Topological lattice field theories from intertwiner dynamics
Abstract
We introduce a class of 2D lattice models that describe the dynamics of intertwiners, or, in a condensed matter interpretation, the fusion and splitting of anyons. We identify different families and instances of triangulation invariant, that is, topological, models inside this class. These models give examples for symmetry protected topologically ordered 1D quantum phases with quantum group symmetries. Furthermore the models provide realizations for anyon condensation into a new effective vacuum. We explain the relevance of our findings for the problem of identifying the continuum limit of spin foam and spin net models.
1 Introduction
We introduce a class of 2D lattice models, describing the dynamics of (quantum) group intertwiners, and present families of triangulation invariant, that is topological, models inside this class. The models, and the investigations of their fixed point structure under coarse graining, are motivated by a research program to understand the phase diagram and continuum limit of spin foam models [1], which are candidates for a quantum theory of gravity [2, 3]. There are however several additional tantalizing connections to quantum gravity as well as other areas of physics. One is the theory of anyon condensation [4, 5], for which we provide Hamiltonians. The condensate states appear as ground states of these Hamiltonians and are given by the topological models. Finding all possible topological models leads to a classification problem of module categories over the category of representations of a quantum group [6, 7, 8]. This connects to the task of classifying phases for condensed matter [9, 10], for which we here provide a broad class of examples based on quantum group symmetries.
Besides these topics there are other reasons why these models and its fixed points are of interest for quantum gravity:

For the construction of spin foam models itself, in particular the intertwiners defining these models. The fixed points of our models define naturally intertwiners for spin foam vertices of arbitrary valency. Such a connection between fixed points and a consistent construction of spin foam vertices for arbitrary valence has been first pointed out by Reisenberger in [11] in connection with the Barrett–Crane model [12] and will be explored in detail in the forth coming work [13].

The fixed points for the intertwiner models will also determine fixed points for spin net models, introduced and investigated in [14, 15, 16, 1]. The motivation for these models is the construction of analogue models for spin foams. The hope is that statistical properties of 2D spin nets and 4D spin foams are similar. This is based on a similar property for 4D lattice gauge models and related edge (Ising like) 2D models, and the fact that spin foams can be seen as generalized lattice gauge models, [16] and references therein.

The 2D triangulation invariant models constructed in this work also allow for a geometrical interpretation of the underlying variables. Similar to the 3D quantum gravity models these models can be used to study and illustrate conceptual questions such as a notion of diffeomorphism invariance in the discrete [17, 18], uniqueness of such diffeomorphism invariant models, the relation between covariant and canonical formalism and a derivation of Hamiltonian constraints, a dynamical notion of cylindrical consistency [19] and the expansions of theories around different vacua (here fixed points). More precise relations of the mentioned topics to the models we introduce here will emerge in the course of this work.

