# Rigidity of acute triangulations of the plane

Tianqi Wu  Department of Mathematics, Clark University, 950 Main St,
Worcester, MA 01610, USA<EMAIL_ADDRESS>
###### Abstract.

We show that a uniformly acute triangulation of the plane is rigid under Luo’s
discrete conformal change, extending previous results on hexagonal
triangulations. Our result is a discrete analogue of the conformal rigidity of
the plane. We followed He’s analytical approach in his work on the rigidity of
disk patterns. The main tools include maximum principles, a discrete Liouville
theorem, smooth and discrete extremal lengths on networks. The key step is
relating the Euclidean discrete conformality to the hyperbolic discrete
conformality, to obtain an $L^{\infty}$ bound on the discrete conformal
factor.

###### Contents

  1. 1 Introduction
    1. 1.1 Other Related Works
    2. 1.2 Notations and Conventions
    3. 1.3 Organization of the Paper
    4. 1.4 Acknowledgement
  2. 2 Preparations for the Proof
    1. 2.1 Extremal Length and Modulus of Annuli
    2. 2.2 Discrete Harmonic Functions
    3. 2.3 Differential of the Curvature Map
    4. 2.4 Maximum Principles
    5. 2.5 Key Estimates on the Conformal Factors for Geodesic Embeddings
  3. 3 Proof of Theorem 1.2
    1. 3.1 Proof of Theorem 1.2 Assuming the Boundedness of $\bar{u}$
    2. 3.2 Boundedness of the Conformal Factor
  4. 4 Discrete Extremal Length and the Discrete Liouville Theorem
    1. 4.1 Electrical Networks and Discrete Extremal Length
    2. 4.2 Proof of the Discrete Liouville Theorem
  5. 5 Hyperbolic Maximum Principles and Proof of Lemma 2.9
    1. 5.1 Proof of the Hyperbolic Maximum Principle
  6. A Proof of Lemma 5.1

##  1\. Introduction

A fundamental property in conformal geometry is that a conformal embedding of
the plane $\mathbb{R}^{2}$ to itself must be a similar transformation. In this
paper we discretize the plane by triangulations and prove a similar rigidity
result under the notion of discrete conformal change introduced by Luo
[Luo04].

Let $T=(V,E,F)$ be an (infinite) simplicial topological triangulation of the
Euclidean plane $\mathbb{R}^{2}$, where $V$ is the set of vertices, $E$ is the
set of edges and $F$ is the set of faces. Given a subcomplex
$T_{0}=(V_{0},E_{0},F_{0})$ of $T$, denote $|T_{0}|$ as the underlying space
of $T$. An embedding (_resp._ homeomorphism)
$\phi:|T_{0}|\rightarrow\mathbb{R}^{2}$ is called _geodesic_ if $\phi$ maps
each edge of $T_{0}$ to a geodesic arc, i.e., a straight closed line segment.
A _piecewise linear metric_ (_PL metric_ for short) on $T_{0}$ is represented
by an edge length function $l\in\mathbb{R}^{E_{0}}_{>0}$ satisfying the
triangle inequalities. A geodesic embedding $\phi$ of $T_{0}$ naturally
induces a PL metric $l=l(\phi)$ on $T_{0}$ by letting
$l_{ij}=|\phi(i)-\phi(j)|_{2}$. Luo [Luo04] introduced the following notion of
discrete conformality.

######  Definition 1.1 (Luo [Luo04]).

Two PL metrics $l,l^{\prime}$ on $T_{0}=(V_{0},E_{0},F_{0})$ are _discretely
conformal_ if there exists some $u\in\mathbb{R}^{V_{0}}$ such that for any
edge $ij\in E_{0}$

$l^{\prime}_{ij}=e^{\frac{1}{2}(u_{i}+u_{j})}l_{ij}.$

In this case, $u$ is called a _discrete conformal factor_ , and we denote
$l^{\prime}=u*l$.

Given a PL metric $l$ on $T_{0}$, let $\theta^{i}_{jk}$ denote the inner angle
at the vertex $i$ in the triangle $\triangle ijk$ under the metric $l$. Then
$l$ is called

  1. (a)

_uniformly nondegenerate_ if there exists a constant $\epsilon>0$ such that
$\theta^{i}_{jk}\geq\epsilon$ for all $\triangle ijk$ in $T_{0}$, and

  2. (b)

_uniformly acute_ if there exists a constant $\epsilon>0$ such that
$\theta^{i}_{jk}\leq\pi/2-\epsilon$ for all $\triangle ijk$ in $T_{0}$, and

  3. (c)

_Delaunay_ if $\theta^{k}_{ij}+\theta^{k^{\prime}}_{ij}\leq\pi$ for any pair
of adjacent triangles $\triangle ijk$ and $\triangle ijk^{\prime}$ in $T_{0}$.

A uniformly acute PL metric is clearly uniformly nondegenerate and Delaunay.
The main result of the paper is the following.

######  Theorem 1.2.

Suppose $\phi$ is a geodesic homeomorphism of $T$ and $\psi$ is a geodesic
embedding of $T$. If $l(\phi),l(\psi)$ are discretely conformal and both
uniformly acute, then they differ by a constant scaling.

Wu-Gu-Sun [WGS15] first proved Theorem 1.2 for the special case where
$\phi(T)$ is a regular hexagonal triangulation. Dai-Ge-Ma [DGM22] and Luo-Sun-
Wu [LSW20] generalized Wu-Gu-Sun’s result by allowing $l(\psi)$ to be only
Delaunay rather than uniformly acute. All these works essentially rely on the
lattice structure of the embedded vertices $\phi(V)$, and apparently cannot be
generalized to triangulations without translational invariance. To prove
Theorem 1.2, we adopted a different approach, which is developed by He [He99]
in his state-of-art work on the rigidity of disk patterns.

###  1.1. Other Related Works

After Luo introducing the Definition 1.1, various properties regarding the
rigidity and convergence of the discrete conformality were discussed in
[BPS15][WGS15][GLW19][WZ20][LSW20][LWZ21a][DGM22]. To solve the problem of
singularity in the discrete Yamabe flow, Gu et al. [GLSW18][GGL+18] proposed a
revised notion of discrete conformality for piecewise Euclidean (or
hyperbolic) metrics on closed surfaces with marked points, and perfectly
solved the prescribed curvature problem. This major improvement in the theory
of discrete conformality inspired new advanced numerical methods in computing
conformal maps [SWGL15][GSC21][CCS+21], as well as further theoretical
investigations [Spr19][LW19]. Gu et al. [GLSW18][GGL+18] proposed to use the
discrete Yamabe flow to numerically compute the target metric in the
prescribed curvature problem. Since the discrete Yamabe flow may pass through
different combinatorial triangulations, diagonal switches might be needed
along the flow. In [Wu14] it is proved that only finitely many diagonal
switches are needed in a Yamabe flow. Other works on discrete geometric flows
or deformations of triangle meshes could be found in [ZGZ+14][GH18]
[ZX19][FLZ20][WX21][LWZ21b][LWZ21c][LWZ22][Luo22].

###  1.2. Notations and Conventions

In the remaining of the paper, we will identify the plane $\mathbb{R}^{2}$ as
the complex plane $\mathbb{C}$. Given $0<r<r^{\prime}$, denote
$D_{r}=\\{z\in\mathbb{C}:|z|<r\\}$ and
$A_{r,r^{\prime}}=\\{z\in\mathbb{C}:r<|z|<r^{\prime}\\}$. We also denote
$D=D_{1}$ as the unit open disk. Given a subset $A$ of $\mathbb{C}$, $A^{c}$
denote the complement $\mathbb{C}\backslash A$ and $\partial A$ denotes the
boundary of $A$ in $\mathbb{C}$. Given two subsets $A,B$ of $\mathbb{C}$, the
diameter of $A$ is denoted by

$\text{diam}(A)=\sup\\{|z-z^{\prime}|:z,z^{\prime}\in A\\},$

and the distance between $A,B$ is denoted by

$d(A,B)=\inf\\{|z-z^{\prime}|:z\in A,z^{\prime}\in B\\}.$

Given a subset $V_{0}$ of $V$, we use the following notations and conventions.

  1. (a)

The _complement_ of $V_{0}$ is denoted as $V_{0}^{c}=V\backslash V_{0}$.

  2. (b)

The _boundary_ of $V_{0}$ is denoted as

$\partial V_{0}=\\{i\in V_{0}:\text{there exists $j\in V_{0}^{c}$ such that
$ij\in E$}\\}.$

  3. (c)

The _interior_ of $V_{0}$ is denoted as

$int(V_{0})=V_{0}\backslash\partial V_{0}=\\{i\in V_{0}:j\in V_{0}\text{ if
}ij\in E\\}.$

  4. (d)

The _closure_ of $V_{0}$ is denoted as

$\overline{V_{0}}=V_{0}\cup\partial(V_{0}^{c})=(int(V_{0}^{c}))^{c}.$

  5. (e)

The subcomplex generated by $V_{0}$ is denoted as $T(V_{0})$.

  6. (f)

Denote $E(V_{0})=\\{ij\in E:i\in int(V_{0})\text{ or }j\in int(V_{0})\\}$.
Notice that $E(V_{0})$ generally is not the set of edges in $T(V_{0})$.

  7. (g)

A real-valued function on $V_{0}$ is often identifies as a vector in
$\mathbb{R}^{V_{0}}$.

Given $i\in V$, the _1-ring neighborhood_ of $i$ is the subcomplex generated
by $i$ and its neighbors. In other words, the 1-ring neighborhood of $i$ is

$T(\\{i\\}\cup\\{j\in V:ij\in E\\}).$

Furthermore, we denote $R_{i}$ as the underlying space of the 1-ring
neighborhood of $i$. Given a subcomplex $T_{0}=(V_{0},E_{0},F_{0})$ of $T$ and
$l\in\mathbb{R}^{E_{0}}$ and $u\in\mathbb{R}^{V_{0}}$, if $u*l$ is a PL metric
then

  1. (a)

$\theta^{i}_{jk}(u)=\theta^{i}_{jk}(u,l)$ denotes the inner angle of
$\triangle ijk$ at $i$ under $u*l$, and

  2. (b)

$K_{i}(u)=K_{i}(u*l)$ denotes the discrete curvature

$K_{i}(u)=2\pi-\sum_{jk:\triangle ijk\in F}\theta^{i}_{jk}(u).$

###  1.3. Organization of the Paper

In Section 2 we introduce necessary properties and tools for the proof of the
main theorem. The proof of main Theorem 1.2 is given in Section 3. Section 4
gives a proof of a discrete Liouville theorem, which is used in proving
Theorem 1.2. Section 5 proves a key estimate for the discrete conformal factor
by relating to the hyperbolic discrete conformality.

###  1.4. Acknowledgement

The work is supported in part by NSF 1760471.

##  2\. Preparations for the Proof

###  2.1. Extremal Length and Modulus of Annuli

We briefly review the notions of extremal length and conformal modulus. The
definitions and properties discussed here are mostly well-known. One may refer
[Ahl10] and [LV73] for more comprehensive introductions.

A _closed annulus_ is a subset of $\mathbb{C}$ that is homeomorphic to
$\\{z\in\mathbb{C}:1\leq|z|\leq 2\\}$. An _(open) annulus_ is the interior of
a closed annulus. Given an annulus $A$, denote $\Gamma=\Gamma(A)$ as the set
of smooth simple closed curves in $A$ separating the two boundary components
of $A$. A real-valued Borel measurable function $f$ on $A$ is called
_admissible_ if $\int_{\gamma}fds\geq 1$ for all $\gamma\in\Gamma$. Here $ds$
denotes the element of arc length. The _(conformal) modulus_ of $A$ is defined
as

$\text{Mod}(A)=\inf\\{\int_{A}f^{2}:f\text{ is admissible}\\},$

where $\int_{A}f^{2}$ denotes the integral of $f(z)^{2}$ against the 2-dim
Lebesgue measure on $A$. From the definition it is straightforward to verify
that $\text{Mod}(A)$ is conformally invariant. Furthermore, if $f:A\rightarrow
A^{\prime}$ is a $K$-quasiconformal homeomorphism between two annuli, then

$\frac{1}{K}\cdot\text{Mod}(A)\leq\text{Mod}(A^{\prime})\leq{K}\cdot\text{Mod}(A).$

Given $0<r<r^{\prime}$, denote $A_{r,r^{\prime}}$ as the annulus
$\\{z\in\mathbb{C}:r<|z|<r^{\prime}\\}$. It is well-known that

$\text{Mod}(A_{r,r^{\prime}})=\frac{1}{2\pi}\log\frac{r^{\prime}}{r}.$

Intuitively, the conformal modulus measures the relative thickness of an
annulus. If an annulus $A$ in $\mathbb{C}\backslash\\{0\\}$ contains
$A_{r,r^{\prime}}$, then it is “thicker” than $A_{r,r^{\prime}}$ and one can
show that

$\text{Mod}(A)\geq\text{Mod}(A_{r,r^{\prime}})=\frac{1}{2\pi}\log\frac{r^{\prime}}{r}.$

On the other hand, we have that

######  Lemma 2.1.

Suppose $A\subseteq\mathbb{C}\backslash\\{0\\}$ is an annulus separating $0$
from the infinity. If $\text{Mod}(A)\geq 100$, then $A\supseteq A_{r,2r}$ for
some $r>0$.

