tricritical-ising

The Tricritical Ising Model (TIM) is not a lattice model per se... this term is used to describe the universal (conjectural) limit of a collection of lattice models, the simplest of which is perhaps the Blume-Capel model, which I like to think about as a kind of 'dilute' Ising model

•

Basically, the spins can be

\pm1

, or they can also 'not be there', in which case their value can be understood to be

0

•

So... we have spins

\sigma_i\in\{0,\pm 1\}

on the vertices

i

of a graph and a probability measure that is proportional to

\exp (\beta \sum_{i\sim j} \sigma_i\sigma _j +\Delta\sum_i \sigma_i^2)

•

The

\Delta

parameter can be understood as a 'chemical potential'... as

\Delta\to\infty

, obviously, we recover the Ising model at inverse temperature

\beta

•

Conjecturally, if we look at the model on a 2d discrete domain

\Omega_\delta

\delta\to0

, if we look at the behavior with respect to

\beta

(as

\beta

gets large, say), we see an order-disorder phase transition at

\beta=\beta_c(\Delta)

The Blume-Capel Model and the Tricritical Ising Model

•

Now, the Tricritical Ising Model (TIM) is the universality class of the tricritical point: what happens at

\Delta=\Delta_{tc}

, at

\beta=\beta_{tc}:=\beta_c(\Delta_c)

•

For large enough

\Delta

, i.e.

\Delta>\Delta_{tc}

, it is conjectured that this phase transition is a second-order, Ising-type phase transition: intuitively, when

\Delta

is large enough 'most of the spins' are

\pm1

, and we just get a regular Ising model on a 2d graph, so if

\beta<\beta_c(\Delta)

, we just have no long-range order, and this emerges as

\beta

increases to

\beta_c(\Delta)

It would be good to have a convincing heuristic explanation for this... perhaps this looks like a large-

q

Potts model?

The reason it is called 'tricritical' is because there is always a magnetic field parameter assumed through which one probes the order-disorder phase transition (but it is set to

0

)

Why is the tricritical Ising model interesting?

The scaling limit as

\delta\to 0

of the 2D Tricritical Ising Model (TIM) should be described by a certain Conformal Field Theory (CFT)

The CFT of the TIM corresponds to Unitary Minimal Model

\mathcal M_4

, which is a particularly interesting CFT

•

It is the only Unitary Minimal Model CFT with predicted supersymmetry

•

It has 6 primary fields:

\mathbb I,\sigma,\epsilon,\sigma',\epsilon', \epsilon''

of (left-) scaling dimensions

0,3/80,1/10,7/16,2

respectively: this means that their lattice counterparts (should they exist) would need to be renormalized by

1,\delta^{-3/40},\delta^{-1/5},\delta^{-7/8},\delta^{-4}

to converge to non-trivial limits

•

\sigma

is the spin field

•

\sigma'

is the so-called subleading spin field

This needs to be validated in various settings

In 2D, the Blume-Capel Model does not have a particular exactly solvable structure (and this is also is reflected by the fact that the specific values of

\Delta_c

and

\beta_{tc}

are only known numerically)

There are then models which are conjecturally in the same universality class, like the ADE models or the dilute loop

O(\sqrt 2)

model (more about this later)

For concreteness (and reasons that should become clearer later), I particularly like to think of the model on a triangular lattice (i.e. on the faces of a honeycomb graph)... this is mostly cosmetic (as we are not proving anything about that model anyway), but I think it makes some things a little bit cleaner in my mind

•

For

\Delta<\Delta_{tc}

, it is conjectured that this transition is first-order, which (I guess) should mean that we abruptly go from a situation where we have tiny clusters of

\pm1

, to a single large cluster with just one of the signs (basically at some point the large

\beta

will win over the small

\Delta

, and we will have something where at the critical point we can't have some co-existence of large

+

cluster and of large

-

clusters)... this is not very intuitive to me

\beta_c(\Delta)

should be a decreasing function of

\Delta

•

\epsilon

is the so-called energy field

•

\epsilon'

is the so-called vacancy field

•

\epsilon''

is the so-called irrelevant field

•

Besides the identity, the primaries are called in the following manner:

We will discuss about the relevance of this naming below

•

This model is sometimes introduced as a model with

\pm 1

spins

\tilde\sigma_i

, together with vacancy variables

t_i\in\{0,1\}

... what we described above should be essentially equivalent, if we think of

\sigma_i=\tilde \sigma _i t_i

As usual, one should be a little bit careful about the presence of extra degrees of freedom in that description... it would be good to make sure that everything is indeed equivalent

One can also note that this description in terms of the vacancies will turn out to be of interest for what is going to be discussed below

The Super-Stress Tensor

To Which Lattice Local Fields Do The Above Fields Correspond?

