# Separating the online and offline DP-chromatic numbers

Peter Bradshaw  Simon Fraser University, Burnaby, BC, Canada<EMAIL_ADDRESS>
###### Abstract.

The DP-coloring problem is a generalization of the list-coloring problem in
which the goal is to find an independent transversal in a certain topological
cover of a graph $G$. In the online DP-coloring problem, the cover of $G$ is
revealed one component at a time, and the independent transversal of the cover
must be constructed in parts based on incomplete information. Kim, Kostochka,
Li, and Zhu asked whether the chromatic numbers corresponding to these two
graph coloring problems can have an arbitrarily large difference in a single
graph. We answer this question in the affirmative by constructing graphs for
which the gap between the online DP-chromatic number and the offline DP-
chromatic number is arbitrarily large.

##  1\. Introduction

We will consider several graph coloring problems. In the _list coloring
problem_ , we have a graph $G$ and a list $L(v)$ of colors at each vertex
$v\in V(G)$. In this setting, we say that an _$L$ -coloring_ of $G$ is a
proper coloring $\varphi:V(G)\rightarrow\bigcup_{v\in V(G)}L(v)$ of $G$ in
which $\varphi(v)\in L(v)$ for every vertex $v\in V(G)$. If $G$ always has an
$L$-coloring whenever $|L(v)|=f(v)$ for each vertex $v\in V(G)$, then we say
that $G$ is _$f$ -choosable_. If $f$ is a constant function $f(v)=k$, then we
say that the _list-chromatic number_ (or _choosability_) of $G$ is at most
$k$, and we write $\operatorname{\chi_{\ell}}(G)\leq k$.

The _DP-coloring problem_ is a generalization of the list coloring problem
introduced by Dvořák and Postle [3], defined as follows. Given a graph $G$ and
a function $f:V(G)\rightarrow\mathbb{N}$, an _$f$ -fold cover_ of $G$ is a
graph $H$ obtained by the following process:

  * •

For each vertex $v\in V(G)$, add a clique $K_{f(v)}$ to $H$, and write $L(v)$
for the vertex set of this clique.

  * •

For each edge $uv\in E(G)$, add a matching between $L(u)$ and $L(v)$.

Then, we say that an independent set in $H$ of size $|V(G)|$ is a _DP-
coloring_ of $G$ with respect to $H$. If $G$ always has a DP-coloring for
every $f$-fold cover $H$ of $G$, then we say that $G$ is DP-$f$-colorable. If
$f$ is a constant function $f(v)=k$, then we say that $H$ is a _$k$ -fold_
cover of $G$, and if $G$ always has a DP-coloring for every $k$-fold cover $H$
of $G$, then we say that the DP-chromatic number of $G$ is at most $k$, and we
write $\chi_{DP}(G)\leq k$. Given a cover $H$ of $G$, we often refer to the
vertices of $H$ as _colors_ , and if $c\in L(v)$ for a vertex $v\in V(G)$,
then we say that the color $c$ is _above_ $v$. Note that when $f(v)=k$ is a
constant function, if the cliques in $H$ corresponding to vertices in $G$ are
replaced with independent sets, and if each matching between sets $L(u)$ and
$L(v)$ is a perfect matching, then $H$ is a $k$-sheeted covering space of $G$,
and a DP-coloring of $G$ is equivalent to an independent transversal of the
fibers in $H$ above the vertices of $G$ (see [4] for an introduction to graphs
as topological spaces).

Every list-coloring problem can be transformed into a DP-coloring problem as
follows. Given a graph $G$ with a color list $L^{\prime}(v)$ at every vertex
$v\in V(G)$, we construct a cover $H$ of $G$ by adding a clique with vertex
set $L(v)$ for every vertex $v\in V(G)$, with elements of $L(v)$ corresponding
to colors in $L^{\prime}(v)$. Then, we consider each edge $uv\in E(G)$, and we
add an edge in $H$ between each pair $(c,c^{\prime})\in L(u)\times L(v)$ for
which $c$ and $c^{\prime}$ both correspond to a common color from
$L^{\prime}(u)\cap L^{\prime}(v)$. When $H$ is constructed this way, a DP-
coloring of $G$ with respect to $H$ is equivalent to an $L^{\prime}$-coloring
of $G$. Therefore, it holds that
$\operatorname{\chi_{\ell}}(G)\leq\chi_{DP}(G)$.

We will also consider two online graph coloring problems. The _online DP-
coloring_ problem takes place in the form of a DP-coloring game between two
players, called Lister and Painter. The game is played on a graph $G$ with a
function $f:V(G)\rightarrow\mathbb{N}$. At the beginning of the game, each
vertex $v\in V(G)$ has $f(v)$ tokens. On each turn $i$, Lister removes some
number $m_{i}(v)$ (possibly zero) of tokens from each vertex $v\in V(G)$ and
then reveals a clique $K_{m_{i}(v)}$ above each vertex $v$. Furthermore, for
each edge $uv\in E(G)$, Lister reveals a matching between the revealed cliques
above $u$ and $v$. The cliques and matchings revealed on this turn $i$ form a
cover $H_{i}$ of $G$. After $H_{i}$ is revealed, Painter chooses an
independent set from the vertices of $H_{i}$. The game ends when $G$ has no
more tokens for Lister to remove. Painter wins the game if she manages to
choose at least one color above each vertex of $G$ before the game is over;
otherwise, Lister wins. If Painter always has a winning strategy in the DP-
coloring game on a graph $G$ when each vertex $v\in V(G)$ begins with $k$
tokens, then we say that the _online DP-chromatic number_ (or _DP-
paintability_) of $G$ is at most $k$, and we write $\chi_{DPP}(G)\leq k$.
Observe that if each vertex of $G$ begins with $k$ tokens, then Lister has the
option of revealing a $k$-fold cover $H$ of $G$ on the first turn and asking
Painter to find a DP-coloring of $G$ with respect to $H$, and therefore
$\chi_{DP}(G)\leq\chi_{DPP}(G)$.

If, in the DP-coloring game, Lister is only allowed to remove at most one
token from each vertex of $G$ during each turn and must always reveals edges
wherever possible, then we call this variant of the game the _list-coloring
game_. For the list-coloring game, we may equivalently imagine that on each
turn $i$, Lister reveals a single color $c_{i}$ above each vertex of some
induced subgraph $G^{\prime}_{i}$ of $G$, and Painter must choose some
independent set $I_{i}$ of $G^{\prime}_{i}$ and color each vertex in $I_{i}$
with $c_{i}$. In this equivalent setting, each vertex $v\in V(G)$ still begins
with $f(v)$ tokens, and a single token is removed from $v$ whenever Lister
reveals a color above $v$. In this setting, Painter wins the game if and only
if she can color every vertex of $G$ before the game ends. If Painter always
has a strategy to win the list-coloring game on a graph $G$ when each vertex
$v\in V(G)$ begins with $f(v)$ tokens, then we say that $G$ is _$f$
-paintable_. If $f$ is a constant function $f(v)=k$, then we say that $G$ is
_$k$ -paintable_, and we write $\chi_{P}(G)\leq k$. The online list-coloring
game was originally invented using this framework of revealing colors above
vertices independently by Schauz [7] and Zhu [8].

At the end of the list-coloring game on $G$ with a constant function $f(v)=k$,
the colors revealed at each vertex $v$ form a set $L(v)$ of $k$ colors, and if
Painter wins the game, then Painter completes a proper $L$-coloring of $G$.
Since Lister is free to form any list assignment $L$ on the vertices of $G$,
it follows that if $G$ is $k$-paintable, then $G$ is also $k$-choosable, and
hence $\chi_{\ell}(G)\leq\chi_{P}(G)$. Also, since the online list-coloring
game is at least as difficult for Lister as the DP-coloring game, it also
follows that $\chi_{P}(G)\leq\chi_{DPP}(G)$.

After putting all of the inequalities between these parameters together, we
obtain two inequality chains:

$\displaystyle\chi_{\ell}(G)\leq$ $\displaystyle\chi_{DP}(G)$
$\displaystyle\leq\chi_{DPP}(G);$ $\displaystyle\chi_{\ell}(G)\leq$
$\displaystyle\chi_{P}(G)$ $\displaystyle\leq\chi_{DPP}(G).$

Given these inequality chains, it is natural to ask whether the differences
between adjacent parameters can be arbitrarily large. For three out of these
four differences, we find an affirmative answer by letting $G=K_{n,n}$.
Indeed, Bernshteyn [1] showed that a graph $G$ of average degree $d$ satisfies
$\chi_{DP}(G)=\Omega\left(\frac{d}{\log d}\right)$, implying that
$\chi_{DP}(K_{n,n})=\Omega\left(\frac{n}{\log n}\right)$. Since it is known
that $\chi_{\ell}(K_{n,n})\leq\chi_{P}(K_{n,n})=\log_{2}n+O(1)$ [2], this
shows that

$\chi_{DP}(K_{n,n})-\chi_{\ell}(K_{n,n})=\Omega\left(\frac{n}{\log
n}\right)\textrm{ \indent and
\indent}\chi_{DPP}(K_{n,n})-\chi_{P}(K_{n,n})=\Omega\left(\frac{n}{\log
n}\right).$

Duraj, Gutowski, and Kozik [2] also showed that

$\chi_{P}(K_{n,n})-\chi_{\ell}(K_{n,n})=\Omega(\log\log n).$

Therefore, by letting $G=K_{n,n}$, we achieve an arbitrarily large difference
for each adjacent pair of parameters except for $\chi_{DPP}(G)-\chi_{DP}(G)$.
For this final difference, Kim, Kostochka, Li, and Zhu [5] showed there exist
graphs $G$ for which
$\chi_{DPP}(G)-\chi_{DP}(G)\geq\chi_{P}(G)-\chi_{DP}(G)\geq 1$, implying that
this final difference can be positive. However, it has not been shown that
this difference can be arbitrarily large.

