# 4-connected 1-planar chordal graphs are Hamiltonian-connected

Licheng Zhang School of Mathematics, Hunan University, China Email:
<EMAIL_ADDRESS>School of Mathematics, Hunan University, China Yuanqiu
Huang College of Mathematics and Statistics, Hunan Normal University, China
Corresponding author. Email<EMAIL_ADDRESS>College of Mathematics and
Statistics, Hunan Normal University, China Shengxiang Lv School of Mathematics
and Statistics,Hunan University of Finance and Economics, China
<EMAIL_ADDRESS>School of Mathematics and Statistics,Hunan University of
Finance and Economics, China Fengming Dong<EMAIL_ADDRESS>National Institute of Education, Nanyang Technological University, Singapore

###### Abstract

Tutte proved that 4-connected planar graphs are Hamiltonian. It is unknown if
there is an analogous result on $1$-planar graphs. In this paper, we
characterize 4-connected 1-planar chordal graphs, and show that all such
graphs are Hamiltonian-connected. A crucial tool used in our proof is a
characteristic of 1-planar 4-trees.

Keywords: 1-planar graph, chordal graph, Hamiltonian-connected

MSC: 05C10, 05C40, 05C45, 05C62

##  1 Introduction

Only finite simple connected graphs are considered in this paper. For a graph
$G$, let $V(G)$ and $E(G)$ denote its vertex set and edge set. For any subset
$V_{0}$ of $V(G)$, the subgraph of $G$ induced by $V_{0}$, denoted by
$G[V_{0}]$, is the graph with vertex set $V_{0}$ and edge set $\\{uv\in
E(G):u,v\in V(G)\\}$. Let $G-V_{0}$ denote the subgraph of $G$ induced by
$V(G)\setminus V_{0}$ when $V_{0}\neq V(G)$. A subset $S$ of $V(G)$ is called
a separator if $G-S$ is disconnected, and a separator $S$ is called a
$k$-separator if $|S|=k$. For a non-complete graph $G$, its connectivity,
denoted by $\kappa(G)$, is defined to be the minimum value of $|S|$ over all
separators $S$ of $G$, and $\kappa(G)=|V(G)|-1$ if $G$ is connected.

A Hamiltonian path (resp. cycle)  in $G$ is a path (resp. cycle) in $G$ that
visits every vertex of $G$ exactly once. A graph $G$ is called Hamiltonian if
$G$ contains a Hamiltonian cycle. A graph $G$ is Hamiltonian-connected if
every two vertices in $G$ are connected by a Hamiltonian path. Clearly, all
Hamiltonian-connected graphs of orders at least three are Hamiltonian.

A drawing of a graph $G=(V,E)$ is a mapping $D(G)$ that assigns to each vertex
in $V$ a distinct point in the plane and to each edge $uv$ in $E$ a continuous
arc connecting $D(u)$ and $D(v)$. A drawing is said to be good if no edge
crosses itself, no two edges cross more than once, and no two edges incident
with a vertex cross each other. All drawings considered in this paper are good
ones. A graph is called planar if has a drawing so that no edges cross each
other. The problem of Hamiltonicity of planar graphs has received considerable
attention. Whitney [33] showed that every 4-connected plane triangulation is
Hamiltonian, and Tutte [31] extended this conclusion to every 4-connected
planar graph. Tutte’s result has been strengthened in various ways. For
example, Thomassen [30] proved that every 4-connected planar graph is
Hamiltonian-connected, and Thomas and Yu [29] proved that 4-connected
projective-planar graphs are Hamiltonian. For more references, the reader may
refer to [12, 21, 24, 25].

A graph is called $1$-planar if has a drawing so that any edge is crossed at
most once, and such a drawing is also called a $1$-plane graph. 1-planar
graphs are introduced by Ringel (1965) [26] in the connection with the problem
of the simultaneous colouring of the vertices and faces of plane graphs. Since
then, many properties of 1-planar graphs have been widely investigated (see
[5, 9, 13, 16, 22, 28] for example). It is known that 1-planar graphs have
minimum degree at most 7, and thus they are at most $7$-connected [14]. People
try to establish a foundational result on Hamiltonicity of highly connected
1-planar graphs, akin to Tutte’s work on 4-connected planar graphs. A 1-planar
graph $G$ with $|V(G)|\geq 3$ vertices has at most $4|V(G)|-8$ edges [6]. A
1-planar graph is called optimal if $|E(G)|=4|V(G)|-8$. Fujisawa et al. [15,
28] showed that every optimal 1-planar graphs has connectivity 4 or 6. Hudák
et al. [17] proved that every optimal 1-planar is Hamiltonian. Fabrici et al.
[13] proved that a 3-connected locally maximal 1-planar graph $G$ is
Hamiltonian if it has at most three 3-vertex-cuts, implying that all
4-connected maximal 1-planar graphs are Hamiltonian. However, there are still
many non-Hamiltonian 1-planar graphs with connectivity 4 or 5 [3, 13].
Actually, it is even unknown if 1-planar graphs with connectivity 6 or 7 are
Hamiltonian.

######  Problem 1 ([3, 13]).

Is every $6$-connected (or $7$-connected) $1$-planar graph is Hamiltonian?

A hole is an induced cycle of length at least four. A graph is chordal if it
does not contain holes. In this article, we will establish the following
conclusion on the Hamiltonicity of $1$-planar graphs.

######  Theorem 1.

Let $G$ be a $1$-planar graph. If $G$ is $4$-connected and choral, then $G$ is
Hamiltonian-connected.

Note that the conclusion in Theorem 1 does not hold if some condition is
replaced by a weaker one. Some details are given in following remark.

######  Remark 1.

  1. (i).

There are non-Hamiltonian $4$-connected $1$-planar graphs (see [17]);

  2. (ii).

It is easy to construct $4$-connected chordal graphs that are not Hamiltonian.
For example, the graph obtained from a graph $H$ isomorphic to $K_{4}$ by
adding $r$ vertices $v_{1},\dots,v_{r}$, where $r\geq 5$, and adding edges
joining each $v_{i}$ to every vertex in $H$ is not Hamiltonian.

  3. (iii).

There exist $3$-connected planar chordal graphs that are not Hamiltonian [23].
We can also construct $3$-connected $1$-planar chordal graphs that are neither
planar nor Hamiltonian. An example is shown in Fig. 1. It is easy to verify
that this graph $G_{0}$ is chordal, $1$-planar and $3$-connected, but not
$4$-connected. This graph is not Hamiltonian due to the fact that the subgraph
of $G_{0}$ obtained by removing vertices $0,1,3,5$ and $7$ has six components.
Taking any one of such $1$-planar graphs with a $3$-region $f$ whose boundary
edges are not crossed, we can obtained a new one by identifying the three
vertices of $f$ with vertices $0,11$ and $12$ in $G_{0}$. Hence there are
infinity many $3$-connected $1$-planar chordal graphs that are neither planar
nor Hamiltonian.

$0$$2$13687945101112 Figure 1: A chordal and 1-planar graph $G_{0}$ that has
connectivity $3$ and is not Hamiltonian.

The remaining sections of this paper are organized as follows. In Section 2,
we explain some terminology, notations and some preliminary lemmas, and in
Section 3, we give some fundamental results on chordal graphs. In Section 4,
we introduce a 4-join operation for generating 1-planar 4-trees. In Section 5,
we characterize $4$-connected and $1$-planar chordal graphs. In Section 6, we
prove Theorem 1 by the result obtained in Section 5. Finally we leave some
interesting open questions.

##  2 Preliminaries

In this section, we introduce some notations and definitions.

Let $G$ be a graph with a drawing $D$. For a subgraph $H$ of graph $G$, the
subdrawing of $D$ induced by $H$ is called a restricted drawing of $D$,
denoted by $D|H$. Two drawings of a graph are isomorphic if there is a
homeomorphism of the sphere that maps one drawing to the other. If $D$ is a
1-planar drawing, an edge of $D$ is crossed if it crosses another edge, and is
uncrossed otherwise. A face in a 1-planar drawing $D$ is given by a cyclic
list of edges and edge segments, where the latter occurs in the case of a
crossing. The planar skeleton $\mathcal{S}(D)$ of $D$ is the subgraph of $G$
by removing all crossed edges of $D$. For a face $f$ of $D$, denote
$\partial(f)$ by the boundary of $f$. We call $f$ is crossed if $\partial(f)$
has a crossing, and is uncrossed otherwise. All ertices and crossings on
$\partial(f)$ are called  corners. A face $f$ is triangular if $\partial(f)$
has exactly three corners. A triangulated 1-plane graph is one in which all
faces are triangular.

For other terminology and notations not defined here, we refer to [8], and the
following are some auxiliary lemmas in this paper.

Some fundamental results on $1$-planar graphs are provided below.

######  Lemma 1 ([28]).

For any $1$-planar graph $G$ of order $n\geq 3$, $|E(G)|\leq 4n-8$, and
moreover, $|E(G)|\leq 4n-9$ if $n=7$ or $9$.

######  Lemma 2 ([28]).

The complete graphs $K_{5}$ and $K_{6}$ have exactly one (up to isomorphism)
1-planar drawings as shown in Fig. 2, respectively.

$0$$2$13687945101112

$0$$2$13687945101112

(a) $K_{5}$ (b) $K_{6}$

Figure 2: The unique 1-planar drawings of $K_{5}$ and $K_{6}$.

######  Lemma 3.

Let $D$ be a 1-planar drawing of $G$, and let $H$ be a subgraph of $G$. If
some vertex of $G-H$ lies inside a crossed triangular face $f$ of $D|H$, then
$\kappa(G)\leq 3$.

###### Proof.

Let $f=[abca]$. Assume that $a$ is the unique crossing in $\partial(f)$. Let
$v$ be a vertex lying inside $f$. If $bc$ is uncrossed, then $\\{b,c\\}$ is a
2-separator of $G$. If $bc$ is crossed by $xy$, where $x\notin\partial(f)$,
then $\\{b,c,x\\}$ is a 3-separator of $G$. In either case, $\kappa(G)\leq 3$.
∎

If $e$ is an uncrossed edge on $\partial(f)$ for some face $f$ in a $1$-planar
drawing $D$, let $\widehat{f}_{e}$ denote the other face in $D$ such that $e$
is also on $\partial(\widehat{f}_{e})$.

######  Lemma 4.

Let $D$ be a $1$-planar drawing of a $1$-planar graph $G$. Assume that vertex
$v\in V(G)$ lies within face $f$ of $D^{\prime}:=D|(G-v)$. Then, the following
conclusions hold:

  1. (i).

the sum of the number of uncrossed corners (i.e., vertices in $G-v$) on
$\partial(f)$ and the number of uncrossed edges $e$ on $\partial(f)$ for which
$\partial(\widehat{f}_{e})$ has at least three uncrossed corners is at least
$d_{G}(v)$; and

  2. (ii).

if $D^{\prime}$ is triangulated and $d_{D}(v)=4$, then $f$ is uncrossed and
$\widehat{f}_{e}$ is also uncrossed for at least one edge $e$ on
$\partial(f)$.

###### Proof.

The result follows directly by the definition of $1$-planar drawings. ∎

A clique of a graph $G$ is a subset $S$ of $V(G)$ such that $G[S]$ is
complete. A $k$-clique is a clique with exactly $k$ vertices.

######  Lemma 5.

For a $1$-planar graph $G$ of order $n\geq 7$, if $G$ contains a $6$-clique,
then $\kappa(G)\leq 3$.

###### Proof.

Let $D$ be a 1-planar drawing of $G$ and let $S$ be a $6$-clique of $G$. By
Lemma 2, $D|{G[S]}$ is unique, which is shown in Fig 2 (b).

Since $n\geq 7$, there is a vertex $u\in V(G)\setminus S$. Then $u$ lies in a
triangular face $f$ in $D|{G[S]}$. If $f$ is crossed, the conclusion follows
from Lemma 3. If $f$ is uncrossed, the three vertices on $\partial(f)$ form a
3-separator of $G$. Hence $\kappa(G)\leq 3$. ∎

##  3 Some fundamental results on chordal graphs

In this section, we introduce some basic results on chordal graphs. By the
definition of chordal graphs, the following result holds.

######  Lemma 6 (p51,[32]).

Let $G$ be a chordal graph. Then every induced subgraph of $G$ is chordal.

A vertex $u$ in $G$ is said to be simplicial if either $d(u)=0$ or $N_{G}(u)$
is a clique. An ordering $(u_{1}u_{2}\cdots u_{n})$ of the vertices in $G$ is
a called perfect elimination order, if $u_{i}$ is a simplicial vertex of
$G-\\{u_{j}:1\leq j<i\\}$ for all $i=1,2,\cdots,n-1$. Rose [27] proved that a
graph is chordal if and only if it has a perfect elimination ordering. Fix an
integer $k\geq 1$, the class of  $k$-trees is defined recursively as follows.
The complete graph $K_{k}$ is the smallest $k$-tree, and a graph $G$ of order
$n$, where $n\geq k+1$, is a $k$-tree if $G$ has a simplicial vertex $u$ of
degree $k$ and $G-\\{u\\}$ is a $k$-tree. Clearly, from the construction of
$k$-trees, they naturally contain perfect elimination orderings, and so
$k$-trees form a subclass of chordal graphs.

