https://doi.org/10.48350/151851 | downloaded: 31.1.2022
Quantum crystals, Kagome lattice, and plane partitions fermion-boson duality
Thiago Araujo ,1,* Domenico Orlando ,1,2,†and Susanne Reffert 1,‡
1Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, ch-3012, Bern, Switzerland
2INFN, sezione di Torino and Arnold-Regge Center, via Pietro Giuria 1, 10125 Torino, Italy
(Received 23 September 2020; accepted 2 January 2021; published 26 January 2021) In this work we study quantum crystal melting in three space dimensions. Using an equivalent description in terms of dimers in a hexagonal lattice, we recast the crystal melting Hamiltonian as an occupancy problem in a Kagome lattice. The Hilbert space is spanned by states labeled by plane partitions, and, writing them as a product of interlaced integer partitions, we define a fermion-boson duality for plane partitions. Finally, based upon the latter result we conjecture that the growth operators for the quantum Hamiltonian can be represented in terms of the affine YangianY½glðb 1Þ.
DOI:10.1103/PhysRevD.103.026020
I. INTRODUCTION
Random partitions appear in many contexts in math- ematics and physics. This omnipresence is partly explained by the fact that they are part of the core of number theory, the queen of mathematics[1]. It is nonetheless remarkable that they appear in a variety of distinct problems—such as the Seiberg-Witten theory, integrable systems, black holes, string theory and others [2–6].
In two dimensions, random partitions can be success- fully realized in terms of fermionic operators living on a chain where particles and holes are labeled by half-integers along the real line [3,7–9]. In this formalism, the empty partition is equivalent to a configuration where all negative holes (or better yet, holes on negative positions) are occupied and all positive holes are vacant. The creation and annihilation of squares are defined as holes and particles hopping as in Fig.1.
The description above has been used in[9,10]to study the two-dimensional version of quantum crystal melting.
Quite remarkably, with a Jordan-Wigner transformation, the crystal melting Hamiltonian is shown to be equivalent to theXXZspin chains with kink boundary conditions, and it automatically implies the integrability of this problem.
In this work, we focus on the three-dimensional version of the quantum crystal melting, which can be rephrased in terms of random plane partitions, and it has been partly addressed in[9]. Plane partitions can be written in terms of interlacedXXZdiagrams or as lattice fermions, see Fig.2.
The empty partition is a configuration of the hexagonal lattice where all positions below the diagonals are occu- pied. Additionally, based on the one- and two-dimensional cases, a conjecture for the mass gap has been made, and further numerical analysis has been performed.
We present a number of equivalent formulations of the 3D quantum crystal melting problem, such as in terms of particle-hole hopping, a fermionic description on a Kagome lattice, and a tensor product representation. We are even- tually lead to a fermion-boson duality, and to a conjecture for an underlying Yangian symmetry. We expect this structure to be instrumental in uncovering a possible hidden integrable structure, suggested by the results in lower dimensions.
The plan of this note is as follows. We start with a short review in Sec. II, where we see how plane partitions are equivalently written in terms of dimers in a hexagonal lattice.
As we said before, we want to find an alternative particle- hole dynamics which is equivalent to the plane partition growth. More explicitly, we search for a statistical system with states labeled by plane partitions. That is the problem we address in Sec.III, where we discuss a construction in terms of particle hopping in a hexagonal lattice. This formalism is considerably simplified if we use a dual description which turns out to be the Kagome lattice. We write the new Hamiltonian, and we provide additional details of it in Sec.IV.
Having discussed characterizations of the crystal melting as occupation problems, we try to understand other
*thiago@itp.unibe.ch
†domenico.orlando@to.infn.it
‡sreffert@itp.unibe.ch
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fundamental aspects of the system, in particular, its Hilbert space. In Sec.V, we write the plane partition states in terms of interlaced integer partitions, and we use this fact to show that the growth operators have nontrivial coproducts. In addition, we define a novel fermion-boson correspondence for plane partitions states in Sec.VI. Finally, based on the action of the growth operators on the bosonized states, we conjecture that this Hamiltonian can represented in terms of the affine Yangian of glð1Þ. We give some concludingb remarks in SectionVII. In the AppendixA, we give present some necessary details on the 2D fermion-boson duality, in AppendixBwe argue the equivalence of the Hamiltonians in the plane-partition language and in the Kagome picture.
In AppendixC, we detail basis transformations for various states and in Appendix D, we give amplitudes for tran- sitions between various states.
II. PLANE PARTITIONS AND HEXAGONAL DIMERS
We start by describing the well known correspondence between the crystal melting problem and dimers in a hexagonal lattice, see [9,11–13] and references therein.
The empty partition (vacuum) is equivalent to an infinity perfect matching with different staggered boundary con- ditions in the regions I, II and III, see Fig. 3, defined as
ϑ∈I¼ ð−π=6;π=2Þ;
ϑ∈II¼ ðπ=2;7π=6Þ;
ϑ∈III¼ ð−5π=6;−π=6Þ: ð1Þ
In the vacuum configuration, the origin is defined as the only plaquette where the three phases, or staggered directions, meet. We say that a plaquette with the three phases is aflippable plaquette.
Adding a box to the vacuum corresponds to thepositive flip , as seen in Fig.4, and as a matter of fact, there is only one flippable plaquette in the vacuum.
Additionally, simple inspection of Fig.4reveals that there are now three plaquettes where we can perform a positive flip, and just one plaquette where the negative flip
can be performed.