That is not an easy question... though we can get a reasonably convincing picture by studying

What does the supersymmetric structure means in term of transfer matrix?

(and of course its anti-holomorphic conjugate

\bar G

, of dimensions

(0,3/2)

)

If we think in terms of supersymmetry as a square root

R

of the transfer matrix

V

, the idea is that we should have fermionic and bosonic sectors of the transfer matrix (we should clarify what this means) that are exchanged by

R

: schematizing a bit, if we want to move 'one lattice step ahead', we move 'half a lattice step ahead' while exchanging bosons and fermions

•

At the CFT level, it contains a form of Kramers-Wannier duality, which in particular should map

\epsilon\to-\epsilon,\epsilon'\to\epsilon',\epsilon''\to-\epsilon''

General Claims

We should understand the tricritical Ising model (in the scaling limit) as follows:

•

A probabilistic construction of the spin field

\sigma

should be obtained by taking a random collection of loops generated by CLE(16/5), understanding these as separating (continuum) clusters of nonzero spins from clusters of zero spins

•

In each cluster of nonzero spins, the spins are all the same, i.e. either all 1 or all -1, with probability 1/2, independently of the other spin clusters

•

In other words: to find the tricritical Ising spin field, we take the recipe that brings us from the CLE(16/3) loops to the continuum Ising spin field, and we apply it to CLE(16/5) loops instead

In a realization of CLE(16/5), we can label each loop as even or odd in terms of the parity of the number of loops surrounding that loop; the nonzero clusters (if we try to understand the model with + boundary conditions) consist of the inside of each even loop minus the inside of each of the odd loops contained in that loop (it is a 'gasket' of Hausdorff dimension 3/40)

To be more specific about that picture, we understand the CLE(16/3) loops as boundaries of continuum FK-Ising clusters, and which consists in assigning a global, independent, uniform sign to each spin of what corresponds to primal cluster

•

This picture naturally carries a symmetry of zero versus nonzero spins, which naturally corresponds (as is the case in the CLE(16/3) world) to the Kramers-Wannier symmetry: we can couple the primal model with a dual model whose nonzero spins are exactly where the zero spins of the primal are, and vice versa

•

This also suggests that in the scaling limit, the tricritical Ising model corresponds to taking the dilute loop

O(\sqrt 2)

model on the honeycomb lattice, and assigning to all the spins on the hexagons contained in each even loop minus the interior of the odd loops within a global random sign, independently of the other signs in the other loops

•

This is the sense in which the tricritical Ising model can really be thought of as a 'dilute' Ising model: if we look at the dense loop

O(\sqrt 2)

model on the honeycomb lattice and do the same, we should, in the scaling limit, get the Ising model universality class (though this is a bit confusing, because the only thing I understand properly is if we do FK-Ising which leads to loops on the medial graph or something like that... if we do the FK thing, we don't need to decide dynamically what is in the primal model and what is in the dual model... they coexist on dual graphs... while with that

O(\sqrt 2)

model, each hexagon is randomly assigned to the primal or the dual randomly)

•

This nuance of primal vs dual either sharing the space or having their own vertices on which to live should in some sense become unimportant in the scaling limit, but while it makes the connection with a local spin model of course harder, if we want to add what we will call duality disorders (or duality defect lines), it will be much easier to do it while staying on the same lattice than connecting across dual lattices

•

Something that has also remained a bit unclear to me is this intuitive idea that the

\sqrt 2

in the

O(\sqrt 2)

model is due to the fact that 'only half of the loops matter', because only half of the loops carry a degree of freedom for the primal model (and there is two possibilities, and that's why gain a factor

2

for every other loop in weight)... there is definitely something there, but I don't really know what to make of it; should we just give a

2

weight for 'every even loop'... would that give the same result as a

\sqrt2

weight for each loop, if we pick the correct renormalization procedure?

