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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Hidden fermions in Z(2) theories

Mark Srednicki

Published May 15, 1980• 15 citations
Publisher

Abstract

I show that Ising models and $Z(2)$ gauge theories, in any number of dimensions, are equivalent to systems of fermions with local interactions only. This is done in the Hamiltonian formalism by rewriting the Hamiltonians in terms of locally coupled fermion operators. One advantage of the fermionic formalism is that duality transformations become trivial. All unsolved models have products of four or more fermion operators as terms in their Hamiltonians.

DOI
10.1103/physrevd.21.2878

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I show that Ising models and Z(2)Z(2)Z(2) gauge theories, in any number of dimensions, are equivalent to systems of fermions with local interactions only. This is done in the Hamiltonian formalism by rewriting the Hamiltonians in terms of locally coupled fermion operators. One advantage of the fermionic formalism is that duality transformations become trivial. All unsolved models have products of four or more fermion operators as terms in their Hamiltonians.
The mechanism underlying any bosonisation or fermionisation is exposed.It is shown that any local theory of fermions on a lattice in any spatial dimension greater than one is equivalent to a local theory of Ising spins coupled to a Z2Z_{2}Z2​ gauge field.There is a close relation to the description of anyons using a Chern-Simons term.
Abstract I solve a quantum chain whose Hamiltonian is comprised solely of local four-fermi operators by constructing free-fermion raising and lowering operators. The free-fermion operators are both non-local and highly non-linear in the local fermions. This construction yields the complete spectrum of the Hamiltonian and an associated classical transfer matrix. The spatially uniform system is gapless with dynamical critical exponent z = 3/2, while staggering the couplings gives a more conventional free-fermion model with an Ising transition. The Hamiltonian is equivalent to that of a spin-1/2 chain with next-nearest-neighbour interactions, and has a supersymmetry generated by a sum of fermion trilinears. The supercharges are part of a large non-abelian symmetry algebra that results in exponentially large degeneracies. The model is integrable for either open or periodic boundary conditions but the free-fermion construction only works for the former, while for the latter the extended symmetry is broken and the degeneracies split.
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
We devise a unitary transformation that replaces the fermionic degrees of freedom of lattice gauge theories by (hard-core) bosonic ones. The resulting theory is local and gauge invariant, with the same symmetry group. The method works in any spatial dimension and can be directly applied, among others, to the gauge groups G=U(N)G=U(N)G=U(N) and SU(2N)SU(2N)SU(2N), where $N\ensuremath{\in}\mathbb{N}$. For SU(2N+1)SU(2N+1)SU(2N+1), one can also carry out the transformation after introducing an extra idle Z2{\mathbb{Z}}_{2}Z2​ gauge field, so that the resulting symmetry group trivially contains Z2{\mathbb{Z}}_{2}Z2​ as a normal subgroup. Those results have implications in the field of quantum simulations of high-energy physics models.
Abstract We show how certain long-range models of interacting fermions in d + 1 dimensions are equivalent to gauge theories in D + 1 dimensions in which gauge fields are defined in a dimension ( D ) larger than the dimension ( d ) of the fermionic theory to be simulated. For d = 1 it is possible to obtain an exact mapping, providing an expression of the fermionic interaction potential in terms of half-integer powers of the Laplacian. An analogous mapping can be applied to the kinetic term of the bosonized theory. A diagrammatic representation of the theories obtained by dimensional mismatch is presented, and consequences and applications of the established duality are discussed. Finally, by using a perturbative approach, we address the canonical quantization of fermionic theories presenting non-locality in the interaction term to construct the Hamiltonians for the effective theories found by dimensional reduction. We conclude by showing that one can engineer the gauge fields and the dimensional mismatch in order to obtain long-range effective Hamiltonians with 1/ r or 1/ r 2 potentials as examples.
State sum constructions, such as Kuperberg's algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights with different parts of a closed triangulated manifold. Here we extend this construction by including matter fields to build partition functions in both two and three space–time dimensions. The matter fields introduce new weights to the vertices and they correspond to Potts spin configurations described by an -module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain transfer matrices which are decomposed into a product of local operators acting on vertices, links and plaquettes. The vertex and plaquette operators are similar to the ones appearing in the quantum double models (QDMs) of Kitaev. The link operator couples the gauge and the matter fields, and it reduces to the usual interaction terms in known models such as gauge theory with matter fields. The transfer matrices lead to Hamiltonians that are frustration-free and are exactly solvable. According to the choice of the initial input, that of the gauge group and a matter module, we obtain interesting models which have a new kind of ground state degeneracy that depends on the number of equivalence classes in the matter module under gauge action. Some of the models have confined flux excitations in the bulk which become deconfined at the surface. These edge modes are protected by an energy gap provided by the link operator. These properties also appear in 'confined Walker–Wang' models which are 3D models having interesting surface states. Apart from the gauge excitations there are also excitations in the matter sector which are immobile and can be thought of as defects like in the Ising model. We only consider bosonic matter fields in this paper.