Dimer geometry, amoebae and a vortex dimer model
We present a geometrical approach and introduce a connection for dimer problems on bipartite and non-bipartite graphs. In the bipartite case the connection is flat but has non-trivial holonomy round certain curves. This holonomy has the universality property that it does not change as the number of vertices in the fundamental domain of the graph is increased. It is argued that the K-theory of the torus, with or without punctures, is the appropriate underlying invariant. In the non-bipartite case the connection has non-zero curvature as well as non-zero Chern number. The curvature does not require the introduction of a magnetic field. The phase diagram of these models is captured by what is known as an amoeba. We introduce a dimer model with negative edge weights which correspond to vortices. The amoebae for various models are studied with particular emphasis on the case of negative edge weights. Vortices give rise to new kinds of amoebae with certain singular structures which we investigate. On the amoeba of the vortex full hexagonal lattice we find the partition function corresponds to that of a massless Dirac doublet.
The subject of dimers has a large literature and has attracted the interest of both mathematicians and physicists. A few useful mathematical and physical sources are (1); (2); (3); (4); (5) and (6); (7); (8); (9); (11); (9); (12); (10); (13) respectively, as well as references therein.
Dimer partition functions can be expressed as a sum of Pfaffians of a Kasteleyn matrix, , which is a signed weighted adjacency matrix. The dimer partition function, with uniform weights, counts the number of perfect matchings of a graph. The model is naturally considered with positive weights and can have a non-trivial phase diagram as the weights are altered. In particular, it has a gapless phase which is described by the amoeba of a certain curve known as the spectral curve (1); (2) (see section III). The Kasteleyn matrix can be thought of as a discrete lattice Dirac operator (3); (4); (14) and the finite size corrections to the partition function in the scaling limit coincide with that of a continuum Dirac–Fermion on a torus (14). If one further adds signs to the weights the model describes a lattice Dirac operator in a fixed gauge field background. The presence of additional signs we refer to as the presence of vortices in the dimer system.
We review the basic construction of dimer models and give a detailed construction of the connection on the determinant line bundle over the positive frequency eigenvector space of the Kasteleyn matrix. We find that
In the bipartite case the determinant line bundle has a flat connection.
The flat connection has non-trivial holonomy in accordance with the -theory of the torus.
When vortices are included the system can describe additional massless Fermions and we present an example where the partition function, in the vortex full case of a hexagonal lattice, corresponds to a massless Dirac Fermion doublet.
For certain vortex configurations, the domain where the system describes a massless Dirac operator, the amoeba can develop a pinch.
The presence of vortices alters the thermodynamic phases, we exhibit a case where the gapped phase—an island, or compact oval in the amoeba corresponding to a massive Dirac phase—can, on introduction of vortices, shrink and even disappear.
The paper is organised as follows: section II describes basic results on dimers and dimer partition functions, focusing on bipartite dimer models. Sections III and IV describe a mathematical object known as an amoebae which describes the gapless parameter domain of the Kasteleyn matrix . In section V we construct the vector bundle of positive eigenvalues of , and show that its determinant line bundle has a flat connection and in section VI we show that this connection has non-trivial -holonomy in accordance with the -theory of the punctured torus. Section VII treats dimers on non-bipartite graphs. Section VIII is devoted to dimer models in to the presence of vortices. In section IX we present our conclusions; this is followed by an appendix on some of the relevant K-theory.
Dimer models are concerned with the set of vertex matchings of a graph, or lattice, . We shall consider to be a bipartite or non-bipartite lattice with sites, or vertices, on the two torus . A perfect matching, , on is a disjoint collection of edges that contains all the vertices: for to exist must have an even number of vertices. An edge belonging to a matching is called a dimer and perfect matchings are the same thing as dimer configurations.
We denote the set of dimer configurations on by . Then to each matching, , we assign a weight ; is normally required to be real in which case all weights are positive. We will find it useful to go beyond this restriction in the latter part of this paper and consider signed weights, but for the moment we take all weights to be positive.