•

For the FK model, the exclusion between a primal and a dual model are via the edges of the model (and so it doesn't affect the presence of spins directly), while for the tricritical model on the hexagon lattice, the exclusion is via the hexagons; an important advantage of the latter case is (as we will discuss below) that it allows us to look at duality defects, which will be important to understand chiral fields

•

The

G

field of the tricritical Ising CFT should be understood in a way that is analogous to the way the Ising fermion should be understood: in terms of order and disorder operators and (correspondingly) in terms of loops of the loop representation (in the same fashion as the Ising fermions can be understood in terms of the FK-Ising loops, as particular cases of the parafermionic observables of spin

\sigma=1/2

)

•

The loop representation has the advantage that it connects more easily to the SLE picture, and that the latter gives us more definite information: there are martingale observables for SLEs, and once we understand

G

as related to loops, we can relate

G

to SLE curves, from which it will naturally emerge as a martingale

Based on the above consideration, a natural thing to postulate about the tricritical Ising CFT is that the

G

field should be understood as 'the result of the same kind of recipe that constructs the free fermion from SLE(16/3), but applied to SLE(16/5), and with spin

\sigma=3/2

instead of

\sigma=1/2

)

The key question that should be asked is: what do we gain in defining this

G

in that way? What is the interesting insight we gain?

An essential point is that

G

is a holomorphic field of scaling dimension

3/2

and that we can act via its (half-integer) modes on the various fields, transforming fermions into bosons and vice versa, with

G_0^2=L_0

and more generally

G_k^2=L_{2k}

, making them really a square root of the Virasoro algebra

What is the difference with the Ising model's fermion modes?

The difference there is perhaps a bit subtle, as the half-integer modes of the fermion, as much as they do indeed map fermions to bosons and vice versa, typically create a lot more mess and we need an infinite sum

\sum_k(k+\frac 1 2):\psi_{n-k}\psi_k:

to create any

L_n

Is there a clear geometric reason why

G_k^2=L_{2k}

Not that I see at the moment; any simple intuitive reason on e.g. CLE(16/5) (e.g. suggesting that

G_0^2=L_0

follows from the fact that 'peeling two loops away from the outside amounts to starting over with the same boundary conditions in a smaller domain') must correspondingly fail for the Ising model and CLE(16/3), for which it is not the case

What can we say about

G

on the boundary of the domain?

This is a place where things get conceivably clearer if we work with something like a loop-based parafermion picture, where the corresponding random variable looks like

\mathbb 1_{z\in\gamma} e^{-i sW(\gamma;a\to z)}

, where

s

is the spin parameter (3/2, here) and

W

is the winding (i.e. turning) number of the curve

\gamma

made from a point

a

z

(let's leave aside what

a

is for the moment: in the case where we would have mixed +1 vs 0 boundary conditions, it would correspond to one of the boundary condition changing operator, which generates the curve

\gamma

)

The idea is that the phase

e^{-i \sigma W}

creates some substantial cancellations in the bulk (the probability of visiting a

\delta

-neighborhood of a point scales like

\delta^{3/5}=\delta^{2-d}

, where

d=1+\frac{16/5}{8}=7/5

is the Hausdorff dimension of SLE(16/5), while the scaling of

G

should be

\delta^{3/2}

), but that on the boundary,

W

is constant, and the phase makes no cancellation (and on the other hand, the probability of visiting a

\delta

-neighborhood of a boundary point scales like

\delta^{3/2}

)

The way I think of this is using the mental model of some lattice-level loop model, e.g. loop

O(\sqrt 2)

on the honeycomb lattice, where these 'neighborhood visits' are bona fide visits of the edges, and where

\delta

is just the mesh size, and the

G

field would probably be defined on each hexagonal face as a sum over the six edges of weighted visit events

Somehow, this way of thinking could be in principle too naive as the limited number of angles that

W

may take on the lattice level may create some undesired effects (given the relatively high value of the spin parameter

s

)... for instance, one would have to ask what is the difference on an edge between

s=3/2

and

s=1/2

, and there wouldn't be much of a difference for any individual edge, so we would have to average over e.g. the six edges bounding a hexagon to have a field that would faithfully represent

G

at the lattice level;

s=1/2

corresponds to another field, a spin-1/2 fermion, see below

The honeycomb lattice model seems reasonable as the definition we would get at the lattice level would at least change sign when making a

