to converge to (10.4) when $L\to\infty$. We finally identify the region of
criticality ${\cal R}$ with the interior of ${\cal B}_{2}$.

##  11 Conclusions and discussion

In this paper, we discovered a precise relation between representation theory
of the aTL algebra and the algebraic structure of the solutions to the Bethe
equations of the XXZ spin chain with a diagonal twist. While such a connection
might seem surprising at the first glance, it turns out to be quite natural.
One of the reasons that this connection appears to have been left unnoticed
hitherto is presumably that it would be rather difficult to see it by solving
the Bethe equation numerically. This demonstrates the power and usefulness of
computational algebraic geometry in the study of integrable models.

Using this relation, we were able to compute the torus partition function of
the O($\mathfrak{m}$) loop model and the $Q$-state Potts model in two very
different ways, which we called the _geometric algebra_ and _algebraic
geometry_ approaches, respectively. The final results of these two methods
match nicely. We studied the distribution of partition-function zeros for
$L=6,8,10$ in the partial thermodynamic limit and found that the zeros
condense on certain curves. The condensation curves can be constructed
directly in the partial thermodynamic limit and match beautifully with our
numerical results of partition function zeros. We would like to highlight the
fact that part of the condensation curves can be understood from conformal
field theory in the full thermodynamic limit, $L\to\infty$. This is quite
remarkable because usually condensation curves can only be determined
numerically, while here we can write down the analytic formulae (10.3)–(10.4)
for these curves.

As a byproduct of the algebraic-geometry approach, we constructed the rational
$Q$-system for the _twisted_ Bethe equations, both for the XXX and the XXZ
spin chains. The main advantage of the rational $Q$-system, compared to the
original Bethe equations, is that it leads to physical solutions only, and
hence we do not need to impose further conditions to exclude unphysical ones.
We expect that our construction generalizes naturally to higher-rank cases,
some of which have recently been shown to harbor random-geometry applications
generalizing those of the loop model [37, 38].

From the perspective of the CAG (Computational Algebraic Geometry)-BAE (Bethe
Ansatz Equations) program, this work is the first step in applying CGA
techniques to XXZ-type BAE, which are not polynomial equations in their
original form, but which can nevertheless be rationalized by a proper change
of variables. These equations contain free parameters, which are the
anisotropic parameter $q$ and the twist $\rho^{2}$ in our case. On the one
hand, these parameters bring richer structures to the equations because the
solutions depend on the parameters. There can be special loci in the parameter
space where the structure of the solutions change drastically and exhibit
noteworthy features. The resonance condition which we analyzed in this paper
is an example. On the other hand, it brings new challenges in the CAG
computations. To start with, the zero-remainder conditions associated with the
rational $Q$-system are more involved. At the same time, although the
computation of Gröbner bases and companion matrices are purely algebraic and
straightforward, the standard algorithm leads to rather bulky analytic
expressions in intermediate steps. This fact slows down the computation
considerably. This prohibits us from obtaining explicit analytic expression
for $L>10$ by the algebraic-geometry approach.

A number of natural future directions can be pursued following the current
work. One important immediate question is to investigate the case where $q$ is
a root of unity. It has been known for a while that the representation theory
of Temperley-Lieb algebra has an intricate structure [39] in this case.
Already in the non-periodic case, this leads to indecomposable structures in
the lattice model [40, 41] which entail strong connections to the
corresponding structures in the logarithmic CFT that govern the continuum
limit [42]. The case of periodic boundary conditions is even richer and, as
yet, the resulting incomposable structures are only partially understood (see
the review [43] and references therein). Given the relation established in the
current paper for generic $q$, we expect that these structures should also be
encoded in the solutions of the corresponding Bethe equations. Indeed, we can
already see that all the Gröbner bases that we constructed in this paper
exhibit vanishing leading coefficients when $q$ is an appropriate root of
unity. Can we see more fine structures by using proper algebro-geometric
tools? We leave this fascinating question for future studies.

Another natural generalization is to consider the XYZ spin chain. This case is
even less investigated, as it is significantly more complicated. Following the
strategy of the current paper, we need to seek for a proper change of
variables to rationalize the Bethe equations which involve elliptic function.
This seems to be possible following the treatment in [44], but with new
features.

Apart from developing algebro-geometric methods for more general types of
Bethe equations, one can also find applications of our method in broader
contexts. So far we have mainly applied our methods to compute partition
functions of important statistical-mechanical models, such as the six-vertex
model [4] and the Potts model. In recent years, it has been discovered that
integrable spin chains have deep connections with supersymmetric gauge
theories in various setups. We give one potential application as an example.
By the Bethe/gauge duality [45, 46, 47], Bethe-ansatz techniques can be
applied to compute twisted superconformal indices [48]. The Bethe equations
arising in this context are precisely those of the twisted XXZ spin chain
which we studied in this paper. The algebraic-geometry approach developed here
can be applied straightforwardly to compute twisted superconformal indices
analytically.

## Acknowledgements

We would like to thank S. Ribault and H. Saleur for inspiring discussions. The
research of JLJ was supported by the European Research Council through the
grant NuQFT and by the Agence Nationale de la Recherche through the grant
CONFICA (grant No. ANR-21-CE40-0003). YZ received support from the NSF of
China through Grants No. 11947301, 12047502, 12075234 and the Key Research
Program of the Chinese Academy of Sciences through Grant No. XDPB15. YZ is
also grateful to the Institute of Theoretical Physics, Chinese Academy of
Sciences, for its hospitality through the “Peng Huanwu visiting professor
program”. Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) -
Projektnummer 286237555 - TRR 195 [Funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Project- ID
286237555 - TRR 195]. The work of JB was supported by Project B5 of SFB-TRR
195.

##  Appendix A More on computational algebraic geometry

In this appendix, we provide an overview of the methods of computational
algebraic geometry used in the main text. Our discussion contains two parts.
In the first part, we introduce the basic notions of computational algebraic
geometry with examples. Most of the contents in this part is standard. For
more details we refer to the textbooks [49, 50]. When the algebraic equations
under consideration contain extra free parameters, like the ZRC of the twisted
XXZ spin chain, direct computations lead to rather complicated analytic
expressions in the intermediate steps. This slows down the computations
considerably. To improve the efficiency, we need to employ a slightly
different strategy, which will be discussed in the second part.

###  A.1 Basic notions of computational algebraic geometry

###### Definition A.1

An affine algebraic set is the common zero set

$V(f_{1},\ldots,f_{r})=\left\\{p\in K^{n}\mid
f_{1}(p)=0,\ldots,f_{r}(p)=0\right\\}$

of polynomials $f_{1},\ldots,f_{r}\in R=K[x_{1},\ldots,x_{n}]$.

Is there a structure in algebra which corresponds to the geometry? If
$f_{1}\left(p\right)=0,\ldots,f_{s}\left(p\right)=0$ for $p\in K^{n}$, then
also every $R$-linear combination of the $f_{i}$ vanishes on $p$, that is

$\left(\sum_{i=1}^{s}r_{i}\cdot
f_{i}\right)\left(p\right)=\sum_{i=1}^{s}r_{i}\left(p\right)f_{i}\left(p\right)=0$

for all $r_{i}\in R$. This leads to a key concept in algebra, the notion of an
ideal:

###### Definition A.2

Let $R$ be a commutative ring with $1$. An ideal is a non-empty subset
$I\subset R$ with $a+b\in I$ and $ra\in I$ for all $a,b\in I$ and $r\in R$.

For $S\subset R$ we write $\left\langle S\right\rangle$ for the ideal
generated by $S$, which consists of all finite $R$-linear combinations of
elements of $S$.

So it makes sense to generalize our definition of a vanishing loci to
arbitrary subsets $S\subset R$, and, as seen above, we have
$V(f_{1},\ldots,f_{s})=V(\left\langle f_{1},\ldots,f_{s}\right\rangle)$. On
the other hand, if $I\subset R$ is an ideal, then $V(I)$ is an algebraic set:
Every ideal $I\subset R$ is finitely generated, that is, of the form
$I=\left\langle f_{1},\ldots,f_{s}\right\rangle$ with $f_{i}\in R$. Rings with
this property are called Noetherian.

###### Definition A.3

A commutative ring $S$ with $1$ is called Noetherian if every ideal $I\subset
S$ is finitely generated, equivalently, if every ascending chain of ideals
$I_{1}\subset I_{2}\subset\ldots$ becomes stationary.

