arXiv subjects

Da-Chuan Lu

Publications and source records attributed to Da-Chuan Lu.

At least 19 recordsLinked to original sources

arXiv2026

Self-dual $S_3$ gauge theory in 2+1d: lattice model and topological phase transitions

Electric-magnetic self-duality of the $\mathbb{Z}_2$ gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the $S_3$ quantum double $\mathcal{D}(S_3)$ on a tensor product Hilbert space in which the $\mathbb{Z}^{\mathrm{em}}_2$ anyon-permutation symmetry, exchanging the non-Abelian chargeon $C$ and fluxon $F$, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent $\mathbb{Z}_2^{\mathrm{em}}$-preserving bosonic perturbations, driving $\mathcal{D}(S_3)$ into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles $\mathcal{D}(D_{2n})$. Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.

cond-mat.str-el↗

arXiv2026

Sp(4,Z) actions on 3d U(1)^2 symmetric theories: Order-five duality and bilayer quantum Hall hierarchies

The $\mathrm{SL}(2,\mathbb{Z})$ electromagnetic duality of 4d Maxwell theory induces theory-generating operations on 3d theories with $U(1)$ global symmetry. For theories with $U(1)^n$ symmetry, this structure generalizes to $\mathrm{Sp}(2n,\mathbb{Z})$. Focusing on the $U(1)^2$ case, we formulate the bulk $\mathrm{Sp}(4,\mathbb{Z})$ action and derive the corresponding boundary operations. We identify an intrinsically two-component element of order five, which generalizes the order-three $ST$ element of the single-$U(1)$ theory. Although its fifth power acts trivially in the bulk, the corresponding boundary operation closes only up to a decoupled $U(1)_1$ invertible phase, suggesting a mixed duality-gravitational anomaly. We realize the resulting theory-generating web for Abelian Chern-Simons theories within the $K$-matrix formalism and apply it to bilayer fractional quantum Hall systems. Recasting the Haldane--Halperin hierarchy construction as a sequence of $\mathrm{SL}(2,\mathbb{Z})$ operations, we generalize it to systems with charge $U(1)_c$ and pseudospin $U(1)_s$ symmetries. The resulting bilayer hierarchies contain branches terminating in an interlayer-correlated bosonic $(221)$ daughter sector, yielding candidate Abelian states at the even-denominator equal-layer fillings $3/8+3/8$ and $5/12+5/12$. Integral changes of anyon basis establish the equivalence of these states to their corresponding Abelian composite-fermion descriptions. We further discuss the spin-charge constraints that arise when the electromagnetic background field is treated as a spin-$c$ connection.

hep-th↗

arXiv2026

Generalized Kramers-Wannier Self-Duality in Hopf-Ising Models

The Kramers-Wannier transformation of the 1+1d transverse-field Ising model exchanges the paramagnetic and ferromagnetic phases and, at criticality, manifests as a non-invertible symmetry. Extending such self-duality symmetries beyond gauging of abelian groups in tensor-product Hilbert spaces has, however, remained challenging. In this work, we construct a generalized 1+1d Ising model based on a finite-dimensional semisimple Hopf algebra $H$ that enjoys an anomaly-free non-invertible symmetry $\mathrm{Rep}(H)$. We provide an intuitive diagrammatic formulation of both the Hamiltonian and the symmetry operators using a non-(co)commutative generalization of ZX-calculus built from Hopf-algebraic data. When $H$ is self-dual, we further construct a generalized Kramers-Wannier duality operator that exchanges the paramagnetic and ferromagnetic phases and becomes a non-invertible symmetry at the self-dual point. This enlarged symmetry mixes with lattice translation and, in the infrared, flows to a weakly integral fusion category given by a $\mathbb{Z}_2$ extension of $\mathrm{Rep}(H)$. Specializing to the Kac-Paljutkin algebra $H_8$, the smallest self-dual Hopf algebra beyond abelian group algebras, we numerically study the phase diagram and identify four of the six $\mathrm{Rep}(H_8)$-symmetric gapped phases, separated by Ising critical lines and meeting at a multicritical point. We also realize all six $\mathrm{Rep}(H_8)$-symmetric gapped phases on the lattice via the $H$-comodule algebra formalism, in agreement with the module-category classification of $\mathrm{Rep}(H_8)$. Our results provide a unified Hopf-algebraic framework for non-invertible symmetries, dualities, and the tensor product lattice models that realize them.