######  Lemma 7 ([11, 27]).

Let $G$ be a graph. The following statements are equivalent:

  1. (i).

$G$ is chordal;

  2. (ii).

every minimal separator of $G$ is a clique; and

  3. (iii).

$G$ has a perfect elimination ordering.

######  Lemma 8.

Let $G$ be a 4-tree with $|V(G)|=6$. Then $G\cong K_{6}\setminus e$. Moreover,
$G$ has exactly three non-isomorphic $1$-planar drawings, as shown in Fig. 3.

###### Proof.

By the definition of $4$-trees, any 4-tree with five vertices is $K_{5}$, and
then $G\cong K_{6}\setminus e$. Due to Korzhik [20], $G$ has exactly three
1-planar drawings $A_{1},A_{2}$ and $A_{3}$ shown in Fig. 3. ∎

$0$$2$13687945101112$A_{1}$$A_{2}$$A_{3}$ Figure 3: Three non-isomorphic
1-planar drawings of $K_{6}\setminus e$.

######  Lemma 9 ([18]).

Let $G$ be a $k$-connected chordal graph. If $G$ has no $(k+2)$-clique, then
$G$ is a $k$-tree.

The following lemma must have appeared somewhere, although we cannot find it.
We provide a simple proof.

######  Lemma 10.

Let $G$ be a chordal graph with $n$ vertices. If $G$ is $k$-connected, then
$|E(G)|\geq kn-\frac{1}{2}k(k+1)$.

###### Proof.

We can prove this result by induction on $n$. Since $G$ is $k$-connected, we
have $n\geq k+1$. When $n=k+1$, $G$ is the complete graph $K_{k+1}$, and the
result holds. Now assume that $n\geq k+2$. By Lemma 7, $G$ contains a
simplicial vertex $u$. Since $G$ is $k$-connected, we have $d(u)\geq k$.
Obviously, $G-\\{u\\}$ is $k$-connected and chordal. By induction, the result
holds for $G-\\{u\\}$, and thus

$|E(G)|=|E(G-\\{u\\})|+d(u)\geq
k(n-1)-\frac{1}{2}k(k+1)+k=kn-\frac{1}{2}k(k+1).$

Hence the result holds. ∎

##  4 4-join operation

In this section, we introduce a graph operation, the 4-join operation, which
will be applied to generate 1-planar 4-trees.

Let $D$ be a $1$-plane graph. If $f_{1}:=[abv_{1}a]$ and $f_{2}:=[abv_{2}a]$
are uncrossed triangular faces in $D$ such that $\partial(f_{1})$ and
$\partial(f_{2})$ share exactly one edge (i.e., $ab$) and
$D[\\{a,b,v_{1},v_{2}\\}]\cong K_{4}$, then $f_{1}$ and $f_{2}$ are called
twin faces in $D$, as shown in Fig. 4 (a).

$0$$2$13687945101112$a$$v_{1}$$b$$v_{2}$$f_{1}$$f_{2}$$a$$v_{1}$$b$$v_{2}$$x$$g_{1}$$g_{2}$(a)(b)$a$$v_{1}$$b$$v_{2}$$x$(c)$g_{1}$$g_{2}$
Figure 4: 4-join operation on twin faces $f_{1}$ and $f_{2}$

######  Definition 1 (4-join operation).

Let $f_{1}:=[abv_{1}a]$ and $f_{2}:=[abv_{2}a]$ be twin faces in a plane graph
$D$, as shown in Fig. 4 (a). A $4$-join operation at the order pair
$(f_{1},f_{2})$ is defined as follows:

  1. (i).

put a new vertex $x$ within face $f_{1}$, and

  2. (ii).

draw edges $xa$,$xb$, $xv_{1}$ inside $f_{1}$, and draw edge $xv_{2}$ that
crosses $ab$, as shown in Fig. 4 (b).

The $1$-plane graph drawing obtained by the 4-join operation at
$(f_{1},f_{2})$ is denoted by $D\oplus(f_{1},f_{2})$, as shown at Fig. 4
(b).111Note that $D\oplus(f_{2},f_{1})$, shown at Fig. 4 (c), is different
from $D\oplus(f_{1},f_{2})$, as $D\oplus(f_{2},f_{1})$ is the one with the
vertex $x$ within $f_{2}$.

######  Lemma 11.

Let $D$ be a $1$-planar drawing of a $1$-planar graph $G$, and let $v\in V(G)$
is a simplicial vertex in $G$ of degree $4$. Assume that $D^{\prime}:=D|(G-v)$
is triangulated and $v$ is within face $f$ of $D^{\prime}$. If $e$ is the only
edge on $\partial(f)$ such that $f^{\prime}:=\widehat{f}_{e}$ is uncrossed,
then $D=D^{\prime}\oplus(f,f^{\prime})$.

###### Proof.

Let $f=[bcab]$ and $f^{\prime}=[bcdb]$, where edge $bc$ is on both
$\partial(f)$ and $\partial(f^{\prime})$. Since $v$ is within $f$ of
$D^{\prime}$, by Lemma 4 (ii), $N_{D}(v)=\\{a,b,c,d\\}$ is a clique of $D$.
Thus, $D=D^{\prime}\oplus(f,f^{\prime})$. ∎

##  5 Characterization of $4$-connected $1$-planar chordal graphs

Clearly, $K_{5}$ is the $4$-connected chordal graph with the smallest order,
and $K_{6}$ and $K_{6}-e$ are the only $4$-connected chordal graph of order
$6$. Note that $\kappa(K_{6})=5>4$. We first prove that $K_{6}$ is the only
$1$-planar chordal graph with connectivity $5$, and every $1$-planar chordal
graph $G$ with $\kappa(G)=4$ is a $4$-tree.

######  Proposition 1.

Let $G$ be a $1$-planar chordal graph that is not $K_{6}$. Then $\kappa(G)\leq
4$. Moreover, if $\kappa(G)=4$, $G$ is a $4$-tree.

###### Proof.

Suppose $\kappa(G)\geq 5$. Clearly, $|V(G)|\geq 6$. Since $G\ncong K_{6}$, we
have $|V(G)|\geq 7$ . By Lemmas 1 and 10, we have

$5|V(G)|-15\leq|E(G)|\leq 4|V(G)|-8,$

implying that $|V(G)|\leq 7$. Thus, $|V(G)|=7$. However, when $|V(G)|=7$,
applying Lemmas 1 and 10 yields that

$20=5\times 7-15\leq|E(G)|\leq 4\times 7-9=19,$

a contradiction. Hence $\kappa(G)\leq 4$.

Now assume that $\kappa(G)=4$. Then $|V(G)|\geq 5$. When $|V(G)|=5$, $G$ is
$K_{5}$ which is a $4$-tree. If $|V(G)|=6$, by Lemma 10, $|E(G)|\geq 4\times
6-\frac{4\times 5}{2}=14$. Since $|E(K_{6})|=15$, we have $|E(G)|=14$. Hence,
$G\cong K_{6}-e$. By Lemma 8, $K_{6}-e$ is a 1-planar 4-tree. For $|V(G)|\geq
7$, since $\kappa(G)=4$, by Lemma 5, $G$ has no 6-clique. Furthermore, by
Lemma 9, $G$ is a $4$-tree. ∎

By Proposition 1, $K_{6}$ is the only 5-connected 1-planar chordal graphs, and
all 1-planar chordal graphs with connectivity four are 4-trees. However, there
exist 4-trees which are not 1-planar. For example, $K_{4}+\overline{K_{3}}$ is
a 4-tree, where $G+H$ is the graph obtained from vertex disjoint copies of $G$
and $H$ by adding edges joining every vertex in $G$ to every vertex in $H$.
But $K_{4}+\overline{K_{3}}$ is not 1-planar (see [20]).

An Apollonian network is a planar graph obtained by starting with a triangle,
and repeatedly picking a triangular face $f$ and subdividing $f$ into three
triangles by inserting a new vertex in it. A planar graph is a 3-tree if and
only if it is a Apollonian network [4]. In the following, we are going to
establish an analogous result for $4$-connected and 1-planar chordal graphs by
the 4-join operation introduced in the previous section.

By Proposition 1, every $1$-planar chordal graph $G$ with $\kappa(G)=4$ is a
$4$-tree. Such graphs of orders at most $6$ are $K_{5}$ and $K_{6}-e$. In the
following, we first characterize $1$-planar $4$-trees of order $7$, and then
all $1$-planar $4$-trees of order more than $7$.

######  Proposition 2.

The $1$-planar drawings of all $1$-planar $4$-trees of order $7$ are shown in
Fig. 5.

$0$$2$13687945101112$a$$v_{1}$$b$$v_{2}$$f_{1}$$f_{2}$$a$$v_{1}$$b$$v_{2}$$x$$g_{1}$$g_{2}$(a)(b)$a$$v_{1}$$b$$v_{2}$$x$(c)$g_{1}$$g_{2}$$B_{2}$$B_{3}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{6}$$B_{1}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$u_{5}$$u_{6}$$u_{6}$$u_{7}$$u_{7}$$u_{7}$
Figure 5: All $1$-planar drawings of $1$-planar $4$-trees of order $7$

###### Proof.

Let $D$ be a $1$-planar drawing of a $1$-planar $4$-tree $T$ of order $7$, and
let $v$ be a simplicial vertex of $T$ with $d_{T}(v)=4$. Then, $N_{T}(v)$ is a
clique in $T$ of size $4$ and $D|(T-v)$ must be a $1$-planar drawing of $T-v$.
Since $\kappa(T)=4$, by Lemma 3, $v$ cannot be within a crossed triangular
face of $D|(T-v)$.

Claim A: $D|(T-v)$ is isomorphic to $A_{1}$ shown in Fig. 3 or 6

Note that $T-v\cong K_{6}-e$ is a $1$-planar $4$-tree of order $6$. By Lemma
8, $D|(T-v)$ must be one of the three non-isomorphic $1$-planar drawings
$A_{1},A_{2}$ or $A_{3}$ shown in Fig. 3 or 6.

$0$$2$13687945101112$a$$v_{1}$$b$$v_{2}$$f_{1}$$f_{2}$$a$$v_{1}$$b$$v_{2}$$x$$g_{1}$$g_{2}$(a)(b)$a$$v_{1}$$b$$v_{2}$$x$(c)$g_{1}$$g_{2}$$B_{2}$$B_{3}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{6}$$B_{1}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$u_{5}$$u_{6}$$u_{6}$$u_{7}$$u_{7}$$u_{7}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{5}$$u_{5}$$u_{4}$$u_{4}$$u_{4}$$u_{6}$$u_{6}$$u_{6}$$u_{1}$$u_{2}$$u_{3}$$A_{2}$$A_{3}$$c_{1}$$c_{2}$$c_{1}$$u_{1}$$u_{2}$$u_{3}$$A_{1}$$f_{1}$$f_{2}$$g_{1}$$g_{2}$
Figure 6: $A_{1}$, $A_{2}$ and $A_{3}$

We shall show that $D|(T-v)$ must be the $1$-planar drawing $A_{1}$. If
$D|(T-v)$ is $A_{2}$, as shown in Fig. 6, then $v$ must be within a uncrossed
triangular face of $A_{2}$ or a face of $A_{2}$ which is not triangular,
implying that $v$ must be within the crossed face
$[u_{1}c_{1}u_{6}c_{2}u_{1}]$, or within one of the uncrossed faces
$[u_{1}u_{2}u_{3}u_{1}],[u_{4}u_{5}u_{6}u_{4}]$, where $c_{1}$ and $c_{2}$ are
crossings of edges, as shown in $A_{2}$ of Fig. 6. However, by Lemma 4 (i), in
any one of three situations, connecting $v$ to any one clique in $D|(T-v)$ of
size $4$ will violate the 1-planarity of $T$, a contradiction.

Similarly, $D|(T-v)$ cannot be $A_{3}$. Hence Claim A holds.

Now we are going to show that $D$ is isomorphic to one of the drawings shown
in Fig. 5. By Claim A, $D$ is obtained by inserting $v$ to a face of $A_{1}$
and then joining $v$ to all vertices in a $4$-clique of $A_{1}$.

Note that $A_{1}$ is triangular. By Lemma 4 (i), $v$ cannot be within any
crossed triangular face of $A_{1}$. Thus, $v$ must be in a triangular face of
$A_{1}$, i.e., $f_{1},f_{2},g_{1}$ or $g_{2}$ (see Fig. 6).

Case 1: $v$ is within face $f_{1}=[u_{4},u_{5},u_{6},u_{4}]$.

In this case, by Lemma 4 (ii), $N_{T}(v)$ must be the set
$\\{u_{4},u_{5},u_{6},u_{1}\\}$, and $D$ is isomorphic to $B_{1}$ shown in
Fig. 5.

Case 2: $v$ is within face $f_{2}=[u_{1},u_{5},u_{6},u_{1}]$.

In this case, by Lemma 4 (ii), $N_{T}(v)$ must be either
$\\{u_{1},u_{4},u_{5},u_{6},u_{1}\\}$ or
$\\{u_{1},u_{3},u_{4},u_{5},u_{1}\\}$, and $D$ is isomorphic to either $B_{2}$
or $B_{3}$ shown in Fig. 5.