###### Proof.

Deonte $B$ as the bounded component of $\mathbb{C}-A$, and $r=\max\\{|z|:z\in
B\\}$ and $R=\min\\{|z|:z\in\mathbb{(}B\cup A)^{c}\\}$. If $R\geq 2r$ we are
done. So we may assume $R<2r$.

Then $D_{2r}\cap\gamma\neq\emptyset$ for all $\gamma\in\Gamma(A)$. Let $f$ be
a function on $A$ such that $f(z)=1/r$ on $A\cap D_{3r}$ and $f(z)=0$ on
$A\backslash D_{3r}$. If $\gamma\in\Gamma$ and $\gamma\subseteq D_{3r}$,

$\int_{\gamma}fds=s(\gamma)\cdot\frac{1}{r}\geq
2\cdot\text{diam}(B)\cdot\frac{1}{r}=2r\cdot\frac{1}{r}>1.$

If $\gamma\in\Gamma$ and $\gamma\subsetneq D_{3r}$, then $\gamma$ is a
connected curve connecting $D_{2r}$ and $D_{3r}^{c}$ and

$\int_{\gamma}fds\geq
d(D_{2r},D_{3r}^{c})\cdot\frac{1}{r}=r\cdot\frac{1}{r}=1.$

So $f$ is admissible and

$\text{Mod}(A)\leq\int_{A}f^{2}=\frac{1}{r^{2}}\cdot\text{Area}(A\cap
D_{3r})\leq\frac{\pi(3r)^{2}}{r^{2}}=9\pi<100.$

This contradicts with our assumption. ∎

######  Remark 2.2.

To some extend, Lemma 2.1 is a consequence of Teichmüller’s result on extremal
annuli (see Theorem 4-7 in [Ahl10]). The constant $100$ is chosen for
convenience and should not be optimal.

###  2.2. Discrete Harmonic Functions

Given $V_{0}\subset V$ and $\eta\in\mathbb{R}^{E(V_{0})}_{>0}$, a discrete
function $f:V_{0}\rightarrow\mathbb{R}$, or equivalently a vector
$f\in\mathbb{R}^{V_{0}}$, is called _harmonic at_ $i\in int(V_{0})$ if

$\sum_{j:ij\in E}\eta_{ij}(f_{j}-f_{i})=0.$

The following result is well-known and easy to prove.

######  Proposition 2.3.

Suppose $V_{0}$ is a finite subset of $V$ and
$\eta_{ij}\in\mathbb{R}^{E(V_{0})}_{>0}$.

  1. (a)

If $f\in\mathbb{R}^{V_{0}}$ is harmonic at $i$ for all $i\in int(V_{0})$, then
for all $i\in V_{0}$

$|f_{i}|\leq\max_{j\in\partial V_{0}}|f_{j}|.$

  2. (b)

Given $g:\partial V_{0}\rightarrow\mathbb{R}$, there exists a unique function
$f:V_{0}\rightarrow\mathbb{R}$ such that

    1. (i)

$f_{i}=g_{i}$ on $\partial V_{0}$, and

    2. (ii)

$f$ is harmonic at any $i\in int(V_{0})$.

Furthermore, such a map $(\eta,g)\mapsto f$ is smooth from
$\mathbb{R}^{E(V_{0})}_{>0}\times\mathbb{R}^{\partial V_{0}}$ to
$\mathbb{R}^{V_{0}}$.

Given $\eta\in\mathbb{R}^{E}_{>0}$, $f\in\mathbb{R}^{V}$ is called _harmonic_
if it is harmonic at all points in $V$. It is well-known by Liouville’s
Theorem that any bounded smooth harmonic function on the plane is constant.
Here we have a discrete version of Liouville’s Theorem.

######  Theorem 2.4.

Suppose $\phi$ is a geodesic embedding of $T$ and $l(\phi)$ is uniformly
nondegenerate. Given $\eta\in\mathbb{R}^{E}_{>0}$ with
$|\eta|_{\infty}<\infty$, then any bounded harmonic function on $(T,\eta)$ is
constant.

The proof of Theorem 2.4 is postponed to Section 4.

###  2.3. Differential of the Curvature Map

The differential of $K_{i}(u)$ has the following elegant formula, first
proposed by Luo [Luo04].

######  Proposition 2.5 (Adapted from Theorem 2.1 in [Luo04]).

Suppose $T_{0}=(V_{0},E_{0},F_{0})$ is a 1-ring neighborhood of $i\in V$ and
$l\in\mathbb{R}^{E_{0}}$. Then $K_{i}=K_{i}(u)$ is a smooth function on an
open set in $\mathbb{R}^{V_{0}}$, and

$dK_{i}=\sum_{j:ij\in E}\eta_{ij}(du_{i}-du_{j}).$

where $\eta_{ij}=\eta_{ij}(u)$ is defined to be

(2.1)
$\eta_{ij}(u)=\frac{1}{2}\left(\cot\theta^{k}_{ij}(u)+\cot\theta^{k^{\prime}}_{ij}(u)\right),$

where $\triangle ijk,\triangle ijk^{\prime}$ are the two triangles in $F$
containing edge $ij$.

###  2.4. Maximum Principles

We need the following maximum principle.

######  Lemma 2.6.

Suppose $V_{0}$ is a finite subset of $V$, and $u*l,u^{\prime}*l$ are Delaunay
PL metrics on $T(V_{0})$. If $K_{i}(u)=K_{i}(u^{\prime})=0$ for all $i\in
int(V_{0})$, then for all $i\in V_{0}$

$|u_{i}^{\prime}-u_{i}|\leq\max_{j\in\partial V_{0}}|u_{j}^{\prime}-u_{j}|.$

Lemma 2.6 is a standard consequence the following local maximum principle,
which is adapted from Lemma 2.12 in [DGM22] (or Theorem 3.1 in [LSW20]).

######  Lemma 2.7.

Suppose $i\in V$ and $T_{0}=(V_{0},E_{0},F_{0})$ is the 1-ring neighborhood of
$i$ in $V$. Given $l\in\mathbb{R}^{E_{0}}$, if $u*l,u^{\prime}*l$ are two
Delaunay PL metrics on $T_{0}$ and $K_{i}(u)=K_{i}(u^{\prime})=0$, then

$u_{i}^{\prime}-u_{i}\leq\max_{j:ij\in E}(u_{j}^{\prime}-u_{j})$

and the equality holds if $(u_{i}^{\prime}-u_{i})=(u_{j}^{\prime}-u_{j})$ for
any neighbor $j$ of $i$.

######  Remark 2.8.

Lemma 2.12 in [DGM22] is a special case of our Lemma 2.7, where
$u_{i}=u_{i}^{\prime}=0$ is further assumed. However, by the scaling
invariance these two Lemmas are really equivalent.

###  2.5. Key Estimates on the Conformal Factors for Geodesic Embeddings

######  Lemma 2.9.

Suppose $\epsilon>0$ and $\phi,\psi$ are two geodesic embeddings of a
subcomplex $T_{0}=(V_{0},E_{0},F_{0})$ of $T$, such that

  1. (i)

$l(\psi)=u*l(\phi)$ for some $u\in\mathbb{R}^{V_{0}}$, and

  2. (ii)

the inner angles in both PL metrics $l(\phi)$ and $l(\psi)$ are at most
$\pi/2-\epsilon$.

Given $r,r^{\prime}>0$ and $i\in V$, if

$\phi(|T_{0}|)\subseteq D_{r}$

and

$\psi(i)\in D_{r^{\prime}/2}\subseteq D_{r^{\prime}}\subseteq\psi(|T_{0}|),$

then

$u_{i}\geq\log(r^{\prime}/r)-M$

for some constant $M=M(\epsilon)>0$.

##  3\. Proof of Theorem 1.2

Assume $l(\psi)=\bar{u}*l(\phi)$, and all the inner angles in
$l(\phi),l(\psi)$ are at most $\pi/2-\epsilon$ for a constant $\epsilon>0$. We
will first prove Theorem 1.2 assuming $\bar{u}:V\rightarrow\mathbb{R}$ is
bounded in Section 3.1, and then prove $\bar{u}$ is bounded in Section 3.2.

###  3.1. Proof of Theorem 1.2 Assuming the Boundedness of $\bar{u}$

Let us prove by contradiction and assume that $\bar{u}$ is not constant.
Without loss of generality, we can do a scaling and assume

$\inf_{i\in V}\bar{u}_{i}<0<\sup_{i\in V}\bar{u}_{i}$

and

$-\inf_{i\in V}\bar{u}_{i}=\sup_{i\in V}\bar{u}_{i}=|\bar{u}|_{\infty}.$

By a standard compactness argument, it is not difficult to see that there
exists a small constant
$\delta=\delta(\epsilon,\bar{u})\in(0,|\bar{u}|_{\infty})$ such that if
$|u|_{\infty}<2\delta$,

$\theta^{i}_{jk}(u)=\theta^{i}_{jk}(u,l(\phi))\geq\pi/2-\epsilon/2$

for all $\triangle ijk\in F$. Pick a sequence of increasing subsets $V_{n}$ of
$V$ such that $\cup_{n=1}^{\infty}V_{n}=V$. For each $n\in\mathbb{Z}_{>0}$, we
will construct a smooth $\mathbb{R}^{V_{n}}$-valued function
$u^{(n)}(t)=[u_{i}^{(n)}(t)]_{i\in V_{n}}$ on $(-2\delta,2\delta)$ such that

  1. (a)

$u^{(n)}(0)=0$, and

  2. (b)

$\dot{u}_{i}^{(n)}(t)=\bar{u}_{i}/|\bar{u}|_{\infty}$ if $i\in\partial V_{n}$,
and

  3. (c)

if $i\in\text{int}(V_{n})$ then

(3.1) $\sum_{j:ij\in
E}\eta_{ij}(u^{(n)}(t))(\dot{u}_{i}^{(n)}(t)-\dot{u}_{j}^{(n)}(t))=0$

where $\eta_{ij}(u)$ is defined for all $ij\in E(V_{n})$ as in equation (2.1).

The conditions (b) and (c) give an autonomous ODE system on

$\mathcal{U}_{n}=\\{u\in\mathbb{R}^{V_{n}}:|u|_{\infty}<2\delta\\}.$

Notice that $\eta_{ij}(u)>0$ if $u\in\mathcal{U}_{n}$. Then by part (b) of
Lemma 2.3, $\dot{u}^{(n)}(t)$ is smoothly determined by $u^{(n)}(t)$ on
$\mathcal{U}_{n}$. Given the initial condition $u^{(n)}(0)=0$, assume the
maximum existence interval for this ODE system on $\mathcal{U}_{n}$ is
$(t_{\min},t_{\max})$ where $t_{\min}\in[-\infty,0)$ and
$t_{\max}\in(0,\infty]$. By the maximum principle (part (a) in Lemma 2.3), for
all $i\in V_{n}$

$|\dot{u}^{(n)}|_{\infty}\leq\max_{j\in\partial
V_{n}}|\dot{u}_{j}^{(n)}|=\max_{j\in\partial
V_{n}}|\bar{u}_{j}|/|\bar{u}|_{\infty}\leq 1.$

So $|u^{(n)}(t)|_{\infty}\leq t\leq t_{\max}$ for all $t\in[0,t_{\max})$. By
the maximality of $t_{\max}$, $t_{\max}=\infty$ or

$|u^{(n)}(t)|_{\infty}\rightarrow 2\delta\quad\text{ as }\quad t\rightarrow
t_{\max}.$

So $t_{\max}\geq 2\delta$ and by a similar reason $t_{\min}\leq-2\delta$.
$u^{(n)}(t)$ is indeed well-defined on $(-2\delta,2\delta)$. By Proposition
2.5 and equation (3.1), $K_{i}(u^{(n)}(t))=0$ for all $i\in int(V_{n})$. Then
by Lemma 2.6, for all $i\in V_{n}$

(3.2) $\displaystyle|\bar{u}_{i}-u_{i}^{(n)}(\delta)|\leq\max_{j\in\partial
V_{n}}|\bar{u}_{j}-u_{j}^{(n)}(\delta)|=\max_{j\in\partial
V_{n}}\left(\bar{u}_{j}-\delta\cdot\frac{\bar{u}_{j}}{|\bar{u}|_{\infty}}\right)$
$\displaystyle\leq$
$\displaystyle(1-\frac{\delta}{|\bar{u}|_{\infty}})|\bar{u}|_{\infty}=|\bar{u}|_{\infty}-\delta.$

By picking a subsequence, we may assume that $u^{(n)}_{i}$ converge to
$u_{i}^{*}$ on $[0,\delta]$ uniformly for all $i\in V$. Then
$u^{*}=[u_{i}^{*}]_{i\in V}$ satisfies the following.

