article
S. Bryson, M. H. Freedman, Z.-X. He, and Z. Wang :
“Möbius invariance of knot energy Bull. Amer. Math. Soc. (N.S.)
28 : 1
(1993 ),
pp. 99–103 .
MR
1168514 Zbl
0776.57003 ArXiv
math/9301212
Abstract
People
BibTeX
A physically natural potential energy for simple closed curves in \( \mathbb{R}^3 \) is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist and are regular. Finally, the number of knot types with energy less than any constant \( M \) is estimated.
@article {key1168514m,
AUTHOR = {Bryson, Steve and Freedman, Michael
H. and He, Zheng-Xu and Wang, Zhenghan},
TITLE = {M{\"o}bius invariance of knot energy},
JOURNAL = {Bull. Amer. Math. Soc. (N.S.)},
FJOURNAL = {American Mathematical Society. Bulletin.
New Series},
VOLUME = {28},
NUMBER = {1},
YEAR = {1993},
PAGES = {99--103},
DOI = {10.1090/S0273-0979-1993-00348-3},
NOTE = {ArXiv:math/9301212. MR:1168514. Zbl:0776.57003.},
ISSN = {0273-0979},
}
article
M. H. Freedman and Z. Wang :
“\( \mathbf{C}P^ 2 \) -stable theoryMath. Res. Lett.
1 : 1
(1994 ),
pp. 45–48 .
MR
1258488 Zbl
0849.57016
Abstract
People
BibTeX
In the topological category, it is shown that the dimension 4 disk theorem holds without fundamental group restriction after stabilizing with many copies of complex projective space. As corollaries, a stable 4-dimensional surgery theorem and a stable 5-dimensional \( s \) -cobordism are obtained. These results contrast with the smooth category where the usefulness of adding \( \mathbb{C}P^2 \) ’s depends on chirality.
@article {key1258488m,
AUTHOR = {Freedman, Michael H. and Wang, Zhenghan},
TITLE = {\$\mathbf{C}P^2\$-stable theory},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {1},
NUMBER = {1},
YEAR = {1994},
PAGES = {45--48},
NOTE = {MR:1258488. Zbl:0849.57016.},
ISSN = {1073-2780},
}
article
M. H. Freedman, Z.-X. He, and Z. Wang :
“Möbius energy of knots and unknots Ann. of Math. (2)
139 : 1
(1994 ),
pp. 1–50 .
MR
1259363 Zbl
0817.57011
People
BibTeX
@article {key1259363m,
AUTHOR = {Freedman, Michael H. and He, Zheng-Xu
and Wang, Zhenghan},
TITLE = {M\"obius energy of knots and unknots},
JOURNAL = {Ann. of Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {139},
NUMBER = {1},
YEAR = {1994},
PAGES = {1--50},
DOI = {10.2307/2946626},
NOTE = {MR:1259363. Zbl:0817.57011.},
ISSN = {0003-486X},
}
incollection
M. H. Freedman and Z. Wang :
“Controlled linear algebra ,”
pp. 138–156
in
Prospects in topology: Proceedings of a conference in honor of William Browder
(Princeton, NJ, March 1994 ).
Edited by F. Quinn .
Annals of Mathematics Studies 138 .
Princeton University Press (Princeton, NJ ),
1995 .
MR
1368657 Zbl
0924.19003
Abstract
People
BibTeX
These notes derive from lectures given at UCSD in 1990. The purpose of those lectures was to elucidate the two main theorems of controlled linear algebra: the vanishing theorem for Whitehead and \( K_0 \) -type obstructions when only “simply connected directions” are left uncontrolled and the squeezing principle — that once a threshold level of geometric control is obtained any finer amount of control is also available. These notes present in detail (and with perhaps some new estimates on the optimal relations between \( \varepsilon \) , \( \delta \) and dimension) the vanishing theorem of Frank Quinn’s Section 8 [1979] in the context of Whitehead torsion.
@incollection {key1368657m,
AUTHOR = {Freedman, Michael H. and Wang, Zhenghan},
TITLE = {Controlled linear algebra},
BOOKTITLE = {Prospects in topology: {P}roceedings
of a conference in honor of {W}illiam
{B}rowder},
EDITOR = {Quinn, Frank},
SERIES = {Annals of Mathematics Studies},
NUMBER = {138},
PUBLISHER = {Princeton University Press},
ADDRESS = {Princeton, NJ},
YEAR = {1995},
PAGES = {138--156},
NOTE = {(Princeton, NJ, March 1994). MR:1368657.