Case 3: $v$ is within face $g_{1}=[u_{1},u_{3},u_{5}]$.

In this case, by Lemma 4 (ii), $N_{T}(v)$ must be either
$\\{u_{1},u_{3},u_{5},u_{2}\\}$ or $\\{u_{1},u_{3},u_{5},u_{4}\\}$, and thus
$D$ is isomorphic to either $B_{2}$ or $B_{3}$ shown in Fig. 5.

Case 4: $v$ is within face $g_{2}=[u_{1},u_{2},u_{3}]$.

In this case, by Lemma 4 (ii), $N_{T}(v)$ must be the set
$\\{u_{1},u_{2},u_{3},u_{5}\\}$, and $D$ is thus isomorphic to $B_{1}$ shown
in Fig. 5.

Hence the result holds. ∎

Let $\Phi$ denote the set of $1$-planar drawings of some $1$-planar graphs
defined below:

  1. (i).

the $1$-planar drawings $B_{1}$ and $B_{2}$ in Fig. 5 are the ones in $\Phi$
with the smallest order; and

  2. (ii).

if $D\in\Phi$, then both $D\oplus(f_{1},f_{2})$ and $D\oplus(f_{2},f_{1})$
belong to $\Phi$ for twin faces $f_{1}$ and $f_{2}$ in $D$.

######  Lemma 12.

Every $1$-planar drawing $D$ in $\Phi$ represents a $1$-planar $4$-tree with
exactly two simplicial vertices.

###### Proof.

We have the following two facts:

  1. (i).

Both $B_{1}$ and $B_{2}$, shown in Fig. 5, represent $1$-planar $4$-trees with
exactly two somplicial vertices and two pairs of twin faces, and each
simplicial vertex is located in one pair of twin faces;

  2. (ii).

if $D=D^{\prime}\oplus(f_{1},f_{2})$ for a pair of twin faces $f_{1}$ and
$f_{2}$ of $D^{\prime}$ and some simplicial vertex of $D$ is on
$\partial(f_{1})\cup\partial(f_{2})$, then $D$ and $D^{\prime}$ have the equal
number of simplicial vertices.

By the definition of $\Phi$, the result holds. ∎

In the following, we will show that the $1$-planar drawings of all $1$-planar
$4$-trees of order at least $8$ belongs to $\Phi$.

For any $1$-planar drawing $D$ with at least one uncrossed face, let
$E_{uf}(D)$ be the set of edges $e$ in $D$ such that $e$ is on $\partial(f)$
for some uncrossed face $f$ of $D$, and let $V_{uf}$ be the set of vertices in
$D$ which are incident with some edges in $E_{uf}(D)$. Let $D_{uf}$ denote the
$1$-planar drawing $D|H$, where $H=(V_{uf},E_{uf}(D))$.

######  Proposition 3.

Any $1$-planar drawing of a $1$-planar $4$-tree of order at least $8$ is
isomorphic to some $1$-planar drawing in $\Phi$.

###### Proof.

Let $T$ be a $1$-planar $4$-tree of order $n$ ($\geq 8$), and let $v$ be a
simplicial vertex of $T$. Thus, $N_{T}(v)$ is a $4$-clique in $T$. Let $D$ be
a $1$-planar drawing of $T$. Then, $D|(T-v)$ must be a $1$-planar drawing of
$T-v$. Since $\kappa(T)=4$, by Lemma 3, $v$ cannot be within a crossed
triangular face of $D|(T-v)$.

In the following, we shall show that $D_{uf}$ is isomorphic to some $D_{j}$ in
the set $\\{D_{j}:1\leq j\leq 6\\}$, shown in Fig. 7 with the following
properties:

  1. P1.

in each $D_{j}$ ($1\leq j\leq 6$), $f_{i}$ and $f_{i+1}$ are twin faces222The
set of vertices in $\partial(f_{i})\cup\partial(f_{i+1})$, where
$i\in\\{1,3\\}$, forms a $4$-clique of $D$, although it does not in $D_{uf}$.
of $D$ for each $i\in\\{1,3\\}$, where $f_{1},f_{2},f_{3}$ and $f_{4}$ are
faces in $D_{j}$, and

  2. P2.

$v_{0}v_{2}$, $v_{1}v_{2}$ and $v_{1}v_{3}$ are not edges in $D$ if $D_{uf}$
is $D_{1}$ or $D_{2}$.333 In each $D_{i}$, $1\leq i\leq 2$, Property 2 implies
that there are two nonadjacent vertices in $D_{i}$ of degree $3$ each of which
is not adjacent to some vertex in $D_{i}$ of degree $2$ in $D$. P2 also
implies that if $v$ is within face $f_{2}$, then $v_{2}\notin N_{D}(v)$.

$0$$2$13687945101112$a$$v_{1}$$b$$v_{2}$$f_{1}$$f_{2}$$a$$v_{1}$$b$$v_{2}$$x$$g_{1}$$g_{2}$(a)(b)$a$$v_{1}$$b$$v_{2}$$x$(c)$g_{1}$$g_{2}$$B_{2}$$B_{3}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{6}$$B_{1}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$u_{5}$$u_{6}$$u_{6}$$u_{7}$$u_{7}$$u_{7}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$D_{1}$$D_{2}$$D_{3}$$D_{4}$$D_{5}$$D_{6}$
Figure 7: Possible $1$-planar drawings of $D^{\prime}_{uf}$ with properties P1
and P2

We first prove the following conclusion. Let $D^{\prime}:=D|(T-v)$.

######  Claim 1.

If $n=8$, then $D^{\prime}$ is triangulated and $D^{\prime}_{uf}$ is
isomorphic to $D_{1}$ or $D_{4}$ in Fig. 7 with properties P1 and P2.

Note that $D^{\prime}$ is a $1$-planar drawing of $T-v$. By Proposition 2,
$D^{\prime}$ is isomorphic to $B_{1},B_{2}$ or $B_{3}$ in Fig. 5. By Lemma 4
(i), $v$ cannot be within any face of $B_{3}$. Thus, $D^{\prime}$ is
isomorphic to $B_{1}$ or $B_{2}$. Clearly, $D^{\prime}$ is triangulated.

If $D^{\prime}$ is isomorphic to $B_{1}$, then $D^{\prime}_{uf}$ is isomorphic
to $D_{4}$ in Fig. 7. If $D^{\prime}$ is isomorphic to $B_{2}$, then
$D^{\prime}_{uf}$ is isomorphic to $D_{1}$ in Fig. 7 with the property that
there is no edge in $D^{\prime}$ joining $v_{2}$ to $v_{0}$ or $v_{1}$, or
$v_{3}$ to $v_{1}$. In both case, properties P1 and P2 hold.

Claim 1 holds.

######  Claim 2.

If $D^{\prime}$ is triangulated and $D^{\prime}_{uf}$ is isomorphic to $D_{i}$
in Fig. 7 for some $i\in\\{1,2,\cdots,6\\}$ with properties P1 and P2, then
$D$ is obtained from $D^{\prime}$ by a 4-join operation and $D_{uf}$ is also
isomorphic to isomorphic to some $D_{i}$ in Fig. 7, where
$i\in\\{1,2,\dots,6\\}$, with properties P1 and P2.

By Lemma 4 (ii), $v$ is within a uncrossed face of $D^{\prime}$.

Case 1: $D^{\prime}_{uf}$ is isomorphic to $D_{1}$.

If $v$ is within $f_{1}$ or $f_{4}$, then, by Lemma 11,
$D=D^{\prime}\oplus(f_{1},f_{2})$ or $D=D^{\prime}\oplus(f_{4},f_{3})$
respectively. Clearly, $D_{uf}$ is isomorphic to $D_{5}$ and properties P1 and
P2 hold (see Fig. 8 (a)).

If $v$ is within $f_{2}$ or $f_{3}$, then, by Lemma 11,
$D=D^{\prime}\oplus(f_{2},f_{1})$ or $D=D^{\prime}\oplus(f_{3},f_{4})$
respectively. Clearly, $D_{uf}$ is isomorphic to $D_{2}$ and properties P1 and
P2 hold (see Fig. 8 (b)).

$0$$2$13687945101112$a$$v_{1}$$b$$v_{2}$$f_{1}$$f_{2}$$a$$v_{1}$$b$$v_{2}$$x$$g_{1}$$g_{2}$(a)(b)$a$$v_{1}$$b$$v_{2}$$x$(c)$g_{1}$$g_{2}$$B_{2}$$B_{3}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{6}$$B_{1}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$u_{5}$$u_{6}$$u_{6}$$u_{7}$$u_{7}$$u_{7}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$D_{1}$$D_{2}$$D_{3}$$D_{4}$$D_{5}$$D_{6}$(a)(b)(c)(d)
Figure 8: $D$ is obtained from $D^{\prime}$ by a $4$-join operation

Case 2: $D^{\prime}_{uf}$ is isomorphic to $D_{2}$.

If $v$ is within $f_{1}$ or $f_{4}$, then, by Lemma 11,
$D=D^{\prime}\oplus(f_{1},f_{2})$ or $D=D^{\prime}\oplus(f_{4},f_{3})$
respectively. Clearly, $D_{uf}$ is isomorphic to $D_{4}$ and properties P1 and
P2 hold (see Fig. 8 (c)).

If $v$ is within $f_{2}$ or $f_{3}$, then, by Lemma 11,
$D=D^{\prime}\oplus(f_{2},f_{1})$ or $D=D^{\prime}\oplus(f_{3},f_{4})$
respectively. Clearly, $D_{uf}$ is isomorphic to $D_{1}$ and properties P1 and
P2 hold (see Fig. 8 (d)).

Case 3: $D^{\prime}_{uf}$ is isomorphic to $D_{i}$ in Fig. 7 for some
$i\in\\{3,4,5,6\\}$.

By Lemma 11, $D=D^{\prime}\oplus(f_{i},f_{i+1})$ or
$D=D^{\prime}\oplus(f_{i+1},f_{i})$ for some $i\in\\{1,3\\}$. Observe that

  * •

if $D^{\prime}_{uf}$ is isomorphic to $D_{3}$, then $D_{uf}$ is also
isomorphic to $D_{3}$ (see Fig. 9 (a));

  * •

if $D^{\prime}_{uf}$ is isomorphic to $D_{4}$, then $D_{uf}$ is isomorphic to
$D_{3}$ or $D_{5}$ (see Fig. 9 (b) and (c));

  * •

if $D^{\prime}_{uf}$ is isomorphic to $D_{5}$, then $D_{uf}$ is isomorphic to
$D_{3}$, $D_{4}$ or $D_{6}$ (see Fig. 9 (d), (e) and (f)), and

  * •

if $D^{\prime}_{uf}$ is isomorphic to $D_{6}$, then $D_{uf}$ is isomorphic to
$D_{5}$ (see Fig. 9 (g)).

In all cases above, properties P1 and P2 hold.

$0$$2$13687945101112$a$$v_{1}$$b$$v_{2}$$f_{1}$$f_{2}$$a$$v_{1}$$b$$v_{2}$$x$$g_{1}$$g_{2}$(a)(b)$a$$v_{1}$$b$$v_{2}$$x$(c)$g_{1}$$g_{2}$$B_{2}$$B_{3}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{6}$$B_{1}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$u_{5}$$u_{6}$$u_{6}$$u_{7}$$u_{7}$$u_{7}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$D_{1}$$D_{2}$$D_{3}$$D_{4}$$D_{5}$$D_{6}$(a)(b)(c)(d)(a)(b)(c)(d)(e)(f)(g)
Figure 9: $D$ is obtained from $D^{\prime}$ by a $4$-join operation

Thus, Claim 2 holds. By Claims 1 and 2, every $1$-planar drawing of a
$1$-planar $4$-tree of order at least $8$ belongs to $\Phi$. The result holds.
∎

##  6 Proof of Theorem 1

We first provides a sufficient condition for a graph to be Hamiltonian-
connected.

######  Proposition 4.

Let $G$ be a graph with two vertices $a$ and $u$ such that $d_{G}(u)\geq 3$,
$N_{G}[u]\subseteq N_{G}[a]$ and $|N_{G}(a)\setminus N_{G}[u]|\leq 1$. If both
$G-u$ and $G-\\{u,a\\}$ are Hamiltonian-connected, then $G$ is also
Hamiltonian-connected.

###### Proof.

By the given condition, we may assume that $\\{a,b,c\\}\subseteq N_{G}(u)$.
Since $N_{G}[u]\subseteq N_{G}[a]$, $\\{u,b,c\\}\subseteq N_{G}(a)$, as shown
in Fig. 10. Let $G^{\prime}=G-u$ and $G^{\prime\prime}=G-\\{u,a\\}$. We only
need to prove that for any two vertices $x$ and $y$ in $G$, there is a
Hamiltonian path in $G$ connecting $x$ and $y$. By the given conditions,
$G^{\prime}$ has a Hamiltonian path $P$ connecting $b$ and $y$.

