Paul Fendley
Work with David Aasen and Roger Mong (2016,2020), � and with Luisa Eck (2023, 2024)
All Souls College, Oxford
Topological defects generate categorical/topological/higher/non-invertible �“symmetries” and “dualities”
Working in a Hilbert-space formalism, the defect-line creation operators necessarily commute with the Hamiltonian/transfer matrix.
Higher-dimensional analogs generate ``higher symmetries”.
Symmetries and dualities are in quotes because these operators are in general non-unitary. Sometimes they are not even invertible.
The partition function in the presence of topological defects is independent of local deformations of the defects, both on the lattice and in the continuum.
Just need to keep track of how they branch/fuse, and how they wrap around cycles. Schematically:
Outline
The setup: Statistical mechanics from (braided) fusion categories
Use #1: Many many generalizations of Kramers-Wannier duality
Use #2: Exact degeneracies for ground states and for kinks
Use #3: Non-obvious non-invertible mappings � (from XXZ to the Rydberg-blockade ladder)
Use #4: Exact lattice computation of g factor ratios and conformal spins
Including braiding gives
Use #5: Baxterization via terminating defect lines
Use #6: 2+1d models for chiral topological order
See Luisa Eck’s poster
Finding lattice topological defects
People found the duality defects in Ising by brute force, and that’s not easy.
Beyond a few other simple examples, it’s very difficult to get much further.
The moral of the story is: draw pictures!
Our work exploits the fact that many interesting statistical mechanical models are conveniently described in terms of graphs and the corresponding knot and link invariants.
A fusion category gives a set of rules that allow for associating isotopy invariants to graphs. We showed how exact topological defects in these models can be found by using data from a fusion category.
Feiguin et al 2006; Aasen-Fendley-Mong 2020
Temperley-Lieb 71; Fortuin-Kasteleyn 71;�Baxter-Kelland-Wu 76; Jones ’80s
A tensor category gives a set of rules to turn a �fusion diagram into a number, an isotopy invariant
Braiding and fusing must obey consistency conditions, such as the Reidemeister moves of knot theory, the statistical properties of fused anyons, the bootstrap equations of rational conformal field theory…
The category allows e.g. Boltzmann weights to be defined precisely via evaluating a fusion diagram, a labelled trivalent graph. To compute the numbers, the rules allow various manipulations of the graphs to relate the topological invariants, e.g
Evaluating a fusion diagram
t'
b
s
r
a
Category is a list of objects: a,b,c,… obeying fusion rules
t
b
s
r
a
F move:
bubble removal:
s
b
r
a
b
F moves preserve the evaluations. Doing them and bubble removal repeatedly�express a fusion diagram as a sum over closed loops, and hence a number.
quantum dimensions
F symbols:
Data:
Non-negative integers
Local statistical-mechanical models
Boltzmann weight depends on interactions among the four heights ``round a face’’
For nearest-neighbor Ising, include degrees of freedom on half the sites:
Discrete degrees of freedom called ``heights’’ living on the sites of a square lattice:
Heights are objects in category, and allowed nearest neighbours obey fusion rules. Hamiltonian/Boltzmann weights are constructed from projection operators in category.
a
b
a
b
The fusion rules are a truncated version of SU(2), with objects
Nearest-neighbours obey fusion with spin 1
0
1
2
Label each site by object so that nearest neighbours obey these spin-1/2 fusion rules
Ising/
Fusion with spin ½ :
0
1
,
FIbonacci/hard squares/ golden chain/
0
1
Rydberg-blockade ladder/ Rep(
Inserting a defect
The defects have a weight depending on the adjacent heights:
In the presence of the defect, the partition function is modified to
For to be invariant under deformations of the defect’s path:
Any lattice model built from category possesses solutions!
Identities between partition functions with different defect configuations
By either the abstract manipulations or using the explicit defect weights, find
Use commutation relations to move defect around. For any path, partition functions on a disc are related as
In a category,
c
δ
d
δ
γ
γ
Two types of topological defects in Ising
spin-flip defect:
For Ising, fluctuating degrees of freedom on only half the sites:
duality defect:
Coupings on one side of duality defect are dual values of those on the other!
Useful application #1: Generalised KW duality
Micro to macro: category tells you how defects fuse:
=
Identity defect
spin-flip defect
+
In Ising category (and CFT):
KW duality is not a symmetry – even at the critical point, it changes boundary conditions.
(At Ising critical point, in the Hamiltonian limit, with periodic b.c., you can mix KW duality with translation symmetry to get an actual symmetry.)
``Topological” symmetry for Fibonacci category is actually a symmetry (but is invertible)
=
+
Feiguin et al 2006, Aasen-Fendley-Mong 2020
Seiberg and Shao 2023
Many well-known models fit in this framework
Seems likely there exists at least at least one (often more than one) such lattice model for each RCFT.
