# Photo-switchable liquid crystalline brush as an aligning surface for liquid
crystals: modeling via mesoscopic computer simulations

Dmytro Yaremchuk  Taras Patsahan  Jaroslav Ilnytskyi<EMAIL_ADDRESS>Institute for Condensed Matter Physics of Nat. Acad. Sci. of Ukraine,
1, Svientsitskii Str., Lviv, Ukraine

###### Abstract

We consider the mesoscopic model for the liquid crystalline brush that might
serve as a photoswitchable aligning surface for preorientation of low
molecular weight liquid crystals in a bulk. The brush is built by grafting the
polymer chains of a side-chain molecular architecture, with the side chains
terminated by a chromophore unit mimicking the azobenzene unit, to a
substrate. When irradiated with ultraviolet light, the chromophores
photoisomerize into a non-mesogenic cis state and the whole system turns into
an ordinary polymer brush with no orientational order and two states: the
collapsed and straightened one, depending on the grafting density. When
irradiated with visible light, the chromophores photoisomerize into a
mesogenic trans state, resulting in formation of a transient network between
chains because of a strong attraction between chromophores. Spontaneous self-
assembly of the brush in these conditions results in a orientationally
isotropic polydomain structure. The desired uniaxial planar ordering of
chromophores within a brush can be achieved in certain temperature and
grafting density intervals, as the result of a two-stage preparation protocol.
An external stimulus orients chromophores uniaxially on the first stage. The
system is equilibrated at the second stage at a given temperature and with the
external stimulus switched off. The preoriented chromophores either keep or
loose their orientations depending on the strength of the memory effect
inherent to a transient network of chains, that are formed during the first
stage, similarly to the case of the liquid crystalline elastomers, where such
effects are caused by the covalent crosslinks.

###### keywords:

polymer brush , azobenzene , molecular dynamics

##  1 Introduction

Polymer brush, typically a flat surface containing grafted polymer chains,
provides a good example of a functional surface [1, 2, 3, 4]. All the factors,
such as: grafting density, chemical structure and molecular architecture of
polymer chains, as well as the presence of specific responsive groups, play a
crucial role in their functionality. One of examples of such groups are
mesogens, that are able to form orientationally and spatially ordered liquid
crystalline (LC) phases [5]. Polymer chains serve as soft embedding matrix for
the mesogens resulting in stabilization or disruption of their LC phases,
similarly to the case of LC elastomers [6]. On the other hand, the presence of
mesogens turns this composite, a liquid crystalline polymer brush (LCPB), into
a light-sensitive functional surface.

Both synthetic protocols [7] and theoretical treatment [8, 9, 10] of LCPBs
have been started at least two decades ago. Their principal features have been
narrowed up to formation the LC phases with the possibility to achieve either
homeotropic or uniaxial planar (UPL) orientation of the mesogens with respect
to the substrate. The latter case is thought to be especially useful for
practical application in the LC displays technology, where such LCPB can serve
as a commanding surface for precise aligning of the bulk LC in a LC cell, see,
e.g. Ref. [11]. Other applications are based on spontaneous tilting transition
in the main chain LCPB [12]; LC ordering as the result of dewetting of the
LCPB [13]; the synergy between the LCPB and nanotubes allowing to fabricate
smart windows with their transparency controlled by the near infrared light
[14] and so on.

To introduce more control over the alignement of mesogens in the LCPB, the
mesogens in a form of the azobenzene chromophores can be used. By illumination
of such chromophores by the light with suitable wavelength, intensity and
polarization, their trans-isomers are able to reorient theirselves
perpendicularly to the polarization vector because of their angular-dependent
absorption oh photons, the so-called “angular hole burning” [15]. This leads
to the possibility of a precise control for the nematic director in the LC
phases of the azobenzene containing LCPBs [16, 17, 18, 19, 20],
photocontrolled nanopatterning [21, 22, 23, 24], surface relief formation [25,
26], designing superadhesive surfaces [27], making waves in such brushes [28,
29], developing smart biointerfaces [30] and to other applications [31].