It is easy to see that this logic remains the same for general configurations, that is, positive and negative flips correspond to adding and removing boxes in a given plane partition state. Furthermore, for any lattice configuration, there is only a finite number of flippable plaquettes, and from the plane partition perspective it obviously means that there is only a finite number of boxes that can be consistently removed, or available places where we can add a box.
FIG. 3. Empty partition.
FIG. 4. One box configuration.
FIG. 1. Fermionic fields and Young diagram relation.
FIG. 2. 3D Young diagram and its fermions lattice.
We want to study the crystal melting Hamiltonian defined in [9]
ð2aÞ
that we conveniently write as H¼−JðΓþΠÞþV
pffiffiffiq
OþðΠ;ΓÞþ 1 ffiffiffiq
p O−ðΠ;ΓÞ
: ð2bÞ
Using and , one can easily see
that the Hamiltonian is Hermitian. The two terms multi- plied byJare kinetic and describe the two flips previously mentioned. More specifically, the operator Γ (Π) adds (removes) boxes in all possible ways. Moreover, the potential termsOþðΠ;ΓÞandO−ðΠ;ΓÞcount the number of positive and negative flippable plaquettes, respectively.
In other words, they determine the number of places where we can add a box, and the number of boxes that can be removed respecting the plane partition rules.
This system has a special point forJ¼V, the so-called Rokhsar-Kivelson point [14,15], where it is natural to understand the model as the stochastic quantization of the classical counting of plane partitions [16]. In the continuum limit this corresponds to a quantum critical point [17]. At this point we expect the system to have special symmetry properties and we will focus on this case.
Some progress in the understanding of this system has been achieved in[9], where it was shown that the ground state of the system is given by a sum over all plane partitions, where a partition of the integer kis weighted byqk=2. If follows that the norm of the ground state is given by
Z≡hgroundjgroundi ¼Y
n≥1
1
ð1−qnÞn; ð3Þ which we recognize as the famous MacMahon function: the generating function for the numberplðkÞof plane partitions
Z¼X∞
k¼0
plðkÞqk: ð4Þ
In the following sections, we want to represent the Hamiltonian (2b) in terms of a statistical mechanics systems of particles and holes, and we start investigating its integrability properties. More specifically, we define a new form of fermion-boson correspondence and we find connections between this problem and the affine Yangians of glð1Þ.b
III. PARTICLE-HOLE BOUND STATES In this section we describe the dynamics of the crystal melting problem in terms of particle hopping in two Bravais lattices(2b). In the next section we build a Hamiltonian for this problem using the dual description of this particle-hole system.
First we need to remember that the hexagons above do not define a Bravais lattice, but we can write it as a superposition of two triangular Bravais lattices, say H whose points are labeled by ∘, and P, labeled by •. We call themholeandparticle sublattices, respectively.
The position vectors in the Psublattice are
⃗
p1¼a 2ð ffiffiffi
p3
;1Þ; p⃗ 2¼a 2ð ffiffiffi
p3
;−1Þ; ð5Þ where a is the lattice spacing, that can, without loss of generality, be set toa¼1, see details in[18]. In Fig.5we have also given the linear combination p⃗ 3¼p⃗ 1þp⃗ 2. Finally, the dimer between two lattice points is represented by the positively oriented bound states∘→•.
The fermionic operatorsψð⃗xÞandψð⃗xÞin an arbitrary lattice point ⃗x satisfy the anticommutation relations
fψð⃗xÞ;ψð⃗x0Þg ¼0;
fψð⃗xÞ;ψð⃗x0Þg ¼0;
fψð⃗xÞ;ψð⃗x0Þg ¼δ⃗x;⃗x0: ð6Þ Thebottom of the Dirac sea is defined by the condition
ψð⃗xÞj0i ¼0; ∀⃗x∈H;P; ð7Þ and the empty partition (vacuum) by
j∅i ¼Y
j∈Z
Y
k∈Z
ψðjp⃗ 1þkp⃗ 2Þj0i: ð8Þ
With these definitions, it is straightforward to see that all points in the sublatticePare occupied. It should be implicit (since it is a visual aid) that the holes and particles are bonded positively,∘→•. A flip in the hexagon (creation or
FIG. 5. SublatticesH (∘) andP (•).
annihilation of a box) is simply a hole-particle hopping which preserves the positive orientation of the bound state.
Furthermore, we set the origin atðj; kÞ ¼0⃗ , and we can identify a generic point ⃗x¼jp⃗ 1þk⃗p2, ðj; kÞ∈Z2 as in Fig. 6.
Points in the Hole sublatticeHcan be identified by linear combinations of ðp⃗ 1;p⃗ 2;p⃗ 3Þ plus the additional offsets
⃗δ ¼a
2 1ffiffiffi
p3;1
; δ⃗ ¼a
− 1ffiffiffi p3;0
: ð9Þ Putting all these facts together, we can try to build a model where the flips in the hexagonal dimers correspond to simultaneous hoppings in the two sublattices.
The creation of one box is described as a counterclock- wise rotation as in Fig. 7, and it is equivalent to three simultaneous hoppings, a 3-hopping, along the positively oriented (the red arrow) hole-particle bound states,
ð10aÞ
This point of view describes the box creation in terms of next-neighbor interactions, but it may be computationally useful to describe the box creation in terms of the horizontal 3-hopping that we define as in Fig.8.
In other words, we reshuffle the expression (10a) ð10bÞ
The 3-hopping above describes the creation and anni- hilation of a state corresponding to a partition with one box.
For configurations with more boxes, some of the relevant bound states are edges of two hexagons. In this case, we
would need to consider a horizontal 2-hopping where there are two free bound states, and the usual hopping when there is just one free bound state.
