Existence of a tricritical point for the Blume-Capel model on Zd Trishen S. Gunaratnam Dmitrii Krachun and Christoforos Panagiotis April 4 2024

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Existence of a tricritical point for the Blume-Capel model on Zd

Trishen S. Gunaratnam∗∗

, Dmitrii Krachun††

, and Christoforos Panagiotis‡‡

April 4, 2024

Abstract

We prove the existence of a tricritical point for the Blume-Capel model on Zdfor every d≥2.

The proof in d≥3relies on a novel combinatorial mapping to an Ising model on a larger graph,

the techniques of Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys, 2015), and

the celebrated infrared bound. In d= 2, the proof relies on a quantitative analysis of crossing

probabilities of the dilute random cluster representation of the Blume-Capel. In particular, we develop

a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which

allows us to obtain a ﬁne picture of the phase diagram in d= 2, including asymptotic behaviour of

correlations in all regions. Finally, we show that the techniques used to establish subcritical sharpness

for the dilute random cluster model extend to any d≥2.

1 Introduction

Let Ld= (Zd,Ed)be the standard d-dimensional hypercubic lattice with vertex set Zdand nearest-

neighbour edges Ed. Given a ﬁnite subgraph G= (V, E)of Ld, the Blume-Capel model on Gwith

inverse temperature β > 0, crystal ﬁeld strength ∆∈R, and boundary condition η∈ {−1,0,1}Zdis the

probability measure µη

G,β,∆deﬁned on spin conﬁgurations σ∈ {−1,0,1}Vby

µη

G,β,∆(σ) = 1

Zη

G,β,∆

e−Hη

G,β,∆(σ),

where

Hη

G,β,∆(σ) = −βX

xy∈E

σxσy−∆X

x∈V

σ2

x−βX

xy∈Ed

x∈V, y∈Zd\V

σxηy

is the Hamiltonian, and Zη

G,β,∆is the partition function. By convention, we write µη

V,β,∆to denote the

Blume-Capel measure on the subgraph of Ldspanned by V, and we denote expectations by ⟨·⟩η

V,β,∆. We

denote by 0(resp. +,−) the boundary conditions ηx= 0 (resp. ηx= +1,ηx=−1) for all x∈Zd.

The Blume-Capel model is closely related to one of the most famous models in statistical physics,

the Ising model. The latter is deﬁned analogously, with spins taking values in {±1}and the Hamiltonian

only consisting of the terms involving β. In particular, one can view the Blume-Capel model as an Ising

model on an annealed random environment, i.e. on the (random) set of vertices for which σx̸= 0. In

the limit ∆→ ∞, the underlying random environment becomes deterministically equal to Zd, and the

Blume-Capel model converges to the classical Ising model on Zd.

∗Universit´

e de Gen`

eve, trishen.gunaratnam@unige.ch

†Princeton University, dk9781@princeton.edu

‡University of Bath, cp2324@bath.ac.uk

arXiv:2210.13394v3 [math.PR] 3 Apr 2024

The model was introduced independently by Blume [Blu66] and Capel [Cap66] in 1966 to study the

magnetisation of uranium oxide and an Ising system consisting of triplet ions, respectively. Both papers

were trying to explain ﬁrst-order phase transitions that are driven by mechanisms other than external

magnetic ﬁelds. Since then, it has been studied extensively by physicists due to its particularly rich phase

diagram, i.e. as an archetypical example of a model that exhibits a multicritical point, see [BP18] and

references therein. Indeed, for each value of the parameter ∆∈R, the Blume-Capel model undergoes a

phase transition at a critical1parameter βc(∆), which is deﬁned as

βc(∆) = inf{β > 0| ⟨σ0⟩+

β,∆>0},

where ⟨·⟩+

β,∆denotes the plus measure at inﬁnite volume, which is deﬁned as the limit of the ﬁnite volume

measures ⟨·⟩+

G,β,∆as Gtends to Ld. It is expected that the critical behaviour of the model depends

strongly on ∆: as ∆increases, the phase transition changes from discontinuous, when ⟨σ0⟩+

βc(∆),∆>0,

to continuous, when ⟨σ0⟩+

βc(∆),∆= 0, and this transition happens as ∆crosses a single point ∆tric. See

Figure 1.