Zbl:0924.19003.},
ISBN = {9780691027289},
}
article
M. H. Freedman, M. J. Larsen, and Z. Wang :
“The two-eigenvalue problem and density of Jones representation of braid groups Comm. Math. Phys.
228 : 1
(2002 ),
pp. 177–199 .
MR
1911253 Zbl
1045.20027
People
BibTeX
@article {key1911253m,
AUTHOR = {Freedman, Michael H. and Larsen, Michael
J. and Wang, Zhenghan},
TITLE = {The two-eigenvalue problem and density
of {J}ones representation of braid groups},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {228},
NUMBER = {1},
YEAR = {2002},
PAGES = {177--199},
DOI = {10.1007/s002200200636},
NOTE = {MR:1911253. Zbl:1045.20027.},
ISSN = {0010-3616},
}
article
M. H. Freedman, M. Larsen, and Z. Wang :
“A modular functor which is universal for quantum computation Comm. Math. Phys.
227 : 3
(2002 ),
pp. 605–622 .
MR
1910833 Zbl
1012.81007 ArXiv
quant-ph/0001108
Abstract
People
BibTeX
@article {key1910833m,
AUTHOR = {Freedman, Michael H. and Larsen, Michael
and Wang, Zhenghan},
TITLE = {A modular functor which is universal
for quantum computation},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {227},
NUMBER = {3},
YEAR = {2002},
PAGES = {605--622},
DOI = {10.1007/s002200200645},
NOTE = {ArXiv:quant-ph/0001108. MR:1910833.
Zbl:1012.81007.},
ISSN = {0010-3616},
}
article
M. H. Freedman, A. Kitaev, and Z. Wang :
“Simulation of topological field theories by quantum computers Comm. Math. Phys.
227 : 3
(2002 ),
pp. 587–603 .
MR
1910832 Zbl
1014.81006 ArXiv
quant-ph/0001071
Abstract
People
BibTeX
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian \( H \) for a time \( t \) . In contrast to this quantum engineering, the most
abstract reaches of theoretical physics has spawned “topological models” having a finite
dimensional internal state space with no natural tensor product structure and in which
the evolution of the state is discrete, \( H \equiv 0 \) . These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold:
TQFTs cannot be used to define a model of computation stronger than the usual
quantum model “BQP”.
TQFTs provide a radically different way of looking at quantum computation. The
rich mathematical structure of TQFTs might suggest a new quantum algorithm.
@article {key1910832m,
AUTHOR = {Freedman, Michael H. and Kitaev, Alexei
and Wang, Zhenghan},
TITLE = {Simulation of topological field theories
by quantum computers},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {227},
NUMBER = {3},
YEAR = {2002},
PAGES = {587--603},
DOI = {10.1007/s002200200635},
NOTE = {ArXiv:quant-ph/0001071. MR:1910832.
Zbl:1014.81006.},
ISSN = {0010-3616},
}
article
M. H. Freedman, K. Walker, and Z. Wang :
“Quantum \( \mathit{SU}(2) \) faithfully detects mapping class groups modulo center Geom. Topol.
6
(2002 ),
pp. 523–539 .
MR
1943758 Zbl
1037.57024 ArXiv
math.GT/0209150
Abstract
People
BibTeX
The Jones–Witten theory gives rise to representations of the (extended) mapping class group of any closed surface \( Y \) indexed by a semi-simple Lie group \( G \) and a level \( k \) . In the case \( G = \mathit{SU}(2) \) these representations (denoted \( V_A(Y) \) ) have a particularly simple description in terms of the Kauffman skein modules with parameter \( A \) a primitive \( 4r \) -th root of unity (\( r = k + 2 \) ). In each of these representations (as well as the general \( G \) case), Dehn twists act as transformations of finite order, so none represents the mapping class group \( \mathcal{M}(Y) \) faithfully. However, taken together, the quantum \( \mathit{SU}(2) \) representations are faithful on non-central elements of \( \mathcal{M}(Y) \) . (Note that \( \mathcal{M}(Y) \) has non-trivial center only if \( Y \) is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2 punctures, or the closed surface of genus \( = 2 \) .) Specifically, for a non-central \( h\in\mathcal{M}(Y) \) there is an \( r_0(h) \) such that if \( r \geq r_0(h) \) and \( A \) is a primitive \( 4r \) -th root of unity then \( h \) acts projectively nontrivially on \( V_A(Y) \) . Jones’ [1987] original representation \( \rho_n \) of the braid groups \( B_n \) , sometimes called the generic \( q \) -analog-\( \mathit{SU}(2) \) -representation, is not known to be faithful. However, we show that any braid \( h \neq \mathrm{id} \in B_n \) admits a cabling \( c = c_1,\dots,c_n \) so that \( \rho_N(c(h)) \neq\mathrm{id} \) , \( N = c_1 + \dots +c_n \) .