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arXiv2025

Defect Bootstrap: Tight Ground State Bounds in Spontaneous Symmetry Breaking Phases

The recent development of bootstrap methods based on semidefinite relaxations of positivity constraints has enabled rigorous two-sided bounds on local observables directly in the thermodynamic limit. However, these bounds inevitably become loose in symmetry broken phases, where local constraints are insufficient to capture long-range order. In this work, we identify the origin of this looseness as order parameter defects which are difficult to remove using local operators. We introduce a $\textit{defect bootstrap}$ framework that resolves this limitation by embedding the system into an auxiliary $\textit{defect model}$ equipped with ancilla degrees of freedom. This construction effectively enables local operators to remove order parameter defects, yielding tighter bounds in phases with spontaneous symmetry breaking. This approach can be applied broadly to pairwise-interacting local lattice models with discrete or continuous internal symmetries that satisfy a property we call $\textit{defect diamagnetism}$, which requires that the ground state energy does not decrease upon adding any finite number of symmetry defects. Applying the method to the transverse field Ising models in 1D and 2D, we obtain significantly improved bounds on energy densities and spin correlation functions throughout the symmetry broken phase in 1D and deep within the phase in 2D. Our results demonstrate that physically motivated constraint sets can dramatically enhance the power of bootstrap methods for quantum many-body systems.

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arXiv2025

Intrinsic NISPT Phases, igNISPT Phases, and Mixed Anomalies of Non-Invertible Symmetries

A bosonic non-invertible Symmetry Protected Topological (NISPT) phase in (1+1)-dim is referred to as $\textit{intrinsic}$ if it cannot be mapped, under discrete gauging, to a gapped phase with any invertible symmetry, that is, if it is protected by a non-group-theoretical fusion category symmetry. We construct the intrinsic NISPT phases by performing discrete gauging in a partial SSB phase with a fusion category symmetry that has a certain mixed anomaly. Sometimes, the anomaly of that symmetry category can be alternatively understood as a self-anomaly of a proper categorical sub-symmetry; when this is the case, the same gauging provides an anomaly resolution of this anomalous categorical sub-symmetry. This allows us to construct intrinsic gapless SPT (igSPT) phases, where the anomalous faithfully acting symmetry is non-invertible; and we refer to such igSPT phases as igNISPT phases. We provide two concrete lattice models realizing an intrinsic NISPT phase and an igNISPT phase, respectively. We also generalize the construction of intrinsic NISPT phases to (3+1)-dim.

hep-th↗

arXiv2025

Exploring $G$-ality defects in 2-dim QFTs

The Tambara-Yamagami (TY) fusion category symmetry $\text{TY}(\mathbb{A},χ,ε)$ describes the enhanced non-invertible self-duality symmetry of a $2$-dim QFT under gauging a finite Abelian group $\mathbb{A}$. We generalize the enhanced non-invertible symmetries by considering twisted gauging which allows stacking $\mathbb{A}$-SPTs before and after the gauging. Such non-invertible symmetries can be obtained from invertible anyon permutation symmetries of the $3$-dim SymTFT. Consider a finite group $G$ formed by (un)twisted gaugings of $\mathbb{A}$, a $2$-dim QFT invariant under topological manipulations in $G$ admits non-invertible \textit{$G$-ality defects}. We study the classification and the physical implication of the $G$-ality defects using the SymTFT and the group-theoretical fusion categories, with three concrete examples. 1) Triality with $\mathbb{A} = \mathbb{Z}_N \times \mathbb{Z}_N$ where $N$ is coprime with $3$. The classification was previously determined by Jordan and Larson where the data is similar to the $\text{TY}$ fusion categories, and we determine the anomaly of these fusion categories. 2) $p$-ality with $\mathbb{A} = \mathbb{Z}_p \times \mathbb{Z}_p$ where $p$ is an odd prime. We consider two such categories $\mathcal{P}_{\pm,m}$ which are distinguished by different choices of the symmetry fractionalization, a new data that does not appear in the TY classification, and show that they have distinct anomaly structures and spin selection rules. 3) $S_3$-ality with $\mathbb{A} = \mathbb{Z}_N \times \mathbb{Z}_N$. We study their classification explicitly for $N < 20$ via SymTFT, and provide a group-theoretical construction for certain $N$. We find $N=5$ is the minimal $N$ to admit an $S_3$-ality and $N=11$ is the minimal $N$ to admit a group-theoretical $S_3$-ality.

hep-th↗

arXiv2025

Strange correlator and string order parameter for non-invertible symmetry protected topological phases in 1+1d