In other words, we write the three configurations with two boxes as
ð11aÞ
ð11bÞ
and
ð11cÞ
This description is rather involved, and somewhat con- trived, for states with more than one box. Fortunately, in the next section we show that there exists a more natural description when we assume that for each dimer, there is a dual description in which this extended object is pointlike.
IV. DUAL KAGOME LATTICE
In order to define this dual description, let us start with the basis
⃗
m1¼að0;1Þ; m⃗ 2¼a 2ð ffiffiffi
p3
;−1Þ; ð12Þ connecting the barycenters of hexagons, as in Fig. 9.
Furthermore, the offset vectors from the center of a hexagon to its edges are
⃗ξ¼a
4ð ffiffiffi p3
;1Þ; ξ⃗ ¼a
2ð0;−1Þ: ð13Þ Given a hexagon in the lattice, let us associate a dual fermion (created by ψ˜) to the midpoint of each edge containing a dimer and adual hole (where the fermion is annihilated by ψ˜) to the midpoint of each empty edge.
Connecting these midpoints generates another lattice made of hexagons and triangles, which is known asArchimedean FIG. 6. Generic points in the lattice.
FIG. 7. Counterclockwise 3-hopping.
FIG. 8. Horizontal 3-hopping.
tessellationf3;6;3;6g[19], orKagome lattice. The empty configuration corresponds to Fig. 10.
In terms of the dual fermions, the empty configuration is given by
j∅i ¼Y
j∈Z
Y
k∈Z
˜
ψðjm⃗ 1þkm⃗ 2þΔj;kÞj0i;˜ ð14aÞ
wherej0i˜ is the Dirac sea in the Kagome lattice, which is defined by putting a dual hole at the midpoint of each honeycomb edge, or equivalently, in the f3;6;3;6g tes- sellation, see Fig. 11.
Moreover, we have defined
Δj;k¼ξ⃗ þδI½argðjm⃗ 1þkm⃗ 2Þ þξ⃗ −δII½argðjm⃗ 1þkm⃗ 2Þ
þξδ⃗ III½argðjm⃗ 1þkm⃗ 2Þ; ð14bÞ where
δA½argð⃗vÞ ¼
0 if argðvÞ⃗ ∉A
1 if argðvÞ⃗ ∈A; A¼I;II;III; ð14cÞ and the regions I, II and III have been defined in Eq.(1), see also Fig. 3. Moreover, it is easy to write an explicit expression for δA½argð⃗vÞ as a difference of Heaviside functions.
Using the dual lattice, the box creation and annihilation are given by the simultaneous action of three pairs of dual hole-particle pairsψ˜ψ˜ at neighboring vertices. The vectors from the origin to the vertices are given by
⃗PðsÞj;k¼j⃗m1þk⃗m2
þa
2ðcosðð2sþ1Þπ=6Þ;sinðð2sþ1Þπ=6ÞÞ;
s¼0;1;2;3;4;5; ð15Þ
where we have introduced an indexs as in Fig.12.
Putting all these facts together, the 1-box configuration is given by
ð16aÞ
with inverse
ð16bÞ
Making the notation lighter with ψ˜ðsÞj;k≡ψ˜ðP⃗ ðsÞj;kÞ, we have
ð17aÞ
ð17bÞ FIG. 9. Dual basis.
FIG. 10. Empty configuration in the Kagome lattice.
FIG. 11. Dual Dirac sea.
FIG. 12. Vector position in the dual lattice.
Observe that the action of the operatorΓj;konj∅ifor any positionj≠0andk≠0naturally vanishes because there is a mismatch of fermions and holes.
As the notation above suggests, we represent the operatorsΓ andΠ in (2b)as
Γ¼X
j;k
Γj;k ¼ X
j;k∈Z
˜
ψð1Þj;k ψ˜ð0Þj;kψ˜ð3Þj;k ψ˜ð2Þj;kψ˜ð5Þj;k ψ˜ð4Þj;k; ð18aÞ Π¼X
j;k
Πj;k¼ X
j;k∈Z
˜
ψð0Þj;k ψ˜ð1Þj;kψ˜ð2Þj;k ψ˜ð3Þj;kψ˜ð4Þj;k ψ˜ð5Þj;k: ð18bÞ
One may also calculate the commutators among the componentsΓij andΠij of the growth operators. It is easy to see that the only nontrivial commutator is ½Γij;Πkl. Although it does not give any new information, it is useful to have its explicit expression since it is important in our calculations. It is straightforward (but laborious) to check the following relation:
½Γij;Πkl ¼δikδjlðψð¯54¯32¯1Þij ψð1¯23¯45Þkl −ψð¯0Þklψð¯54¯32Þij ψð¯23¯45Þkl ψð0Þij þψðkl¯01Þψðij¯54¯3Þψð3¯45Þkl ψðij¯10Þ−ψðkl¯01¯2Þψðij¯54Þψðkl¯45Þψð2¯10Þij þψð¯01¯23Þkl ψð¯5Þij ψð5Þkl ψðij¯32¯10Þ−ψð¯01¯23¯4Þkl ψð4¯32¯10Þij Þ:
ð19Þ We use the notationψðaijb…Þ¯ ¼ψðaÞij ψðbÞij to make the commutator slightly friendlier.
The operatorsOðΠ;ΓÞ are built from the components Γi;jandΠi;j. We can defineOþðΠ;ΓÞby locally adding and removing a box in all possible vector positionsðj; kÞ, whilst O−ðΠ;ΓÞis defined by locally removing and adding a box in all possible vector positions ðj; kÞ.
