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SciPost Physics Vol. 18 issue 6 (June 2025)

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Generalizations of Kitaev’s honeycomb model from braided fusion categories

Luisa Eck, Paul Fendley

SciPost Phys. 18, 170 (2025) · published 2 June 2025 | · pdf

Fusion surface models, as introduced by Inamura and Ohmori, extend the concept of anyon chains to 2+1 dimensions, taking fusion 2-categories as their input. In this work, we construct and analyze fusion surface models on the honeycomb lattice built from braided fusion 1-categories. These models preserve mutually commuting plaquette operators and anomalous 1-form symmetries. Their Hamiltonian is chosen to mimic the structure of Kitaev's honeycomb model, which is unitarily equivalent to the Ising fusion surface model. In the anisotropic limit, where one coupling constant is dominant, the fusion surface models reduce to Levin-Wen string-nets. In the isotropic limit, they are described by weakly coupled anyon chains and are likely to realize chiral topological order. We focus on three specific examples: (i) Kitaev's honeycomb model with a perturbation breaking time-reversal symmetry that realizes chiral Ising topological order, (ii) a $\mathbb{Z}_N$ generalization proposed by Barkeshli et al., which potentially realizes chiral parafermion topological order, and (iii) a novel Fibonacci honeycomb model featuring a non-invertible 1-form symmetry.
Minimal areas from entangled matrices

Jackson R. Fliss, Alexander Frenkel, Sean A. Hartnoll, Ronak M. Soni

SciPost Phys. 18, 171 (2025) · published 2 June 2025 | · pdf

We define a relational notion of a subsystem in theories of matrix quantum mechanics and show how the corresponding entanglement entropy can be given as a minimisation, exhibiting many similarities to the Ryu-Takayanagi formula. Our construction brings together the physics of entanglement edge modes, noncommutative geometry and quantum internal reference frames, to define a subsystem whose reduced state is (approximately) an incoherent sum of density matrices, corresponding to distinct spatial subregions. We show that in states where geometry emerges from semiclassical matrices, this sum is dominated by the subregion with minimal boundary area. As in the Ryu-Takayanagi formula, it is the computation of the entanglement that determines the subregion. We find that coarse-graining is essential in our microscopic derivation, in order to control the proliferation of highly curved and disconnected non-geometric subregions in the sum.
Electroweak double-box integrals for Møller scattering

Niklas Schwanemann, Stefan Weinzierl

SciPost Phys. 18, 172 (2025) · published 3 June 2025 | · pdf

We present for Møller scattering planar and non-planar two-loop double-box integrals where three electroweak gauge bosons are exchanged between the fermion lines, among which at least one is a photon. These integrals are relevant for the NNLO electroweak corrections to Møller scattering.
Higher spin swampland conjecture for massive AdS$_{3}$ gravity

Rajae Sammani, El Hassan Saidi

SciPost Phys. 18, 173 (2025) · published 3 June 2025 | · pdf

In this paper, we show that a possible version of the swampland weak gravity conjecture for higher spin (HS) massive topological AdS$_{3}$ gravity can be expressed in terms of mass M$_{\mathrm{hs}}$, charge Q$_{\mathrm{hs}}$ and coupling constant ${\large g}_{\mathrm{hs}}$ of 3D gravity coupled to higher spin fields as M$_{\mathrm{hs}}≤ \sqrt{2}$Q$_{\mathrm{hs}}{\large g}_{\mathrm{hs}}M_{\mathrm{Pl}}$. The higher spin charge is given by the $SO(1,2)$ quadratic Casimir Q$_{\mathrm{hs}}^{2}=s\left( s-1\right) $ and the HS coupling constant by ${\large g}_{\mathrm{hs}}^{2}=2/\left( M_{\mathrm{Pl}}^{2}l_{\mathrm{AdS}_{3}}^{2}\right) $ while the mass expressed like $\left( l_{\mathrm{AdS}_{3}}\text{M}_{\mathrm{hs}}\right) ^{2}$ is defined as $\left( 1+\mathrm{\mu }l_{\mathrm{AdS}_{3}}\right) ^{2}s\left( s-1\right) +\left[1-\left( \mathrm{\mu }l_{\mathrm{AdS}_{3}}\right) ^{2}\left( s-1\right) \right]$.
Vacua, symmetries, and Higgsing of Chern-Simons matter theories

Fabio Marino, Marcus Sperling

SciPost Phys. 18, 174 (2025) · published 3 June 2025 | · pdf

Three-dimensional supersymmetric Chern–Simons matter (CSM) theories typically preserve $\mathcal{N}=3$ supersymmetry but can exhibit enhanced $\mathcal{N}=4$ supersymmetry under special conditions. A detailed understanding of the moduli space of CSM theories, however, has remained elusive. This paper addresses this gap by systematically analysing the maximal branches of the moduli space of $\mathcal{N}=3$ and $\mathcal{N}=4$ CSM realised via Type IIB brane constructions. Firstly, for $\mathcal{N}=4$ theories with Chern–Simons levels equal $1$, the $SL(2,\mathbb{Z})$ dualisation algorithm is employed to construct dual Lagrangian 3d $\mathcal{N}=4$ theories without CS terms. This allows the full moduli space to be determined using quiver algorithms that compute Higgs and Coulomb branch Hasse diagrams and associated RG flows. Secondly, for $\mathcal{N}=4$ theories with CS-levels greater $1$, where $SL(2,\mathbb{Z})$ dualisation does not yield CS-free Lagrangians, a new prescription is introduced to derive two magnetic quivers, $\mathsf{MQ}_A $ and $\mathsf{MQ}_B$, whose Coulomb branches capture the maximal A and B branches of the original $\mathcal{N}=4$ CSM theory. Applying the decay and fission algorithm to $ \mathsf{MQ}_{A/B}$ then enables the systematic analysis of A/B branch RG flows and their geometric structures. Thirdly, for $\mathcal{N}=3$ CSM theories, one magnetic quiver for each maximal (hyper-Kähler) branch is derived from the brane system. This provides an efficient and comprehensive characterisation of these previously scarcely studied features.
Cosmology of the companion-axion model: Dark matter, gravitational waves, and primordial black holes

Zhe Chen, Archil Kobakhidze, Ciaran O'Hare, Zachary S. C. Picker, Giovanni Pierobon