In this paper, we construct strange correlators and string order parameters for non-invertible symmetry protected topological phases (NISPTs) in 1+1d quantum lattice spin models. The strange correlator exhibits long-range order when evaluated between two distinct NISPTs and decays exponentially otherwise. We show that strange charged operators inserted into the strange correlator are linked to the interface algebra (boundary tube algebra) and are non-trivial when all its irreducible representations have dimensions greater than one. We discuss the generalization to higher dimensions. The string order parameter is obtained by contracting the truncated symmetry operator with charge decoration operators, which are determined by the NISPT action tensors. We illustrate the above construction using the three NISPTs of $\text{Rep}(D_8)$ and demonstrate the extraction of categorical data via tensor networks, particularly through the ZX calculus. Finally, we show that the entanglement spectrum degeneracy is determined by the irreducible representations of the interface algebra when assuming non-invertible symmetry on-site condition.

cond-mat.str-el↗

arXiv2025

Deconfined criticality as intrinsically gapless topological state in one dimension

Deconfined criticality and gapless topological states have recently attracted growing attention, as both phenomena go beyond the traditional Landau paradigm. However, the deep connection between these two critical states, particularly in lattice realization, remains insufficiently explored. In this Letter, we reveal that certain deconfined criticality can be regarded as an intrinsically gapless topological state without gapped counterparts in a one dimensional lattice model. Using a combination of field-theoretic arguments and large-scale numerical simulations, we establish the global phase diagram of the model, which features deconfined critical lines separating two distinct spontaneous symmetry breaking ordered phases. More importantly, we unambiguously demonstrate that the mixed anomaly inherent to deconfined criticality enforces topologically robust edge modes near the boundary, providing a general mechanism by which deconfined criticality manifests as a gapless topological state. Our findings not only offer a new perspective on deconfined criticality but also deepen our understanding of gapless topological phases of matter.

cond-mat.str-el↗

arXiv2025

SymSETs and self-dualities under gauging non-invertible symmetries

The self-duality defects under discrete gauging in a categorical symmetry $\mathcal{C}$ can be classified by inequivalent ways of enriching the bulk SymTFT of $\mathcal{C}$ with $\mathbb{Z}_2$ 0-form symmetry. The resulting Symmetry Enriched Topological (SET) orders will be referred to as $\textit{SymSETs}$ and are parameterized by choices of $\mathbb{Z}_2$ symmetries, as well as symmetry fractionalization classes and discrete torsions. In this work, we consider self-dualities under gauging $\textit{non-invertible}$ $0$-form symmetries in $2$-dim QFTs and explore their SymSETs. Unlike the simpler case of self-dualities under gauging finite Abelian groups, the SymSETs here generally admit multiple choices of fractionalization classes. We provide a direct construction of the SymSET from a given duality defect using its $\textit{relative center}$. Using the SymSET, we show explicitly that changing fractionalization classes can change fusion rules of the duality defect besides its $F$-symbols. We consider three concrete examples: the maximal gauging of $\operatorname{Rep} H_8$, the non-maximal gauging of the duality defect $\mathcal{N}$ in $\operatorname{Rep} H_8$ and $\operatorname{Rep} D_8$ respectively. The latter two cases each result in 6 fusion categories with two types of fusion rules related by changing fractionalization class. In particular, two self-dualities of $\operatorname{Rep} D_8$ related by changing the fractionalization class lead to $\operatorname{Rep} D_{16}$ and $\operatorname{Rep} SD_{16}$ respectively. Finally, we study the physical implications such as the spin selection rules and the SPT phases for the aforementioned categories.

hep-th↗

arXiv2024

Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems

In this paper, we study the twisted gauging on the (1+1)d lattice and construct various non-local mappings on the lattice operators. To be specific, we define the twisted Gauss law operator and implement the twisted gauging of the finite group on the lattice motivated by the orbifolding procedure in the conformal field theory, which involves the data of non-trivial element in the second cohomology group of the gauge group. We show the twisted gauging is equivalent to the two-step procedure of first applying the SPT entangler and then untwisted gauging. We use the twisted gauging to construct the triality (order 3) and $p$-ality (order $p$) mapping on the $\mathbb{Z}_p\times \mathbb{Z}_p$ symmetric Hamiltonians, where $p$ is a prime. Such novel non-local mappings generalize Kramers-Wannier duality and they preserve the locality of symmetric operators but map charged operators to non-local ones. We further construct quantum process to realize these non-local mappings and analyze the induced mappings on the phase diagrams. For theories that are invariant under these non-local mappings, they admit the corresponding non-invertible symmetries. The non-invertible symmetry will constrain the theory at the multicritical point between the gapped phases. We further give the condition when the non-invertible symmetry can have symmetric gapped phase with a unique ground state.

