SciPost Phys. 16, 003 (2024)

On correlation functions in models related
to the Temperley-Lieb algebra
Kohei Fukai1 , Raphael Kleinemühl2 , Balázs Pozsgay3⋆ and Eric Vernier4
1 The Institute for Solid State Physics, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8581 Japan
2 Fakultät für Mathematik und Naturwissenschaften,
Bergische Universität Wuppertal, Wuppertal, Germany
3 MTA-ELTE “Momentum” Integrable Quantum Dynamics Research Group,
Department of Theoretical Physics, Eötvös Loránd University, Budapest, Hungary
4 Laboratoire de Probabilités, Statistique et Modélisation
CNRS - Univ. Paris Cité - Sorbonne Univ. Paris, France⋆ pozsgay.balazs@ttk.elte.hu

Abstract
We deal with quantum spin chains whose Hamiltonian arises from a representation of theTemperley-Lieb algebra, and we consider the mean values of those local operators which
are generated by the Temperley-Lieb algebra. We present two key conjectures which
relate these mean values to existing literature about factorized correlation functions in
the XXZ spin chain. The first conjecture states that the finite volume mean values of thecurrent and generalized current operators are given by the same simple formulas as in
the case of the XXZ chain. The second conjecture states that the mean values of productsof Temperley-Lieb generators can be factorized: they can expressed as sums of productsof current mean values, such that the coefficients in the factorization depend neither
on the eigenstate in question, nor on the selected representation of the algebra. The
coefficients can be extracted from existing work on factorized correlation functions in
the XXZ model. The conjectures should hold for all eigenstates that are non-degeneratewith respect to the local charges of the models. We consider concrete representations,where we check the conjectures: the so-called golden chain, the Q-state Potts model,
and the trace representation. We also explain how to derive the generalized current
operators from concrete expressions for the local charges.
Copyright K. Fukai et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 25-09-2023
Accepted 20-12-2023
Check forupdatesPublished 05-01-2024
doi:10.21468/SciPostPhys.16.1.003

Contents
1 Introduction

2

2 Temperley-Lieb algebra and its representations
2.1 Irreducible representations of the (periodic) TL algebra
2.2 The XXZ model

4
5
6

1

SciPost Phys. 16, 003 (2024)

2.3
2.4
2.5

The Potts model
The golden chain
The trace representation

7
8
9

3 Charges and currents in the Temperley-Lieb algebra
3.1 Integrability
3.2 Local charges and currents in Temperley-Lieb models
3.3 Construction of the current operators
3.4 Construction of the generalized current operators

9
10
11
12
12

4 Factorized correlation functions in XXZ
4.1 Hidden Grassmann structure
4.2 Specific results for mean value of the current operators

13
13
14

5 Correlation functions
5.1 Main results and numerical checks
5.2 Special bi-linear combinations

15
16
18

6 Conclusions and outlook

18

A Current operators of Temperley-Lieb Hamiltonian

19

B Numerical checks
B.1 Definition of the transfer matrix in the XXZ representation
B.2 Numerical data

20
21
21

References

29

1

Introduction

One dimensional quantum integrable models are special systems, which allow for exact solutions, at least for certain physical quantities and in certain physical situations [1, 2]. The
Bethe Ansatz is a central method which is used to solve many such models. The strength of
the Bethe Ansatz lies in diagonalizing the Hamiltonian: in finding the eigenvectors and theassociated eigenvalues. However, the computation of correlation functions is a notoriously
difficult problem, already in equilibrium situations.
In this work, we contribute to the computation of correlation functions in a selected classof integrable models: those quantum spin chains, which are related to the Temperley-Liebalgebra.
The Temperley-Lieb algebra was discovered in the paper [3], where it was used to computethe partition function of selected 2D statistical physical models by relating them to the XXZ
Heisenberg spin chain. The key observation is that the defining relations of the algebra arestrong enough to guarantee equivalences in the spectrum of different models. These models
are then seen as different representations of the same algebra. The precise statement is thatthe eigenvalues of the Hamiltonians in some representation of the Temperley-Lieb algebra
are included in the spectrum of the XXZ model (with the same volume, and open boundary
conditions). However, the degeneracies of the states can differ in the various representations[4–6] (see also [7]).

2SciPost Phys. 16, 003 (2024)

As opposed to the many studies dealing with the spectrum and the representation theory,the correlation functions of these models have received far less attention. In fact, we are notaware of any work dealing systematically with this problem. Specific correlation functions in
Temperley-Lieb models have been investigated in selected cases [8, 9], but we are not awareof a general treatment. Meanwhile, correlation functions of the XXZ spin chain have been
investigated for multiple decades (see the book [10] and the habilitation thesis [11]). This
leads to the idea of utilizing the Temperley-Lieb algebra to make connections between the XXZ
chain and other representations also on the level of the correlation functions, thus yielding
useful results for many models for which there are no results for correlators in the literature.
It is our goal to use the theory of factorized correlation functions of the XXZ chain (alsoknown as the “hidden Grassmann structure”) to compute correlations for other TemperleyLieb models. In the Heisenberg chain, this theory was initiated in [12], where it was observedthat certain multiple integrals for correlation functions in the XXX chain factorize: they can be
expressed as sums of products of single integrals. This led to the development of a complete
algebraic theory, which led to convenient factorized formulas for the mean values of shortrange operators [13–20]. On a practical level, the theory states that mean values of shortrange correlators can be expressed using the Taylor coefficients of a few number of functionsof one and two variables. The theory consists of two parts: the algebraic part expresses thecorrelation functions using sums of products of these building blocks, whereas the physical partgives concrete values to the building blocks, depending on the concrete physical situations. In
the first works, the ground state and finite temperature [21–23] mean values were considereddirectly in the infinite volume limit. An extension to the finite volume ground state was given
in [24]. Excited states (with arbitrary Bethe root distributions) were later treated in [25], andfinally, an extension to arbitrary finite volume excited states was given in [26].
As an alternative method towards factorized correlations, functional relations for reduced
density matrices were considered in [27], and later for higher spin versions of the XXZ chain in[28,29]. Higher rank models were considered in [30–32]. The work [33] raises the question ofwhether all correlation functions of higher rank models can be expressed in a simple factorized
form, and [34] treated factorized correlations in the XYZ model.
A new contribution to the theory was given in the works [35–37], where a connection wasestablished to Generalized Hydrodynamics (GHD), a theory describing the large-scale transport properties of integrable models [38,39]. The papers [35–37] considered the mean valuesof current operators (including the so-called generalized current operators), which describe
the flow of conserved charges during real-time evolution. It was found that these mean values
are exceptionally simple: they are given simply by the Taylor coefficients of the so-called ω
function. In the finite volume case, these building blocks can be expressed in a very simple
way: they involve the one-particle charges corresponding to the Bethe roots and a single copyof the inverse of the so-called Gaudin matrix. The final expression has a semi-classical interpretation [35,40]: the currents of the conserved charges are given by a sum over the one-particle
charge eigenvalues, multiplied by a certain effective velocity describing particle propagation in
the background of the other particles. This simple result demonstrates the completely elastic
and factorized scattering characteristic of integrable models, and it underlies the formulationof Generalized Hydrodynamics. The result for the current mean values was later extended to
the XYZ model in [41].
It is then very natural to ask whether some of the above results about the currents and thefactorized correlators can be worked out also for other integrable quantum spin chains. The
algebraic construction of [37] appears rather general, but so far it has been applied only to
the XXZ and XYZ spin chains. However, there is motivation to study other types of models as
well.

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SciPost Phys. 16, 003 (2024)

As examples, we mention the integrable quantum spin chains acting on constrained Hilbert
spaces with the Rydberg blockade [42]. The Hamiltonians of these models are related to
the Restricted Solid On Solid (RSOS) models of Andrews, Baxter, and Forrester [43, 44], andthey have been studied for example in [45, 46]. Interest in these models also comes fromthe various studies on the so-called PXP model and its relatives, see [47–52]. The transport
in these integrable models has been studied numerically in [51], but GHD has not yet been
established. At the same time, short-range correlation functions in these models were treatedin [9, 53], although general formulas have not yet been found. A specific model in this familyis the so-called “golden chain” introduced in [54]. It is a special point in the integrable family
where the Hamiltonian density satisfies the Temperley-Lieb algebra.
The goal of this work is to use the connections to the XXZ chain to find exact correlation
functions in the golden chain and other Temperley-Lieb models. Such an approach cannot
yield the factorized correlation functions of all short-range operators, but it is natural to expect
exact relations for correlators built from the Temperley-Lieb algebra. This is the task that we set
ourselves in this work. Apart from the golden chain, we will consider the Q-state Potts modeland also the so-called trace representation of the Temperley-Lieb algebra, which appeared
recently in the study of Hilbert space fragmentation [55].
In the main text, we formulate the conjecture that for operators constructed in terms of
the Temperley-Lieb algebra, the factorization of the mean values works in essentially the sameway in all representations of the Temperley-Lieb algebra, and only minor modifications need
to be added. This seems to hold for all states with singlet eigenvalues of commuting set oftransfer matrices. Thus, we claim that both the algebraic part and the physical part of theconstruction will be essentially the same. In particular, we conjecture that the physical part canbe computed in finite volume using the formulas first derived in [26]. We test our conjectures inseveral representations. We also develop a method to compute generalized current operatorsfrom local charges, and afterward, we present a few concrete generalized current operators
expressed via the Temperley-Lieb generators. These concrete formulas are then used to check
our conjectures. We should note that in [56], explicit and exact formulas were found for all
local conserved charges in the Temperley-Lieb models. However, we use a different basis ofcharges and, therefore, do not directly use the results of [56].
This paper consists of the following Sections: In Section 2 we review the Temperley-Liebalgebra and its several representations and explain the decomposition of the spectrum. In
Section 3, we discuss the local charges in the Temperley-Lieb algebra and derive a way toconstruct the current operators. Section 4 gives a review of the factorization of correlation
functions in the XXZ chain. In Section 5, we formulate our main conjectures and give some
explicit formulas for short-range correlators. Section 6 contains our conclusions. Finally, someexamples of the expressions for the local charges and currents in the Temperley-Lieb algebra
are given in Appendix A, and some numerical checks are presented in Appendix B.

2

Temperley-Lieb algebra and its representations

The Temperley-Lieb (TL) algebra is defined as follows. There are generators e j with index
j = 1, . . . , L − 1, which satisfy the relations
e2j = d e j ,e j e j±1 e j = e j ,[e j , ek ] = 0 ,for| j − k| > 1 .

(1)Here, d ∈ C is a fixed parameter of the algebra, which is also commonly parametrized as
d = q + q−1 = 2 cos γ ,4

q = e iγ .