We introduce techniques to construct triangulation invariant models via recursion relations. In 2D these recursion relations are derived from the 2–2 Pachner move invariance or crossing symmetry. Furthermore we will argue that the 2–2 Pachner move invariance leads to Hamiltonian and Diffeomorphism constraints in these 2D models. We believe that these techniques can be also applied to higher dimensional models. The technique illustrates nicely how larger building blocks are constructed from some basic (smallest) building block using the principle of triangulation invariance. In this sense the microscopic theory determines macroscopic physics even for single building blocks. Furthermore such recursion relations lead to the Hamiltonian constraints [20, 21]. The methods here are restricted to triangulation invariant models with local couplings only and therefore topological field theories. We however believe that the recursion relations, representing (the Ward identities of) diffeomorphism symmetry, admit a generalization to theories with propagating degrees of freedom.
In the following section we will elaborate on the relation to spin net models and comment more in sections 10,11 and the discussion section 12 on the relation to anyon condensation, the classification of phases and the role of diffeomorphism symmetry. The reader not interested in spin net models can directly go to the next section 3, where the models will be introduced.
This section will also specify the fixed point conditions – which leads here to the requirement of triangulation invariance of the model. We will express this requirement as conditions on the amplitudes or weights of the model. From these conditions we derive in section 4 recursion relations for the amplitudes, for which we find a family of solutions in section 5. In section 6 we apply an alternative method to find fixed point amplitudes and give additional examples.
We then explore the physical interpretation of these fixed point models, first by defining and computing the partition function of the torus in section 7. This will give the number of ground states of the Hamiltonians associated to a given fixed point model. In section 8 we in particular reconsider the fixed point models obtained by solving the recursion relations and show that these models originate from a so called Corner Double Line structure or valence bond construction. We comment on the general structure of the fixed point amplitudes, which in particular depends on the ground state degeneracy, and relate to the classification of phases for 1D quantum systems (that is space time dimensions).
The results of section 8 allow us to interpret this particular family of fixed point models as a boundary theory of a 3D topological model (the Tuarev–Viro model) in section 9 and moreover to relate our 2D recursion relations to 3D recursion relations, obtained from the Biedenharn–Elliot identity for these models. This makes the notion of diffeomorphism symmetry obvious, as the 3D recursion relations are the Ward identities of this symmetry for the 3D models.
Section 10 explains the notion of matrix product states and specifies the matrix product state representation of the ground states associated to the fixed point models as well as the Hamiltonians. In quantum gravity language this provides the link between the covariant and canonical description of the models, with the partition function providing the projector on the physical states (i.e. ground states), which are specified by the Hamiltonian (constraints). Here the problem to understand the symmetry of the ground states motivates the consideration of a notion of finite subgroups in a quantum group.
2 Spin net models
Spin foams [2, 22, 3] provide a proposal for a non–perturbative path integral for quantum gravity, based on fundamental building blocks. A key open question [2] to validate this proposal is the continuum limit of the models, which we here understand as the limit including a large number of these building blocks. The complexity of the models made progress difficult, in addition coarse graining and renormalization had to be adapted to a background independent framework [23]. In [14, 15, 16, 1] a program was started to tackle the problem of the continuum limit first in simpler analogue models. This helped to develop tools and techniques and most of all approximations and truncations, that can be tested on increasingly more complicated models. In this way we can hope to build up to the full models. Indeed, in this work we will consider models which in their algebraic structure (i.e. the structure group) almost reach the full models.
Spin net models have been introduced in [14, 15, 1] as analogues to spin foams. They can be interpreted as dimensional reductions of spin foams and for this reason are much easier to investigate, in particular with numerical techniques, as done in [1]. The results in [1] motivate us to introduce a further variant of these models in this work, which indeed is closely related to the so called spin net evaluation and the definition of the spin foam dynamics via these spin net evaluations prescribing intertwiner degrees of freedom [12, 11].
Here we will define this class of intertwiner models and present a large family of triangulation invariant models inside this class.
Spin foams and spin nets are based on some structure group, which for the full models is taken to be or . This however precludes numerical investigations, as the models based on such groups involve infinite summations and potential infinities [24]. This led to the introduction of finite group models in [14, 15, 1]. The use of the group structure simplifies the definition of the so called simplicity constraints, central to the dynamics of spin foams, for the analogue models. A parametrization of possible simplicity constraints can be found in [16]. However, there does not exist e.g. a family of subgroups of the rotation group which would allow to reach the full models in a limiting procedure^{1}^{1}1One could consider but here a geometrical interpretation is not obvious..
For this reason we introduce in this work intertwiner models based on the quantum group . Here denotes the so–called level of the quantum group at root of unity. These quantum groups have a finite number (namely ) of irreducible finite dimensional representations with non–vanishing quantum trace. We will see however that some of the results generalize to the classical group as well as to the quantum groups with deformation parameter real.