(a) $u^{*}_{i}(t)$ is 1-Lipschitz for all $i\in V$. As a consequence, for all
$i\in V$, $u_{i}^{*}(t)$ is differentiable at a.e. $t\in[0,\delta]$.

(b) For all $\triangle ijk\in F$,
$\theta^{i}_{jk}(u^{*}(t))\leq\frac{\pi}{2}-\frac{\epsilon}{2}$. As a
consequence $\theta^{i}_{jk}(u^{*}(t))\geq\epsilon$ for all $\triangle ijk\in
F$ and $\eta_{ij}(u^{*}(t))\leq 2\cot\epsilon$ for all $ij\in E$.

(c) For all $i\in V$, $K_{i}(u^{*}(t))=0$. As a consequence for a.e.
$t\in[0,\delta]$,

$0=\frac{d}{dt}K_{i}(u^{*}(t))=\sum_{j:ij\in
E}\eta_{ij}(u^{*}(t))(\dot{u}^{*}_{i}(t)-\dot{u}^{*}_{j}(t)),$

for all $i\in V$.

(d) By Theorem 2.4, $\dot{u}^{*}(t)$ is constant on $V$ for a.e.
$t\in[0,\delta]$. As a consequence $u_{i}^{*}(\delta)$ equals to a constant
$c$ independent on $i\in V$.

(f) By equation (3.2),

$|\bar{u}_{i}-c|=|\bar{u}_{i}-u^{*}_{i}(\delta)|\leq|\bar{u}|_{\infty}-\delta$

for all $i\in V$. As a consequence we get the following contradiction

$2|\bar{u}|_{\infty}=|\sup_{i\in V}\bar{u}_{i}-\inf_{i\in
V}\bar{u}_{i}|\leq|\sup_{i\in V}\bar{u}_{i}-c|+|\inf_{i\in
V}\bar{u}_{i}-c|\leq 2|\bar{u}|_{\infty}-2\delta.$

###  3.2. Boundedness of the Conformal Factor

Without loss of generality, we may assume that $\psi\circ\phi^{-1}$ is linear
on each triangle $\phi(\triangle ijk)$. Then $\psi\circ\phi^{-1}$ is
$K$-quasiconformal for some constant $K=K(\epsilon)>0$. We will prove the
boundedness of $\bar{u}$ by showing that for any $j,j^{\prime}\in V$,

$|\bar{u}_{j}-\bar{u}_{j^{\prime}}|\leq 2M+2\log C+\log C^{\prime}-\log 2,$

where $M=M(\epsilon)$ is the constant given in Lemma 2.9 and $C=C(\epsilon)$
is the constant given in Lemma 4.3 and
$C^{\prime}=C^{\prime}(\epsilon)=e^{200\pi K}$.

Assume $j,j^{\prime}\in V$. For convenience, let us assume $\phi(j)=\psi(j)=0$
by translations. Pick $r>0$ sufficiently large such that
$|\phi(j^{\prime})|<r/(2C)$ and $\phi(R_{j})\subseteq D_{r}$. Let
$V_{1}=\\{i\in V:\phi(i)\in D_{r}\\}$ and $V_{2}=\\{i\in V:\phi(i)\in
D_{CC^{\prime}r}\\}$ and $T_{1}=T(V_{1})$ and $T_{2}=T(V_{2})$. Then by Lemma
4.3 we have

(3.3) $\\{\phi(j),\phi(j^{\prime})\\}\subseteq D_{r/(2C)}\subseteq
D_{r/C}\subseteq\phi(|T_{1}|),$

and

$\phi(|T_{1}|)\subseteq D_{r}\subseteq D_{C^{\prime}r}\subseteq\phi(|T_{2}|)$

and

(3.4) $\phi(|T_{2}|)\subseteq D_{CC^{\prime}r}.$

So $A=A_{r,C^{\prime}r}$ separates $\phi(|T_{1}|)$ and $\phi(|T_{2}|)^{c}$,
and then $A^{\prime}=\psi\circ\phi^{-1}(A)\ni\psi(j)=0$ separates
$\psi(T_{1})$ and $\psi(T_{2})^{c}$. Furthermore

$\text{Mod}(A^{\prime})\geq\frac{1}{K}\cdot\text{Mod}(A)=\frac{1}{K}\cdot\frac{1}{2\pi}\log\frac{C^{\prime}r}{r}=100.$

Then by Lemma 2.1 there exists $r^{\prime}>0$ such that
$A_{r^{\prime},2r^{\prime}}\subseteq A^{\prime}$. So
$A_{r^{\prime},2r^{\prime}}$ separates $\psi(T_{1})$ and $\psi(T_{2})^{c}$ and
then

(3.5) $\psi(|T_{1}|)\subseteq D_{r^{\prime}}$

and

(3.6) $\\{\psi(j),\psi(j^{\prime})\\}\subseteq D_{r^{\prime}}\subseteq
D_{2r^{\prime}}\subseteq\psi(|T_{2}|).$

By Lemma 2.9 and equations (3.4) and (3.6), both
$\bar{u}_{j},\bar{u}_{j^{\prime}}$ are at least

$\log\frac{2r^{\prime}}{CC^{\prime}r}-M=\log\frac{r^{\prime}}{r}+\log\frac{2}{CC^{\prime}}-M.$

Again by Lemma 2.9 and equations (3.5) and (3.3), both $-\bar{u}_{j}$ and
$-\bar{u}_{j^{\prime}}$ are at least

$\log\frac{r/C}{r^{\prime}}-M=\log\frac{r}{r^{\prime}}-\log C-M.$

So both $\bar{u}_{j}$ and $\bar{u}_{j^{\prime}}$ are in the interval

$[\log\frac{r^{\prime}}{r}+\log\frac{2}{CC^{\prime}}-M,\log\frac{r^{\prime}}{r}+\log
C+M],$

and $|\bar{u}_{j}-\bar{u}_{j^{\prime}}|$ is bounded by the length of this
interval

$2M+\log C-\log\frac{2}{CC^{\prime}}=2M+2\log C+\log C^{\prime}-\log 2.$

##  4\. Discrete Extremal Length and the Discrete Liouville Theorem

###  4.1. Electrical Networks and Discrete Extremal Length

Discrete harmonic functions are closely related to the theory of electrical
networks. Here the 1-skeleton $(V,E)$ of the triangulation $T$ could be viewed
as an electrical network, and $\eta_{ij}$ denotes the conductance of the edge
$ij$, and the function $f$ denotes the electric potentials at the vertices.
Then $f$ is harmonic at $i$ if and only if the outward electric flux at $i$ is
$0$. The theory of electrical networks is closely related to discrete (edge)
extremal length, originally introduced by Duffin [Duf62]. Here we briefly
review the theory of discrete (edge) extremal length, adapted to our setting.
All the definitions and properties here are well-known and one may read
[Duf62][He99] for references.

Assume $V_{1},V_{2}$ are two nonempty disjoint subsets of $V$ such that
$V_{0}=(V_{1}\cup V_{2})^{c}$ is finite. A _path_ $p$ between $V_{1}$ and
$V_{2}$ is a finite set of edges in

$E_{0}=E_{0}(V_{1},V_{2})=\\{ij\in E:i\in V_{0}\text{ or }j\in V_{0}\\}$

such that $\gamma_{p}=\cup\\{e:e\in p\\}$ is a simple curve connecting $V_{1}$
and $V_{2}$. Denote $P=P(V_{1},V_{2})$ as the set of paths between $V_{1}$ and
$V_{2}$. A _cut_ $q$ between $V_{1}$ and $V_{2}$ is a finite set of edges in
$E_{0}$ such that $q$ separates $V_{1}$ and $V_{2}$, i.e., for any path $p\in
P$, $p\cap q\neq\emptyset$. Denote $Q=Q(V_{1},V_{2})$ as the set of cuts
between $V_{1}$ and $V_{2}$.

Given $\mu\in\mathbb{R}^{E}_{>0}$, the _discrete (edge) extremal length_
$EL=EL(V_{1},V_{2},\mu)$ is defined as

(4.1) $EL=\min\\{\sum_{e\in
E_{0}}\mu_{e}w_{e}^{2}:w\in\mathbb{R}^{E_{0}},\sum_{e\in q}w_{e}\geq 1\text{
for all }q\in Q\\},$

and the _discrete (edge) extremal width_ $EW=EW(V_{1},V_{2},\mu)$ is defined
as

$EW=\min\\{\sum_{e\in E_{0}}\mu_{e}w_{e}^{2}:w\in\mathbb{R}^{E_{0}},\sum_{e\in
p}\mu_{e}w_{e}\geq 1\text{ for all }p\in P\\}.$

Here $\mu_{e}$ should be viewed as the resistance of edge $e\in E$. Then the
conductance of edge $e\in E$ should be $\eta_{e}=1/\mu_{e}$. If
$f:V\rightarrow\mathbb{R}$ is harmonic on $V_{0}$ with respect to $\eta$ and
$f|_{V_{1}}=0$ and $f|_{V_{2}}=1$, then $w_{ij}=|f_{j}-f_{i}|/\mu_{ij}$ gives
the unique minimizer in the quadratic minimization problem in equation (4.1).
If we view such $f$ as an electric potential, then $w_{e}$ represents the
current on edge $e\in E_{0}$ and $EL=\sum_{e\in E_{0}}\mu_{e}w_{e}^{2}$ is the
electrical power in the network, which is equal to the _(equivalent)
resistance_ between $V_{1}$ and $V_{2}$. The discrete extremal length and
width satisfy the following reciprocal theorem.

######  Theorem 4.1 (Adapted from Corollary 1 in [Duf62]).

$EL(V_{1},V_{2},\mu)\cdot EW(V_{1},V_{2},\mu)=1$.

Now assume $\emptyset\neq V_{0},V_{1},V_{2}...$ is an increase sequence of
subsets of $V$ and $\cup_{k=0}^{\infty}V_{k}=V$. Then the electric network
$(T,\mu)$ is called _recurrent_ if

$EL(V_{0},V_{n}^{c},\mu)\rightarrow\infty$

as $n\rightarrow\infty$. The recurrency of the network does not depend on the
choice of $V_{n}$’s. Intuitively the recurrency means that the equivalent
resistance between a finite set and the infinity is infinite. Discrete
extremal length is a useful tool to prove the discrete Liouville theorem,
since the recurrency implies the discrete Liouville property.

######  Lemma 4.2 (Lemma 5.5 in [He99]).

Assume $(T,\mu)$ is recurrent, and let $\eta_{e}=1/\mu_{e}$ for all $e\in E$.
Then any bounded harmonic function on $(T,\eta)$ is bounded.

###  4.2. Proof of the Discrete Liouville Theorem

We need the following lemma for the proof.

######  Lemma 4.3.

Suppose $\phi:|T|\rightarrow\mathbb{R}^{2}$ is a geodesic homeomorphism and
any inner angle in $l(\phi)$ is at least $\epsilon>0$. Let $a\in V$ be a
vertex and assume $\phi(a)=0$. Given $r>0$, denote $V_{r}=\\{i\in
V:|\phi(i)|<r\\}$ and $T_{r}=T(V_{r})$. Then there exists a constant
$C=C(\epsilon)>0$ such that if $\phi(R_{a})\subseteq D_{r}$,

  1. (a)

$D_{r/C}\subseteq\phi(|T_{r}|),$ and

  2. (b)

as a consequence $|\phi(i)|\geq r/C$ for all $i\in\partial V_{r}$.