(2)

SciPost Phys. 16, 003 (2024)

Here, we are interested in the case of periodic boundary conditions, where another generator e L is introduced, satisfying [5]:e2L = d e L ,eL eb eL = eL ,eb eL eb = eb ,[e L , e j ] = 0 ,(3)where b = 1, L − 1 and j ̸= 1, L − 1. The TL Hamiltonian with periodic boundary conditions isthen defined as
L
X
H=
ej .
(4)
j=1

The Temperley-Lieb algebra has multiple representations that are of physical interest. In
particular, we will be interested in representations such as the XXZ spin chain, the quantum
Potts chain, the RSOS representations (also known as anyon chains), and the so-called “trace
representation”.
A crucial fact is that since the Hamiltonian (4) is part of the (periodic) TL algebra, its
spectrum in a given representation is entirely determined by the way the latter decomposes
as a sum of irreducible representations. Starting from a given model, an important task is,therefore, to see how its Hilbert space decomposes as a direct sum of the TL irreducible representations.
Below, we discuss a few key statements about the representation theory of the TL algebra,
and concrete representations will be considered in the following sections.

2.1

Irreducible representations of the (periodic) TL algebra

The TL algebra (1) can be viewed as an algebra of diagrams acting on L vertical strands, byassigning to the generators ei the following graphical representation [57]
...

ei =

...

1

i

(5)

i+1

L

Multiplication of two generators corresponds to stacking the corresponding diagrams on topof each other, and the rules (1) translate into the fact that diagrams are identified modulo
smooth stretching of strands, or removing of closed loops at the cost of a multiplicative scalar
factor d.
A natural basis for representations of the TL algebra is constructed in terms of “reduced
states”, obtained by cutting in half horizontally all possible diagrams formed by products ofTL generators. Those are constituted of a set of non-intersecting arcs joining pairs of strands,
and “through lines”, or “strings”, which propagate vertically without the possibility of being
contracted with one another by the action of TL generators. Irreducible representations arecharacterized by their number 2 j of through-lines (where 0 ≤ j ≤ L/2 is an integer when L iseven, a half-integer when L is odd), and denoted as W j [L]. For instance, for L = 4 there arethree irreducible modules with respective basis states

W0 [4] = {,},W1 [4] = {,,W2 [4] = {

}.

},

(6)

Turning to periodic boundary conditions by introducing the generator e L ≡ e0 , the TL
algebra becomes infinite-dimensional, due to the possibility for through-lines to wind an arbitrary number of times around the cylinder, as well as, in the sector with no through-lines,
the possibility for an arbitrary number of non-contractible loops wrapping around the cylinder.

5

SciPost Phys. 16, 003 (2024)

Finite-dimensional quotients can be recovered by allowing to undo the winding of throughlines at the price of multiplication by a complex factor z ±1 (one such factor per through-line
per turn around the cylinder), and to eliminate non-contractible loops at the price of multiplication by a factor z + z −1 [58, 59]. The corresponding irreducible representations are denotedby W j,z [L], and generically have more states than the open TL ones. For instance, for L = 4,basis states for the representations with j ̸= 0 are

W1,z [4] = {,W2,z [4] = {

}.

,

},

,

(7)

For j = 0, we have to distinguish arcs which connect two strands by going one way or theother around the cylinder, therefore, a basis isW0,z [4] = {

,

,

,

,

,

},

(8)

where an arc marked by a dot means that it goes around the periodic boundary conditions of
the cylinder. All these representations have dimension


L
,
(9)
dim W j,z =L/2 − jirrespectively of z.
These representations are irreducible for generic q and z. For specific cases, however (qequal to a root of unity or z equal to some integer power of q), they become reducible but
indecomposable, as a result of the TL algebra becoming non-semisimple (similar conclusions
hold for the open case at root of unity). We refer to the existing literature [57, 60, 61] for amore detailed exposition.

2.2

The XXZ model

In this case, the generators act on the Hilbert space of a spin-1/2 chain, and we havee j = hXXZ
j, j+1 ,(10)

where hXXZ
j, j+1 is a two-site operator given by [62]
1  iφ/L + −2eσ j σ j+1 + 2e−iφ/L σ−j σ+j+1 + cos γ σzj σzj+1 − 1 + i sin γ σzj − σzj+1 ,2(11)
where γ is the parameter introduced above (see eq. (2)), in practice, restricted to be realor pure imaginary. It is related to the so-called anisotropy parameter ∆ of the XXZ chain by∆ = cos γ. Furthermore, φ ∈ C is the so-called twist parameter. In our representation, we
chose to apply a homogeneous distribution of the twist and focus on the periodic case. Notethat the normalization of the Hamiltonian density is now different from the usual one: now it
includes an overall factor of −1/2 so that hXXZ
j, j+1 satisfies the Temperley-Lieb algebra.
The XXZ model can be solved by the Bethe Ansatz. Eigenstates are constructed using
interacting spin waves, that are created on top of a reference state. The resulting eigenstates
are characterized by a set of rapidities λN = {λ1 , . . . , λN }, which satisfy the Bethe equations
hXXZ
j, j+1 = −

e

iφ



sinh(λ j + iγ/2)

L Ysinh(λ j − λk − iγ)sinh(λ j − iγ/2)k̸= jsinh(λ j − λk + iγ)

= 1.

(12)

Here ∆ = cos(γ). If |∆| < 1 then γ ∈ R and the ground state configuration consists of realroots.
6

SciPost Phys. 16, 003 (2024)

The energies of the states (eigenvalues of H with the normalization given above) are
given byNX
E=ϵ(λ j ) ,
(13)
j=1

with
ϵ(λ) =

sin2 γ.
sinh(λ + iγ/2) sinh(λ − iγ/2)(14)

Decomposition ofP
the spectrum The Hilbert space can be split into sectors of fixed magL1z
netization S = 2 i=1 σzi , and any operator written in terms of the TL generators (11) isblock-diagonal in this decomposition. In fact, for generic γ and φ, we can identify each ofthese sectors with one of the irreducible representations of the periodic TL algebra. Recallingfrom Section 2.1 that those are indexed by a number 2 j of “through lines” and by a “twist”
parameter z, and denoted W j,z , we find that the XXZ sector of magnetization S z and twist φidentifies as the representation W j,z , with the following correspondence
j = |S z | ,z = e iφ .

(15)Note in particular that sectors of opposite magnetization correspond to the same representation of TL and have the same spectrum.
Note that for a given TL representation, we can always find the same eigenenergy withina sector of the XXZ chain with the same parameter d and with an appropriate twist up to thedegeneracy [5].

2.3

The Potts model

The Q-states quantum Potts model is defined on a chain of L/2 sites carrying Q-dimensional
spins. On each sites one defines matrices X and Z, generalizing the Pauli matrices σ x and σz ,which satisfy the following “ZQ clock” algebra
X † = X Q−1 ,Z † = Z Q−1 ,X Q = ZQ = 1 ,X Z = ωZ X ,

(16)

2π

where ω = e i Q . A concrete representation can be obtained, for instance, by taking


Z =


1
ω
..

.






,




X =


ωQ−1

0

1

1
..
.


..

.



.
..
. 1 
0

(17)

From there, the generators e1 , . . . e L defined as [6]Q−1a1 X †
e2 j = p
X j X j+1 ,
Q a=0

Q−1

1 X a
e2 j+1 = p
Zj ,
Q a=0

(18)

p
satisfy the Temperley-Lieb algebra with d = Q, and periodic boundary conditions. Furthermore, they are manifestly Hermitian operators.

7

SciPost Phys. 16, 003 (2024)

Decomposition of the spectrum As for the XXZ case, we can decompose the Hilbert spaceof the periodic Potts chain in terms of the representations W j,z , where we recall the correspondence (15) between the parameters j, z and the magnetization andp twist in the XXZ chain.
We start with the case Q = 3, which corresponds to d = 3 = 2 cos(π/6) and hence
γ = π6 . At such “roots of unity” cases (namely, whenever q is a root of unity), the TL algebra is
known to be non-semisimple, which means that the representations W j,z become reducible but
indecomposable: they cannot be decomposed as a direct sum of irreducible representations.
For our matters, we will not need to go into the details of this complicated subject, and werefer the interested reader to the existing literature [57, 59–61]. In practice, we should only
stick to the observation that at the root of unity, the spectrum of the TL Hamiltonian (or moregeneral operators built out of the TL algebra) in representation W j ′ ,z ′ may arise as a subset ofthe spectrum in a larger W j,z , and we note W j,z /W j ′ ,z ′ the remaining subspace. For L = 4, we
find that the Potts chain Hilbert space decomposes as

HQ=3 = W0,−1 ⊕ (W0,q2 /W1,1 ) ⊕ W2,1 .

(19)

Similar decompositions can be written for other system sizes L (note that for j = 2 here, or
j = L/2 in general, W L/2,z is a one-dimensional representation where all TL generators cancel,
so the precise value of the twist z does not matter).
p
We now turn to Q = 4 and Q = 5, which correspond to γ = 0 and γ=arccos( 5/2)≃0.48i ,respectively. Those are not “root of unity” points of the kind discussed above, however we shallstill encounter quotients of the form W j,z /W j ′ ,z ′ . The reason is that, as explained in [8] (seealso [58, 62]), even for generic q representations W j,z become reducible when z = q2 j+2k ,where k is some positive integer, and contain some irreducible submodule isomorphic to
W j+k,q2 j . For Q = 5 we find accordingly, for L = 4

HQ=5 = (W0,q2 /W1,1 ) ⊕ 2W0,−1 ⊕ 11W2,1 ,(20)

1
HQ=4 = W0,1 ⊕ (W0,−1 /W1,1 ) ⊕ W0,−1 ⊕ 5W2,1
2(21)

while for Q = 4, L = 4(in this last case, other subtleties arise, leading in particular to the 12 factor which results fromthe fact that W0,−1 contains two copies of the same irreducible representation, but we shallnot discuss these further here).

2.4

The golden chain

The Hamiltonian of the so-called golden chain was published in [54]. It is a special case of thefamily of Hamiltonians related to the RSOS models [45, 46].
In this case, the Hilbert space is constrained: in a volume L it is spanned by the states of
the computational basis, which do not have two neighboring down spins.
Let us define the local projectors
P j = (1 + σzj )/2 ,N j = (1 − σzj )/2 .

(22)

Then, the constraint can be formalized as
N j N j+1 = 0 .