The model at the so-called tricritical point (∆tric, βc(∆tric)) is of particular interest. Indeed, in some

dimensions it is expected to exhibit vastly diﬀerent behaviour from the points on the critical line when

∆>∆tric, even though in both cases the phase transition is expected to be continuous. In particular,

in d= 2, the scaling limit of the model at criticality for ∆>∆tric is expected to be in the Ising and

ϕ4universality class. On the other hand, at ∆tric, the scaling limit of the model is expected to be in a

distinct universality class corresponding to the ϕ6minimal conformal ﬁeld theory with central charge

c= 7/10 (whilst the Ising universality class is of central charge c= 1/2) – see Mussardo [Mus10]. A

rigorous glimpse of this distinct universality class appears in the work of Shen and Weber [SW18], where

near-critical scaling limits of related models with so-called Kac interactions are considered. On the other

hand, in d= 3, the scaling limit of the Blume-Capel model at ∆>∆tric is expected to be nontrivial, as

conjecturally for Ising, which is supported by e.g. conformal bootstrap methods – see Rychkov, Simmons-

Duﬃn, and Zan [RSDZ17]. Whereas for ∆tric,d= 3 is predicted to be the upper-critical dimension

for the model and one expects triviality of the scaling limit. This is supported by renormalisation group

heuristics, which have recently been made rigorous for the ϕ6model in the weak coupling regime by

Bauerschmidt, Lohmann, and Slade [BLS20]. See also [BS20] for a mean-ﬁeld random walk model

exhibiting a tricritical behaviour. In dimensions d≥5, one expects that the model is trivial throughout

the continuous phase transition regime ∆≥∆tric – c.f. the case of Ising, where triviality was shown by

Aizenman [Aiz82] and Fr¨

ohlich [Fr¨

o82]. We also refer to the review book [BBS19] for an account of

renormalisation group approaches to this problem. In d= 4, for ∆=∆tric one also expects a triviality

result, whereas for ∆>∆tric one expects a marginal triviality result as in the case for Ising, which was

recently shown by Aizenman and Duminil-Copin [ADC21].

Despite the interest of this model in physics and the interesting predictions about the tricritical point,

there is a lack of rigorous understanding of the phase diagram of the Blume-Capel model. Indeed, many

rigorous results about the model have been focused well within the discontinuous transition regime, where

it is a good test case for Pirogov-Sinai theory – see [FV17, Chapter 7] and [BS89]. This is in contrast to

the case of the Ising model, where much of the phase diagram is now well-understood [DC17]. Stochastic

geometric methods have been at the heart of many recent developments, in particular probabilistic

representations of spin correlations via the random cluster and random current representations. For

the Blume-Capel model, an analogous random cluster representation, called the dilute random cluster

representation, has been developed by Graham and Grimmett [GG06]. In this article, we show that the

underlying philosophy of the recent techniques used to analyse the Ising model and its random cluster

1In the physics literature, βcis sometimes called a transition point or triple point when the phase transition is discontinuous,

to distinguish from a critical point corresponding to continuous phase transition. In this article we adopt percolation terminology

and call βca critical point.

representation can be adapted to rigorously analyse the phase diagram of the Blume-Capel model in

dimensions d≥2.

Our ﬁrst result establishes fundamental properties of the phase diagram of the model that has been

up until now folklore.

Theorem 1.1. Let d≥2. For every ∆∈R, we have that βc(∆) ∈(0,∞). In other words, there is a

nontrivial phase transition. Moreover, the function ∆7→ βc(∆) is continuous and decreasing with limits

lim∆→−∞ βc(∆) = +∞and lim∆→+∞=βc(Ising), where βc(Ising) is the critical point of the Ising

model on Zd.

The next theorem is the main result of this paper. It establishes the existence of at least one separation

point between the points of continuous and discontinuous phase transitions, i.e. the existence of a tricritical

point. Namely, we prove the following.

Theorem 1.2. Let d≥2. Then, there exist ∆−(d)≤∆+(d)such that

•for any ∆<∆−(d),⟨σ0⟩+

βc(∆),∆>0

•for any ∆≥∆+(d),⟨σ0⟩+

βc(∆),∆= 0.

Moreover, one can take ∆+(d) := −log 2 for d≥3, and ∆+(d) := −log 4 for d= 2.

The proof of the existence of a discontinuous critical phase is a standard application of Pirogov-Sinai

theory and is given in Section 3. The crux of the article is to establish the existence of a continuous

critical phase, for which we use diﬀerent techniques depending on the dimension. In the case d≥3, the

proof relies on a new representation of the model as an Ising model on a larger (deterministic) graph,

which is ferromagnetic only when ∆≥ −log 2 – see Section 4. This representation has the advantage

that it naturally relates the correlation functions between the Blume-Capel and the Ising model, and

allows us to use techniques and tools developed for the latter to study the former. We use the celebrated

infrared/Gaussian domination bound of Fr¨

ohlich, Simon, and Spencer [FSS76] to show that under the free

boundary conditions, the two-point correlations decay, in an averaged sense, to 0for every β≤βc(∆).