######  Theorem A.4 (Hilbert Basis Theorem)

If $S$ is a Noetherian ring, then also $S\left[x\right]$ is Noetherian. In
particular, if $K$ is a field then, by induction,
$K\left[x_{1},\ldots,x_{n}\right]$ is Noetherian.

###### Definition A.5

Let $K$ be a field. An element

$x^{\alpha}:=x_{1}^{\alpha_{1}}\cdot\ldots\cdot x_{n}^{\alpha_{n}}\in R$

is called a monomial, and an element $c\cdot x^{\alpha}$ with $c\in K$ is
called a term. Every polynomial $f\in R$ is a finite sum
$f={\textstyle\sum\nolimits_{\alpha}}c_{\alpha}x^{\alpha}$ of terms. The
degree $\deg(f)$ of $f$ is the maximal
$\left|\alpha\right|:=\alpha_{1}+\ldots+\alpha_{n}$ with $c_{\alpha}\neq 0$.

If $>$ is a total ordering on the set of monomials of $R$ and $0\neq f\in R$,
then the largest term $\operatorname{LT}(f)=c\cdot x^{\alpha}$ is called the
lead term of $f$, $\operatorname{LC}(f)=c$ the lead coefficient of $f$, and
$L(f)=x^{\alpha}$ the lead monomial of $f$. For $f=0$ we set
$L(0)=LT(0)=LC(0)=0$.

###### Definition A.6

A monomial ordering on the semigroup of monomials $x^{\alpha}$ is a total
ordering $>$ which respects multiplication, that is,
$x^{\alpha}>x^{\beta}\Rightarrow x^{\alpha}x^{\gamma}>x^{\beta}x^{\gamma}$ for
all $\alpha,\beta,\gamma$. A monomial ordering is global if it is a well
ordering,777That is, any non-empty set of monomials has a smallest element.
equivalently if $x_{i}>1$ for all $i$.

###### Definition A.7

The most important global monomial orderings are the lexicographical ordering
(lp), where $x^{\alpha}>x^{\beta}$ if the leftmost nonzero entry of
$\alpha-\beta$ is positive, and the degree reverse ordering (dp), where we
prefer the monomial of larger degree, and break a tie via
$x^{\alpha}>x^{\beta}$ if there is an $1\leq i\leq n$ such that
$\alpha_{n}=\beta_{n},\ldots,\alpha_{i+1}=\beta_{i+1},\alpha_{i}<\beta_{i}$.
While the former is the key to solve multivariate polynomial systems, the
latter usually leads to the best performance in computations.

###### Example A.8

Using $lp$ we divide $x^{2}y+x$ by $y-1$ and $x^{2}-1$:

$\boldsymbol{x}^{2}\boldsymbol{y}+x=x^{2}\left(\boldsymbol{y}-1\right)+1\cdot\left(\boldsymbol{x}^{2}-1\right)+\underline{x+1}\text{.}$

Here the lead terms are shown in bold and the remainder is underlined.

###### Definition A.9

Given a monomial ordering $>$ and a subset $G\subset R$, we define the leading
ideal of $G$ as

$L(G)=\left\langle L(f)\mid f\in G\backslash\\{0\\}\right\rangle\subset
R\text{.}$

######  Definition A.10 (Gröbner bases)

Given an ideal $I$ and a global monomial ordering $>$, then for a finite
subset $G\subset I$ with $0\notin G$ obviously $L(G)\subset L(I)$, and we call
$G$ a Gröbner basis of $I$ with respect to $>$, if $L(G)=L(I)$.

###### Theorem A.11

Every ideal $I\subset K[x_{1},\ldots,x_{n}]$ has a Gröbner basis.

###### Definition A.12

Given a set $G=\\{g_{1},\ldots,g_{s}\\}\subset R$, a normal form is a map
$\operatorname*{NF}(-,G):R\rightarrow R$ with

  1. 1.

$\operatorname*{NF}(0,G)=0$.

  2. 2.

If $\operatorname*{NF}(f,G)\neq 0$ then $L(\operatorname*{NF}(f,G))\notin
L(G)$.

  3. 3.

For every $0\neq f\in R$ there are $a_{i}\in R$ with

$f-\operatorname*{NF}(f,G)={\textstyle\sum\nolimits_{i=1}^{s}}a_{i}g_{i}$

and $L(f)\geq L(a_{i}g_{i})$ for all $i$ with $a_{i}g_{i}\neq 0$. Such an
expression we call a standard represenation of $f$.

###### Lemma A.13

Given a set $G=\\{g_{1},\ldots,g_{s}\\}\subset R$ and any order of preference
of the $g_{i}$ in the division, the Buchberger division yields a normal form
$\operatorname*{NF}(-,G)$, the Buchberger normal form.

###### Example A.14

If in Example A.8, we prefer the divisor $x^{2}-1$ over $y-1$, we obtain:

$\boldsymbol{x^{2}y}+x=y\cdot\left(\boldsymbol{x^{2}}-1\right)+\underline{x+y}\
\ $

So depending on the choice made, the remainder will be $x+1$ or $x+y$.

######  Theorem A.15 (Ideal membership)

Let $I\subset R$ be an ideal and $f\in R$. If
$G=\left\\{g_{1},\ldots,g_{s}\right\\}$ is a Gröbner basis of $I$ and
$\operatorname*{NF}$ is a normal form, then

$f\in I\Longleftrightarrow\operatorname*{NF}(f,G)=0\text{.}$

###### Corollary A.16

If $G\ $is a Gröbner basis of $I$, then $I=\left\langle G\right\rangle$.

###### Definition A.17

An element $f\in R$ is called reduced with respect to a set $G\subset R$, if
no term of $f$ is contained in $L(G)$. A normal form $\operatorname*{NF}(-,G)$
is called a reduced normal form if $\operatorname*{NF}(f,G)$ is reduced with
respect to $G$ for all $f$.

###### Example A.18

If we in the Buchberger normal form put non-divisible terms into the remainder
also in intermediate steps, we obtain a reduced normal form, the reduced
Buchberger normal form.

###### Example A.19

We can continue the Example A.14:

$\boldsymbol{x}^{2}\boldsymbol{y}+x=y\cdot\left(\boldsymbol{x}^{2}-1\right)+\underline{x}+1\cdot(\boldsymbol{y}-1)+\underline{1}$

Indeed, the remainder is now unique provided we divide by a Gröbner basis:

###### Theorem A.20

Let $>$ be a global ordering, $I\subset R$ an ideal, $f\in R$ and $G$ a
Gröbner basis of $I$. If $\operatorname*{NF}(-,G)$ is a reduced normal form,
then the map $\operatorname*{NF}(-,G)$ is uniquely determined by $>$ and $I$.
We then also write $\operatorname*{NF}(-,I)$.

###### Remark A.21

Even when using a reduced normal form and a Gröbner basis, although the
remainder is unique, the generated expression in the generators may not be. In
the Examples A.8 and A.19 we obtain

$\boldsymbol{x}^{2}\boldsymbol{y}+x=y\cdot\left(\boldsymbol{x}^{2}-1\right)+1\cdot(\boldsymbol{y}-1)+\underline{x+1}\vspace{-0.1cm}$

and

$\boldsymbol{x}^{2}\boldsymbol{y}+x=x^{2}\left(\boldsymbol{y}-1\right)+1\cdot\left(\boldsymbol{x}^{2}-1\right)+\underline{x+1}$

respectively.

Gröbner bases can be computed via Buchberger’s algorithm, which is derived
from

######  Theorem A.22 (Buchberger’s criterion)

If $I\subset R$ is an ideal, $\operatorname*{NF}$ a normal form and $0\notin
G=\\{g_{1},\ldots,g_{s}\\}\subset I$, then $G$ is a Gröbner basis of $I$, if
and only if $I=\left\langle G\right\rangle$ and
$\operatorname*{NF}(\operatorname*{spoly}(g_{i},g_{j}),G)=0$ for all $i\neq
j$, where

$\operatorname*{spoly}(f,g)=\frac{\operatorname{lcm}(L(f),L(g))}{LT(f)}f-\frac{\operatorname{lcm}(L(f),L(g))}{LT(g)}g$

is constructed to cancel the lead terms.