Given this data the dimer partition function is given by:
Each matching, , consists of dimers with positive edge weights, whose relation to is that where are the edges of the matching , and is the weight associated with the edge , then
More generally one can consider dimers: i.e. matched edges where is less than or equal to the total number of edges; when one has a dimer model, otherwise one has a monomer-dimer system. Monomer-dimer systems have not yet yielded to exact solution methods.
An alternative generalisation is to consider some of the weights being negative, we will refer to such a system as containing vortices. It is relatively straightforward to solve for the partition function of such systems and as we shall see they have a rich physics. We shall only concern ourselves with dimer models, and dimer models with vortices, in this paper.
One can also investigate the probability of one, two, or more dimers (or edges), belonging to a matching : to do this one uses, the characteristic function of an edge , and the dimer-dimer correlation functions : their joint definitions are that
For a planar region in , and a graph with vertices, Kasteleyn (6) showed that
where is the Pfaffian of the Kasteleyn matrix which is a signed, antisymmetric, weighted adjacency matrix for : Kasteleyn’s sign assignments in are precisely what is needed to convert into the sum of positive terms that constitute the partition function .
We specialise, for the moment, to the case where is bipartite, so that we can colour vertices black and the other white. This allows us to write in the form
with an real matrix (we write rather than since, later on, we want to use to denote a connection) and denotes transpose; note, too, that .
Let and be vector spaces generated by the sets of black and white vertices respectively, then is the linear map
where is the sign associated to the edge whose weight is .
The signs are computed by the clockwise odd rule (7): arrows are placed on the edges and when following an arrow and when opposing one, and the product of the signs associated with any fundamental plaquette is when the plaquette is circulated in an anticlockwise direction; such an assignment of signs is called a Kasteleyn orientation.
Note that all closed paths on bipartite graphs have an even number of edges: thus clockwise odd is also anticlockwise odd. This is false for non-bipartite graphs which, when Kasteleyn oriented, possess an orientation which can be detected—cf. below where we discuss Chern numbers.
where denotes the set of equivalence classes of the spin structures on , is the Arf-invariant of the spin structure labeled by , and is the Kasteleyn matrix with these boundary conditions.
When the graph is on a torus , the weighted sum is over the four different Kasteleyn matrices corresponding to the four choices of periodic and anti-periodic boundary conditions around the cycles of the torus and
Each term in the sum corresponds to one of the discrete spin structures of on and, writing , one has
Note that since is on a torus it is doubly periodic and can be realised as a quotient: one has
where is the graph in the plane consisting of all possible translations by elements of of an appropriate fundamental domain contained in . However, itself may be multiple copies of this fundamental domain where the weights are repeated as translates. Then, by Fourier transforming we obtain a matrix with off-diagonal block .
On the torus corresponding to the translates of a
fundamental tile, the spin structure decomposition for the partition function is given by
The determinant of results in a polynomial with
A key property of the polynomial is that it has a pair of zeros , related by complex conjugation.
We now describe the construction of . Let us label an arbitrary translated copy of the fundamental domain from which is built by with . Here denotes the number of horizontal translations to the right, and the number of translations to the left, the integer labels vertical translations in a similar way; the fundamental domain is .
Thus one has
Now to each we associate the matrix ; so that, if denotes the signed weight of an edge joining vertex to , then we define
With this data the fundamental domain containing the basic graph is and we define by writing
so that then is a Laurent polynomial in and with real coefficients.
Note that if we defined using a domain other than the fundamental domain, e.g. if we wrote then would just be multiplied by a monomial in and and the formula for the partition function would still hold true.
Now both the infinite graph , and its bipartite black-white assignment, are translation invariant, and the decomposition
can be viewed as an indexing of by the characters of the translation group ; in other words is simply the Fourier transform of the Kasteleyn matrix of .
A pair of examples illustrating the above process can be quite simply given: consider to be the graphs tiled by the fundamental domains of figure 1.