(23)

A representation of the Temperley-Lieb algebra is the following:e j = h j, j+1, j+2 ,8(24)

SciPost Phys. 16, 003 (2024)

where h j, j+1, j+2 is a three-site operator acting on the constrained Hilbert space, given explicitly by
h j, j+1, j+2 = −ϕ (P j + P j+2 − 1) − P j P j+2 ϕ −3/2 σ xj+1 + ϕ −3 P j+1 + ϕ −2 + 1 ,(25)
where ϕ = (1 +d = ϕ.

p5)/2 = 2 cos(π/5) is the golden ratio. The Temperley-Lieb parameter is

Decomposition of the spectrum As for the XXZ and Potts representations, the RSOS Hilbert
space can be decomposed in terms of the TL standard representations. We have here q = e iπ/5 ,again a root of unity, so similar comments to those made above for the Potts chain can beaddressed here. We find, for L = 4:

HRSOS = (W0,q2 /W1,1 ) ⊕ (W0,q4 /W2,1 ) ,where again the correspondence with the XXZ parameters is encoded in (15).

2.5

The trace representation

In this representation, we are dealing with local Hilbert spaces Cd with d ≥ 2, and the generators are given by
e j = K j, j+1 ,(26)
where K is the so-called trace operator, given explicitly byK=dX|aa〉〈b b| .

(27)

a,b=1

In this case, the parameter of the Temperley-Lieb algebra is equal to the local dimension d(more generally, representations where d is a positive or negative integer can be constructedby using graded vector spaces [63]).
These models appeared recently in the study of Hilbert space fragmentation [55].
Decomposition of the spectrum We take d = 3 for the example. Here γ = arccos(3/2) ispure imaginary, and q = e iγ is not a root of unity. In terms of the modules W j,z , with again(15), we find for L = 2
Hd=3 = W0,q2 ⊕ 7W1,1 .
(28)
For L = 4, we find

Hd=3 = W0,q2 ⊕ 7W1,1 ⊕ 47W2,1 .

(29)

Hd=3 = W0,q2 ⊕ 7W1,1 ⊕ 27W2,1 ⊕ 20W2,−1 ⊕ 322W3,1 .

(30)

For L = 6, we find3

Charges and currents in the Temperley-Lieb algebra

In this Section we discuss the family of conserved charges and their currents in the models
related to the Temperley-Lieb algebra. To this order we first discuss the integrability properties,
and afterward we turn to the charges and currents.

9

SciPost Phys. 16, 003 (2024)

3.1

Integrability

The models defined by (4) are integrable, both in the periodic case and in the open case (where
the additional generator e L is absent in (4)). The Hamiltonians can be embedded into a familyof commuting transfer matrices. It is possible, by going to a specific representation, to constructcommuting transfer matrices in finite volume with periodic or open boundary conditions. For
the XXZ representation such transfer matrices are related to the six-vertex model [64]; in thiscase we give concrete formulas in the Appendix B.1. For the golden chain the transfer matrices
are related to integrable RSOS models [65]; for the Potts representation, they are based on
the Star-Triangle Relation [66,67]. However we do not know of a representation-independent
formulation in finite volume, and will therefore restrict to a definition in infinite volume, valid
for any representation.
First, we defineŘ j (u) = 1 + ue j ,(31)
where u ∈ C is a spectral parameter. Then, the formal definition of the transfer matrices canbe given asYt(u) =
Ř j (u) .
(32)
jFor the ordering of the operators, we choose a convention that operators with lower indicesact first. These operators should be regarded as a formal power series in u. Their finite volume
counterparts can also be specified.
It can be shown by formal manipulation that[t(u), t(v)] = 0 .

(33)

Let us now define the operators with k > 1
Ak = (∂u )k−1 log(t(u)) u=0 .

(34)

A2 = H .

(35)

We have
It follows from (33) that the Ak form a commuting family, and their construction ensures thatthey are extensive operators with a short-range operator density.
The alternative to the transfer matrix is the boost operator formalism [68–70]. This method
for obtaining the charges is also limited to the infinite volume case. We define formally

B=

∞
X

je j ,(36)

Q k+1 = [Q k , B ] ,(37)

Q2 = H .

(38)

j=−∞

and define a series of charges via the formal rulewith the initial condition
These charges are linear combinations of the Ak given by (34). The definition (37) does notyield Hermitian charges, in fact with this definition every second charge is anti-Hermitian.
However, we use this convention because it is convenient for our purposes.
The Hamiltonians (4) also enjoy translational invariance, either in finite volume with periodic boundary conditions, or formally in the infinite volume case. However, there is a subtlety
in connection with the translations.
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SciPost Phys. 16, 003 (2024)

For a given model, let us define U as the one-site translation operator acting on the physicalHilbert space. In all cases we have
[U , H] = 0 .
(39)Furthermore, U also commutes with the higher charges. It is then very natural to consider thesimultaneous diagonalization of U and the extensive local charges. The Bethe states will beeigenvectors of U as well, therefore U is the natural translational symmetry of both the charges
and the states.
However, the situation is more delicate in the Potts model. In that case, we can also define
the operator U based on its action in real space, but then U will shift the Temperley-Liebgenerators by two indices, due to the staggering introduced in (18). On the level of operators,
we have an additional symmetry V , which shifts the indices of the operators by one. However,on the level of the Hilbert space, the operation V is seen as a duality, and it is not guaranteedthat the Bethe states will be eigenvectors of V .

3.2

Local charges and currents in Temperley-Lieb models

The charges introduced above are extensive and are expressed asXqα ( j) ,
Qα =

(40)

j

where qα ( j) is a short-range operator density which can be expressed using the generators of
the Temperley-Lieb algebra.
We choose our conventions such that qα ( j) spans α sites in the XXZ representation, andwe choose q2 ( j) = e j , which also implies that Q 2 = H in our conventions.
The first non-trivial charge above the Hamiltonian isq3 ( j) = e j e j+1 − e j+1 e j .

(41)Further examples are found in the Appendix A.
The general explicit expressions for the local charges of the Temperley-Lieb models were
derived by Nienhuis and Huijgen [56]. However, these expressions are some linear combination of the charges obtained from expanding the transfer matrix constructed from the usual
6-vertex R-matrix in the XXZ representation, or obtained through the boost operation. Therefore, the concrete formulas of [56] coincide neither with our Ak nor with our Q k .
The current operators Jα (x) are defined through the continuity equations[H, qα (x)] = Jα (x) − Jα (x + 1) .

(42)

We also introduce the generalized currents Jα,β (x) [35,37] that describe the flow of qα (x)under the time evolution generated Q β by
Q β , qα (x) = Jα,β (x) − Jα,β (x + 1) .

(43)

The locality of the charge densities and the global relation [Q α , Q β ] = 0 imply that the operator
equation (43) can always be solved with a short-range operator Jα,β (x).
The definition (43) holds for α, β ≥ 2, but it will turn out convenient to extend it to β = 1
by setting
Jα,1 (x) ≡ qα (x) .
(44)

11

SciPost Phys. 16, 003 (2024)

3.3

Construction of the current operators

Below, we demonstrate the procedure for constructing the expressions of current operators andgeneralized current operators from the corresponding expressions of local conserved quantities. The construction of the current operators was originally derived for the spin-1/2 XYZchain [71]. Here, we generalize this result of [71] to the construction of the generalized current operators. Notably, this method applies to general quantum integrable spin chains of local
Hamiltonian. We first show the general procedure, which does not involve the Temperley-Liebalgebra. Afterwards, we apply the result to the Temperley-Lieb Hamiltonian.
We start by reviewing the procedure for constructing the current operators [71]. In thefollowing, we refer to an operator that acts on the x-th site and also locally on sites beyond thex-th site as “starting from the x-th site”. We assume qα (x) is starting from x-th site. Since the
Hamiltonian is a two-site operator, the left-hand side of (42) is constructed from the operator
starting from the (x − 1)-th, x-th, and (x + 1)-th sites, which we denote by Fα−1 (x − 1), Fα0 (x),and Fα1 (x + 1), respectively:[H, qα (x)] = Fα−1 (x − 1) + Fα0 (x) + Fα1 (x + 1) .

(45)

The current operator is given byJα (x) = Fα−1 (x − 1) − Fα1 (x) .

(46)

The proof of (46) is as follows: substituting (46) to the RHS of (42), we haveJα (x) − Jα (x + 1) = [H, qα (x)] − δα (x) ,(47)

where we defined δα (x) ≡ Fα−1 (x) + Fα0 (x) + Fα1 (x). Summing over x of (45), we havePL
0 = x=1 δα (x). Since {δα (x)} x=1,2,...,L are mutually linear independent, each δα (x) shouldbe zero itself. Now, we have proved (46) satisfies the continuity equation (42).
Concrete examples for current operators are found in the Appendix A.

3.4

Construction of the generalized current operators

We generalize the result for the current operator [71] to the generalized current operator.
Since Q β is a β-site operator, the left-hand side of (42) is written as
β−1
X



Q β , qα (x) =yFα,β (x + y) ,(48)

y=−(β−1)ywhere Fα,β (x + y) is an operator starting from x + y-th site. By summing up the x in (48) andin the same manner as the usual current operator case, we have
β−1
X

y

Fα,β (x) = 0 .

(49)

y=−(β−1)

The generalized current operator is given by
Jα,β (x) =

β−1
b 
XX


−b
b
Fα,β
(x − y) − Fα,β(x + y − 1) .

b=1 y=1

The β = 2 case of (50) recovers (46).

12

(50)

SciPost Phys. 16, 003 (2024)

Pβ−1 ±by
We prove (50) in the following. We define E± (x) ≡ b= y Fα,β(x), and we can write thePβ−1  yygeneralized current as Jα,β (x) = y=1 E− (x − y) − E+ (x + y − 1) . Substituting (50) to the
RHS of (43), we haveJα,β (x) − Jα,β (x + 1) =

β−1
X




y
y
E− (x − y) + E+ (x + y) −y=1

F y (x + y) +β−1≥| y|>0

−

β−1
X




y
y
E− (x − y + 1) + E+ (x + y − 1)

y=1

X

=

β−1
X



β−2
X

y+1

E−

y+1

(x − y) + E+


(x + y)y=1yyE− (x − y + 1) + E+ (x + y − 1)

y=1



0
= Q β , qα (x) − Fα,β(x) − E+1 (x) − E−1 (x)
β−1
X



= Q β , qα (x) −yFα,β (x)y=−β+1= Q β , qα (x) ,

(51)
y

y+1

y

β

where in the second equality, we have used E± (x) = E± (x) + Fα,β (x) and E± (x) = 0, and in
the last equality we have used (49). Thus, we have proved (50) actually satisfies (43).
In principle, we can obtain the explicit expressions of the generalized currents by evaly
uating the commutator in the continuity equation and subsequently obtaining Fα,β (x) andusing (50), provided that we have the explicit expressions of the local charges Q α .
The aforementioned index x need not strictly denote a physical site index; the only requirement is that the operators with different indices are to be linearly independent. Thus,
the above construction of the generalized currents is also applicable to the Potts representation where the indices of the Temperley-Lieb generators do not correspond to the physical site
indices.
Concrete examples for generalized current operators in the Temperley-Lieb algebra arefound in the Appendix A.