In this note, we will show that the difference $\chi_{DPP}(G)-\chi_{DP}(G)$
can also be arbitrarily large, answering a question of Kim, Kostochka, Li, and
Zhu [5]. Rather than choosing $G=K_{n,n}$, we will construct a graph $G_{t}$
for each $t\geq 1$ that satisfies $\chi_{DPP}(G_{t})-\chi_{DP}(G_{t})\geq t$.
We construct our graphs $G_{t}$ by generalizing an idea from the original
paper of Kim, Kostochka, Li, and Zhu [5]. Our graphs $G_{t}$ will also satisfy
the additional property that $\chi_{P}(G_{t})-\chi_{DP}(G_{t})\geq t$. By
combining this equality with the already-known estimate
$\chi_{DP}(K_{n,n})-\chi_{P}(K_{n,n})=\Omega\left(\frac{n}{\log n}\right)$, we
hence see that the difference $\chi_{DP}(G)-\chi_{P}(G)$ can achieve both
positive and negative values of arbitrarily large magnitude.

##  2\. The construction

For each integer $t\geq 1$, we will construct a graph $G_{t}$ that satisfies
$\chi_{P}(G_{t})-\chi_{DP}(G_{t})\geq t$. Since $\chi_{DPP}(G)\geq\chi_{P}(G)$
for all graphs $G$, our graphs $G_{t}$ will also satisfy
$\chi_{DPP}(G_{t})-\chi_{DP}(G_{t})\geq t$. Our construction is based on a
generalization of an idea of Kim, Kostochka, Li, and Zhu [5].

As we are concerned with showing that the paintability of each graph $G_{t}$
is large enough, we begin with an observation about the online list-coloring
game. If Lister and Painter play the online list-coloring game on a graph $G$
with some initial token assignment, then Lister wins if and only if he can
reach a position in which each uncolored vertex $v\in V(G)$ has some $g(v)$
remaining tokens, and the uncolored subgraph of $G$ is not $g$-choosable. In
the original paper of Kim, Kostochka, Li, and Zhu [5], the authors take
advantage of this idea in order to construct a graph $G$ satisfying
$\chi_{P}(G)\geq\chi_{DP}(G)+1$. In order to show that their graph $G$ has a
large enough paintability, these authors show that in the online list-coloring
game on their graph $G$, Lister always has a strategy to create an uncolored
$K_{1,k}$ subgraph of $G$ in which each leaf $\ell$ has $g(\ell)=1$ token and
the center vertex $v$ has $g(v)=k$ tokens. Since $K_{1,k}$ is not
$g$-choosable, it follows that Lister has a strategy to win the online list-
coloring game on their graph $G$.

In our construction, we will use a similar idea. We first fix the value
$k=2^{8t^{3}}$. (With more careful calculation, our proof would work with a
smaller value of $k$, but we use this larger value for clearer presentation.)
In each of our graphs $G_{t}$, we will show that Lister can always create an
uncolored $K_{t,k^{t}}$ subgraph in which each $t$-degree vertex $u$ has
$g(u)=t$ tokens and each $k^{t}$-degree vertex $v$ has $g(v)=k$ tokens. The
following lemma shows that if Lister manages to create such a subgraph of
$G_{t}$, then Lister wins the online list-coloring game.

######  Lemma 2.1.

Given the function $g:V(K_{t,k^{t}})\rightarrow\mathbb{N}$ defined above,
$K_{t,k^{t}}$ is not $g$-choosable.

###### Proof.

We let the $t$ vertices $v_{1},\dots,v_{t}$ of degree $k^{t}$ have pairwise
disjoint color lists $L(v_{1}),\dots,L(v_{t})$ of size $k$. Then, for each of
the $k^{t}$ elements $L\in L(v_{1})\times\dots\times L(v_{t})$, we let $L$ be
the list of a vertex $u$ of degree $t$. Then, for any coloring of
$v_{1},\dots,v_{t}$ using colors from their lists, some vertex $u$ of degree
$t$ has no available color in its list, and hence $K_{t,k^{t}}$ is not
$g$-choosable. ∎

The most important piece of our construction of $G_{t}$ will be the following
gadget $H_{t}$. We construct our gadget $H_{t}$ along with a function
$h:V(H_{t})\rightarrow\mathbb{N}$ as follows. We let $H_{t}$ contain
$(t+1)k^{t}$ copies $K^{1},\dots,K^{(t+1)k^{t}}$ of the clique $K_{t+1}$, and
we write $u^{\ell}_{1},\dots,u^{\ell}_{t+1}$ for the vertices of each clique
$K^{\ell}$. We write $U$ for the set of all of these vertices of the form
$u_{j}^{\ell}$; in other words, we let $U$ consist of all vertices that we
have introduced so far. Then, for each value $1\leq j\leq t+1$, we add $t$
independent vertices $x_{j}^{1},\dots,x_{j}^{t}$, and we make each of these
vertices adjacent to $u^{1}_{j},u^{2}_{j},\dots,u^{(t+1)k^{t}}_{j}$. We write
$X$ for the set consisting of all of these vertices of the form $x_{j}^{i}$.
For each vertex $u_{j}^{\ell}\in U$, we let $h(u_{j}^{\ell})=t+1$, and for
each vertex $x_{j}^{i}\in X$, we let $h(x_{j}^{i})=k-t+1$. We now prove two
lemmas that show that under appropriate circumstances, winning the online
list-coloring game on $H_{t}$ as Painter is much harder than finding a DP-
coloring on $H_{t}$.

######  Lemma 2.2.

$H_{t}$ is not $(h+t-1)$-paintable.

###### Proof.

We give each vertex $v\in V(H_{t})$ exactly $h(v)+t-1$ tokens, and we show
that Painter cannot win the list-coloring game on $H_{t}$.

On each of the first $t$ turns, Lister reveals a color at each vertex of each
clique $K^{\ell}$ and reveals an edge wherever possible. After these first $t$
turns, each clique $K^{\ell}$ must have an uncolored vertex $u_{j}^{\ell}$
with exactly $h(u_{j}^{\ell})-1=t$ tokens. Furthermore, since we have
$(t+1)k^{t}$ cliques $K^{\ell}$, each with at least one uncolored vertex,
there must exist some value $1\leq j^{*}\leq t+1$ for which at least $k^{t}$
vertices of the form $u_{j^{*}}^{\ell}$ are uncolored. However, the $k^{t}$
vertices of the form $u_{j^{*}}^{\ell}$ along with the $t$ vertices
$x^{1}_{j^{*}},\dots,x^{t}_{j^{*}}$ form an uncolored $K_{t,k^{t}}$ subgraph
in which each $t$-degree vertex $v$ has only $g(v)=t$ remaining tokens, and
each $k^{t}$-degree vertex $v$ has only $g(v)=k$ remaining tokens. Since Lemma
2.1 shows that this $K_{t,k^{t}}$ subgraph is not $g$-choosable, Lister has a
strategy to win the game. ∎

######  Lemma 2.3.

$H_{t}$ is DP-$h$-colorable.

###### Proof.

Consider an $h$-fold cover $H^{\prime}$ of $H_{t}$. Recall that given a vertex
$v\in V(H_{t})$, we say that $v$ corresponds to a clique $K_{h(v)}$ in
$H^{\prime}$ with a vertex set $L(v)$, and we say that $L(v)$ contains $h(v)$
_colors_. Using this terminology, we say that each color in $H^{\prime}$
appears above only one vertex of $H_{t}$, as the sets $L(v)$ forming the
cliques of $H^{\prime}$ are pairwise disjoint.

We will first color the vertices $x^{i}_{j}\in X$. For each vertex
$x^{i}_{j}\in X$ and color $c\in L(x^{i}_{j})$, we assign a set
$S_{c}\subseteq[(t+1)k^{t}]$ that consists of those indices $\ell$ for which
$L(u_{j}^{\ell})$ contains a color adjacent to $c$. We would like to color the
$t(t+1)$ vertices of $X$ using $t(t+1)$ colors $c_{1},\dots,c_{t(t+1)}$ that
correspond to a family $\mathcal{S}=\\{S_{c_{1}},\dots,S_{c_{t(t+1)}}\\}$ such
that for each value $1\leq q\leq t+1$, the following property holds:

($\star$) The intersection of any $q$ sets of $\mathcal{S}$ contains at most
$k^{t-\frac{qt}{t+1}}-1$ elements.

In particular, the intersection of any $t+1$ sets of $\mathcal{S}$ is empty.

We show that we may greedily color each vertex $x_{j}^{i}\in X$ while
satisfying ($\star$ ‣ 2). Suppose we wish to color some vertex $x\in X$ and
that we have already colored some subset $Y\subseteq X$ while satisfying
($\star$ ‣ 2). For each subset $A\subseteq Y$ of size $q\in[0,t]$ whose
vertices are colored with colors $a_{1},\dots,a_{q}$, we must choose some
color $c\in L(x)$ for which

($\bullet$) $|S_{c}\cap S_{a_{1}}\cap\dots\cap S_{a_{q}}|\leq
k^{t-\frac{(q+1)t}{t+1}}-1.$

Note that since $h(u^{\ell}_{j})=t+1$ for each vertex $u^{\ell}_{j}\in U$,
each value $\ell\in[(t+1)k^{t}]$ appears in at most $t+1$ sets $S_{c}$ for
colors $c\in L(x)$. Furthermore, since $|S_{a_{1}}\cap\dots\cap
S_{a_{q}}|<k^{t-\frac{qt}{t+1}}$ whenever $q\geq 1$, the number of colors
$c\in L(x)$ that do not satisfy ($\bullet$ ‣ 2) for a given $A\subseteq X$ is
at most

$\frac{(t+1)k^{t-\frac{qt}{t+1}}}{k^{t-\frac{(q+1)t}{t+1}}}=(t+1)k^{\frac{t}{t+1}}.$

Furthermore, since fewer than $2^{t(t+1)}$ subsets $A\subseteq Y$ can be
chosen, the number of colors $c\in L(x)$ that would cause ($\star$ ‣ 2) to be
violated is less than

$\displaystyle 2^{t(t+1)}(t+1)k^{\frac{t}{t+1}}$ $\displaystyle=$
$\displaystyle 2^{t(t+1)+\frac{8t^{4}}{t+1}}(t+1)$ $\displaystyle=$
$\displaystyle 2^{t(t+1)+8(t^{3}-t^{2}+t-1+\frac{1}{t+1})}(t+1)$
$\displaystyle<$ $\displaystyle 2^{8t^{3}-t}$ $\displaystyle<$ $\displaystyle
k-t+1=h(x).$

Therefore, some color $c\in L(x)$ can be used to color $x$ without violating
($\star$ ‣ 2).

