The two- and three-dimensional Z2-symmetric phi^4 quantum field theories admit strong-weak self-dualities (Chang duality in d=2, Magruder duality in d=3) whose exact renormalization-scheme dependence was determined by Sberveglieri, Serone and Spada (JHEP 2021). Since these theories are the continuum scaling limits of the Ising model, whose Kramers-Wannier duality (and, in 3d, the duality between the Ising model and a Z2 lattice gauge theory) is exact, it is natural to conjecture that the continuum self-dualities are the scaling limits of the lattice dualities. The open problem, stated explicitly in the conclusions of that work, is to construct the precise correspondence: the action of the lattice duality maps on the renormalized phi^4 parameters and fields in the scaling limit, and whether a proper finite-volume definition of Chang duality requires Z2 gauge fields as Kramers-Wannier duality does. No construction of this correspondence is known in the literature to date.

Background

Euclidean scalar theories with quartic interaction and O(N) symmetry, ϕ4\phi^4 vector models, are the simplest interacting relativistic quantum field theories. In two and three spacetime dimensions they are super-renormalizable: only the mass term and the vacuum energy require counterterms, while the quartic coupling is finite. This makes the exact dependence of the theory on the renormalization scheme - the finite redefinition of the mass parameter relating two regularizations, parametrized by a real number κ\kappa in the analysis of Sberveglieri, Serone and Spada (2021) - computable in closed form.

In this setting two unexpected strong-weak dualities are known. Chang duality (1976) states, in its modern form, that the two-dimensional N=1N=1 (Z2\mathbb{Z}_2-symmetric) ϕ4\phi^4 theory in the phase where the Z2\mathbb{Z}_2 symmetry is classically broken (negative squared mass) is equivalent to the same theory in the classically unbroken phase (positive squared mass) at the same coupling, the two mass parameters being related by an explicit non-linear function built from the Lambert-W function (the inverse of x↦xexx \mapsto x e^x), which depends on the scheme; for κ=0\kappa=0 it reduces to Chang's original normal-ordering derivation. Magruder duality (1976) is the three-dimensional analogue. In 3d O(N) models these relations imply that the classically unbroken phase possesses two critical points - a weak and a strong branch exchanged by the self-duality - which merge at the self-dual point and, in schemes with κ\kappa below a critical value, move into the complex coupling plane as a complex-conjugate pair of fixed points (complex CFTs, i.e. conformal field theories with complex, non-unitary data). The dualities are exact to all orders in perturbation theory and have been verified numerically by Borel-resummed high-order perturbative series (Borel resummation reconstructs a function from its divergent asymptotic series by analytically continuing the series in an auxiliary Borel variable and integrating it; Serone, Spada and Villadoro 2019; Sberveglieri, Serone and Spada 2021) and by Hamiltonian truncation at finite volume (a numerical method that diagonalizes the field-theory Hamiltonian in a state basis truncated below an energy cutoff; Rychkov and Vitale 2016; Elias-Miró and Hardy 2020).

Independently, the Ising lattice model possesses Kramers-Wannier (KW) duality (1941): an exact equivalence exchanging the ordered and disordered phases, mapping the model to a model on the dual lattice with inverted coupling (in 2d, sinh⁡(2K)sinh⁡(2K∗)=1\sinh(2K)\sinh(2K^*)=1). The dual (disorder) variables are non-local in the original spins, and a proper implementation of the duality - especially at finite volume or on manifolds with non-trivial topology - is formulated most naturally in terms of a Z2\mathbb{Z}_2 lattice gauge theory (a lattice theory whose fundamental variables are Z2\mathbb{Z}_2-valued fields living on lattice links rather than spins on sites); in three dimensions the KW dual of the Ising model is precisely such a Z2\mathbb{Z}_2 lattice gauge theory. The ϕ4\phi^4 theories above are precisely the continuum scaling limits of the Ising model near criticality: the critical 2d Ising model flows to the Ising conformal field theory (a free Majorana fermion), and the near-critical scaling limit is the 2d N=1N=1 ϕ4\phi^4 theory at its critical coupling (the Landau-Ginzburg description). The continuum version of KW duality is defined starting from the critical theory and extends to the massive theory, where it exchanges the order field with the disorder (twist) field.

It is therefore natural to conjecture that KW duality and Chang duality are two descriptions of the same structure, and to ask for the precise map between them; the analogous question relates Magruder duality to the continuum limit of the 3d Ising-Z2-gauge duality. A related motivation noted by the same authors is particle-vortex duality - the conjectural 3d equivalence between the Wilson-Fisher O(2) critical theory (the infrared fixed point of the 3d O(2) model, i.e. the XY universality class) and the Abelian Higgs model (a complex scalar field coupled to a U(1)U(1) gauge field), formulated on the lattice by Peskin (1978) and reviewed by Senthil, Son, Wang and Xu (2019) - whose ultraviolet completion is not understood; a Chang/Magruder-like duality for a gauged 3d O(2) model would be a natural candidate, but constructing it is a separate open problem.

Problem Statement

Construct the precise correspondence between the lattice dualities of the Ising model and the continuum self-dualities of phi^4 theory, as posed in the conclusions of Sberveglieri, Serone and Spada (JHEP 02 (2021) 098):

  1. In two dimensions, determine how Kramers-Wannier duality of the 2d Ising lattice model acts, in the critical scaling limit, on the parameters of the continuum N=1N=1 ϕ4\phi^4 theory (the squared mass, the quartic coupling, and the renormalization-scheme parameter κ\kappa) and on its fields; in particular, whether the induced map coincides with the Chang duality relation - reproducing Chang's normal-ordering relation at κ=0\kappa=0 and the scheme-dependent relation of the cited work at general κ\kappa - together with the corresponding operator map between order and disorder fields.

  2. Determine whether a proper definition of Chang duality at finite volume requires the introduction of Z2\mathbb{Z}_2 gauge fields, as Kramers-Wannier duality does, and construct the finite-volume duality accordingly (or prove that no gauge fields are needed).

  3. Establish, or refute, the analogous three-dimensional connection between Magruder duality of the 3d ϕ4\phi^4 theory and the continuum limit of the duality between the 3d Ising model and the Z2\mathbb{Z}_2 lattice gauge theory.

Each of the three parts is independently resolvable: settling any one of them, by construction or by refutation, constitutes a complete answer, and the parts need not be settled together. An answer may consist of an explicit construction with derivation, a demonstration that the conjectured identification fails (exhibiting the precise obstruction), or a combination of the two, e.g. a construction valid in a specified regime together with a proof of breakdown outside it.

References

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