4Factorized correlation functions in XXZ

4.1

Hidden Grassmann structure

The theory of the factorized correlation functions [13–20] concerns the mean values of local
operators in the XXZ chain in a variety of physical situations. The goal is to compute meanvalues of the form
〈Ψ|O|Ψ〉 ,(52)
in finite volume or in the thermodynamic limit. The papers [13–20] considered the groundstate and finite temperature situations in infinite volume. The ground state in finite volume
was treated in [24], and an extension to arbitrary finite volume excited states was given in [26].
Excited states in thermodynamic limit were treated earlier in [25].
The results of the theory for the homogeneous limit can be summarized as follows. The
statements below hold for all the physical situations mentioned above, thus also for the finite
volume excited states considered in [26].
The mean values can be expressed as a sum of products of certain building blocks, which
are the Taylor expansion coefficients of a few number of functions of one and two variables.
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SciPost Phys. 16, 003 (2024)

For example, in the XXZ spin chain the mean values of spin reflection invariant operators are
expressed using the Taylor coefficients of just two functions ω(x, y) and ω′ (x, y), both ofwhich are symmetric with respect to the exchange of the two variables. The construction consists of the algebraic part and the physical part. The algebraic part describes the expressions of
the mean values of a given operator as a combination of the building blocks, and this computation is independent of the physical situation. The physical part gives concrete values to thefunctions involved, depending on the physical situation considered.
Simple examples of short-range correlators are:σ1z σ2z = coth(η)ω0,0 + W1,0 ,ω0,0cosh(η)σ1x σ2x = −−W1,0 ,
2 sinh(η)
2

σ1x σ3xω2,0 − 2ω1,1

sinh2 (η)
W2,1 ,
44
ω2,0 − 2ω1,1
W2,1
cosh(2η)1
ω0,0 −
W1,0 − tanh(η) cosh(2η)+ sinh2 (η).
=−
sinh(2η)
2
8
8
(53)

σ1z σ3z = 2 coth(2η)ω0,0 + W1,0 + tanh(η)−

Hereωa,b = ω b,a = (∂ x )a (∂ y ) b ω(x, y) x, y=0 ,(54)

and the function ω(x, y) coincides with the one defined in [21, 22], and Wa,b are given similarly by the coefficients of the function W (x, y) = ω′ (x, y)/η, where ω′ (x, y) is definedin [21, 22] and ∆ = cosh(η). The concrete values of these functions in the finite temperatureensembles were given in [21,22], whereas for finite volume excited states they were computedin [26].
For operators which are invariant under the action of the quantum group Uq (sl(2)) themean values involve only the function ω [18, 72]. For the closely related SU(2)-symmetriccase see [20, 24, 27, 73]. In the XXZ chain the operators invariant under the quantum groupare generated by the Temperley-Lieb algebra. Therefore, the works [18, 72] compute thosemean values which we are interested in, specifically in the XXZ chain.

4.2

Specific results for mean value of the current operators

The key result of the works [35–37] is that the mean values of the generalized currents are
given by the coefficients of the ω function. This was proven in [35–37] by realizing that theformulas for the current mean values are identical to the formulas for the ω function foundoriginally in [26].
Let us now summarize these results. For our purposes it is convenient to define a new
function ψ(x, y) = ψ( y, x) which is related to ω via1iψ(x, y) = −i sin(γ) ω(i sin(γ)x, i sin(γ) y) + K(i sin(γ)(x − y)) ,
(55)
2
4
where
K(u) =sin(2γ).
sinh(u + iγ) sinh(u − iγ)(56)

Similarly to (54) we defineψa,b = ψ b,a = (∂ x )a (∂ y ) b ψ(x, y) x, y=0 .
They are related to the coefficients ωa,b via


1
a i
a+b
ψa,b = − ωa,b + (−1)(∂u ) K(u) |u=0 sinha+b+1 (iγ) .
2
4
14

(57)

(58)

SciPost Phys. 16, 003 (2024)

The main result of [35–37] is that for any Bethe state |λN 〉 given by the set of Bethe roots
λN = {λ1 , . . . , λN }, the current mean values are given by〈λN |Jα,β |λN 〉 = ψα−2,β−1 ,(59)where ψ(x, y) is now defined by the ω function for the Bethe state |λN 〉, and it is written using
the Bethe roots by
ψ(x, y) = h(i sin(γ)x) · G −1 · h(i sin(γ) y) × (− sin(γ)) ,(60)

where h(x) is a parameter dependent vector of length N with elements h j (x) = h(λ j − x) withh(λ) given byh(λ) = coth(λ − iγ/2) − coth(λ + iγ/2) ,(61)
and G is the Gaudin matrix, defined asNXsin(γ)
G jk = δ jk L−
K(λ jl ) + K(λ jk ) ,sinh(λ j + iγ/2) sinh(λ j − iγ/2) l=1(62)

where we denote λ jk ≡ λ j − λk .
In the specific case of the charge densities we get〈{λ}N |qα |{λ}N 〉 = ψ0,α−2 .

(63)Note that the quantities ψα,β are symmetric with respect to α, β, but the current meanvalues involve certain shifts in the indices. These shifts appear due to our definitions of thecharges and currents. The symmetry of the ψα,β leads to the symmetry〈λN |Jα,β |λN 〉 = 〈λN |Jβ+1,α−1 |λN 〉 ,(64)

for the mean values. This equation does not hold on the level of the operators, only on thelevel of the mean values.

5

Correlation functions

In this section, we formulate two conjectures about the correlation functions for the quantum
integrable models, which are representations of the (affine) TL algebra (4).
We are interested in correlation functions of the form〈Ψ L |O|ΨR 〉.
〈Ψ L |ΨR 〉(65)

Here, |ΨR 〉 is an eigenvector of the Temperley-Lieb Hamiltonian in a selected representation.
For the correlation functions, we will consider only the periodic case. Furthermore, we will
focus on states which are singlets of the transfer matrix. The vector 〈Ψ L | is the left eigenvectorof the transfer matrices corresponding to the same eigenvalue.
The operator O is chosen to be a word from the Temperley-Lieb algebra. We will focus onshort-range operators, which include products of Temperley-Lieb generators with indices close
to each other.

15

SciPost Phys. 16, 003 (2024)

5.1

Main results and numerical checks

We first introduce the linear map T over the TL algebra. This map generates a shift-invariant
operator, such that the result is translationally invariant with respect to the Temperley-Lieb
indices. More concretely, the action of T is defined by:L

1X
T [ei1 ei2 · · · eim ] ≡e i + j e i2 + j · · · e i m + j .
L j=1 1(66)

In many representations, T generates operators that are invariant with respect to a shift in thephysical sites. However, in the case of the Potts model, we are dealing with the invariance withrespect to shifts in the Temperley-Lieb indices, which is not identical to the physical shifts.
In the following, we assume |E〉 is an eigenstate of a Hamiltonian of TL representation withan eigenenergy E. We assume that |E〉 is a singlet of the commuting set of charges. We define
the translational mean value 〈·〉 ≡ 〈E|T [ · ]|E〉.
Below, we present two conjectures, which we tested in multiple representations and various
volumes and singlet eigenstates.
Conjecture 1 The translationally invariant mean values of the generalized current operators are
given byJβ+2,α+1 ( j) = ψα,β (α, β ≥ 0) ,(67)
where ψα,β are the Taylor coefficients of the function ψ(x, y), which is expressed via the Betheroots in eq. (60), corresponding to the eigenstate with the same value of local charges in the XXZ
representation with the same TL parameter d, and appropriate twist φ.
The expression for ψα,β is the same in all different representations with the same TL parameters d.
Conjecture 2 In eigenstates |E〉 which are singlets, the translationally invariant mean values ofoperators generated by the TL algebra are universally factorized as:
!
pf
p
X
X Y
ψα j ,β j Cα,β (d) ,(68)
e i1 e i2 · · · e i m =p=1 α,βj=1where α = {α1 , α2 , . . . , α p } and β = {β1 , β2 , . . . , β p } satisfies 0 ≤ αn ≤ βn ≤ M and αn ≤ αn+1
and βn ≤ βn+1 , and M and p f are some positive integers. The coefficients Cα,β (d) are independentof the choice of the eigenstate |E〉 and the representation of the TL algebra. The information about
our choice of the operator ei1 . . . eim is encoded in Cα,β (d). They depend only on the parameter dof the TL algebra. This implies that all the Cα,β (d) can be found in the concrete example of the
XXZ spin chain, and they can be extracted from the existing works [18, 22, 72].
We stress that the conjectures refer to the shift-invariant mean values generated by theoperator T . In many models, it is not necessary to introduce T , because all mean values
are shift-invariant. However, in the Potts model, there is a distinction between translational
invariance and shift invariance, and while all mean values are translationally invariant (withrespect to physical translations), they are not shift-invariant. We observed that the conjectures
hold only for the shift-invariant combinations.
We computed several examples of factorized formulas, using the results of [22]. Theseare presented in the following. Furthermore, in Subsection 5.2 below we consider special
combinations of charges and currents, whose mean values are always bi-linear in ψa,b .