This allows us to use the breakthrough result of Aizenmann, Duminil-Copin and Sidoravicius [ADCS15]

on the continuity of the Ising model on Zd(see also [Rao20] for a relevant result for general amenable

graphs) to conclude that the phase transition of the Blume-Capel model is continuous in d≥3when the

corresponding Ising model is ferromagnetic, namely, when ∆≥ −log 2.

In dimension 2, the infrared bound is not suﬃcient to deduce that the two-point correlations under free

boundary conditions decay. Instead, we prove this by using a representation of the correlation function in

terms of connection events in the dilute random cluster model. It is well-known that in two-dimensional

percolation theory the relevant observables that keep track of continuous versus discontinuous phase

transitions are crossing probabilities of macroscopic rectangles. We analyse crossing probabilities in

the dilute random cluster model using the renormalisation strategy of [DCT20]. As output, we obtain

that the two-point correlation functions under free boundary conditions (associated to the Blume-Capel

model) decay to 0for every β≤βc(∆). In turn, by the same arguments discussed in the preceding

paragraph, we obtain that in dimension d= 2 the phase transition is continuous when ∆≥ −log 2. The

estimate on ∆+can be improved in d= 2 to obtain continuity for −log 4 ≤∆<−log 2, where the

corresponding Ising model is not ferromagnetic. Instead we can directly analyse the diﬀerence between

the spin models with free and plus boundary conditions by interpolating between them using arbitrarily

small positive boundary conditions (that we call weak plus). The comparison between the free and the

weak plus boundary conditions is enabled by our quantitative analysis of crossing probabilities, whereas

the comparison between weak plus and plus boundary conditions is enabled by a Lee-Yang type theorem

on the complex zeros of the Blume-Capel partition function.

Remark 1.3. There does not seem to be an explicit formula for the tricritical point in any dimension. In

d= 2 numerics [MFH+24] suggest that in our parametrisation it is located approximately at ∆≈ −3.23

and β≈1.64. Our bound of ∆+=−log 4 ≈ −1.39 is therefore not sharp. In dimensions d≥3we

do not expect our current bound ∆+=−log 2 to be sharp. However, we note that ∆ = −log 4 and

β= 1/N correspond to the location of the tricritical point for the Blume-Capel model on the complete

graph on Nvertices in the limit N→ ∞, see [EOT05, SW18]. It would be interesting to see whether

the improved threshold ∆+(2) = −log 4 can be extended to any dimension. Indeed, we expect that the

tricritical point on Zdconverges to that of the complete graph as d→ ∞, i.e. in the mean-ﬁeld limit. In

support of this, we point out that the Lee-Yang type theorem that we establish holds for ∆≥ −log 4 on

any graph.

Proving uniqueness of the tricritical point, which is the main omission of Theorem 1.2, amounts to

showing that one can take ∆−(d) = ∆+(d)in Theorem 1.2. Unfortunately, it is unclear whether there

is monotonicity along the critical line and, in the absence of integrability, to the best of our knowledge

all known techniques for showing discontinuity are intrinsically perturbative. Nevertheless, in dimension

2, the quantitative estimates on crossing probabilities, and other considerations that we describe shortly,

allow us to obtain a more precise picture of the phase diagram, although we fall short of proving uniqueness

of the tricritical point.

To state our next result, we ﬁrst recall the deﬁnition of the truncated 2-point correlation:

⟨σ0;σx⟩+=⟨σ0σx⟩+− ⟨σ0⟩+⟨σx⟩+.

In dimension d= 2 we obtain a ﬁne picture of the phase diagram by showing that the truncated 2-point

correlation decays exponentially everywhere except for the continuous critical regime, where it decays

polynomially. We also give an alternative characterisation of the points of continuity in terms of the

percolation of 0and −spins (equivalently, ∗-percolation properties of +spins, where ∗-percolate refers

to percolation on Z2union the diagonals).

Theorem 1.4. Let d= 2. Then the following hold.

(OﬀCrit) For all β > 0and ∆∈Rsuch that β̸=βc(∆), there exists c=c(β, ∆) >0such that

⟨σ0;σx⟩+

β,∆≤e−c∥x∥∞,∀x∈Z2.

