We determine exact finite-volume spin form factors and two-point correlations for a massive Kramer--Wannier interface in the scaling Ising model, separating ordered and disordered phases. The interface admits two complementary lattice realizations: a self-dual inhomogeneous Ising chain whose continuum limit has a sign-changing Majorana mass profile, and, after exchanging Euclidean space and time, a mixed thermal average amenable to Toeplitz/Fredholm methods. Combining these descriptions, we obtain the exact Neveu-Schwarz-Ramond spin matrix elements and a factorized Fredholm determinant for the interface correlator. Defect TCSA and independent fermionic spin-chain calculations confirm the predicted spectrum and matrix elements.
We determine exact finite-volume spin form factors and two-point correlations for a massive Kramer--Wannier interface in the scaling Ising model, separating ordered and disordered phases. The interface admits two complementary lattice realizations: a self-dual inhomogeneous Ising chain whose continuum limit has a sign-changing Majorana mass profile, and, after exchanging Euclidean space and time, a mixed thermal average amenable to Toeplitz/Fredholm methods. Combining these descriptions, we obtain the exact Neveu-Schwarz-Ramond spin matrix elements and a factorized Fredholm determinant for the interface correlator. Defect TCSA and independent fermionic spin-chain calculations confirm the predicted spectrum and matrix elements.
In this work, we provide a self-contained derivation of the spin-operator matrix elements in the fermionic basis, for the critical periodic Ising chain at a generic system length N∈2Z≥2. The approach relies on the near-Cauchy property of certain matrices formed by the Toeplitz symbols in the critical model, and leads to a few square-root products for the leg functions. The square root products allow simple integral representations, that further reduce to the Binet's second integral and its generalization by Hermite, in the finite volume scaling limit. This leads to product formulas for the spin operator matrix elements in the scaling limit, providing explicit expressions for the spin-operator form factors of the Ising CFT in the fermionic basis, that were computed iteratively in Yurov:1991my. They are all rational numbers up to 2. We also determine the normalization factor of the spin-operator and show explicitly how the coefficient G(21)G(23) appear through a ground state overlap. Moreover, by expanding the spin-spin two point correlator in the fermionic basis, we observed a Fredholm determinant identity that allows to show the convergence of the rescaled two-point correlator to the CFT version on a cylinder.
We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The nth summand of the series is a multiple integral accounting for all n-particle–n-hole excitations of the quantum transfer matrix. In previous works, the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here, we obtain a representation which reduces, in the zero-temperature limit, essentially to a product of two determinants of finite matrices whose entries are known special functions. This will facilitate the further analysis of the correlation function.
A non-invertible topological line fixes global defect data but does not select a reduced density matrix (RDM) in a spatially twisted sector. For the Kramers-Wannier defect of the critical Ising chain, we choose the equal-weight incoherent mixture of the two charge-sector ground states and reduce it to the ordinary full spin algebra of a complete non-wrapping prefix anchored at the defect endpoint. The finite-size RDM is the convex average of two charge-sector RDMs rather than a sign-zero Gaussian proxy, and an exact descended antiunitary enforces many-body Kramers pairing without determining the entropy. At fixed prefix, the RDMs converge in trace norm to a common Gaussian state as the circumference grows. Convergence is uniform over all ground-doublet preparations, including coherent pure states. The subsequent large-prefix entropy limit follows from the odd-even Toeplitz/Fisher-Hartwig endpoint, yielding half the natural logarithm of two above the homogeneous chain. The matched spin-flip defect has zero excess. The exact joint energy-modified-translation character of the same twisted Hamiltonian records the finite-size roots and their multiplicities, retains chirality in the marked scaling limit, and, using the standard Ising character identities, resolves the four Virasoro towers of the Ising duality-twisted sector.
It is well known that the noninteracting Majorana chain is dual to the one-dimensional transverse-field Ising model, either through the Jordan–Wigner transformation or by gauging fermion parity. In this correspondence, the minimal translation of the Majorana chain maps to the celebrated Kramers–Wannier (KW) duality of the spin model, with the critical point mapped to the self-dual point. In this work, we generalize this mapping to two and higher dimensions by constructing a unitary equivalence between the parity-gauged fermionic system and a spin system defined on arbitrary polyhedral decompositions of space. Imposing the flatness condition on the gauge field yields a bosonization duality between the original (ungauged) fermionic system and a gauged spin system obeying a Gauss law. The dependence of the Gauss law in the spin system on the Kasteleyn orientation (and the discrete spin structure) of the fermionic system is made explicit. Applying this bosonization to one or two copies of Majorana fermions on translationally invariant lattices, we derive higher-dimensional analogs of KW (self-) dualities in spin systems arising from fermionic minimal translations. The KW (self-) dualities are non-invertible due to projections onto eigenspaces of higher-form symmetries in the associated symmetry operators. The bosonization framework we present is intuitive, general, and systematic, encompassing other known exact bosonization methods while offering a novel approach to establish new connections between fermionic and spin systems in arbitrary dimensions.
Explicit factorized formulas for the matrix elements (form factors) of the spin operators σ x and σ y between the eigenvectors of the Hamiltonian of the finite quantum periodic XY -chain in a transverse field were derived. The derivation is based on the relations between three models: the model of quantum XY -chain, Ising model on 2D lattice and N = 2 Baxter–Bazhanov–Stroganov τ (2) -model. Due to these relations we transfer the formulas for the form factors of the latter model recently obtained by the use of separation of variables method to the model of quantum XY -chain. Hopefully, the formulas for the form factors will help in analysis of multipoint dynamic correlation functions at a finite temperature. As an example, we re-derive the asymptotics of the two-point correlation function in the disordered phase without the use of the Toeplitz determinants and the Wiener–Hopf factorization method.
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 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 Z2 extension of Rep(H). Specializing to the Kac-Paljutkin algebra H8, the smallest self-dual Hopf algebra beyond abelian group algebras, we numerically study the phase diagram and identify four of the six Rep(H8)-symmetric gapped phases, separated by Ising critical lines and meeting at a multicritical point. We also realize all six Rep(H8)-symmetric gapped phases on the lattice via the H-comodule algebra formalism, in agreement with the module-category classification of Rep(H8). Our results provide a unified Hopf-algebraic framework for non-invertible symmetries, dualities, and the tensor product lattice models that realize them.