16

SciPost Phys. 16, 003 (2024)

The simplest short-range correlation function (involving at least two TL generators) is


1 1
1
〈e1 e2 〉 =ψ0,2 − 2ψ0,0 − ψ1,1 + ψ0,1 .
(69)
d 2
2
In this case, there is no actual factorization happening because the decomposition above works
on the level of the operators, guaranteed by the operator identity
§
ª
1 1
1
e1 e2 =
q4 (1) − 2q2 (2) − J3,2 (2) + q3 (1) .
(70)
d 2
2
Actual factorization is observed in more complicated cases. For exampled 2 − 401112〈e1 e3 〉 =(d − 4)ψ0,0 + 2ψ0,2 +ψ1,1 + ψ1,3 − ψ2,2d(d 2 − 1)
12
6
4
 2
1
1
d − 28 2+ 2(ψ0,1 − ψ0,0 ψ1,1 ) − (ψ20,2 − ψ0,0 ψ2,2 )
d −1
12
4

1
1
+ (ψ0,2 ψ1,1 − ψ0,1 ψ1,2 ) + (ψ0,1 ψ0,3 − ψ0,0 ψ1,3 ) .
2
6

(71)

Furthermore



1
3
2
2
5d
ψ
+
11d
ψ
−
ψ
ψ
1,1
0,0
1,1
1,0
6d (d 2 − 1)+d(36ψ0,0 + 34ψ1,1 − 24ψ2,0 + 3ψ2,2 − 2ψ3,1 ) + 3ψ2,0 (ψ2,0 − 2ψ1,1 )+2ψ1,0 (8ψ1,0 + 3ψ2,1 − ψ3,0 ) + ψ0,0 (−16ψ1,1 − 3ψ2,2 + 2ψ3,1 ) , (72)〈e1 e2 e3 + e3 e2 e1 〉 =

and
e1 e4


1
11d 2 − 92
d 4 + 40d 2 − 392
d 2 + 56
2
=
2(d
−
4)ψ
−
ψ
+
ψ
+
ψ1,3
0,0
0,2
1,1
d(d 2 − 1)(d 2 − 2)
12
36
36

d 2 + 441
11−
ψ2,2 −
ψ0,4 +
ψ2,4 −
ψ3,3
24
12
24
18
 4

1
5d − 28d 2 − 220  2
+ 2
ψ
−
ψ
ψ
0,0
1,1
0,1
(d − 1)(d 2 − 2)182¦©
2d + 16+2(ψ0,1 ψ0,3 − ψ0,0 ψ1,3 ) + 3(ψ0,0 ψ2,2 − ψ20,2 )9
7d 4 + 226d 2 + 928+
ψ0,2 ψ1,1 − ψ0,1 ψ1,2
144©
5d 2 + 22 ¦+
ψ0,1 ψ1,4 − ψ1,1 ψ0,4 − 2ψ0,1 ψ2,3 + 6(ψ1,1 ψ2,2 − ψ21,2 )
144
 2
 2
10d 2 + 1401+
ψ0,3 ψ1,2 +
ψ0,2 ψ0,4 − ψ0,0 ψ2,4 +ψ0,0 ψ3,3 − ψ20,3 − ψ0,2 ψ1,3
144
6
9
3



1
1 
1
2
+
ψ1,3 ψ2,2 − ψ1,2 ψ2,3 +ψ1,1 ψ3,3 − ψ1,3 +ψ1,2 ψ1,4 − ψ1,1 ψ2,4
12
18
24



1
1
1
+
ψ0,3 ψ2,3 − ψ0,2 ψ3,3 +ψ0,2 ψ2,4 − ψ0,4 ψ2,2 +
ψ0,4 ψ1,3 − ψ0,3 ψ1,4 .
(73)
36
48
72

We numerically confirmed the above equations in the XXZ chain, the golden chain, thePotts chain with 2 ≤ Q ≤ 5, and for the trace representation, in various finite volumes. A
selection of concrete numerical data is presented in Appendix B. We note that the denominatorof (73) diverges for Q = 2 Potts (Ising case), thus we checked the formula (73) works in the
Potts chain for 2 < Q ≤ 5. However, we checked in the representations where d can betuned continuously that this divergence is compensated by a cancellation of the numerator as
17

SciPost Phys. 16, 003 (2024)

d 2 → 2. A similar phenomenon, which has also been observed in the works [9,53] occurs also
for the above correlators as d 2 → 1, and ensures that all correlation functions remain finite.
Here, we only consider the eigenstates which are singlets of the commuting set of charges,
i.e., the eigenstates that can be distinguished by the values of the local charges and also thefundamental charges, such as the momentum operator. In the setups where we used to check
Conjecture 1 and Conjecture 2 up to L = 10, almost all eigenstates are singlets, and we confirmed Conjecture 1 and Conjecture 2 holds for singlet eigenstates. For the case that q is aroot of unity, we observed that non-singlet eigenstates can appear. In the case of non-singlet
eigenstates, there remains the freedom to select a basis within the degenerate eigensubspace,
and the value of the correlation function might vary depending on this choice. We leave thetreatment for non-singlet eigenstates to future research.

5.2

Special bi-linear combinations

The work [74] found a special family of operators, whose mean values always factorize in
integrable quantum spin chains. These operators are bi-linear combinations of (generalized)
currents and charge densities. Now we also summarize these results here.
Let us introduce the operators Kα,β,γ ( j, k), which depend on three indices α, β, γ and twocoordinates (site indices) j, k:
Kα,β,γ ( j, k) = Jα,γ ( j)qβ (k) − qα ( j)Jβ,γ (k + 1) ,(74)

It can be shown that the mean values of these operators do not depend on k; the proof relies on
the continuity equations. Afterwards, it can be shown that the mean values factorize, namely
for the mean values in all eigenstates we haveKα,β,γ ( j, k) = Jα,γqβ − 〈qα 〉 Jβ,γ .

(75)

In the present conventions this means that
Kα,β,γ ( j, k) = ψα−2,γ−1 ψβ−2,0 − ψα−2,0 ψβ−2,γ−1 .

(76)These special operator combinations can be seen as the lattice version of the famous T T̄ operator known in conformal field theories [75, 76].

6

Conclusions and outlook

It has been known for many decades that the Temperley-Lieb algebra connects the spectra ofdifferent models, which are seen as different representations of the same algebra. However, nosystematic study of correlation functions of Temperley-Lieb models existed before our work.
The main conclusion of this work is that the TL algebra also connects the correlation functions,and that the existing results of the XXZ spin chain can be used to compute correlation functions
in many other models. It is important that our results are not limited to the ground states,
instead they can be applied to any state which is a singlet of the local charges in the TemperleyLieb algebra.
We presented our results in the form of conjectures, an actual proof of which is desirable.
A possible route to follow is to exploit the algebraic construction of the conserved current
in integrable models [37] in order to derive algebraic relations relating the latter with local
correlators. We plan to explore this direction in a future work.
At present, our conjectures hold for the translationally invariant mean values. In thosecases when the translational invariance is broken by the representation of the TemperleyLieb algebra (for example the Potts chain), we observed that the conjectures hold for the18

SciPost Phys. 16, 003 (2024)

translationally invariant averages. At the moment there is no theoretical explanation as towhy the conjectures still work for the averages. This is a question that also deserves further
study.
Also, it would be useful to investigate those states which are not singlets of the TemperleyLieb algebra. In such cases it was observed that the mean values can depend on the choice ofthe basis within the degenerate sub-space. A more detailed investigation is beyond the scopeof the present paper.
As a by-product of our computations, we found an apparently new way to construct thegeneralized current operators from the expressions for the local charges, see Sections 3.3-3.4.
This result is independent of the Temperley-Lieb algebra, therefore it could be applied to theother integrable spin chains, and it might facilitate the search for factorized correlators in othercircumstances.

Acknowledgments
We are thankful to Hermann Boos, Frank Göhmann, Holger Frahm, Arthur Hutsalyuk, Andreas
Klümper, Yuan Miao, Levente Pristyák and Fedor Smirnov for useful discussions.
Funding information K. F. was supported by FoPM, WINGS Program, JSR Fellowship, theUniversity of Tokyo, and KAKENHI Grants No. JP21J20321 from the Japan Society for the
Promotion of Science.

A

Current operators of Temperley-Lieb HamiltonianHere, we provide examples of the current and generalized current operators. We use theconventions given in the main text; this convention corresponds to using the boost operator,
or equivalently, the usual R-matrix in the 6-vertex model.
Examples of the lower-order charges areq4 ( j) = 2(e j+2 e j+1 e j − e j+1 e j e j+2 − e j e j+2 e j+1 + e j e j+1 e j+2 ) + d e j+1 e j + d e j e j+1 ,(A.1)

q5 ( j) = 6(e j e j+1 e j+2 e j+3 − e j+3 e j+2 e j+1 e j + e j+2 e j+1 e j e j+3 + e j+1 e j e j+3 e j+2 + e j e j+3 e j+2 e j+1− e j+1 e j e j+2 e j+3 − e j e j+2 e j+1 e j+3 − e j e j+1 e j+3 e j+2 − d e j+2 e j+1 e j + d e j e j+1 e j+2 )+ (2 + d 2 ) e j e j+1 − e j+1 e j ,(A.2)
q6 ( j) = 24(e j+4 e j+3 e j+2 e j+1 e j − e j+3 e j+2 e j+1 e j e j+4 − e j+2 e j+1 e j e j+4 e j+3 − e j+1 e j e j+4 e j+3 e j+2− e j e j+4 e j+3 e j+2 e j+1 + e j+2 e j+1 e j e j+3 e j+4 + e j+1 e j e j+3 e j+2 e j+4 + e j+1 e j e j+2 e j+4 e j+3+ e j e j+3 e j+2 e j+1 e j+4 + e j e j+2 e j+1 e j+4 e j+3 + e j e j+1 e j+4 e j+3 e j+2 − e j+1 e j e j+2 e j+3 e j+4− e j e j+2 e j+1 e j+3 e j+4 − e j e j+1 e j+3 e j+2 e j+4 − e j e j+1 e j+2 e j+4 e j+3 + e j e j+1 e j+2 e j+3 e j+4 )+ 36d(e j+3 e j+2 e j+1 e j + e j e j+1 e j+2 e j+3 ) − 12d(e j+2 e j+1 e j e j+3 + e j+1 e j e j+3 e j+2+ e j e j+3 e j+2 e j+1 + e j+1 e j e j+2 e j+3 + e j e j+2 e j+1 e j+3 + e j e j+1 e j+3 e j+2 )+ (16 + 14d 2 )(e j+2 e j+1 e j + e j e j+1 e j+2 ) + (−16 − 2d 2 )(e j+1 e j e j+2 + e j e j+2 e j+1 )+ (8d + d 3 )(e j+1 e j + e j e j+1 ) − 24d e j e j+2 .