We will restrict our focus here to the application of the LCPBs as the
photocontrolled commanding surfaces for alignment of bulk LCs. After an early
work [32] involving the azobenzene-treated substrate, the later research was
concentrated on the LCPBs comprising polymers of the side-chain architecture,
where the side-chains were terminated by an azobenzene unit. As first
demonstrated in Ref. [16], such LCPB organize into a smectic phase
characterized by the planar orientation of azobenzenes, in contrary to the
spin-casted film, where the homeotropic orientation prevails. The $2D$ in-
plane orientation of azobenzenes is characterized, in general, by a polydomain
structure. It can be transformed into a monodomain smaple once the system is
illuminated with the linearly polarized beam with its incidence perpendicular
to the substrate [17, 18]. The nematic director in this case is perpendicular
to the polarization vector of a beam. This level of the in-plane dichroism is
achieved at the temperatures slightly above the glass transition temperature
and is not attained for the spin-cast prepared films. In another study [33],
the azobenzene-containing fragment was grafted to the substrate via linear
spacer, and, upon the increase of its length, “the smectic layer structure was
also become assembled more intrinsically with the cooperative wiggling
ascribed to the flexible buffer layer” [33]. Yet another studies, based on the
block-copolymer LCPB, showed that the surface segregation of the low surface
free energy block also leads the planar orientation for the azobenzenes [34,
19]. A gradual alignment of bulk mesogens from homeotropic to planar can be
achieved by changing the grafting density of the azobenzene-containing LCPB,
using the interplay between the preferentially perpendicular orientation of
azobenzenes and the backbone and the stretching director of the backbone,
controlled by the grafting density [20].

Summarizing these experimental findings, it is quite evident, that for the
application of the LCPB as commanding surfaces for alignement of bulk
mesogens, their orientational order should be (a) controllable, and (b)
stable. Requirement (a) can be satisfied by incorporating the azobenzene
groups into its molecular architecture and providing that they can
photoisomerize. Requirement (b) needs deeper insight towards both formation
and stability of the ordered LC phases, that are inherent for the LCPB of a
given molecular architecture at particular grafting density and temperature.
These issues will be addressed in the current study using mesoscale computer
simulations. In this respect, it continues a series of our previous works on
self-assembly of decorated nanoparticles [35, 36, 37, 38, 39] and of their
adsorption on the LCPB [40], by using similar type of modelling.

The outline of a study is as follows: section 2 contains the description of
the model, section 3 covers the properties of the model azobenzene-containing
LCPB under ultraviolet light, section 4 addresses such a brush under visible
light, conclusions are given in section 6.

##  2 Mesoscopic model for the photo-switchable liquid crystralline brush

Figure 1: Snapshot for the LCPB containing $N_{br}=9$ polymer molecules.
Polymer spherical beads are shown in gray (the ones grafted to the substrate
are shown in pink), chromophores are shown in blue, solvent and substrate are
not shown.

The model LCPB considered here, contains a set of $N_{br}$ polymer molecules
of the side-chain architecture, grafted to the bottom surface of the
simulation box using an uniform pattern, see Fig. 1. Following our previous
works [35, 36, 37, 38, 39, 40], we employ a coarse-grained modelling approach.
It utilizes splitting each polymer molecule into a set of connected beads,
each representing a specific atomic group, e.g. a fragment of three $CH_{2}$
units, ester group, the azobenzene unit, etc. As the result, each molecule
contains a backbone of $36$ spherical beads and $12$ side chains, where each
side chain contains a spacer of two spherical beads and a terminal
spherocylinder bead mimicking the azobenzene chromophore. Because of the
coarse-grained nature of a model, the same spherocylinder shape is used for
both trans- and cis-isomers of azobenzene, whereas the differences between
them are reflected in different interaction potentials involving these
isomers. We consider LCPB in an explicit solvent, represented by spherical
beads. To simplify following notations, we will denote spherical beads of a
polymer and of a solvent via subscripts $p$ and $s$, respectively;
spherocylinder bead representing trans- or cis-isomer via subscripts $t$ and
$c$, respectively, whereas the case of any of these isomers will be denoted by
$a$. All spherical beads are chosen to have the same diameter of
$\sigma_{p}=\sigma_{s}=4.59\mathrm{\AA}$, spherocylinder chromophore beads are
characterised by the diameter $\sigma_{a}=3.74\mathrm{\AA}$ of their spherical
cap and by the elongation of $L/\sigma_{a}=3$, where $L$ is the separation
between the centers of their spherical caps.