\pi/3

rotation of its definition, and that would capture the essence of a spin-3/2 field (and things would be substantially more symmetric and clearer than on the square lattice)

And anyway, on the boundary of the domain, we would have something that's just a boundary visit has an easy interpretation in terms of (regular) disorders for the model, once we use the fact (seen experimentally and justified from the theoretical considerations that lead us to study the model in terms of CLE(16/5) clusters, which never touch each other) that

+1

and

-1

spin clusters never touch each other

If we don't think of

O(\sqrt 2)

for a second and go back to the Blume-Capel model at the tricritical point, we see that

+1

spins never touch

-1

spins macroscopically (i.e. there are no macroscopic clusters of

+1

touching macroscopic clusters of

-1

), because there are always 'buffers' of

0

spins in between; again this is important to allow us to think of CLE(16/5) clusters as above as being correct

What happens for the Blume-Capel model?

But the point here is: if we introduce defect lines like for the Ising model, across which the spins pretend that their neighbors are the opposite of what they are we see that this automatically forces a cluster of 0 spins to 'cover' the resulting frustration, and basically to link both ends of the defect line

We also see this (somehow more elegantly) if we work with spins on a line bundle (like in the 'colored pictures' where we see the Ising model pick a random branch cut for the square root function, made of the frustration induced by the nontriviality of the bundle

Hence inserting a disorder line in a correlator between two points

x

and

y

is the same as inserting into the correlator the variable

\mathbb 1_{\{x \leftrightarrow y\}}

where

\{x\leftrightarrow y\}

denotes the event that

x

and

y

are in the same 0 cluster

If we think of the picture where there are dual spins

\tilde \sigma

living on the 0 clusters, this is the same as inserting

\tilde \sigma _x \tilde \sigma_y

into the correlator (because if they are in the same cluster they give one and otherwise they are independent and hence give 0), and this is exactly what we would call Kramers-Wannier duality: disorders being dual spins

Thinking of Kramers-Wannier, we can also see that free boundary conditions for the Ising model should correspond to 0 boundary conditions (and the corresponding dual spins will 'magically' all take the same values); it is a priori slightly surprising to think that if we let the spins free on the boundary for the Blume-Capel model, there would be no macroscopic +1 or -1 cluster touching the boundary (as the CLE(16/5) picture would suggest)... the intuition would conceivably be that on the boundary, there is not enough space to make it profitable to be a group of +1 spins versus a group of 0 spins (there is an intrinsic 'reward' for being 0 modulated by

\Delta

(remember that

\Delta_{tc}<0

, so we encourage spins 'not to be there'), and the reward for being +1 (or -1) is that one can be together with a lot of other +1 and benefit from the 'group effect', but that's not as easy on the boundary because there is not enough space if there is not an 'encouragement' (a forced nonzero boundary condition)

•

From the paper of Beale (Finite-size scaling study of the two-dimensional Blume-Capel model), on the square lattice we should have

\Delta_{tc}\approx-1.97/0.61\approx-3.23

and

\beta_{tc}\approx1.64

How does Kramers-Wannier work for boundary conditions?

The above reasoning is definitely an advantage of the Blume-Capel view, because we would probably get things wrong without a nice local spin model in terms of qualitative picture

Difference between Ising and Tricritical Ising concerning supersymmetry?

In terms of loop pictures, the only qualitative difference between CLE(16/5) and CLE(16/3) is that the loops don't touch each other, which makes the sampling of spins much more straightforward for tricritical

Could it be that because there is less mess, things are a bit more straightforward and that this would cause the cleaner relations leading from

G

T

modes?

The supersymmetry generator in terms of order and disorders

For the Ising case, the fermion viewed as a parafermion with

\sigma=1/2

is of course a fermion that is the product of an order and an adjacent disorder (so it technically lives on the corner graph), that we take the appropriate complex linear combinations of (otherwise, we just get a projection of the fermion on some line of the complex plane)

It is natural to want to the same for the

G

field, and what we will have should be very much the same: a product of orders and disorders, except that this would be living on the medial graph if we think of the honeycomb loop model, since both the order and the disorder would be living on the faces of the honeycomb lattice... and it would seem natural to take the appropriate linear combinations to have the desired spin, etc.

What happens with the (6/10,1/10) fermion?