###### Corollary A.23

By computing $r=\operatorname*{NF}(\operatorname*{spoly}(g_{i},g_{j}),G)$ for
all $i,j$, if $r\neq 0$ adding $r$ to the generating set and iterating until
all $r$ vanish, one obtains a Gröbner basis of $\left\langle G\right\rangle$.
This process terminates in finitely many steps since $R\ $is Noetherian, and
is called Buchberger’s algorithm.

Buchberger’s algorithm generalizes both Gaussian elimination (for reducing
linear systems of equations) and the Euclidean algorithm (for computing
greatest common divisors of univariate polynomials) to higher-degree
multivariate polynomial equations. Alternative algorithms to obtain Gröbner
bases include Faugère’s F4/F5 algorithms [51]. For our computations, we make
use of the implemenation of Buchberger’s algorithm in the open-source computer
algebra system Singular [24].

###### Example A.24

Buchberger’s algorithm using the ordering $lp$ transforms the following given
system

$\begin{tabular}[c]{r}$2x^{2}-xy+2y^{2}-2=0$\\\
$2x^{2}-3xy+3y^{2}-2=0$\end{tabular}\ \ \ \ \longmapsto\ \ \ \
\begin{tabular}[c]{r}$3y+8x^{3}-8x=0$\\\ $4x^{4}-5x^{2}+1=0$\end{tabular}\ \ \
\ $

such that the resulting system allows us to read off the four solutions
$(x,y)=\left(\pm 1,0\right),(\pm\frac{1}{2},\pm 1)$, see Figure A.4. This
behaviour is typical for systems with zero-dimensional solution sets when
using $lp$.

Figure A.4: Buchberger’s algorithm for the intersection of two ellipses.

###### Definition A.25

Let $S$ be a commutative ring with $1$ and $I\subset S$ an ideal. The set
$S/I=\\{\overline{f}=f+I\mid f\in S\\}$ is a commutative ring with
one888$\overline{f}=f+I=\\{f+g\mid g\in I\\}$. Note that $\overline{f}$ is the
class of $f$ with respect to the equivalence relation $f\sim g\Leftrightarrow$
$f-g\in I$. with representative-wise addition
$\overline{f}+\overline{g}:=\overline{f+g}$ and multiplication
$\overline{f}\cdot\overline{g}:=\overline{f\cdot g}$.

###### Definition A.26

For our usual notation, and $I\subset R$ an ideal, the quotient ring $R/I$ is
called the coordinate ring of $V(I)$.

In particular, if, as in Example A.24, the vanishing locus of a given ideal
$I$ is finite, the coordinate ring $R/I$ allows us to directly analyize
$V(I)$. The coordinate ring becomes computationally accessible via Gröbner
bases.

###### Lemma A.27

If $I\subset R$ is an ideal and $\operatorname*{NF}(-,I)$ a reduced normal
form, then

$\phi:R/I\longrightarrow V=_{K}\left\langle x^{\alpha}\mid x^{\alpha}\notin
L(I)\right\rangle\text{, \quad}\overline{f}\longmapsto\operatorname*{NF}(f,I)$

is an isomorphism of $K$-vector spaces. With the multiplication on $V$ induced
by applying $\operatorname*{NF}(-,I)$ after the respective operation in $R$,
it becomes an isomorphism of $K$-algebras.

###### Theorem A.28

Let $K=\overline{K}$, $I\subset R$. Then the following are equivalent:

  1. 1.

$\left|V\left(I\right)\right|<\infty$

  2. 2.

If $G$ is a Gröbner basis of $I$, then for every $i$ there is a $g\in G$ with
$L(g)=x_{i}^{\alpha_{i}}$ and $\alpha_{i}\geq 0$.

  3. 3.

$\dim_{K}(R/I)<\infty$.

###### Theorem A.29

In the setting of the previous theorem,

$\left|V\left(I\right)\right|\leq\dim_{K}\left(R/I\right)$

and equality holds if $I$ is a radical ideal.999The radical of $I$ is
$\sqrt{I}=\left\\{f\in R\mid\exists a\in\mathbb{N}\text{ with }f^{a}\in
I\right\\}$, and $I$ is racial if $I=\sqrt{I}$. If $K=\overline{K}$ and $V(I)$
is finite, this amounts to $I$ not having multiple zeros.

###### Definition A.30

With respect to the basis $\mathcal{B}=\left\\{x^{\alpha}\mid x^{\alpha}\notin
L(I)\right\\}=:\\{m_{1},\ldots,m_{r}\\}$ of $V$, multiplication
$\overline{f}\cdot$ by $\overline{f}$ in $R/I$ is given by matrix
multiplication with the representing matrix

$M_{f}:=M_{\mathcal{B}}^{\mathcal{B}}(\overline{f}\cdot)=(a_{ij})_{i,j=1,\ldots,r}$

defined by

$\overline{f}\cdot\bar{m}_{j}=\sum_{i=1}^{r}a_{ij}\bar{m}_{i}$ (A.1)

The matrix $M_{f}$ is called the companion matrix of $f$.

###### Remark A.31

Passing from $\overline{f}\cdot$ to $M_{f}$ is a well-defined isomorphism of
$K$-algebras from $R/I$ to a subalgebra of $\operatorname*{Mat}_{n\times
n}(K)$: We have $M_{f}=M_{g}$ if an only if $\overline{f}=\overline{g}$, and
$M_{f+g}=M_{f}+M_{g}$ and $M_{f\cdot g}=M_{f}\cdot M_{g}=M_{g}\cdot M_{f}$.

###### Definition A.32

If $M_{g}$ is invertible, then the companion matrix of the rational function
$f/g$ $\in\operatorname{quot}(R)$ is $M_{f/g}=M_{f}\cdot M_{g}^{-1}$.

###### Remark A.33

Assume that $K=\overline{K}$ and $I$ is radical, and write
$V(I)=\left\\{p_{1},\ldots,p_{r}\right\\}$. Then the sum of the values of
$f\in R$ over the $p_{i}$ can be computed as the trace

$\sum_{i=1}^{r}f(p_{i})=\operatorname{Tr}M_{f}\text{.}$ (A.2)

###  A.2 Implementation notes

From the general introduction above, we see that if a physical quantity is
expressed as a rational function summed over all Bethe roots, it can be
calculated as the trace of the companion matrix (A.2). In practice, the
efficient implementation of this method is a non-trivial task, especially for
our case with two parameters (anisotropy parameter $q$ and the twist
$\rho^{2}$). This is mainly due to two reasons. To start with, the ZRC from
the rational $Q$-system, which are obtained by taking polynomial remainders,
involve complicated coefficients which are rational functions of $q$ and
$\rho$. Second, the algebro-geometric computation makes such complications
proliferate and lead to bulky expressions in the intermediate steps. On the
other hand, we know the final results that we are after have some simple
structures, _e.g._ they are a polynomial in $\mathfrak{m}$ with integer
coefficients. Therefore our strategy is to perform the computation for
different sets of fixed numbers $(q,\rho)$ and then use proper interpolation
formulae to find the final result.

In more details, when computing the Gröbner basis, we consider
$\mathbb{Q}$-valued $\mathfrak{m}$ and a constant $\rho$, and perform the
Gröbner-basis computation. Here $\mathfrak{m}=-q-q^{-1}$ in the usual
parametrization. Note that the choice of $\mathfrak{m}$ should avoid the
singular values of $q$. The Gröbner-basis computation with differnt
$\mathbb{Q}$-valued $\mathfrak{m}$ and a constant $\rho$ can be parallelized
on different cores or computer nodes. We observe that the computation time for
different values is roughly the same, so the parallelization is very
efficient. The Gröbner-basis computation is powered by the computer algebra
system Singular [24].

Secondly, for each value of $\mathfrak{m}$ and $\rho$, once the Gröbner basis
is obtained, we use (A.1) to get the companion matrix. Then the formula (A.2)
is used to compute the corresponding value of the partition function. In this
step, the linear-algebra related computations were done using Singular. Here
we emphasize one subtlety in this step: the companion matrix of a polynomial’s
power, $M_{f^{n}}$, is needed for our computation. In practice, $f^{n}$ may be
a huge polynomial with a high degree and to handle such an object requires a
lot of RAM. One might seek the help from the property of a companion matrix,

$M_{f^{n}}=(M_{f})^{n},$ (A.3)

and carry out the power computation on matrices instead of on polynomials.
However, in practice, we find that $M_{f}$ may be a dense matrix and such a
matrix-power computation is not particularly efficient. Therefore, we consider
the remainder $\bar{f}$ of $f$ divided by the Gröbner basis and the following
relation,

$M_{f^{n}}=M_{\bar{f}^{n}}\,.$ (A.4)

For the computation of $\bar{f}$’s power, after each multiplication, we divide
the intermediate result towards the Gröbner basis and take the remainder. In
the way, the power computation is both efficient and RAM economical.