19

(A.3)

SciPost Phys. 16, 003 (2024)

We give a few examples of the corresponding current operators:
J3,2 ( j) = e j+1 e j e j−1 − e j e j−1 e j+1 − e j−1 e j+1 e j + e j−1 e j e j+1 − 2e j ,(A.4)

J3,3 ( j) = −e j+2 e j+1 e j e j−1 − e j+1 e j e j−1 e j−2 + e j+1 e j e j−1 e j+2 + e j e j−1 e j+2 e j+1 + e j e j−1 e j−2 e j+1+ e j−1 e j+2 e j+1 e j + e j−1 e j−2 e j+1 e j + e j−2 e j+1 e j e j−1 − e j e j−1 e j+1 e j+2 − e j−1 e j+1 e j e j+2− e j−1 e j e j+2 e j+1 − e j−1 e j−2 e j e j+1 − e j−2 e j e j−1 e j+1 − e j−2 e j−1 e j+1 e j + e j−1 e j e j+1 e j+2+ e j−2 e j−1 e j e j+1 − d e j+1 e j e j−1 + d e j−1 e j e j+1 + e j+1 e j + e j e j−1 − e j e j+1 − e j−1 e j ,(A.5)
J4,2 ( j) = −2e j+2 e j+1 e j e j−1 + 2e j+1 e j e j−1 e j+2 + 2e j e j−1 e j+2 e j+1 + 2e j−1 e j+2 e j+1 e j− 2e j e j−1 e j+1 e j+2 − 2e j−1 e j+1 e j e j+2 − 2e j−1 e j e j+2 e j+1 + 2e j−1 e j e j+1 e j+2− d e j+1 e j e j−1 − d e j e j−1 e j+1 + d e j−1 e j+1 e j + d e j−1 e j e j+1 + 2e j+1 e j − 2e j e j+1 , (A.6)
J4,3 ( j) = 2(e j+3 e j+2 e j+1 e j e j−1 + e j+2 e j+1 e j e j−1 e j−2 − e j+2 e j+1 e j e j−1 e j+3 − e j+1 e j e j−1 e j+3 e j+2− e j+1 e j e j−1 e j−2 e j+2 − e j e j−1 e j+3 e j+2 e j+1 − e j e j−1 e j−2 e j+2 e j+1 − e j−1 e j+3 e j+2 e j+1 e j− e j−1 e j−2 e j+2 e j+1 e j − e j−2 e j+2 e j+1 e j e j−1 + e j+1 e j e j−1 e j+2 e j+3 + e j e j−1 e j+2 e j+1 e j+3+ e j e j−1 e j+1 e j+3 e j+2 + e j e j−1 e j−2 e j+1 e j+2 + e j−1 e j+2 e j+1 e j e j+3 + e j−1 e j+1 e j e j+3 e j+2+ e j−1 e j e j+3 e j+2 e j+1 + e j−1 e j−2 e j+1 e j e j+2 + e j−1 e j−2 e j e j+2 e j+1 + e j−2 e j+1 e j e j−1 e j+2+ e j−2 e j e j−1 e j+2 e j+1 + e j−2 e j−1 e j+2 e j+1 e j − e j e j−1 e j+1 e j+2 e j+3 − e j−1 e j+1 e j e j+2 e j+3− e j−1 e j e j+2 e j+1 e j+3 − e j−1 e j e j+1 e j+3 e j+2 − e j−1 e j−2 e j e j+1 e j+2 − e j−2 e j e j−1 e j+1 e j+2− e j−2 e j−1 e j+1 e j e j+2 − e j−2 e j−1 e j e j+2 e j+1 + e j−1 e j e j+1 e j+2 e j+3 + e j−2 e j−1 e j e j+1 e j+2 )+ d(+3e j+2 e j+1 e j e j−1 + e j+1 e j e j−1 e j−2 − e j+1 e j e j−1 e j+2 − e j e j−1 e j+2 e j+1+ e j e j−1 e j−2 e j+1 − e j−1 e j+2 e j+1 e j − e j−1 e j−2 e j+1 e j − e j−2 e j+1 e j e j−1 − e j e j−1 e j+1 e j+2− e j−1 e j+1 e j e j+2 − e j−1 e j e j+2 e j+1 − e j−1 e j−2 e j e j+1 − e j−2 e j e j−1 e j+1 + e j−2 e j−1 e j+1 e j+ 3e j−1 e j e j+1 e j+2 + e j−2 e j−1 e j e j+1 − 2e j e j−1 e j+1 e j + 3e j+1 e j + e j e j−1+ 3e j e j+1 + e j−1 e j ) + d 2 (e j−1 e j e j+1 + e j+1 e j e j−1 + 2e j ) + 4(e j + e j+1 ) .

(A.7)

The higher-order generalized currents have a more complicated structure.
The explicit expressions for the local conserved quantities of the Temperley-Lieb Hamiltonian were obtained by Nienhuis and Huijgen [56]. However, the local conserved quantities
obtained in [56] differ from those obtained by the boost operation; they are linearly dependent, but the choice of the linear combination is different.

B

Numerical checks

We performed exact diagonalization to check our conjectures. Our procedure included thefollowing steps:• We numerically constructed the transfer matrix of the XXZ representation, local charges
and generalized current operators which are relevant to our factorization formula (69)–(73) using (50). Our definitions and conventions are listed in Section B.1 Then, we
exactly diagonalize the transfer matrix and store the eigenstates for the local charges.
• For each eigenstate, we numerically computed the current mean values to obtain ψα,βand then calculated the prediction of short-range correlation functions from the factorization formula (69)–(73), and compared them to the numerics from exact diagonalization.

20

SciPost Phys. 16, 003 (2024)• We also checked the current mean values are actually reproduced by the formula (60)using the corresponding Bethe roots. We utilized the famous TQ relation and calculated
the Q function from the eigenvalue of the transfer matrix, and then we obtained thecorresponding Bethe roots. This strategy is described in detail in [77].
• For other representations, the Potts, the golden, and the trace representation, we construct the eigenstates simultaneously diagonalizing the local charges and additional symmetries such as momentum and check the factorization formula in the same way as thecase of the XXZ rep. If there are common eigenstates with the XXZ representation withthe same energies and the same eigenvalues of the higher local charges, we checked thecurrent mean values also coincide.
It was observed that our factorization formula (69)–(73) holds for all the eigenstates for thefinite size system in all representations we give in this work.

B.1

Definition of the transfer matrix in the XXZ representation

The transfer matrix for the XXZ model with periodic boundary conditions and twist φ is defined as:
T (u) = Tr0 (R0L (u) . . . R01 (u)) ,(B.1)
where each index i = 1, . . . L stands for a spin-1/2, and 0 is an auxiliary spin-1/2 which istraced over in the definition of T (u). The R matrix acting on the i th spin and the auxiliary spinis defined as:


γ
γ
γ
γ
cos 2 sin(u + 2 ) sin 2 cos(u + 2 ) z z
φi 2L σ0z+ −− +R0i (u) = e(B.2)+
σ0 σi + (σ0 σi + σ0 σi ) .
sin γ
sin γ

B.2

Numerical data

We list some concrete numerical data in Table 1–14. In the upper table on each page, i.e.
in Table 1, 3, 5, 7, 9, 11, 13, we list the values of the correlation functions considered in thefactorization formula (69)– (73) w.r.t. several eigenstates. The detailed setups are explained in
the next paragraphs. In the upper table on each page, we denote 〈e1 e2 e3 〉′ ≡ 〈e1 e2 e3 + e3 e2 e1 〉.
In the lower table on each page, i.e., in Table 2, 4, 6, 8, 10, 12, 14, we list the values of thecurrent mean values used in the factorization formula (69)– (73) w.r.t. the same eigenstates
as those referred to in the corresponding upper tables. The errors of the correlation functions
listed in the upper tables and the predicted values from the factorization formula (69)– (73)calculated by the current mean values listed in the lower tables are at most of the order of
10−9 .
We list the values of the correlation functions related to the factorization formula (69)–(73)for several eigenstates in the XXZ representation for L = 8 and the corresponding Bethe rootsin Table 1, 3, 5, 7, and list the corresponding current mean values in Table 2, 4, 6, 8. Table 1
and 2 are the zero-twist case. Table 3 and 4 correspond to the Q = 3 Potts case. Table 5 and 6
correspond to the golden chain case. Table 7 and 8 correspond to the trace representation
with d = 3.
We also list examples of numerical data for the other representations. We consider the
3-state Potts representation, the golden chain, and the trace representation for L = 8, andconsider Q = 3 for the Potts representation and d = 3 for the trace representation. We show
the values of the correlation functions and the current mean values of the 3-state Potts representation in Table 11 and 12, and for the golden chain representation in Table 9 and 10, andfor trace representation with d = 3 in Table 13 and 14.

21

SciPost Phys. 16, 003 (2024)

We note that for the trace representation, the Hamiltonian is non-Hermitian, and we haveto use the left and right eigenvectors for the calculation of the current mean values and thecorrelation functions.
Table 1: List of the correlation functions in the first 8 eigenstates of the XXZ representation calculated by exact diagonalization for d = 0.35, φ = 0, total magnetization
sector of M = 0, and L = 8. E is the eigenenergy and k is the overall momentum
quantum number. We list the eigenstates with 0 ≤ k ≤ 4 to resolve degeneracies
caused by inversion symmetry. We also list the corresponding Bethe roots. We denote 〈e1 e2 e3 〉′ ≡ 〈e1 e2 e3 + e3 e2 e1 〉. The numerical errors of the correlation functions
compared to those computed from the factorization formula (69)–(73), using thecurrent mean values in Table 2, are at most 10−13 . The third eigenstate marked witha star includes singular rapidities iγ/2.

1
2
3∗
4
5
6
7
8

E

k

6.23684.74744.67474.55833.55263.47033.40833.3539

0
4
4
1
3
2
3
2

〈e1 e3 〉〈e1 e2 e3 〉′

e1 e4

λ1

0.80200.57820.6817−0.09140.48450.47470.5222 + 0.2063i 0.1942
0.5550 − 0.0321i −0.0375
0.4184 + 0.1691i 0.2523
0.2689 − 0.0114i 0.2701
0.3032 + 0.2103i 0.0395

1.6032
0.9435
1.26790.92980.63840.60000.49070.20040.60980.8249−0.11420.3235−0.0798
0.04460.23830.1721−0.7220
1.5708i−0.6974i−0.7688 + 1.5708i−0.3035 + 1.5708i−0.4989 + 1.5708i−0.2845 − 0.6975i−0.4630 − 0.6977i〈e1 e2 〉

λ2

λ3

λ4

−0.1795
0.1795−0.25510−0.18810.1881−0.15380.1915−0.3514−0.0980−0.4435
0.2022−0.1751 −0.2845 + 0.6975i
0.1901 −0.4630 + 0.6977i

0.72200.25510.6974i
0.73110.75290.74020.74420.7358

Table 2: List of the current mean values in the XXZ representation used in the factorization formula (69)–(73). The parameters and the setup are the same as Table 1.
The numerical errors of the current mean value formula (60) against the above numerics are at most 10−10 except for the third eigenstate with singular Bethe roots.
ψ0,0
1
2
3∗
4
5
6
7
8

ψ0,1

ψ0,2

ψ0,3

0.779601.081500.593401.807200.58430
1.819400.5698 0.4126i 1.6891 0.3878i
0.4441 −0.0641i 0.7184 −5.0049i
0.4338 0.3381i 0.3301 3.2589i
0.4260 −0.0228i 0.6272 −4.7661i
0.4192 0.4205i 0.5331 4.0813i