The corresponding spin net models will be defined in [13]. In the full theory such quantum group models are argued to lead to general relativity with a cosmological constant [25]. Thus we deal almost with the same algebraic structures as in the full theory, which would involve and related groups. The replacement of groups with quantum groups opens up the question of how to describe the simplicity constraints. We will propose a way to construct models with simplicity constraints in [13], based on an idea of Reisenberger [11] and the fixed points constructed in this work.
Let us shortly explain spin net models. These are versions of vertex models, i.e. models defined on graphs, with weights or amplitudes attached to vertices and depending on labels attached to the adjacent edges. Here the edges are decorated with representation labels , as well as two magnetic indices, labelling a basis in and (the dual representation space) respectively. Thus every edge carries a Hilbert space
(1) 
where the sum is over all irreducible representations. The contraction of magnetic indices associated to one edge, i.e. between two vertex weights, and the sum over the representation label attached to this edge corresponds then to integrating out all degrees of freedom associated to this edge.
Thus the partition function for a spin net is defined as
(2) 
where we made the index structure of the vertex weights explicit.
We considered coarse graining of such models with non–trivial simplicity constraints and for the permutation group of three elements in [1]. Under coarse graining the models flow in two ways: the weights change as well as the variables. The coarse graining procedure summarizes a set of edges to effective edges. In terms of the associated Hilbert spaces we just take the tensor product of the ’bare’ Hilbert spaces associated to the original edge. Performing a reduction to a sum over irreducibles again we obtain
(3) 
with in general.
The results of [1] indicate that indeed the models (2) have the potential to flow to fixed points which carry a notion of simplicity constraints. We will confirm this conjecture in providing an interpretation of the fixed points in terms of imposing simplicity constraints.
Most of the fixed points found [26] were of a particular factorizing structure. These fixed points are not included in the initial phase space of models decribed by (1) but require also for pairs . Indeed for these fixed points we find the factorizing structure
(4) 
This motivates us to introduce models with edge Hilbert spaces
(5) 
and vertex weights
(6) 
so that these vertex weights are invariant under the action of the group on the associated tensor product of representation spaces. We will term these models intertwiner models, as the work [1] led to the conclusion that this choice of intertwiners encode the relevant parameters for spin net (and in this way also for spin foam) dynamics.
The question which arose after the work [1] was whether the appearance of these fixed points was specific to using the group or a more universal feature. Thus we will investigate the case with quantum group here. Not only do we find a large family of fixed points, these also generalize to the classical group , or a real quantum deformation parameter. Moreover we confirm the interpretation of the fixed points carrying a notion of the simplicity constraints. We leave open here the investigation of the stability of these fixed points and hence the question in which sense and whether these fixed points define phases. From the experience with the models one needs to finetune first to the phase transition between the equivalent of and strong coupling phase to allow a flow to these fixed points. This fine tuning can be understood as imposing a certain weak notion of triangulation invariance (and hence diffeomorphism symmetry [18]) in the sense, that one demands invariance of the partition function under edge (and for spin foams face) subdivisions [27, 28, 29].
The appearance of further fixed points opens up new perspectives for spin foams. So far the discussion for spin foam renormalization and phases has been largely confined to theory and the (equivalent of) strong coupling fixed point, see for instance the second reference in [23]. However there might be more topological theories, which even implement a notion of simplicity constraints. A theory with propagating degrees of freedom might then arise as a perturbation of one of these topological field theories (a similar conjecture has been formulated in [30], however not specifying the topological theory). An alternative scenario is to take the continuum limit at a phase transitions between these fixed points.
3 Quantum group intertwiner models
Some basic facts on the representation theory of the quantum group and ‘diagrammatic calculus’ are summarized in appendix A.
Models with edge Hilbert spaces of the type
(7) 
where denotes an irreducible finite dimensional representation of come up in the description of anyon fusion and splitting [31]. Indeed the models we introduce here can be seen as state sum versions of anyon fusion and splitting dynamics. More abstractly we can describe these models as the nesting of intertwiner maps of type and , where one sums over intertwiner (i.e. representation) labels. (We will often replace indices denoting the representation associated to an edge with just the index .) In both cases a direction is involved – in the graphical representation we will take this direction as upwards. The intertwiner maps can then be described by two different types of three–valent vertices
(8) 
representing Clebsch Gordan coefficients, which give the components of the maps in the chosen basis. (The indices label the vectors of the basis in the representation space .) Here denotes a root of unity^{2}^{2}2We adopt the conventions of [32]. specified by the level of the quantum group .
The intertwiner maps can be combined to give maps between tensor products with more factors. This can be all nicely represented graphically leading to ‘diagrammatic calculus’ [33, 34]. This diagrammatic calculus also includes replacement rules which follow from certain identities for the Clebsch Gordan coefficients (or from the fact that one is considering a certain type of category). A particular important replacement rule is that the following two maps are equal:
(9) 
Hence we can think of any diagonal edge also as a horizontal edge. This allows to put the model also on (threevalent) lattices with horizontal edges.
To introduce a dynamics we allow the vertices to be dressed with weights and which we will indicate graphically by fat vertices
(10) 
We then define an intertwining map between the spaces and by (see figure 1)