In AppendixB we build states corresponding to 2- and 3-boxes plane partitions and we also describe the action of the operatorsΓ,ΠandOðΠ;ΓÞ. Using this construction, one can rewrite the crystal melting Hamiltonian (2b)as
H¼−JX
j;k
Γj;kþΠj;k
þVX
j;k
ffiffiffiq
p Πj;kΓj;kþ 1 ffiffiffiq
p Γj;kΠj;k: ð20Þ
The ground state of this model has been defined in[9]. It is the sum of the states corresponding to plane partitions Λ weighted byqjΛj, wherejΛj ¼#BoxesðΛÞ. In the notation above we have
jgroundi ¼X
Λ
qjΛj2jΛi;
¼X
n
X
ðj1;j2;…;jnÞ∈Zn ðk1;k2;…;jnÞ∈Zn
Yn
a¼0
qn2Γja;kaj∅i; ð21Þ
where the first term in the sum, n¼0, is simply j∅i.
Therefore, one can verify thatZ¼ hgroundjgroundiis precisely the MacMahon function(3).
The Hamiltonian (20) can be thought of as the limit J0¼0of the more general model
Hhexagon ¼J0X
h⃗r;⃗r0i
ψð⃗rÞψð⃗r0Þ þH; ð22Þ
where the first term describesnearest-neighbor interactions.
In other words, the Hamiltonian (20) is an interaction Hamiltonian of 6th and 12th degrees, and if we assume a slightly more general setting with lowest-order hopping, we obtain the Hamiltonian(22).
V. TENSOR PRODUCT REPRESENTATION In the remainder of this work, we try to understand some universal features of the Hamiltonian(2b). In other words, we investigate aspects that do not depend on the lattice descriptions above, in particular, its Hilbert space and underlying symmetries. The fact that the states in our system are labeled by plane partitions suggests an inter- pretation in terms of the MacMahon representation of the affine Yangian ofglð1Þb [20–24]. In this part of the paper we start our program of rewriting and interpreting the Hamiltonian (2b) in terms of Yangian invariants, which we believe to be the key to understand the integrability properties of the problem. We begin by identifying a nontrivial coproduct structure of the operators that we have defined in the previous sections.
Plane partitions can be written as stacks of interlaced integer partitions[9,25], that is
Λ¼ fΛð1Þ;Λð2Þ;…;ΛðNÞjΛðaÞ≻Λðaþ1Þg; where ΛðaÞ¼ ðλðaÞ1 ;λðaÞ2 ;…;λðaÞlaÞ;λðaÞi ≥λðaÞiþ1;
∀i¼1;2;…;la; a¼1;2;…; N−1: ð23Þ The number of boxes in the plane partitionΛis given by
jΛj ¼#Boxes¼XN
a¼1
jΛðaÞj; ð24Þ
in other words, it is the sum of squares of each integer partition layer. In this notation, a generic state jΛi asso- ciated to a given plane partitionΛcan be represented as the tensor product
jΛi ¼ jΛð1Þi⊗jΛð2Þi⊗ ⊗jΛðNÞi; ð25Þ and the empty plane partition is the product of empty integer partitions
j∅i ¼ j∅i⊗j∅i⊗ : ð26Þ
As we discuss in Appendix A, the usefulness of this structure is that the integer-partition layers can de described in terms of free fermions with Neveu-Schwarz boundary conditions.
As we have seen before, the operatorΓadds a box in all consistent places of a given partitionΛ. Let us denote the space of allowed places as QþðΛÞ, therefore
ΓjΛi ¼ X
□∈QþðΛÞ
jΛþ□i: ð27Þ
Additionally, the operatorΓcan be described in terms of its action on the integer partitions:
ΓjΛi≡ΔðγÞ
⨂N
b¼1jΛðbÞi⊗j∅i
;
¼Nþ1⨁
a¼1RaðγÞ
⨂N
b¼1jΛðbÞi⊗j∅i
; ð28aÞ
whereγ is the growth operator on integer partitions. In the definition above, for each layer labeled byathere exists an action of the integer partition growth operatorγ∈A, where Ais the algebra of operators acting on the space of integer partitionsHðaÞ. In other words, given thatjΛi∈⨂aHðaÞ, the operator Γ can be represented as the coproductΔðγÞ.
Moreover, we assume a decomposition ΔðγÞ ¼⨁aRaðγÞ that denotes the growth of each integer partition level.
A trivial action of the coproduct ΔðΓÞ is given by RaðγÞ ¼1⊗ ⊗RðaÞðγÞ⊗ ⊗1, where the operator RðaÞðγÞis the (representation of the) growth operator in the tensor product a slot. This condition clearly violates the constrained growth of plane partitions; therefore we impose a nontrivial coproduct Δ∶A→A⊗A by
RaðγÞ≔1⊗⊗Mða−1ÞðγÞ⊗RðaÞðγÞ⊗⊗1; ð28bÞ where the operator Mða−1ÞðγÞ imposes that the a-slot growth does not violate the plane partition conditions, and its eigenvalues are þ1, if the a-slot integer partition grows consistently, and 0 otherwise. The existence of this nontrivial coproduct is the first sign of a quantum group structure underlying the tensor representation above.