###### Proof.

By a standard compactness argument, it is not difficult to show that there
exists a constant $\delta=\delta(\epsilon)>0$ such that for all $\triangle
ijk\in F$,

$d(U_{ijk}^{c},\phi(\triangle ijk))\geq\delta\cdot\text{diam}(\phi(\triangle
ijk))$

where

$U_{ijk}=int(\phi(R_{i}))\cup int(\phi(R_{j}))\cup
int(\phi(R_{k}))\supseteq\phi(\triangle ijk).$

We claim that $C=1+2/\delta$ is a desired constant. Let us prove by
contradiction. Suppose $r>\max\\{|\phi(i)|:ai\in E\\}$ and
$D_{r/C}\not\subseteq\phi(|T_{r}|).$ Then there exists $z\in
D_{r/C}\backslash\phi(|T_{r}|)$. Since $\phi$ is a geodesic homeomorphism,
there exists a triangle $\triangle ijk\in F$ such that $z\in\phi(\triangle
ijk)$. Then $\triangle ijk$ is not a triangle in $T_{r}$ and we may assume
$i\notin V_{r}$. So $|\phi(i)|\geq r$ and $ai\notin E$ and $0=\phi(a)\notin
U_{ijk}$. Then

${r}/{C}\geq|0-z|\geq d(U_{ijk}^{c},\phi(\triangle
ijk))\geq\delta\cdot\text{diam}(\phi(\triangle ijk))$
$\geq\delta\cdot|\phi(i)-z|\geq\delta\cdot(r-r/C)=(r/C)\cdot\delta(C-1)=2r/C$

and we get a contradiction. ∎

###### Proof of Theorem 2.4.

By replacing $\eta$ by $\eta/|\eta|_{\infty}$ we may assume that
$|\eta|_{\infty}=1$. Assume $\mu\in\mathbb{R}^{E}$ is defined as
$\mu_{e}=1/\eta_{e}\geq 1$ for all $e\in E$. Then by Lemma 4.2 we only need to
show that $(T,\mu)$ is recurrent. Let
$\mathbf{1}=(1,1,...,1)\in\mathbb{R}^{E}$. Then by the definition (equation
(4.1)) $EL(V_{1},V_{2},\mu)\geq EL(V_{1},V_{2},\mathbf{1})$ whenever well-
defined. So we only need to show that $(T,\mathbf{1})$ is recurrent.

Suppose $a\in V$ is a vertex and without loss of generality we may assume that
$\phi(a)=0$. Let $\epsilon>0$ be the infimum of the inner angles in the PL
metric $l(\phi)$, and $C=C(\epsilon)>1$ be the constant given in Lemma 4.3.

Let $r_{0}=\max\\{|\phi(i)|:ai\in E\\}$ and $r_{n}=(2C)^{n}r_{0}$ and
$V_{n}=\\{i\in V:\phi(i)\in D_{r_{n}}\\}$ for all $n\in\mathbb{Z}_{\geq 0}$.
Clearly $V_{n}$ is an increasing sequence of subsets of $V$ and
$\cup_{n=1}^{\infty}V_{n}=V$. We will prove the recurrency of $(T,\mathbf{1})$
by showing that $EL(V_{0},V_{n}^{c},\mathbf{1})\rightarrow\infty$ as
$n\rightarrow\infty$.

By Lemma 4.3 (b), $|\phi(i)|\geq r_{n}/C=2r_{n-1}$ if $i\in\partial V_{n}\cup
V_{n}^{c}=\overline{V_{n}^{c}}$. So
$V_{n-1}\cap\overline{V_{n}^{c}}=\emptyset$, i.e.,
$V_{n-1}\subseteq(\overline{V_{n}^{c}})^{c}=int(V_{n})$. It is easy to see

(4.2) $E_{0}(V_{n-1},\overline{V_{n}^{c}})\subseteq E(V_{n})\backslash
E(V_{n-1}).$

From the definition of extremal length, we have

$EL(V_{0},V_{n}^{c})\geq
EL(V_{0},\overline{V_{1}^{c}})+EL(V_{1},\overline{V_{2}^{c}})+...+EL(V_{n-1},\overline{V_{n}^{c}})$

since

  1. (1)

$E_{0}(V_{0},\overline{V_{1}^{c}}),E_{0}(V_{1},\overline{V_{2}^{c}}),...,E_{0}(V_{n-1},\overline{V_{n}^{c}})$
are disjoint by equation (4.2), and

  2. (2)

$Q(V_{0},\overline{V_{1}^{c}}),Q(V_{1},\overline{V_{2}^{c}}),...Q(V_{n-1},\overline{V_{n}^{c}})$
are all subsets of $Q(V_{0},V_{n}^{c})$.

So it suffices to show that for all $n$,

$EL(V_{n-1},\overline{V_{n}^{c}},\mathbf{1})\geq\frac{\sin^{2}\epsilon}{12\pi
C^{2}},$

which by Theorem 4.1 is equivalent to

$EW(V_{n-1},\overline{V_{n}^{c}},\mathbf{1})\leq\frac{12\pi
C^{2}}{\sin^{2}\epsilon}.$

In the remaining of the proof we denote
$E_{0}=E_{0}(V_{n-1},\overline{V_{n}^{c}})$. Pick $w_{e}=l_{e}/r_{n-1}$, and
then for any $p\in P=P(V_{n-1},\overline{V_{n}^{c}})$,

$\sum_{e\in P}w_{e}=\frac{1}{r_{n-1}}\sum_{e\in
P}l_{e}\geq\frac{1}{r_{n-1}}\cdot
d(\phi(V_{n-1}),\phi(\overline{V_{n}^{c}}))\geq\frac{1}{r_{n-1}}\cdot(2r_{n-1}-r_{n-1})=1.$

So

$EW(V_{n-1},\overline{V_{n}^{c}},\mathbf{1})\leq\sum_{e\in
E_{0}}w_{e}^{2}=\frac{1}{r_{n-1}^{2}}\sum_{e\in E_{0}}l_{e}^{2}$

and it remains to show

$\sum_{e\in E_{0}}l_{e}^{2}\leq\frac{12\pi C^{2}}{\sin^{2}\epsilon}\cdot
r_{n-1}^{2}.$

Given $e\in E$, denote $\triangle_{e},\triangle_{e}^{\prime}$ as the two
triangles in $T$ containing $e$. If $e\in E_{0}$, then $e$ contains at least 1
vertex in $(\overline{V_{n}^{c}})^{c}=int(V_{n})$ and
$\triangle_{e},\triangle_{e}^{\prime}$ are both triangles in $T_{n}$, i.e.,
$\phi(\triangle_{e}),\phi(\triangle_{e}^{\prime})$ are both in $D_{r_{n}}$.
Given a triangle $\triangle\in F$, we denote $|\triangle|$ as the area of
$\phi(\triangle)$. Then by the sine law

$|\triangle
ijk|=\frac{1}{2}l_{ij}l_{jk}\sin\theta^{j}_{ik}\geq\frac{1}{2}l_{ij}^{2}\cdot\frac{\sin\theta^{i}_{jk}}{\sin\theta^{k}_{ij}}\cdot\sin\theta^{j}_{ik}\geq
l_{ij}^{2}\cdot\frac{\sin^{2}\epsilon}{2}.$

Notice that a triangle $\triangle\in F$ is counted for at most $3$ times in
$\sum_{e\in E_{0}}(|\triangle_{e}|+|\triangle_{e}^{\prime}|)$ and then

$\sum_{e\in E_{0}}l_{e}^{2}\leq\frac{1}{\sin^{2}\epsilon}\sum_{e\in
E_{0}}(|\triangle_{e}|+|\triangle_{e}^{\prime}|)\leq\frac{1}{\sin^{2}\epsilon}\sum_{\triangle:\phi(\triangle)\subseteq
D_{r_{n}}}3|\triangle|=\frac{3\pi r_{n}^{2}}{\sin^{2}\epsilon}=\frac{12\pi
C^{2}}{\sin^{2}\epsilon}\cdot r_{n-1}^{2}.$

∎

##  5\. Hyperbolic Maximum Principles and Proof of Lemma 2.9

Given $z_{1},z_{2}\in D$, we denote $d_{h}(z_{1},z_{2})$ as the hyperbolic
distance between $z_{1},z_{2}$ in the Poincaré disk model. The (Euclidean)
discrete conformal change is related with the hyperbolic discrete conformal
change as follows.

######  Lemma 5.1.

Suppose $z_{1},z_{2},z_{1}^{\prime},z_{2}^{\prime}\in D$ and
$u_{1},u_{2},u_{1}^{h},u_{2}^{h}\in\mathbb{R}$ are such that

$u_{i}^{h}=u_{i}+\log\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}$

for $i=1,2$. Then

$|z_{1}^{\prime}-z_{2}^{\prime}|=e^{\frac{1}{2}(u_{1}+u_{2})}|z_{1}-z_{2}|$

if and only if

(5.1)
$\sinh\frac{d_{h}(z_{i}^{\prime},z_{j}^{\prime})}{2}=e^{\frac{1}{2}(u_{i}^{h}+u_{j}^{h})}\sinh\frac{d_{h}(z_{i},z_{j})}{2}.$

######  Remark 5.2.

Equation (5.1) is indeed the formula of the discrete conformal change for
piecewise hyperbolic metric. This formula was first proposed by Bobenko-
Pinkall-Springborn [BPS15], and $u^{h}_{i}$ in the formula is called the
hyperbolic discrete conformal factor at $i$.

Lemma 5.1 could be verified by elementary computations. The proof is given in
Appendix. The hyperbolic discrete conformal factor $u^{h}$ also satisfies a
maximum principle.

######  Lemma 5.3.

Suppose $V_{0}$ is a subset of $V$ and $u\in\mathbb{R}^{V_{0}}$ and
$\phi,\psi$ are Euclidean geodesic embeddings of $T(V_{0})$, such that
$\phi(|T(V_{0})|),\psi(|T(V_{0})|)\subseteq D$ and $l(\phi),l(\psi)$ are both
uniformly acute and $l(\psi)=u*l(\phi)$. For all $i\in V_{0}$, denote
$z_{i}=\phi(i)$ and $z_{i}^{\prime}=\psi(i)$ and

$u_{i}^{h}=u_{i}+\log\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}.$

  1. (a)

If $i\in int(V_{0})$ and $u_{i}^{h}<0$, then there exists a neighbor $j$ of
$i$ such that

$u_{j}^{h}<u_{i}^{h}.$

  2. (b)

If $u_{i}^{h}\geq 0$ for all $i\in\partial V_{0}$, then $u_{i}^{h}\geq 0$ for
all $i\in V_{0}$.

We first prove Lemma 2.9 using the hyperbolic maximum princple and then prove
Lemma 5.3.

###### Proof of Lemma 2.9.

For any $\triangle ijk\in F$,

$e^{\frac{1}{2}(u_{j}-u_{i})}=\frac{e^{(u_{j}+u_{k})/2}}{e^{(u_{i}+u_{k})/2}}=\frac{l_{jk}(\psi)/l_{jk}(\phi)}{l_{ik}(\psi)/l_{ik}(\phi)}=\frac{l_{jk}(\psi)}{l_{ik}(\psi)}\cdot\frac{l_{ik}(\phi)}{l_{jk}(\phi)}\geq\sin^{2}\epsilon.$

So there exists a constant $C=C(\epsilon)>0$ such that $|u_{j}-u_{i}|\leq 2C$
for all $ij\in E$. We will show that $M(\epsilon)=C(\epsilon)+3$ is a
satisfactory constant. By a scaling, we only need to prove for the special
case where $r^{\prime}=1$ and $r=e^{-C-2}$.