Category of lattice model is typically a subcategory of the category describing RCFT.
Important to emphasise that generically, these lattice models are neither critical, nor integrable, even with spatially uniform couplings. Very rich phase diagrams, e.g.
Any ``categorical” lattice has a family of topological defects!
O’Brien & Fendley 2019
Use #2: Deriving exact degeneracies
A certain integrable massive field theory has effective potential with two asymmetric degenerate minima. Initial observation numerical, then from quantum-group algebra.
Lassig-Mussardo-Cardy 90;�Zamolodchikov 90; Smirnov 91
In the corresponding non-integrable spin chain (staggered ``golden chain’’), this degeneracy follows immediately from the existence of a topological defect – no detailed analysis is required.
Aasen-Fendley-Mong 2020
FIbonacci/hard squares/golden chain/
Nearest-neighbour spins obey fusion with spin 1:
0
1
Along a line, can’t have two zeroes in a row, e.g.
Fibonacci symmetry commutes with each individually! Take
get ground states exactly:
Hamiltonian chosen to commute with symmetry
Exact degeneracy must be preserved throughout phase!
and
where
These two g.s. are related by but no ordinary symmetry!
Rydberg blockade
x
Kinks interpolating between the two minima
Kink/antikink bound state only in shallow well:
x
The kink and the bound state are degenerate!
Low-lying states in the potential are comprised of:
In spin chain,
Still more:
Use #3: Non-invertible, non-obvious mappings, e.g.
The Hamiltonians obey e.g.
Integrable Rydberg-blockade ladder
Eck & Fendley 2023; see also Lootens et al
3-state antiferromagnet
XXZ chain
Ising zig-zag ladder
Rep(
The maps obey e.g.
0
1
2
1 2 1 1 0 1 0 1 1 2 1 2 1 1 1 2
Interpret as a ladder: 0 is particle on top rung, 2 on bottom, 1 is empty rung
Very well-understood phase diagram of XXZ:
Δ
-1
1
KT transition
Exact ferromagnetic� ground states
Antiferromagnetic� broken order
Critical phase
Free-boson CFT
Integrable Rydberg ladder:
Δ
Exact ground states�– e e – e e – e e – e e – e
Three-phase coexistence
Orbifold CFT
Ground states as
e e e e e e e e e e �e + e + e + e + e + �+ e + e + e + e + e
A non-invertible symmetry guarantees three ground states for all Δ > 1 !
Easiest way to implement this symmetry is by transforming to 3-state antiferromagnet,�do ordinary transformation, then transform back. Sheep in wolf’s clothing?
Another non-invertible symmetry is remnant of U(1) from XXZ. Survives away from this�integrable line!
Eck & Fendley 2023
Use #4: Exact g-factor ratios from lattice
Consider vector space of all configurations of spins/heights near edge. Each vector corresponds to a boundary condition, e.g.
Acting with absorbs the defect into edge, changing
e.g. duality defect in Ising changes b.c. as
where dual coupling is defined by
Exact!!
Aasen-Fendley-Mong 2020
For conformal boundary conditions, this ratio of partition functions is
where is the boundary free energy. This Affleck-Ludwig g-factor is universal, and computable in RCFT.
Calculation is direct and easy here! Moreover, gives precise lattice expressions for boundary states. Provides precise way of comparing lattice and CFT.
Another way to compare is to compute Dehn twist (i.e. modular transformation) in the presence of twisted boundary conditions (vertical topological defect). Yields conformal dimensions up to half-integer shift!
Use #5: Baxterization from fractional-spin� conserved currents
Conclusions
Defining the currents by terminating a defect
Choose an object so that there is a vertex
ρ
ρ
φ
eval
Current is non-local: need braiding for string to go over intervening edges.
w
z
ρ
ρ
φ
Independent of path except for
ρ
ρ
φ
h
1
2
h
h’
1
Conserved-current relation in a braided tensor category
u
u
u
u
=
+ μ
+ μ
Cardy et al
Smirnov et al
In all known cases, the weights solving the linear equation also solve the YBE!
u
u’
u+u’
=
u
u’
u+u’
All the data save μ and the amplitudes are specified by the category
Solving the conserved-current relation
χ
u
χ
φ
χ
φ
Plug
into each of the terms
Then manipulate
to a common form:
a
χ
φ
χ
φ
a
χ
ρ
ρ
ρ
ρ
χ
χ
a
χ
a
Solving the conserved-current relation
u
u
u
u
=
+ μ
+ μ
χ
a
Find all F symbols cancel, leaving only twist factors and
for all
such that