@article {key1943758m,
AUTHOR = {Freedman, Michael H. and Walker, Kevin
and Wang, Zhenghan},
TITLE = {Quantum \$\mathit{SU}(2)\$ faithfully
detects mapping class groups modulo
center},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry \& Topology},
VOLUME = {6},
YEAR = {2002},
PAGES = {523--539},
DOI = {10.2140/gt.2002.6.523},
NOTE = {ArXiv:math.GT/0209150. MR:1943758.
Zbl:1037.57024.},
ISSN = {1465-3060},
}
article
M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang :
“Topological quantum computation Bull. Amer. Math. Soc. (N.S.)
40 : 1
(2003 ),
pp. 31–38 .
MR
1943131 Zbl
1019.81008 ArXiv
quant-ph/0101025
Abstract
People
BibTeX
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten–Chern–Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and \( 2D \) -magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like \( e^{-\alpha l} \) , where \( l \) is a length scale, and \( \alpha \) is some positive constant. In contrast, the “presumptive” qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about \( 10^{-4} \) ) before computation can be stabilized.
@article {key1943131m,
AUTHOR = {Freedman, Michael H. and Kitaev, Alexei
and Larsen, Michael J. and Wang, Zhenghan},
TITLE = {Topological quantum computation},
JOURNAL = {Bull. Amer. Math. Soc. (N.S.)},
FJOURNAL = {American Mathematical Society. Bulletin.
New Series},
VOLUME = {40},
NUMBER = {1},
YEAR = {2003},
PAGES = {31--38},
DOI = {10.1090/S0273-0979-02-00964-3},
NOTE = {ArXiv:quant-ph/0101025. MR:1943131.
Zbl:1019.81008.},
ISSN = {0273-0979},
}
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},
}
article
M. H. Freedman, A. Kitaev, C. Nayak, J. K. Slingerland, K. Walker, and Z. Wang :
“Universal manifold pairings and positivity Geom. Topol.
9
(2005 ),
pp. 2303–2317 .
MR
2209373 Zbl
1129.57035 ArXiv
math/0503054
Abstract
People
BibTeX
Gluing two manifolds \( M_1 \) and \( M_2 \) with a common boundary \( S \) yields a closed manifold \( M \) . Extending to formal linear combinations \( x = \sum a_iM_i \) yields a sesquilinear pairing \( p = \langle\,\cdot\,,\cdot\,\rangle \) with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing \( p \) onto a finite dimensional quotient pairing \( q \) with values in \( \mathbb{C} \) which in physically motivated cases is positive definite. To see if such a “unitary” TQFT can potentially detect any nontrivial \( x \) , we ask if \( \langle x,x\rangle\neq 0 \) whenever \( x\neq 0 \) . If this is the case, we call the pairing \( p \) positive. The question arises for each dimension \( d = 0,1,2,\dots\, \)
We find \( p(d) \) positive for \( d = 0,1 \) , and 2 and not positive for \( d = 4 \) . We conjecture that \( p(3) \) is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The results in dimension 4 imply that unitary TQFTs cannot distinguish homotopy equivalent simply connected 4-manifolds, nor can they distinguish smoothly \( s \) -cobordant 4-manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for \( d = 3 + 1 \) . There is a further physical implication of this paper. Whereas 3-dimensional Chern–Simons theory appears to be well-encoded within 2-dimensional quantum physics, e.g. in the fractional quantum Hall effect, Donaldson–Seiberg–Witten theory cannot be captured by a 3-dimensional quantum system. The positivity of the physical Hilbert spaces means they cannot see null vectors of the universal pairing; such vectors must map to zero.
