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International Journal of Modern Physics A
Vol. 31, Nos. 28 & 29 (2016) 1645044 (38 pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0217751X16450445
Spin TQFTs and fermionic phases of matter
Davide Gaiotto
Perimeter Institute for Theoretical Physics,
Waterloo, Ontario, Canada N2L 2Y5
Anton Kapustin
Simons Center for Geometry and Physics,
Stony Brook, NY 11790
Published 19 October 2016
We study lattice constructions of gapped fermionic phases of matter. We show that
the construction of fermionic Symmetry Protected Topological orders by Gu and Wen
has a hidden dependence on a discrete spin structure on the Euclidean space-time. The
spin structure is needed to resolve ambiguities which are otherwise present. An identical
ambiguity is shown to arise in the fermionic analog of the string-net construction of
2D topological orders. We argue that the need for a spin structure is a general feature
of lattice models with local fermionic degrees of freedom and is a lattice analog of the
spin-statistics relation.
1. Introduction and Summary
1.1. Bosonic and fermionic gapped phases
In condensed matter physics, topological phases of matter are often defined as
equivalence classes of local gapped bosonic Hamiltonians, usually defined on a lat-
tice, which can be deformed into each other without ever becoming gapless.
1, 2
The
notion of topological phase can be enriched by imposing additional constraints on
the theories, such as a choice of global symmetry preserved by all the Hamiltoni-
ans. On the other hand, topological quantum field theories
a
can be thought of as
describing the far infrared behavior of gapped bosonic quantum field theories (see,
e.g., Section 4 of Ref. 3 or the monograph
4
).
There is a close relation between topological phases of matter and topological
quantum field theories, which can be thought of as a map from a topological phase
a
This is a somewhat looser notion of TQFT compared to some formal definitions. For example,
we consider Chern-Simons theory to be a TQFT, even though it has a partition function which
depends on a choice of metric on space-time. In other words, we allow the stress tensor to be
non-zero, but proportional to the identity operator.
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D. Gaiotto & A. Kapustin
of matter to the TQFT which encodes the low energy continuum limit of the corre-
sponding Hamiltonian. In principle, one may imagine the map being many-to-one:
it is not obvious that two local gapped Hamiltonians which map to the same TQFT
will always be deformable into each other. Still, in practice we do not know of any
observable which can distinguish two phases of matter, but cannot be formulated
in terms of the TQFT data.
b
In condensed matter physics, one also encounters the notion of a fermionic topo-
logical phase of matter, defined as an equivalence class of local gapped Hamiltoni-
ans which can involve fermionic degrees of freedom.
5, 6
Perhaps surprisingly, some
fermionic phases of matter are not expected to admit a purely bosonic realization.
This is expected to be due to the difference in the notion of locality for bosonic
and fermionic systems. Intuitively, if we partition a bosonic system in two parts,
the total Hilbert space factors uniquely in the tensor product of the Hilbert spaces
for the two parts. If we partition a fermionic system, though, the factorization has
an intrinsic ambiguity, as observables in the tensor product of the Hilbert spaces
for the two parts are defined up to a sign in the sector where both factors have odd
fermion number.
1.2. Spin structure dependence
In unitary quantum field theory, fermions are naturally spinors and thus the low
energy physics of a gapped fermionic theory is a spin-TQFT: a topological field
theory defined on manifolds which can be equipped with a spin structure, whose
correlation functions possibly depend on the choice of spin structure. The purpose of
this paper is to explore the relation between fermionic topological phases of matter
and spin-TQFTs. It is not obvious that such a relation should exist, as a lattice
Hamiltonian involving fermionic degrees of freedom is usually written down without
any reference to a spin structure on the manifold which is discretized by the lattice.
One also cannot appeal to the spin-statistics relation, because the lattice destroys
Lorenz and even rotational invariance which are the conditions of the spin-statistics
theorem.
c
The first step of our analysis is to look carefully at the fermionic SPT phases
constructed by Gu and Wen in Ref. 7. We find that the prescription used to de-
fine the partition function of such theories runs into an obstruction if applied to
b
It is also conceivable, perhaps, that some topological phase of matter may not give rise to a
TQFT at low energy, i.e., that some anomaly/obstruction may prevent the definition of TQFT
amplitudes on general manifolds in terms of the Hamiltonian data. But in all cases known to
us one circumvent such obstructions by postulating that the TQFT depends on some additional
geometric data, such as metric or framing.
c
Taking the continuum limit and then applying the spin-statistics relation does not ameliorate
the problem. The TQFT itself is, of course, Lorentz invariant, but the spin-statistics relation is a
property of Lorenz invariant particle excitations. The continuum limit from the lattice theory to
the low-energy TQFT only concerns the ground states of the system. A priori, massive excitations
above these ground states do not need to transform properly under the Lorentz group.