Denote $V_{1}=\\{i\in V:\psi(i)\in D\\}$ and $z_{i}=\phi(i)$ and
$z_{i}^{\prime}=\psi(i)$. Define $u^{h}\in\mathbb{R}^{V_{1}}$ as

$u_{i}^{h}=u_{i}+\log\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}$

for all $i\in V_{1}$. Assume $i\in\partial V_{1}$, then there exists $j\in
V_{0}-V_{1}$ such that $ij\in E$. We claim that $u_{i}^{h}\geq 0$, i.e.,

$e^{u_{i}}\cdot\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}\geq 1.$

Notice that

$1-|z_{i}^{\prime}|\leq|z_{i}^{\prime}-z_{j}^{\prime}|=e^{\frac{1}{2}(u_{i}+u_{j})}|z_{i}-z_{j}|\leq
e^{u_{i}+C}\cdot 2r=2e^{-2}e^{u_{i}}.$

So

$e^{u_{i}}\cdot\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}\geq\frac{e^{2}}{2}\cdot\frac{1-|z_{i}|^{2}}{1+|z_{i}^{\prime}|}\geq\frac{e^{2}}{2}\cdot\frac{1-r^{2}}{2}\geq\frac{e^{2}}{2}\cdot\frac{1-(e^{-2})^{2}}{2}>1.$

By the hyperbolic maximum principle Lemma 5.3 (b), $u_{i}^{h}\geq 0$ for all
$i\in V_{1}$. Then for all $i\in V_{0}$ with $|z_{i}^{\prime}|<1/2$,

$u_{i}=u_{i}^{h}-\log\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}\geq-\log\frac{1-|z_{i}|^{2}}{1-|z_{i}^{\prime}|^{2}}\geq\log(1-|z_{i}^{\prime}|^{2})\geq-1=\log(r^{\prime}/r)-M.$

∎

###  5.1. Proof of the Hyperbolic Maximum Principle

For the proof of Lemma 5.3, we need to briefly review the notion of hyperbolic
Delaunay. Given a subcomplex $T_{0}$ of $T$, an embedding
$\phi:|T_{0}|\rightarrow D$ is called a _hyperbolic geodesic embedding_ if
$\phi_{h}$ maps each edge of $T_{1}$ to a hyperbolic geodesic arc in
$(D,d_{h})$. Given a triangle $\triangle{ijk}$ in $T_{0}$ and a Euclidean or
hyperbolic geodesic embedding $\phi$ of $T_{0}$, denote
$C_{ijk}=C_{ijk}(\phi)$ as the circumcircle of $\phi(\triangle ijk)$, i.e., a
round circle in the Riemann sphere $\hat{\mathbb{C}}$ passing through the
three vertices of $\phi(\triangle ijk)$. Furthermore, we denote
$D_{ijk}=D_{ijk}(\phi)$ as the circumdisk of $\phi(\triangle ijk)$, i.e., the
closed round disk in $\hat{\mathbb{C}}$ such that $\partial D_{ijk}=C_{ijk}$
and $\phi(\triangle ijk)\subseteq D_{ijk}$. For a Euclidean geodesic embedding
$\phi$ of $T_{0}$, it is well-known that $l(\phi)$ is Delaunay if and only if
that for any pair of adjacent triangles $\triangle ijk,\triangle ijk^{\prime}$
in $T_{0}$,

$\phi(k^{\prime})\notin int(D_{ijk}).$

So here we naturally call a Euclidean or hyperbolic geodesic embedding $\phi$
_Delaunay_ if

$\phi(k^{\prime})\notin int(D_{ijk})$

for any pair of adjacent triangles $\triangle ijk,\triangle ijk^{\prime}$ in
$T_{0}$.

###### Proof of Lemma 5.3 (a).

Assume $i\in int(V_{0})$ and $T_{1}=(V_{1},E_{1},F_{1})$ is the 1-ring
neighborhood of $i$. Then by Lemma 5.4 below there exists a hyperbolic
Delaunay geodesic embedding $\phi_{h}$ (_resp._ $\psi_{h}$) of $T_{1}$ such
that $\phi_{h}(j)=z_{j}$ (_resp._ $\psi_{h}(j)=z_{j}^{\prime}$) for all $j\in
V_{1}$. By Lemma 5.1,

$\sinh\frac{d_{h}(z_{j}^{\prime},z_{k}^{\prime})}{2}=e^{\frac{1}{2}(u_{j}^{h}+u_{k}^{h})}\sinh\frac{d_{h}(z_{j},z_{k})}{2}$

for all $jk\in E_{1}$. Suppose $f_{1},f_{2}:D\rightarrow D$ are hyperbolic
isometries such that $f_{1}(z_{i})=0$ and $f_{2}(z_{i}^{\prime})=0$. Then
$\tilde{\phi}_{h}=f_{1}\circ\phi_{h}$ (_resp._
$\tilde{\psi}_{h}=f_{2}\circ\psi_{h}$) is a hyperbolic Delaunay geodesic
embedding. Denote $\tilde{z}_{j}=\tilde{\phi}_{h}(j)$ (_resp._
$\tilde{z}_{j}^{\prime}=\tilde{\psi}_{h}(j)$) for all $j\in V_{1}$. Then
$z_{i}=z_{i}^{\prime}=0$ and

$\sinh\frac{d_{h}(\tilde{z}_{j}^{\prime},\tilde{z}_{k}^{\prime})}{2}=e^{\frac{1}{2}(u_{j}^{h}+u_{k}^{h})}\sinh\frac{d_{h}(\tilde{z}_{j},\tilde{z}_{k})}{2}$

for all $jk\in E_{1}$. It is not hard to see that there exists a Euclidean
Delaunay geodesic embedding $\tilde{\phi}$ (_resp._ $\tilde{\psi}$) of $T_{1}$
such that $\tilde{\phi}(j)=\tilde{z}_{j}$ (_resp._
$\tilde{\psi}(j)=\tilde{z}_{j}^{\prime}$). By Lemma 5.1
$l(\tilde{\psi})=\tilde{u}*l(\tilde{\phi})$ where

$\tilde{u}_{j}=u_{j}^{h}-\log\frac{1-|\tilde{z}_{j}|^{2}}{1-|\tilde{z}_{j}^{\prime}|^{2}}.$

By the Euclidean maximum principle Lemma 2.7,
$\tilde{u}_{j}\leq\tilde{u}_{i}<0$ for some neighbor $j$ of $i$. Then

$|\tilde{z}_{j}^{\prime}|=l_{ij}(\tilde{\psi})=e^{\frac{1}{2}(\tilde{u}_{i}+\tilde{u}_{j})}l_{ij}(\tilde{\phi})=e^{\frac{1}{2}(\tilde{u}_{i}+\tilde{u}_{j})}|\tilde{z}_{j}|<|\tilde{z}_{j}|$

and

$u_{j}^{h}=\tilde{u}_{j}+\log\frac{1-|\tilde{z}_{j}|^{2}}{1-|\tilde{z}_{j}^{\prime}|^{2}}<\tilde{u}_{j}\leq\tilde{u}_{i}=u_{i}^{h}-\frac{1-|\tilde{z}_{i}|^{2}}{1-|\tilde{z}_{i}^{\prime}|^{2}}=u_{i}^{h}.$

∎

###### Proof of Lemma 5.3 (b).

If not, assume $u_{i}^{h}=\min_{j:j\in V_{0}}u_{j}^{h}<0$ and then $i\in
int(V_{0})$. By the minimality of $u_{i}^{h}$, $u_{j}^{h}\geq u_{i}^{h}$ for
any neighbor $j$ of $i$. This contradicts with part (a). ∎

######  Lemma 5.4.

Suppose $i\in V$ and $T_{1}=(V_{1},E_{1},F_{1})$ is a 1-ring neighborhood of
$i$. If $\phi$ is a geodesic embedding of $T_{1}$ such that
$\phi(|T_{1}|)\subseteq D$ and $l(\phi)$ is uniformly acute, then there exists
a hyperbolic geodesic embedding $\phi_{h}$ of $T_{1}$ such that
$\phi_{h}(j)=\phi(j)$ for all $j\in V_{1}$. Furthermore, such $\phi_{h}$ is
Delaunay.

###### Proof.

Let $j_{1},j_{2},...,j_{m}$ be the neighbors of $i$ listed counterclockwise in
$\phi(|T_{1}|)$. Denote $z_{0}=\phi(i)$ and $z_{k}=\phi(j_{k})$ for
$k=1,...,m$. If $\gamma(t):[0,1]\rightarrow D$ is a smooth curve such that
$\gamma(0)=z_{0}$, then $\dot{\gamma}(t)$ could be viewed as not only a
complex number but also a vector in the tangent space $T_{z_{0}}D$ of
$(D,d_{h})$ at $z_{0}$. By this way we naturally identify $T_{z_{0}}D$ with
$\mathbb{C}$.

Given $z\in D$, let $v(z)=\exp_{z_{0}}^{-1}z\in T_{z_{0}}D=\mathbb{C}$ where
$\exp_{z_{0}}:T_{z_{0}}D\rightarrow D$ is the exponential map at $z_{0}$ on
the hyperbolic plane $D$. We first show that $v(z_{1}),...,v(z_{m})$ are
counterclockwise around 0 and wrap around $0$ once. More specifically, we will
show that

(5.2) $\arg\left(\frac{v({z_{k+1}})}{v({z_{k})}}\right)\in(0,\pi)$

and

(5.3) $\sum_{k=1}^{m}\arg\left(\frac{v({z_{k+1}})}{v({z_{k})}}\right)=2\pi$

where $z_{m+1}=z_{1}$ and $\arg(z)$ denotes the argument of $z$.

Assume $k\in\\{1,...,m\\}$. Denote $\gamma$ (_resp._ $\gamma_{h}$) as the
Euclidean straight line in $\mathbb{C}$ (hyperbolic geodesic in $D$)
containing $z_{0},z_{k}$. Then $\gamma$ (_resp._ $\gamma_{h}$) cuts
$\mathbb{C}$ (_resp._ D) into two open subsets $P,P^{\prime}$ (_resp._
$P_{h},P^{\prime}_{h}$). We may assume

$P=\\{z\in\mathbb{C}:\arg\left(\frac{z-z_{0}}{z_{k}-z_{0}}\right)\in(0,\pi)\\}$

and

$P_{h}=\\{z\in D:\arg\left(\frac{v(z)}{v(z_{k})}\right)\in(0,\pi)\\}.$

Then $z_{k+1}\in P$. If $\gamma_{h}$ is a straight line, $P_{h}=P\ni z_{k+1}$
and we have proved equation (5.2). If $\gamma_{h}$ is a round circular arc
orthogonal to $\\{|z|=1\\}$, there are two different cases.

Case 1: assume $z_{0},z_{k}$ are counterclockwise on $\gamma_{h}$ (see Figure
1 (A)). If $z_{k+1}\in P\backslash P_{h}$, $\angle z_{0}z_{k}z_{k+1}>\pi/2$ or
$\angle z_{k}z_{0}z_{k+1}>\pi/2$ and it is contradictory to the acuteness
assumption. So $z_{k+1}\in P_{h}$.

Case 2: assume $z_{0},z_{k}$ are clockwise on $\gamma_{h}$ (see Figure 1 (B)).
If $z_{k+1}\in P\backslash P_{h}$, $\angle z_{0}z_{k+1}z_{k}>\pi/2$ and it is
contradictory to the acuteness assumption. So $z_{k+1}\in P_{h}$.

(a) Case 1

(b) Case 2

Figure 1.

So we proved equation (5.2) and now prove equation (5.3). It is easy to see
that

$\arg\left(\frac{v(z_{k})}{z_{k}-z_{0}}\right)\in(-\frac{\pi}{2},\frac{\pi}{2}).$

for all $k=1,...,m$. We claim that

(5.4)
$\arg\left(\frac{v({z_{k+1}})}{v({z_{k})}}\right)+\arg\left(\frac{v(z_{k})}{z_{k}-z_{0}}\right)=\arg\left(\frac{z_{k+1}-z_{0}}{z_{k}-z_{0}}\right)+\arg\left(\frac{v(z_{k+1})}{z_{k+1}-z_{0}}\right).$

Since

$\exp(\sqrt{-1}\cdot LHS)=\exp(\sqrt{-1}\cdot
RHS)=\frac{v(z_{k+1})}{z_{k}-z_{0}},$

we have that

$LHS=RHS+2n\pi$

for some integer $n$. On the other hand $LHS$ and $RHS$ are both bounded in

$(0-\frac{\pi}{2},\pi+\frac{\pi}{2})=(-\frac{\pi}{2},\frac{3\pi}{2}),$

so $LHS=RHS$. Now by adding up equation (5.4) for $k=1,...,m$ we have that

$\sum_{k=1}^{m}\arg\left(\frac{v({z_{k+1}})}{v({z_{k})}}\right)=\sum_{k=1}^{m}\arg\left(\frac{z_{k+1}-z_{0}}{z_{k}-z_{0}}\right)=2\pi$

since $\phi$ is a geodesic embedding. So we proved equations (5.2) and (5.3),
and as a consequence there exists a hyperbolic embedding $\phi_{h}$ of
$|T_{1}|$ such that $\phi_{h}(j)=z_{j}$ for all $j\in V_{1}$.

By equation (5.2) it is not difficult to see that the two circumdisks
$D_{ij_{k}j_{k+1}}(\phi)$ and $D_{ij_{k}j_{k+1}}(\phi_{h})$ are the same for
$k=1,...,m$. So $\phi_{h}$ is Delaunay since $\phi$ is Delaunay.