As an instructive example, consider the configuration
, withΓ . In the tensor product
notation we have
ð29aÞ
The action of R1ðγÞreads
ð29bÞ
since the first layer is unconstrained. On the other hand ð29cÞ
since any growth of the second layer violates the plane partition conditions. Finally
ð29dÞ
More generally, the first layer can always grow freely as long as it satisfies the integer partition rules. Using the equivalence between 2D Young diagrams and free fer- mions, one can representRð1ÞðγÞas
Rð1ÞðγÞ≡γ ¼ X
r∈Zþ12
Ψrþ1Ψr: ð30aÞ
Consequently,
R1ðγÞ ¼γ⊗1⊗ ⊗1: ð30bÞ In fact, γ acts as
γjΛðaÞi ¼ jðλðaÞ1 þ1;λðaÞ2 ;…;λðaÞlaÞi þ þ jðλðaÞ1 ;λðaÞ2 ;…;λðaÞla þ1Þi
þ jðλðaÞ1 ;λðaÞ2 ;…;λðaÞla;1Þi: ð30cÞ For the subsequent layers, the partitions are subject to the interlacing constraint, then
Mða−1Þ ⊗RðaÞðjΛða−1Þi⊗jΛðaÞiÞ
¼ jΛða−1Þi⊗ X
ΛðaÞþ□≺Λða−1Þ
jΛðaÞþ□i; a≥2; ð31Þ
which means that the resulting partitionðΛ˜ðaÞÞ≡ðΛðaÞþ□Þ satisfies λ˜ðaÞj ≤λða−1Þj ∀j¼1;…;la. The first, and almost trivial, observation is that the operatorsMact diagonally, whilstRðaÞ is a constrained growth operator.
We propose the form
Mða−1Þ⊗RðaÞ¼X
i
Mða−1Þi ⊗RðaÞi ; ð32Þ
where RðaÞi acts as
RðaÞi jðλðaÞ1 ;…;λðaÞi ;…Þi ¼ jðλðaÞ1 ;…;λðaÞi þ1;…Þi: ð33Þ The operatorMða−1Þi tests whether the partitionðΛðaÞþ□Þ interlaces ðΛða−1ÞÞ, and it can be defined as
Mða−1Þi jðλða−1Þ1 ;…;λða−1Þi ;…Þi
¼ri;ajðλða−1Þ1 ;…;λða−1Þi ;…Þi; ð34aÞ where i¼1;…;la;laþ1 and
ri;a ¼
0 if λai þ1>λa−1i
1 otherwise : ð34bÞ
Observe that the operatorMða−1Þis nonlocal, since it acts on the (a−1) slot, but additional information on the states of theaslot is required. It would very be interesting to find representation for the operatorsMða−1Þ ⊗RðaÞ, fora≥2, using the free fermions formalism.
Similarly, the operatorΠremoves a box in all consistent places, denoted byQ−ðΛÞ, in a given partitionΛ. Then
ΠjΛi ¼ X
□∈Q−ðΛÞ
jΛ−□i: ð35Þ
As before, we write ΠjΛi≡ΔðπÞ
⊗N
b¼1jΛðbÞi⊗j∅i
;
¼⨁N
a¼1SaðπÞ
⊗N
b¼1jΛðbÞi⊗j∅i
; ð36aÞ where π acts on integer partitions. In addition
SaðπÞ ¼1⊗⊗SðaÞðπÞ⊗Nðaþ1ÞðπÞ⊗1⊗1: ð36bÞ The final layer can shrink freely as long as it satisfies the integer partition rules, therefore
SðNÞðπÞ≡π¼ X
r∈Zþ12
ΨrΨrþ1 ð37Þ
and
πjΛðaÞi ¼ jðλðaÞ1 −1;λðaÞ2 ;…;λðaÞlaÞi þ
þ jðλðaÞ1 ;λðaÞ2 ;…;λðaÞla −1Þi: ð38Þ
For the precedent layers, the partitions are subject to the interlacing constraint, therefore
SðaÞ⊗Nðaþ1ÞðjΛðaÞi⊗jΛðaþ1ÞiÞ
¼ X
ΛðaÞ−□≻Λðaþ1Þ
jΛðaÞ−□i⊗jΛðaþ1Þi; a < N: ð39Þ
We propose the form
SðaÞðπÞ⊗Nðaþ1Þ ¼X
i
SðaÞi ðπÞ⊗Nðaþ1Þi ; ð40Þ
whereNðaþ1Þ is defined by
Nðaþ1Þi jðλðaþ1Þ1 ;…;λðaþ1Þi ;…Þi
¼si;ajðλðaþ1Þ1 ;…;λðaþ1Þi ;…Þi; ð41aÞ wherei¼1;…;la;la and
si;a¼
0 if λai −1<λaþ1i
1 otherwise : ð41bÞ
It would be interesting to study the fermionic representation of the operatorsMandNand the quantum group structure of the operatorsΓandΠin this context. On the other hand, in the next section we will see that the growth of plane partitions is better described in terms of the Yangian generators, and in this new context some of the underlying algebraic structure becomes more transparent. It is worth mentioning that although the Yangian algebra has its defining coproduct structure, it is different from the coproduct structure we consider in this section.
VI. FERMION-BOSON DUALITY AND AFFINE YANGIAN
The important observation of the previous section is the nontrivial coproduct structure of the operatorsΓandΠin the tensor product representation. In this section, we study other algebraic structure of these operators in terms of the affine Yangians of glð1Þ, and we also define a fermion- boson correspondence for the plane partition states.