ψ0,4

ψ1,1

ψ1,2

10.4254
13.7250
14.778119.29549.4013−7.8149
6.42992.5638−1.2991−0.5219−0.4285−0.4778−0.7232−0.8489−0.6326−0.6781000−1.2087i−1.2983i
0.1415i−1.5619i−0.0447i

22

ψ1,3

ψ1,4

ψ2,2

ψ2,3

ψ2,4

−7.021308.8692085.4961−8.582205.9757083.4127−8.415106.0217078.4259−6.4624 −17.0919i 6.9130 0.0233i 72.5437−3.8280 −3.8111i 5.5405 −11.0802i 90.1072−5.4473 12.3512i 3.7061 15.5615i 28.0494−3.5612 −11.3436i 4.5125 −14.1238i 65.2281−4.2107 2.4632i 3.8831 17.0993i 45.2646

ψ3,3−153.1087−141.1362−143.1089−140.0925−56.6526−93.2620−63.3650−71.6897

SciPost Phys. 16, 003 (2024)

Table 3: List of the correlation functions in the 8 eigenstatesof the XXZ represenp
tation calculated by exact diagonalization for d = 3, z = −1, total magnetization
sector of M = 0, and L = 8. E is the eigenenergy and k is the overall momentum
quantum number. We list the eigenstates for which the TQ equation can be solved,
with 0 ≤ k ≤ 2 to resolve the degeneracy, and if there are still degeneracies, only
one of the degenerate states was listed. We also list the corresponding Bethe roots.
The numerical errors of the correlation functions compared to those computed fromthe factorization formula (69)–(73), using the current mean values in Table 4, are at
most 10−9 .
Ek〈e1 e3 〉 〈e1 e2 e3 〉′e1 e4

1.70590.83670.62850.82850.30100.54670.25940.1972

1.5374−0.1783−0.02220.1171
0.49270.9696−0.5795−0.0068
0.1306
0.52580.7205 −0.5658 − 0.3534i −0.5658 + 0.3534i−0.0603
0.06570.5595−0.5470−0.13810.13810.54700.5750 0.3458 − 0.2891i
0.05640.27470.3458 + 0.2891i
0.3386−0.5792
0.1972 − 0.2614i−0.0033
0.1972 + 0.2614i
0.0679 −0.4953 − 0.3368i −0.4953 + 0.3368i−0.24210.22680.1012 0.2167 − 0.5333i
0.03690.2167 + 0.5333i
0.2313〈e1 e2 〉1 10.0809 0
1.20532 7.9421 1 0.8162 + 0.2823i
3 7.4797 00.81924 7.0595 2
0.65895 6.1962 2 0.5458 + 0.4208i
6 5.8054 00.4385
7 4.5898 00.2697
8 4.4528 1 0.4040 + 0.1477i

2.45001.35741.68020.58200.42001.0597−0.44390.8463

λ1

λ2

λ3

λ4

Table 4: List of the current mean values in the XXZ representation with the parameters being the same as Table 3. The numerical errors of the current mean value
formula (60) against the above numerics are at most 10−8 .
ψ0,0
1
2
3
4
5
6
7
8

ψ0,1

ψ0,2

ψ0,3

ψ0,4

ψ1,1

ψ1,2

ψ1,3

ψ1,4

ψ2,2

ψ2,3

ψ2,4

ψ3,3

1.260103.3463031.6763 −2.9346
0−39.9194034.6966
01172.5214 −931.3222
0.9928 0.5646i 3.3830 10.0458i 95.2807 −1.7078 −0.0231i −18.8898 −137.1473i 34.2995 0.1941i 1198.6269 −528.0020
0.935004.94510
98.2431 −0.8162
0−31.1809027.92660614.1580 −1106.1864
0.8824000−121.1016 −2.9061
0−40.3672
03.71370−171.0383 −910.7546
0.7745 0.8415i 2.0736 15.6585i 35.9072 −1.4575 0.2653i −16.7290 −32.4849i 21.4550 79.1153i 712.4336 −429.6805
0.725703.35950
159.4623 −0.5310
02.6678034.255901208.7936 −126.7140
0.57370−1.51820−65.9139 −2.3736
0−8.177606.43920272.2224−8.7733
0.5566 0.2953i 2.9808 9.1305i 115.1619 −0.3225 1.2303i −4.7323 63.9288i 23.5032 74.0467i 836.7167 −226.6202

23

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Table 5: List of the correlation functions in 8peigenstates of the XXZ representation
calculated by exact diagonalization for d = 1+2 5 , z = e i2γ , total magnetization sectorof M = 0, and L = 8. E is the eigenenergy and k is the overall momentum quantum number. We list the eigenstates for which the TQ equation can be solved, with0 ≤ k ≤ 4 to resolve the degeneracy, and if there are still degeneracies, only one ofthe degenerate states was listed. We also list the corresponding Bethe roots. The
numerical errors of the correlation functions compared to those computed from thefactorization formula (69)–(73), using the current mean values in Table 6, are at
most 10−11 .
Ek〈e1 e2 〉〈e1 e3 〉 〈e1 e2 e3 〉′1 10.0259 01.2264
1.7212
2 7.9546 4
0.80901.0181
3 6.2683 3 0.5180 − 0.0180i 0.6133
4 6.2025 2 0.5312 + 0.3708i 0.3187
5 4.6077 4
0.23870.2136
6 4.1221 00.38510.1706
7 3.4399 1 0.2320 − 0.2680i 0.0412
8 2.4143 0−0.15930.14902.4912
1.86250.75810.3476−0.4355
1.02680.0510−0.0909

λ1

e1 e4

λ2

λ3

1.5016−0.2745−0.0580
0.6663 0.3081 − 0.3184i−0.09670.6905 0.0797 − 0.3142i−0.0868
0.6375 −0.0397 − 0.3142i −0.0397 + 0.3142i
0.0837 0.1632 − 0.3142i−0.3152
0.0217 0.4136 − 0.6247i−0.00100.1185 0.4503 − 0.6307i−0.1456
0.0357 −0.1660 − 0.3142i 0.4358 − 0.3290i

λ40.10550.4036
0.06650.3081 + 0.3184i
0.0797 + 0.3142i
0.3956
0.0966
0.40250.1632 + 0.3142i
0.38730.4136 + 0.6247i
0.45840.4503 + 0.6307i
0.4963−0.1660 + 0.3142i 0.4358 + 0.3290i

Table 6: List of the current mean values in the XXZ representation with the parameters being the same as Table 5. The numerical errors of the current mean value
formula (60) against the above numerics are at most 10−8 except for the eigenstates
indexed by 3 and 4. The current mean values of the eigenstates indexed by 3 and 4
calculated by (60) have larger numerical errors up to the order of 10−3 because thecorresponding Bethe roots include (small value) + iγ/2 and cause loss of digits in thecalculation of the Gaudin matrix.
ψ0,0
1
2
3
4
5
6
7
8

ψ0,1

ψ0,2

ψ0,3

ψ0,4

ψ1,1

ψ1,2

ψ1,3

ψ1,4

ψ2,2

ψ2,3

ψ2,4

ψ3,3

1.253202.7575040.1762 −3.1122
0−30.4413
035.72920699.9198 −1041.1975
0.994304.4673058.7948 −1.0641
0−35.8730
024.23810636.4054 −977.7322
0.7835 −0.0360i 1.6763 −14.6067i 23.9194 −1.5671 −4.8238i −15.9397 −64.9254i 17.9797 −77.5466i 514.1511 −421.2121
0.7753 0.7416i 1.4832 12.7286i 10.2488 −1.6686 −0.1011i −18.3237 9.9267i 15.8433 90.9083i 386.2803 −477.2798
0.57600−1.26770−24.5509 −2.1720
01.51240
11.09470359.2093 133.1447
0.515303.40660139.3342 0.0497
00.2926024.970501033.3667 −13.3474
0.4300 −0.5360i 0.7507 −12.0616i −46.6899 −0.8600 −0.8968i −19.5390 79.3132i 1.3187 −20.1408i −95.3254 −444.0795
0.301800.34450−7.5902 −0.1736
0−4.21200−1.65880141.3088 93.4591

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SciPost Phys. 16, 003 (2024)

Table 7: List of the correlation functions in 8 eigenstates of the XXZ representation
calculated by exact diagonalization for d = 3, z = e i2γ , total magnetization sectorof M = 0, and L = 8. E is the eigenenergy and k is the overall momentum quantum number. We list the eigenstates for which the TQ equation can be solved, with0 ≤ k ≤ 4 to resolve the degeneracy, and if there are still degeneracies, only oneof the degenerate states was listed. We also list the corresponding Bethe roots. We
denote 〈e1 e2 e3 〉′ ≡ 〈e1 e2 e3 + e3 e2 e1 〉. The numerical errors of the correlation functions compared to those computed from the factorization formula (69)–(73), using
the current mean values in Table 8, are at most 10−12 .
E
1
2
3
4
5
6
7
8

14.7275
13.2170
10.414210.0000
8.90388.00007.58585.3687k〈e1 e2 〉〈e1 e3 〉 〈e1 e2 e3 〉′

λ1

e1 e4

0
1.66554.3998 3.6248 2.6996
4
1.0245
4.1074 3.1863 1.1395
3 0.6402 − 0.1402i 1.6553 1.1036 2.5303
2 0.7000 + 0.5000i 1.0500 0.3000 2.1000
00.1595
1.8310 2.1352 0.6915
4
0.2727
0.6364 −0.5455 0.6364
1 0.1098 − 0.3902i 0.5947 0.3964 1.4697
0−0.2000
0.1442 −0.2600 0.1089

λ2

λ3

λ4

−0.1177 − 0.4815i −0.1177 + 0.4815i −0.0474 + 0.1232i −0.0474 − 0.1232i−1.5652 − 1.5708i −0.0084 + 0.2508i −0.0084 − 0.2508i−0.0070−0.9135 − 0.2344i −0.0550 + 0.5535i 0.0313 − 0.0726i 0.0968 − 0.2466i−0.7742 − 0.3414i −0.0764 + 0.5237i −0.0175 + 0.1471i 0.1867 − 0.3294i−0.5422 + 1.5708i−0.6884
0.01120.2740−0.7277−0.0668 − 0.5371i −0.0668 + 0.5371i
0.2349−0.6223 − 1.5383i −0.6377 + 0.1239i −0.0117 − 0.2804i 0.3247 + 0.1239i−0.6759 + 0.4202i −0.6759 − 0.4202i 0.2793 + 0.4224i 0.2793 − 0.4224i

Table 8: List of the current mean values in the XXZ representation with the parameters being the same as Table 7. The numerical errors of the current mean value
formula (60) against the above numerics are at most 10−10 .
ψ0,0
1
2
3
4
5
6
7
8

