4\boxtimes \overline{\mathrm{JK}}^{\,}_4$. The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual $S^{\,}_3$-symmetric and $\mathsf{Rep}(S^{\,}_3)$-symmetric models.
Although condensed matter systems usually do not have higher-form symmetries, we show that, unlike 0-form symmetry, higher-form symmetries can emerge as exact symmetries at low energies and long distances. In particular, emergent higher-form symmetries at zero temperature are robust to arbitrary local UV perturbations in the thermodynamic limit. This result is true for both invertible and non-invertible higher-form symmetries. Therefore, emergent higher-form symmetries are $\textit{exact emergent symmetries}$: they are not UV symmetries but constrain low-energy dynamics as if they were. Since phases of matter are defined in the thermodynamic limit, this implies that a UV theory without higher-form symmetries can have phases characterized by exact emergent higher-form symmetries. We demonstrate this in three lattice models, the quantum clock model and emergent ${\mathbb{Z}_N}$ and ${U(1)}$ ${p}$-gauge theory, finding regions of parameter space with exact emergent (anomalous) higher-form symmetries. Furthermore, we perform a generalized Landau analysis of a 2+1D lattice model that gives rise to $\mathbb{Z}_2$ gauge theory. Using exact emergent 1-form symmetries accompanied by their own energy/length scales, we show that the transition between the deconfined and Higgs/confined phases is continuous and equivalent to the spontaneous symmetry-breaking transition of a $\mathbb{Z}_2$ symmetry, even though the lattice model has no symmetry. Also, we show that this transition line must $\textit{always}$ contain two parts separated by multi-critical points or other phase transitions. We discuss the physical consequences of exact emergent higher-form symmetries and contrast them to emergent ${0}$-form symmetries. Lastly, we show that emergent 1-form symmetries are no longer exact at finite temperatures, but emergent $p$-form symmetries with ${p\geq 2}$ are.
A generalized symmetry (defined by the algebra of local symmetric operators) can go beyond group or higher group description. A theory of generalized symmetry (up to holo-equivalence) was developed in terms of symmetry-TO -- a bosonic topological order (TO) with gappable boundary in one higher dimension. We propose a general method to compute the 2+1D symmetry-TO from the local symmetric operators in 1+1D systems. Our theory is based on the commutant patch operators, which are extended operators constructed as products and sums of local symmetric operators. A commutant patch operator commutes with all local symmetric operators away from its boundary. We argue that topological invariants associated with anyon diagrams in 2+1D can be computed as contracted products of commutant patch operators in 1+1D. In particular, we give concrete formulae for several topological invariants in terms of commutant patch operators. Topological invariants computed from patch operators include those beyond modular data, such as the link invariants associated with the Borromean rings and the Whitehead link. These results suggest that the algebra of commutant patch operators is described by 2+1D symmetry-TO. Based on our analysis, we also argue briefly that the commutant patch operators would serve as order parameters for gapped phases with finite symmetries.
Two global symmetries are holo-equivalent if their algebras of local symmetric operators are isomorphic. Holo-equivalent classes of global symmetries are classified by gappable-boundary topological orders (TO) in one higher dimension (called symmetry TO), which leads to a symmetry/topological-order (Symm/TO) correspondence. We establish that: (1) For systems with a symmetry described by symmetry TO $M$, their gapped and gapless states are classified by condensable algebras $A$, formed by elementary excitations in $M$ with trivial self/mutual statistics. Such classified states (called $A$-states) can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc., in a unified way. (2) The local low-energy properties of an $A$-state can be calculated from its reduced symmetry TO $M_{/A}$, using holographic modular bootstrap (holoMB) which takes $M_{/A}$ as an input. Here $M_{/A}$ is obtained from $M$ by condensing excitations in $A$. Notably, an $A$-state must be gapless if $M_{/A}$ is nontrivial. This provides a unified understanding of the emergence and symmetry protection of gaplessness that applies to symmetries that are anomalous, higher-form, and/or non-invertible. (3) The relations between condensable algebras constrain the structure of the global phase diagram. (4) 1+1D bosonic systems with $S_3$ symmetry have four gapped phases with unbroken symmetries $S_3$, $\mathbb{Z}_3$, $\mathbb{Z}_2$, and $\mathbb{Z}_1$. We find a duality between two transitions $S_3 \leftrightarrow \mathbb{Z}_1$ and $\mathbb{Z}_3 \leftrightarrow \mathbb{Z}_2$: they are either both first order or both (stably) continuous, and in the latter case, they are described by the same conformal field theory (CFT).