@article {key2209373m,
AUTHOR = {Freedman, Michael H. and Kitaev, Alexei
and Nayak, Chetan and Slingerland, Johannes
K. and Walker, Kevin and Wang, Zhenghan},
TITLE = {Universal manifold pairings and positivity},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry \& Topology},
VOLUME = {9},
YEAR = {2005},
PAGES = {2303--2317},
DOI = {10.2140/gt.2005.9.2303},
NOTE = {ArXiv:math/0503054. MR:2209373. Zbl:1129.57035.},
ISSN = {1465-3060},
}
article
M. Freedman, A. Feiguin, S. Trebst, A. Ludwig, M. Troyer, A. Kitaev, and Z. Wang :
“Interacting anyons in topological quantum liquids: The golden chain Phys. Rev. Lett.
98
(2007 ),
pp. 160409 .
ArXiv
cond-mat/0612341
Abstract
People
BibTeX
We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (“identity”) channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent \( z=1 \) , and described by a two-dimensional (\( 2D \) ) conformal field theory with central charge \( c=7/10 \) . An exact mapping of the anyonic chain onto the \( 2D \) tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley–Lieb algebra. The gaplessness of the chain is shown to have topological origin.
@article {keycond-mat/0612341a,
AUTHOR = {Freedman, M. and Feiguin, A. and Trebst,
S. and Ludwig, A. and Troyer, M. and
Kitaev, A. and Wang, Z.},
TITLE = {Interacting anyons in topological quantum
liquids: {T}he golden chain},
JOURNAL = {Phys. Rev. Lett.},
FJOURNAL = {Physical Review Letters},
VOLUME = {98},
YEAR = {2007},
PAGES = {160409},
DOI = {10.1103/PhysRevLett.98.160409},
NOTE = {ArXiv:cond-mat/0612341.},
ISSN = {0031-9007},
}
article
M. H. Freedman and Z. Wang :
“Large quantum Fourier transforms are never exactly realized by braiding conformal blocks Phys. Rev. A (3)
75 : 3
(2007 ),
pp. 032322 .
MR
2312110 ArXiv
cond-mat/0609411
Abstract
People
BibTeX
Fourier transform is an essential ingredient in Shor’s factoring algorithm. In the standard quantum circuit model with the gate set \( \{U(2) \) , controlled-NOT\( \} \) , the discrete Fourier transforms \( F_N=(\omega^{ij})_{N\times N} \) for \( i,j=0,1,\dots,N{-}1 \) and \( \omega=e^{2\pi i/N} \) can be realized exactly by quantum circuits of size \( O(n^2) \) with \( n=\ln N \) , and so can the discrete sine or cosine transforms. In topological quantum computing, the simplest universal topological quantum computer is based on the Fibonacci \( (2+1) \) -topological quantum field theory (TQFT), where the standard quantum circuits are replaced by unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms \( F_N \) and the discrete sine or cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that an approximation is unavoidable in the implementation of Fourier transforms by braiding conformal blocks.
@article {key2312110m,
AUTHOR = {Freedman, Michael H. and Wang, Zhenghan},
TITLE = {Large quantum {F}ourier transforms are
never exactly realized by braiding conformal
blocks},
JOURNAL = {Phys. Rev. A (3)},
FJOURNAL = {Physical Review. A. Third Series},
VOLUME = {75},
NUMBER = {3},
YEAR = {2007},
PAGES = {032322},
DOI = {10.1103/PhysRevA.75.032322},
NOTE = {ArXiv:cond-mat/0609411. MR:2312110.},
ISSN = {1050-2947},
}
incollection
M. Freedman :
“Freedman’s writings on Lin ,”
pp. 430–431
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 .
Several letters written by Freedman to Lin.
MR
2489601
Abstract
People
BibTeX
Xiao-Song studied under Michael Freedman from 1984–1988 at UCSD. In September 2006, Freedman wrote Xiao-Song on the inside cover of the Soviet book, How the Steel Was Tempered , to encourage him. Then in December 2006, he wrote Xiao-Song again during the critical juncture of Xiao-Song’s life. These are the letters.
@incollection {key2489601m,
AUTHOR = {Freedman, Michael},
TITLE = {Freedman's writings on {L}in},
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 = {430--431},
NOTE = {(Tianjin, China, 27--31 July 2007).
Several letters written by Freedman
to Lin. MR:2489601.},
ISBN = {9789812819109},
}
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},
}