SciPost Phys. 18, 175 (2025) · published 4 June 2025 | · pdf

The companion-axion model introduces a second QCD axion to rescue the Peccei-Quinn solution to the strong-CP problem from the effects of colored gravitational instantons. As in single-axion models, the two axions predicted by the companion-axion model are attractive candidates for dark matter. The model is defined by two free parameters, the scales of the two axions' symmetry breaking, so we can classify production scenarios in terms of the relative sizes of these two scales with respect to the scale of inflation. We study the cosmological production of companion-axion dark matter in order to predict windows of preferred axion masses, and calculate the relative abundances of the two particles. Additionally, we show that the presence of a second axion solves the cosmological domain wall problem automatically in the scenarios in which one or both of the axions are post-inflationary. We also suggest unique cosmological signatures of the companion-axion model, such as the production of a $\sim$10 nHz gravitational wave background, and $\sim 100\,M_\odot$ primordial black holes.
Cosmological infrared subtractions & infrared-safe computables

Paolo Benincasa, Francisco Vazão

SciPost Phys. 18, 176 (2025) · published 4 June 2025 | · pdf

Cosmological observables in perturbation theory turn out to be plagued with infrared divergences, which represents both a conceptual and computational challenge. In this paper we present a proof of concept for a systematic procedure to remove these divergences in a large class of scalar cosmological integrals and consistently define an infrared safe computable in perturbation theory. We provide diagrammatic rules which are based on the nestohedra underlying the asymptotic structure of such integrals.
Universal thermalization dynamics in (1+1)d QFTs

Richard A. Davison, Luca V. Delacretaz

SciPost Phys. 18, 177 (2025) · published 4 June 2025 | · pdf

We identify the universal mechanism behind the thermalization of (1+1)d QFTs at high and low temperatures. Viewing these theories as CFTs perturbed by relevant or irrelevant deformations, we show that conformal perturbation theory in the thermal state breaks down at late times allowing for the emergence of hydrodynamics. This breakdown occurs universally due to the unsuppressed exchange of stress tensors near the lightcone. Furthermore, for theories with central charge $c→∞$ we solve for the emergent hydrodynamic theory to all orders in the gradient expansion by arguing that all transport parameters appearing in two-point functions have universal expressions in terms of the scaling dimension $\Delta$ of the perturbation. The radius of convergence of the hydrodynamic dispersion relations provides an early time cutoff for hydrodynamics, which agrees with the time scale at which conformal perturbation theory breaks down.
Quantum transport theory for unconventional magnets: Interplay of altermagnetism and p-wave magnetism with superconductivity

Tim Kokkeler, Ilya Tokatly, F. Sebastian Bergeret

SciPost Phys. 18, 178 (2025) · published 5 June 2025 | · pdf

We present a quantum transport theory for generic magnetic metals, in which magnetism occurs predominantly due to exchange interactions, such as ferromagnets, antiferromagnets, altermagnets and p-wave magnets. Our theory is valid both for the normal and the superconducting state and is based on the generalization of nonlinear sigma model to such systems. We derive the effective low-energy action for each of these materials, where the symmetries of the corresponding spin space group are used to determine the form of the tensor coefficients appearing in the action. The transport equations, which are obtained as the saddle point equations of this action, describe a wider range of phenomena than the usual quasiclassical equations. In ferromagnets, in addition to the usual exchange field and spin relaxation effects, we identify a spin-dependent renormalization of the diffusion coefficient, and a correction to the Hanle effect. The former provides a description of spin-polarized currents in both the normal and superconducting equal spin-triplet states. In the normal state, our equations provide a complete description of the spin-splitting effect in diffusive systems, recently predicted in ideal clean altermagnets. In the superconducting state, our equations predict a proximity induced magnetization, the appearance of a spontaneous magnetic moment in hybrid superconductor-altermagnet systems. The distribution and polarization direction of this magnetic moment depend on the symmetry of the structure, thus measurements of such polarization reveal the underlying microscopic symmetry of the altermagnet. Finally, for inversion-symmetry-broken antiferromagnets, such as the p-wave magnet, we show that spin-galvanic effects may emerge. This effect is indistinguishable from the spin-galvanic effect induced by spin-orbit coupling in the normal state, but they can be distinguished through the temperature dependence in the superconducting state. Besides these examples, our model applies to arbitrary magnetic systems, providing a complete theory for nonequilibrium transport in diffusive nonconventional magnets at arbitrary temperatures.
Lightcone Hamiltonian for Ising field theory I: $T < T_c$

Andrew Liam Fitzpatrick, Emanuel Katz, Yuan Xin

SciPost Phys. 18, 179 (2025) · published 5 June 2025 | · pdf

We study 2d Ising Field Theory (IFT) in the low-temperature phase in lightcone quantization, and show that integrating out zero modes generates a very compact form for the effective lightcone interaction that depends on the finite volume vacuum expectation value of the $\sigma$ operator. This form is most naturally understood in a conformal basis for the lightcone Hilbert space. We further verify that this simple form reproduces to high accuracy results for the spectra, the $c$-function, and the form-factors from integrability methods for the magnetic deformation of IFT. For generic non-integrable values of parameters we also compute the above observables and compare our numeric results to those of equal-time truncation. In particular, we report on new measurements of various bound-state form-factors as well as the stress-tensor spectral density. We find that the stress tensor spectral density provides additional evidence that certain resonances of IFT are surprisingly narrow, even at generic strong coupling. Explicit example code for constructing the effective Hamiltonian is included in an appendix.
2d dualities from 4d

Jiaqun Jiang, Satoshi Nawata, Jiahao Zheng

SciPost Phys. 18, 180 (2025) · published 6 June 2025 | · pdf

We find new $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,2)$ dualities through the twisted compactifications of 4d supersymmetric theories on $S^2$. Our findings include dualities for both $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,2)$ non-Abelian gauge theories, as well as $\mathcal{N}=(0,2)$ Gauge/Landau-Ginzburg duality.
Classical weight-four L-value ratios as sums of Calabi-Yau invariants