In AppendixAwe review the necessary tools we use in this section. In order to define a fermion boson-correspondence for plane partition states, let us write the statesjΛi, that we callfermionic plane partition state, as
jΛi ¼⨂N
a¼1
X
⃗kðaÞ
χΛðaÞ½Cðk⃗ ðaÞÞ zk⃗ðaÞ
jk⃗ ðaÞi;
¼X
⃗K
χΛ½CðKÞ⃗ ZK⃗
jKi;⃗ ð42aÞ
where we use
jKi ¼ j⃗ k⃗ ð1Þi⊗ ⊗jk⃗ ðNÞi; ð42bÞ
and
χΛ½CðKÞ ¼⃗ χΛð1Þ½Cðk⃗ ð1ÞÞ χΛðNÞ½Cðk⃗ ðNÞÞ ð42cÞ is a product of characters of the symmetric group. Using the results of AppendixAand Eq.(24), the number of boxes inΛ is simply
jΛj ¼XN
a¼1
X
j
jkðaÞj : ð43Þ
The states j⃗ki can be labeled in terms of partitions themselves, that is
jki ¼ j…;⃗ 2;|fflfflffl{zfflfflffl}…;2
k2times
;1;|fflfflffl{zfflfflffl}…;1
k1times
i; ð44Þ
and we denote these states asjki⃗ ≡jNk⃗⟫. For example, the state associated to the vector k⃗ ¼ ð0;1;2;1;0;0;…Þ is α−4α2−3α−2j0iand we denote it as abosonic plane partition statejð4;3;3;2Þ⟫. Using the Frobenius formula(A10a), we can write the plane partition state in terms of jNk⃗ðaÞ⟫,
jΛi ¼⨂N
a¼1
X
⃗kðaÞ
χΛðaÞ½Cðk⃗ ðaÞÞ zk⃗ðaÞ jNk⃗ðaÞ⟫;
¼X
⃗K
χΛ½CðKÞ⃗
ZK⃗ jΛðNK⃗ Þ⟫: ð45Þ
Its inverse is
jΛðNK⃗ Þ⟫¼X
Λ
χΛ½CðKÞjΛi:⃗ ð46Þ
The relations (45) and (46) define the plane partition fermion-boson correspondence. From (43), it is clear that jΛj ¼ jΛðNK⃗ Þj, therefore, the correspondence between the plane partition basis preserves thenumber of boxes grad- ing. It can be seen form the fact that the only nontrivial characters associated to the Young diagram Λ are those related to the conjugacy classes ofjΛj. In AppendixCwe write these relations for a few states.
We would like to represent the growth operators, and consequently the Hamiltonian (2b), in terms of the new bosonic basisjΛðNk⃗ðaÞÞ⟫. It is easy to see that since the map preserves the number of boxes
ΓjΛðNK⃗ Þ⟫¼X
Λ
χΛ½CðKÞ⃗ X
□∈QþðΛÞ;L⃗
χΛþ□½CðLÞ⃗
ZL⃗ jΛðNL⃗ Þ⟫;
≡ X
□∈QþðΛðNK⃗ÞÞ
AþðΛðNK⃗ Þ↦ΛðNK⃗ Þ þ□Þ
×jΛðNK⃗ Þ þ□⟫; ð47aÞ
where jΛðNL⃗ Þj ¼ jΛðNK⃗ Þj þ1 and P
⃗L¼P
□∈QþðΛðNK⃗ÞÞ. Therefore, we have the amplitudes
AþðΛðNK⃗ Þ↦ΛðNK⃗ Þ þ□Þ
¼X
Λ
X
□∈QþðΛÞ
χΛ½CðKÞ⃗ χΛþ□½CðLÞ⃗
ZL⃗ ; ð47bÞ
where L⃗ is the conjugacy class associated to the plane partitionΛðNL⃗ Þ ¼ΛðNK⃗ Þ þ□.
Similarly,
ΠjΛðNK⃗ Þ⟫¼X
Λ
χΛ½CðKÞ⃗ X
□∈Q−ðΛÞ;J⃗
χΛ−□½CðJÞ⃗ Z⃗J
jΛðNJ⃗Þ⟫
≡ X
□∈Q−ðΛðNK⃗ ÞÞ
A−ðΛðNK⃗ Þ↦ΛðNK⃗ Þ−□Þ
×jΛðNK⃗ Þ−□⟫; ð48aÞ
where jΛðNJ⃗Þj ¼ jΛðNK⃗ Þj−1 and P
⃗J ¼P
□∈Q−ðΛðNK⃗ÞÞ. The amplitudes are
A−ðΛðNK⃗ Þ↦ΛðNK⃗ Þ−□Þ
¼X
Λ
X
□∈Q−ðΛÞ
χΛ½CðKÞ⃗ χΛ−□½CðJÞ⃗ Z⃗J
; ð48bÞ
where J⃗ is the conjugacy class associated to the plane partitionΛðNJ⃗Þ ¼ΛðNK⃗ Þ−□. We give the amplitudes for the 0-, 1- and 2-box transitions in AppendixD.
Working with the (bosonic) plane partition statesjNK⃗ ⟫ seems to be much more involved, but it, in fact, has some conceptual advantages. In particular, the action of the growth operators on jNK⃗ ⟫ suggests a relation between these states and the MacMahon representation of the affine Yangian ofglð1Þb [20–24]. More specifically, we have the following conjecture:
The growth operators Γ, Π and OðΓ;ΠÞ are linear combinations of the generators of the algebraY½glð1Þ,b and the states jNK⃗ ⟫ fulfill the Y½glð1Þb MacMahon representation.
The algebraY½glð1Þb is an associative algebra defined by the generatorsfej; fj;φjjj≥0gand a set of (anti)commu- tations that can be found in[20]. One of the most important features of the affine Yangian of glð1Þb is its triangular decomposition Y½glð1Þ ¼b Yþ ⊕B⊕Y− which is gen- erated, respectively, by ej,φjand fj.