Finally, from the previous two steps, we get the partition function’s values
for a list of $\mathbb{Q}$-values of $\mathfrak{m}$ and the constant $\rho$.
Note that the partion function is not a polynomial in $q$ but a polynomial in
$\mathfrak{m}$. Then we do a standard Newton-polynomial interpolation in
$\mathfrak{m}$ to get the fully analytic partition function. This step is also
powered by Singular.

We observe that these computations over finite fields are much faster than
over rational numbers. The results over finite fields can be further lifted to
rational numbers. Although in this paper, our computation is done with
rational numbers, the finite-field method is promising for future
computations.

##  Appendix B Partition function results

In the appendix, we give some results for the partition functions on $L\times
L$ tori.

###  B.1 Loop model

$\displaystyle Z(2,2)$ $\displaystyle=$ $\displaystyle
8\mathfrak{m}+8\mathfrak{m}^{2}\,,$ (B.1a) $\displaystyle Z(4,4)$
$\displaystyle=$ $\displaystyle
8192\mathfrak{m}+21480\mathfrak{m}^{2}+21632\mathfrak{m}^{3}+10870\mathfrak{m}^{4}+2912\mathfrak{m}^{5}+416\mathfrak{m}^{6}+32\mathfrak{m}^{7}+2\mathfrak{m}^{8}\,,$
(B.1b) $\displaystyle Z(6,6)$ $\displaystyle=$ $\displaystyle
933120000\mathfrak{m}+4805838000\mathfrak{m}^{2}+11273241408\mathfrak{m}^{3}+16045483644\mathfrak{m}^{4}+15559888944\mathfrak{m}^{5}+10937711238\mathfrak{m}^{6}+$
(B.1c) $\displaystyle
5789102208\mathfrak{m}^{7}+2366852316\mathfrak{m}^{8}+761946512\mathfrak{m}^{9}+196445178\mathfrak{m}^{10}+41330040\mathfrak{m}^{11}+7264344\mathfrak{m}^{12}+$
$\displaystyle
1093680\mathfrak{m}^{13}+142350\mathfrak{m}^{14}+15504\mathfrak{m}^{15}+1296\mathfrak{m}^{16}+72\mathfrak{m}^{17}+2\mathfrak{m}^{18}\,,$
$\displaystyle Z(8,8)$ $\displaystyle=$ $\displaystyle
11399736556781568\mathfrak{m}+98753367284702880\mathfrak{m}^{2}+408215404199487488\mathfrak{m}^{3}+1073458486777028536\mathfrak{m}^{4}+$
(B.1d) $\displaystyle
2019645508709576192\mathfrak{m}^{5}+2898901544581297904\mathfrak{m}^{6}+3305315996372032512\mathfrak{m}^{7}+3077975536301413966\mathfrak{m}^{8}+$
$\displaystyle
2388746188868407168\mathfrak{m}^{9}+1568981821963717920\mathfrak{m}^{10}+882883392305392000\mathfrak{m}^{11}+429902632210623280\mathfrak{m}^{12}+$
$\displaystyle
182694107341546624\mathfrak{m}^{13}+68277701699851360\mathfrak{m}^{14}+22602553762014464\mathfrak{m}^{15}+6675367972759608\mathfrak{m}^{16}+$
$\displaystyle
1772015524475520\mathfrak{m}^{17}+426101046596224\mathfrak{m}^{18}+93529351075584\mathfrak{m}^{19}+18862409602432\mathfrak{m}^{20}+$
$\displaystyle
3507873891968\mathfrak{m}^{21}+601033262944\mathfrak{m}^{22}+94278773888\mathfrak{m}^{23}+13385143248\mathfrak{m}^{24}+1693508480\mathfrak{m}^{25}+$
$\displaystyle
187378704\mathfrak{m}^{26}+17724928\mathfrak{m}^{27}+1392704\mathfrak{m}^{28}+87296\mathfrak{m}^{29}+4096\mathfrak{m}^{30}+128\mathfrak{m}^{31}+2\mathfrak{m}^{32}\,,$

###  B.2 Potts model

$\displaystyle Z(2,2)$ $\displaystyle=$ $\displaystyle
4\mathfrak{m}^{3}+7\mathfrak{m}^{4}+4\mathfrak{m}^{5}+\mathfrak{m}^{6}\,,$
(B.2a) $\displaystyle Z(4,4)$ $\displaystyle=$ $\displaystyle
4096\mathfrak{m}^{9}+14056\mathfrak{m}^{10}+20064\mathfrak{m}^{11}+15866\mathfrak{m}^{12}+7904\mathfrak{m}^{13}+2716\mathfrak{m}^{14}+688\mathfrak{m}^{15}+129\mathfrak{m}^{16}+16\mathfrak{m}^{17}+\mathfrak{m}^{18}\,,$
(B.2b) $\displaystyle Z(6,6)$ $\displaystyle=$ $\displaystyle
466560000\mathfrak{m}^{19}+2787706800\mathfrak{m}^{20}+7672413504\mathfrak{m}^{21}+2973997052\mathfrak{m}^{22}+15177608016\mathfrak{m}^{23}+13124121918\mathfrak{m}^{24}+$
(B.2c) $\displaystyle
8762270208\mathfrak{m}^{25}+4665716646\mathfrak{m}^{26}+2034402892\mathfrak{m}^{27}+742926501\mathfrak{m}^{28}+231451420\mathfrak{m}^{29}+62280555\mathfrak{m}^{30}+$
$\displaystyle
14527800\mathfrak{m}^{31}+2918460\mathfrak{m}^{32}+496704\mathfrak{m}^{33}+69786\mathfrak{m}^{34}+7788\mathfrak{m}^{35}+649\mathfrak{m}^{36}+36\mathfrak{m}^{37}+\mathfrak{m}^{38}\,,$
$\displaystyle Z(8,8)$ $\displaystyle=$ $\displaystyle
5699868278390784\mathfrak{m}^{33}+54114294708625056\mathfrak{m}^{34}+245768814667966464\mathfrak{m}^{35}+712035958563348408\mathfrak{m}^{36}+$
$\displaystyle
1480662138337422848\mathfrak{m}^{37}+2357818409374775024\mathfrak{m}^{38}+2995910569524062208\mathfrak{m}^{39}+3125648118282379662\mathfrak{m}^{40}+$
$\displaystyle
2735085239079547520\mathfrak{m}^{41}+2040940779811857696\mathfrak{m}^{42}+1316441555189606912\mathfrak{m}^{43}+742458511348946736\mathfrak{m}^{44}+$
$\displaystyle
369838333858415104\mathfrak{m}^{45}+164188589066409952\mathfrak{m}^{46}+65497210088055296\mathfrak{m}^{47}+23650757646619036\mathfrak{m}^{48}+$
$\displaystyle
7779745847048704\mathfrak{m}^{49}+2343002229704412\mathfrak{m}^{50}+648253398180864\mathfrak{m}^{51}+165020839221248\mathfrak{m}^{52}+$
$\displaystyle
38632494033536\mathfrak{m}^{53}+8297778619424\mathfrak{m}^{54}+1628522160640\mathfrak{m}^{55}+290394241736\mathfrak{m}^{56}+46707916288\mathfrak{m}^{57}+$
$\displaystyle
6714987800\mathfrak{m}^{58}+853018752\mathfrak{m}^{59}+94339168\mathfrak{m}^{60}+8906112\mathfrak{m}^{61}+698400\mathfrak{m}^{62}+43712\mathfrak{m}^{63}+2049\mathfrak{m}^{64}+64\mathfrak{m}^{65}+\mathfrak{m}^{66}\,.$

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# Thermoelectric power of overdoped Tl2201 crystals: Charge density waves and
$T^{1}$ and $T^{2}$ resistivities