Philip Candelas, Xenia de la Ossa, Joseph McGovern

SciPost Phys. 18, 181 (2025) · published 6 June 2025 | · pdf

We revisit the series solutions of the attractor equations of 4d $\mathcal{N}=2$ supergravities obtained by Calabi–Yau compactifications previously studied in [P. Candelas et al., J. High Energy Phys. 11, 032 (2021)]. While only convergent for a restricted set of black hole charges, we find that they are summable with Padé resummation providing a suitable method. By specialising these solutions to rank-two attractors, we obtain many conjectural identities of the type discovered in [P. Candelas et al., J. High Energy Phys. 11, 032 (2021)]. These equate ratios of weight-four special L-values with an infinite series whose summands are formed out of genus-0 Gromov–Witten invariants. We also present two new rank-two attractors which belong to moduli spaces each interesting in their own right. Each of these moduli spaces possess two points of maximal unipotent monodromy. One has already been studied by Hosono and Takagi, and we discuss issues stemming from the associated L-function having nonzero rank.
Geometric constructions of generalized dual-unitary circuits from biunitarity

Michael A. Rampp, Suhail A. Rather, Pieter W. Claeys

SciPost Phys. 18, 182 (2025) · published 10 June 2025 | · pdf

We present a general framework for constructing solvable lattice models of chaotic many-body quantum dynamics with multiple unitary directions using biunitary connections. We show that a network of biunitary connections on the kagome lattice naturally defines a multi-unitary circuit, where three 'arrows of time' directly reflect the lattice symmetry. These models unify various constructions of hierarchical dual-unitary and triunitary gates and present new families of models with solvable correlations and entanglement dynamics. Using multilayer constructions of biunitary connections, we additionally introduce multilayer circuits with monoclinic symmetry and higher level hierarchical dual-unitary solvability and discuss their (non-)ergodicity. Our work demonstrates how different classes of solvable models can be understood as arising from different geometric structures in spacetime.
Infrared properties of two-dimensional SU(N)/H nonlinear $\sigma$ models at nonzero $\theta$ angles

Philippe Lecheminant, Yuya Tanizaki, Keisuke Totsuka

SciPost Phys. 18, 183 (2025) · published 10 June 2025 | · pdf

A general strategy is proposed to explore the low-energy properties of two-dimensional nonlinear $\sigma$ models with $\theta$ terms. We demonstrate its application to nonlinear $\sigma$ models with the target space $SU(N)/H$, which include $\mathbb{C}P^{N-1}$, complex Grassmannian manifolds as well as the flag $SU(N)/U(1)^{N-1}$ and $SU(N)/SO(N)$ manifolds. By analyzing the symmetry and its anomaly content, we realize these nonlinear $\sigma$ models by considering specific deformations of the $SU(N)_1$ conformal field theory. For the flag-manifold $SU(N)/U(1)^{N-1}$ and $SU(N)/SO(N)$ models, those deformations are shown to correspond to the marginal current-current operator with the specific sign which leads to a massless renormalization group flow to the $SU(N)_1$ fixed point. In contrast, a massive regime with a two-fold ground-state degeneracy is found for the $\mathbb{C}P^{N-1}$ ($N >2$) and the Grassmannian nonlinear $\sigma$ models at $\theta=\pi$.
Emergent dipole field theory in atomic ladders

Hernan B. Xavier, Poetri Sonya Tarabunga, Marcello Dalmonte, Rodrigo G. Pereira

SciPost Phys. 18, 184 (2025) · published 10 June 2025 | · pdf

We study the dynamics of hard-core bosons on ladders, in the presence of strong kinetic constrains akin to those of the Bariev model. We use a combination of analytical methods and numerical simulations to establish the phase diagram of the model. The model displays a paired Tomonaga-Luttinger liquid phase featuring an emergent dipole symmetry, which encodes the local pairing constraint into a global, nonlocal quantity. We scrutinize the effect of such emergent low-energy symmetry during quench dynamics including single-particle defects. We observe that, despite being approximate, the dipole symmetry still leads to very slow relaxation dynamics, which we model via an effective field theory. The model we discuss is amenable to realization in both cold atoms in optical lattices and Rydberg atom arrays with dynamics taking place solely in the Rydberg manifold. To observe the unusual dynamics of excitations in such experimental platforms, we propose a two-step protocol, which starts with the quasi-adiabatic preparation of low-energy states, followed by the local creation of defects and their study under quench dynamics.
Efficient on-shell matching

Mikael Chala, Javier L. Miras, José Santiago, Fuensanta Vilches

SciPost Phys. 18, 185 (2025) · published 11 June 2025 | · pdf

We propose an efficient method to perform on-shell matching calculations in effective field theories. The standard off-shell approach to matching requires the use of a Green's basis that includes redundant and evanescent operators. The reduction of such a basis to a physical one is often highly non-trivial, difficult to automate and error prone. Our proposal is based on a numerical solution of the corresponding on-shell matching equations, which automatically implements in a trivial way the delicate cancellation between the non-local terms in the full theory and those in the effective one. The use of rational on-shell kinematics ensures an exact analytic solution despite the numerical procedure. In this way we only need a physical basis to perform the matching. Our procedure can be used to reduce any Green's basis to an arbitrary physical one, or to translate between physical bases; to renormalize arbitrary effective Lagrangians, directly in terms of a physical basis; and to perform finite matching, including evanescent contributions, as we discuss with explicit examples.
Chapman-Enskog theory and crossover between diffusion and superdiffusion for nearly integrable quantum gases

Maciej Łebek, Miłosz Panfil

SciPost Phys. 18, 186 (2025) · published 12 June 2025 | · pdf

Integrable systems feature an infinite number of conserved charges and on hydrodynamic scales are described by generalised hydrodynamics (GHD). This description breaks down when the integrability is weakly broken and sufficiently large space-time-scales are probed. The emergent hydrodynamics depends then on the charges conserved by the perturbation. We focus on nearly-integrable Galilean-invariant systems with conserved particle number, momentum and energy. Basing on the Boltzmann approach to integrability-breaking we describe dynamics of the system with GHD equation supplemented with a collision term. The limit of large space-time-scales is addressed using Chapman-Enskog expansion adapted to the GHD equation. For length scales larger than $\sim \lambda^{-2}$, where $\lambda$ is integrability-breaking parameter, we recover Navier-Stokes equations and find transport coefficients: viscosity and thermal conductivity. At even larger length scales, this description crosses over to Kardar-Parisi-Zhang universality class, characteristic to generic non-integrable one-dimensional fluids. Employing nonlinear fluctuating hydrodynamics we estimate this crossover length scale as $ \sim \lambda^{-4}$.
Bethe Ansatz, quantum circuits, and the F-basis

