Fractionalization in a square-lattice model with time-reversal symmetry

Seradjeh, Babak; Weeks, Conan; Franz, Marcel

doi:10.1103/physrevb.77.033104

Cited by 57 publications

(88 citation statements)

References 22 publications

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“…This might not be the only route -as noted in a similar context in Ref. [11] another approach might be through an appropriately engineered semiconductor heterostructure.…”

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Birefringent breakup of Dirac fermions on a square optical lattice

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We generalize a proposal by Sørensen et al. [Phys. Rev. Lett. 94, 086803 (2005)] for creating an artificial magnetic field in a cold atom system on a square optical lattice. This leads us to an effective lattice model with tunable spatially periodic modulation of the artificial magnetic field and the hopping amplitude. When there is an average flux of half a flux quantum per plaquette the spectrum of low-energy excitations can be described by massless Dirac fermions in which the usually doubly degenerate Dirac cones split into cones with different "speeds of light" which can be tuned to give a single Dirac cone and a flat band. These gapless birefringent Dirac fermions arise because of broken chiral symmetry in the kinetic energy term of the effective low energy Hamiltonian. We characterize the effects of various perturbations to the low-energy spectrum, including staggered potentials, interactions, and domain wall topological defects. PACS numbers: 71.10.Fd, 37.10.Jk, 05.30.Fk, 71.10.Pm With the discovery of graphene [1] and topological insulators [2] there has been much recent interest in systems in which low energy excitations can be described using Dirac fermions. A parallel area of interest has been the exploration of the possibility of generating artificial magnetic fields for cold atoms confined in an optical lattice. Neutral bosonic cold atoms cannot couple to a magnetic field directly, so there have been numerous proposals [3-6] of approaches to couple atoms to an artificial magnetic field, several of which have been implemented experimentally [7,8].The problem of the spectrum of quantum particles in a uniform magnetic field on a lattice has the well-known Hofstadter spectrum [9]. Our modification of the proposal by Sørensen et al. [6] leads to an effective Hamiltonian with a tunable Hofstadter-like spectrum that arises from the combination of hopping and an artificial magnetic field with a non-zero average that are both periodically modulated in the x and the y directions. The presence of spatial periodicity in the amplitude as well as the phase of the hopping is the key difference between the model we consider here and previous work on the spectrum of particles in the presence of magnetic fields that are periodic in both the x and y directions [10]. This difference facilitates the unusual Dirac-like spectrum that we discuss in this Letter. In our effective model, when there is an average of half a flux quantum per plaquette, and at half-filling, the low energy degrees of freedom can be described by a Dirac Hamiltonian with the unusual property that chiral symmetry is broken in the kinetic energy rather than via mass terms. This has the consequence that the doubly degenerate Dirac cone for massless fermions splits into two cones with tunable distinct slopes, analagous to a situation in which there are two speeds of light for fermionic excitations, similar to birefringence of light in crystals such as calcite. We discuss the meaning of broken chiral symmetry in our effective model and explore the ef...

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“…This might not be the only route -as noted in a similar context in Ref. [11] another approach might be through an appropriately engineered semiconductor heterostructure.…”

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“…When α = 1/2 there is an average of half a flux quantum per plaquette, and the theory is time reversal symmetric as fermions cannot detect the sign of the flux [11]. The effective Hamiltonian Eq.…”

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Birefringent breakup of Dirac fermions on a square optical lattice

et al. 2011

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“…Second, fractional statistics have recently captured attention due to their relevance to topologically protected quantum information processing [7]. Since the honeycomb lattice is found in natural graphene [8], and the square lattice model could be realized in artificially engineered structures [9], the possibility of realizing anyons in time-reversal invariant systems has both theoretical and practical significance.In this Letter we construct the low-energy effective theory for the fractional particles in models [3,4]. We find that they are indeed anyons, albeit of a very special kind, described by a doubled U(1) 2 × U(1) 2 Chern-Simons (CS) theory previously discussed by Freedman et al [10].…”

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“…This phenomenon is known to occur in the fractional quantum Hall (FQH) liquids [1], quintessential strongly correlated systems with broken time-reversal symmetry, where fractionally-charged excitations also obey fractional exchange statistics [2]. Recently, two model systems have been introduced on the honeycomb and square lattices [3,4] that exhibit charge fractionalization without breaking of the time-reversal symmetry. These models, in essence, generalize the concept of fractionalization in polyacetylene [5] to two dimensions and, remarkably, can be considered weakly correlated.…”

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Fractional Statistics of Topological Defects in Graphene and Related Structures

Seradjeh

Franz

2008

Phys. Rev. Lett.

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We show that fractional charges bound to topological defects in the recently proposed time-reversal-invariant models on honeycomb and square lattices obey fractional statistics. The effective low-energy description is given in terms of a 'doubled' level-2 Chern-Simons field theory, which is parity and time-reversal invariant and implies two species of semions (particles with statistical angle ±π/2) labeled by a new emergent quantum number that we identify as the fermion axial charge.Introduction -When the excitations of a many-body system carry electric charge that is smaller than the charge of its constituent particles (e.g., electrons) the charge is said to be fractionalized. This phenomenon is known to occur in the fractional quantum Hall (FQH) liquids [1], quintessential strongly correlated systems with broken time-reversal symmetry, where fractionally-charged excitations also obey fractional exchange statistics [2]. Recently, two model systems have been introduced on the honeycomb and square lattices [3,4] that exhibit charge fractionalization without breaking of the time-reversal symmetry. These models, in essence, generalize the concept of fractionalization in polyacetylene [5] to two dimensions and, remarkably, can be considered weakly correlated. The experience with FQH systems suggests that the exchange statistics of these fractionally charged excitations could be anomalous. This question is interesting for several reasons. First, a very general argument can be made [6] that would seem to prohibit the existence of anyons in systems that obey time-reversal symmetry. Second, fractional statistics have recently captured attention due to their relevance to topologically protected quantum information processing [7]. Since the honeycomb lattice is found in natural graphene [8], and the square lattice model could be realized in artificially engineered structures [9], the possibility of realizing anyons in time-reversal invariant systems has both theoretical and practical significance.In this Letter we construct the low-energy effective theory for the fractional particles in models [3,4]. We find that they are indeed anyons, albeit of a very special kind, described by a doubled U(1) 2 × U(1) 2 Chern-Simons (CS) theory previously discussed by Freedman et al. [10]. In its topological sector the theory contains two species of semions, which transform into each other under parity and time reversal, thus escaping the constraints imposed by the argument of Ref. 6. Systems under consideration here [3,4] represent the first explicit example of models for which such a gauge structure emerges as the lowenergy effective theory.Fractional charge -The low-energy theory for fermions on the graphene honeycomb lattice [11] and the square lattice threaded with π flux per plaquette [12] is the Dirac Lagrangian

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