J R Cooper1, J C Baglo2, C Putzke3111Present address: Max-Planck-Institute for
the Structure and Dynamics of Matter, Luruper Chaussee 149, Geb. 99 (CFEL),
22761 Hamburg, Germany, A Carrington4 1 Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom  2
Département de physique and Institut quantique, Université de Sherbrooke,
Sherbrooke, Québec, Canada  3 Institute of Materials, École Polytechnique
Fédéral de Lausanne, Lausanne, Switzerland  4 H. H. Wills Physics Laboratory,
University of Bristol, Bristol, BS8 1TL, United Kingdom<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We report measurements of the in-plane thermoelectric power (TEP) for an
overdoped (OD) crystal of the single layer cuprate superconductor Tl2Ba2CuO6+x
(Tl2201) at several hole concentrations ($p$), from 300 or 400 K to below the
superconducting transition temperature ($T_{c}$). For $p$ = 0.192 and 0.220,
small upturns in the TEP below 150 K are attributed to the presence of charge
density waves (CDW) detected by resonant inelastic X-ray scattering studies.
This suggests that measurement of the TEP could provide a simple and effective
guide to the presence of a CDW. Over a certain temperature range, often
strongly restricted by the CDW, the TEP is consistent with the Nordheim-Gorter
rule and the $T^{1}$ and $T^{2}$ terms in the in-plane resistivity of similar
crystals observed below 160 K. Two scenarios in which the $T^{1}$ scattering
term is uniform or non-uniform around the Fermi surface are discussed. As
found previously by others, for uniform scattering the $T^{1}$ terms give
scattering rates ($\tau^{-1}$) at lower $p$ that are somewhat larger than the
Planckian value $k_{B}T/\hbar$ and fall to zero for heavily OD crystals. Near
160 K, $\tau^{-1}$ from the $T^{2}$ terms corresponds to the Planckian value.

††: Supercond. Sci. Technol.

Cuprates, Thermoelectric power, Charge density waves, Electrical resistivity

##  1 Introduction

Identifying the pairing mechanism in cuprate superconductors and the origin of
the characteristic $T$-linear behaviour of the in-plane electrical
resistivity, are still controversial and unsolved questions that may turn out
to be closely related. In this regard single crystals of Tl2Ba2CuO6+x (Tl2201)
are of great interest because their superconducting transition temperature
($T_{c}$) can be progressively reduced from 90 K to zero by increasing the
oxygen content, $x$, and they have lower in-plane residual resistivity than
other cuprates such as La2-xSrxCuO4 (LSCO) and Bi2Sr2CuO6+δ that can be
overdoped (OD). Furthermore, any van Hove singularity is well away from the
Fermi energy [1]. Quantum oscillation (QO) experiments, e.g. [2] show that
heavily OD crystals with $T_{c}\leqslant$ 26 K have a large Fermi surface.
There appears to be no work on strongly underdoped (UD) Tl2201 crystals, with
difficulties caused, at least partly, by the volatility and toxicity of
thallium compounds, but this does not affect our conclusions. Recently, charge
density waves (CDW) have been detected below 150 K in a resonant inelastic
X-ray scattering (RIXS) study of OD Tl2201 crystals [3] with higher $T_{c}$.
The question as to whether this can account for the unusual evolution of the
Hall number as the hole doping ($p$) is reduced [1] is still being debated
[4]. In this paper we report measurements of the in-plane thermoelectric power
(TEP), $S_{a}(T)$ for a single crystal from the same preparation batch (Oct.
2014) as in the RIXS and Hall effect work. We argue that the CDW gives a small
but distinct upturn in $S_{a}(T)$ below 150 K. This conclusion is reinforced
by the observation of similar upturns in all polycrystalline samples [5] with
$T_{c}\geq 29$ K. Such data usually agree reasonably well with measurements of
$S_{a}(T)$ on single crystals.

We discuss in-plane electrical resistivity $\rho_{a}(T)$ data [1] for Tl2201
crystals and show that the TEP is consistent with the $T^{1}$ and $T^{2}$
terms in $\rho_{a}(T)$ observed below 160 K and the Nordheim-Gorter (NG) rule.
Corresponding scattering rates $\tau^{-1}$ are compared with the Planckian
rate $k_{B}T/\hbar$ [6, 7, 8]. We estimate $\tau^{-1}$ at low $T$ from
$\rho_{a}(0)$ and raise the question as to whether the residual low-$T$
specific heat of OD Tl2201 [9] is caused by standard pair-breaking in a
$d$-wave superconductor [10, 11] combined with a decrease in the strength of
the pairing interaction [12, 13], or whether it is caused by the pairing
interaction $V_{kk^{\prime}}$ being sufficiently anisotropic so that some
electrons are not paired. Namely if the $T^{1}$ scattering term is much weaker
on some parts of the Fermi surface does this imply that $V_{kk^{\prime}}$ is
much smaller there?

##  2 Results and analysis

###  2.1 Effect of CDW on the TEP

Above $1.1~{}T_{c}$ the TEP of OD cuprates generally varies approximately as
$a+b(T/300)$, where $a\simeq 1$ to $2~{}\mu$V/K and $b$ changes from
$\simeq-1$ to $-10~{}\mu$V/K as $p$ is increased from 0.16, the value for
maximum $T_{c}$, to the upper edge of the superconducting dome [5] while for
UD samples it is usually positive and much larger [5]. TEP data were taken in
an earlier, independent research project before the RIXS studies, using the
method described in the appendix. One crystal was used for the TEP work and
re-annealed to give successive $T_{c}$ values of 26.5 K, 29 K, 67 K and 88 K,
annealing conditions are given as Supplementary Information [14]. A small
correction $\simeq+0.1\mu$V/K below 50 K has been made for the $T_{c}$ = 29 K
and 26.5 K data using the phosphor bronze results shown in the appendix. AC
susceptibility data for various crystals of Tl2201 with small AC fields
parallel and perpendicular to the CuO2 layers after different annealing
treatments are reported as Supplementary Information [14]. They reveal an
unexpectedly complex behaviour and give evidence for two oxygen diffusion
processes with very different time scales. It is probable that one or both of
these are influenced by the concentration of Cu ions on Tl sites. This data
could be useful for future studies. TEP data for polycrystalline samples [5]
with $T_{c}$ = 84 K, 82 K, 59 K, 29 K and less than 4 K, provide important
confirmation of the present work.

Fig. 1(a) shows that the in-plane TEP $S_{a}(T)$ for the same Tl2201 crystal
with $T_{c}$ values of 88 K and 67 K has a small upturn below $\sim$150 K,
where there is also a clear change in the derivative $dS_{a}/dT$, as shown in
Fig. 1(b), but there is no such effect for $T_{c}=26.5$ K. For $T_{c}=29$ K
there is marginal evidence for a CDW with an onset near 40 K as indicated in
Fig. 3. All this correlates well with RIXS measurements on Tl2201 crystals
from the same preparation batch for which CDWs were detected for $T_{c}$
values of 56 K and 45 K but not for 22 K [3]. TEP data for $T_{c}$ = 88 K, on
an expanded scale in Fig.1(c), show a particularly clear anomaly near 145 K.

Figure 1: (a) In-plane thermoelectric power $S_{a}(T)$ for the same Tl2201
crystal with 4 different $T_{c}$ values. $p$ =0.192, 0.220, 0.271 and 0.274
for $T_{c}$ = 88 K, 67 K, 29 K and 26.5 K respectively determined from the
same modified formula used in Ref. [1] that accounts for the Fermi surface
area obtained from quantum oscillation studies [2] for $T_{c}\leq 26$ K. (b)
Left-hand scale, corresponding derivative plots $dS_{a}/dT$, and right-hand
scale RIXS intensity vs $T$ for a crystal with $T_{c}$ = 45 K from [3]. The
CDW causes a downturn in $dS_{a}/dT$ below 150 K, upturns at lower $T$ are
caused by the onset of superconductivity. Before forming the derivatives,
$S_{a}(T)$ and $T$ data were smoothed over 5 points, typically $\pm 10$ K, 10
or 20 K above $T_{c}$ and $\pm 4$ K nearer $T_{c}$. This reduced the scatter
without causing other changes. (c) Inset, expanded view for $T_{c}$ = 88 K
showing a clear anomaly at 145 K. Figure 2: (a) In-plane resistivity data
$\rho_{a}(T)$ for Tl2201 crystals from the same batch, taken from [1]. The
crystal with $T_{c}$ = 85 K is probably underdoped [1] and not considered
further here. The $p$ values for the other crystals are 0.183, 0.227, 0.256
and 0.274 for $T_{c}$ = 90 K, 65 K, 40 K and 26.5 K respectively. (b)
Corresponding derivative plots and straight lines expected when
$\rho_{a}(T)=\alpha_{0}+\alpha_{1}T+\alpha_{2}T^{2}$. Data above 175 K for
$T_{c}$ = 85 K were too noisy to give accurate derivatives. The straight lines
show the results of least squares fits.