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Spin TQFTs and fermionic phases of matter
space-time manifolds of general topology, unless the second Stiefel–Whitney class
[w
2
] of the manifolds vanishes, i.e., the manifold admits a spin structure. If the
manifolds admits a spin structure, the obstruction can be eliminated, but the final
answer will depend on the choice of spin structure η. In other words, these fermionic
SPT phases define (invertible) spin-TQFTs.
Next, we look at other known constructions of fermionic phases of matter which
are expected to admit a state-sum-like definition of their partition function: the
construction of fermionic toric code in Ref. 8 and the general fermionic Turaev-Viro
construction in Ref. 9. These references focus on the construction of a fixed-point
Hamiltonian and wave-function for these fermionic phases of matter, rather than a
partition function. It is straightforward, though, to assemble the same ingredients
into a partition sum, borrowing some ideas from the Gu–Wen fermionic SPT phase
construction. Again, we find an obstruction to define the partition sum unless the
space-time manifolds admits a spin structure, in which case one can remove the
obstruction and define a well defined partition function which depends on the choice
of spin structure η. Thus these fermionic phases of matter are associated to spin-
TQFTs.
We can describe the obstruction schematically here, referring the reader to
Secs. 2 and 6 for further details. State-sum models assemble the partition func-
tion from a triangulation of the space-time manifold X: each simplex is associated
to some tensor in the tensor product of vector spaces associated to the faces and
the legs of these tensors are contracted together as the simplices are glued along the
corresponding faces of the triangulation. In a fermionic model, the vector spaces
may be Grassmann-odd and Koszul signs occur when re-organizing and contracting
the factors of the tensor products.
These Koszul signs, arising from the anti-commutation of fermionic variables,
are of course a key element of the problem. The non-local nature of these signs is
precisely what should allow these fermionic phases of matter to be distinct from any
bosonic phase. In order for the partition sum to be invariant under local changes
in the triangulation of the manifold, one needs to cancel the change in the Koszul
signs against the change in the local data attached to the simplices. The obstruction
arises precisely when this cancellation is not possible.
We can express the obstruction neatly by encoding the fermion number of the
vector spaces attached to faces in a Z
2
-valued (d − 1)-cochain β
d−1
. The cochain
β
d−1
is actually a cocycle, as the total fermion number of the tensors attached
to simplices is even. It is useful to decompose the partition sum into a sum of
terms Z[X, β
d−1
], which contain the parts of the state sum due to states of fermion
number β
d−1
.
We can encode a general change of triangulation of X into a triangulation
of the (d + 1)-dimensional manifold X × [0, 1]. Intuitively, we are gluing a se-
quence of (d + 1)-dimensional simplices on top of our initial triangulation to get
the final triangulation. We find that the triangulation invariance of the partition
function is obstructed by some irreducible sign mismatch, which can be written
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D. Gaiotto & A. Kapustin
schematically as
(−1)
R
X×[0,1]
w
2
∪β
d−1
. (1)
Here w
2
is a 2-cocycle with values in Z
2
representing the second Stiefel–Whitney
class of X×[0, 1] and β
d−1
is a lift to X×[0, 1] of the cocycle β
d−1
. If the cohomology
class of [w
2
] is non-trivial and the theory involves choices of fermion numbers β
d−1
which are non-trivial in cohomology, this sign mismatch cannot be absorbed by
a redefinition of the local part of the partition function. This prevents us from
constructing a well-defined partition sum and ruins the state-sum construction.
If we restrict X to be a spin manifold then w
2
(X) is exact and we can write
w
2
= δη for some 1-cochain η, which represents a choice of spin structure. This
allows us to thus cancel the obstruction (1) by the variation of a local term,
(−1)
R
X
η∪β
d−1
, (2)
so that the improved state sum,
Z[X, η] =
X
β
d−1
Z[X, β
d−1
](−1)
R
X
η∪β
d−1
, (3)
is fully invariant under changes of triangulations and defines a good theory. This
theory is a spin-TQFT: it can only be defined on a spin manifold and depends on
a choice of spin structure.
In Secs. 3 and 4 we will look in further detail at the properties of the Koszul
signs which occur in the state sum. The definition the partition function requires
specific choices of how to order the factors in the tensor product associated to each
simplex, and the two factors in the contraction of vector spaces at each face. Given
some ordering choices Π, the permutations of the vector spaces involved in the state
sum will produce some overall Koszul sign σ
Π
(X, β
d−1
), which depends only on the
triangulation, on Π and on β
d−1
.