$0$$2$13687945101112$a$$v_{1}$$b$$v_{2}$$f_{1}$$f_{2}$$a$$v_{1}$$b$$v_{2}$$x$$g_{1}$$g_{2}$(a)(b)$a$$v_{1}$$b$$v_{2}$$x$(c)$g_{1}$$g_{2}$$B_{2}$$B_{3}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{6}$$B_{1}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$u_{5}$$u_{6}$$u_{6}$$u_{7}$$u_{7}$$u_{7}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{1}$$f_{2}$$f_{3}$$f_{4}$$D_{1}$$D_{2}$$D_{3}$$D_{4}$$D_{5}$$D_{6}$(a)(b)(c)(d)(a)(b)(c)(d)(e)(f)(g)$b$$c$$\cdots$$a$$u$$N(u)\setminus\\{a\\}$
Figure 10: $\\{a,b,c\\}\subseteq N_{G}(u)\subseteq N_{G}[a]$ and
$|N_{G}(a)\setminus N_{G}[u]|\leq 1$

We divide the proof into three cases.

######  Case 1.

$u\in\\{x,y\\}$.

By symmetry, assume that $x=u$. Then $y\in V(G^{\prime})$. Obviously, $y\neq
a$ or $y\neq b$. Assume that $y\neq b$. Combining $P$ and edge $ub$ produces a
Hamiltonian path in $G$ between $u$ (i.e., $x$) and $y$.

######  Case 2.

$u\notin\\{x,y\\}$ and $a\notin\\{x,y\\}$.

Since $|N_{G}(a)\setminus N_{G}[u]|\leq 1$, at least one edge $e^{*}$ in the
set $\\{av:v\in N_{G}(u)\\}$ is on $P$. Without loss of generality, assume
that $e^{*}=ab$. Let $P^{\prime}$ be the path in $G$ obtained from $P$ by
replacing edge $ab$ by the path $aub$. Clearly, $P^{\prime}$ is a Hamiltonian
path in $G$ connecting $x$ and $y$.

######  Case 3.

$u\notin\\{x,y\\}$ and $a\in\\{x,y\\}$.

By symmetry, assume that $a=x$. By the given condition, $G^{\prime\prime}$ is
Hamiltonian-connected. Then, $G^{\prime\prime}$ has Hamiltonian paths $P_{1}$
connecting $b$ and $y$ when $b\neq y$, and $P_{2}$ connecting $c$ and $y$ when
$c\neq y$. If $b\neq y$, then combining path $P_{1}$ and path $aub$ yields a
Hamiltonian path in $G$ connecting $a$ (i.e., $x$) and $y$. If $b=y$,
combining path $P_{2}$ and path $auc$ yields a Hamiltonian path in $G$
connecting $a$ (i.e., $x$) and $y$.

Hence the result holds. ∎

By the definition of $k$-trees, some conclusions on simplicial vertices of
$k$-trees can be obtained.

######  Lemma 13.

Let $G$ be a $k$-tree of order $n$. If $n\geq k+2$, then $G$ has the following
properties:

  1. (i).

$G$ has at least two simplicial vertices; and

  2. (ii).

any two simplicial vertices in $G$ are not adjacent, and

  3. (iii).

for any simplicial vertex $v$ of $G$, if $n\geq k+3$, then $G-v$ does not have
more simplcial vertices than $G$ has.

###### Proof.

The result is obvious when $n=k+2$. For $n>k+2$, let $v$ be a simplicial
vertex. Then $G-v$ is a $k$-tree of order $n-1\geq k+2$. By induction, $G-v$
has all these three properties.

Let $S(G)$ denote the set of simplicial vertices in $G$. If no vertex in
$N_{G}(v)$ belongs to $S(G-v)$, then $S(G)=S(G-v)\cup\\{v\\}$. If some vertex
$u\in N_{G}(v)$ belongs to $S(G-v)$, then
$S(G)=(S(G-v)\setminus\\{u\\})\cup\\{v\\}$.

Therefore $G$ also has these three properties. ∎

Applying Proposition 4, we can obtain the following conclusion on $k$-trees.

######  Proposition 5.

For any $k$-tree $G$, where $k\geq 3$, if $|V(G)|\geq k+2$ and $G$ has only
two simplicial vertices, then $G$ is Hamiltonian-connected.

###### Proof.

Let $G$ be a $k$-trees. Then $|V(G)|\geq k+1$. If $k\leq k+1$, then $G$ is a
complete graph and thus it is Hamiltonian-connected. If $|V(G)|=k+2$, then
$G=K_{k+2}-e$, which is also Hamiltonian-connected.

Now assume the result holds for all $k$-trees of order less than $n$, where
$n\geq k+3$. Let $G$ be any $k$-tree of order $n$. By Lemma 13 (i), $G$ has at
least two non-adjacent simplicial vertices. By the given conditions, $G$ has
exactly two simplicial vertices, say $u$ and $v$.

By the definition of $k$-tree, $G^{\prime}:=G-u$ is a $k$-tree of order $n-1$
($\geq k+2$). By Lemma 13 (iii), $G^{\prime}$ has exactly two simplcial
vertices and one of them must be $v$. Let $a$ be another simplicial vertex of
$G^{\prime}$. Clearly, $a\in N_{G}(u)$. By the inductive assumption, both
$G-u$ and $G^{\prime}-a$ ($=G-\\{u,a\\}$) are Hamiltonian connected.

Since $ua\in E(G)$, $u$ is a simplicial vertex of $G$ with $d_{G}(u)=k\geq 3$
and $a$ is a simplicial vertex of $G^{\prime}$ with $d_{G^{\prime}}(a)=k$, we
have $N_{G}[u]\subseteq N_{G}[a]$ and $|N_{G}(a)\setminus N_{G}[u]|=1.$ By
Proposition 4, $G$ is Hamiltonian-connected.

Hence the result holds. ∎

Now we are going to prove Theorem 1.

_Proof of Theorem 1_: Let $G$ be a $1$-planar graph which is $4$-connected and
chordal.

Let $n$ be the order of $G$. Then $n\geq 5$. If $n=5$, then $G\cong K_{5}$. If
$n=6$, then $G\cong K_{6}-e$ or $K_{6}$. Clearly, $G$ is Hamiltonian-connected
when $n\in\\{5,6\\}$. Now assume that $n\geq 7$. By Proposition 1, $G$ is a
$4$-tree. In the following, we first show that $G$ has exactly two simplicial
vertices.

If $n=7$, by Proposition 2, $G$ has a $1$-planar drawing isomorphic to
$B_{1},B_{2}$ or $B_{3}$ shown in Fig. 5, implying that $G$ is a $4$-tree with
exactly two simplicail vertices.

When $n\geq 8$, by Lemma 12 and Proposition 3, $G$ has exactly two two
simplicail vertices.

The result then follows from Proposition 5. $\Box$

##  7 Unsolved problems

The toughness of a graph is closely associated with Hamiltonicity. The
toughness of a graph $G$, denoted by $\tau(G)$, is the minimum value of
$\frac{|X|}{c(G-X)}$ over all non-empty subsets $X$ of $V(G)$ with $c(G-X)>1$,
where $c(H)$ is the number of components of a graph $H$. The toughness of a
complete graph is defined as being infinite. We say that a graph is $t$-tough
if its toughness is at least $t$. There are some known results on the
Hamiltonicity of chordal graphs in terms of their toughness. For example,
every 10-tough chordal graph is Hamiltonian [19], and every chordal planar
graph of order at least three and toughness greater than one is Hamiltonian
[7].

Note that a $t$-tough graph is always $\lceil 2t\rceil$-vertex-connected [10].
So the corollary below follows directly from Theorem 1.

######  Corollary 1.

Every $1$-planar chordal graph $G$ with $\tau(G)>\frac{3}{2}$ is Hamiltonian-
connected.

Chvátal (1973) [10] conjectured that all graphs which are more than
$\frac{3}{2}$-tough are Hamiltonian, but this was disproved by Bauer et al.
(2000), who showed that not every 2-tough graph is Hamiltonian [2]. Chvátal’s
toughness conjecture posits that there exists a toughness threshold $t_{0}$
above which $t_{0}$-tough graphs are always Hamiltonian; its truth remains
unresolved.

Corollary 1 states that every 1-planar chordal graph with toughness greater
than $\frac{3}{2}$ is Hamiltonian-connected. Naturally, we pose the following
problem:

######  Problem 2.

Is there a $1$-planar chordal graph with toughness $\frac{3}{2}$ non-
Hamiltonian?

Notice that neither the examples in [23] nor our examples in Remark 1 can be
used for Problem 2. This is because the toughness of every chordal planar non-
Hamiltonian graph in [23] is always 1, while our examples can be shown to be
at most $\frac{5}{6}$.

##  8 Acknowledgment

The work is supported by the National Natural Science Foundation of China
(Grant No. 12271157, 12371340, 12371346), the Natural Science Foundation of
Hunan Province, China (Grant No. 2022JJ30028), Hunan Provincial Department of
Education(No. 21A0590) and Natural Science Foundation of Changsha (No.
kq2208001).

We claim that there is no conflict of interest in our paper. No data was used
for the research described in the article.

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# Spin dynamics of the $E_{8}$ particles

Xiao Wang  Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai,
201210, China  Konrad Puzniak  Helmholtz-Zentrum Berlin für Materialien und
Energie GmbH, Hahn-Meitner Platz 1, D-14109 Berlin, Germany Institut für
Festkörperphysik, Technische Universität Berlin, Hardenbergstraße 36, D-10623
Berlin, Germany Karin Schmalzl  Jülich Centre for Neutron Science JCNS,
Forschungszentrum Jülich GmbH, Outstation at ILL, 38042 Grenoble, France C.
Balz  ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory,
Didcot OX11 0QX, United Kingdom  M. Matsuda  Neutron Scattering Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Akira Okutani
Center for Advanced High Magnetic Field Science, Graduate School of Science,
Osaka University, Osaka 560-0043, Japan  M. Hagiwara  Center for Advanced High
Magnetic Field Science, Graduate School of Science, Osaka University, Osaka
560-0043, Japan  Jie Ma<EMAIL_ADDRESS>Tsung-Dao Lee Institute, Shanghai
Jiao Tong University, Shanghai, 201210, China  School of Physics & Astronomy,
Shanghai Jiao Tong University, Shanghai, 200240, China  Collaborative
Innovation Center of Advanced Microstructures, Nanjing 210093, China  Jianda
Wu<EMAIL_ADDRESS>Tsung-Dao Lee Institute, Shanghai Jiao Tong University,
Shanghai, 201210, China  School of Physics & Astronomy, Shanghai Jiao Tong
University, Shanghai, 200240, China  Shanghai Branch, Hefei National
Laboratory, Shanghai 201315, China  Bella Lake<EMAIL_ADDRESS>Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Hahn-Meitner Platz
1, D-14109 Berlin, Germany Institut für Festkörperphysik, Technische
Universität Berlin, Hardenbergstraße 36, D-10623 Berlin, Germany

###### Abstract

In this article, we report on inelastic neutron scattering measurements on a
quasi-1D antiferromagnet BaCo2V2O8 under a transverse magnetic field applied
along the (0,1,0) direction. Combining results of inelastic neutron scattering
experiments, analytical analysis, and numerical simulations, we precisely
studied the $E_{8}$ excitations appearing in the whole Brillouin zone at
$B_{c}^{1D}\approx 4.7$ T. The energy scan at $Q=(0,0,2)$ reveals a match
between the data and the theoretical prediction of energies of multiple
$E_{8}$ excitations. Furthermore, dispersions of the lightest three $E_{8}$
particles have been clearly observed, confirming the existence of the $E_{8}$
particles in BaCo2V2O8. Our results lay down a concrete ground to
systematically study the physics of the exotic $E_{8}$ particles.

###### pacs:

75.10.Pq,75.30.Ds,75.50.Ee,78.70.Nx

Unlike the classical phase transition driven by thermal fluctuations, the
quantum phase transition arises at zero temperature when the system is tuned
by a non-thermal parameter Sachdev (2011). For a continuous quantum phase
transition, novel physics with higher symmetry may emerge at the quantum
critical point (QCP), classified by a set of critical exponents manifested
into scaling form Sachdev (2011). Moreover, when the system is driven away
from the QCP with a relevant perturbation, exotic physics may further emerge
due to the strong renormalization of the almost infinite low-lying
excitations, which is “emergence of emergence” Zamolodchikov (1989); Dorey
(1997); Braden  _et al._ (1990).

One such paradigmatic model is the transverse-field Ising chain (TFIC) Sachdev
(2011); Pfeuty (1970). When an Ising chain is tuned to its QCP by a magnetic
field applied transverse to its Ising anisotropy, a central charge 1/2
conformal field theory emerges with corresponding scaling exponents falling
into the class of Ising universality see Fig. 1 Pfeuty (1970); Boyanovsky
(1989). Surprisingly, when it is further perturbed by a longitudinal field
parallel to the Ising direction, the quantum $E_{8}$ integrable model emerges
- a massive relativistic quantum field theory containing eight massive $E_{8}$
particles whose relative masses have precise values as listed in the first row
of Table I. The physics of the model is described by scattering of the $E_{8}$
particles, which is characterized by the maximal exceptional Lie $E_{8}$
algebra Zamolodchikov (1989); Delfino and Mussardo (1995); Zou  _et al._
(2021); Zhang  _et al._ (2020); Wang  _et al._ (2021).