∎

##  Appendix A Proof of Lemma 5.1

###### Proof of Lemma 5.1.

It suffices to show that for all $z_{1},z_{2}\in D$,

$\sinh\frac{d_{h}(z_{1},z_{2})}{2}=\frac{|z_{1}-z_{2}|}{\sqrt{(1-|z_{1}|^{2})(1-|z_{2}|^{2})}}.$

We first consider a special case where $z_{1}=0$ and $z_{2}=r\in(0,1)$ is
real. Then

$d_{h}(z_{1},z_{2})=\ln\frac{1\cdot(1+r)}{1\cdot(1-r)}=\ln\frac{1+r}{1-r}$

and

$\sinh\frac{d_{h}(z_{1},z_{2})}{2}=\frac{1}{2}\sqrt{\frac{1+r}{1-r}}-\frac{1}{2}\sqrt{\frac{1-r}{1+r}}=\frac{r}{\sqrt{1-r^{2}}}=\frac{|z_{1}-z_{2}|}{\sqrt{(1-|z_{1}|^{2})(1-|z_{2}|^{2})}}.$

For general $z_{1},z_{2}\in D$, we can find a hyperbolic isometric map
$f(z)=\frac{z-a}{1-\bar{a}z}$ such that $f(z_{1})=0$ and $f(z_{2})$ is a
positive real number. We only need to verify that

$\frac{|f(z_{1})-f(z_{2})|^{2}}{{(1-|f(z_{1})|^{2})(1-|f(z_{2})|^{2})}}=\frac{|z_{1}-z_{2}|^{2}}{{(1-|z_{1}|^{2})(1-|z_{2}|^{2})}}.$

This equality can be derived from

$\displaystyle\frac{|f(z_{1})-f(z_{2})|^{2}}{{(1-|f(z_{1})|^{2})(1-|f(z_{2})|^{2})}}$
$\displaystyle=$
$\displaystyle\frac{\big{|}(z_{1}-a)(1-\bar{a}z_{2})-(1-\bar{a}z_{1})(z_{2}-a)\big{|}^{2}}{\big{(}|1-\bar{a}z_{1}|^{2}-|z_{1}-a|^{2}\big{)}\cdot\big{(}|1-\bar{a}z_{2}|^{2}-|z_{2}-a|^{2}\big{)}}$
$\displaystyle=$
$\displaystyle\frac{\big{|}(1-a\bar{a})(z_{1}-z_{2})\big{|}^{2}}{\big{(}|1-\bar{a}z_{1}|^{2}-|z_{1}-a|^{2}\big{)}\cdot\big{(}|1-\bar{a}z_{2}|^{2}-|z_{2}-a|^{2}\big{)}}$

and

$\displaystyle|1-\bar{a}z_{1}|^{2}-|z_{1}-a|^{2}=(1-\bar{a}z_{1})(1-a\bar{z}_{1})-(z_{1}-a)(\bar{z}_{1}-\bar{a})$
$\displaystyle=$
$\displaystyle(1-a\bar{a})(1-z_{1}\bar{z}_{1})=(1-a\bar{a})(1-|z_{1}|^{2}).$

and similarly

$|1-\bar{a}z_{2}|^{2}-|z_{2}-a|^{2}=(1-a\bar{a})(1-|z_{2}|^{2}).$

∎

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# Quantum revivals in HgTe/CdTe quantum wells and topological phase
transitions

Alberto Mayorgas<EMAIL_ADDRESS>Department of Applied Mathematics,
University of Granada, Fuentenueva s/n, 18071 Granada, Spain  Manuel Calixto
Department of Applied Mathematics, University of Granada, Fuentenueva s/n,
18071 Granada, Spain  Institute Carlos I for Theoretical for Theoretical and
Computational Physics (iC1), Fuentenueva s/n, 18071 Granada, Spain  Nicolás A.
Cordero  Department of Physics, University of Burgos, 09001 Burgos, Spain
International Research Center in Critical Raw Materials for Advanced
Industrial Technologies (ICCRAM), University of Burgos, 09001 Burgos, Spain
Institute Carlos I for Theoretical for Theoretical and Computational Physics
(iC1), Fuentenueva s/n, 18071 Granada, Spain  Elvira Romera  Department of
Atomic, Molecular and Nuclear Physics, University of Granada, Fuentenueva s/n,
18071 Granada, Spain  Institute Carlos I for Theoretical for Theoretical and
Computational Physics (iC1), Fuentenueva s/n, 18071 Granada, Spain  Octavio
Castaños  Institute of Nuclear Sciences, National Autonomous University of
Mexico, Apdo. Postal 70-543, 04510, CDMX, Mexico

###### Abstract

The time evolution of a wave packet is a tool to detect topological phase
transitions in two-dimensional Dirac materials, such as graphene and silicene.
Here we extend the analysis to HgTe/CdTe quantum wells and study the evolution
of their electron current wave packet, using 2D effective Dirac Hamiltonians
and different layer thicknesses. We show that the two different periodicities
that appear in this temporal evolution reach a minimum near the critical
thickness, where the system goes from normal to inverted regime. Moreover, the
maximum of the electron current amplitude changes with the layer thickness,
identifying that current maxima reach their higher value at the critical
thickness. Thus, we can characterize the topological phase transitions in
terms of the periodicity and amplitude of the electron currents.

###### pacs:

03.65.Vf, 03.65.Pm,

##  I Introduction

The time evolution of wave packets can have interesting behaviors due to
quantum interference. Revivals occur when a well-localized wave-packet evolves
in time to recover, at least approximately, its initial waveform. This event
occurs periodically and the period is known as the revival period. The
phenomenon of quantum wave packet revivals has been investigated theoretically
in atomic systems, molecules, many body systems or 2D Materials [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11] and observed experimentally in among others, Rydberg
atoms or molecular systems [12, 13, 14, 15, 16]. Recently, it has been shown
how revival and classical periods reveal quantum phase transitions in many-
body systems [5, 6]. Furthermore, it has also been seen how both periods are
capable of detecting topological phase transitions (TPTs for short) in two-
dimensional materials such as graphene [4] and silicene [7].

In this work, we focus on a particular zincblende heterostructure, the mercury
telluride-cadmium telluride (HgTe/CdTe) quantum wells (QWs). They have been
widely used to study the quantum spin Hall effect and new types of topological
phases [17, 18, 19, 20], and traditionally are part of optical and transport
experiments involving spin-related observations [21, 22, 23]. At present,
HgTe/CdTe QWs appear together with other topological insulators to construct
low-dimensional quantum devices, which can experimentally realize quantum
anomalous Hall effects [24, 25, 26, 27, 28]. One of the most interesting
properties of these materials is that we can switch between normal or inverted
band structures by simply changing the QW width (layer thickness in our
jargon). In particular, we study the time evolution of electron current wave
packets in HgTe/CdTe QWs in magnetic fields, for different values of the HgTe
layer thickness to characterize TPTs. We analyze the periodicities in the
dynamics of the wave packets and the amplitude of the electron currents. There
are other ways to detect topological-band insulator phase transitions, such as
information or entropic measures [29, 30, 31, 32, 33], or magneto-optical
properties [34, 35, 36, 37, 38, 39]. In contrast to these methods, quantum
revivals provide an straightforward approach to TPTs that has not been applied
to HgTe/CdTe QWs so far.

This paper is organized as follows. In the next section we will describe the
2D effective Dirac Hamiltonian for surface states in HgTe/CdTe QWs. In the
third section we will study the relation between wave packet revivals and
classical periodicities with topological phase transition. The relation
between the evolution of the electron currents and the TPTs will be described
in section IV. Finally we will present some conclusions.

##  II $\text{HgTe}/\text{CdTe}$ quantum wells low-energy Hamiltonian

We shall use a 2D effective Dirac Hamiltonian to describe the surface states
in HgTe/CdTe QWs, following the prescription of the references [17, 18, 19,
20],

$H=\left(\begin{array}[]{cc}H_{+1}&0\\\
0&H_{-1}\end{array}\right),\,H_{s}(\bm{k})=\epsilon_{0}(\bm{k})\tau_{0}+\bm{d}_{s}(\bm{k})\cdot\bm{\tau},$
(1)

where $\tau_{i}$ are the Pauli matrices, $s=\pm 1$ is the spin and
$H_{-1}(\bm{k})=H_{+1}^{*}(-\bm{k})$ (temporarily reversed). It is convenient
to expand the Hamiltonian $H_{s}(\bm{k})$ around the center $\Gamma$ of the
first Brillouin zone [18],

$\epsilon_{0}(\bm{k})=\gamma-\delta\bm{k}^{2},\quad\bm{d}_{s}(\bm{k})=(\alpha
sk_{x},\alpha k_{y},\mu-\beta\bm{k}^{2}),$ (2)

where $\alpha,\beta,\gamma,\delta$ and $\mu$ are expansion parameters that
depend on the HgTe layer thickness $\lambda$, as can be found in [40] and in
Table 1. Among all these parameters, we highlight the mass or gap term $\mu$
related to the magnetic moment, and the Wilson term $\beta\bm{k}^{2}$
(introduced to avoid the Fermion doubling problem [41]).

$\lambda$(nm) |  $\alpha$(meV.nm) |  $\beta$(meV.nm2) |  $\delta$(meV.nm2) |  $\mu$(meV)
---|---|---|---|---
5.5 | 387 | -480 | -306 | 9
6.1 | 378 | -553 | -378 | -0.15
7.0 | 365 | -686 | -512 | -10
Table 1: Different values of the HgTe/CdTe QW expansion parameters depending
on the layer thicknesses $\lambda$ [40].

For $s=\pm 1$, the energy of the valence and conduction bands is

$\epsilon_{\pm}(\bm{k})=\epsilon_{0}(\bm{k})\pm\sqrt{\alpha^{2}\bm{k}^{2}+(\mu-\beta\bm{k}^{2})^{2}}.$
(3)

To differentiate between band insulator and topological insulator phases, one
can use the Thouless-Kohmoto-Nightingale-Nijs (TKNN) formula [42] providing
the Chern-Pontryagin number $\mathcal{C}$. In the case of the HgTe QWs (see
[39] for more details),

$\mathcal{C}_{s}=s[\mathrm{sign}(\mu)+\mathrm{sign}(\beta)].$ (4)

The Chern number depends on the sign of the material parameters $\mu$ and
$\beta$, and on the spin $s$. Considering Table 1, only $\mu$ changes sign for
different layer thicknesses $\lambda$, thus, the TPT is governed by
$\mathrm{sign}(\mu)$, or by $\mathrm{sign}(\mu/\beta)$ as can be found in the
literature [40]. Namely, around the critical HgTe layer thickness
$\lambda_{\textrm{c}}\approx 6.1$ nm, the system goes from normal
($\lambda<\lambda_{\textrm{c}}$ or $\mu/\beta<0$) to the inverted
($\lambda>\lambda_{\textrm{c}}$ or $\mu/\beta>0$) regimes.

We introduce the interaction with a perpendicular magnetic field $B$ along the
$z$-axis using minimal coupling $\bm{p}\to~{}\bm{P}=\bm{p}+e\bm{A}$, where
$\bm{A}=(A_{x},A_{y})=(-By,0)$ is the electromagnetic potential in the Landau
gauge, $e$ the electron charge, and $\bm{p}$ the momentum operator
($\bm{k}\to\bm{p}/\hbar$). Using Peierls’ substitution [43, 44], the
Hamiltonian (1) is written in terms of creation $a^{\dagger}$ and annihilation
$a$ operators [39],

$\displaystyle H_{+1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\gamma+\mu-\frac{(\delta+\beta)(2N+1)}{\ell_{B}^{2}}&\frac{\sqrt{2}\alpha}{\ell_{B}}a\\\
\frac{\sqrt{2}\alpha}{\ell_{B}}a^{\dagger}&\gamma-\mu-\frac{(\delta-\beta)(2N+1)}{\ell_{B}^{2}}\end{array}\right),$
(7) $\displaystyle H_{-1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\gamma+\mu-\frac{(\delta+\beta)(2N+1)}{\ell_{B}^{2}}&-\frac{\sqrt{2}\alpha}{\ell_{B}}a^{\dagger}\\\
-\frac{\sqrt{2}\alpha}{\ell_{B}}a&\gamma-\mu-\frac{(\delta-\beta)(2N+1)}{\ell_{B}^{2}}\end{array}\right),$
(10)

with $N=a^{\dagger}a$ and $\ell_{B}=\sqrt{\hbar/(eB)}$ the magnetic length.

Figure 1:  HgTe/CdTe quantum well low-energy spectrum $E_{n}^{s}$ for $B=0.05$
T, as a function of the HgTe layer thickness $\lambda$. The thin solid lines
represent Landau levels $n=\pm 1,\pm 2,\pm 3$ (valence $(-)$ and conduction
$(+)$) for spin $s=-1$ (blue) and $s=+1$ (red), and the thick lines represent
edge states ($n=0$). A vertical dashed black line indicates the HgTe thickness
$\lambda_{\mathrm{inv}}(0.05)=6.173$ nm $\simeq\lambda_{\textrm{c}}$ where the
band inversion for edge states occurs for $B=0.05$ T according to (28).