The choice of order Π can be given independently of the other data in the state
sum. The combined sign,
z
Π
[X, η, β
d−1
] = σ
Π
(X, β
d−1
)(−1)
R
X
η∪β
d−1
, (4)
appears to be a very useful object, which captures the intrinsically fermionic part of
the full partition function. From now on we will drop the subscript Π. Our formulae
will refer to the specific choice of order used in the Gu–Wen definition of fermionic
SPT phases.
7
We will comment briefly on other choices of order in Sec. 3.
We can think about z[X, η, β
d−1
] as defining an effective action for a (d−1)-form
Z
2
gauge field with a very specific anomaly, or a very simple invertible spin-TQFT
K
d
equipped with an anomalous (d − 2)-form Z
2
global symmetry.
Under changes of triangulation, z[X, η, β
d−1
] changes by another interesting
cocycle, the Steenrod square of β
d−1
:
(−1)
R
X×[0,1]
Sq
2
[β
d−1
]
≡ (−1)
R
X×[0,1]
β
d−1
∪
d−3
β
d−1
. (5)
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Spin TQFTs and fermionic phases of matter
We refer to Appendix B for the explicit definition of the higher cup products ∪
a
.
Their basic property is
A ∪
a
B + B ∪
a
A = δ(A ∪
a+1
B) + δA ∪
a+1
B + A ∪
a+1
δB , (6)
with ∪
0
≡ ∪.
Under gauge transformations, we find the precise form of the ’t Hooft anomaly,
z[X, η, β
d−1
+ δλ
d−2
] = z[X, η, β
d−1
](−1)
R
X
β∪
d−3
λ+λ∪
d−3
β+λ∪
d−3
δλ+λ∪
d−4
λ
. (7)
Although z[X, η, β
d−1
] does not appear to admit a d-dimensional bosonic de-
scription, we also find that it is a quadratic refinement of a bosonic pairing:
z[X, η, β
d−1
+ β
0
d−1
] = z[X, η, β
d−1
]z[X, η, β
0
d−1
](−1)
R
X
β
d−1
∪
d−2
β
0
d−1
. (8)
Finally, if X is a boundary of a compact oriented (d + 1)-manifold Y and d > 2, we
find an explicit WZW-like expression for z[X, η, β
d−1
]:
z[X, η, β
d−1
] = (−1)
R
X
η∪β
b−1
+
R
Y
Sq
2
[β
d−1
]+w
2
∪β
d−1
. (9)
Here we use the fact that for d > 2 the cocycle β
d−1
can be extended to Y . The
action is independent of the choice of Y or of the way β
d−1
is extended from X too
Y because the expression Sq
2
[β
d−1
] + w
2
∪ β
d−1
is exact for closed oriented Y. This
formula is particularly useful for d = 3, since any closed oriented 3-manifold X is a
boundary of a compact oriented 4-manifold Y .
With a bit of extra work, we can rewrite the partition function Z[X, η] of our
spin-TQFT as the partition function of a (d − 1)-form Z
2
gauge theory,
Z[X, η] =
X
β
d−1
˜
Z[X, β
d−1
]z[X, η, β
d−1
] , (10)
where the gauge fields are coupled to two sets of degrees of freedom: a standard
bosonic TQFT equipped with a (d − 2)-form Z
2
global symmetry and partition
function
˜
Z[X, β
d−1
], and the spin-TQFT K
d
. The bosonic theory associated to
˜
Z[X, β
d−1
] must have a ’t Hooft anomaly which cancels the ’t Hooft anomaly of
K
d
, controlled by Sq
2
[β
d−1
].
In order to make contact with concepts which are more familiar in condensed
matter physics, it is useful to replace the notion of a TQFT with an anomalous
global symmetry with the notion of a gapped boundary condition for a (d + 1)-
dimensional SPT phase, protected by a (d − 2)-form Z
2
global symmetry, with
partition function
(−1)
R
Y
Sq
2
[β
d−1
]
. (11)
Then
˜
Z[X, β
d−1
] defines a bosonic gapped boundary condition for the (d + 1)-
dimensional SPT phase, while z[X, η, β
d−1
] defines a fermionic gapped boundary
condition. The original spin TQFT can be recovered by gauging the (d − 2)-form
Z
2
global symmetry on a slab, with one of these boundary conditions at either end.
In Sec. 7 we will argue that this construction has a close relation to the notion
of fermionic anyon condensation. The generators of a non-anomalous (d − 2)-form
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