For the experimental realization of the exotic $E_{8}$ physics, two conditions
need to be satisfied: accessing the Ising universality and the presence of a
small perturbation field along the Ising direction. An early inelastic neutron
scattering (INS) experiment on the ferromagnetic chain compound CoNb2O6
provided evidence of the existence of the lightest two particles: the ratio of
the energies of the lowest two peaks echos the Golden ratio of the two
lightest $E_{8}$ particles’ masses Coldea  _et al._ (2010). However, there is
an apparent deviation in the spectrum continuum region between the recent THz
experiment on the material Zhang  _et al._ (2020) and the analytical result
Wang  _et al._ (2021), which implies more efforts Morris  _et al._ (2021);
Fava  _et al._ (2020) are needed for confirming the existence of the $E_{8}$
physics in the material CoNb2O6.

Recently, the quasi-1D Heisenberg-Ising antiferromagnetic (AFM) material
BaCo2V2O8 (a member of a family of materials with formula $AM_{2}$V2O8 in
which: $A$ = Sr, Ba, and the magnetic ions are $M$ = Cu, Ni, Co, Mn), has
attracted the attention of several experimental studies Faure  _et al._
(2018); Niesen  _et al._ (2013). When the materials are tuned by a transverse
magnetic field, quantum criticality with Ising universality is observed Zou
_et al._ (2021). Then the $E_{8}$ excitations with zero transfer momentum have
been sought at low temperatures where the interchain coupling and long-range
magnetic order provide the longitudinal field Zou  _et al._ (2021); Zhang  _et
al._ (2020). These $E_{8}$ excitations are measured by inelastic neutron
scattering Zou  _et al._ (2021) (in this case all the $E_{8}$ particles were
found) and terahertz spectroscopy Zhang  _et al._ (2020) (in the latter case
the excitations up to the fifth $E_{8}$ particles were found) and compared
successfully to theory.

However, all these measurements were confined to only the AFM zone center
where the precise $E_{8}$ masses have already been calculated. Furthermore,
other peaks due to combinations of the $E_{8}$ particles and zone folding were
also observed, which, while fully explainable, resulted in many overlapping
excitations decreasing the certainty of the results. Considering that the
$E_{8}$ model is a massive relativistic quantum field theory, a full
investigation of the relativistic dispersion of the $E_{8}$ particles in the
whole Brillouin zone is necessary for a complete realization of the $E_{8}$
physics in the material BaCo2V2O8.

Figure 1: Schematic phase diagram of BaCo2V2O8 in a transverse magnetic field,
light blue indicates the AFM order phase while the deep blue area covers the
location of emergent exotic $E_{8}$ particles around the putative 1D QCP. The
inset shows one of the CoO6 screw chains.

In this work, we combine and compare experimental, analytical, and numerical
approaches for the dispersion of the lightest three branches of excitations
and unambiguously demonstrate the existence of the exotic $E_{8}$ particles in
the AFM material BaCo2V2O8. BaCo2V2O8 crystallizes in tetragonal symmetry
(space group $I4_{1}$/acd No. 142) with the lattice parameters: $a$ = $b$ =
12.40 Å and $c$ = 8.375 Å Niesen  _et al._ (2013). The magnetic Co2+ ions have
effective spin of $S=\frac{1}{2}$ and are arranged in edge-sharing CoO6
octahedra forming 4-fold screw chains running along the ${\bf c}$-axis (see
inset of Fig. 1). There are four screw chains per unit cell, two of which
rotate clockwise while the other two rotate anticlockwise Canévet  _et al._
(2013). The Co2+ ions are coupled by strong AFM interactions within the screw
chains which have partial Ising (XXZ) anisotropy favouring spin directions
parallel to the ${\bf c}$-axis. In the absence of an external magnetic field,
BaCo2V2O8 develops a long-range magnetic Néel order below $T_{\mathrm{N}}=5.5$
K due to weak interchain coupling, where neighboring spins are aligned
antiferromagnetically along the screw chains and ferromagnetically
(antiferromagnetically) between chains along the ${\bf a}$ (${\bf b}$)
direction respectively. The spins point almost parallel to the ${\bf c}$-axis.

Under an external transverse magnetic field applied perpendicular to the spin
direction (e.g. parallel to the ${\bf b}$-axis), BaCo2V2O8 undergoes a three-
dimensional (3D) quantum phase transition at $\mu_{0}H_{\bot}^{c,3D}\approx
10.3$ T (Fig. 1) that was identified, by a combination of field theory,
numerical analysis, and neutron scattering experiments, as a spin-flop
transition from the ${\bf c}$ to ${\bf a}$ directions Faure  _et al._ (2018).
This transition originates from the complex $g$-factor of BaCo2V2O8. A one-
dimensional (1D) quantum phase transition was also discovered at the lower
transverse magnetic field of $\mu_{0}H_{\bot}^{c,1D}=4.7$ T Cui  _et al._
(2019); Zou  _et al._ (2021). This transition lies hidden within the dome of
the 3D magnetic order (see Fig. 1) which provides the (staggered) longitudinal
magnetic field required for the emergence of the $E_{8}$ quantum particles.

The aim of this paper is to investigate BaCo2V2O8 at its putative critical
transverse field $\mu_{0}H_{\bot}^{c,1D}\approx 4.7$ T Zou  _et al._ (2021) by
combining an inelastic neutron scattering study of the dispersion of $E_{8}$
particles with a theoretical analysis based on the infinite time-evolving
block decimation (iTEBD) technique Vidal (2007). The plan of this paper is as
follows: firstly we describe our experimental and theoretical approaches, then
the neutron measurements of the dispersion of $E_{8}$ particles are presented
and compared to the numerical simulations based on the iTEBD technique.
Excellent agreement is achieved as a function of wavevector and energy for the
lowest three $E_{8}$ particles.

Table 1: Predicted ratios of the energies of the single $E_{8}$ particle (first row) and the multi-particle (second row) excitations along with their expected values in BaCo2V2O8 (third row). The experimental excitation energies at various wavevectors from LET and IN12 are then listed along with the values obtained from iTEBD. Single |  $m_{1}$ |  $m_{2}$ |  $m_{3}$ |  |  $m_{4}$ |  |  $m_{5}$ |  |  |  $m_{6}$ |  |  $m_{7}$
---|---|---|---|---|---|---|---|---|---|---|---|---
Multi |  |  |  |  2$m_{1}$ |  |  $m_{1}$+$m_{2}$ |  |  $m_{1}$+$m_{3}$ |  3$m_{1}$ |  |  $2m_{2}$ |
Theoretical ratio $m_{i}$/$m_{1}$ |  1 |  1.618 |  1.989 |  2 |  2.405 |  2.618 |  2.956 |  2.989 |  3 |  3.218 |  3.236 |  3.891
BaCo2V2O8 theory [meV] |  1.26 |  2.04 |  2.50 |  2.52 |  3.03 |  3.30 |  3.72 |  3.73 |  3.75 |  4.05 |  4.08 |  4.90
LET (0,0,2) [meV] |  1.28 |  2.04 |  2.52 |  2.58 |  3.04 |  3.30 |  - |  - |  - |  - |  - |  -
IN12 (0,0,2) [meV] |  1.26 |  2.05 |  2.49 |  - |  - |  - |  - |  - |  - |  - |  - |  -
iTEBD (0,0,2) [meV] |  1.26 |  2.06 |  2.50 |  2.52 |  3.06 |  3.32 |  3.64 |  3.70 |  3.78 |  4.04 |  4.10 |  4.84

Two large, high-quality single crystals of BaCo2V2O8 were grown using the
floating-zone technique at Osaka University, Japan, and at the Core Lab for
Quantum Materials, Helmholtz Zentrum Berlin für Materialien und Energie (HZB),
Germany. Inelastic neutron scattering was performed to measure the magnetic
excitations on the cold neutron multichopper spectrometer, LET (at the ISIS
Facility, Rutherford Appleton Laboratory, UK) using the HZB crystal (mass 4.13
g) Lake  _et al._ (2022). INS experiments were also performed on the cold
neutron triple-axis spectrometer, IN12 from the Forschungszentrum Jülich
Collaborating Research Group (FZJ-CRG) installed at Institut Laue Langevin
(ILL) France, using the Osaka crystal (mass 3.66 g).

For the LET experiment, the single crystal was aligned in the (0,K,L)
horizontal scattering plane and a vertical field cryomagnet was used to apply
a constant magnetic field of $B=4.7$ T along the ${\bf a}$-axis to reach the
1D QCP. These measurements were carried out at $T=0.3$ K using a 3He-insert.
This temperature is well below the Néel temperature ($T_{\mathrm{N}}=5.5$ K)
ensuring the presence of the effective longitudinal perturbing field necessary
to stabilize $E_{8}$ physics. Using repetition rate multiplication and the
chopper frequencies 280/140 Hz, incident neutron energies of $E_{i}$ = 22.69,
13.21, 8.51, 6.00, 4.42, 3.42, 2.70 meV were achieved with corresponding
elastic energy resolutions of $\Delta E$ = 0.91, 0.41, 0.22, 0.14, 0.094,
0.065, 0.048 meV. The INS data were processed using the MANTID and HORACE
software packages and converted to absolute units. The spectrum for the
incident energy of 6 meV is displayed in Fig. 2 as a function of energy and
wavevector transfer along the chain direction, (0,0,L).

For the IN12 experiment, the crystal was aligned with the ${\bf a}$\- and
${\bf c}$-axes within the horizontal instrumental scattering plane and a
vertical DC magnetic field of 4.7 T was applied parallel to the ${\bf
b}$-axis. A fixed final wavevector of $k_{f}=1.15$ Å-1 was used, giving an
energy resolution of $\Delta E\approx 0.114$ meV and wavevector resolution of
$\approx 0.067$ r.l.u. A Beryllium filter was used to suppress higher-order
wavelengths and spurious scattering. A series of energy scans at constant
wavevector in the range from ${\bf Q}$ = (0,0,1.5) to (0,0,2.5) were performed
over the energy range from $E=05$ meV with steps of at least 0.05 meV.
Constant-energy scans were also performed within this range. These
measurements took place at a temperature of $T=1.50$ K($\ll T_{\mathrm{N}}$).
The constant-energy and constant-wavevector scans are combined together to
make the energy-wavevector map in Fig. 3 (c).

When applying a transverse field along (0,1,0) direction, the effective
Hamiltonian for BaCo2V2O8 is described by a 1D spin-1/2 Heisenberg-Ising model
Kimura  _et al._ (2013); Cui  _et al._ (2019); Zou  _et al._ (2021); Zhang
_et al._ (2020); Zou  _et al._ (2019):

$\displaystyle\mathcal{H}$ $\displaystyle=H_{XXZ}+H_{t}+H_{s}$ (1)
$\displaystyle H_{XXZ}$
$\displaystyle=J\sum_{n}[S^{z}_{n}S^{z}_{n+1}+\epsilon(S^{x}_{n}S^{x}_{n+1}+S^{y}_{n}S^{y}_{n+1})]$
$\displaystyle H_{t}$
$\displaystyle=-\mu_{B}g_{yy}H\sum_{n}[S^{y}_{n}+h_{x}(-1)^{n}S^{x}_{n}$
$\displaystyle\;\;\;\;+h_{z}\cos(\pi\frac{2n-1}{4})S^{z}_{n}]$ $\displaystyle
H_{s}$ $\displaystyle=-\mu_{B}H^{\prime}\sum_{n}(-1)^{n}S^{z}_{n}$

where $S^{\alpha}_{n}=\frac{1}{2}\sigma^{\alpha}_{n}$($\alpha=x,y,z$) are
spin-1/2 operators at site $n$ with Pauli matrices $\sigma^{\alpha}$.
$J=5.8\;$meV, $\epsilon=0.46$, $h_{x(z)}=0.4(0.14)$, $g_{yy}=2.75$. The
applied transverse field is set $\mu_{0}H=4.7\;$T, which is the critical field
of the putative 1D QCP Zou  _et al._ (2019, 2021). The effective staggered
longitudinal field $\mu_{B}H^{\prime}=0.018J$ comes from a mean-field
treatment of the inter-chain coupling in the 3D ordering region below
$T_{\mathrm{N}}$ Faure  _et al._ (2018); Zou  _et al._ (2021). $H_{s}$
provides a necessary relevant perturbation for realizing the quantum $E_{8}$
physics Zou  _et al._ (2021). Focusing on the parameter region around the
putative 1D QCP, in the scaling limit, the effective Hamiltonian of the spin
chain becomes Delfino and Mussardo (1995); Zou  _et al._ (2021); Wang  _et
al._ (2021)

$\displaystyle\mathcal{H}_{E_{8}}=\mathcal{H}_{c=1/2}+h\int dx\sigma(x).$ (2)

$\mathcal{H}_{c=1/2}$ is the Hamiltonian for a central charge 1/2 conformal
field theory, which describes the quantum critical physics of the TFIC. $h$
and $\sigma(x)$ corresponding to the scaling limits of $\mu_{B}H^{\prime}$ and
$\sigma_{j}^{z}$ are the strengths of the perturbation field and the relevant
primary field, respectively.

