An algebraic theory of dualities based on the notion of bond algebras is developed. This framework deals with classical and quantum dualities in a unified fashion and its implementation includes discrete lattice, continuum field, and gauge theories. Dualities are revealed to be structure-preserving local mappings between model-specific bond algebras, and can be implemented as unitary (or projective-unitary) transformations. This characterization allows to search systematically for dualities or self-dualities in, finite or infinite, quantum and classical models of arbitrary complexity and number of spatial dimensions. Non-local transformations, like dual or disorder variables and Jordan-Wigner dictionaries, are algorithmically derived from these local duality mappings of bonds. An exact connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or path integrals), as well as the connection between quantum dual and classical disorder variables, is thus established. Our bond-algebraic approach goes beyond the former standard approach to classical dualities, and could help resolve the long standing problem of non-Abelian dualities. In particular, we present new dualities in any spatial dimension for the quantum Heisenberg model. Remarkably, the elimination of gauge constraints can be handled as a bond-algebraic duality (a gauge-reducing duality). In this way, we show that the 2 Higgs model is dual to the extended toric code model in any space dimension. We also show how, in specific models, the phenomenon of dimensional reduction can be accomplished by exact duality mappings. Finally, we discuss various applications including location of phase boundaries, spectral behaviour and, notably, show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.
An algebraic theory of dualities based on the notion of bond algebras is developed. This framework deals with classical and quantum dualities in a unified fashion and its implementation includes discrete lattice, continuum field, and gauge theories. Dualities are revealed to be structure-preserving local mappings between model-specific bond algebras, and can be implemented as unitary (or projective-unitary) transformations. This characterization allows to search systematically for dualities or self-dualities in, finite or infinite, quantum and classical models of arbitrary complexity and number of spatial dimensions. Non-local transformations, like dual or disorder variables and Jordan-Wigner dictionaries, are algorithmically derived from these local duality mappings of bonds. An exact connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or path integrals), as well as the connection between quantum dual and classical disorder variables, is thus established. Our bond-algebraic approach goes beyond the former standard approach to classical dualities, and could help resolve the long standing problem of non-Abelian dualities. In particular, we present new dualities in any spatial dimension for the quantum Heisenberg model. Remarkably, the elimination of gauge constraints can be handled as a bond-algebraic duality (a gauge-reducing duality). In this way, we show that the 2 Higgs model is dual to the extended toric code model in any space dimension. We also show how, in specific models, the phenomenon of dimensional reduction can be accomplished by exact duality mappings. Finally, we discuss various applications including location of phase boundaries, spectral behaviour and, notably, show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.
An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ℤ2 Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.
We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in 2 dimensions, with electromagnetic duality for finite gauge theories in 3 dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.
The ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other. This duality is remarkable inasmuch as it connects systems commonly associated to different areas of physics -- that of long range entangled topological order, (topological) band insulators, and classical statistical mechanics, respectively. Connecting fermionic and bosonic systems, the duality construction is intrinsically non-local, a complication that has been addressed in a plethora of different approaches, including dimensional reduction to one dimension, conformal field theory methods, and operator algebra. In this work, we propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator. Introducing a fourth node into the net of dualities offers several advantages: the formulation is integrative in that all links of the duality are treated on an equal footing, (unlike in field theoretical approaches) it is formulated with lattice precision, a feature that becomes key in the mapping of correlation functions, and their possible numerical implementation. Finally, the passage from bosons to fermions is formulated entirely within the two-dimensional TN framework where it assumes an intuitive and technically convenient form. We illustrate the predictive potential of the formalism by exploring the fate of phase transitions, point and line defects, topological boundary modes, and other structures under the mapping between system classes. Having condensed matter readerships in mind, we introduce the construction pedagogically in a manner assuming only minimal familiarity with the concept of TNs.
The exact analytical solution of the three-dimensional (3D) Ising model remains a central unresolved problem in statistical mechanics. While exact solutions exist in one and two dimensions, the three-dimensional case has resisted closed-form treatment. While standard Kramers-Wannier duality fails to yield a self-dual spin model in 3D, instead mapping to a gauge theory, we introduce a quasi-duality transformation by perturbing the isotropic cubic lattice with a slight, localized spatial inhomogeneity. Specifically, this is introduced via the boundary correction term A n ( ? )= δ η,nN ( s η ? s η?N+1 ? ? s η ? s η+1 ? ) , which arises from flattening the 2D planes into 1D row-continuous chains. By enforcing a condition where the introduced inhomogeneity algebraically absorbs the leading-order topological mismatch of the dual gauge fields, we recover an effective self-dual relation. This algebraic solution yields a critical coupling of K c = 1 4 ln( 1+ 2 )≈0.22033 , which remarkably deviates by only ~0.6% from the widely accepted homogeneous Monte Carlo estimation of K c ( MC ) ?0.22165 . We discuss the derivation and outline future applications of this method to study phase transitions in complex, structurally disordered systems. The method provides insight into how weak inhomogeneity restores a form of dual symmetry that approximates the true critical manifold. Implications for critical phenomena, renormalization structure, and complex biological systems operating near criticality are discussed.
The ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other. This duality is remarkable in as much as it connects systems commonly associated to different areas of physics—that of long-range entangled topological order, (topological) band insulators, and classical statistical mechanics, respectively. Connecting fermionic and bosonic systems, the duality construction is intrinsically nonlocal, a complication that has been addressed in a plethora of different approaches, including dimensional reduction to one dimension, conformal field theory methods, and operator algebra. In this paper, we propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator. Introducing a fourth node into the net of dualities offers several advantages: the formulation is integrative in that all links of the duality are treated on an equal footing, (unlike in field theoretical approaches) it is formulated with lattice precision, a feature that becomes key in the mapping of correlation functions, and their possible numerical implementation. Finally, the passage from bosons to fermions is formulated entirely within the two-dimensional TN framework where it assumes an intuitive and technically convenient form. We illustrate the predictive potential of the formalism by exploring the fate of phase transitions, point and line defects, topological boundary modes, and other structures under the mapping between system classes. Having condensed-matter readerships in mind, we introduce the construction pedagogically in a manner assuming only minimal familiarity with the concept of TNs. Published by the American Physical Society 2024
The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's original paradigm. Here, we develop an algebraic framework for systematically deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries, and focus on a (2+1)-dimensional quantum Ising plaquette model as a concrete illustration. We show that bond-algebraic automorphisms, when combined with the so-called half-gauging procedure, reveal the structure of the non-invertible duality symmetry operators, which can be explicitly represented as a sequential quantum circuit. The resulting duality defects are constrained by the model's rigid higher symmetries (lower-dimensional subsystem symmetries), leading to restricted mobility. We establish the fusion algebra of these defects. Finally, in constructing the non-invertible duality transformation, we explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.