Then, for the hexagonal graph, we readily calculate that
while for the rectangular one we have
Observe that varying the dimer weights and in (18) is equivalent to moving and off the unit torus. In general, for any polynomial arising from a bipartite dimer construction, one can always absorb combinations of dimer weights into and to move them off the unit torus.
Let with be a polynomial, then its zero locus
is a curve in . The image of in under the logarithmic map defined by
is known as the amoeba of so that .
The polynomial above is a special case of such an having real coefficients and is central to much of what follows; its zero locus is known as the spectral curve of the graph .
The amoebae for the two graphs of figure 1 are displayed in figure 2; the closed curve in figure 2 (b) is called a compact oval—we have moved off the unit torus and used weights , , for figure 2 (a); and for example , , with and all remaining weights unity for figure 2 (b).
Although an amoeba is unbounded in it has finite area; in fact its area is bounded above by an irrational multiple of the area of the Newton polygon of —the Newton polygon being the convex hull of the (integer) points for which . More precisely one has
After multiplication of by a suitable monomial to eliminate its negative powers, one obtains a polynomial of degree , say. Since has real coefficients, it determines the real homogeneous polynomial in —where now —and thus a real algebraic curve in , this being natural geometrical data possessed by the spectral curve . We denote this real algebraic curve by . When plotting the amoeba the signs of and play an essential role in determining all its components which, in turn, constitute the amoeba boundary.
For dimer models on bipartite graphs is a Harnack curve. Harnack curves are very special curves possessing the maximal number of components: i.e. . The integer is the genus of the curve and is equal to the number of compact ovals of the amoeba.
It will be convenient for us to abuse terminology slightly and often refer to a curve (rather than ) as being Harnack, the context should prevent any confusion.
A fundamental result of Kenyon, Okounkov, and Sheffield (1) and Kenyon and Okounkov (2) is that every Harnack curve arises in this way so that the correspondence between the spectral curves of periodic hexagonal dimer models and Harnack curves is a bijection. Also all bipartite planar graphs can be realised inside a large enough hexagonal lattice by a combination of setting some dimer edge weights to zero and bond contraction (7); (2); thus no bipartite graph is excluded. In addition, the most general hexagonal dimer model yields a generic Harnack curve of genus .
Positive rescaling of and gives a free action of on the set of Harnack curves; if one quotients by this action one obtains what is called in (2) the moduli space of Harnack curves. Amoebae provide natural coordinates for this moduli space: these coordinates being the areas of the holes and the distances between the tentacles (2).
When a curve is Harnack the area of its amoeba is maximal and saturates the area inequality above—i.e.
The converse of this equality also holds in the sense that implies that the curve of (after possible rescaling of , and by complex constants) is invariant under complex conjugation and possesses a real part which determines a Harnack curve in —cf. (15) for more details.
Iv Phases and amoebae
The amoeba can be viewed as the massless or gapless phase with its bounding curves as the phase boundaries in a dimer model phase diagram (1): the complement of the amoeba consists of both compact and non-compact regions. In the terminology of dimer models, as models of melting crystals, the non-compact regions exterior to the bounding ovals constitute the frozen regions. The amoeba itself is referred to as the liquid phase, and the interior of the compact ovals as the gaseous phase. There are also useful applications of these ideas to the Kitaev model (16) and topological phase transitions (14).
These different phases arise naturally when one calculates the correlation functions between the edges of . The correlation functions possess three types of decay (1)—where a decay is measured by the fall off of with distance between and —these types being exponential, polynomial, or no decay, and they correspond to the gaseous, liquid and frozen phases respectively. In the context of Kasteleyn matrices as Dirac operators, the amoeba is the massless phase while the interiors of the compact ovals correspond to massive Dirac operators (14).
V Dimer connections and curvatures
Now let once again denote coordinates on , rather than on , and let be a bipartite graph on whose Kasteleyn matrix , when Fourier transformed, gives where
With these conventions the matrix is Hermitian. Here we use for the number of vertices in the fundamental tile, in contrast to the usage in the introduction where referred to the total number of vertices in the graph .