The eigenvalues of both Hamiltonians $H_{+1}$ and $H_{-1}$ are

$E_{n}^{s}=\gamma-\tfrac{2\delta|n|-s\beta}{\ell_{B}^{2}}+\mathrm{sgn}(n)\Delta_{n}^{s}$
(12)

with

$\Delta_{n}^{s}=\sqrt{\tfrac{2\alpha^{2}|n|}{\ell_{B}^{2}}+\left(\mu-\tfrac{2\beta|n|-s\delta}{\ell_{B}^{2}}\right)^{2}}\,,$
(13)

for Landau level (LL) index $n=\pm 1,\pm 2,\pm 3,\dots$ [valence $(-)$ and
conduction $(+)$] , and

$E_{0}^{s}=\gamma-s\mu-\frac{\delta-s\beta}{\ell_{B}^{2}}\,,$ (14)

for the edge states $n=0$ [45, 46, 34]. The associated eigenvectors are
spinors containing Fock states $||n|\rangle$, that is,

$|\bm{n}\rangle_{s}=\left(\begin{array}[]{l}A_{n}^{s}\left||n|-\frac{s+1}{2}\right>\\\
B_{n}^{s}\left||n|+\frac{s-1}{2}\right>\end{array}\right),$ (15)

with coefficients

$\displaystyle A_{n}^{s}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\frac{\mathrm{sgn}(n)}{\sqrt{2}}\sqrt{1+\mathrm{sgn}(n)\cos\theta_{n}^{s}},\quad
n\neq 0,\\\ (1-s)/2,\quad n=0,\end{array}\right.$ (18) $\displaystyle
B_{n}^{s}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\frac{s}{\sqrt{2}}\sqrt{1-\mathrm{sgn}(n)\cos\theta_{n}^{s}},\quad
n\neq 0,\\\ (1+s)/2,\quad n=0,\end{array}\right.$ (23)

where

$\theta_{n}^{s}=\arctan\left(\frac{\sqrt{2|n|}\,\alpha/\ell_{B}}{\mu-\frac{2\beta|n|-s\delta}{\ell_{B}^{2}}}\right).$
(24)

Depending on $\mathrm{sgn}(n)$, the coefficients $A_{n}^{s}$ and $B_{n}^{s}$
can be written as $\cos(\theta_{n}^{s}/2)$ and $\sin(\theta_{n}^{s}/2)$ [47].

The two zero Landau levels $E_{0}^{+1}$ and $E_{0}^{-1}$ belong to different
Hamiltonians, that is to spin $s=+1$ and $s=-1$ respectively. The level cross
condition

$E_{0}^{+1}=E_{0}^{-1}\Rightarrow B_{\text{inv}}=\frac{\mu}{e\beta/\hbar},$
(25)

gives the critical magnetic field $B_{\text{inv}}$ which separates the quantum
spin Hall and quantum Hall regimes [46]. For instance, for a QW thickness
$\lambda=7.0$ nm (see Table 1), one obtains $B_{\mathrm{inv}}\simeq 9.60$ T.
This band inversion is also graphically represented in Figure 1.

It is convenient to perform a linear fit of the parameters in Table 1, in
order to analyze the HgTe QWs spectrum and properties for a continuous range
of the thicknesses $\lambda$,

$\displaystyle\mu(\lambda)$ $\displaystyle=$ $\displaystyle
77.31\,-12.53\lambda,$ $\displaystyle\alpha(\lambda)$ $\displaystyle=$
$\displaystyle 467.49-14.65\lambda,$ $\displaystyle\beta(\lambda)$
$\displaystyle=$ $\displaystyle 283.58-138.16\lambda,$
$\displaystyle\delta(\lambda)$ $\displaystyle=$ $\displaystyle
458.46-138.25\lambda,$ (26)

where we use the Table 1 units and $\lambda$ is in nanometers. The coefficient
of determination is $R^{2}>0.99$ in all cases. Using $\mu(\lambda)$ in (II),
we can estimate the critical HgTe thickness where the TPT occurs in the
absence of magnetic field, according to the criteria in eq.(4),

$\mu=0\Rightarrow\lambda_{\textrm{c}}=6.17~{}\mathrm{nm}.$ (27)

In addition, the linear fit (II) let us plot the low energy spectra (12,14) as
a function of the HgTe layer thickness $\lambda$. Namely, in Figure 1, we
extrapolate the linear fit (II) to the interval [4 nm, 8 nm]. The band
inversion formula (25) together with the linear fit (II) yield the relation

$\lambda_{\mathrm{inv}}(B)=\frac{368.31-2.05B}{59.7-B}$ (28)

between the applied magnetic field $B$ (in Tesla) and the HgTe layer thickness
$\lambda_{\mathrm{inv}}(B)$ (in nanometers) at which the band inversion
$E_{0}^{+1}=E_{0}^{-1}$ takes place. Note that
$\lambda_{\mathrm{inv}}(B)\simeq\lambda_{\textrm{c}}=6.17$ nm for low $B\ll 1$
T, and that $E_{0}^{+1}=E_{0}^{-1}\simeq 0$ meV at this point as shows Figure
1. The thickness $\lambda_{\mathrm{inv}}(B)$, where the band inversion
happens, is a deviation of the critical thickness $\lambda_{\textrm{c}}$ in
eq.(27) for $B>0$, so it is a way to characterize TPTs when the external
magnetic field is non-null.

Higher Landau levels with $|n|>0$ have a structural change when the spinor
components in eq.(LABEL:coefHg) have equal module $|A_{n}^{s}|=|B_{n}^{s}|$,
that is, when the angle (24) is $\theta_{n}^{s}=\pi/2$, which implies
$\mu=(2\beta|n|-s\delta)/\ell_{B}^{2}$. The valence and conduction band
contributions interchange their roles at this point, hence this is a way to
define a band inversion for higher Landau levels, or equivalently, we can
introduce the concept higher Landau level topological phase transition (HTPT,
see [47] for more details). The condition $\theta_{n}^{s}=\pi/2$ fixes a
relationship between the layer thickness and the magnetic field as it happens
in eq.(28),

$\lambda_{\textrm{HTPT}}(B,n,s)=\frac{77.31-0.86B|n|+0.7Bs}{12.53-0.42B|n|+0.21Bs}\,\quad
n\neq 0\,.$ (29)

This layer thickness is higher than $\lambda_{\text{inv}}(B)$ in eq.(28) for
all $B,\>n$ and $s$, and for low magnetic fields tends to
$\lambda_{\textrm{HTPT}}(B<<1,n,s)\simeq\lambda_{\mathrm{c}}$. In Ref.[47] it
has been shown that quantum fluctuations and entanglement in higher Landau
levels grow at the layer thickness $\lambda_{\textrm{HTPT}}$. The scope of the
next section is to use the periodicities in the wave packet evolution as TPT
and HTPT markers, and compare the critical thicknesses of this method with the
ones in eq.(29).

##  III Classical and revival times in the topological phase transitions
detection

a

Figure 2: Classical period $T_{\text{Cl}}$ (top, eq.(33)) and revival time
$T_{\text{R}}$ (bottom, eq.(34)) as a function of the layer thickness
$\lambda$, for three different initial wave packets $n_{0}=5,10,15$. In both
figures, we set $B=0.05$ T and $s=+1$, and log-lin scale. The vertical dashed
line indicates the critical thickness $\lambda_{\textrm{c}}~{}=~{}6.17~{}$nm.

a

Figure 3: Layer thicknesses $\lambda_{\textrm{Cl}}(B,n)$ and
$\lambda_{\textrm{R}}(B,n)$ in which $T_{\textrm{Cl}}$ and $T_{\textrm{R}}$
achieve their minimum value respectively, as a function of the external
magnetic field $B$ and for different initial wave packets $n_{0}\in[0,30]$. In
both figures, we set $s=+1$ and a horizontal dashed line indicating the
critical thickness $\lambda_{\textrm{c}}=6.17~{}$nm. Figure 4: Layer thickness
$\lambda_{\textrm{Cl}}(B,n)$ in which $T_{\textrm{Cl}}$ achieve its minimum
value (solid lines) and layer thickness of HTPT (dashed lines, see eq.(29)),
as functions of the external magnetic field $B$ and for different initial wave
packets $n_{0}\in[1,10]$. In both figures, we set $s=+1$ and we mark the
critical thickness $\lambda_{\textrm{c}}=6.17~{}$nm in the vertical axis.

The time evolution of a wave packet for the time-independent Hamiltonian of
the HgTe QW (see eq.(LABEL:hamHgTe)) is given by

$|\Psi(t)\rangle_{s}=\sum_{n=-\infty}^{\infty}c_{n}^{s}|\bm{n}\rangle_{s}e^{-iE_{n}^{s}t/\hbar}\,,$
(30)

where $|n\rangle_{s}$ are the eigenvectors in (15), $E_{n}^{s}$ the energies
in (12), and $c_{n}^{s}={}_{s}\langle n|\Psi(0)\rangle$ with $|\Psi(0)\rangle$
the initial wave packet. For the sake of simplicity, we shall take $s$ fixed,
and $|\Psi(t)\rangle_{s}$, $E_{n}^{s}$, $c_{n}^{s}$ and
$\lambda_{\text{HTPT}}(B,n,s)$ will be referred to as $|\Psi(t)\rangle$,
$E_{n}$, $c_{n}$ and $\lambda_{\text{HTPT}}(B,n)$. We also select a Gaussian-
like initial wave packet, distributed around a given energy $E_{n_{0}}$ of the
spectrum $E_{n}$, so that

$c_{n}=\frac{1}{\sigma\sqrt{2\pi}}e^{-(n-n_{0})^{2}/2\sigma^{2}}\,,$ (31)

and we can Taylor expand the energy $E_{n}$ around the energy level $n_{0}$
[3]. Therefore, the exponential $\exp(-iE_{n}^{s}t/\hbar)$ in (30) yields

$\displaystyle\exp\left(-iE_{n_{0}}\frac{t}{\hbar}-2\pi
i(n-n_{0})\frac{t}{T_{\text{Cl}}}-2\pi
i(n-n_{0})^{2}\frac{t}{T_{\text{R}}}+\ldots\right)\,$ (32)

obtaining different time scales characterized by the classical period
$T_{\text{Cl}}=2\pi\hbar/|E^{\prime}_{n_{0}}|$ and the revival time
$T_{\text{R}}=4\pi\hbar/|E^{\prime\prime}_{n_{0}}|$ up to second order in the
series (the first term $\exp(iE_{n_{0}}t/\hbar)$ becomes an irrelevant global
phase in eq.(30)). In fact, the classical period is the time that the wave
packet needs to follow the expected semiclassical trajectory, and the revival
time is the time that the wave packet needs to return approximately to its
initial shape [3]. Quantum revivals are a consequence of the quantum beats
[48], representing interference effects of the terms in (30). Notice that
$T_{\text{Cl}}<<T_{\text{R}}$, and thus, a signal can be analyzed in these
different regimes. Both periods have been previously studied in 2D gapped
Dirac materials [4], and now we shall put the spotlight on HgTe QWs. In
particular, for the energies in (12), the classical and revival periods are

$\displaystyle T_{\text{Cl}}=$
$\displaystyle\,2\pi\hbar\ell_{B}^{2}\left[-2\mathrm{sgn}(n_{0})\delta+\tfrac{1}{\Delta_{n_{0}}^{s}}(\alpha^{2}-2\beta\chi_{n_{0}}^{s})\right]^{-1}\,,$
(33) $\displaystyle T_{\text{R}}=$
$\displaystyle\,4\pi\hbar\ell_{B}^{4}\,\mathrm{sgn}(n_{0})\left[-\tfrac{4\beta^{2}}{\Delta_{n_{0}}^{s}}+\tfrac{1}{(\Delta_{n_{0}}^{s})^{3}}(\alpha^{2}-2\beta\chi_{n_{0}}^{s})^{2}\right]^{-1}\,,$
(34)

where $\chi_{n}^{s}=\mu-(2\beta|n|-s\delta)/\ell_{B}^{2}$ is the denominator
in (24) and $\Delta_{n}^{s}$ is defined in eq.(13). Both periods are functions
of the magnetic field $B$, the wave packet center $n_{0}$, the spin (fixed to
$s=+1$ in this section), and the layer thickness $\lambda$ through the
parameters in equation (II).

The wave packets time evolution is visualized with the autocorrelation
function $A(t)=\langle\Psi(t)|\Psi(0)\rangle$, which turns into

$A(t)=\sum_{n=-\infty}^{\infty}|c_{n}|^{2}e^{-iE_{n}t/\hbar}$ (35)

for a Gaussian wave packet like the one in eq.(30).

a

Figure 5: Autocorrelation function amplitude $|A(t)|^{2}$ as a function of
time in $T_{\text{Cl}}=3.75$ ps (top) and $T_{\text{R}}=79.80$ ps (bottom)
units, for an initial wave packet with $n_{0}=5$ and
$\sigma=\sqrt{n_{0}}/5=0.47$. We set the HgTe parameters
$\lambda=\lambda_{\textrm{c}}$, $B=0.05$ T, and $s=+1$.