As a first evidence in favor of our conjecture, it has been shown that in the MacMahon representation studied by Procházka[20], the operatorsejandfjact precisely as in (47a)–(48a), that is
ejjΛ⟫¼ X
□∈Qþ
EjðΛ↦Λþ□ÞjΛþ□⟫;
fjjΛ⟫¼ X
□∈Q−
FjðΛ↦Λ−□ÞjΛ−□⟫: ð49Þ
Furthermore, the character of this representation is pre- cisely the MacMahon function(3), and the operatorsφjact diagonally as
φjjΛ⟫¼φj;ΛjΛ⟫: ð50Þ Finally, if the conjectured correspondence between the generators ofY½glð1Þb and with the growth operatorsΓand Π is indeed correct, we immediately conclude that OðΠ;ΓÞ∈B since they can be written as linear combi- nations ofφj. In other words, both systems are precisely the same once we identify the amplitudes A with linear combinations of the coefficients Ej and Fj. We then conclude that the Hamiltonian(2b) can be written as H¼−JX
j∈Zþ
ajðej−fjÞ þVX
j∈Zþ
ffiffiffi pq
bjþ 1 ffiffiffiq p cj
φjþVðc0φ0þc1φ1Þ; ð51Þ
where the coefficientsaj,bjandcjare necessary to define the equivalence, including combinatorial terms, without introducing new constraints to the Yangian Y½glð1Þ.b Observe that we write the generatorsφ0andφ1separately since they are central elements.
VII. CONCLUSIONS AND OUTLOOK In this work we have studied the 3D quantum crystal melting of[9], governed by the Hamiltonian(2b)in various guises. An unanswered question is whether or not this system is quantum integrable, as is the case for its 2D cousin. In the current paper, we push this study further. We have rewritten the dimer model as a fermionic system with sixth-order interactions in a Kagome lattice. We can think of this relation as a duality, where the extended object (the dimer) in the hexagonal lattice corresponds to a particle in
the dual Kagome lattice. This description and its possible advantages have not yet been fully explored.
Using the tensor product representation of the plane partitions, we have argued that the growth operators have a nontrivial coproduct, which may point to the existence of underlying quantum group structures. In fact, it is interest- ing to explore the conjectured relation between this structure and the coproduct of the affine Yangian of glð1Þ. Additionally, using the fermion-boson correspon-b dence, we have been able to extend the duality to the 3D plane partition states. Putting all these facts together, we managed to write the crystal melting Hamiltonian in terms of Yangian generators, although the conjecture is yet to be proved.
There is a myriad of interesting problems still to be addressed. As mentioned, the Kagome lattice description is still poorly understood and it may be an interesting description from the viewpoint of condensed matter physics. It would be very desirable to study the dynamics of the more general Hamiltonian (22), and see how the states classified by plane partitions emerge from the limit J0¼0.
We have not specified details of the YangianY½glð1Þ,b but we should mention that we have been able to use its representation because there are isomorphisms relating the Yangian Y½glð1Þ, theb W1þ∞ and the spherical Hecke algebra SHc, see [20,21]. More specifically, the linear W1þ∞ has a well-known representation in terms of free fermions, and in Sec. VI we have used the Heisenberg subalgebra uð1Þˆ ⊂W1þ∞ to define the bosonic plane partition states.
We should also observe that the maps relating the Heisenberg and the Yangian generators are of the form α−m ∝adm−1e1 e0andαm ∝adm−1f
1 f0; and that the Heisenberg subalgebra is the only ingredient we used in the definitions of the bosonized expressions. Therefore, it is possible that the only nontrivial coefficients in (51) are a0 and a1. Finally, we have specifically realized the construction in terms of the linearW1þ∞, but one can try to generalize it to the nonlinearW1þ∞ domain assuming, for example, that the Hamiltonian(51) is the defining problem.
In conclusion, despite the evidence in favor of the integrability of the Hamiltonian (2b), we have not yet been able to close this question. Let us hope nature does not disappoint us.
ACKNOWLEDGMENTS
T. A. is supported by the Swiss National Science Foundation under Grant No. PP00P2_183718/1.
D. O. acknowledges partial support by the NCCR 51nf40141869 The Mathematics of Physics (Swissmap).
The work of S. R. is supported by the Swiss National Science Foundation under Grant No. PP00P2_183718/1.
APPENDIX A: 2D FERMION-BOSON DUALITY Bosonization in two dimensions can be rephrased as a duality between the fermionic and bosonic Hilbert spaces.
In this section we would like to review the necessary ingredients for the construction of the fermion-boson duality for plane partitions of Sec.VI, in this section we follow [3,7–9,26,27].
Neveu-Schwarz free fermions states can be classified in terms of integer partitions. Using the algebra
fψm;ψng ¼ fψm;ψng ¼0;
fψm;ψng ¼δmn; ðA1Þ the fermionic state jλi in the Hilbert spaceH is given by
jλi ¼εYr
i¼1
ψ−mi−1=2ψ−ni−1=2j0i; ðA2Þ where the vacuumj0i is defined by the conditions
ψnj0i ¼ψnj0i ¼0 ∀ n >0; ðA3Þ andε¼ ð−1ÞPr
niþrðr−1Þ=2.