Roberto Ruiz, Alejandro Sopena, Esperanza López, Germán Sierra, Balázs Pozsgay

SciPost Phys. 18, 187 (2025) · published 12 June 2025 | · pdf

The Bethe Ansatz is a method for constructing exact eigenstates of quantum-integrable spin chains. Recently, deterministic quantum algorithms, referred to as "algebraic Bethe circuits", have been developed to prepare Bethe states for the spin-$1/2$ XXZ model. These circuits represent a unitary formulation of the standard algebraic Bethe Ansatz, expressed using matrix-product states that act on both the spin chain and an auxiliary space. In this work, we systematize these previous results, and show that algebraic Bethe circuits can be derived by a change of basis in the auxiliary space. The new basis, identical to the "F-basis" known from the theory of quantum-integrable models, generates the linear superposition of plane waves that is characteristic of the coordinate Bethe Ansatz. We explain this connection, highlighting that certain properties of the F-basis (namely, the exchange symmetry of the spins) are crucial for the construction of algebraic Bethe circuits. We demonstrate our approach by presenting new quantum circuits for the inhomogeneous spin-$1/2$ XXZ model.
Four no-go theorems on the existence of spin and orbital angular momentum of massless bosons

Eric Palmerduca, Hong Qin

SciPost Phys. 18, 188 (2025) · published 16 June 2025 | · pdf

The past decades have seen substantial interest in the so-called orbital angular momentum (OAM) of light, driven largely by its diverse range of applications. However, there are fundamental theoretical issues with decomposing the angular momentum of massless particles, such as photons, into spin (SAM) and orbital angular momentum parts. While the angular momentum of massive particles has a natural splitting into the Wigner SAM and OAM, there are numerous proposed splittings for photons and no consensus about which is correct. Moreover, it has been shown that most of the proposed SAM and OAM operators do not satisfy the defining commutation relations of angular momentum operators and are thus not legitimate splittings. Here, we prove that it is generally impossible to split the total angular momentum operator of massless bosons, such as photons and gravitons, into spin and orbital parts. We prove two further generalizations of this result, showing that there are no SAM-OAM splittings even if (1) the SAM operator generates non-internal symmetries or (2) if one allows the SAM and OAM operators to generate non-SO(3) symmetries.
Characteristic features of an active polar filament pushing a load

Prabhakar Maurya, Shalabh Kumar Anand, Sunil Pratap Singh

SciPost Phys. 18, 189 (2025) · published 16 June 2025 | · pdf

We present the structural and dynamical behavior of an active polar filament that is pushing a load using overdamped Langevin dynamics simulations. By varying the bending rigidity and the connectivity between the filament and the load, we smoothly transition the boundary condition of the filament from pivoted to clamped. In the clamped state, the load remains strongly aligned with the filament, whereas in the pivoted state, the load is free to rotate at its attachment point. Under the pivoted boundary condition, the active polar filament buckles and exhibits various fascinating dynamical phases, including snake-like motion, rotational motion, and helical conformations. However, under the clamped boundary condition, the helical phase disappears, and the filament attains either an extended or a bent conformation. The transition from the extended state to the helical phase is characterized using a global helical order parameter in the parameter space of active force and a physical quantity associated with the boundary condition. We have obtained various power laws relating the curvature radius of the helical phase, effective diffusivity, and rotational motion of the monomers to the active force. Furthermore, we demonstrate that the filament’s effective diffusivity in the helical phase exhibits a non-monotonic dependence on the active force: it initially increases linearly but decreases sharply at high active force strengths.
Quantum many-body thermal machines enabled by atom-atom correlations

Raymon S. Watson, Karen V. Kheruntsyan

SciPost Phys. 18, 190 (2025) · published 16 June 2025 | · pdf

Particle-particle correlations, characterized by Glauber's second-order correlation function, play an important role in the understanding of various phenomena in radio and optical astronomy, quantum and atom optics, particle physics, condensed matter physics, and quantum many-body theory. However, the relevance of such correlations to quantum thermodynamics has so far remained illusive. Here, we propose and investigate a class of quantum many-body thermal machines whose operation is directly enabled by second-order atom-atom correlations in an ultracold atomic gas. More specifically, we study quantum thermal machines that operate in a sudden interaction-quench Otto cycle and utilize a one-dimensional Lieb-Liniger gas of repulsively interacting bosons as the working fluid. The atom-atom correlations in such a gas are different to those of a classical ideal gas, and are a result of the interplay between interparticle interactions, quantum statistics, and thermal fluctuations. We show that operating these thermal machines in the intended regimes, such as a heat engine, refrigerator, thermal accelerator, or heater, would be impossible without such atom-atom correlations. Our results constitute a step forward in the design of conceptually new quantum thermodynamic devices which take advantage of uniquely quantum resources such as quantum coherence, correlations, and entanglement.
The asymmetric Fermi surface of Bi2201

Steef Smit, Kourosh L. Shirkoohi, Saumya Mukherjee, Sergio Barquero Pierantoni, Lewis Bawden, Erik van Heumen, Arnaud P. N. Tchiomo, Jans Henke, Jasper van Wezel, Yingkai Huang, Takeshi Kondo, Tsunehiro Takeuchi, Timur K. Kim, Cephise Cacho, Marta Zonno, Sergey Gorovikov, Stephen B. Dugdale, Jorge I. Facio, Mariia Roslova, Laura Folkers, Anna Isaeva, Nigel E. Hussey, Mark S. Golden

SciPost Phys. 18, 191 (2025) · published 17 June 2025 | · pdf

High-resolution angle-resolved photoemission spectroscopy (ARPES) performed on the single-layered cuprate (Pb$_{1-y}$,Bi$_y$)$_2$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$ (Bi2201) reveals a 6-10% difference in the nodal $k_F$ vectors along the $\Gamma$Y and $\Gamma$X directions. This asymmetry is notably larger than the 2% orthorhombic distortion in the CuO$_2$ plane lattice constants determined using X-ray crystallography from the same samples. First principles calculations indicate that crystal-field splitting of the bands lies at the root of the $k_{F}$ asymmetry. Concomitantly, the nodal Fermi velocities for the $\Gamma$Y quadrant exceed those for $\Gamma$X by 4%. Momentum distribution curve widths for the two nodal dispersions are also anisotropic, showing identical energy dependencies, bar a scaling factor of $\sim$ 1.17$± 0.05$ between $\Gamma$Y and $\Gamma$X. Consequently, the imaginary part of the self-energy is found to be 10-20% greater along $\Gamma$Y than $\Gamma$X. These results emphasize the need to account for Fermi surface asymmetry in the analysis of ARPES data on Bi-based cuprate high temperature superconductors such as Bi2201. To illustrate this point, an orthorhombic tight-binding model (with twofold in-plane symmetry) was used to fit ARPES Fermi surface maps spanning all four quadrants of the Brillouin zone, and the ARPES-derived hole-doping (Luttinger count) was extracted. Comparison of the Luttinger count with one assuming four-fold in-plane symmetry strongly suggests the marked spread in previously-reported Fermi surface areas from ARPES on Bi2201 results from the differences in $k_F$ along $\Gamma$Y and $\Gamma$X. Using this analysis, a new, linear relationship emerges between the hole-doping derived from ARPES ($p_{\text{ARPES}}$) and that derived using the Presland ($p_{\text{Presland}}$) relation such that $p_{\text{ARPES}} = p_{\text{Presland}}+0.11$. The implications for this difference between the ARPES- and Presland-derived estimates for $p$ are discussed and possible future directions to elucidate the origin of this discrepancy are presented.
Cons-training tensor networks: Embedding and optimization over discrete linear constraints