Now, with every vertex $x_{j}^{i}\in X$ colored, and with ($\star$ ‣ 2)
satisfied, no index $\ell$ belongs to the intersection of more than $t$ sets
$S_{c}$, where $c$ is a color used at a vertex $x_{j}^{i}$, and hence after
coloring the vertices in $X$, at most $t$ colors are unavailable at each
clique $K^{\ell}$. Therefore, the vertices of each clique $K^{\ell}$ can be
ordered so that the first vertex has at least one available color, the second
vertex has at least two available colors, and so forth, until the last vertex
has $t+1$ available colors. Therefore, each remaining clique $K^{\ell}$ can be
$DP$-colored with its available colors, and the lemma is proven. ∎

Now, we construct our graph $G_{t}$. First, we make $k^{k-2t}$ copies of the
graph $H_{t}$, and we index them by the $(k-2t)$-tuples in $[k]^{k-2t}$. We
also add $k-2t$ vertices $y_{1},\dots,y_{k-2t}$ that are adjacent to all
vertices of $U$ in each copy of $H_{t}$. We write $\tilde{U}$ for this set of
neighbors of $y_{1},\dots,y_{k-2t}$, that is, the set of vertices belonging to
a set $U$ in some copy of $H_{t}$. The following two theorems show that
$\chi_{P}(G_{t})-\chi_{DP}(G_{t})\geq t$.

######  Theorem 2.4.

$\chi_{DP}(G_{t})\leq k-t+1$.

###### Proof.

We may give $G_{t}$ a DP-coloring with lists of size $(k-t+1)$ as follows.
First, we arbitrarily color the vertices $y_{1},\dots,y_{k-2t}$. Next, we
observe that the vertices in $\tilde{U}$ have lost at most $k-2t$ available
colors, so for each vertex $v$ in a copy of $H_{t}$, $v$ has at least $h(v)$
available colors remaining. Therefore, by Lemma 2.3, we may finish our DP-
coloring of $G_{t}$ by giving each remaining copy of $H_{t}$ a
DP-$h$-coloring. ∎

######  Theorem 2.5.

$\chi_{P}(G_{t})>k$.

###### Proof.

Suppose that the online list-coloring game is played on $G_{t}$ with $k$
tokens at each vertex. We will show that Lister has a winning strategy. For
each pair $(i,j)$ satisfying $1\leq i\leq k$ and $1\leq j\leq k-2t$, Lister
executes the following command. When Lister executes the command for a given
pair $(i,j)$, we say that this takes place on _turn $(i,j)$_.

> Reveal a color $c_{i,j}$ above $y_{j}$ and above each vertex of $\tilde{U}$
> that belongs to a copy of $H_{t}$ indexed by a $(k-2t)$-tuple with the value
> $i$ in the $j$th coordinate.

For each value $j\in[k-2t]$, we write $L(y_{j})=\\{c_{1,j},\dots,c_{k,j}\\}$
for the set of colors revealed above $y_{j}$.

For each $j\in[k-2t]$, we may assume that for some value $i_{j}\in[k]$,
Painter colors $y_{j}$ during turn $(i_{j},j)$ and hence does not color any
vertex of $\tilde{U}$ during turn $(i_{j},j)$. Indeed, if this is not the
case, then $y_{j}$ is never colored, and Painter will not have another chance
to color $y_{j}$. Therefore, for each value $j\in[k-2t]$, we may assume that
no vertex in a copy of $H_{t}$ indexed by a $(k-2t)$-tuple with an $i_{j}$
entry in the $j$th coordinate is colored using a color in $L(y_{j})$.

Now, let $H$ be the copy of $H_{t}$ indexed by the $(k-2t)$-tuple
$(i_{1},\dots,i_{k-2t})$. By our observation above, no vertex of $H$ has been
colored by a color in $L(y_{1})\cup\dots\cup L(y_{k-2t})$, and hence no vertex
of $H$ has been colored. Furthermore, since $k-2t$ tokens have been removed
from each vertex in $\tilde{U}\cap V(H)$, it follows that for each vertex
$v\in V(H)$, only $h(v)+t-1$ tokens remain at $v$. Therefore, Lister can
follow the strategy in Lemma 2.2 on $H$ in order to win the game, and thus
$\chi_{P}(G_{t})>k$. ∎

##  3\. Conclusion

While we have shown for each $t\geq 1$ the existence of a graph $G_{t}$ for
which $\chi_{P}(G_{t})-\chi_{DP}(G_{t})\geq t$, it is still open whether there
exists a sequence $\\{G_{t}\\}_{t\geq 1}$ of graphs for which

$\lim_{t\rightarrow\infty}\frac{\chi_{DPP}(G_{t})}{\chi_{DP}(G_{t})}>1\textrm{
\indent or
\indent}\lim_{t\rightarrow\infty}\frac{\chi_{P}(G_{t})}{\chi_{\ell}(G_{t})}>1.$

On the other hand, it is unknown whether $\chi_{P}(G)$ can be bounded above by
a linear or even polynomial function of $\chi_{\ell}(G)$, and it is unknown
whether $\chi_{DPP}(G)$ can be bounded above by a linear function of
$\chi_{DP}(G)$. Duraj, Gutowski, and Kozik [2] have pointed out that
currently, the best known bound for $\chi_{P}(G)$ in terms of $\chi_{\ell}(G)$
comes from the relationship between a graph’s choosability and minimum degree.
Namely, a result of Saxton and Thomason [6] states that a graph $G$ of minimum
degree $\delta$ satisfies $\chi_{\ell}(G)\geq(1+o(1))\log_{2}\delta$. Writing
$d$ for the degeneracy of a graph $G$, we observe that $G$ must have a
subgraph of minimum degree $d$, and hence we may use this result to observe
that

$\chi_{P}(G)\leq d+1\leq 2^{(1+o(1))\chi_{\ell}(G)}.$

For $\chi_{DPP}(G)$, we may use a result of of Bernshteyn showing that a graph
$G$ of minimum degree $\delta$ satisfies
$\chi_{DP}(G)\geq\frac{\delta/2}{\log(\delta/2)}$ in order to bound
$\chi_{DPP}(G)$ in terms of $\chi_{DP}(G)$ in a similar way. Using the same
observation as above, if $d$ is the degeneracy of $G$, then

$\chi_{DPP}(G)\leq d+1\leq(2+o(1))\chi_{DP}(G)\log\chi_{DP}(G).$

It is likely that a deeper understanding of these coloring parameters is
necessary to determine tight bounds between them.

##  4\. Acknowledgment

I am grateful to Alexandr Kostochka for offering helpful feedback on an
earlier version of this paper.

## References

  * [1] Anton Bernshteyn.  The asymptotic behavior of the correspondence chromatic number.  Discrete Math., 339(11):2680–2692, 2016.
  * [2] Lech Duraj, Grzegorz Gutowski, and Jakub Kozik.  Chip games and paintability.  Electron. J. Combin., 23(3):Paper 3.3, 12, 2016.
  * [3] Zdeněk Dvořák and Luke Postle.  Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8.  J. Combin. Theory Ser. B, 129:38–54, 2018.
  * [4] Allen Hatcher.  Algebraic topology.  Cambridge University Press, Cambridge, 2002.
  * [5] Seog-Jin Kim, Alexandr Kostochka, Xuer Li, and Xuding Zhu.  On-line DP-coloring of graphs.  Discrete Appl. Math., 285:443–453, 2020.
  * [6] David Saxton and Andrew Thomason.  Hypergraph containers.  Invent. Math., 201(3):925–992, 2015.
  * [7] Uwe Schauz.  Mr. Paint and Mrs. Correct.  Electron. J. Combin., 16(1):Research Paper 77, 18, 2009.
  * [8] Xuding Zhu.  On-line list colouring of graphs.  Electron. J. Combin., 16(1):Research Paper 127, 16, 2009.


# Incoherent transport in a classical spin liquid

Yao Wang  Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  University of Chinese Academy of Sciences, Beijing 100049, China  Yuan
Wan<EMAIL_ADDRESS>Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  University of Chinese Academy of Sciences, Beijing
100049, China  Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808,
China

###### Abstract

We study the energy and spin transport of the classical spin liquid hosted by
the pyrochlore Heisenberg antiferromagnet in the large $S$ limit. Molecular
dynamics calculation suggests that both the energy and spin diffusion
constants approach finite limits as the temperature tends to zero. We explain
our results in terms of an effective disorder model, where the energy/spin-
carrying normal modes propagate in a quasi-static disordered spin background.
The finite zero temperature limits of the diffusion constants are then
naturally understood as a result of the finite mean free path of the normal
modes due to the effective disorder.

##  I Introduction

Understanding the transport properties of materials is a major theme of
condensed matter physics. Historically, the Drude model of electrical
conduction heralded the beginning of condensed matter physics at the turn of
the last century [1, 2]. The Drude model foreshadowed the modern kinetic
theory [3], a powerful formalism for analyzing transport phenomena in
materials [4, 5]. What pillars the kinetic theory is the notion of _elementary
excitations_ — The elementary excitations are the carriers of the conserved
quantities such as energy, charge, or spin, and their scattering processes
determine the transport properties of these conserved quantities in materials.