cond-mat.str-el↗

arXiv2024

Optical Conductivity in Symmetric Mass Generation Insulators

Symmetric mass generation (SMG) insulators are interaction-driven, featureless Mott insulating states in quantum many-body fermionic systems. Recent advancements suggest that zeros in the fermion Green's function could lead to non-vanishing negative optical conductivity in SMG insulators, even below the charge excitation gap. This study explores the origin of this unusual behavior through the lens of pole-zero duality, highlighting a critical issue where the current operator becomes unbounded, rendering the response function unphysical. By employing a lattice model, we derive a well-behaved lattice regularization of the current operator, enabling a detailed study of optical conductivity in SMG insulators. Utilizing both analytical and numerical methods, including strong-coupling expansions, we confirm that SMG insulators exhibit no optical conductivity at low energies below the charge gap, effectively resolving the paradox. This work not only deepens our understanding of quantum many-body phenomena but also lays a robust theoretical groundwork for future experimental explorations of SMG materials.

cond-mat.str-el↗

arXiv2023

Emergent self-duality in long range critical spin chain: from deconfined criticality to first order transition

Over the past few decades, tremendous efforts have been devoted to understanding self-duality at the quantum critical point, which enlarges the global symmetry and constrains the dynamics. In this letter, we employ large-scale density matrix renormalization group simulations to investigate the critical spin chain with long-range interaction $V(r) \sim 1/r^α$. Remarkably, we reveal that the long-range interaction drives the deconfined criticality towards a first-order phase transition as $α$ decreases. More strikingly, the emergent self-duality leads to an emergent symmetry and manifests at these first-order critical points. This discovery is reminiscent of self-duality protected multicritical points and provides the example of the critical line with generalized symmetry. Our work has far-reaching implications for ongoing experimental efforts in Rydberg atom quantum simulators.

cond-mat.str-el↗

arXiv2023

Self-duality under gauging a non-invertible symmetry

We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of $c=1$ CFTs, it is known that the theory is self-dual under gauging a $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry, and has $\mathsf{Rep}(H_8)$ and $\mathsf{Rep}(D_8)$ fusion category symmetries as a result. We find that gauging the entire $\mathsf{Rep}(H_8)$ fusion category symmetry maps the orbifold theory at radius $R$ to that at radius $2/R$. At $R=\sqrt{2}$, which corresponds to two decoupled Ising CFTs (Ising$^2$ in short), the theory is self-dual under gauging the $\mathsf{Rep}(H_8)$ symmetry. This implies the existence of a topological defect line in the Ising$^2$ CFT obtained from half-space gauging of the $\mathsf{Rep}(H_8)$ symmetry, which commutes with the $c=1$ Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the $c=1$ Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the $\mathsf{Rep}(H_8)$ symmetry to form a bigger fusion category which is a $\mathbb{Z}_2$-extension of $\mathsf{Rep}(H_8)$. We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising$^2$ CFT. Finally, we show that the torus partition functions of the Monster$^2$ CFT and Ising$\times$Monster CFT are also invariant under gauging the $\mathsf{Rep}(H_8)$ symmetry.

hep-th↗

arXiv2023

Superconductivity from Doping Symmetric Mass Generation Insulators: Application to La$_3$Ni$_2$O$_7$ under Pressure

We investigate the bilayer nickelates as a platform to realize the symmetric mass generation (SMG) insulator, a featureless Mott insulator that arises due to the Lieb-Schultz-Mattis (LSM) anomaly cancellation in bilayer spin-1/2 lattice systems. Through a single-orbital bilayer square lattice model involving intralayer hopping $t$ and interlayer superexchange interaction $J$, we demonstrate the emergence of high-temperature superconductivity (SC) upon doping the SMG insulator. The SC phase features $s$-wave interlayer spin-singlet pairing and exhibits a crossover between the BCS and BEC limits by tuning the $J/t$ ratio. We estimate the SC transition temperature $T_c$ from both the weak and strong coupling limits at the mean-field level. Our findings offer insights into the experimentally observed decrease in $T_c$ with pressure and the strange metal behavior above $T_c$. Additionally, we propose that both Ni $3d_{z^2}$ and $3d_{x^2-y^2}$ orbitals can exhibit superconductivity in La$_3$Ni$_2$O$_7$ under pressure, but their $T_c$ should vary in opposite ways under doping. This characteristic difference suggests a potential experimental pathway to identify which electronic orbital plays the principal role in the formation of superconductivity in this system.