Javier Lopez-Piqueres, Jing Chen

SciPost Phys. 18, 192 (2025) · published 17 June 2025 | · pdf

In this study, we introduce a novel family of tensor networks, termed constrained matrix product states (MPS), designed to incorporate exactly arbitrary discrete linear constraints, including inequalities, into sparse block structures. These tensor networks are particularly tailored for modeling distributions with support strictly over the feasible space, offering benefits such as reducing the search space in optimization problems, alleviating overfitting, improving training efficiency, and decreasing model size. Central to our approach is the concept of a quantum region, an extension of quantum numbers traditionally used in $U(1)$ symmetric tensor networks, adapted to capture any linear constraint, including the unconstrained scenario. We further develop a novel canonical form for these new MPS, which allow for the merging and factorization of tensor blocks according to quantum region fusion rules and permit optimal truncation schemes. Utilizing this canonical form, we apply an unsupervised training strategy to optimize arbitrary objective functions subject to discrete linear constraints. Our method's efficacy is demonstrated by solving the quadratic knapsack problem, achieving superior performance compared to a leading nonlinear integer programming solver. Additionally, we analyze the complexity and scalability of our approach, demonstrating its potential in addressing complex constrained combinatorial optimization problems.
Mode-Shell correspondence, a unifying phase space theory in topological physics - Part II: Higher-dimensional spectral invariants

Lucien Jezequel, Pierre Delplace

SciPost Phys. 18, 193 (2025) · published 18 June 2025 | · pdf

The mode-shell correspondence relates the number $\mathcal{I}_M$ of gapless modes in phase space to a topological \textit{shell invariant} $\mathcal{I}_S$ defined on a closed surface - the shell - surrounding those modes, namely $\mathcal{I}_M=\mathcal{I}_S$. In part I, we introduced the mode-shell correspondence for zero-modes of chiral symmetric Hamiltonians (class AIII). In this part II, we broaden the correspondence to arbitrary dimension and to both symmetry classes A and AIII. This allows us to include, in particular, $1D$-unidirectional edge modes of Chern insulators, $2D$ massless Dirac and $3D$-Weyl cones, within the same formalism. We provide an expression for $\mathcal{I}_M$ that only depends on the dimension of the dispersion relation of the gapless mode, and does not require a translation invariance. Then, we show that the topology of the shell (a circle, a sphere, a torus), that must account for the spreading of the gapless mode in phase space, yields specific expressions of the shell index. Semi-classical expressions of those shell indices are also derived and reduce to either Chern or winding numbers depending on the parity of the mode’s dimension. In that way, the mode-shell correspondence provides a unified and systematic topological description of both bulk and boundary gapless modes in any dimension, and in particular includes the bulk-boundary correspondence. We illustrate the generality of the theory by analyzing several models of semimetals and insulators, both on lattices and in the continuum, and also discuss weak and higher-order topological phases within this framework. Although this paper is a continuation of Part I, the content remains sufficiently independent to be mostly read separately.
Fermionic non-invertible symmetries in (1+1)d: Gapped and gapless phases, transitions, and symmetry TFTs

Lakshya Bhardwaj, Kansei Inamura, Apoorv Tiwari

SciPost Phys. 18, 194 (2025) · published 18 June 2025 | · pdf

We study fermionic non-invertible symmetries in (1+1)d, which are generalized global symmetries that mix fermion parity symmetry with other invertible and non-invertible internal symmetries. Such symmetries are described by fermionic fusion supercategories, which are fusion $\pi$-supercategories with a choice of fermion parity. The aim of this paper is to flesh out the categorical Landau paradigm for fermionic symmetries. We use the formalism of Symmetry Topological Field Theory (SymTFT) to study possible gapped and gapless phases for such symmetries, along with possible deformations between these phases, which are organized into a Hasse phase diagram. The phases can be characterized in terms of sets of condensed, confined and deconfined generalized symmetry charges, reminiscent of notions familiar from superconductivity. Many of the gapless phases also serve as phase transitions between gapped phases. The associated fermionic conformal field theories (CFTs) can be obtained by performing generalized fermionic Kennedy-Tasaki (KT) transformations on bosonic CFTs describing simpler transitions. The fermionic non-invertible symmetries along with their charges and phases discussed here can be obtained from those of bosonic non-invertible symmetries via fermionization or Jordan-Wigner transformation, which is discussed in detail.
Choose your diffusion: Efficient and flexible ways to accelerate the diffusion model in fast high energy physics simulation

Cheng Jiang, Sitian Qian, Huilin Qu

SciPost Phys. 18, 195 (2025) · published 18 June 2025 | · pdf

The diffusion model has demonstrated promising results in image generation, recently becoming mainstream and representing a notable advancement for many generative modeling tasks. Prior applications of the diffusion model for both fast event and detector simulation in high energy physics have shown exceptional performance, providing a viable solution to generate sufficient statistics within a constrained computational budget in preparation for the High Luminosity LHC. However, many of these applications suffer from slow generation with large sampling steps and face challenges in finding the optimal balance between sample quality and speed. The study focuses on the latest benchmark developments in efficient ODE/SDE-based samplers, schedulers, and fast convergence training techniques. We test on the public CaloChallenge and JetNet datasets with the designs implemented on the existing architecture, the performance of the generated classes surpass previous models, achieving significant speedup via various evaluation metrics.
A minimal tensor network beyond free fermions