While a highly successful theoretical framework, the kinetic theory is silent
about the transport phenomena in systems _without coherent elementary
excitations_ [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In the context of
frustrated magnetism, a prominent example is the classical spin liquid hosted
by the spin-$S$ pyrochlore Heisenberg antiferromagnet in the limit
$S\to\infty$ [18, 19, 20, 21, 22, 23]. The classical spin liquid phase appears
in the temperature regime $k_{B}T/(JS^{2})\lessapprox 1$, where $J$ is the
exchange constant 111To make the classical limit $S\to\infty$ meaningful, we
must scale the energy $E$ with $JS^{2}$ and time $t$ with $\hbar/JS$.
Mathematically, we are taking the limit $S\to\infty$ whilst keeping
$E/(JS^{2})$ and $JSt/\hbar$ fixed.. It is characterized by a diverging spin
correlation length $\xi\propto\sqrt{JS^{2}/(k_{B}T)}$ [20, 21, 22] in the low
temperature limit and a “Planckian” spin correlation time $\tau\propto\hbar
S/(k_{B}T)$ [18, 19, 23]. Crucially, the classical spin liquid does _not_
support magnons or paramagnons in that the system is neither magnetically
ordered nor proximate to any magnetically ordered state. Its dynamic spin
structure factor possesses no sharp features that are indicative of coherent
elementary excitations [23]. As a result, the transport phenomena in this
system falls outside of the purview of the ordinary kinetic theory.

In this work, we explore the transport phenomena in this system. For
conceptual simplicity, we omit at the outset the phonon contributions. The
pyrochlore Heisenberg antiferromagnet conserves both energy and magnetization.
We thus focus on the thermal conductivity $\kappa$ and the spin conductivity
$\sigma$. Our molecular dynamics calculation suggests that the energy and spin
current correlation functions decay rapidly on the time scale of order
$\hbar/(JS)$. Furthermore, both $\kappa$ and $\sigma$ approach finite limits
as the temperature tends to zero, i.e. $k_{B}T/(JS^{2})\to 0$. Since the heat
capacity and the magnetic susceptibility also approach finite limits, the
Einstein relation immediately implies that the energy diffusion constant
$D_{E}$ and the spin diffusion constant $D_{M}$ are both finite as
$k_{B}T/(JS^{2})\to 0$. This is in sharp contrast with the more familiar
magnon transport in a clean, ordered classical magnet, where $\kappa$ and
$D_{E}$ diverge in this limit owing to the divergent mean free path [24, 25].
The fast decay of current correlation functions and the finite zero
temperature diffusion constants are hallmarks of incoherent transport in this
classical spin liquid. We note that the saturation of spin diffusion constant
in the low temperature limit was previously observed in a related, two-
dimensional classical spin liquid [26, 27].

We also investigate the magnetic field dependence of the thermal and spin
conductivity. At low temperature, the system remains a classical spin liquid
up to the saturation field, beyond which point the system is fully polarized.
In the classical spin liquid phase, both the thermal and spin conductivity
approach finite limits as $k_{B}T/(JS^{2})\to 0$. At fixed temperature, we
find the thermal conductivity grows as the field approaches the saturation
field, whereas the spin conductivity decreases.

We interpret our results by using an _effective disorder model_. The classical
pyrochlore Heisenberg antiferromagnet possesses a high-dimensional degenerate
ground state manifold, where each point of the manifold represents a classical
ground state. The system’s motion may be decomposed into slow (with the time
scale on the order of $\hbar S/(k_{B}T)$) drifting modes in the tangent space
of the ground state manifold, and the fast (with the time scale on the order
of $\hbar/(JS)$) normal modes away from the manifold [18, 19]. We identify the
normal modes as the carrier of the energy and spin, which immediately implies
the transport of energy and spin are relatively fast processes comparing to
the change in the ground state spin configurations. We therefore may
approximately describe the energy and spin transport in terms of an effective
disorder model, where the normal modes propagate in a static, disordered spin
background. We stress that the effective disorder model is valid on time
scales shorter than $\hbar S/(k_{B}T)$ and that the system’s Hamiltonian is
manifestly invariant under lattice translations.

Using the effective disorder model, we are able to compute semi-analytically
the thermal and spin conductivity and find excellent agreement with the
molecular dynamics calculation. In particular, the mean free path of the
normal modes is finite thanks to the effective disorder, which naturally
explains the finite energy/spin diffusion constants in the low temperature
limit. The effective disorder model reveals yet another aspect of the
multifaceted link between geometric frustration and disorder physics.

The rest of this work is organized as follows. In Section II, we describe the
model and the molecular dynamics method. In Sec. III, we present the results
from molecular dynamics calculations. In Section IV, we discuss the effective
disorder model. In Section V, we discuss a few outstanding questions.

##  II Model and method

Figure 1: Relationship between the diamond lattice and the pyrochlore lattice.
Spins (yellow arrows) reside on the midpoint of neighboring diamond links,
which form the pyrochlore lattice. The up (blue) and down (pink) tetrahedra of
the pyrochlore lattice correspond to the A (mauve spheres) and B (green
spheres) sites of the diamond lattice. Dark solid arrows show the four real-
space vectors that point from a diamond A site to the four neighboring B
sites: $\vec{r}_{0}=\mathrm{a}(1,1,1)/4$, $\vec{r}_{1}=\mathrm{a}(1,-1,-1)/4$,
$\vec{r}_{2}=\mathrm{a}(-1,1,-1)/4$, $\vec{r}_{3}=\mathrm{a}(-1,-1,1)/4$,
where $\mathrm{a}$ is the size of the cubic crystallographic unit cell.

It is convenient for our purpose to view the pyrochlore lattice as the median
of the diamond lattice (Fig. 1). The up and down tetrahedra of the pyrochlore
lattice then naturally map to the A and B sublattices of the diamond lattice.
The classical Hamiltonian reads [18, 19],

$\displaystyle H=\frac{J}{2}\sum_{i}(\sum_{j\in
N_{i}}\mathbf{S}_{ij})^{2}-\mathbf{B}\cdot\sum_{\langle
ij\rangle}\mathbf{S}_{ij}.$ (1)

Here, $i,j$ label the diamond sites. The spin $\mathbf{S}_{ij}$ of length $S$
resides on the midpoint of the diamond link $ij$. The first term in Eq. (1)
describes the the Heisenberg exchange interaction between the neighboring
spins. $J>0$ is the exchange constant. The summation inside the bracket is
over the four neighboring sites $j$ of a given diamond site $i$. We add to the
Hamiltonian Eq. (1) an external magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$
as a handle to tune the spin fluctuations in the system, where
$\hat{\mathbf{z}}$ is the unit vector in the spin-$z$ direction. The Bohr
magneton and Landé $g$-factors are subsumed in $B$.

As the model Eq. (1) possesses independent spin and spatial rotation
symmetries, it is necessary to distinguish a vector in the _spin_ space and a
vector in the _real_ space to avoid any potential confusion. Throughout this
work, we write spin space vectors in boldface (e.g. Eq. (1)), and accent real
space vectors with an arrow (e.g. the caption of Fig. 1). When written in
components, we label the spin space directions in Greek alphabet, and real
space directions in Latin alphabet.

The thermodynamic phase diagram of Eq. (1) is well understood [18, 19, 20, 21,
22]. When $0\leq B/(JS)<8$, the spins remain disordered in the limit of
$k_{B}T/(JS^{2})\to 0$ and show algebraic long-range correlations
characteristic of the classical spin liquid. At finite temperature, the
algebraic spin correlation is cut off by a finite spin correlation length
$\xi\propto\sqrt{JS^{2}/(k_{B}T)}$. At $k_{B}T/(JS^{2})\sim 1$, the system
crosses over from the low temperature classical spin liquid phase to the high
temperature trivial paramagnetic phase. By contrast, when $B/(JS)\geq 8$, the
spins are fully polarized by the external field.

We endow the spins with Landau-Lifshitz precessional dynamics. Following Refs.
18, 19, we define:

$\displaystyle\mathbf{L}_{i}\equiv\sum_{j\in
N_{i}}\mathbf{S}_{ij}-\frac{\mathbf{B}}{2J}.$ (2)

The first term on the right hand side of Eq. (2) is the total magnetization of
the diamond site $i$ (or, equivalently, the corresponding pyrochlore
tetrahedron). The second term is the average magnetization per diamond site in
thermal equilibrium. Therefore, $\mathbf{L}_{i}$ is the magnetization
fluctuation on the diamond site $i$. Using $\mathbf{L}_{i}$, the classical
Hamiltonian can be recast in the following form:

$\displaystyle H=\frac{J}{2}\sum_{i}\mathbf{L}^{2}_{i},$ (3)

The ground state manifold is characterized by the condition
$\mathbf{L}_{i}=0$, $\forall i$. Thus, the set of $\mathbf{L}_{i}$ constitute
the normal modes that bring the system out of the ground state manifold. At
low temperature, the equipartition theorem implies
$\langle\mathbf{L}^{2}_{i}\rangle\approx 3k_{B}T/J$.

Using $\mathbf{L}_{i}$, we may succinctly write the equation of motion for
spin $\mathbf{S}_{ij}$ as [18, 19]:

$\displaystyle\dot{\mathbf{S}}_{ij}=\frac{J}{\hbar}(\mathbf{L}_{i}+\mathbf{L}_{j})\times\mathbf{S}_{ij}.$
(4a) Eq. (4a) is completed by a “dual” equation of motion for $\mathbf{L}_{i}$
[18, 19]:
$\displaystyle\dot{\mathbf{L}}_{i}=\frac{1}{\hbar}(\mathbf{L}_{i}\times\frac{\mathbf{B}}{2}+J\sum_{j\in
N_{i}}\mathbf{L}_{j}\times\mathbf{S}_{ij}).$ (4b)

Recall $|\mathbf{L}_{i}|\sim\sqrt{k_{B}T/J}$ in the low temperature limit. Eq.
(4a) suggests that the precession of the spin $\mathbf{S}_{ij}$ has a slow
component as $T\to 0$. By contrast, Eq. (4b) shows the precession frequency of
$\mathbf{L}_{i}$ is of order 1. We shall return to this point in Sec. IV.