To determine the dispersion of $E_{8}$ particles and compare with the spectrum
measured by INS, we calculate the spin dynamic structure factor (DSF) in the
field theory frame,
$D^{\alpha\alpha}(\omega,q)=\sum_{n=1}^{\infty}\frac{(2\pi)^{2}}{\prod_{a_{i}=1}^{8}n_{a_{i}}!}\int_{-\infty}^{\infty}\prod_{j=1}^{n}\frac{d\theta_{j}}{2\pi}|\langle
0|\sigma^{\alpha}|A_{a_{1}}(\theta_{1})...A_{a_{n}}(\theta_{n})\rangle|^{2}$
$\delta(\omega-\sum_{j=1}^{n}E_{j})\delta(q-\sum_{j=1}^{n}P_{j}),$ where
$\alpha=x,z$, and $a_{i}=1...8$ are quasi-particles obtained from the quantum
$E_{8}$ integrable theory Zamolodchikov (1989); Delfino and Mussardo (1995);
Zou  _et al._ (2021); Wang  _et al._ (2021). $n_{a_{i}}$ is the number of
particle $a_{i}$ involved in the corresponding channel.
$E_{j}=m_{a_{j}}\cosh\theta_{j}$ and $P_{j}=m_{a_{j}}\sinh\theta_{j}$ are the
energy and momentum of particle $a_{j}$ in terms of the rapidity $\theta$. The
two Dirac $\delta$-functions reflect the energy and momentum conservation of
the scattering. The DSF of $\sigma^{x,z}$ can be directly calculated from
quantum $E_{8}$ integrable field theory Wang  _et al._ (2021), and the DSF of
$\sigma^{y}$ can be obtained from DSF of $\sigma^{z}$ Wu  _et al._ (2014). The
analytical result for the dispersion of the lightest three $E_{8}$ particles
is shown in Fig. 3(a). For a better comparison of the theoretical prediction
from quantum $E_{8}$ field theory with the INS experimental result, two subtle
issues are worth noting.

Figure 2: Magnetic intensity of BaCo2V2O8 measured in a transverse magnetic
field of $\mu_{0}H_{\bot}^{c,1D}=4.7$ T at $T=0.3$ K using the LET
spectrometer. The data is displayed in absolute units as a function of
wavevector ${\bf Q}$ = (0,0,L) and energy, for neutron incident energy $E_{i}$
= 6 meV (integration range: $-1.0\leq H\leq 1.0$ & $-2\leq K\leq 2$).  Figure
3: (a) Analytical calculation of dispersion of three lightest $E_{8}$
particles. (b) Numerical simulation of dispersion using the iTEBD method,
based on Eq. (9). The dashed lines illustrate dispersions of the three
lightest $E_{8}$ particles given by Eq. (8). (c) Energy-wavevector map of the
magnetic excitations constructed from constant-energy and constant-wavevector
scans measured on the IN12 spectrometer at $B^{1D}_{c}$ = 4.7 T and $T=1.5$ K.
The black symbols are the peak positions extracted from fitting the individual
scans, whereas the dashed lines are fits to the dispersions to the lowest
three $E_{8}$ particles using Eq. (8). (d) shows an energy scan measured at
the wavevector $Q$ = (0,0,2) on LET and IN12, with the theoretically predicted
energies of the first $E_{8}$ excitations indicated by the red solid lines.
Identified peaks are labelled $m_{n}$ (single $E_{8}$ excitations),
$m_{n}+m_{m}$ (multi-$E_{8}$ excitations) and $F_{n}$ (zone-folding peaks).

1\. In the above field theory frame calculation, the speed of light is set as
$c=1$. For the quantum $E_{8}$ model, as a massive relativistic quantum field
theory, the dispersion of the $E_{8}$ particles follows the massive
relativistic dispersion $E_{i}^{2}=\Delta_{i}^{2}+p_{i}^{2}{c^{2}}$, where
$\Delta_{i}=m_{i}c^{2}$ and $p_{i}=m_{i}c$ with the “rest mass” of the
$i^{th}$ $E_{8}$ particle $m_{i}$ and the “speed of light” $c$. When coming to
a real material which actually is a lattice discrete in space, we need to re-
scale the dispersion of the $E_{8}$ particles with the proper energy scale and
length (momentum) scale serving as IR cutoffs. The theoretically expected
energy peak corresponding to the lightest $E_{8}$ particle $m_{1}$ can be
estimated from $E_{m_{1}}^{\text{theory}}=C_{\text{lattice}}H^{\prime
8/15}\approx 1.2\,\text{meV}$ Yang  _et al._ (2023), where
$C_{\text{lattice}}=4.010\cdots$ is a modified constant for the lattice which
originally comes from quantum $E_{8}$ field theory Caselle and Hasenbusch
(2000). The value of $E_{m_{1}}^{\text{iTEBD}}$ matches the minimum gap
$\Delta_{1}=1.26\,\text{meV}$ observed at the zone center (corresponding to
zero transfer momentum), thus $\Delta_{1}$ can naturally serve as IR cutoff of
the energy scale for the experimental data. Since $\Delta_{1}=m_{1}c^{2}$ then
we can pick up the corresponding IR momentum cutoff $p_{1}=m_{1}c$. By
applying these two IR cutoffs scales we arrive at

$\frac{{E_{i}^{2}}}{{{\Delta_{1}^{2}}}}=\frac{{\Delta_{i}^{2}}}{{\Delta_{1}^{2}}}+\frac{{p_{i}^{2}}}{(\Delta_{1}/c)^{2}}=\frac{{\Delta_{i}^{2}}}{{{{\Delta_{1}}^{2}}}}+\left({\frac{{\hbar(L-2)\pi/2d}}{{{m_{1}}c}}}\right)^{2},$
(3)

where $p_{i}=\hbar(L-2)\pi/2d$ is the momentum transfer with respect to ${\bf
Q}$ = (0,0,2) and $d=8.4192/4=2.105$ is the nearest neighbor distance between
Co2+ ions projected onto the chain direction. We need to determine the value
of $c$ to obtain the IR cutoff for the momentum, whose value cannot be
uniquely determined by the analytical theory but actually depends on the
microscopic details of the material.

2\. The four-fold periodicity of BaCo2V2O8 leads to the sizable zone-folding
effect of the experimental measurement, which makes the $E_{8}$ particles’
dispersion shadowed by additional spectra. Such an effect cannot be obtained
from the field theory calculation, instead, we need to go back to the original
effective lattice model. By comparing spectra obtained from the lattice model
and the field theory, the $E_{8}$ particles’ dispersion will be extracted.

To make these two subtle issues clear, we carry out iTEBD simulation for the
effective Hamiltonian Eq. (6) with $J=1$, $\epsilon=0.47$, and critical field
$\mu_{B}g_{yy}H=0.15$ Zou  _et al._ (2019, 2021),

$\displaystyle\begin{split}D_{\rm{lat}}^{\alpha\alpha}(\omega,q)=\frac{1}{N}&\sum_{j,j^{\prime}=1}^{N}\exp\\{-iq(j-j^{\prime})\\}\\\
&\times\int_{-\infty}^{\infty}dt\exp(i\omega t)\langle
S_{j}^{\alpha}(t)S_{j^{\prime}}^{\alpha}(0)\rangle,\end{split}$ (4)

with total number of lattice sites $N$ ($N\to\infty$ in iTEBD), and spin-1/2
operators $S^{\alpha},\alpha=x,y,z$. The iTEBD simulation result is shown in
Fig. 3 (b), where the value of the ”speed of light” is found to be
$c\approx(1.441\pm 0.096)\times 10^{3}\leavevmode\nobreak\ m/s$ and the zone-
folding effect can be identified as well.

The procedure of the iTEBD calculation is as follows: 1. Generate a four-
periodic ground state wave function of the effective Hamiltonian Eq. (6) with
the parameters. The imaginary time-evolution is done with fifth-order Trotter-
Suzuki decomposition Hatano and Suzuki (2005), where the imaginary time slide
is set as $d\tau=0.01$. The convergence condition is chosen as the difference
of the norm of singular values in the matrix product states being smaller than
$10^{-12}$. The truncated dimension is chosen as $\chi=45$ Vidal (2007);
Danshita and Naidon (2009). 2. Calculate the DSF [Eq. (9)] for $S^{x}$ and
$S^{z}$, while the DSF of $S^{y}$ can be obtained from DSF of $S^{z}$ by using
$D^{yy}(\omega,q)=\omega^{2}D^{zz}(\omega,q)/(4J^{2})$ Wu  _et al._ (2014).
For calculating this DSF with iTEBD algorithm, we first do real time and space
propagation in Heisenberg picture, then by using Fourier transformation we
transform the obtained result into momentum and energy space to obtain the
final spectrum. The real time evolution is done by a second order Trotter-
Suzuki decomposition with $t=200,\leavevmode\nobreak\ dt=0.02$ for obtaining a
relatively high accurate result near the Brillouin zone center. 3. A zone-
folding effect is necessary to consider when obtaining the final spectrum for
comparison with INS experimental results.

The magnetic excitations of BaCo2V2O8, measured using the LET time-of-flight
spectrometer at $T$ = 0.3 K and $\mu_{0}H=\mu_{0}H_{\bot}^{c,1D}(=4.7$ T) with
incident neutron energy of 6 meV, is shown in Fig. 2 as a function of energy
and wavevector transfer along the chain direction. To complement this, Fig. 3
(c) shows the low-energy excitations built from the energy- and wavevector-
scans measured at $T=1.5$ K and $\mu_{0}H=4.7$ T on the IN12 spectrometer. An
incredibly rich series of modes is found with complex dispersions and
intensity modulations. First of all, the 4-fold screw-chain structure about
the c-axis gives rise to four independent chain spectra shifted consecutively
in wavevector along the chain by $\Delta L=1$ r.l.u. Each individual spectrum
is periodic over an interval of $\Delta L=4$ r.l.u. due to the fact that there
are four Co2+ ions along each chain per unit cell (because of the screw-chain
structure). Together these four spectra ensure that an antiferromagnetic zone
center where the excitations are minimum is found at every integer value of
$L$. At low energies, each spectrum is expected to consist of the
$E_{8}$-particles observed as sharp (resolution-limited) gapped modes. In
addition, multi-particle excitations are expected, due to the simultaneous
creation of two or more $E_{8}$ particles such as $m_{1}+m_{2}$. These
excitations form a continuum with a sharp lower boundary. Finally, zone-
folding modes are expected which are a consequence of the screw chain
structure of BaCo2V2O8. Unlike the other excitations, these can be
incommensurate with minima that do not always occur at integer values of $L$.

Figure 4: (a) Comparison of DSF results between the iTEBD and the LET
experimental data for constant momentum cuts. (b) Comparison of DSF results
between the iTEBD data, the analytical data, and the experimental IN12 and LET
data for constant energy cuts. The presented LET experimental constant
momentum and constant energy cuts are shifted by an offset of 0.004 r.l.u (the
offset was caused by experimental conditions). All DSF intensities are
normalized up to the maximum intensity of experimental data.

According to theory, the ratios of the energies of the eight $E_{8}$-particles
have precise values Borthwick and Garibaldi (2011). These values are given in
the first row of Table I along with the expected values of the multiparticle
excitations. The excitations were indeed observed previously at these energies
at the antiferromagnetic zone centers (integer $L$-wavevectors) of BaCo2V2O8
Zou  _et al._ (2021). Here we reconfirm this observation using our higher
resolution (but lower intensity) data. The energy scans through our LET and
IN12 datasets are shown in Fig. 3(d). For both datasets, the first peak is
observed at 1.26 meV. Multiplying the $E_{8}$ particle ratios by this value
gives the theoretically expected peak positions for BaCo2V2O8 (second row of
Table I) which are also indicated by the solid vertical red lines in Fig.
3(d). The first three peaks in the LET and IN12 experimental data (at
$m_{1}=1.26$, $m_{2}=2.04$, $m_{3}=2.52$ meV and at $m_{1}=1.26$,
$m_{2}=2.05$, $m_{3}=2.49$ meV respectively) clearly lie at the expected
positions of the first three $E_{8}$ peaks, (it should be noted that the
multi-particle peak $2m_{1}$ coincides with $m_{3}$). It should be also noted
that close to $m_{2}$ there is a zone-folding peak of 1.98 meV (indicated by
F0). There exists also another zone-folding peak at 2.75 meV (indicated by
F1). The fourth $E_{8}$ excitation at $m_{4}=3.04$ meV is very weak, while the
feature at 3.30 meV is the lower boundary of the $m_{1}+m_{2}$ multi-particle
continuum. At higher energies, it is difficult to distinguish the $E_{8}$
peaks due to overlapping continuua and zone-folding modes. The positions of
peaks were found by fitting Gaussians. They are listed in Table I and are in
agreement with the results of H. Zou et. al. Zou  _et al._ (2021). Because we
collected high-precision data over a wide range of wavevectors rather than at
just the antiferromagnetic zone centers, we now have the opportunity to
observe the behavior of the $E_{8}$-particles as well as the other excitations
as a function of wavevector as well as energy. Returning to the energy-
wavevector plots in Fig. 2 and 3(c), it is clear that the three lowest $E_{8}$
excitations actually form dispersive modes with parabolic curvature and a
minimum at ${\bf Q}$ = (0,0,2). The zone-folding modes are now easily
identified, such as the dispersive excitations which have minima at the
incommensurate wavevectors (0,0,1.875) and (0,0,2.15) at $E$ = 1.9 meV and
overlap with $m_{2}$ at (0,0,2). Finally above 3 meV broad diffuse scattering
is observed due to the multi-particle continua and overlapping modes. The
dispersions of the $E_{8}$ particles are expected to follow the theoretical
expression given by Eq. (8), which can be modified as

$E_{i}=\sqrt{\Delta_{i}^{2}+\left(\gamma\cdot(L-2)\right)^{2}},$ (5)

where the single parameter $\gamma=\frac{\hbar\pi c}{2d}$ can be extracted. We
simultaneously fit the lowest three $E_{8}$ dispersions and get the value
$\gamma=8.07$ meV. The fitted dispersions are given by the red lines in Fig. 3
(c) and show good agreement with the data. The ‘speed of light’ was extracted
from $\gamma$ and found to be $c\approx(1.643\pm 0.041)\times
10^{3}\leavevmode\nobreak\ m/s$ (using $d=2.105$ Å - the projection of the
nearest neighbor Co2+-Co2+ distance onto the c-axis). The value of $\gamma$,
found from fitting the lowest three $E_{8}$ dispersions in the case of LET
data, is $\gamma=8.81$ meV and the ‘speed of light’ is found to be
$c\approx(1.794\pm 0.008)\times 10^{3}\leavevmode\nobreak\ m/s$. This value
agrees well with the value of $c\approx(1.441\pm 0.096)\times
10^{3}\leavevmode\nobreak\ m/s$ found from iTEBD.