Carolin Wille, Maksimilian Usoltcev, Jens Eisert, Alexander Altland

SciPost Phys. 18, 196 (2025) · published 19 June 2025 | · pdf

This work proposes a minimal model extending the duality between classical statistical spin systems and fermionic systems beyond the case of free fermions. A Jordan-Wigner transformation applied to a two-dimensional tensor network maps the partition sum of a classical statistical mechanics model to a Grassmann variable integral, structurally similar to the path integral for interacting fermions in two dimensions. The resulting model is simple, featuring only two parameters: one governing spin-spin interaction (dual to effective hopping strength in the fermionic picture), the other measuring the deviation from the free fermion limit. Nevertheless, it exhibits a rich phase diagram, partially stabilized by elements of topology, and featuring three phases meeting at a multicritical point. Besides the interpretation as a spin and fermionic system, the model is closely related to loop gas and vertex models and can be interpreted as a parity-preserving (non-unitary) circuit. Its minimal construction makes it an ideal reference system for studying non-linearities in tensor networks and deriving results by means of duality.
On the relation between fractional charge and statistics

Thors Hans Hansson, Rodrigo Arouca, Thomas Klein Kvorning

SciPost Phys. 18, 197 (2025) · published 19 June 2025 | · pdf

We revisit an argument, originally given by Kivelson and Roček, for why the existence of fractional charge necessarily implies fractional statistics. In doing so, we resolve a contradiction in the original argument, and in the case of a $\nu = 1/m$ Laughlin holes, we also show that the standard relation between fractional charge and statistics is necessary by an argument based on a t'Hooft anomaly in a one-form global $\mathcal{Z}_m$ symmetry.
Quantum reservoir probing: An inverse paradigm of quantum reservoir computing for exploring quantum many-body physics

Kaito Kobayashi, Yukitoshi Motome

SciPost Phys. 18, 198 (2025) · published 19 June 2025 | · pdf

Quantum reservoir computing (QRC) is a brain-inspired computational paradigm that exploits the natural dynamics of a quantum system for information processing. To date, a multitude of quantum systems have been utilized in QRC, with diverse computational capabilities demonstrated accordingly. This study proposes a reciprocal research direction: Probing quantum systems themselves through their information processing performance in the QRC framework. Building upon this concept, here we develop quantum reservoir probing (QRP), an inverse extension of the QRC. The QRP establishes an operator-level linkage between physical properties and computational performance. A systematic scan of this correspondence reveals the intrinsic quantum dynamics of the reservoir system from both computational and informational perspectives. Unifying quantum information and quantum matter, the QRP holds great promise as a potent tool for exploring various aspects of quantum many-body physics. In this study, we specifically apply it to analyze information propagation in a one-dimensional quantum Ising chain. We demonstrate that the QRP not only distinguishes between ballistic and diffusive information propagation, reflecting the system's dynamical characteristics, but also identifies system-specific information propagation channels, a distinct advantage over conventional methods.
Non-linear photoconductivity of strongly driven graphene

Lukas Broers, Ludwig Mathey

SciPost Phys. 18, 199 (2025) · published 20 June 2025 | · pdf

We present the non-linear DC photoconductivity of graphene under strong infra-red (IR) radiation. The photoconductivity is obtained as the response to a strong DC electric field, with field strengths outside of the linear-response regime, while the IR radiation is described by a strong AC electric field. The conductivity displays two distinct regimes in which either the DC or the AC field dominates. We explore these regimes and associate them with the dynamics of driven Landau-Zener quenches in the case of a large DC field. In the limit of a large AC field, we describe the conductivity in a Floquet picture and compare the results to the closely related Tien-Gordon effect. We present analytical calculations for the non-linear differential photoconductivity, for both regimes based on the corresponding mechanisms. As part of this discussion of the non-equilibrium state of graphene, we present analytical estimates of the conductivity of undriven graphene as a function of temperature and DC bias field strength that show very good agreement with our simulations.
Generative unfolding with distribution mapping

Anja Butter, Sascha Diefenbacher, Nathan Huetsch, Vinicius Mikuni, Benjamin Nachman, Sofia Palacios Schweitzer, Tilman Plehn

SciPost Phys. 18, 200 (2025) · published 20 June 2025 | · pdf

Machine learning enables unbinned, highly-differential cross section measurements. A recent idea uses generative models to morph a starting simulation into the unfolded data. We show how to extend two morphing techniques, Schrödinger Bridges and Direct Diffusion, in order to ensure that the models learn the correct conditional probabilities. This brings distribution mapping (DM) to a similar level of accuracy as the state-of-the-art conditional generative unfolding methods. Numerical results are presented with a standard benchmark dataset of single jet substructure as well as for a new dataset describing a 22-dimensional phase space of $Z+2$-jets.
Exploring the flavor symmetry landscape

Alfredo Glioti, Riccardo Rattazzi, Lorenzo Ricci, Luca Vecchi

SciPost Phys. 18, 201 (2025) · published 20 June 2025 | · pdf

We explore flavor dynamics in the broad scenario of a strongly interacting light Higgs (SILH). Our study focuses on the mechanism of partial fermion compositeness, but is otherwise as systematic as possible. Concretely, we classify the options for the underlying flavor (and CP) symmetries, which are necessary in order to bring this scenario safely within the range of present or future explorations. Our main goal in this context is to provide a practical map between the space of hypotheses (the models) and the experimental ground that will be explored in the medium and long term, in both indirect and direct searches, in practice at HL-LHC and Belle II, in EDM searches and eventually at FCC-hh. Our study encompasses scenarios with the maximal possible flavor symmetry, corresponding to minimal flavor violation (MFV), scenarios with no symmetry, corresponding to the so-called flavor anarchy, and various intermediate cases that complete the picture. One main result is that the scenarios that allow for the lowest new physics scale have intermediate flavor symmetry rather than the maximal symmetry of MFV models. Such optimal models are rather resilient to indirect exploration via flavor and CP violating observables, and can only be satisfactorily explored at a future high-energy collider. On the other hand, the next two decades of indirect exploration will significantly stress the parameter space of a large swath of less optimal but more generic models up to mass scales competing with those of the FCC-hh.
BV formalism and partition functions

Pietro Antonio Grassi, Ondrej Hulik

SciPost Phys. 18, 202 (2025) · published 23 June 2025 | · pdf

The BV formalism is a well-established method for analyzing symmetries and quantizing field theories. In this paper, we use BV formalism to derive partition functions and the space of gauge invariant operators implementing the equations of motions and their redundancies for selected theories. We discuss various interpretations of the results, some dualities, and relation to the first quantized models.
Missing odd-order Shapiro steps do not uniquely indicate fractional Josephson effect