The CDW onset temperature for $T_{c}$ = 45 K is 150 K as shown by the RIXS
data on the right hand scale of Fig. 1(b) but unfortunately the onset
temperature of 160 K shown in Ref. [3] for $T_{c}$ = 56 K is only an upper
bound. Although better overlap of $T_{c}$ values would have been desirable it
is very probable that the changes in $S_{a}(T)$ and $dS_{a}/dT$ near 150 K
shown in Fig. 1(a),(b) and (c) as well as at various temperatures for the
polycrystalline data in Ref. [5] are caused by a CDW. This agreement is
potentially useful because it shows that the TEP is a sensitive probe of the
CDW and could be used to study its pressure dependence and possibly its
magnetic-field dependence. Data for the OD88 crystal and the OD84
polycrystalline sample [5] show that the CDW is still present at lower $p$,
which is a new finding.

###  2.2 $T^{1}$ and $T^{2}$ terms in the resistivity

Resistivity data [1] for five different crystals from the same preparation
batch are shown in Fig. 2(a). As noted in [3] there appears to be no effect of
the CDW on the in-plane resistivity $\rho_{a}(T)$. This is unusual since other
CDW materials do show resistivity anomalies, but it is reinforced by the
derivative plots in Fig. 2(b). Although there is a change in slope of
$d\rho_{a}(T)/dT$ near 160 K, we do not believe that this is caused by a CDW.
It is also present for the crystal with $T_{c}=$ 26.5 K and a crystal from
another batch with $T_{c}=$ 17 K as well as for older data scanned in from
Ref. [15].

Such derivative plots [16, 17] are a convenient way of applying the formula

${\rho_{a}(T)=\alpha_{0}+\alpha_{1}T+\alpha_{2}T^{2}}$ (1)

because the range of fit is easily seen by inspection. They suggest that there
are two distinct values for the $T^{1}$ and $T^{2}$ terms, below and above 160
K where both the slopes and the intercepts in Fig. 2(b) change. For the
crystals with $T_{c}$ = 65 K, 40 K and 26.5 K, $\alpha_{1}$ and $\alpha_{2}$
were found from the intercepts at $T=0$ and the slopes of the straight lines
in Fig. 2(b), while the residual resistivity $\alpha_{0}$ was then found by
fitting the data in Fig. 2(a) to a second order polynomial in which
$\alpha_{1}$ and $\alpha_{2}$ were fixed. The coefficients $\alpha$ are given
in Table 1. (A small correction for a measurement offset of $\sim$1
$\mu\Omega$-cm below $T_{c}$ has been included where appropriate). Here we
focus on the values below 160 K, expecting them to be more closely related to
superconductivity. The coefficient $\alpha_{1}$ is presently of great interest
because of the suggestion that the Planckian scattering rate
$\hbar/\tau\approx k_{B}T$ [6] is the maximum allowed for any material [8],
and that it is an important intrinsic property of superconducting cuprates [7]
and many other correlated metals [8]. The Drude formula for the in-plane
conductivity of an isotropic cylindrical Fermi surface,
$\sigma=ne^{2}\tau/m^{*}$, which is a reasonable first approximation for
Tl2201, shows that $\alpha_{1}=$ 1 $\mu\Omega$-cm/K corresponds to
$\hbar/\tau\approx 3.3k_{B}T$, if the carrier density $n$ = 1.3 holes per unit
cell and $m^{*}=5.2m_{e}$. Although this value of $m^{*}$ has been measured by
QO only for OD Tl2201 [2], specific heat data [9] show that it only varies
weakly with $p$. So $\alpha_{1}=$ 0.3 $\mu\Omega$-cm/K in Fig. 2(b)
corresponds to $\hbar/\tau=k_{B}T$. For crystals with $T_{c}<$ 50 K,
$\hbar/\tau$ falls below this value because, as found previously [16, 17], the
size of the $T^{1}$ term falls linearly to zero as $p$ approaches the edge of
the superconducting dome. This could be an indication that the anomalous
scattering becomes confined to certain parts of the Fermi surface, in contrast
to the results for some other cuprates [7]. The value of $\alpha_{0}$ for
$T_{c}$= 26.5 K in Table 1, gives $\hbar/(k_{B}\tau)=19$ K and for a Fermi
velocity $v_{F}\equiv\hbar k_{F}/m^{*}=1.7\times 10^{7}$ cm/s, a mean free
path of 68 nm. Specific heat data for Tl2201 [9] show a large residual linear
term which increases as $p$ approaches the edge of the superconducting dome,
as expected when $\hbar/(k_{B}\tau)$ becomes comparable with $T_{c}$.

Previously [16] the deviations from a straight line above 160 K for the three
most overdoped crystals in Fig 2(b). were interpreted in terms of a parallel
conducting channel and the maximum resistivity of 1800 $\mu\Omega$-cm obtained
when the carrier mean free path is equal to the in-plane lattice spacing $a$.
However they are still clearly visible for the two most overdoped crystals in
Fig. 2(a) that have resistivities as low as 60 $\mu\Omega$-cm at 160 K and as
mentioned already for another crystal from a different preparation batch with
$T_{c}=17$ K. Therefore we have no clear interpretation of this effect. It is
worth noting that $\alpha_{2}$ is rather insensitive to $p$ as also shown in
[16]. At 160 K it corresponds to a $T^{2}$ term in the resistivity of 40
$\mu\Omega$-cm and, making the same assumptions as before, a scattering rate
$\hbar/\tau=k_{B}T$, the Planck condition [6]. This observation could have
ramifications for interpretations of the $T^{1}$ term arising from the Planck
condition. Perhaps $\hbar/\tau\geq k_{B}T$ might cause the deviations from
linearity in Fig. 2(b) above 160 K. Alternatively the deviations might be
connected with the effect of lattice vibrations on electron-electron Umklapp
scattering. Electron-electron scattering is rarely studied up to such high
temperatures. It is known that the Debye-Waller factor, which reduces the
intensity of X-ray diffraction lines, causes the energy gap of semiconductors
such as Si to fall as $T$ is increased. It is possible that Umklapp scattering
between Brillouin zones is reduced by a similar $T$-dependent factor that
becomes more significant as the amplitude of the lattice vibrations increases.

Data for all hole-doped cuprates clearly show that the curvature of in-plane
$\rho(T)$ plots increases on the OD side. It is widely accepted that there is
a crossover line on the $T-p$ phase diagram [18], which although ill-defined,
increases linearly with $p$ from zero for $p\geq 0.19$. It can be interpreted
as a coherence temperature $T_{\textrm{coh}}(p)$ for the charged carriers and
when $T\lesssim T_{\textrm{coh}}$ the $T^{2}$ term dominates. The results for
$T\leq$ 160 K in Fig. 2(b) and Table 1 are _qualitatively_ consistent this is
because $\alpha_{1}$ falls with $p$ while $\alpha_{2}$ stays constant. For
$T\geq$ 160 K this requires the slope of $d\rho_{a}/dT$ to become constant
which is the case above 200 K for $T_{c}=$ 30 K [16] and for data in Ref. [15]
above 250 K for several $T_{c}$ values between 0 and 80 K. It is puzzling that
this does not hold for $T_{c}$ = 65 K in Fig.2(b) suggesting that more data
are needed to clarify this point. We note that there can be complications from
Cu ions on the Tl sites affecting $p$ [14] as well as time dependent changes
in $T_{c}$ when a Tl2201 crystal is held at room temperature [19] that were
also seen in our TEP data for $T_{c}$ = 26.5 K.