Throughout the article, we have selected an initial wave packet localized
around $n_{0}=5$ and with standard deviation $\sigma=\sqrt{n_{0}}/5\simeq
0.45$ in order to analyze the wave packet evolution in (30). We have also set
an external magnetic field of $B=0.05$ T in order to observe TPT phenomena
around $\lambda_{\text{inv}}(B)\simeq\lambda_{\textrm{c}}=6.17$ nm [39]. The
last restriction will be more evident later when we present Figures 3 and 9.

In Figure 2, we plot $T_{\text{Cl}}$ and $T_{\text{R}}$ as a function of the
layer thickness $\lambda$ for spin $s=+1$. Both periods reach a minimum near
the critical thickness $\lambda_{\textrm{c}}=6.17$ nm, hence they are useful
magnitudes to identify TPTs. These minima separate from $\lambda_{c}$ for
larger values of the magnetic field $B$ as shows Figure 3, where we present
the values of the thickness $\lambda_{\textrm{Cl}}(B,n_{0})$ and
$\lambda_{\textrm{R}}(B,n_{0})$ in which $T_{\textrm{Cl}}$ and
$T_{\textrm{R}}$ achieve their minimum value respectively, as a function of
the external magnetic field $B$ and the center of the wave packet $n_{0}$
(spin $s=+1$ fixed). For instance, for $n_{0}=5$ we obtain the bounds
$|\lambda_{\textrm{Cl}}(B,n_{0})-\lambda_{\textrm{c}}|<0.14$ nm and
$|\lambda_{\textrm{R}}(B,n_{0})-\lambda_{\textrm{c}}|<0.25$ nm in a magnetic
field range $B\leq 0.5$ T. For wave packets centered in high-energy states
(blue lines in Figure 3), the deviation from $\lambda_{\textrm{c}}$ is even
higher, representing a criticality of the system which differs from the band-
insulator phase transition. In order to characterize the criticality of
$T_{\text{Cl}}$ for magnetic fields $B>0.5$ T, in Figure 4 we compare the
minimum thicknesses $\lambda_{\textrm{Cl}}(B,n_{0})$ (solid lines) with the
ones $\lambda_{HTPT}(B,n_{0})$ in eq.(29) (dashed lines), where the HTPTs
occurs. Both solid and dashed lines exhibit different behaviors when varying
the magnetic field $B$ and the wave packet center $n_{0}$. Therefore, it seems
that there is no correlation between the minimum thicknesses
$\lambda_{\textrm{Cl}}$ and the HTPT at $\lambda_{HTPT}$. Nevertheless, when
the magnetic field approximates to zero, both thicknesses tend to
$\lambda_{\mathrm{c}}=6.17$ nm at zero field, that is,
$\lambda_{\mathrm{HTPT}}(B,n_{0})\simeq\lambda_{\textrm{Cl}}(B,n_{0})\simeq\lambda_{\mathrm{c}}$
for all $n_{0}\neq 0$ and $B<<1$ T.

In Figure 5, we present the squared modulus of the autocorrelation function
for $\lambda=\lambda_{\textrm{c}}$ and $s=+1$ in two different time scales.
The top panel displays the time in units of the classical period
$T_{\text{Cl}}=3.75$ ps, where each oscillation corresponds to one unit of the
scale; whereas the bottom panel shows the wave packet revivals at half of the
revival time $T_{\text{R}}/2=39.90$ ps, and the time scale is in
$T_{\text{R}}$ units.

##  IV Electron current revivals and topological phase transitions

We have also identified topological phase transition by analyzing how the
electron current changes with the layer thickness and the time evolution. The
electron currents of the HgTe QWs have been previously studied in reference
[39], where the current operators are

$\displaystyle j_{x}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\left(s\alpha\tau_{x}-\sqrt{2}\frac{a^{\dagger}+a}{\ell_{B}}(\beta\tau_{z}+\delta\tau_{0})\right),$
$\displaystyle j_{y}^{s}$ $\displaystyle=$
$\displaystyle\frac{e}{\hbar}\left(\alpha\tau_{y}+\mathrm{i}\sqrt{2}\frac{a^{\dagger}-a}{\ell_{B}}(\beta\tau_{z}+\delta\tau_{0})\right)\,,$
(36)

Figure 6:  Currents expected values $\langle j_{a}\rangle$ as a function of
time in $T_{\text{Cl}}=3.75$ ps (top) and $T_{\text{R}}=79.80$ ps (bottom)
units, for an initial wave packet with $n_{0}=5$ and
$\sigma=\sqrt{n_{0}}/5=0.47$. The red (blue) line correspond to the current in
the $x$ ($y$) axis. We set the HgTe parameters $\lambda=\lambda_{\textrm{c}}$,
$B=0.05$ T, and $s=+1$.

and the matrix elements in the eigenstate basis (15) are

$\displaystyle\langle\bm{m}|j_{x}^{s}|\bm{n}\rangle_{s}=$
$\displaystyle\,\frac{es\alpha}{\hbar}\Xi_{m,n}^{s,+}-\frac{\sqrt{2}e}{\hbar\ell_{B}}\Phi_{m,n}^{s,+}\,,$
$\displaystyle\langle\bm{m}|j_{y}^{s}|\bm{n}\rangle_{s}=$
$\displaystyle\,-\mathrm{i}\frac{e\alpha}{\hbar}\Xi_{m,n}^{s,-}+\mathrm{i}\frac{\sqrt{2}e}{\hbar\ell_{B}}\Phi_{m,n}^{s,-}\,,$
(37)

where

$\displaystyle\Xi_{m,n}^{s,\pm}=$
$\displaystyle\,(A_{m}^{s}B_{n}^{s}\delta_{|m|-s,|n|}\pm
A_{n}^{s}B_{m}^{s}\delta_{|m|+s,|n|})\,,$ (38)
$\displaystyle\Phi_{m,n}^{s,\pm}=$
$\displaystyle\,((\delta+\beta)A_{m}^{s}A_{n}^{s}+(\delta-\beta)B_{m}^{s}B_{n}^{s})$
$\displaystyle{\scriptstyle\times\left(\sqrt{|n|+1+\tfrac{s-1}{2}}\delta_{|m|-1,|n|}\right.\left.\pm\sqrt{|n|-\tfrac{s+1}{2}}\delta_{|m|+1,|n|}\right)}\,.$

For a Gaussian wave packet (30), the electron current expected value is

${}_{s}\langle\Psi(t)|j_{a}^{s}|\Psi(t)\rangle_{s}=\sum_{m,n=-\infty}^{\infty}\overline{c_{m}^{s}}c_{n}^{s}e^{-i(E_{n}^{s}-E_{m}^{s})t/\hbar}\langle\bm{m}|j_{a}^{s}|\bm{n}\rangle\,,$
(39)

where $a=x,y$ and the bar indicates complex conjugation. From now on we
identify $j_{a}^{s}\equiv j_{a}$ and choose $s=+1$ for simplicity. We plot
both currents in Figure 6, for the same values $\lambda=\lambda_{\textrm{c}}$,
$B=0.05$ T, $s=+1$, $n_{0}=5$, $\sigma=0.47$, as in the previous section. The
results are similar to the autocorrelation in Figure 5. We observe
oscillations in two different time scales, the classical ones (top panel) and
the revivals (bottom panel). After half of the revival time $T_{\mathrm{R}}/2$
in the bottom panel, the electron currents reach again their maximum initial
values revealing the quantum revival phenomenon. This is more evident in the
phase space plot of Figure 7, where both currents decrease to zero at
$t=T_{\mathrm{R}}/4$, and then they grow reaching their initial value at
$t=T_{\mathrm{R}}/2$. Notice that there is a phase difference of $\pi/2$ rad
between the currents $\langle j_{x}\rangle$ and $\langle j_{y}\rangle$, which
is also depicted in Figure 7. The behavior shown in Figure 6 is also found in
graphene [4, 49], and in 2D gapped Dirac materials under magnetic fields [39],
as silicene [50].

Figure 7:  Parametric plot of the currents expected values $(\langle
j_{x}\rangle$,$\langle j_{y}\rangle)$ in the time intervals
$t\in[0,T_{\text{R}}/4]$ (blue) and $t\in[T_{\text{R}}/4,T_{\text{R}}/2]$
(yellow), for an initial wave packet with $n_{0}=5$ and
$\sigma=\sqrt{n_{0}}/5=0.47$. We set the HgTe parameters
$\lambda=\lambda_{\textrm{c}}$, $B=0.05$ T, and $s=+1$, so that the revival
time is $T_{\text{R}}=79.80$ ps.  Figure 8:  Maximum values of the current
amplitudes $\text{Max}_{t\in[0,T_{\text{R}}/2]}|\langle j_{a}\rangle|$ as a
function of the layer thickness $\lambda$, for an initial wave packet with
$n_{0}=5$ and $\sigma=\sqrt{n_{0}}/5=0.47$. The red (blue) dots correspond to
the current in the $x$ ($y$) axis. We set the parameters $B=0.05$ T, and
$s=+1$.

We have repeated the calculations of Figure 6 for different values of the
layer thickness $\lambda$, in order to study the impact of this parameter on
the electric currents. We select the maximum of the electron current
amplitudes $\text{Max}_{t\in[0,T_{\text{R}}/2]}|\langle j_{a}\rangle|$
(maximum in the time domain) for different thicknesses $\lambda$, and plot
them in Figure 8. Both current maxima reach their higher value at the critical
thickness. Therefore, measuring the amplitudes of the electron currents is
another way to characterize TPTs in HgTe QWs. For higher magnetic fields, this
maximal behavior deviates from the critical thickness $\lambda_{\textrm{c}}$.

In Figure 9, we plot the layer thickness $\lambda_{j_{a}}(B,n_{0})$ in which
the dots $\text{Max}_{t\in[0,T_{\text{R}}/2]}|\langle j_{a}\rangle|$ of Figure
8 achieve a maximum in the $\lambda$ domain, against the external magnetic
field $B$ and for an initial wave packet with $n_{0}=5$. The maxima
$\lambda_{j_{a}}$ are close to the critical thickness $\lambda_{\textrm{c}}$
in a region of the magnetic field, i.e.
$|\lambda_{j_{a}}(B,n_{0})-\lambda_{\textrm{c}}|<0.1$ nm for all $B<0.5$ T and
$n_{0}=5$. When increasing the magnetic field above $B\simeq 0.5$ T, the
maxima $\lambda_{j_{a}}$ (red and blue dots in Figure 9) start growing in a
similar way to the thickness $\lambda_{\mathrm{Cl}}$ where $T_{\mathrm{Cl}}$
achieves its minimum in Figure 3.

Figure 9: Layer thickness $\lambda_{j_{a}}(B,n_{0})$ in which the dots
$\text{Max}_{t\in[0,T_{\text{R}}/2]}|\langle j_{a}\rangle|$ of Figure 8
achieve a maximum in the $\lambda$ domain, as a function of the external
magnetic field $B$. The red and blue dots correspond to the directions $a=x$
and $a=y$ respectively, and the yellow line depicts the thicknesses where
$T_{\mathrm{Cl}}$ achieves its minimum (retrieved from Figure 3). We set
$s=+1$ and an initial wave packet with $n_{0}=5$ and
$\sigma=\sqrt{n_{0}}/5=0.47$. The horizontal dashed line indicates the
critical thickness $\lambda_{\textrm{c}}=6.17~{}$nm.

##  V Conclusions

In summary, we have shown that the time evolution of a wave packet is useful
to detect TPTs in HgTe QWs, which corroborates the results previously found in
[7] for other 2D materials (silicene, germanene, tinene and indinene). Using
the 2D effective Dirac Hamiltonian for surface states in HgTe/CdTe QWs, it is
possible to analyze the time evolution of electron current wave packets. As a
general result, the classical and revival time appear as two different
periodicities in this temporal evolution, and reach their minima at different
values of the layer thickness, depending on the external magnetic field and
the Landau level where the packet is centered at. In addition, we have
investigated how the maximum of the electron current amplitude changes with
the thickness $\lambda$, identifying that current maxima reach their higher
value at the critical thickness, so we can characterize the TPTs in terms of
the amplitude of the electron currents. As a proposal for future work, this
quantum revival analysis could be extended to non-topological anisotropic
materials like phosphorene, which also present criticality when its energy gap
is closed by an external electric field [39].

## Acknowledgments

We thank the support of Junta de Andalucía through the project
PID2022-138144NB-I00. AM thanks the Spanish MIU for the FPU19/06376
predoctoral fellowship. OC is on sabbatical leave at Granada University,
Spain, since the 1st of September 2023. OC thanks support from the program
PASPA from DGAPA-UNAM.

a

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