P. Zhang, S. Mudi, M. Pendharkar, J. S. Lee, C. P. Dempsey, A. P. McFadden, S. D. Harrington, J. T. Dong, H. Wu, A. -H. Chen, M. Hocevar, C. J. Palmstrøm, S. M. Frolov

SciPost Phys. 18, 203 (2025) · published 23 June 2025 | · pdf

Topological superconductivity is expected to spur Majorana zero modes-exotic states that are also considered a quantum technology asset. Fractional Josephson effect is their manifestation in electronic transport measurements, often under microwave irradiation. A fraction of induced resonances, known as Shapiro steps, should vanish, in a pattern that signifies the presence of Majorana modes. Here we report patterns of Shapiro steps expected in topological Josephson junctions, such as the missing first Shapiro step, or several missing odd-order steps. But our junctions, which are InAs quantum wells with Al contacts, are studied near zero magnetic field, meaning that they are not in the topological regime. We also observe other patterns such as missing even steps and several missing steps in a row, not relevant to topological superconductivity. Potentially responsible for our observations is rounding of not fully developed steps superimposed on non-monotonic resistance versus voltage curves, but several origins may be at play. Our results demonstrate that any single pattern, even striking, cannot uniquely identify topological superconductivity, and a multifactor approach is necessary to unambiguously establish this important phenomenon.
Timelike and gravitational anomalous entanglement from the inner horizon

Qiang Wen, Mingshuai Xu, Haocheng Zhong

SciPost Phys. 18, 204 (2025) · published 24 June 2025 | · pdf

In the context of the AdS$_3$/CFT$_2$, the boundary causal development and the entanglement wedge of any boundary spacelike interval can be mapped to a thermal CFT$_2$ and a Rindler $\widetilde{\text{AdS}_3}$ respectively via certain boundary and bulk Rindler transformations. Nevertheless, the Rindler mapping is not confined in the entanglement wedges. While the outer horizon of the Rindler $\widetilde{\text{AdS}_3}$ is mapped to the RT surface, we also identify the pre-image of the inner horizon in the original AdS$_3$, which we call the inner RT surface. In this paper we give some new physical interpretation for the inner RT surface. First, the inner RT surface breaks into two pieces which anchor on the two tips of the causal development. Furthermore, we can take the two tips as the endpoints of a certain timelike interval and the inner RT surface is exactly the spacelike geodesic that represents the real part of the so-called holographic timelike entanglement entropy (HTEE). We also identify a timelike geodesic at boundary of the extended entanglement wedge, which represents the imaginary part of the HTEE. Second, in the duality between the topological massive gravity (TMG) and gravitational anomalous CFT$_2$, the entanglement entropy and the mixed state correlation that is dual to the entanglement wedge cross section (EWCS) receive correction from the Chern-Simons term in the TMG. We find that, the correction to the holographic entanglement entropy can be reproduced by the area of the inner RT surface with a proper regulation, while the mixed state correlation can be represented by the saddle geodesic chord connecting the two pieces of the inner RT surface of the mixed state we consider, which we call the inner EWCS. The equivalence between the twist on the RT surface and the length of inner RT surface is also discussed.
The chiral SYK model in three-dimensional holography

Alexander Altland, Dmitry Bagrets, Nele Callebaut, Konstantin Weisenberger

SciPost Phys. 18, 205 (2025) · published 24 June 2025 | · pdf

A celebrated realization of the holographic principle posits an approximate duality between the $(0+1)$-dimensional quantum mechanical SYK model and two-dimensional Jackiw-Teitelboim gravity, mediated by the Schwarzian action as an effective low energy theory common to both systems. We here propose a generalization of this correspondence to one dimension higher. Starting from different microscopic realizations of effectively chiral $(1+1)$-dimensional generalizations of the SYK model, we derive a reduction to the Alekseev-Shatashvilli (AS)-action, a minimal extension of the Schwarzian action which has been proposed as the effective boundary action of three-dimensional gravity. In the bulk, we show how the same action describes fluctuations around the Euclidean BTZ black hole configuration, the dominant stationary solution of three-dimensional gravity. These two constructions allow us to match bulk and boundary coupling constants, and to compute observables. Specifically, we apply semiclassical techniques inspired by condensed matter physics to the computation of out-of-time-order correlation functions (OTOCs), demonstrating maximal chaos in the chiral SYK chain and its gravity dual.
Interaction-induced strong zero modes in short quantum dot chains with time-reversal symmetry

A. Mert Bozkurt, Sebastian Miles, Sebastiaan L. D. ten Haaf, Chun-Xiao Liu, Fabian Hassler, Michael Wimmer

SciPost Phys. 18, 206 (2025) · published 25 June 2025 | · pdf

We theoretically explore the emergence of strong zero modes in a two-site chain consisting of two quantum dots coupled due to a central dot that mediates electron hopping and singlet superconducting pairing. In the presence of time-reversal symmetry, the on-site Coulomb interaction leads to a three-fold ground-state degeneracy when tuning the system to a sweet spot as a function of the inter-dot couplings. This degeneracy is protected against changes of the dot energies in the same way as "poor man's" Majorana bound states in short Kitaev chains. In the limit of strong interactions, this protection is maximal and the entire spectrum becomes triply degenerate, indicating the emergence of a "poor man's" version of a strong zero mode. We explain the degeneracy and protection by constructing corresponding Majorana Kramers-pair operators and $\mathbb{Z}_3$-parafermion operators. The strong zero modes share many properties of Majorana bound states in short Kitaev chains, including the stability of zero-bias peaks in the conductance and the behavior upon coupling to an additional quantum dot. However, they can be distinguished through finite-bias spectroscopy and the exhibit a different behavior when scaling to longer chains.
Theorems for the lightcone bootstrap

Balt C. van Rees

SciPost Phys. 18, 207 (2025) · published 26 June 2025 | · pdf

Consider a conformally covariant four-point function of identical scalar operators with a discrete spectrum, a twist gap, and compatible with the unitarity conditions. We give a mathematical proof confirming that the spectrum and OPE coefficients at large spin and fixed twist always become that of a generalized free field theory.
Quasiperiodic quadrupole insulators