cond-mat.str-el↗

arXiv2023

Green's Function Zeros in Fermi Surface Symmetric Mass Generation

The Fermi surface symmetric mass generation (SMG) is an intrinsically interaction-driven mechanism that opens an excitation gap on the Fermi surface without invoking symmetry-breaking or topological order. We explore this phenomenon within a bilayer square lattice model of spin-1/2 fermions, where the system can be tuned from a metallic Fermi liquid phase to a strongly-interacting SMG insulator phase by an inter-layer spin-spin interaction. The SMG insulator preserves all symmetries and has no mean-field interpretation at the single-particle level. It is characterized by zeros in the fermion Green's function, which encapsulate the same Fermi volume in momentum space as the original Fermi surface, a feature mandated by the Luttinger theorem. Utilizing both numerical and field-theoretical methods, we provide compelling evidence for these Green's function zeros across both strong and weak coupling regimes of the SMG phase. Our findings highlight the robustness of the zero Fermi surface, which offers promising avenues for experimental identification of SMG insulators through spectroscopy experiments despite potential spectral broadening from noise or dissipation.

cond-mat.str-el↗

arXiv2023

Definition and Classification of Fermi Surface Anomalies

We propose that the Fermi surface anomaly of symmetry group $G$ in any dimension is universally classified by $G$-symmetric interacting fermionic symmetry-protected topological (SPT) phases in $(0+1)$-dimensional spacetime. The argument is based on the perspective that the gapless fermions on the Fermi surface can be viewed as the topological boundary modes of Chern insulators in the phase space (position-momentum space). Given the non-commutative nature of the phase space coordinates, we show that the momentum space dimensions should be counted as negative dimensions for SPT classification purposes. Therefore, the classification of phase-space Chern insulators (or, more generally fermionic SPT phases) always reduces to a $(0+1)$-dimensional problem, which can then be answered by the cobordism approach. In addition to the codimension-1 Fermi surface case, we also discuss the codimension-$p$ Fermi surface case briefly. We provide concrete examples to demonstrate the validity of our classification scheme, and make connections to the recent development of Fermi surface symmetric mass generation.

cond-mat.str-el↗

arXiv2023

On Triality Defects in 2d CFT

We consider the triality fusion category discovered in the $c = 1$ KT theory \cite{Thorngren:2021yso}. We analyze this fusion category using the tools from the group theoretical fusion category and describe how to compute the simple lines, fusion rules and $F$-symbols. We then study the physical implication of this fusion category including deriving the spin selection rule, computing the asymptotic density of states of irreps of the fusion category symmetries, and analyzing its anomaly and constraints on the renormalization group flow. There is another set of $F$-symbols for the fusion categories with the same fusion rule known in the literature \cite{teo2015theory} which we compare with, and find the two are different as they lead to different spin selection rules. This gives a complete list of the fusion categories with the same fusion rule by the classification result in \cite{jordan2009classification}.

hep-th↗

arXiv2022

Correlated metals and unconventional superconductivity in rhombohedral trilayer graphene: A renormalization group analysis

Motivated by recent experimental observations of correlated metallic phases and superconductivity in rhombohedral trilayer graphene (RTG), we perform an unbiased study of electronic ordering instabilities in hole-doped RTG. Specifically, we focus on electronic states energetically proximate to Van Hove singularities (VHSs), where a large density of states promotes different interaction-induced symmetry-breaking electronic orders. To resolve the Fermi surface near VHSs, we construct a fermionic hot-spot model and demonstrate that a perpendicular electric field can tune different nesting structures of the Fermi surface. Subsequently, we apply a renormalization group analysis to describe the low-energy phase diagrams of our model under both short-range repulsive interactions as well as realistic (long-range) Coulomb interactions. Our analysis shows instabilities towards either intervalley coherent metallic phases or superconducting phases. The dominant pairing channel depends crucially on the nature of Fermi surface nesting -- repulsive Coulomb interaction favors spin-singlet $d$-wave pairing for relatively small displacement field and spin-singlet $i$-wave pairing for larger displacement field. We argue that the phase diagram of RTG can be well-understood by modeling the realistic Coulomb interaction as the sum of repulsive density-density interaction and ferromagnetic spin-triplet intervalley coherence (IVC) Hund's coupling, while phonon-mediated electronic interactions have a negligible effect on this system, in sharp contrast to twisted graphene multilayers.

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