###  2.3 Pair breaking vs “not-pairing”

There seems to be two possible ways of explaining the variation of
$\alpha_{1}$ and $T_{c}$ with $p$, which we will refer to as scenarios A and
B. The more standard one, scenario A, based on the usual $d$-wave form for the
pairing interaction, $V_{kk^{\prime}}$ $\propto$ $(\cos k_{x}-\cos k_{y})(\cos
k^{\prime}_{x}-\cos k^{\prime}_{y})$, has been thoroughly analysed and
discussed in Refs. [10, 11] and related papers where the importance of small
angle scattering was emphasised. Here $T_{c}$ falls as $p$ is increased
because the pairing interaction falls [12, 13] and hence $(\hbar/\tau)/\Delta$
where $\Delta$ is the superconducting energy gap, is larger. The theory is
very similar to that used to understand the effect of magnetic impurities in
classical superconductors. Experimental evidence for the relation between
$T_{c}$ and $\rho_{res}$ is summarised in Ref. [20] for both UD and OD
cuprates. A second, less conventional one, scenario B, involves a different
variation of the pairing interaction around the Fermi surface and is
considered later. Scenario A could be referred to as “pair-breaking” and B as
“not-pairing”. For our data set, the values of $\alpha_{0}$ for the OD65, OD40
and OD26 crystals in Table 1 are very close to that in Ref. [11] for their
OD25 crystal measured using microwaves.

###  2.4 Nordheim-Gorter analyis

In standard transport theory, when there are two or more independent
scattering mechanisms, for example electron-impurity and electron-phonon
scattering, giving contributions $\rho_{0}$ and $\rho_{1}$ to the resistivity,
the total resistivity $\rho=\rho_{0}+\rho_{1}$, which is known as
Matthiessen’s rule (MR). For ordinary metallic alloys MR is a good first
approximation although there are systematic deviations from MR that are well-
documented [21]. There is considerable experimental evidence for Zn-doped YBCO
and LSCO that MR applies to the $T$ \- independent impurity scattering and the
$T^{1}$ scattering terms in the cuprates [22, 23, 24]. The equivalent way of
calculating the TEP is via the Nordheim-Gorter (NG) [25] rule:

${S_{NG}=(\rho_{0}S_{0}+\rho_{1}S_{1}+\rho_{2}S_{2}+\cdots)/(\rho_{0}+\rho_{1}+\rho_{2}+\cdots)}$
(2)

where $S_{0}$ and $S_{1}$ etc. are the corresponding contributions to the TEP.
If the electron diffusion TEP is dominant, and there are no small energy
scales, these will be proportional to $T$, but with different signs and
magnitudes governed by the energy dependence of the particular scattering
mechanism. As shown in Table 1 the $T_{c}$ values in Fig. 1 are close enough
to some of those in Fig. 2(a) so the coefficients $\alpha$ can be used
directly to calculate the TEP via Eq. 2.

Table 1: Values of $\alpha$ given by the straight lines below 160 K in Fig. 2(b) plus a constrained fit to the data in Fig. 2(a) (see text), i.e. $\rho(T)=\alpha_{0}+\alpha_{1}T+\alpha_{2}T^{2}$. The carrier mean free path (mfp) is obtained from $\alpha_{0}$. $T_{c}^{\textrm{TEP}}$ values are from Fig. 1(a). $S_{0}$, $S_{1}$ and $S_{2}$ are from fits to Eq. 2, but cannot be determined independently (see text). The errors in $\alpha$ values are $\pm 10\%$ from geometrical uncertainties.  $T_{c}$ | $\alpha_{0}$ | $\alpha_{1}$ |  $\alpha_{2}$ x 1000 | mfp | $T_{c}^{\textrm{TEP}}$ | S${}_{0}/T$ | S${}_{1}/T$ | S${}_{2}/T$
---|---|---|---|---|---|---|---|---
(K) | ($\mu\Omega$-cm) | ($\mu\Omega$-cm/K) | ($\mu\Omega$-cm)/K2) | (nm) | (K) | ($\mu$V/K2) | ($\mu$V/K2) | ($\mu$V/K2)
90 | $(6.4\pm 0.4)$ | 1.37 | 0.125 | – | 88 | 0.132 | $-0.0115\pm 6\%$ | –
65 | 6.3 | 0.313 | 1.82 | 65 | 67 | 0.128 | – | $-0.030\pm 10\%$
40 | 6.9 | 0.17 | 1.34 | 60 | – | – | – | –
26.5 | 6.0 | 0.053 | 1.6 | 68 | 26.5 | 0.016 | – | $-0.033\pm 3\%$
Figure 3: $S_{a}(T)/T$ vs $T$ for $T_{c}$= 88 K, 67 K, 29 K and 26.5 K
together with fits to Eq. 2 for $T_{c}=$88 K, 67 K and 26.5 K, solid lines,
with parameters given in Table 1.

For $T_{c}\leq$ 67 K the parameters $S_{0}$, $S_{1}$, and $S_{2}$, all assumed
to be proportional to $T$, were found by fitting the TEP data in Fig. 1(a) to
Eq. 2. The range of fit was very limited, to below 200 K because of deviations
from Eq. 1 at higher $T$ and for the two higher $T_{c}$ values to above 150 K,
because of the onset of the CDW. Partly as a result of this, only the
parameter $S_{2}$ representing the contribution to the TEP from the $T^{2}$
resistivity term is well-defined, although the term $S_{0}$ from impurity
scattering must be positive. This procedure did not work for $T_{c}=$90 K
where it gave the very small value of $\alpha_{2}$ shown in Table 1 and a
negative value for $\alpha_{0}$. For this crystal we therefore set
$\alpha_{0}=6.4\pm 0.4\mu\Omega$-cm, the average for the other three $T_{c}$
values, giving the values of $S_{1}$ and $S_{0}$ shown in Table 1. For $T_{c}$
= 88 K and 67 K $S_{0}$ must be at least 0.1$\mu$V/K2 which is unexpectedly
large. In Fig. 3 it can seen that the calculated $S_{NG}$ curves for $T_{c}$ =
67 K and 26.5 K, agree well with the data giving $S_{2}=-0.033T\mu$V/K, of the
same sign and approximately 3 times larger than $S_{1}$ for $T_{c}$= 90 K. We
can therefore conclude that the energy dependence of the $T^{1}$ scattering
term has the same sign as that of the $T^{2}$ term, which is generally
attributed to electron-electron scattering, but is significantly smaller.
Measurements in which the superconductivity is suppressed by a magnetic field
and on crystals with different residual resistivity, possibly controlled by
doping with in-plane impurities such as Zn, would provide a further test of
this picture. The magnitude of $S_{2}$ is discussed below. Because of momentum
conservation the electron-electron scattering contribution to $\rho(T)$
usually depends on Umklapp scattering processes which will generally tend to
be stronger for larger energies. This implies that for electrons, the
electron-electron scattering term $S_{2}$ will be positive and for holes it
will be negative in agreement with experiment for Tl2201. Within this NG
picture the changes in TEP below 160 K in Fig. 1(a) as $p$ is increased are
caused by the increase in the ratio $\alpha_{2}T^{2}/\alpha_{1}T$ with doping
$p$ and the fact that $-S_{2}$ is larger than $-S_{1}$.

###  2.5 Possible anisotropy around the Fermi surface, scenario B

In the previous sections it was assumed implicitly that the $T^{1}$ and
$T^{2}$ scattering rates do not vary appreciably around the Fermi surface, so
that their contributions to $\rho(T)$ can be added in the same way as for
electron-phonon and electron-impurity scattering. This is only approximate
because there is experimental evidence from angle-dependent magnetoresistance
(ADMR) of the $c$-axis resistivity [27] that for crystals with $T_{c}$ = 20 K,
17 K and 15 K, the anisotropy, (the parameter $\alpha$ in Ref. [27]) increases
from 15$\%$ at 20 K to 40$\%$ at 110 K. Recently, new short-range CDW signals
have been identified in RIXS studies of many cuprate families [28]. These are
present for a wide range of $p$, including the “strange metal” region,
$p>0.19$, that is relevant for Tl2201 although such short range signals have
not yet been observed for Tl2201. They are short-ranged in real space, and
therefore broader in reciprocal space and are visible to higher $T$ than the
CDWs that have reasonably well-defined, often incommensurate, wave vectors
$\underline{Q}$. It has been argued that [29] that scattering from these
short-range charge density fluctuations (CDF) are responsible for the $T^{1}$
term in $\rho(T)$ and because the CDF have a wide range of $\underline{Q}$
vectors they would indeed give isotropic scattering. So if CDF could be
observed in Tl2201 crystals a good experimental test of this approach would be
to compare their intensities with the size of the $T^{1}$ terms in $\rho(T)$
for different values of $p$.

