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
Abstract
People
BibTeX
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.
@article {key2044743m,
AUTHOR = {Freedman, Michael and Nayak, Chetan
and Shtengel, Kirill and Walker, Kevin
and Wang, Zhenghan},
TITLE = {A class of \$P,T\$-invariant topological
phases of interacting electrons},
JOURNAL = {Ann. Physics},
FJOURNAL = {Annals of Physics},
VOLUME = {310},
NUMBER = {2},
YEAR = {2004},
PAGES = {428--492},
DOI = {10.1016/j.aop.2004.01.006},
NOTE = {MR:2044743. Zbl:1057.81053.},
ISSN = {0003-4916},
}
techreport
M. Freedman, C. Nayak, and K. Walker :
Tilted interferometry realizes universal quantum computation in the Ising TQFT without overpasses .
Preprint ,
December 2005 .
ArXiv
cond-mat/0512072
Abstract
People
BibTeX
We show how a universal gate set for topological quantum computation in the Ising TQFT, the non-Abelian sector of the putative effective field theory of the \( \nu=5/2 \) fractional quantum Hall state, can be implemented. This implementation does not require overpasses or surgery, unlike the construction of Bravyi and Kitaev, which we take as a starting point. However, it requires measurements of the topological charge around time-like loops encircling moving quasiaparticles, which require the ability to perform ‘tilted’ interferometry measurements
@techreport {keycond-mat/0512072a,
AUTHOR = {Freedman, M. and Nayak, C. and Walker,
K.},
TITLE = {Tilted interferometry realizes universal
quantum computation in the {I}sing {TQFT}
without overpasses},
TYPE = {Preprint},
MONTH = {December},
YEAR = {2005},
NOTE = {ArXiv:cond-mat/0512072.},
}
M. Freedman, C. Nayak, and K. Walker :
“Towards universal topological quantum computation in the \( \nu=5/2 \) fractional quantum Hall state Phys. Rev. B
73 : 24
(2006 ),
pp. 245307 .
Abstract
People
BibTeX
The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction \( \nu = 5/2 \), can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry [cond-mat/0512072].
@article {key51905847,
AUTHOR = {Freedman, M. and Nayak, C. and Walker,
K.},
TITLE = {Towards universal topological quantum
computation in the \$\nu=5/2\$ fractional
quantum {H}all state},
JOURNAL = {Phys. Rev. B},
FJOURNAL = {Physical Review B},
VOLUME = {73},
NUMBER = {24},
YEAR = {2006},
PAGES = {245307},
NOTE = {Available at
http://dx.doi.org/10.1103/PhysRevB.73.245307.},
ISSN = {1098-0121},
}
incollection
M. Freedman, C. Nayak, K. Walker, and Z. Wang :
“On picture \( (2+1) \) -TQFTs 19–106
in
Topology and physics
(Tianjin, China, 27–31 July 2007 ).
Edited by K. Lin, Z. Weng, and W. Zhang .
Nankai Tracts in Mathematics 12 .
World Scientific (Hackensack, NJ ),
2008 .
MR
2503392 Zbl
1168.81024 ArXiv
0806.1926
People
BibTeX
@incollection {key2503392m,
AUTHOR = {Freedman, Michael and Nayak, Chetan
and Walker, Kevin and Wang, Zhenghan},
TITLE = {On picture \$(2+1)\$-{TQFT}s},
BOOKTITLE = {Topology and physics},
EDITOR = {Kevin Lin and Zhenghan Weng and Weiping
Zhang},
SERIES = {Nankai Tracts in Mathematics},
NUMBER = {12},
PUBLISHER = {World Scientific},
ADDRESS = {Hackensack, NJ},
YEAR = {2008},
PAGES = {19--106},
DOI = {10.1142/9789812819116_0002},
NOTE = {(Tianjin, China, 27--31 July 2007).