We compute the thermal and spin conductivity by using the Kubo formula. To
this end, we derive the expression for the energy flux and the spin flux on
diamond link $ij$. We begin with the energy flux. Eq. (3) suggests the energy
associated with the diamond site $i$ is given by:

$\displaystyle E_{i}=\frac{J}{2}\mathbf{L}^{2}_{i}.$ (5)

Taking its time derivative, and using the equation of motion for
$\mathbf{L}_{i}$ (Eq. (4b)), we obtain:

$\displaystyle\dot{E}_{i}=\frac{J^{2}}{\hbar}\sum_{j\in
N_{i}}\mathbf{L}_{i}\cdot(\mathbf{L}_{j}\times\mathbf{S}_{ij}).$ (6)

Comparing the above with the energy continuity equation
$\dot{E}_{i}+\sum_{j\in N_{i}}I_{E,i\to j}=0$, where $I_{E,i\to j}$ denotes
the energy flux from the diamond site $i$ to $j$, we obtain:

$\displaystyle I_{E,i\to
j}=-\frac{J^{2}}{\hbar}\mathbf{S}_{ij}\cdot(\mathbf{L}_{i}\times\mathbf{L}_{j}).$
(7)

The above expression fulfills the symmetry requirements for the energy flux,
namely it is odd under time reversal and spatial inversion $i\leftrightarrow
j$.

The spin flux may be found in the same vein. Since Eq. (1) conserves the $z$
component of the total magnetization, only the $S^{z}$ flux is meaningful. The
$z$-component of the magnetization of the diamond site $i$ is given by:

$\displaystyle M_{i}=\hat{\mathbf{z}}\cdot\mathbf{L}_{i}+\frac{B}{2J},$ (8)

where $\hat{\mathbf{z}}$ is the spin-space unit vector in the $S^{z}$
direction. Taking its time derivative yields:

$\displaystyle\dot{M}_{i}=\frac{J}{\hbar}\sum_{j\in
N_{i}}(\mathbf{L}_{j}\times\mathbf{S}_{ij})\cdot\hat{\mathbf{z}},$ (9)

Comparing the above with the spin continuity equation $\dot{M}_{i}+\sum_{j\in
N_{i}}I_{M,i\to j}=0$, where $I_{M,i\to j}$ denotes the spin flux from the
diamond site $i$ to $j$, we find:

$\displaystyle I_{M,i\to
j}=-\frac{J}{\hbar}(\hat{\mathbf{z}}\times\mathbf{S}_{ij})\cdot(\mathbf{L}_{i}-\mathbf{L}_{j}).$
(10)

We may check that the above is even under time reversal and odd under spatial
inversion, consistent with the symmetry requirements for the spin flux.

We are now ready to write down the Kubo formula for the thermal conductivity
tensor $\kappa^{ab}$ and the spin conductivity tensor $\sigma^{ab}$ [28]:

$\displaystyle\kappa^{ab}$
$\displaystyle=\lim_{t\to\infty}\lim_{V\to\infty}\frac{1}{k_{B}T^{2}V}\int^{t}_{0}\langle
J^{a}_{E}(s)J^{b}_{E}(0)\rangle ds;$ (11a) $\displaystyle\sigma^{ab}$
$\displaystyle=\lim_{t\to\infty}\lim_{V\to\infty}\frac{1}{k_{B}TV}\int^{t}_{0}\langle
J^{a}_{M}(s)J^{b}_{M}(0)\rangle ds.$ (11b)

Here, $V$ is the volume of the system. $\langle\cdots\rangle$ denotes thermal
average. Note the extra power of $T$ in the Kubo formula for the thermal
conductivity. $J^{a}_{E}$ and $J^{a}_{M}$ are respectively the zero-wave-
vector component of the spatial Fourier transform of the energy and spin
current density:

$\displaystyle J^{a}_{E}$ $\displaystyle=\sum_{i\in A}\sum_{j\in
N_{i}}r^{a}_{i\to j}I_{E,i\to j};$ (12a) $\displaystyle J^{a}_{M}$
$\displaystyle=\sum_{i\in A}\sum_{j\in N_{i}}r^{a}_{i\to j}I_{M,i\to j},$
(12b)

where the first summation is over the A sublattice of the diamond lattice.
$r^{a}_{i\to j}$ are the real-space vectors pointing from an $A$ site to
neighboring $B$ sites (Fig. 1). As the Hamiltonian Eq. (1) possesses the cubic
lattice symmetry, the thermal conductivity and the spin conductivity tensors
are all diagonal: $\kappa^{ab}=\kappa\delta^{ab}$, and
$\sigma^{ab}=\sigma\delta^{ab}$. We shall present only the diagonal component
in this work.

We compute $\kappa$ and $\sigma$ by using the spin molecular dynamics method
[23]. We draw random initial spin configurations from the Boltzmann
distribution by using the Markov chain Monte Carlo. We then evolve each
initial spin configuration according to the equations of motion Eq. (4a) and
Eq. (4b). This produces an ensemble of evolution trajectories. We estimate the
thermal average in Eq. (11) by averaging over this ensemble.

We use in our calculation a system of $L\times L\times L$ primitive unit cells
with periodic boundary conditions. We monitor the convergence of the
integration in Eq. (11) by plotting the integral as a function of the
termination time $t$, which we interpret as the effective thermal conductivity
$\kappa(t)$ and the effective spin conductivity $\sigma(t)$ on that time
scale. We deem the integration has converged when the difference between
$\kappa(t)$ and $\kappa(2t)$ ($\sigma(t)$ and $\sigma(2t)$) is smaller than
the sampling noise.

It is convenient to embed the time integration that appears in the Kubo
formula into the numerical integration of the equation of motion. This is done
by exchanging the order of the time integration and the thermal average in Eq.
(11). To this end, we define an observable
$Q^{a}_{E,M}(t)=\int_{0}^{t}J^{a}_{E,M}(s)ds$, which obeys the equation of
motion $\dot{Q}^{a}_{E,M}=J^{a}_{E,M}$ with the initial condition
$Q^{a}_{E,M}(0)=0$. We solve this equation on the fly along with the spin
equation of motion. We can find $\kappa(t)$ and $\sigma(t)$ straightforwardly
by computing the correlation functions between $Q^{a}_{E,M}(t)$ and
$J^{a}_{E,M}(0)$.

In our Markov chain Monte Carlo, we obtain more than $7\times 10^{4}$ samples
from 144 independent runs. Each Monte Carlo step (MCS) consists of 1 lattice
sweep of heat bath update and 10 lattice sweeps of over-relaxation update. We
discard at least 500 MCS between two conseutive samples to reduce the sample
correlation. We integrate the equations of motion by using the 4th order
Runge-Kutta method. We set the step width to $0.02\hbar/(JS)$ with the
relative energy drift $<3\times 10^{-6}$ upon the termination of integration.

Figure 2: (a) Energy current correlation function as a function of time $t$ in
zero magnetic field, computed from molecular dynamics for various system sizes
$L$ (labeled “MD”) and from the effective disordered model for system size
$L=8$ (labeled “Disorder”). The temperature $k_{B}T/(JS^{2})=10^{-3}$. (b)
Similar to (a) but in magnetic field $B/(JS)=6$. Inset shows the short time
behavior of the correlation function. (c) Thermal conductivity $\kappa$ as a
function of the termination time $t$ in zero magnetic field. $\kappa_{0}\equiv
k_{B}JS/(\hbar\mathrm{a})$ is the natural unit for thermal conductivity, where
$\mathrm{a}$ is the size of the cubic crystallographic unit cell. (d) Similar
to (c) but for magnetic field $B/(JS)=6$. Figure 3: (a) Spin current
correlation function as a function of time $t$ in zero magnetic field,
computed from molecular dynamics (labeled “MD”) and from the effective
disorder model (labeled “Disorder”). The temperature
$k_{B}T/(JS^{2})=10^{-3}$. (b) Similar to (a) but in magnetic field
$B/(JS)=6$. (c) Spin conductivity $\sigma$ as a function of termination time
$t$ in zero magnetic field. $\sigma^{(0)}\equiv\hbar S/\mathrm{a}$ is the
natural unit for spin conductivity. (d) Similar to (c) but in magnetic field
$B/(JS)=6$.

##  III Results

In this section, we present results obtained from the molecular dynamics
calculation.

Fig. 2a shows the energy current correlation function $\langle
J^{a}_{E}(t)J^{a}_{E}(0)\rangle$ as a function of time $t$ at temperature
$k_{B}T/(JS^{2})=10^{-3}$ and in zero magnetic field. The correlation function
decays to 0 on the time scale of order $\hbar/(JS)$, indicating the energy
transport is a fast process comparing to the change in the spin
configurations. This fast time scale is consistent with the fact that the
$\mathbf{L}_{i}$ modes are the carrier of energy (Eq. (5)). In addition, the
correlation function shows weak dependence on the system size $L$.
Accordingly, the thermal conductivity $\kappa$ (Fig. 2c) quickly converges as
$t$ increases and shows little system size dependence, which allows us to use
the $L=16$ result to estimate the value of $\kappa$ in the thermodynamic
limit.

Fig. 2b and Fig. 2d show respectively the energy current correlation function
and the thermal conductivity at temperature $k_{B}T/(JS^{2})=10^{-3}$ and
magnetic field $B/(JS)=6$. The correlation function decays more slowly
comparing to the zero field case, but the decay is nonetheless fast in
comparison to the spin correlation time scale at this temperature, namely
$\hbar S/(k_{B}T)=10^{3}\hbar/(JS)$. Likewise, it takes longer time for the
thermal conductivity $\kappa$ to converge. We note $\kappa$ increases slightly
at late time, i.e. showing a small slope for large $t$. Nevertheless, for
$L=16$, we find the difference between the value of $\kappa$ at termination
time $JSt/\hbar=200$ and $100$ is statistically insignificant. We thus deem
the integral has converged at $JSt/\hbar=200$ within the statistical error.

Figure 4: Top panel: thermal conductivity $\kappa$ (a) and spin conductivity
$\sigma$ (b) as functions of temperature for various values of magnetic field.
The inset of panel (a) shows the thermal conductivity data for $B/(JS)=8$.
Error bars are smaller than the size of the symbol. Bottom panels: energy
diffusion constant $D_{E}$ (c) and spin diffusion constant $D_{M}$ (d)
inferred from the Einstein relation. $D^{(0)}\equiv JS\mathrm{a}^{2}/\hbar$ is
the natural unit of diffusion constant. Figure 5: Top panels: thermal
conductivity $\kappa$ (a) and spin conductivity $\sigma$ (b) as functions of
magnetic field $B/(JS)$ for the fixed temperature $k_{B}T/(JS^{2})=10^{-3}$.
Error bars are smaller than the size of the symbol. Bottom panels: energy
diffusion constant $D_{E}$ (c) and spin diffusion constant $D_{M}$ (d) as
functions of magnetic field, inferred from the Einstein relation.