Calculations by Rice _et al_ [30] are more consistent with scenario B. These
are based on the theory of Yang, Rice and Zhang [31] and show that Umklapp
scattering between hot spots gives rise to a $T^{1}$ scattering rate in a very
limited region of $\underline{k}$-space and a corresponding $T^{1}$ term in
$\rho(T)$. They are not based on the usual Boltzmann transport equation.
Somewhat paradoxically, in an earlier paper using the variational principle
[32] to solve the Boltzmann equation, Hlubina and Rice [33] showed that the
$T^{2}$ term will dominate, although of course at low enough $T$ any non-zero
$T^{1}$ term must become larger than a term going as $T^{2}$. In this scenario
$1/\tau$ at or near the hot spots is likely to be close to the Planck value
because for $p\simeq$ 0.19 where any $T^{2}$ term is very small, as shown for
example in Table 1, $1/\tau$ is somewhat larger than the Planck value. It may
be possible to distinguish between scenarios A and B by measuring the
resistivity and London penetration depth on several crystals with similar
$T_{c}$ values. Scenario B provides a more direct link between the $T^{1}$
term and the superfluid density, emphasised recently in Ref. [17], than
scenario A. However unlike scenario A it would not necessarily account for
Matthiessen’s rule, the Nordheim-Gorter rule or the ADMR results. One possible
test would be to see if either scenario can account for the “vertical scaling”
of the electronic specific heat [34] in applied magnetic fields.

###  2.6 Comparison with the TEP of other correlated electron materials

The low $T$ TEP of many correlated electron materials, including some where
the $T^{2}$ term in $\rho(T)$ is dominant, has been shown [26] to obey the
following simple empirical formula:

$\frac{S}{T}=\frac{\gamma q}{N_{AV}e}$ (3)

Here $\gamma$ is the Sommerfeld coefficient, $N_{AV}$ Avogadro’s number and
$e$ the magnitude of the electronic charge. The Faraday number
$N_{AV}e=9.6\times{}10^{4}$ Coulombs/mole. For the simple case of free
electron dispersion and 1 electron per formula unit, $q=-1$ so taking $\gamma$
= 0.6 mJ/gm-at/K2 [9] gives $S=-13~{}\mu$V/K at 160 K, the end of the linear
regions in Fig. 2(b). This will be reduced by a factor $1+p\simeq 1.3$ by the
larger carrier concentration and by a factor 2/3 for a constant carrier mean
free path [26] giving $-6.7~{}\mu$V/K at 160 K. The value of
$-S_{2}=0.033T\mu$V/K in Table 1 gives a TEP of $-5.3~{}\mu$V/K at 160 K. So
the $T^{2}$ electron-electron scattering term in our TEP data for Tl2201
crystals agrees well with findings [26] for many other correlated electron
materials, while the contribution from the $T^{1}$ term is approximately a
factor 3 smaller.

##  3 Summary

We have argued that the presence of CDWs can be seen in the in-plane TEP of
Tl2201 crystals and in earlier data [5] for polycrystalline samples where the
measured TEP is very similar to $S_{a}(T)$ for similar values of $p$. The
Nordheim-Gorter rule seems to provide an adequate description of the TEP
contributions arising from the $T^{1}$ and $T^{2}$ terms in the in-plane
electrical resistivity. The magnitude of the TEP contribution from the $T^{2}$
term agrees with that for other correlated electron conductors [26] while the
contribution from the $T^{1}$ term is $\sim{}\\!3$ times smaller. If the
$T^{1}$ term were truly anomalous one might perhaps have expected a larger
difference. Within a simple cylindrical Fermi surface approximation, taking
into account the known value of $m^{*}$, the $T^{1}$ terms for the crystals
with $T_{c}$ of 40 K and 26.5 K correspond to scattering rates $1/\tau$ that
are significantly smaller than the Planckian value $k_{B}T/\hbar$, possibly
indicating that $\tau$ is anisotropic. A strongly anisotropic scattering rate
could provide a direct way of understanding the apparent correlation between
the magnitude of the $T^{1}$ term and the superfluid density emphasised
recently [17], namely only those carriers with a $T^{1}$ scattering rate would
be paired. But there could be difficulties in accounting for Matthiessen’s
rule, the Nordheim-Gorter rule and the ADMR work which suggests a small
anisotropy at low $T$ [27]. Also an extremely strong variation in
$V_{kk^{\prime}}$ might not be consistent with $d$-wave behaviour. Somewhat
unexpectedly, near 160 K, the $T^{2}$ term corresponds to a scattering rate
$1/\tau\gtrsim k_{B}T/\hbar$ for all values of the hole concentration.

###  A.1 Method used for TEP

Figure 4: Photo of the 1.5 mm long Tl2201 crystal used for TEP measurements.
50 $\mu$m diameter Au wires and 25 $\mu$m diameter Chromel wires are spot-
welded together to form differential thermocouples that are attached to Au
pads on the crystal with DuPont 4929N silver paint. The Au wires are heat sunk
to sapphire plates with General Electric GE-7031 low temperature varnish.

The basic method used for the TEP is similar to that originally used for
sintered polycrystalline bars in [5], but because of the small size of the
crystals, and the fact that thicker crystals or thin films on a crystalline
substrate have high thermal conductance, Au reference wires and Chromel wires
forming two differential thermocouples are attached to Au pads on the ends of
the crystal as shown in the photograph in Fig. A1. These wires extend for a
few centimetres before being soldered to Cu wires heat sunk to a high thermal
conductivity Cu block. A temperature difference ($\Delta T\lesssim 1$ K) is
produced by alternately heating two sapphire blocks to which the thermocouple
wires are attached with General Electric GE-7031 low temperature varnish and
the Seebeck coefficient of the sample $S_{\textrm{sample}}$ is given by
eliminating $\Delta T$ from the equations:

${V_{\textrm{Au-Au}}=(S_{\textrm{sample}}-S_{\textrm{Au}})\Delta T}$ (4)
${V_{\textrm{Ch-Ch}}=(S_{\textrm{sample}}-S_{\textrm{Ch}})\Delta T}$ (5)

Here the voltages across the Au wires, $V_{\textrm{Au-Au}}$ and the Chromel
wires, $V_{\textrm{Ch-Ch}}$ are measured using a Keithley Instruments digital
nanovoltmeter and a home-built nanovolt amplifier respectively, and their
ratio obtained by making a least squares linear fit to a plot of
$V_{\textrm{Ch-Ch}}$ vs $V_{\textrm{Au-Au}}$ for $\pm\Delta T$ and two points
with $\Delta T=0$. $S_{\textrm{Ch}}$ is the Seebeck coefficient of Chromel
taken from published standards and shown in Fig. A2. The same reel of
$50~{}\mu$m diameter Au wire was always used and $S_{\textrm{Au}}$, also shown
in Fig. A2 was obtained by measuring a strip of pure Pb which has a low TEP
[35].

Figure 5: Thermopower of Chromel and Au wires used in the present work. Figure
6: (a) and (b) TEP of Lake Shore Cryotronics phosphor bronze alloy wire on
logarithmic and linear temperature scales respectively. The most recent data
labelled JB PBronze is compared with earlier measurements, including one data
set where the superconducting Bi2212 sample had zero TEP below 85 K. (c) Low
$T$ data for the Tl2201 crystal after making a small correction of
$\sim{}\\!0.1~{}\mu$V/K, given by the data in (a). The dashed lines show how
$T_{c}$ values of 26 K and 29 K are determined.

In the past further checks have been made by measuring a superconducting
cuprate, Bi2212 with $T_{c}\simeq$ 90 K, and a piece of 127 $\mu$m diameter
phosphor bronze alloy wire purchased from Lake Shore Cryotronics which has a
conveniently low TEP. In the present work a piece of the same wire was
measured again and compared with earlier data, as shown in Figs. A3(a) and
(b). Small changes in the TEP of Au can be induced by bending (cold working)
the wire which will change the phonon drag component and the electron
diffusion component from any Fe impurities. Measuring the phosphor bronze wire
is a convenient method of correcting for such changes. Fig. A3(c) shows the
TEP data from Fig. 1(a) after making a small correction of
$\sim{}0.1~{}\mu$V/K corresponding to the difference in the TEP of the
phosphor bronze wire at low $T$ in earlier and more recent measurements shown
in Fig. A3(a). It can be seen that $T_{c}$ rises from $26\pm{}1$ to $29\pm{}1$
K after one month at room temperature.

AcknowledgmentsThis work was funded by the EPSRC (U.K.) grant number
EP/K016709/1. We are grateful to Prof. N. E. Hussey for helpful comments on
the first draft of the manuscript.

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