specifying a three–valent graph ‘in a box’ with edges entering at the bottom of the box and edges emerging at the top of the box representing the corresponding tensor product of representation spaces (no edges enter at the vertical sides of the box)

specifying weights and for the vertices

summing over all representation labels attached to the inner edges, i.e. those not entering at the bottom or emerging at the top.
Thus we define a partition function with two boundary components
(12) 
Considering graphs with many edges we can ask how the amplitudes behave under coarse graining: this can be described as taking a given graph in a box and partitioning either the edges on the top or the bottom into two sets. This gives three sets of edges defining three Hilbert spaces which we associate to three ‘effective edges’ respectively (if one wants to stay with a model based on three–valent vertices). There is also an effective vertex weight given by the intertwiner map defined by the original diagram in the box.
Now the effective edges rather carry a Hilbert space labelled by a tensor product of representation spaces. This tensor product can be reduced to a sum over representations, which in general might appear with multiplicities larger than one. At this point a truncation is required (alternatively one enlarges the space of models and allows higher multiplicities, i.e. additional multiplicity labels on the edges). Ideally this truncation should pick up the part of the amplitude most relevant for the next coarse graining step. This in particular includes stacking boxes on top of each other, which corresponds to take powers of the corresponding intertwiner maps. Hence one can expect that a truncation determined by the largest singular value (or the largest singular values) would lead to satisfying results.
This leads to the tensor network coarse graining algorithm [35, 36]. The (quantum) group structure allows to introduce symmetry preserving algorithms as described and utilized in [37, 15, 1]. This symmetry preserving algorithm has several advantages:^{3}^{3}3Another advantage is that it takes nicely care of the problem that quantum groups at the root of unity require to factor out trace zero representations. besides improving the efficiency of the algorithm it allows to extract a lot of information on the structure of the fixed point models. In particular the singular values are attached with representation labels and thus one can follow which representations still appear (are excited) or not by tracking these singular values [1].
Most importantly this allows to define easily truncations which stay inside a certain space of models. For the space of models we are considering this would be implemented by keeping only the largest singular value per representation label, so all edges keep carrying a Hilbert space . (In a non–symmetry preserving algorithm this already corresponds to a bond dimension with giving the (classical) dimension of the representation space associated to and the sum is over all representations that appear, i.e. in we have and .) Larger spaces of models, or less severe truncations, can be obtained by allowing higher multiplicities and keeping the corresponding number of singular values.
3.1 Fixed points and triangulation invariant models
We implemented the Levin–Nave algorithm [35] shortly explained below in its symmetry preserving version, truncating to the largest singular value per representation label. The aim was to identify fixed points of the coarse graining flow. This also allowed us to choose considerably large (i.e. with ) and to still obtain a fast (on a laptop) algorithm, which is certainly advantageous if searching ‘randomly’ for fixed points.
These investigations showed a rich structure of fixed points^{4}^{4}4Actually most of these fixed points are (small) fixed point cycles where signs still alternate from one coarse graining step to the next. In the course of the work we will see that this is due to the fixed points requiring complex amplitudes. Astonishingly the algorithm still converges to the fixed point cycles even with real amplitudes.. Although we considered coarse graining on a fixed hexagonal lattice, the fixed points define triangulation invariant models: We defined our models on three–valent graphs, embedded into the plane. Dualization of such graphs (with certain regularity requirements) lead to 2D triangulations. Triangulations (of a given topology) can be changed into each other by a sequence of Pachner moves. In 2D there are three such Pachner moves, which we will call (2–2) (self–inverse) and (3–1) and its inverse (1–3), indicating the number of triangles involved before and after the move. Translating back to the original graphs these numbers indicate the number of vertices involved in this move.
These findings support the strategy advertised in [38, 16, 19, 15, 1] to look for triangulation invariant models (in the gravitational context tight to diffeomorphism symmetry [17, 18]) by coarse graining. In particular it seems to be sufficient to consider a regular lattice, which indeed is the only way to actually perform coarse graining in praxis.
However having established the existence of these fixed points we might ask whether it is possible to find these also by other means. In the following we will present a strategy to this end. This will allow to consider much more easily more complicated models, which might be not accessible numerically (for instance with structure group ).
We will look for triangulation invariant models right away – these will automatically be fixed points of the coarse graining flow. Hence we ask for the amplitudes to satisfy the (a) tilting condition and to be invariant under (b) the 2–2 move and (c) the two versions of the 3–1 move (as we have two types of three–valent vertices) depicted as follows.
(13) 
(14) 
Here is a scaling constant, which, if not equal to infinity^{5}^{5}5This might only happen for , i.e. symmetry., can be put to one by rescaling the amplitudes. Note that there are other versions of the 3–1 move (where all three edges are vertical or diagonal) which are however equivalent to the ones in (14) due to the tilting condition. For the same reason we only need to consider one type of 2–2 move.
These conditions translate into certain equations that have to hold for the amplitude functions written out below. The Clebsch Gordon coefficients describing the ‘bare’ intertwiner maps on the left hand side of the graphical equations in (13,14) can be contracted with an intertwiner basis and then expanded into the Clebsch Gordan coefficients appearing on the right hand sides in (13,14) again. This leads to the appearance of (or –) symbols. The symbol is defined graphically in (15) as the transformation matrix (modulo dimension factors) between two different bases of intertwiner maps.