Celebratio Mathematica

techreport M. H. Freedman, C. Nayak, and K. Shtengel: Non-Abelian topological phases in an extended Hubbard model. Preprint, September 2003. ArXiv cond-mat/0309120

We describe four closely related Hubbard-like models (models A, B, C and D) of particles that can hop on a \( 2D \) Kagome lattice interacting via Coulomb repulsion. The particles can be either bosons (models A and B) or (spinless) fermions (models C and D). Models A and C also include a ring exchange term. In all four cases we solve equations in the model parameters to arrive at an exactly soluble point whose ground state manifold is the extensively degenerate “\( d \)-isotopy space” \( \bar{V}_d \), \( 0 < d < 2 \). Near the “special” values, \( d = 2 \cos \pi/k+2 \), \( \bar{V}_d \) should collapse to a stable topological phase with anyonic excitations closely related to \( \mathit{SU}(2) \) Chern–Simons theory at level \( k \). We mention simplified models \( A^- \) and \( C^- \) which may also lead to these topological phases.

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article M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang: “A class of \( P,T \)-invariant topological phases of interacting electrons,” Ann. Physics 310 : 2 (2004), pp. 428–492. MR 2044743 Zbl 1057.81053

We describe a class of parity- and time-reversal-invariant topological states of matter which can arise in correlated electron systems in \( 2+1 \)-dimensions. These states are characterized by particle-like excitations exhibiting exotic braiding statistics. \( P \) and \( T \) invariance are maintained by a ‘doubling’ of the low-energy degrees of freedom which occurs naturally without doubling the underlying microscopic degrees of freedom. The simplest examples have been the subject of considerable interest as proposed mechanisms for high-\( T_c \) superconductivity. One is the ‘doubled’ version of the chiral spin liquid. The chiral spin liquid gives rise to anyon superconductivity at finite doping and the corresponding field theory is \( U(1) \) Chern–Simons theory at coupling constant \( m=2 \). The ‘doubled’ theory is two copies of this theory, one with \( m=2 \) the other with \( m=-2 \). The second example corresponds to \( Z_2 \) gauge theory, which describes a scenario for spin-charge separation. Our main concern, with an eye towards applications to quantum computation, are richer models which support non-Abelian statistics. All of these models, richer or poorer, lie in a tightly organized discrete family indexed by the Baraha numbers, \( 2\cos(\pi/(k+2)) \), for positive integer \( k \). The physical inference is that a material manifesting the \( Z_2 \) gauge theory or a doubled chiral spin liquid might be easily altered to one capable of universal quantum computation. These phases of matter have a field-theoretic description in terms of gauge theories which, in their infrared limits, are topological field theories. We motivate these gauge theories using a parton model or slave-fermion construction and show how they can be solved exactly. The structure of the resulting Hilbert spaces can be understood in purely combinatorial terms. The highly constrained nature of this combinatorial construction, phrased in the language of the topology of curves on surfaces, lays the groundwork for a strategy for constructing microscopic lattice models which give rise to these phases.

Kirill Shtengel	Related
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M. H. Freedman, C. Nayak, and K. Shtengel: “Line of critical points in \( 2+1 \) dimensions: Quantum critical loop gases and non-abelian gauge theory,” Phys. Rev. Lett. 94 : 14 (2005), pp. 147205.

In this Letter, we
(1) construct a one-parameter family of lattice models of interacting spins;
(2) obtain their exact ground states;
(3) derive a statistical-mechanical analogy which relates their ground states to \( O(n) \) loop gases;
(4) show that the models are critical for \( d\leq\sqrt{2} \), where \( d \) parametrizes the models;
(5) note that, for the special values \( d=2\cos[\pi/(k+2)] \), they are related to doubled level-\( k \) \( \mathit{SU}(2) \) Chern–Simons theory;
(6) conjecture that they are in the universality class of a nonrelativistic \( \mathit{SU}(2) \) gauge theory; and
(7) show that its one-loop \( \beta \) function vanishes for all values of the coupling constant, implying that it is also on a critical line.

Kirill Shtengel	Related
Chetan Nayak	Related

M. Freedman, C. Nayak, and K. Shtengel: “An extended Hubbard model with ring exchange: A route to a non-abelian topological phase,” Phys. Rev. Lett. 94 : 6 (2005), pp. 066401.

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Chetan Nayak	Related

M. Freedman, C. Nayak, and K. Shtengel: “Lieb–Schultz–Mattis theorem for quasitopological systems,” Phys. Rev. B 78 (2008), pp. 174411.

In this paper we address the question of the existence of a spectral gap in a class of local Hamiltonians. These Hamiltonians have the following properties: their ground states are known exactly; all equal-time correlation functions of local operators are short-ranged; and correlation functions of certain nonlocal operators are critical. A variational argument shows gaplessness with \( \omega \propto k^2 \) at critical points defined by the absence of certain terms in the Hamiltonian, which is remarkable because equal-time correlation functions of local operators remain short ranged. We call such critical points, in which spatial and temporal scaling are radically different, quasitopological. When these terms are present in the Hamiltonian, the models are in gapped topological phases which are of special interest in the context of topological quantum computation.

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