We then turn to the spin transport. Fig. 3a shows the spin current correlation
function $\langle J^{a}_{M}(t)J^{a}_{M}(0)\rangle$ at temperature
$k_{B}T/(JS^{2})=10^{-3}$ and in zero magnetic field. Similar to the energy
current correlation function at the same temperature and field (Fig. 2a), the
spin current correlation function decays rapidly on the time scale of order
$\hbar/(JS)$ and shows little dependence on the system size $L$. We also note
it is more oscillatory than its energy current counterpart. Mirroring the
behavior of the spin current correlation function, the spin conductivity
$\sigma$ (Fig. 3c) converges quickly and shows weak finite size effects. This
indicates that the spin transport is also a fast process, which is consistent
with the fact that the $\mathbf{L}_{i}$ modes are the carriers of spin as well
(Eq. (8)). Similar to the thermal conductivity data, the small increase in
$\sigma$ at late time $t$ is a finite size effect in that the slope is
suppressed with larger system size.

At higher field $B/(JS)=6$, the spin current correlation function (Fig. 3b)
and the spin conductivity (Fig. 3d) show similar behaviors as the zero field
case. However, finite size effects are more pronounced. At system size $L=16$,
we find the difference between the value of $\sigma$ at termination time
$JSt/\hbar=30$ and $15$ is smaller than the statistical error, and we accept
its value at $JSt/\hbar=30$ as the estimate for the spin conductivity in the
thermodynamic limit.

Having established the methodology for estimating the transport coefficients,
we are ready to present their systematic dependence on temperature and
magnetic field. Fig. 4a shows the thermal conductivity $\kappa$ as a function
of temperature $T$ for various value of magnetic field. Throughout the
classical spin liquid phase ($B/(JS)=0,2,4,6$), $\kappa$ exhibits clear
signature of saturation as $T$ decreases by three orders of magnitude from
$k_{B}T/(JS^{2})=1$ to $10^{-3}$. We deduce the energy diffusion constant from
the Einstein relation: $D_{E}=\kappa/C_{V}$, where $C_{V}$ is the heat
capacity per unit volume. Recall the $C_{V}=12k_{B}/\mathrm{a}^{3}$ in the
limit of $k_{B}T/(JS^{2})\to 0$ [18, 19]. It follows that the energy diffusion
constant $D_{E}$ saturates in the low temperature limit (Fig. 4c).

The saturation of $D_{E}$ in the low temperature limit found in the classical
spin liquid is markedly different from clean, ordered classical magnets. In
the latter, the magnons are the energy carrier, and the kinetic theory
suggests $D_{E}\sim vl$ where $v$ is the characteristic spin wave velocity and
$l$ is the mean free path. As $l\to\infty$ as $T$ decreases due to the
suppression of scattering events, $D_{E}\to\infty$ as $k_{B}T/(JS^{2})\to 0$.
Therefore, the fact that $D_{E}$ approaches a finite value as
$k_{B}T/(JS^{2})\to 0$ is a hallmark of incoherent transport of the classical
spin liquid.

We may contrast the saturation of both $\kappa$ and $D_{E}$ in the classical
spin liquid phase with the data in the saturation field $B/(JS)=8$ (Fig. 4a&c,
inset). As the spins are now polarized by the external field, the transport is
due to magnons. As a result, both $\kappa$ and $D_{E}$ show rapid increase as
the temperature decreases.

We observe similar temperature dependence from the spin conductivity $\sigma$
in the classical spin liquid phase (Fig. 4b), namely $\sigma$ saturates to
finite value as $k_{B}T/(JS^{2})\to 0$. Similarly, we deduce the spin
diffusion constant $D_{M}$ by using the Einstein relation $D_{M}=\sigma/\chi$,
where $\chi$ is the magnetic susceptibility per unit volume in the field
direction (Fig. 4d). In particular, $\chi=2/(J\mathrm{a}^{3})$ in the limit of
$k_{B}T/(JS^{2})\to 0$, implying $D_{M}$ approaches a finite limit. By
contrast, at the saturation field $B/(JS)=8$, we find $\sigma$ and $D_{M}$
grows as the temperature decreases, suggesting they diverge in the low
temperature limit.

We thus have demonstrated that the classical spin liquid’s thermal and spin
conductivity, and likewise its energy and spin diffusion constants, approach
finite limits as the temperature tends to zero. We now focus on this low
temperature limit and study the magnetic field dependence. To this end, we fix
$T$ to the lowest simulated temperature $k_{B}T/(JS^{2})=10^{-3}$. Fig. 5a&c
show respectively the thermal conductivity $\kappa$ and the energy
conductivity $D_{E}$ as a function of field $B$. We find both show weak
dependence on $B$ for $B/(JS)\lessapprox 4$ and then a rapid increase as $B$
approaches the saturation field $B/(JS)=8$. Note $\kappa$ and $D_{E}$ for
$B/(JS)>6$ at this temperature are not determined due to high computational
cost. We can also infer this rapid growth of $\kappa$ and $D_{E}$ with
increasing field from Fig. 4a& c, where $\kappa$ and $D_{E}$ at $B/(JS)=6$ are
much larger than that of $B/(JS)=0,2,4$.

Interestingly, the spin conductivity $\sigma$ and the spin diffusion constant
$D_{M}$ show the opposite trend as both decrease as $B$ approaches the
saturation field (Fig. 5b&d). This decrease in the magnitude of $\sigma$ and
$D_{M}$ is also observed in the data shown in Fig. 4b&d.

##  IV Effective disorder model

In this section, we provide a semi-analytic understanding of the molecular
dynamics results by approximately mapping the energy/spin transport problem in
the classical spin liquid phase of Eq. (1) to a model of wave propagation in a
disorder medium. We dub the latter model the _effective_ disorder model to
stress that the disorder is not generated by quenched disorder in the spin
Hamiltonian Eq. (1) but the slow stochastic spin fluctuations idiosyncratic to
the classical spin liquid phase. We find that the effective disorder model
reproduces quantitatively the low temperature transport properties uncovered
previously by the molecular dynamics calculation.

The starting point of the mapping is the observation that the equations of
motion Eq. (4b) and Eq. (4a) display a separation of time scales [18, 19]. The
Hamiltonian Eq. (1) carves out a high-dimensional degenerate ground state
manifold from the full many-body phase space. At low temperature, the system’s
motion is in the proximity of the said ground state manifold. The
$\mathbf{L}_{i}$ modes bring the system out of the ground state manifold, and
constitute the fast degrees of freedom. By contrast, the drifting motion
tangential to the manifold is slow. Pevious molecular dynamics calculation has
confirmed that the spin correlation time diverges as $\hbar S/(k_{B}T)$ as the
temperature $T\to 0$ [18, 19, 23].

As both the magnetization and the energy are carried by the $\mathbf{L}_{i}$
modes, the spin and energy transport are fast processes comparing to the
change in the ground state configuration. We therefore may approximate the
energy flux and the spin flux as:

$\displaystyle I_{E,i\to j}$
$\displaystyle\approx-\frac{J^{2}}{\hbar}\mathbf{S}^{(0)}_{ij}\cdot(\mathbf{L}_{i}\times\mathbf{L}_{j});$
(13a) $\displaystyle I_{M,i\to j}$
$\displaystyle\approx-\frac{J}{\hbar}(\hat{\mathbf{z}}\times\mathbf{S}^{(0)}_{ij})\cdot(\mathbf{L}_{i}-\mathbf{L}_{j}).$
(13b)

The equation of motion of $\mathbf{L}_{i}$ is approximated as:

$\displaystyle\dot{\mathbf{L}}_{i}\approx\frac{1}{\hbar}(\mathbf{L}_{i}\times\frac{\mathbf{B}}{2}+J\sum_{j\in
N_{i}}\mathbf{L}_{j}\times\mathbf{S}^{(0)}_{ij}).$ (14)

Here, we have replaced $\mathbf{S}_{ij}$ by its projection into the ground
state manifold $\mathbf{S}^{(0)}_{ij}$. The error is on the order of
$\sqrt{k_{B}T/(JS^{2})}$. We further take $\mathbf{S}^{(0)}_{ij}$ to be
static, an approximations valid on time scales shorter than $\hbar
S/(k_{B}T)$.

We may view the above set of equations as a model of wave propagation in a
disorder medium. $\mathbf{L}_{i}$ is analogous to the wave field, whereas
$\mathbf{S}^{(0)}_{ij}$, drawn from the degenerate ground states, play the
role of the disorder medium. However, the analogy should not be taken too
literally; the term “wave propagation” sometimes implies the presence of a
Goldstone mode (e.g. spin wave) or hydrodynamic mode (e.g. sound wave in
liquid). Here, the $\mathbf{L}_{i}$ modes are neither.

We stress that the mapping to the effective disorder model crucially relies on
the separation of time scales between the normal modes and the ground state
drifting modes, a condition fulfilled only in the classical spin liquid phase
in the limit $k_{B}T/(JS^{2})\to 0$. This mapping is no longer valid when
$k_{B}T/(JS^{2})$ is not small or when the field is at or above the saturation
field.

We now compute the thermal and spin conductivity. We recast the equation of
motion for $\mathbf{L}_{i}$ in matrix form:

$\displaystyle\dot{L}_{i\alpha}(t)=-\sum_{j\beta}\mathrm{H}_{i\alpha,j\beta}L_{j\beta}(t).$
(15)

Here, the dynamical matrix $\mathrm{H}$ is a $3N\times 3N$ real skew-symmetric
matrix, where $N$ is the number of diamond lattice sites. $i,j$ run over the
diamond lattice sites, whereas $\alpha,\beta$ run over the three _spin_
components. Importantly, the matrix elements of $\mathrm{H}$ depend on the
ground state configuration $\mathbf{S}^{(0)}_{ij}$. The explicit form of
$\mathrm{H}$ is given in Appendix A. The equation of motion for
$\mathbf{L}_{i}$ admits the formal solution:

$\displaystyle
L_{i\alpha}(t)=\sum_{j\beta}\mathrm{G}_{i\alpha,j\beta}(t)L_{j\beta}(0),$ (16)

where the “Green’s function” $\mathrm{G}(t)=\exp(-t\mathrm{H})$ is a $3N\times
3N$ orthogonal matrix. $\mathrm{G}(t)$ depends on the spin configuration
through $\mathrm{H}$.

