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. 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},
}
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 ,”
pp. 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
D. Calegari, M. H. Freedman, and K. Walker :
“Positivity of the universal pairing in 3 dimensions ,”
J. Amer. Math. Soc.
23 : 1
(2010 ),
pp. 107–188 .
MR
2552250
Zbl
1201.57024
ArXiv
0802.3208

Abstract
People
BibTeX

Associated to a closed, oriented surface \( S \) is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along \( S \) defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is that this pairing is positive , i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary \( (2+1) \) -dimensional TQFTs.

The proof involves the construction of a suitable complexity function \( c \) on all closed 3-manifolds, satisfying a gluing axiom which we call the topological Cauchy–Schwarz inequality , namely that
\[ c(AB)\le \max(c(AA),c(BB)) \]
for all \( A,B \) which bound \( S \) , with equality if and only if \( A=B \) .

The complexity function \( c \) involves input from many aspects of 3-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite-volume hyperbolic 3-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3-manifolds due to Agol–Storm–Thurston.

@article {key2552250m,
AUTHOR = {Calegari, Danny and Freedman, Michael
H. and Walker, Kevin},
TITLE = {Positivity of the universal pairing
in {3} dimensions},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical
Society},
VOLUME = {23},
NUMBER = {1},
YEAR = {2010},
PAGES = {107--188},
DOI = {10.1090/S0894-0347-09-00642-0},
NOTE = {ArXiv:0802.3208. MR:2552250. Zbl:1201.57024.},
ISSN = {0894-0347},
}
article
M. Freedman, R. Gompf, S. Morrison, and K. Walker :
“Man and machine thinking about the smooth 4-dimensional Poincaré conjecture ,”
Quantum Topol.
1 : 2
(2010 ),
pp. 171–208 .
MR
2657647
Zbl
1236.57043

Abstract
People
BibTeX

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincaré conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Rasmussen’s \( s \) -invariant, a slice obstruction within the general framework of Khovanov homology, changes this state of affairs. We studied a class of knots \( K \) for which nonzero \( s(K) \) would yield a counterexample to SPC4. Computations are extremely costly and we had only completed two tests for those \( K \) , with the computations showing that \( s \) was 0, when a landmark posting of Akbulut [2009] altered the terrain. His posting, appearing only six days after our initial posting, proved that the family of “Cappell–Shaneson” homotopy spheres that we had geared up to study were in fact all standard. The method we describe remains viable but will have to be applied to other examples. Akbulut’s work makes SPC4 seem more plausible, and in another section of this paper we explain that SPC4 is equivalent to an appropriate generalization of Property R (“in \( \mathbb{S}^3 \) , only an unknot can yield \( \mathbb{S}^1\times\mathbb{S}^2 \) under surgery”). We hope that this observation, and the rich relations between Property R and ideas such as taut foliations, contact geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to look at SPC4.

@article {key2657647m,
AUTHOR = {Freedman, Michael and Gompf, Robert
and Morrison, Scott and Walker, Kevin},
TITLE = {Man and machine thinking about the smooth
4-dimensional {P}oincar{\'e} conjecture},
JOURNAL = {Quantum Topol.},
FJOURNAL = {Quantum Topology},
VOLUME = {1},
NUMBER = {2},
YEAR = {2010},
PAGES = {171--208},
DOI = {10.4171/QT/5},
NOTE = {MR:2657647. Zbl:1236.57043.},
ISSN = {1663-487X},
}
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},
}