We further compare experimental, analytical, and numerical data with constant
momentum scans [Fig. 4(a)] and constant energy scans [Fig. 4 (b)]. For the
former comparison in Fig. 4(a) the analytical data are not included due to the
presence of zone-folding effects for energies above 2 meV. The iTEBD data
agree very well with the experimental data. For the latter comparison Fig.
4(b) in order to avoid mixture from zone-folding effect the energy window is
chosen to match with the dispersion of the lightest $E_{8}$ particle. The
analytical and iTEBD data show excellent agreement with each other, and both
show good agreement with the experimental data. We note that there is about
$1\%$ deviation in momentum for the peak position corresponding to $1\%$
$m_{1}$ energy shift. This possibly is because the transverse field, applied
during the experiment, is slightly smaller than the exact critical field which
can result in slightly heavier $E_{8}$ particles which implies a slightly
larger minimum gap $m_{1}$. $1\%$ $m_{1}$ shift corresponds to about 0.1 T to
0.2 T shift from the exact critical field whose value lies in the range of the
identified putative QCP $4.7\pm 0.3$ T Zou  _et al._ (2021).

To conclude, the quasi-1D antiferromagnet BaCo2V2O8 is a very important
material in the field of quantum magnetism. Among its unique properties are:
Ising-like anisotropy, large intrachain versus weak but non-negligible
interchain interactions, an anisotropic $g$ tensor producing easy-axis
anisotropy, and effective staggered fields under the application of an
external magnetic field. Combining the results of INS experiments,
theoretical, and numerical iTEBD simulations, we have precisely studied the
$E_{8}$ excitation spectrum appearing at the one-dimensional quantum critical
point of $B^{1D}_{c}$ = 4.7 T. The observation of dynamical spectra through
INS, together with excellent agreement with analytical analysis and iTEBD
numerical simulations, enabled us to observe the dispersion of the first three
$E_{8}$ particles and several multi-particles modes, paving the way toward
possible manipulation of the $E_{8}$ particles.

The crystal growth and characterization took place at the Core Laboratory
Quantum Materials, Helmholtz Zentrum Berlin für Materialien und Energie,
Germany and at the Center for Advanced High Magnetic Field Science, Graduate
School of Science, Osaka University. The authors would like to thank ISIS and
ILL facilities for the allocation of neutron beam-time. We would like to also
thank C. Fritsche for his help in the preparation of the sample holder. K. P.
is also very grateful to Dr. C. Rohr for all his important suggestions about
the general structure of the article. The LET data
(https://doi.org/10.5286/ISIS.E.RB2210086) were reduced using Mantid and were
analyzed using the Horace-MATLAB software package. This work has, in part,
been supported by National Natural Science Foundation of China No. U2032213
(J. M.), 12274288 (X. W. and J. W.) and the Innovation Program for Quantum
Science and Technology Grant No. 2021ZD0301900 (X. W. and J. W.),
2022YFA1402702 (J. M.), and the Natural Science Foundation of Shanghai with
grant No. 20ZR1428400 and Shanghai Pujiang Program with grant No. 20PJ1408100
(X.W. and J.W.), and Grants-in-Aid for Scientific Research (Nos. 25220803 and
24244059) from MEXT.

X. W. and K. P contributed equally to this study. J. W., J. M., and B. L.
conceived and coordinate the project. J. M., C. B, and B. L. designed the
experiment. X. W. and J. W. carry out analytical and iTEBD calculations and
provide theoretical analysis. X. W., K. P., J. M., J. W., and B. L. wrote the
manuscript.

## References

  * Sachdev (2011) S. Sachdev,  _Quantum Phase Transitions_ (Cambridge University Press, Cambridge, England, 2011) pp. 1–521.
  * Zamolodchikov (1989) A. B. Zamolodchikov, Int. J. Mod. Phys. A 4, 4235 (1989).
  * Dorey (1997) P. Dorey, Lect. Notes Phys. 498, 85 (1997).
  * Braden  _et al._ (1990) H. Braden, E. Corrigan, P. Dorey, and R. Sasaki, Nucl. Phys. B 338, 689 (1990).
  * Pfeuty (1970) P. Pfeuty, Ann. Phys. 59, 79 (1970).
  * Boyanovsky (1989) D. Boyanovsky, Phys. Rev. B 39, 6744 (1989).
  * Delfino and Mussardo (1995) G. Delfino and G. Mussardo, Nucl. Phys. B 455, 724 (1995).
  * Zou  _et al._ (2021) H. Zou, Y. Cui, X. Wang, Z. Zhang, J. Yang, G. Xu, A. Okutani, M. Hagiwara, M. Matsuda, G. Wang, and et al., Phys. Rev. Lett. 127 (2021).
  * Zhang  _et al._ (2020) Z. Zhang, K. Amelin, X. Wang, H. Zou, J. Yang, U. Nagel, T. Rõ om, T. Dey, A. A. Nugroho, T. Lorenz, J. Wu, and Z. Wang, Phys. Rev. B 101, 220411 (2020).
  * Wang  _et al._ (2021) X. Wang, H. Zou, K. Hódsági, M. Kormos, G. Takács, and J. Wu, Phys. Rev. B 103, 235117 (2021).
  * Coldea  _et al._ (2010) R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177–180 (2010).
  * Morris  _et al._ (2021) C. M. Morris, N. Desai, J. Viirok, D. Hüvonen, U. Nagel, T. Rõõm, J. W. Krizan, R. J. Cava, T. M. McQueen, S. M. Koohpayeh, R. K. Kaul, and N. P. Armitage, Nature Phys. 17, 832–836 (2021).
  * Fava  _et al._ (2020) M. Fava, R. Coldea, and S. A. Parameswaran, Proc. Nat. Acad. Sci. 117, 25219 (2020).
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  * Niesen  _et al._ (2013) S. K. Niesen, G. Kolland, M. Seher, O. Breunig, M. Valldor, M. Braden, B. Grenier, and T. Lorenz, Phys. Rev. B 87, 224413 (2013).
  * Canévet  _et al._ (2013) E. Canévet, B. Grenier, M. Klanjšek, C. Berthier, M. Horvatić, V. Simonet, and P. Lejay, Phys. Rev. B 87, 054408 (2013).
  * Cui  _et al._ (2019) Y. Cui, H. Zou, N. Xi, Z. He, Y. X. Yang, L. Shu, G. H. Zhang, Z. Hu, T. Chen, R. Yu, J. Wu, and W. Yu, Phys. Rev. Lett. 123, 067203 (2019).
  * Vidal (2007) G. Vidal, Phys. Rev. Lett. 98, 070201 (2007).
  * Lake  _et al._ (2022) B. Lake, K. Puzniak, J. Ma, and C. Balz,  _Dispersion of $E_{8}$ particles in the spin-1/2 antiferromagnetic XXZ chain BaCo2V2O8 in a transverse magnetic field_, Ph.D. thesis (2022).
  * Kimura  _et al._ (2013) S. Kimura, K. Okunishi, M. Hagiwara, K. Kindo, Z. He, T. Taniyama, M. Itoh, K. Koyama, and K. Watanabe, J. Phys. Soc. Jpn. 82, 033706 (2013).
  * Zou  _et al._ (2019) H. Zou, R. Yu, and J. Wu, J. Phys.: Condens. Matt. 32, 045602 (2019).
  * Wu  _et al._ (2014) J. Wu, M. Kormos, and Q. Si, Phys. Rev. Lett. 113, 247201 (2014).
  * Yang  _et al._ (2023) J. Yang, X. Wang, and J. Wu, J. Phys. A: Math. Theor. 56, 013001 (2023).
  * Caselle and Hasenbusch (2000) M. Caselle and M. Hasenbusch, Nucl. Phys. B 579, 667 (2000).
  * Hatano and Suzuki (2005) N. Hatano and M. Suzuki,  _Quantum Annealing and Other Optimization Methods_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005) pp. 37–68.
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## Supplemental Material—Spin dynamics of the $E_{8}$ particles

##  Appendix A Details of neutron experiments

Two large, high-quality single crystals of BaCo2V2O8 were grown using the
floating-zone technique at Osaka University, Japan, and at the Core Lab for
Quantum Materials, Helmholtz Zentrum Berlin für Materialien und Energie (HZB),
Germany. Inelastic neutron scattering was performed to measure the magnetic
excitations on the cold neutron multichopper spectrometer, LET (at the ISIS
Facility, Rutherford Appleton Laboratory, UK) using the HZB crystal (mass 4.13
g) Lake  _et al._ (2022). INS experiments were also performed on the cold
neutron triple-axis spectrometer, IN12 from the Forschungszentrum Jülich
Collaborating Research Group (FZJ-CRG) installed at Institut Laue Langevin
(ILL) France, using the Osaka crystal (mass 3.66 g).

For the LET experiment, the single crystal was aligned in the (0,K,L)
horizontal scattering plane and a vertical field cryomagnet was used to apply
a constant magnetic field of $B=4.7$ T along the ${\bf a}$-axis to reach the
1D QCP. These measurements were carried out at $T=0.3$ K using a 3He-insert.
This temperature is well below the Néel temperature ($T_{\mathrm{N}}=5.5$ K)
ensuring the presence of the effective longitudinal perturbing field necessary
to stabilize $E_{8}$ physics. Using repetition rate multiplication and the
chopper frequencies 280/140 Hz, incident neutron energies of $E_{i}$ = 22.69,
13.21, 8.51, 6.00, 4.42, 3.42, 2.70 meV were achieved with corresponding
elastic energy resolutions of $\Delta E$ = 0.91, 0.41, 0.22, 0.14, 0.094,
0.065, 0.048 meV. The INS data were processed using the MANTID and HORACE
software packages and converted to absolute units.

For the IN12 experiment, the crystal was aligned with the ${\bf a}$\- and
${\bf c}$-axes within the horizontal instrumental scattering plane and a
vertical DC magnetic field of 4.7 T was applied parallel to the ${\bf
b}$-axis. A fixed final wavevector of $k_{f}=1.15$ Å-1 was used, giving an
energy resolution of $\Delta E\approx 0.114$ meV and wavevector resolution of
$\approx 0.067$ r.l.u. A Beryllium filter was used to suppress higher-order
wavelengths and spurious scattering.