ψ0,1

ψ0,2

ψ0,3

ψ0,4

ψ1,1

ψ1,2

ψ1,3

ψ1,4

ψ2,2

ψ2,3

ψ2,4

ψ3,3

1.840904.77640146.8741 −6.2902
0−77.62880114.31190
1840.2901 −5749.4570
1.652107.4127076.7147 −2.6713
0−132.1650
074.127404818.2829 −3336.7489
1.3018 −0.2803i 3.8410 −37.3575i 145.4547 −2.6036 −12.0533i −34.1685 42.6595i 60.8878 −225.5422i 3698.3625 −1340.4055
1.2500 1.0000i 2.0000 26.0000i −50.0000 −3.6000 −1.2000i −68.4000 147.6000i 28.4000 397.2000i 650.8000 −2559.6000
1.113007.15150417.1100 0.8714
0−1.6636085.649804379.9362 −1014.5115
1.00000−2.0000
0
14.0000 −3.8182
026.7273046.0000
02198.0000 1324.9091
0.9482 −0.7803i 0.6590 −18.8575i −158.9547 −1.8964 1.4467i −60.3315 566.1595i 3.6122 57.9578i −711.8625 −1976.0945
0.671100.32200−55.7341 −0.5812
0−22.70750−14.46170545.2737 385.9685

25

SciPost Phys. 16, 003 (2024)

Table 9: List of the correlation functions in 8 eigenstates of the golden chain representation calculated by exact diagonalization for L = 8. E is the eigenenergy and kis the overall momentum quantum number. We list the eigenstates with 0 ≤ k ≤ 4.
The numerical errors of the correlation functions compared to those computed fromthe factorization formula (69)–(73), using the current mean values in Table 10, areat most 10−13 . The eigenstates indexed 1, 4 correspond to the eigenstates of the XXZ
chain indexed 1, 2 in Table 5 respectively. The errors between the corresponding correlation functions are at most of the order 10−13 . The eigenstates other than thoseindexed 1, 4 also correspond to eigenstates in the XXZ representation with different
parametrization. For example, the eigenstates indexed 2, 3 correspond to the XXZ
representation with the twist z = e4iγ and the magnetization M = 0.

1
2
3
4
5
6
7
8

E

k

〈e1 e2 〉〈e1 e3 〉〈e1 e2 e3 〉′e1 e4

10.02599.84739.5601
7.95467.7141
7.5275
7.39346.7599

0
4
0
4
3
1
0
2

1.2264
1.1926
1.14410.80900.8007 + 0.2901i
0.7862 − 0.2483i
0.79900.6397 + 0.0425i

1.7212
1.62791.47111.01810.77350.73640.67500.75902.49122.41832.30271.8625
1.34791.2794
1.67990.57681.5016
1.46631.4244
0.66630.93680.82430.67940.5047

Table 10: List of the current mean values in the golden chain representation withthe parameters being the same as Table 9. The eigenstates indexed 1, 4 correspondto the eigenstates of the XXZ chain indexed 1, 2 in Table 6 respectively. The errors
between the corresponding mean values are at most of the order 10−12 .
ψ0,0
1
2
3
4
5
6
7
8

ψ0,1

ψ0,2

ψ0,3

ψ0,4

ψ1,1

ψ1,2

ψ1,3

ψ1,4

ψ2,2

ψ2,3

ψ2,4

ψ3,3

1.253202.7575040.1762 −3.1122
0−30.4413
035.72920699.9198 −1041.1975
1.230902.9948032.2644 −2.8941
0−34.0877032.89610921.9507 −876.4761
1.195003.3370025.1091 −2.5726
0−38.1117
029.376401094.4553 −746.0619
0.994304.4673058.7948 −1.0641
0−35.8730
024.23810636.4054 −977.7322
0.9643 0.5801i 3.3212 7.8011i
86.8173 −1.5634 −0.6982i −18.2058 −120.7357i 31.1574 −1.7555i 966.7497 −546.4995
0.9409 −0.4965i 3.0183 −10.7810i 83.9916 −1.6448 −0.6943i −16.0361 117.2059i 30.5524 −3.5068i 1088.4021 −376.7953
0.924204.6091080.9245 −0.8367
0−30.1360024.96170537.6382 −981.7392
0.8450 0.0850i 0.0164 1.3714i −107.6978 −2.7168 0.0721i −36.1226 23.1560i 3.7302 19.0219i −139.8274 −787.5306

26

SciPost Phys. 16, 003 (2024)

Table 11: List of the correlation functions in 8 eigenstates of the Potts representation
calculated by exact diagonalization for Q = 3 and L = 8. E is the eigenenergy andk is the overall momentum quantum number, and zq is the ZQ quantum number. We
list the eigenstates with 0 ≤ k ≤ 2 and zq = 0, 1. We note that the physical systemsize here is L/2 = 4. The numerical errors of the correlation functions compared tothose computed from the factorization formula (69)–(73), using the current meanvalues in Table 12, are at most 10−11 . The eigenstates indexed 2, 4, 5, 6 correspondto the eigenstates of the XXZ chain indexed 1, 2, 3, 4 in Table 3 respectively. The errors between the corresponding correlation functions are at most of the order 10−10 .
The eigenstates other than those indexed 2, 4, 5, 6 also correspond to eigenstates in
the XXZ representation with different parametrization. For example, the eigenstates
indexed 1, 3 correspond to the XXZ representation with the twist z = e2iγ and themagnetization M = 0.

1
2
3
4
5
6
7
8

E

k

zq

〈e1 e2 〉〈e1 e3 〉〈e1 e2 e3 〉′e1 e4

10.4050
10.08098.38497.9421
7.47977.0595
6.79706.6104

0
0
0
1
0
2
1
1

0
1
0
1
1
1
0
0

1.26661.20530.82960.8162 + 0.2823i
0.81920.65890.7641 + 0.2602i
0.5281 + 0.0281i

1.88301.70591.20290.83670.62850.82850.31950.68122.58272.45001.9706
1.35741.68020.58201.20120.7866

1.59951.53740.70670.96960.72050.55950.66150.8062

Table 12: List of the current mean values in the Potts representation with the parameters being the same as Table 11. The eigenstates indexed 2, 4, 5, 6 correspond to
the eigenstates of the XXZ chain indexed 1, 2, 3, 4 in Table 4 respectively. The errors
between the corresponding mean values are at most of the order 10−7 .
ψ0,0
1
2
3
4
5
6
7
8

ψ0,1

ψ0,2

ψ0,3

ψ0,4

ψ1,1

ψ1,2

ψ1,3

ψ1,4

ψ2,2

ψ2,3

ψ2,4

ψ3,3

1.300602.9079045.7228 −3.3411
0−33.2258039.9918
0771.4239 −1229.7023
1.260103.3463031.6763 −2.9346
0−39.9194034.6966
01172.5214 −931.3222
1.048104.7128060.2540 −1.17680−41.0506
026.75200775.6237 −1112.7968
0.9928 0.5646i 3.3830 10.0458i 95.2807 −1.7078 −0.0231i −18.8898 −137.1473i 34.2995 0.1941i 1198.6269 −528.0020
0.935004.94510
98.2431 −0.8162
0−31.1809027.92660614.1580 −1106.1864
0.8824000−121.1016 −2.9061
0−40.3672
03.71370−171.0383 −910.7546
0.8496 0.5205i 4.1880 11.2664i 98.8764 −0.9287 0.4442i −22.1164 −114.5883i 28.0232 2.6114i 1096.6866 −528.9631
0.8263 0.0562i 1.8294 16.1408i 30.2047 −1.6526 5.3119i −17.0107 63.9912i 20.3713 85.8017i 631.6663 −468.8777

27

SciPost Phys. 16, 003 (2024)

Table 13: List of the correlation functions in 8 eigenstates of the trace representation calculated by exact diagonalization for d = 3. We restrict the eigensubspace of
the U(1) charges: Na = 0 (a = 1, 2, 3) where Na is defined in the Eq. (50) in [55].
E is the eigenenergy and k′ is the overall momentum quantum number concerning
the two-site shift. We list the eigenstates with 0 ≤ k′ ≤ 2. The numerical errorsof the correlation functions compared to those computed from the factorization formula (69)–(73), using the current mean values in Table 14, are at most 10−13 . The
eigenstates indexed 1, 2, 7 correspond to the eigenstates of the XXZ chain indexed
1, 2, 3 in Table 7 respectively. The errors between the corresponding correlation functions are at most of the order 10−13 .

1
2
3
4
5
6
7
8

E

k′

〈e1 e2 〉〈e1 e3 〉〈e1 e2 e3 〉′e1 e4

14.7275
13.2170
13.0858
11.7967
10.9631
10.5358
10.4142
10.3673

0
0
0
1
1
0
1
2

1.66551.02451.4722
1.0923 + 0.5002i
1.0293 + 0.2706i
1.01300.6402 + 0.1402i
0.8388 + 0.2242i

4.39984.10742.67312.0904
1.97181.55711.65532.04263.62483.18632.8585
1.9444
1.71792.30801.10360.86512.6996
1.13953.02512.44911.3104
1.29672.53031.0503

Table 14: List of the current mean values in the trace representation with the parameters being the same as Table 13. The numerical errors of the current mean value
formula (60) against the above numerics are at most 10−10 .
ψ0,0
1
2
3
4
5
6
7
8

ψ0,1

ψ0,2

ψ0,3

ψ0,4

ψ1,1

ψ1,2

ψ1,3

ψ1,4

ψ2,2

ψ2,3

ψ2,4

ψ3,3

1.840904.77640146.8741 −6.2902
0−77.62880114.31190
1840.2901 −5749.4570
1.652107.4127076.7147 −2.6713
0−132.1650
074.127404818.2829 −3336.7489
1.63570
7.5679083.0324 −3.9041
0−118.2845
073.0298
04623.7150 −3583.7330
1.4746 1.0003i 6.8665 11.5587i 255.2844 −2.7927 −4.9296i −61.9843 −402.8328i 91.8175 −10.2651i 3458.0998 −3489.5801
1.3704 0.5411i 4.8218 33.9784i 212.4193 −3.4178 5.5017i −44.7668 −283.1773i 85.5345 73.1007i 4790.6704 −1252.5544
1.317008.75490307.2956 −1.29560−72.0023
077.429202530.5220 −4001.5256
1.3018 0.2803i 3.8410 37.3575i 145.4547 −2.6036 12.0533i −34.1685 −42.6595i 60.8878 225.5422i 3698.3625 −1340.4055
1.2959 0.4484i 0.3683 11.3196i −267.3614 −4.9239 0.7804i −96.6653 257.4877i 23.2054 210.6901i −237.9226 −3551.3820

28

SciPost Phys. 16, 003 (2024)

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