If is non-bipartite its Kasteleyn matrix also has a Fourier transformed component of the form
with at least one of the diagonal blocks and being non-zero and such that is still Hermitian.
We now describe how to use this data to construct a certain connection on . The -dimensional space of eigenvectors of with positive eigenvalues form a rank bundle over which we denote by .
Let , then the fibre , at , has a basis consisting of the corresponding positive eigenvectors which we denote by
With respect to the standard complex inner product, fixed as varies, let each eigenvector have unit norm and, in an orthonormal basis, have components . When taken together the form the non-square matrix where
giving one a map
As varies the map embeds the fibres of —and thus the whole bundle—in the trivial bundle ; conversely, if is the adjoint of , the map is an orthogonal projection from to on which rests the non-triviality of .
Summarising, and abbreviating and by and respectively, yields
where denotes the identity matrix on and ; is called a partial isometry—it is not a real isometry since is not a square matrix.
A section of is then a map taking values in —i.e. one has
However, as usual, derivatives of such as may not, as varies, still be -valued, but we can project them back onto to take care of this problem thereby creating a covariant derivative on . Thus the covariant derivative of is where
Our choice of inner product above means that is the covariant derivative corresponding to a connection , say, which we can identify by direct calculation as follows: if is a map
then the product gives us the section
and the covariant derivative formula gives
Hence the connection is the matrix where
and its curvature is
So the covariant derivative and curvature on are and respectively, with . A routine calculation shows that
We shall examine the connection and its curvature in the subsequent sections but note that its introduction has not required the presence of a magnetic field.
Vi Holonomy and Flat connections
We will be interested in the connection associated with as it is this that gives the first Chern class of the connection. It turns out that for bipartite graphs the curvature is zero so that the associated connection is flat: nevertheless this connection is non-trivial as it has non-trivial holonomy as we shall now show.
It is instructive to first study a case where and we do this for the graph shown in figure 3. For the bipartite case, when , one punctures the torus at the points and since vanishes there, we denote the punctured torus by . One finds that the connection and its curvature are given by
so that we have a flat connection; however the connection is not trivial as it has non-trivial holonomy for some curves on . In other words for such curves
As an example, for figure 3 choose so that
and thus at the points
One then immediately discovers by direct calculation that if is a small circle
Hence we obtain non-trivial holonomy and it is easy to choose a different and obtain other results: indeed if does not contain or , but is non-contractible because it is a non-trivial homology cycle, then can also have non-trivial holonomy.
For example if is the curve , constant, then
Non-trivial flat connections require to have a non vanishing fundamental group but one knows that is homotopic to a bouquet of three circles (meaning three circles sharing one common point) and therefore
so all is satisfactory.
These properties of flatness and non-trivial holonomy persist—for the appropriate curvature and connection—when the bipartite graph is enlarged as we now demonstrate.
First let be the curvature coming from the Kasteleyn matrix for a general bipartite graph. One has
with curvature then it is easy to see that
For, abbreviating to , and setting ,
can be written
with the identity matrix. Hence
and so we have,
Now, just as when , there is non-trivial holonomy when : we shall also find the interesting result that the holonomy obtained is universal and is independent of .
However first we must identify an appropriate line bundle with connection and to this end we shall use the following notation: we denote the connection and curvature on any bundle by and respectively.
So taking our bundle —whose connection and curvature were formerly denoted by and above—we now denote these quantities by
and respectively. As we will see below, the line bundle that we seek is simply the determinant line bundle of , that is
Let denote the unit normalised positive eigenvectors of , then this bundle has projection where
and the associated connection is therefore where
Here it may be useful to recall that, if is a vector space, the inner product on , which for orthonormal renders the vectors orthonormal, is given by
To see this note first that
and for we observe that is a column vector and is a row vector with
where and the projection is given by . Hence
One can check that is indeed a connection; it is also flat since its curvature satisfies
Now we are ready to calculate the holonomy of round some curve : let
then is a unit norm eigenvector of with eigenvalue where
Using this orthogonal decomposition for each we calculate that