We express $J^{a}_{E}$ as a quadratic form,

$\displaystyle
J^{a}_{E}=\frac{1}{2}\sum_{i\alpha,j\beta}\mathrm{X}^{a}_{i\alpha,j\beta}L_{i\alpha}L_{j\beta},$
(17a) where $\mathrm{X}^{a}$ is a $3N\times 3N$ real-symmetric matrix. Note
$a$ runs over _spatial_ components, whereas $\alpha,\beta$ run over _spin_
components. By the same token, we write $J^{a}_{M}$ as a linear function,
$\displaystyle J^{a}_{M}=\sum_{i\alpha}\mathrm{Y}^{a}_{i\alpha}L_{i\alpha},$
(17b)

where $\mathrm{Y}^{a}$ is a $3N$ dimensional real vector. Similar to the
dynamical matrix $\mathrm{H}$, the matrix elements of $\mathrm{X}^{a}$ and
$\mathrm{Y}^{a}$ depend on the ground state spin configuration. They are given
explicitly in Appendix A.

The next step is to find the current correlation functions. When performing
the thermal average, we average over the thermal fluctuations in the
$\mathbf{L}_{i}$ modes, and then the ground state configurations. We find:

$\displaystyle\langle J^{a}_{E}(t)J^{b}_{E}(0)\rangle$
$\displaystyle=\frac{(k_{B}T)^{2}}{2J^{2}}\overline{\mathrm{Tr}(\mathrm{G}^{T}(t)\mathrm{X}^{a}\mathrm{G}(0)\mathrm{X}^{b})}.$
(18a) $\displaystyle\langle J^{a}_{M}(t)J^{b}_{M}(0)\rangle$
$\displaystyle=\frac{k_{B}T}{J}\overline{(\mathrm{Y}^{a})^{T}\mathrm{G}(t)\mathrm{Y}^{b}}.$
(18b)

In deriving the above, we have used the fact that the thermal fluctuations of
$\mathbf{L}_{i}(0)$ are Gaussian and employed the Wick theorem. The overline
denotes the average with respect to the ground state spin configurations.

Substituting the current correlation functions into the Kubo formula (Eq.
(11)), we obtain the following formal expression of the thermal and spin
conductivity:

$\displaystyle\kappa^{ab}$
$\displaystyle=\lim_{t,V\to\infty}\frac{k_{B}}{2J^{2}V}\int^{t}_{0}\overline{\mathrm{Tr}(\mathrm{G}^{T}(s)\mathrm{X}^{a}\mathrm{G}(0)\mathrm{X}^{b})}ds.$
(19a) $\displaystyle\sigma^{ab}$
$\displaystyle=\lim_{t,V\to\infty}\frac{1}{JV}\int^{t}_{0}\overline{(\mathrm{Y}^{a})^{T}\mathrm{G}(s)\mathrm{Y}^{b}}ds.$
(19b)

Note the temperature factors that appear in the current correlation functions
cancel with those in the Kubo formula. As $\mathrm{X,Y,G(t)}$ are all
independent of temperature, an immediate consequence of Eq. (19) is that the
thermal and spin conductivity of the effective disorder model is temperature
independent.

The final step is to evaluate Eq. (19) numerically. We use the same lattice
geometry and boundary conditions as the molecular dynamics calculation. We
generate independent realizations of the ground state spin configuration
$\mathbf{S}^{(0)}_{ij}$ by using the minimization method of Walker and
Walstedt [29, 30]. The resulted energy density is less than $10^{-10}JS^{2}$
per spin. With each realization, we construct numerically the matrices
$\mathrm{X,Y,G(t)}$ and find the corresponding contribution to $\kappa$ and
$\sigma$ by using Eq. (19). We average over 40 independent ground state
configurations. As we obtain the Green’s function $\mathrm{G}$ by an exact
diagonalization procedure, the system size is limited to $L=8$.

The calculated energy current correlation function in zero magnetic field and
in $B/(JS)=6$ are shown in dark blue in Fig. 2a and Fig. 2b, respectively. We
find almost perfect agreement between the effective disorder model and the
molecular dynamics calculation at $k_{B}T/(JS^{2})=10^{-3}$ on the same system
size ($L=8$, cyan). Likewise, the thermal conductivity $\kappa$ computed from
the effective disorder model and from the molecular dynamics calculations also
agree very well except for a small difference at late time for $B/(JS)=6$.
Empirically, we find this difference tends to decrease as the system size $L$
increases; the difference between the effective disorder model and the
molecular dynamics is in fact larger for $L=4$ (data not shown).

We find similar good agreement between the effective disorder model and the
molecular dynamics for the spin current correlation function (Fig. 3a&b) and
the spin conductivity (Fig. 3c&d) at system size $L=8$. Remarkably, the
effective disorder model seems to capture all the oscillatory details of the
molecular dynamics data.

We thus conclude that the effective disorder model captures the essential
features of the transport phenomena in the classical spin liquid phase. Within
the effective disorder model, the finite zero temperature thermal and spin
conductivity, and likewise the finite energy and spin diffusion constants, are
easily understood — the quasi-static, disordered spin background results in
the finite mean free path of the $\mathbf{L}_{i}$ modes, and therefore these
transport coefficients do not diverge.

The effective disorder model can also explain the field dependence of the
thermal and spin conductivity. On one hand, as the field increases toward the
saturation field, the spins are more polarized along $\hat{\mathbf{z}}$, which
effectively reduces the disorder. As a result, the mean free path, and hence
the thermal conductivity, increases with the field. On the other hand, for the
spin conductivity, although polarizing the spins reduces the disorder, it also
suppresses the overall magnitude of the spin current fluctuations.
Mathematically, this can be seen from the expression of the spin flux (Eq.
10): $I_{M,i\to j}\propto|\mathbf{z}\times\mathbf{S}_{ij}|$. As the spins are
more aligned with $\hat{\mathbf{z}}$, the magnitude of $I_{M,i\to j}$
decreases. This explains the opposite field dependence of the spin
conductivity.

##  V Discussion

In this work, we find that both the thermal and spin conductivity approach
finite limits as $k_{B}T/(JS^{2})\to 0$ in the classical spin liquid phase of
the pyrochlore Heisenberg antiferromagnet. We may compare this behavior with
other classical magnetic systems. In clean, ordered classical magnets, the
thermal conductivity diverges in the zero temperature limit due to the
divergent magnon mean free path [24, 25]. In low dimensional systems where the
Mermin-Wagner theorem forbids magnetic ordering, the thermal diffusion
constant also diverges as the temperature tends to zero [31, 32, 24, 33]. This
occurs because the system is proximate to an ordered state, and consequently
the transport is due to paramagnons, whose mean free path diverges in the zero
temperature limit. We also note an interesting report on the spin diffusion
constant of the classical kagome Heisenberg antiferromagnet [34]. As
temperature decreases, it first shows saturation-like behavior in the
intermediate classical spin liquid regime and then grows rapidly in the spin
nematic regime.

We find the energy and spin current correlation functions decay rapidly on the
time scale of order $\hbar/(JS)$, which is much faster than the spin
correlation time $\hbar S/(k_{B}T)$. Viewing from the hydrodynamic
perspective, this suggests the energy and spin currents do not mix with any
long-lived quantities in this system [16, 35]. The fact that the energy and
spin diffusion processes are not the slowest dynamical processes of this
system also sets it apart from the ordered magnets.

So far our discussion is limited to the classical limit $S\to\infty$. At
finite but large $S$, we expect that the thermal and the spin conductivity
shows near saturation in the parametrically large temperature window $JS\ll
k_{B}T\ll JS^{2}$, and, upon further reducing the temperature, start deviating
from the classical behavior as the quantum fluctuations set in. It has been
suggested that the quantum order by disorder effect selects ground states with
complex magnetic orders [36, 37]. We speculate that the transport would then
be due to magnons in the temperature regime $k_{B}T\lessapprox JS$.

The effective disorder model reveals an interesting connection between the
frustrated magnets and the disorder physics. Viewing from the latter angle,
one may ask if the normal modes are extended or localized. Our preliminary
analysis of the inverse partition ratio of the eigenmodes of the dynamical
matrix $\mathrm{H}$ suggest that almost all modes are extensive except the
modes at the band edge for all magnetic field $B/JS<8$ [38], consistent with
the finite thermal and spin conductivity. It may be interesting to further
explore this aspect in future.

##  Appendix A Mathematical details of the effective disorder model

In this appendix, we give the explicit form of the various matrices defined in
Sec. IV.

The dynamical matrix $\mathrm{H}$ is a $3N\times 3N$ real skew-symmetric
matrix, where $N$ is the number of diamond lattice sites. $i,j$ run over the
diamond lattice sites, whereas $\alpha,\beta$ run over the three _spin_
components. It is given by:

$\displaystyle\mathrm{H}_{i\alpha,j\beta}=\frac{1}{\hbar}\left\\{\begin{array}[]{cc}(\mathbf{B}^{\times})_{\alpha\beta}/2&(i=j)\\\
J(\mathbf{S}^{(0),\times}_{ij})_{\alpha\beta}&(i,j\in n.n.)\\\
0&(\mathrm{otherwise})\end{array}\right..$ (23)

Here, we have used a short hand notation for $3\times 3$ matrix:

$\displaystyle\mathbf{A}^{\times}\equiv\begin{bmatrix}0&-A_{z}&A_{y}\\\
A_{z}&0&-A_{x}\\\ -A_{y}&A_{x}&0\end{bmatrix},$ (24)

where $\\{A_{x},A_{y},A_{z}\\}$ form the three components of the _spin_ space
vector $\mathbf{A}$.