For the tricritical Ising model, there should be another fermion

\psi

that would scale like

\delta^{7/10}

, of spin

1/2

... it is important to distinguish that guy from

G

The product of an order and an adjacent disorder will indeed force the appearance of a loop passing through the edge between the two hexagons, and taking into account the sign should yield something like

\exp(-i sW)

, where

s

would be a half-integer

My guess is that

\psi

would be obtained in a similar way, by picking, around a hexagon a different linear combination of product of order and disorder, ensuring that the spin of the said field would be 1/2: this could be ensured by making sure that under the lattice rotation by

\pi/3

, the field gets multiplied by

e^{i\pi/6}

(and not by

e^{i\pi/2}

, like

G

)

So, we could say that

\psi

and

G

are different Fourier modes of the product of adjacent spin and disorders around a point

So, we have a scaling

\delta^{6/10}

for being visited by a loop, a scaling

\delta^{7/10}

for the fermion

\psi

(and so, conceivably, for each edge, the contribution of the two possible windings modulo

2\pi

coming with opposite signs would have a small cancellation leading already to a factor

\delta^{7/10}

), and then if we were to take the linear combination with factors

1,-1,1,-1,1,-1

for the six edges going around a hexagon corresponding to

G

, we would get a further cancellation leading to something of order

\delta^{3/2}

(the fermion

\psi

would have factors

1,e^{i\pi / 6},e^{i\pi / 3},e^{i\pi /2},e^{2\pi i / 3},e^{5 \pi i / 6}

going around the hexagon)

It is perfectly conceivable, but that's not clear at this point

In terms of actions on the fields, if the fields are spin-even (like

\epsilon,\epsilon',\epsilon''

), the action

Is the energy density simpler for the tricritical Ising model?

For the Ising model, if we look at the bare square of a spin, this is a constant field, and it's not interesting

For the tricritical Ising model, if we look at the square of a spin, we get something that is meaningful, and I would like to say that this scales, if we subtract

1/2

(because roughly half of time it will be

0

and the rest of the time it will be

1

) like

\delta^{1/5}

and that it converges to the energy density field

\epsilon

Like the energy density field for the Ising model, it has the Kramers-Wannier duality, it blows up to

+\infty

when approaching a

+1

or a

-1

boundary, and to

-\infty

when approaching a free boundary

It has the same complicated relation with CLE(16/5) as the Ising spin field has with CLE(16/3): they both describe the level lines of these fields, and it in principle their values can be recovered from the parity of the number of loops surrounding the points, but it is not obvious how to do that

In principle, we could guess their correlation functions with Dobrushin boundary conditions (that would mean +/- for Ising, and +/0 for tricritical) and check that these are martingales (for SLE(16/3) for Ising, for SLE(16/5) for tricritical Ising)

For Ising, there is a nice thing, which is that one can introduce spinors that change the boundary conditions from + to -, that count the parity of the number of loops surrounding a point... maybe that can be done with

G

instead of the Ising fermion?

An important additional difficulty is that the OPE of

G

with the energy field is of degree

3/2

, so there are two singular terms to determine, and that's more than we would want to have uniqueness a priori, but if we know the energy is the square of the spin, that could help determining what we have

The way things were happening for the Ising model was that one could work with multiply-connected domains, and these domains would have boundary conditions, and we could find the unique solutions to boundary value problems in these domains, and basically look at the ratio of loop parity around a point with some boundary conditions by loop parity around another point with + boundary conditions, and we could get something quite explicit... and then we could let the sizes of holes go to zero and we would have that, at least

And then we could do some kind of limits with some spinors, and get something meaningful, and get something like what was done for the energy density

Why would the line integral

G_{-1/2}

be the square root of

L_{-1}

To some extent we can get some intuition that we make local fields become also functions of the dual variables and then when we apply again, we go back to the full functions of the primal

What gets mapped to what?

That's not completely clear at the moment, in fact

Why would this not work with the Ising model?

A plausible argument would be that we can't get a clean FK-like picture that would allow for a nice dual coupling from a CLE(3) picture (basically, if we see a CLE(3), the dual of the model is a somehow uncertain thine), and that if we work with a CLE(16/3) picture, the spins are not clear in there... doing the FK percolation on CLE(3) to recover CLE(16/3) in unfeasible, etc... and this could lead to the idea that we can't move 'forward' (i.e. preserve boundary conditions so easily)...