Raul Liquito, Miguel Gonçalves, Eduardo V. Castro

SciPost Phys. 18, 208 (2025) · published 26 June 2025 | · pdf

Higher-order topological insulators are an intriguing new family of topological states that host lower-dimensional boundary states. Concurrently, quasiperiodic systems have garnered significant interest due to their complex localization and topological properties. In this work we study the impact of chiral symmetry preserving quasiperiodic modulations on the paradigmatic Benalcazar-Bernevig-Hughes model, which hosts topological insulating phases with zero-energy sublattice-polarized modes. We find that the topological properties are not only robust to the quasiperiodic modulation, but can even be enriched. In particular, we unveil the first instance of a quasiperiodic induced second-order topological insulating phase. Furthermore, in contrast with disorder, we find that quasiperiodic modulations can induce multiple reentrant topological transitions, showing an intricate sequence of localization properties. Our results open a promising avenue for exploring the rich interplay between higher-order topology and quasiperiodicity.
Hybrid symmetry class topological insulators

Sanjib Kumar Das, Bitan Roy

SciPost Phys. 18, 209 (2025) · published 27 June 2025 | · pdf

Traditional topological materials belong to different Altland-Zirnbauer symmetry classes (AZSCs) depending on their non-spatial symmetries. Here we introduce the notion of hybrid symmetry class topological insulators (HSCTIs): A fusion of two different AZSC topological insulators (TIs) such that they occupy orthogonal Cartesian hyperplanes and their universal massive Dirac Hamiltonian mutually anticommute, a mathematical procedure we name hybridization. The boundaries of HSCTIs can also harbor TIs, typically affiliated with an AZSC that is different from the ones for the parent two TIs. As such, a fusion or hybridization between planar class AII quantum spin Hall and vertical class BDI Su-Schrieffer-Heeger insulators gives birth to a three-dimensional class A HSCTI, accommodating quantum anomalous Hall insulators (class A) of opposite Chern numbers and quantized Hall conductivity of opposite signs on the top and bottom surfaces. Such a response is shown to be stable against weak disorder. We extend this construction to encompass crystalline HSCTI and topological superconductors (featuring half-quantized thermal Hall conductivity of opposite sings on the top and bottom surfaces), and beyond three spatial dimensions. Non-trivial responses of three-dimensional HSCTIs to crystal defects (namely edge dislocations) in terms of mid-gap bound states at zero energy around its core only on the top and bottom surfaces are presented. Possible (meta)material platforms to harness and engineer HSCTIs are discussed.
Boundary operator expansion and extraordinary phase transition in the tricritical O(N) model

Xinyu Sun, Shao-Kai Jian

SciPost Phys. 18, 210 (2025) · published 27 June 2025 | · pdf

We study the boundary extraordinary transition of a three-dimensional (3D) tricritical $O(N)$ model. We first compute the mean-field Green's function with a general coupling of $|\vec \phi|^{2n}$ (with $n=3$ corresponding to the tricritical model) at the extraordinary phase transition. Then, using layer susceptibility, we obtain the boundary operator expansion for the transverse and longitudinal modes within the $\epsilon=3 - d$ expansion. Based on these results, we demonstrate that the tricritical point exhibits an extraordinary transition characterized by an ordered boundary for any $N$. This provides the first nontrivial example of continuous symmetry breaking in 2D in the context of boundary criticality.
On symmetry-resolved generalized entropies

Fei Yan, Sara Murciano, Pasquale Calabrese, Robert Konik

SciPost Phys. 18, 211 (2025) · published 27 June 2025 | · pdf

Symmetry-resolved entanglement, capturing the refined structure of quantum entanglement in systems with global symmetries, has attracted a lot of attention recently. In this manuscript, introducing the notion of symmetry-resolved generalized entropies, we aim to develop a computational framework suitable for the study of excited state symmetry-resolved entanglement as well as the dynamical evolution of symmetry-resolved entanglement in symmetry-preserving out-of-equilibrium settings. We illustrate our framework using the example of (1+1)-d free massless compact boson theory, and benchmark our results using lattice computation in the XX chain. As a byproduct, our computational framework also provides access to the probability distribution of the symmetry charge contained within a subsystem and the corresponding full counting statistics.
Knizhnik-Zamolodchikov equations and integrable hyperbolic Landau-Zener models

Suvendu Barik, Lieuwe Bakker, Vladimir Gritsev, Emil A. Yuzbashyan

SciPost Phys. 18, 212 (2025) · published 27 June 2025 | · pdf

We study the relationship between integrable Landau-Zener (LZ) models and Knizhnik-Zamolodchikov (KZ) equations. The latter are originally equations for the correlation functions of two-dimensional conformal field theories, but can also be interpreted as multi-time Schrödinger equations. The general LZ problem is to find probabilities of tunneling from eigenstates at $t=t_\text{in}$ to eigenstates at $t\to+∞$ for an $N× N$ time-dependent Hamiltonian $\hat H(t)$. A number of such problems are exactly solvable in the sense that their tunneling probabilities are elementary functions of Hamiltonian parameters. Recently, it has been proposed that exactly solvable LZ models of this type map to KZ equations. Here we use this connection to identify and solve a class of integrable LZ models with hyperbolic time dependence, $\hat H(t)=\hat A+\hat B/t$, for $N=2, 3$, and $4$, where $\hat A$ and $\hat B$ are time-independent matrices.
Topological holography, quantum criticality, and boundary states

Sheng-Jie Huang, Meng Cheng

SciPost Phys. 18, 213 (2025) · published 30 June 2025 | · pdf

Topological holography is a holographic principle that describes the generalized global symmetry of a local quantum system in terms of a topological order in one higher dimension. This framework separates the topological data from the local dynamics of a theory and provides a unified description of the symmetry and duality in gapped and gapless phases of matter. In this work, we develop the topological holographic picture for (1+1)$d$ quantum phases, including both gapped phases as well as a wide range of quantum critical points, including phase transitions between symmetry protected topological (SPT) phases, symmetry enriched quantum critical points, deconfined quantum critical points, and intrinsically gapless SPT phases. Topological holography puts a strong constraint on the emergent symmetry and the anomaly for these critical theories. We show how the partition functions of these critical points can be obtained from dualizing (orbifolding) more familiar critical theories. The topological responses of the defect operators are also discussed in this framework. We further develop a topological holographic picture for conformal boundary states of (1+1)$d$ rational conformal field theories. This framework provides a simple physical picture to understand conformal boundary states and also uncovers the nature of the gapped phases corresponding to the boundary states.