One can now associate the plane partitionλ¼ðλ1;λ2;…Þ to the state above, which in Frobenius coordinates reads
λ¼ fðmijniÞji¼1;…; rg: ðA4Þ The number of boxes associated to this Young diagram is
jλj ¼X
j
λj¼X
i
ðmiþniÞ: ðA5Þ
In two dimensions, one can expand the same Hilbert spaceHin terms of a chiral bosonic field∂ϕðzÞdefined by the components
αm ¼ X
j∈Zþ1=2
∶ψ−jψjþm∶; ðA6aÞ
which satisfy the Heisenberg algebra
½αm;αn ¼mδmþn;0: ðA6bÞ
Using these operators, states in H are of the form jki⃗ ≡jk1; k2;…i ¼Y∞
j¼1
ðα−jÞkjj0i: ðA7Þ
The relation between the two basesjki⃗ andjλiis given by
jλi ¼X
⃗k
χλ½CðkÞ⃗ zk⃗
jki;⃗ ðA8aÞ
and its inverse is
j⃗ki ¼X
λ
χλ½Cð⃗kÞjλi; ðA8bÞ
wherez⃗k¼Q∞
j¼1kj!jkj, andχλ½CðkÞ⃗ is the character of the symmetric groupSn evaluated in the representationλand conjugacy classCðkÞ. Moreover, the number of boxes in⃗ the partitionλ is the level of the statejki, that is⃗
jλj ¼X
j≥1
jkj: ðA9Þ
The characters can be easily calculated using the Frobenius character formula[28]
χλ½CðkÞ ¼⃗
Δð⃗xÞYr
j¼1
Pjð⃗xÞkj
ðl1;…;lmÞ
; ðA10aÞ
with
Δð⃗xÞ ¼Y
i<j
ðxi−xjÞ; Pjð⃗xÞ ¼Xm
i¼1
xji: ðA10bÞ
The vector⃗kis assumed to have a finite number of nonzero terms, that isk⃗ ¼ ðk1;…; krÞ. The auxiliary coordinates are
⃗x¼ ðx1;…; xmÞ and m is the number rows of the Young diagram λ¼ ðλ1;…;λmÞ. Furthermore, we also have the coefficients
lj¼λjþm−j; j¼1;…; m; ðA11Þ and given a generic polynomialfð⃗xÞ, we denote
fð⃗xÞjðl1;…;lmÞ ≔coeff of xl11 xlmm: ðA12Þ For completeness, we give here the character tables we use in the text:
χð1Þ½Cð1;0;…Þ ¼1; ðA13Þ
ð0;1;0;…Þ ð2;0;…Þ
χð1;1Þ −1 1
χð2Þ 1 1
ð0;0;1;…Þ ð1;1;0;…Þ ð3;0;0;…Þ
χð1;1;1Þ 1 −1 1
χð2;1Þ −1 0 2
χð3Þ 1 1 1
APPENDIX B: KAGOME LATTICE HAMILTONIAN
In this section we use the operatorsΓandΠto argue that the Hamiltonian (20) in the Kagome lattice is equivalent to (2b).
Let us first write some states. The 2-boxes configura- tions are
ðB1aÞ
and inversely,
ðB1bÞ
Additionally, the 3-box configurations are
ðB2aÞ
and inversely,
ðB2bÞ
Terms with more boxes can be written in a similar manner. In Fig. 13we give the 5-box configuration
ðB3Þ
in the dual lattice.
In summary, we write the kinetic term of the Hamiltonian (2b)as
ðB4Þ
For the potential term, we need to write operators that give the number of places where we can add a box, and the number of boxes that can be consistently removed. These operators can be written as products of Γ and Π. For example, given the empty configuration, it is easy to see that
X
i;j;k;l
Γi;jΠk;lj∅i ¼0; X
i;j;k;l
Πi;jΓk;lj∅i ¼ j∅i; ðB5Þ
that, naturally, means that we have one available place to add a box, and we do not have any box to be removed.
For two boxes, we see that there must be relations between the two sets of indices ði; jÞ andðk; lÞ. Initially we have
ðB6Þ
and it is obvious that we have one box to be removed.
Furthermore,
ðB7Þ
Therefore
ðB8Þ
and now we have three places to add a box to this configuration. It is straightforward to see that this logic remains the same for other configurations.
FIG. 13. A 5-box configuration.
Furthermore, we also have
ðB9Þ
and the other 2-box states, and , work similarly.
For three boxes we have two distinct cases. The first case is
ðB10Þ
and the 3-box states and , work similarly.
Additionally,
ðB11Þ
and the states and work similarly.
Using this logic, one may see that the Hamiltonian(20)is equivalent to(2b).
APPENDIX C: BASIS TRANSFORMATION Using the characters(A13)we can find the basis trans- formation (45)–(46) for some states. For example, the 1-box state is simply
ðC1aÞ
The states with two boxes are
ðC1bÞ
and the states with three boxes are
ðC1cÞ
The nontrivial inverse relations are
ðC2aÞ
and
ðC2bÞ
and finally
ðC2cÞ
APPENDIX D: AMPLITUDES
Using formulas (47b) and (48b) we can find the amplitude A for the first states. It is straightforward to see that
ðD1Þ
where χ∅ðC½⃗0Þ ¼1. The transitionsð1↦2Þ-boxes are
ðD2aÞ
where we have used (42c)to see that χð1Þ;ð1Þ½C½ð0;1;0;…Þ
¼χð1Þ;½C½ð0;1;0…Þχð1Þ½C½ð0;…Þ ¼0: ðD2bÞ Similarly,
ðD2cÞ
and
ðD2dÞ
It is also immediate to see that the transition ð1↦0Þ- boxes is
ðD2eÞ
Therefore
ðD2fÞ
We can repeat the analysis above to consider the transitions ð2↦3Þ-boxes and (2↦1)-box. These are given by
ðD2gÞ
ðD2hÞ
ðD2iÞ
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