Mean-field theory of spin-liquid states with finite energy gap and topological orders

Abstract

The mean field theory of a T and P symmetric spin liquid state is developed. The quasiparticle excitations in the spin liquid state are shown to be spin 1 2 neutral fermions (the spinons) and charge e spinless bosons (the holons). The spin liquid state is shown to be characterized by a non-trivial topological order. Although our discussions are based on the mean field theory, the concept of the topological order and the associated universal properties (e.g., the quantum number of the quasiparticles) are expected to be valid beyond the mean field theory. We also discuss the dynamical stability of the mean field theory.

Topologically protected quantum bits using Josephson junction arrays

“…Choosing the two qubit states from groundstates in different sectors protects these states from unwanted mixing through noise; protection from leakage within the sector has to be secured through a gapped excitation spectrum. As no local operator can interfere with these states, global operators must be found (and implemented) allowing for the manipulation of the qubit state.A promising candidate fulfilling the above requirements is the quantum dimer system [14][15][16]: recent quantum Monte Carlo simulations of the dimer model on a triangular lattice provide evidence for a gapped liquid groundstate [17] (see [18,19] for a discussion of similar exotic groundstates in spin models) and we will discuss its topological robustness below. The physical implementation of such a dimer system can be realized with the help of quantum Josephson junction arrays.…”

“…A promising candidate fulfilling the above requirements is the quantum dimer system [14][15][16]: recent quantum Monte Carlo simulations of the dimer model on a triangular lattice provide evidence for a gapped liquid groundstate [17] (see [18,19] for a discussion of similar exotic groundstates in spin models) and we will discuss its topological robustness below. The physical implementation of such a dimer system can be realized with the help of quantum Josephson junction arrays.…”

“…While the order characterizing the crystal phases is obvious, the topological order present in the dimer liquid phase is more subtle [14][15][16]: we consider a cylindrical geometry with periodic boundary conditions along the x-axis, leaving the boundaries along the y-direction free (an additional periodic boundary condition along the ydirection defines a torus, see later). In the absence of diagonal dimers (the limit µ → ∞ in (3)) the sum φ counting directed dimers cut by the reference line γ parallel to the y-axis defines a topological order parameter (dimers pointing into opposite directions are counted with opposite sign): the action of the Hamiltonian H = H t + H v leaves φ invariant and the Hilbert space splits into topological sectors H φ characterized by the integers φ.…”

Quantum Field Theory of Many-Body Systems

“…Fermions can emerge as collective excitations of purely bosonic models! (The emergence of deconfined fermions/anyons from purely bosonic models was first studied in 2+1 dimensional models [Arovas et al (1984); Kalmeyer and Laughlin (1987); Wen et al (1989); Read and Sachdev (1991); Wen (1991a); Moessner and Sondhi (2001); Kitaev (2003)]. In 2+1 dimension, one can understand the emergent fermions using a flux binding picture.…”

“…Much of this progress has occurred in three areas of research: (1) the study of topological phases in condensed matter systems such as FQH systems [Wen and Niu (1990); Blok and Wen (1990); Read (1990); Fröhlich and Kerler (1991)], quantum dimer models [Rokhsar and Kivelson (1988); Read and Chakraborty (1989); Moessner and Sondhi (2001); Ardonne et al (2004)], quantum spin models [Kalmeyer and Laughlin (1987) ;Wen et al (1989); Wen (1990); Read and Sachdev (1991); Wen (1991a); Senthil and Fisher (2000); Wen (2002b); Sachdev and Park (2002) ;Balents et al (2002)], or even superconducting states [Wen (1991b); Hansson et al (2004)], (2) the study of lattice gauge theory [Wegner (1971); Banks et al (1977); Kogut and Susskind (1975); Kogut (1979)], and (3) the study of quantum computing by anyons [Kitaev (2003); Ioffe et al (2002); Freedman et al (2002)]. The phenomenon of string condensation is important in all of these fields, though the string picture is often de-emphasized.…”