ArXiv:0806.1926. MR:2503392. Zbl:1168.81024.},
ISBN = {9789812819109},
}
article
L. Fidkowski, M. Freedman, C. Nayak, K. Walker, and Z. Wang :
“From string nets to nonabelions Comm. Math. Phys.
287 : 3
(2009 ),
pp. 805–827 .
MR
2486662 Zbl
1196.82072 ArXiv
cond-mat/0610583
Abstract
People
BibTeX
We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an \( \mathit{SO}(3)_3\times\mathit{SO}(3)_3 \) doubled Chern–Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.
@article {key2486662m,
AUTHOR = {Fidkowski, Lukasz and Freedman, Michael
and Nayak, Chetan and Walker, Kevin
and Wang, Zhenghan},
TITLE = {From string nets to nonabelions},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {287},
NUMBER = {3},
YEAR = {2009},
PAGES = {805--827},
DOI = {10.1007/s00220-009-0757-9},
NOTE = {ArXiv:cond-mat/0610583. MR:2486662.
Zbl:1196.82072.},
ISSN = {0010-3616},
}
article
M. Freedman, M. B. Hastings, C. Nayak, X.-L. Qi, K. Walker, and Z. Wang :
“Projective ribbon permutation statistics: A remnant of non-Abelian braiding in higher dimensions Phys. Rev. B
83 : 11
(2011 ),
pp. 115132 .
ArXiv
1005.0583
Abstract
People
BibTeX
In a recent paper, Teo and Kane Phys. Rev. Lett. 104 046401 (2010) proposed a three-dimensional (3D) model in which the defects support Majorana fermion zero modes. They argued that exchanging and twisting these defects would implement a set \( R \) of unitary transformations on the zero-mode Hilbert space which is a “ghostly” recollection of the action of the braid group on Ising anyons in two dimensions. In this paper, we find the group \( T_{2n} \) , which governs the statistics of these defects by analyzing the topology of the space \( K_{2n} \) of configurations of \( 2n \) defects in a slowly spatially varying gapped free-fermion Hamiltonian: \( T_{2n}\equiv \pi_1(K_{2n}) \) . We find that the group \( T_{2n}=\mathbb{Z}\times T_{2n}^r \) , where the “ribbon permutation group” \( T_{2n}^r \) is a mild enhancement of the permutation group
\[ S_{2n}: T_{2n}^r\equiv \mathbb{Z}_2\rtimes E((\mathbb{Z}_2)^{2n}\rtimes S_{2n}) .\]
Here, \( E((\mathbb{Z}_2)^{2n}\rtimes S_{2n}) \) is the “even part” of \( (\mathbb{Z}_2)^{2n}\rtimes S_{2n} \) , namely, those elements for which the total parity of the element in \( (\mathbb{Z}_2)^{2n} \) added to the parity of the permutation is even. Surprisingly, \( R \) is only a projective representation of \( T_{2n} \) , a possibility proposed by Wilczek [hep-th/9806228]. Thus, Teo and Kane’s defects realize projective ribbon permutation statistics, which we show to be consistent with locality. We extend this phenomenon to other dimensions, codimensions, and symmetry classes. We note that our analysis applies to 3D networks of quantum wires supporting Majorana fermions; thus, these networks are not required to be planar. Because it is an essential input for our calculation, we review the topological classification of gapped free-fermion systems and its relation to Bott periodicity.
@article {key1005.0583a,
AUTHOR = {Freedman, Michael and Hastings, Matthew
B. and Nayak, Chetan and Qi, Xiao-Liang
and Walker, Kevin and Wang, Zhenghan},
TITLE = {Projective ribbon permutation statistics:
{A} remnant of non-{A}belian braiding
in higher dimensions},
JOURNAL = {Phys. Rev. B},
FJOURNAL = {Physical Review B},
VOLUME = {83},
NUMBER = {11},
YEAR = {2011},
PAGES = {115132},
DOI = {10.1103/PhysRevB.83.115132},
NOTE = {ArXiv:1005.0583.},
ISSN = {1098-0121},
}