There could be something like: if we insert a CLE(16/5) loop at some point, it won't touch anything else, and so we won't have to deal with the higher-order effects

Duality Disorders and Chiral Fields

An important idea to explore is the presence of chiral spin fields, which would be nonlocal holomorphic companions of the spin field for the Ising model and of the subleading spin field for the tricritical Ising model

What hints at a chiral spin field for critical and tricritical Ising?

If we look at the

O(n)

model (such that the probability of a configuration is

\propto x^{|\omega|} n^{\ell}

for

n\in(0,2)

, there are two integrable values of the

x

parameter:

x_c

(which corresponds to the critical point) and

x_d>x_c

(which is just one point of the dense regime which happens to be integrable)

Conjecturally, the dense

O(\sqrt 2)

model has the same scaling limit as the critical FK-Ising model (described by CLE(16/3)) and the dilute

O(\sqrt 2)

model has the scaling limit as the set of clusters of the tricritical Ising model (described by CLE(16/5))

These two integrable points

x_c

and

x_d

both come with a corresponding parafermionic correlator

F(a,z)

(often called parafermionic observable) of a certain spin parameter value, which satisfies (as a function of its second argument

z

) a lattice counterpart to Cauchy-Riemann equations

Now, what happens to the parafermions is that the

O(\sqrt 2)

model at

x_d

has spin parameter

s=1/16

(suggesting a field that would scale like

\delta^{1/16}

\delta\to 0

) and at

x_c

has a spin parameter

s=7/16

(suggesting a field that would scale like

\delta^{7/16}

\delta \to0

)

How are these related to duality defects?

If we look at the parafermionic observables with

a\in\partial\Omega

and

z\in\partial\Omega

, then we see that

W

becomes configuration-independent, and we are in fact looking at the ratio of the partition function of the model with a boundary change operator divided by the one of the model without

The definition of these fields should be something like

The boundary change operator can be identified with the one between +1 and free (or 0 for tricritical)

What is a promising way to study these things?

Using SLE in multiply-connected domains, and loop counting that is compatible with holomorphicity (following the strategy of Dima and Kostya in their first paper on spinors for the Ising model), we could probably construct some holomorphic counterparts of the parafermionic observables, and some interesting things could be proven for them

Why is it promising?

The idea is we have boundary value problems with solutions that are very likely to be unique (at least if we keep 'macroscopic holes'), and as a result this would lead to unique characterization of correlators...

Then we could probably understand many correlators of interest by collapsing the macroscopic holes into singularities (this strategy worked for the Ising model quite well... there could be problems with it, but in principle, we should get the correct singularities, at least for the

G

correlator

F(a,z)=\sum_{\gamma} e^{-i s W(\gamma:a \to z)} x^{|\gamma|}n^{\ell}/\mathcal Z

, where

\mathcal Z=\sum_\omega x^{|\omega|} n^{\ell}

For the Ising model, a difficulty with the spinors was the 'species of square roots', i.e. there was a spinor for the singularities of type

1/\sqrt z

and one for the singularities of type

i/\sqrt z

(the boundary value problems are not

\mathbb C

-linear, which complicates everything)... and of course we should have the same thing; everything is just

\mathbb R

-linear

So, let's say we have the

G

correlators that can interact with the loop parity, for instance (we can think of the chiral fields later)... what would this give us?

The idea is that this should give us ratio of

\epsilon

correlations with various boundary conditions... and it is in itself quite interesting

But could we also get more information from this kind of calculation?

The idea is as before that these techniques allow one to take a ratio of correlators, with proper normalization due to the OPE, and so on...

If we follow where the scheme developed for the Ising model leads us, we see that collapsing

G

onto

\bar G

will lead to the irrelevant field correlator, for instance, so we could get

\lang\epsilon \epsilon''\rang/\lang\epsilon\rang

correlations

Collapsing

\epsilon''

onto

\epsilon

should probably give something of some interest (it would likely be something involving

\epsilon'

, at least)

If we have

\lang G G \epsilon \cdots \epsilon\rang/\lang \epsilon \cdots \epsilon \rang

, what can we do?