The dynamical matrix $\mathrm{H}$ is real, skew-symmetric, and even-
dimensional. It can be brought to the canonical form by an orthogonal
transformation:

$\displaystyle\mathrm{H}=\mathrm{O}\left(\sum_{i,\oplus}\begin{bmatrix}0&\lambda_{i}\\\
-\lambda_{i}&0\end{bmatrix}\right)\mathrm{O}^{T},$ (25)

where $\lambda_{i}>0$. $\mathrm{O}$ is a real orthogonal matrix. The Green’s
function is then given by:

$\displaystyle\mathrm{G}(t)=\mathrm{O}\left(\sum_{i,\oplus}\begin{bmatrix}\cos(\lambda_{i}t)&-\sin(\lambda_{i}t)\\\
\sin(\lambda_{i}t)&\cos(\lambda_{i}t)\end{bmatrix}\right)\mathrm{O}^{T}.$ (26)

The matrix $\mathrm{X}^{a}$ is a $3N\times 3N$ matrix:

$\displaystyle\mathrm{X}^{a}_{i\alpha,j\beta}=\frac{J^{2}}{\hbar}\left\\{\begin{array}[]{cc}r^{a}_{i\to
j}(\mathbf{S}^{(0),\times}_{ij})_{\alpha\beta}&(i,j\in n.n.)\\\
0&(\mathrm{otherwise})\end{array}\right..$ (29)

$\mathrm{Y}^{a}$ is a $3N$ dimensional vector:

$\displaystyle
Y^{a}_{i\alpha}=\frac{J}{\hbar}\left\\{\begin{array}[]{cc}\sum_{j\in
N_{i}}r^{a}_{i\to j}S^{(0)}_{ij,y}&(\alpha=x)\\\ -\sum_{j\in N_{i}}r^{a}_{i\to
j}S^{(0)}_{ij,x}&(\alpha=y)\\\ 0&(\alpha=z)\end{array}\right..$ (33)

Note $a$ runs over the _spatial_ components, whereas $\alpha,\beta$ run over
_spin_ components. $r^{a}_{i\to j}$ is the real space vector that points from
diamond site $i$ to site $j$.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant No. 11974396) and the Strategic Priority Research Program of the
Chinese Academy of Sciences (Grant No. XDB33020300).

## References

  * Drude [1900a] P. Drude, Zur Elektronentheorie der Metalle, Annalen der Physik 306, 566 (1900a).
  * Drude [1900b] P. Drude, Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomagnetische Effecte, Annalen der Physik 308, 369 (1900b).
  * Lifshitz and Pitaevski [1981] E. M. Lifshitz and L. P. Pitaevski,  _Physical Kinetics_ (Butterworth-Heinemann, 1981).
  * Ziman [1960] J. M. Ziman,  _Electrons and Phonons: The Theory of Transport Phenomena in Solids_ (Oxford, 1960).
  * Abrikosov [1988] A. A. Abrikosov,  _Fundamentals of the Theory of Metals_ (North-Holland, 1988).
  * Mukerjee  _et al._ [2006] S. Mukerjee, V. Oganesyan, and D. Huse, Statistical theory of transport by strongly interacting lattice fermions, Phys. Rev. B 73, 035113 (2006).
  * Shekhter and Varma [2009] A. Shekhter and C. M. Varma, Long-wavelength correlations and transport in a marginal fermi liquid, Phys. Rev. B 79, 045117 (2009).
  * Lindner and Auerbach [2010] N. H. Lindner and A. Auerbach, Conductivity of hard core bosons: A paradigm of a bad metal, Phys. Rev. B 81, 054512 (2010).
  * Hartnoll  _et al._ [2011] S. A. Hartnoll, D. M. Hofman, M. A. Metlitski, and S. Sachdev, Quantum critical response at the onset of spin-density-wave order in two-dimensional metals, Phys. Rev. B 84, 125115 (2011).
  * Wölfle and Abrahams [2011] P. Wölfle and E. Abrahams, Quasiparticles beyond the Fermi liquid and heavy fermion criticality, Phys. Rev. B 84, 041101 (2011).
  * Xu  _et al._ [2013] W. Xu, K. Haule, and G. Kotliar, Hidden Fermi Liquid, Scattering Rate Saturation, and Nernst Effect: A Dynamical Mean-Field Theory Perspective, Phys. Rev. Lett. 111, 036401 (2013).
  * Syzranov and Schmalian [2012] S. V. Syzranov and J. Schmalian, Conductivity Close to Antiferromagnetic Criticality, Phys. Rev. Lett. 109, 156403 (2012).
  * Mahajan  _et al._ [2013] R. Mahajan, M. Barkeshli, and S. A. Hartnoll, Non-Fermi liquids and the Wiedemann-Franz law, Phys. Rev. B 88, 125107 (2013).
  * Hartnoll  _et al._ [2014] S. A. Hartnoll, R. Mahajan, M. Punk, and S. Sachdev, Transport near the Ising-nematic quantum critical point of metals in two dimensions, Phys. Rev. B 89, 155130 (2014).
  * Limtragool and Phillips [2015] K. Limtragool and P. Phillips, Power-law optical conductivity from unparticles: Application to the cuprates, Phys. Rev. B 92, 155128 (2015).
  * Hartnoll [2015] S. A. Hartnoll, Theory of universal incoherent metallic transport, Nature Physics 11, 54 (2015).
  * Werman  _et al._ [2017] Y. Werman, S. A. Kivelson, and E. Berg, Non-quasiparticle transport and resistivity saturation: a view from the large-N limit, npj Quantum Materials 2, 7 (2017).
  * Moessner and Chalker [1998a] R. Moessner and J. T. Chalker, Properties of a Classical Spin Liquid: The Heisenberg Pyrochlore Antiferromagnet, Phys. Rev. Lett. 80, 2929 (1998a).
  * Moessner and Chalker [1998b] R. Moessner and J. T. Chalker, Low-temperature properties of classical geometrically frustrated antiferromagnets, Phys. Rev. B 58, 12049 (1998b).
  * Canals and Garanin [2001] B. Canals and D. A. Garanin, Spin-liquid phase in the pyrochlore anti-ferromagnet, Canadian Journal of Physics 79, 1323 (2001).
  * Isakov  _et al._ [2004] S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Dipolar Spin Correlations in Classical Pyrochlore Magnets, Phys. Rev. Lett. 93, 167204 (2004).
  * Henley [2005] C. L. Henley, Power-law spin correlations in pyrochlore antiferromagnets, Phys. Rev. B 71, 014424 (2005).
  * Conlon and Chalker [2009] P. H. Conlon and J. T. Chalker, Spin Dynamics in Pyrochlore Heisenberg Antiferromagnets, Phys. Rev. Lett. 102, 237206 (2009).
  * Aoyama and Kawamura [2019] K. Aoyama and H. Kawamura, Effects of magnetic anisotropy on spin and thermal transport in classical antiferromagnets on the square lattice, Phys. Rev. B 100, 144416 (2019).
  * Harris  _et al._ [1971] A. B. Harris, D. Kumar, B. I. Halperin, and P. C. Hohenberg, Dynamics of an antiferromagnet at low temperatures: Spin-wave damping and hydrodynamics, Phys. Rev. B 3, 961 (1971).
  * Bilitewski  _et al._ [2018] T. Bilitewski, S. Bhattacharjee, and R. Moessner, Temperature Dependence of the Butterfly Effect in a Classical Many-Body System, Phys. Rev. Lett. 121, 250602 (2018).
  * Rehn  _et al._ [2017] J. Rehn, A. Sen, and R. Moessner, Fractionalized ${\mathbb{Z}}_{2}$ Classical Heisenberg Spin Liquids, Phys. Rev. Lett. 118, 047201 (2017).
  * Kubo  _et al._ [1991] R. Kubo, M. Toda, and N. Hashitsume,  _Statistical Physics II: Nonequilibrium Statistical Mechanics_ (Springer, 1991).
  * Walker and Walstedt [1977] L. R. Walker and R. E. Walstedt, Computer Model of Metallic Spin-Glasses, Phys. Rev. Lett. 38, 514 (1977).
  * Walker and Walstedt [1980] L. R. Walker and R. E. Walstedt, Computer model of metallic spin-glasses, Phys. Rev. B 22, 3816 (1980).
  * Ty and Halperin [1990] S. Ty and B. I. Halperin, Damping of spin waves in a two-dimensional heisenberg antiferromagnet at low temperatures, Phys. Rev. B 42, 2096 (1990).
  * Savin  _et al._ [2005] A. V. Savin, G. P. Tsironis, and X. Zotos, Thermal conductivity of a classical one-dimensional heisenberg spin model, Phys. Rev. B 72, 140402 (2005).
  * Aoyama and Kawamura [2020] K. Aoyama and H. Kawamura, Spin Current as a Probe of the ${\mathbb{Z}}_{2}$-Vortex Topological Transition in the Classical Heisenberg Antiferromagnet on the Triangular Lattice, Phys. Rev. Lett. 124, 047202 (2020).
  * Taillefumier  _et al._ [2014] M. Taillefumier, J. Robert, C. L. Henley, R. Moessner, and B. Canals, Semiclassical spin dynamics of the antiferromagnetic heisenberg model on the kagome lattice, Phys. Rev. B 90, 064419 (2014).
  * Forster [1990] D. Forster,  _Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions_, 1st ed. (CRC Press, 1990).
  * Henley [2006] C. L. Henley, Order by Disorder and Gaugelike Degeneracy in a Quantum Pyrochlore Antiferromagnet, Phys. Rev. Lett. 96, 047201 (2006).
  * Hizi and Henley [2006] U. Hizi and C. L. Henley, Effective hamiltonian for the pyrochlore antiferromagnet: Semiclassical derivation and degeneracy, Phys. Rev. B 73, 054403 (2006).
  * Zhang  _et al._ [2019] S. Zhang, H. J. Changlani, K. W. Plumb, O. Tchernyshyov, and R. Moessner, Dynamical Structure Factor of the Three-Dimensional Quantum Spin Liquid Candidate ${\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}$, Phys. Rev. Lett. 122, 167203 (2019).