##  Appendix B Theoretical model and dispersion of $E_{8}$ particles

When applying a transverse field along (0,1,0) direction, the effective
Hamiltonian for BaCo2V2O8 is described by a 1D spin-1/2 Heisenberg-Ising model
Kimura  _et al._ (2013); Cui  _et al._ (2019); Zou  _et al._ (2021, 2019):

$\displaystyle\mathcal{H}$ $\displaystyle=H_{XXZ}+H_{t}+H_{s}$ (6)
$\displaystyle H_{XXZ}$
$\displaystyle=J\sum_{n}[S^{z}_{n}S^{z}_{n+1}+\epsilon(S^{x}_{n}S^{x}_{n+1}+S^{y}_{n}S^{y}_{n+1})]$
$\displaystyle H_{t}$
$\displaystyle=-\mu_{B}g_{yy}H\sum_{n}[S^{y}_{n}+h_{x}(-1)^{n}S^{x}_{n}$
$\displaystyle\;\;\;\;+h_{z}\cos(\pi\frac{2n-1}{4})S^{z}_{n}]$ $\displaystyle
H_{s}$ $\displaystyle=-\mu_{B}H^{\prime}\sum_{n}(-1)^{n}S^{z}_{n}$

where $S^{\alpha}_{n}=\frac{1}{2}\sigma^{\alpha}_{n}$($\alpha=x,y,z$) are
spin-1/2 operators at site $n$ with Pauli matrices $\sigma^{\alpha}$.
$J=5.8\;$meV, $\epsilon=0.46$, $h_{x(z)}=0.4(0.14)$, $g_{yy}=2.75$. The
applied transverse field is set $\mu_{0}H=4.7\;$T, which is the critical field
of the putative 1D QCP Zou  _et al._ (2019, 2021). The effective staggered
longitudinal field $\mu_{B}H^{\prime}=0.018J$ comes from a mean-field
treatment of the inter-chain coupling in the 3D ordering region below
$T_{\mathrm{N}}$ Faure  _et al._ (2018); Zou  _et al._ (2021). $H_{s}$
provides a necessary relevant perturbation for realizing the quantum $E_{8}$
physics Zou  _et al._ (2021). Focusing on the parameter region around the
putative 1D QCP, in the scaling limit, the effective Hamiltonian of the spin
chain becomes Delfino and Mussardo (1995); Zou  _et al._ (2021); Wang  _et
al._ (2021)

$\displaystyle\mathcal{H}_{E_{8}}=\mathcal{H}_{c=1/2}+h\int dx\sigma(x).$ (7)

$\mathcal{H}_{c=1/2}$ is the Hamiltonian for a central charge 1/2 conformal
field theory, which describes the quantum critical physics of the TFIC. $h$
and $\sigma(x)$ corresponding to the scaling limits of $\mu_{B}H^{\prime}$ and
$\sigma_{j}^{z}$ are the strengths of the perturbation field and the relevant
primary field, respectively. To determine the dispersion of $E_{8}$ particles
and compare with the spectrum measured by INS, we calculate the DSF in the
field theory frame,
$D^{\alpha\alpha}(\omega,q)=\sum_{n=1}^{\infty}\frac{(2\pi)^{2}}{\prod_{a_{i}=1}^{8}n_{a_{i}}!}\int_{-\infty}^{\infty}\prod_{j=1}^{n}\frac{d\theta_{j}}{2\pi}|\langle
0|\sigma^{\alpha}|A_{a_{1}}(\theta_{1})...A_{a_{n}}(\theta_{n})\rangle|^{2}$
$\delta(\omega-\sum_{j=1}^{n}E_{j})\delta(q-\sum_{j=1}^{n}P_{j}),$ where
$\alpha=x,z$, and $a_{i}=1...8$ are quasi-particles obtained from the quantum
$E_{8}$ integrable theory Zamolodchikov (1989); Delfino and Mussardo (1995);
Zou  _et al._ (2021); Wang  _et al._ (2021). $n_{a_{i}}$ is the number of
particle $a_{i}$ involved in the corresponding channel.
$E_{j}=m_{a_{j}}\cosh\theta_{j}$ and $P_{j}=m_{a_{j}}\sinh\theta_{j}$ are the
energy and momentum of particle $a_{j}$ in terms of the rapidity $\theta$,
respectively. The two Dirac $\delta$-functions reflect the energy and momentum
conservation of the scattering. The DSF of $\sigma^{x,z}$ can be directly
calculated from quantum $E_{8}$ integrable field theory Wang  _et al._ (2021),
and the DSF of $\sigma^{y}$ can be obtained from DSF of $\sigma^{z}$ Wu  _et
al._ (2014). For a better comparison of the theoretical prediction from
quantum $E_{8}$ field theory with the INS experimental result, two subtle
issues are worth noting.

1\. In the above field theory frame calculation, the speed of light is set as
$c=1$. For the quantum $E_{8}$ model, as a massive relativistic quantum field
theory, the dispersion of the $E_{8}$ particles follows the massive
relativistic dispersion $E_{i}^{2}=\Delta_{i}^{2}+p_{i}^{2}{c^{2}}$, where
$\Delta_{i}=m_{i}c^{2}$ and $p_{i}=m_{i}c$ with the “rest mass” of the
$i^{th}$ $E_{8}$ particle $m_{i}$ and the “speed of light” $c$. When coming to
a real material which actually is a lattice discrete in space, we need to re-
scale the dispersion of the $E_{8}$ particles with the proper energy scale and
length (momentum) scale serving as IR cutoffs. The theoretically expected
energy peak corresponding to the lightest $E_{8}$ particle $m_{1}$ can be
estimated from $E_{m_{1}}^{\text{theory}}=C_{\text{lattice}}H^{\prime
8/15}\approx 1.2\,\text{meV}$ Yang  _et al._ (2023), where
$C_{\text{lattice}}=4.010\cdots$ is a modified constant for the lattice which
originally comes from quantum $E_{8}$ field theory Caselle and Hasenbusch
(2000). The value of $E_{m_{1}}^{\text{iTEBD}}$ matches the minimum gap
$\Delta_{1}=1.26\,\text{meV}$ observed at the zone center (corresponding to
zero transfer momentum), thus $\Delta_{1}$ can naturally serve as IR cutoff of
the energy scale for the experimental data. Since $\Delta_{1}=m_{1}c^{2}$ then
we can pick up the corresponding IR momentum cutoff $p_{1}=m_{1}c$. By
applying these two IR cutoffs scales we arrive at

$\frac{{E_{i}^{2}}}{{{\Delta_{1}^{2}}}}=\frac{{\Delta_{i}^{2}}}{{\Delta_{1}^{2}}}+\frac{{p_{i}^{2}}}{(\Delta_{1}/c)^{2}}=\frac{{\Delta_{i}^{2}}}{{{{\Delta_{1}}^{2}}}}+\left({\frac{{\hbar(L-2)\pi/2d}}{{{m_{1}}c}}}\right)^{2},$
(8)

where $p_{i}=\hbar(L-2)\pi/2d$ is the momentum transfer with respect to ${\bf
Q}$ = (0,0,2) and $d=8.4192/4=2.105$ is the nearest neighbor distance between
Co2+ ions projected onto the chain direction. We need to determine the value
of $c$ to obtain the IR cutoff for the momentum, whose value cannot be
uniquely determined by the analytical theory but actually depends on the
microscopic details of the material.

2\. The four-fold periodicity of BaCo2V2O8 leads to the sizable zone-folding
effect of the experimental measurement, which makes the $E_{8}$ particles’
dispersion shadowed by additional spectra. Such an effect cannot be obtained
from the field theory calculation, instead, we need to go back to the original
effective lattice model. By comparing spectra obtained from the lattice model
and the field theory, the $E_{8}$ particles’ dispersion will be extracted. To
make these two subtle issues clear, we carry out iTEBD simulation for the
effective Hamiltonian Eq. (6) with $J=1$, $\epsilon=0.47$, and critical field
$\mu_{B}g_{yy}H=0.15$ Zou  _et al._ (2019, 2021),

$\displaystyle\begin{split}D_{\rm{lat}}^{\alpha\alpha}(\omega,q)=\frac{1}{N}&\sum_{j,j^{\prime}=1}^{N}\exp\\{-iq(j-j^{\prime})\\}\\\
&\times\int_{-\infty}^{\infty}dt\exp(i\omega t)\langle
S_{j}^{\alpha}(t)S_{j^{\prime}}^{\alpha}(0)\rangle,\end{split}$ (9)

with total number of lattice sites $N$ ($N\to\infty$ in iTEBD), and spin-1/2
operators $S^{\alpha},\alpha=x,y,z$. The procedure of the iTEBD calculation is
as follows: 1. Generate a four-periodic ground state wave function of the
effective Hamiltonian Eq. (6) with the parameters. The imaginary time-
evolution is done with fifth-order Trotter-Suzuki decomposition Hatano and
Suzuki (2005), where the imaginary time slide is set as $d\tau=0.01$. The
convergence condition is chosen as the difference of the norm of singular
values in the matrix product states being smaller than $10^{-12}$. The
truncated dimension is chosen as $\chi=45$ Vidal (2007); Danshita and Naidon
(2009). 2. Calculate the DSF [Eq. (9)] for $S^{x}$ and $S^{z}$, while the DSF
of $S^{y}$ can be obtained from DSF of $S^{z}$ by using
$D^{yy}(\omega,q)=\omega^{2}D^{zz}(\omega,q)/(4J^{2})$ Wu  _et al._ (2014).
For calculating this DSF with iTEBD algorithm, we first do real time and space
propagation in Heisenberg picture, then by using Fourier transformation we
transform the obtained result into momentum and energy space to obtain the
final spectrum. The real time evolution is done by a second order Trotter-
Suzuki decomposition with $t=200,\leavevmode\nobreak\ dt=0.02$ for obtaining a
relatively high accurate result near the Brillouin zone center. 3. A zone-
folding effect is necessary to consider when obtaining the final spectrum for
comparison with INS experimental results.

## References

  * Sachdev (2011) S. Sachdev,  _Quantum Phase Transitions_ (Cambridge University Press, Cambridge, England, 2011) pp. 1–521.
  * Zamolodchikov (1989) A. B. Zamolodchikov, Int. J. Mod. Phys. A 4, 4235 (1989).
  * Dorey (1997) P. Dorey, Lect. Notes Phys. 498, 85 (1997).
  * Braden  _et al._ (1990) H. Braden, E. Corrigan, P. Dorey, and R. Sasaki, Nucl. Phys. B 338, 689 (1990).
  * Pfeuty (1970) P. Pfeuty, Ann. Phys. 59, 79 (1970).
  * Boyanovsky (1989) D. Boyanovsky, Phys. Rev. B 39, 6744 (1989).
  * Delfino and Mussardo (1995) G. Delfino and G. Mussardo, Nucl. Phys. B 455, 724 (1995).
  * Zou  _et al._ (2021) H. Zou, Y. Cui, X. Wang, Z. Zhang, J. Yang, G. Xu, A. Okutani, M. Hagiwara, M. Matsuda, G. Wang, and et al., Phys. Rev. Lett. 127 (2021).
  * Zhang  _et al._ (2020) Z. Zhang, K. Amelin, X. Wang, H. Zou, J. Yang, U. Nagel, T. Rõ om, T. Dey, A. A. Nugroho, T. Lorenz, J. Wu, and Z. Wang, Phys. Rev. B 101, 220411 (2020).
  * Wang  _et al._ (2021) X. Wang, H. Zou, K. Hódsági, M. Kormos, G. Takács, and J. Wu, Phys. Rev. B 103, 235117 (2021).
  * Coldea  _et al._ (2010) R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177–180 (2010).
  * Morris  _et al._ (2021) C. M. Morris, N. Desai, J. Viirok, D. Hüvonen, U. Nagel, T. Rõõm, J. W. Krizan, R. J. Cava, T. M. McQueen, S. M. Koohpayeh, R. K. Kaul, and N. P. Armitage, Nature Phys. 17, 832–836 (2021).
  * Fava  _et al._ (2020) M. Fava, R. Coldea, and S. A. Parameswaran, Proc. Nat. Acad. Sci. 117, 25219 (2020).
  * Faure  _et al._ (2018) Q. Faure _et al._ , Nature Phys. 14, 716 (2018).
  * Niesen  _et al._ (2013) S. K. Niesen, G. Kolland, M. Seher, O. Breunig, M. Valldor, M. Braden, B. Grenier, and T. Lorenz, Phys. Rev. B 87, 224413 (2013).
  * Canévet  _et al._ (2013) E. Canévet, B. Grenier, M. Klanjšek, C. Berthier, M. Horvatić, V. Simonet, and P. Lejay, Phys. Rev. B 87, 054408 (2013).
  * Cui  _et al._ (2019) Y. Cui, H. Zou, N. Xi, Z. He, Y. X. Yang, L. Shu, G. H. Zhang, Z. Hu, T. Chen, R. Yu, J. Wu, and W. Yu, Phys. Rev. Lett. 123, 067203 (2019).
  * Vidal (2007) G. Vidal, Phys. Rev. Lett. 98, 070201 (2007).
  * Lake  _et al._ (2022) B. Lake, K. Puzniak, J. Ma, and C. Balz,  _Dispersion of $E_{8}$ particles in the spin-1/2 antiferromagnetic XXZ chain BaCo2V2O8 in a transverse magnetic field_, Ph.D. thesis (2022).
  * Kimura  _et al._ (2013) S. Kimura, K. Okunishi, M. Hagiwara, K. Kindo, Z. He, T. Taniyama, M. Itoh, K. Koyama, and K. Watanabe, J. Phys. Soc. Jpn. 82, 033706 (2013).
  * Zou  _et al._ (2019) H. Zou, R. Yu, and J. Wu, J. Phys.: Condens. Matt. 32, 045602 (2019).
  * Wu  _et al._ (2014) J. Wu, M. Kormos, and Q. Si, Phys. Rev. Lett. 113, 247201 (2014).
  * Yang  _et al._ (2023) J. Yang, X. Wang, and J. Wu, J. Phys. A: Math. Theor. 56, 013001 (2023).
  * Caselle and Hasenbusch (2000) M. Caselle and M. Hasenbusch, Nucl. Phys. B 579, 667 (2000).
  * Hatano and Suzuki (2005) N. Hatano and M. Suzuki,  _Quantum Annealing and Other Optimization Methods_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005) pp. 37–68.
  * Danshita and Naidon (2009) I. Danshita and P. Naidon, Phys. Rev. A 79, 043601 (2009).
  * Borthwick and Garibaldi (2011) D. Borthwick and S. Garibaldi, Not. Amer. Math. Soc. 58, 1055 (2011).