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arXiv:cond-mat/0411752 · Physical Review B

String and membrane condensation on three-dimensional lattices

Alioscia Hamma, Paolo Zanardi A, Xiao-Gang Wen

Published Jul 6, 2005• 103 citations
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Abstract

In this paper, we investigate the general properties of lattice spin models that have string and/or membrane condensed ground states. We discuss the properties needed to define a string or membrane operator. We study three three-dimensional spin models which lead to ${Z}_{2}$ gauge theory at low energies. All the three models are exactly soluble and produce topologically ordered ground states. The first model contains both closed-string and closed-membrane condensations. The second model contains closed-string condensation only. The ends of open strings behave like fermionic particles. The third model also has condensations of closed membranes and closed strings. The ends of open strings are bosonic while the edges of open membranes are fermionic. The third model contains a different type of topological order.

DOI
10.1103/physrevb.72.035307

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In this paper, we investigate the general properties of lattice spin models that have string and/or membrane condensed ground states. We discuss the properties needed to define a string or membrane operator. We study three three-dimensional spin models which lead to Z2{Z}_{2}Z2​ gauge theory at low energies. All the three models are exactly soluble and produce topologically ordered ground states. The first model contains both closed-string and closed-membrane condensations. The second model contains closed-string condensation only. The ends of open strings behave like fermionic particles. The third model also has condensations of closed membranes and closed strings. The ends of open strings are bosonic while the edges of open membranes are fermionic. The third model contains a different type of topological order.
This book has three types of theories: boson, fermion, and gauge theories. It asks which theories describe the elementary particles in our universe. Surprisingly, elementary particles are described by the two more complicated and non-local theories: fermion theory and gauge theory. In the last chapter and in this chapter it is demonstrated that fermion and gauge theory are not really needed. Boson theory may be able to explain everything, since both fermion theory and gauge theory can emerge from local bosonic models (or qbit models), if bosons (qbits) are organized to have a string-net condensed order. This chapter studies soluble quantum spin models to explain string-net condensation. It finds that gauge bosons are simply collective excitations associated with string density wave (with the electric field corresponding to the string density). The fermons can appear as the ends of open strings. Thus the string-net order explains the emergence of gauge bosons and fermions. It unifies gauge interaction and Fermi statistics. The chapter shows that non-orientable strings give rise to Z_2 gauge theory while the orientable string give rise to U(1) gauge theory.
We construct an exactly solvable Hamiltonian acting on a three-dimensional lattice of spin-12\frac{1}{2}21​ systems that exhibits topological quantum order. The ground state is a string net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends on the homology of the 3-manifold. We generalize the system to $D\ensuremath{\geqslant}4$, where different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.
We show that quantum systems of extended objects naturally give rise to a large class of exotic phases---namely topological phases. These phases occur when extended objects, called ``string-nets,'' become highly fluctuating and condense. We construct a large class of exactly soluble 2D spin Hamiltonians whose ground states are string-net condensed. Each ground state corresponds to a different parity invariant topological phase. The models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltonians---a spin-1∕21∕21∕2 system on the honeycomb lattice---is a simple theoretical realization of a universal fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.
Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be ``twisted,'' in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.
We investigate the general properties of lattice spin models with emerging fermionic excitations. We argue that fermions always come in pairs and their creation operator always has a stringlike structure with the newly created particles appearing at the end points of the string. The physical implication of this structure is that the fermions always couple to a nontrivial gauge field. We present exactly soluble examples of this phenomenon in two and three dimensions. Our analysis is based on an algebraic formula that relates the statistics of a lattice particle to the properties of its hopping operators. This approach has the advantage in that it works in any number of dimensions---unlike the flux-binding picture developed in fractional quantum Hall theory.
We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang. While these are not models of interacting bosons or fermions, they may well capture the topological behaviour of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a υ=1/2 bosonic fractional quantum Hall effect in its topological limit. Our special example of the 3D semion model captures much of the behaviour of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk. We discuss how more general Walker-Wang models can have a combination of fermionic or bosonic deconfined excitations in the bulk, and excitations which are deconfined on the surface but confined in the bulk. Having classified the spectra and surface properties of a number of these models, we construct a topological field theory which describes a subset of them exactly. Moreover, we argue that some Walker-Wang models capture the low-energy long-wavelength behaviour of certain gapped phases of axion electrodynamics which, in turn, are believed to be germane to certain condensed matter systems with strong spin-orbit coupling.