(Discont) For all ∆∈Rsuch that ⟨σ0⟩+

βc(∆),∆>0, there exists c=c(∆) >0such that

⟨σ0;σx⟩+

βc(∆),∆≤e−c∥x∥∞,∀x∈Z2.

(Cont) For all ∆∈Rsuch that ⟨σ0⟩+

βc(∆),∆= 0, there exist c1=c1(∆), c2=c2(∆), such that

for all x∈Z2with ∥x∥∞large enough

∥x∥c1

∞≤ ⟨σ0σx⟩+

βc(∆),∆≤1

∥x∥c2

∞

(TriCrit) The set of ∆∈Rsuch that ⟨σ0⟩+

βc(∆),∆>0is open.

(Perc) For all ∆∈R,⟨σ0⟩+

βc(∆),∆= 0 if and only if there is no inﬁnite site cluster of {0,−1}

spins under ⟨·⟩0

βc(∆),∆.

∆

βc(∆tric)

∆tric

Figure 1: Conjectural phase diagram of the Blume-Capel model. The dark grey region is the supercritical

phase β > βc(∆), where the spontaneous magnetisation does not vanish, i.e. ⟨σ0⟩+

β(∆),∆>0. The

light grey region is the subcritical phase β < βc(∆), where the spontaneous magnetisation vanishes, i.e.

⟨σ0⟩+

β(∆),∆= 0. The union of the dashed and solid lines, together with the black dot, indicate the critical

curve ∆7→ βc(∆). Note that it is a continuous decreasing curve (the convexity of the curve is purely

for illustration purposes and the actual curve may not be convex). The curve of points of continuous

phase transition is depicted solid, and the curve of discontinuous phase transition appears as dashed.

The tricritical point is the boundary point between these two behaviours along the critical curve and is

marked as a solid black dot. Theorem 1.2 rigorously establishes all parts of this phase diagram except

for the behaviour on the critical curve in the neighbourhood of the (unique) tricritical point. We establish

thresholds ∆−<∆+such that the curve is dashed for ∆≤∆−and solid for ∆≥∆+, but we cannot

say what happens along the critical curve for ∆∈(∆−,∆+).

The proof of the behaviour at the critical points is based on the quadrichotomy for crossing probabilities

for the dilute random cluster representation of the Blume-Capel model mentioned earlier. The fact that

(TriCrit) holds has the implication that at any separation point (i.e. tricritical point) on the line of critical

points, the phase transition is continuous and hence satisﬁes (Cont). The percolation characterisation

(Perc) is an adaptation of an elegant geometric argument for the continuity of nearest-neighbour Ising on

Z2, see [Wer09]. The proof of the subcritical behaviour relies on a generalisation of the OSSS inequality

for monotonic measures by Duminil-Copin, Raouﬁ and Tassion [DCRT19] to the dilute random cluster

model, and more generally to weakly monotonic measures, which we deﬁne in Section 8.1. The original

OSSS inequality was obtained by O’Donell et. al. [OSSS05] for product measures. In fact, the technique

for showing subcritical sharpness is robust enough to extend to all dimensions:

Theorem 1.5. Let d≥2and ∆∈R. Then for every β < βc(∆) there exists c=c(β, ∆, d)>0such

that

⟨σ0σx⟩+

β,∆≤e−c∥x∥∞

for every x∈Zd.

We end on a natural follow up question on Theorem 1.4, which relates the percolative properties of

the underlying random environment to the nature of the critical point.

Question. In d= 2, is it true that, for all ∆∈R,⟨σ0⟩+

βc(∆),∆= 0 if and only if {0}spins do not

∗-percolate under ⟨·⟩0

βc(∆),∆? Is there a higher dimensional analogue of this?

1.1 Paper organisation

In Section 2 we deﬁne the dilute random cluster model and develop the necessary tools that we require

for the rest of this article. In Section 3 we establish fundamental folklore facts about the phase transition

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ExistenceofatricriticalpointfortheBlume-CapelmodelonZdTrishenS.Gunaratnam∗∗,DmitriiKrachun††,andChristoforosPanagiotis‡‡April4,2024AbstractWeprovetheexistenceofatricriticalpointfortheBlume-CapelmodelonZdforeveryd≥2.Theproofind≥3reliesonanovelcombinatorialmappingtoanIsingmodelonalargergraph,thetechni...

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Existence of a tricritical point for the Blume-Capel model on Zd Trishen S. Gunaratnam Dmitrii Krachun and Christoforos Panagiotis April 4 2024

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