The way to construct this would be to first fix one

G

on the boundary, look at the function we get as a function of the other

G

's location, and then to bring the other

G

inside, and then to collapse the two

G

's and to obtain the stress tensor, which would lead us to the logarithmic derivatives of

\epsilon

, and that would be already something interesting

How would we approach the correlations of

\sigma

That is less clear: we can certainly consider the correlations of

G

with

\sigma

, but how to interpret

\sigma

in terms of loops is more difficult... at least we can't replicate the ratio of correlations trick directly

This looks like a harder question, though the 'qualitative' probabilistic construction of the field using CLE(16/5) is easier

There is a boundary condition that makes particularly clear sense for the Ising model in multiply-connected domains, which is 'locally monochromatic', but there are also tricks to deal with '+' or '-' boundary conditions on the inner components

What should the correlations of

G

with

\sigma

be?

In principle, if we fuse

G

with a

\sigma

, then this is not very different from what happens when we fuse a fermion with a spin for the Ising model (though the loop picture looks quite different)

If we think at the critical FK(q=2) picture, then we can definitely consider a spin correlation in that picture, corresponding to a connectivity indicator insertion (if things are not in the same FK cluster, they are independent), together with the fermion field, corresponding to an indicator that a loop passes through a point, weighted by the complex exponential of the curve's winding

Now, the point is that in the FK world, this could be viewed as a fairly complex object, but if we normalize in the right way, and we fuse the spin and the fermion, we get something that is very nice... even though it may not be obvious from the FK-loop picture

Still, something very nice comes out of that picture, and so the hope would be that something that is similarly nice comes out from the tricritical world picture

Clarifying What We Have at This Point

There is a world where we should look at the

\lang\epsilon\cdots\epsilon G\rang/\lang\epsilon\cdots\epsilon\rang

correlations, because we think of the

\epsilon

field as being related to the parity of the number of CLE(16/5) loops, and because we have something that deeply resembles the work of Dima and Kostya when they started looking at spinors: they were looking at the parity of the number of loops around a hole, and added the monodromy as a way to preserve the holomorphicity of the original fermionic correlator; and I don't see why we couldn't have the same

And then there is a world where we understand the tricritical loops as FK-loops, and we understand Ising fermions and spinors within the FK(q=2) picture, we get a sense of what that could mean for the CLE(16/3) loops, and we transfer that understanding to the CLE(16/5) loops

In other words, we can think of the tricritical Ising loops (in the context of parafermions) as CLE(3.2) and CLE(16/5): in the first context, we generalize the low-temperature construction of fermions, and in the second context, we generalize the FK random-cluster construction of such fermions

Of course for the Ising model, these coincide, but they have the potential to give two distinct classes of quantities for the tricritical Ising model, perhaps one for the spin field and one for the energy field

In either case, we can expect to have a boundary value problem; there is a singularity, and we have some prescriptions for that singularity, and we can expect to learn something from there, even though the higher order of the singularity (we should get something like

(z-x)^{3/2}

) leaves a priori some unfixed degrees of freedom

The Boundary Value Problem!

It is still reasonable to hope that we should be able to nail down the relevant degrees of freedom based on symmetry considerations; at least it is very reasonable to expect that something could be done based on the many symmetries that we have, and that some information could be bootstrapped

Boundary value problems with unique solutions

If instead of working with singularities, we work with macroscopic holes, it is reasonable to expect that we have some boundary conditions like

\parallel\frac 1{\nu^{3/2}}

, where

\nu

is the normal to the boundary (viewed as a complex vector), oriented either towards the inside or the outside

And this problem is likely to have a unique solution

This at least gives a lot of hope to generalize the construction of Dima and Kostya to compute the tricritical energy density field

Still, similar techniques, e.g. the modified FK observables used to compute FK crossing probabilities, could also give some meaningful insight about the tricritical Ising spin field

What about the Tricritical Spin Field?

I feel here a good question to ask is: how can we understand (an exact computation of) Ising spin correlation functions within the FK picture?

How do the spinor techniques look from within FK?

In some sense, if a satisfactory answer to these questions can be found, then it is reasonable that it would generalize to the tricritical picture...

Within FK, we can't count the parity of loops, etc. and that's good, because for tricritical magnetization, the parity of loops is not important... the only question is whether we are linked with the boundary cluster, like for FK... so if we could do something where CLE(16/3) is magically replaced by CLE(16/5), that would work