The expression for the intramolecular bonded interactions

$\begin{array}[]{cc}V^{\mathrm{BON}}=&\displaystyle\sum_{i=1}^{m_{b}}k_{b}(l_{i}-l_{0})^{2}+\sum_{i=1}^{m_{a}}k_{a}(\theta_{i}-\theta_{0})^{2}+\sum_{i=1}^{m_{z}}k_{z}(\zeta_{i}-\zeta_{0})^{2}.\end{array}$
(1)

involves $m_{b}$ harmonic bonds and $m_{a}+m_{z}$ pseudo-valent angle bending
terms that control stiffness of the side chains. The backbone contains no bond
bending terms but is semiflexible implicitly, via the presence of the side
chains. The values of $l_{0}=3.617\mathrm{\AA}$ and $8.59\mathrm{\AA}$ are
used as the refernce bonds for the sphere-sphere and sphere-spherocylinder
pairs. The reference angle $\theta_{0}$ is equal to $\pi$ for the linear
fragment of a side chain, and is equal to $\pi/2$ where the attachment of the
side chain to the backbone occurs (the triplet includes a branching bead of a
backbone, the backbone bead that precedes it, and a first bead of a side
chain), to enforce their mutually perpendicular arrangement. The last term in
Eq. (1) involves the angle $\zeta_{0}=0$, and forces a parallel orientation of
the terminal spherocylinder to its host side chain [41]. For the sake of
brevity we will introduce the following energy unit: $\textrm{
j}=10^{-20}\textrm{ J}$. Force constants, expressed via this unit are: of
$k_{b}=50\textrm{ j}/\mathrm{\AA}^{2}$ and $k_{a}=k_{z}=20\textrm{
j}/\mathrm{rad}^{2}$.

Let us consider the nonbonded interactions now. Following the ideology of
coarse-grained models (e.g. these employed in the dissipative particle
dynamics simulations [42]), most of them are assumed to be short-ranged soft
repulsive, with the strength of the repulsion reflecting the chemical nature
of each bead. We assume the polymer $p$ beads and the $t$ chromophores to be
non-polar, whereas the solvent $s$ beads and the $c$ chromophores are polar
(dipole moment of trans- and cis-isomers of a typical azobenzene chromophore
is discussed in Ref. [43]). Higher magnitude for the repulsion parameter is
used then to describe the segregation effect between the polar and non-polar
beads, see, e.g. Ref. [42]. The same principle can be used for the $\\{i,j\\}$
pair of interacting spherocylinders by defining their “internal core” (the
line joining the centers of its two spherical caps) and introducing the
minimal separation $d(\mathrm{\bf q}_{ij})$ between their cores. Here
$\mathrm{\bf q}_{ij}=\\{\hat{\mathrm{\bf e}}_{i},\hat{\mathrm{\bf
e}}_{j},\mathrm{\bf r}_{ij}\\}$ contains all characteristics of their mutual
arrangement: the vector $\mathrm{\bf r}_{ij}$ between their centers and their
respective orientations $\hat{\mathrm{\bf e}}_{i}$, $\hat{\mathrm{\bf e}}_{j}$
in space. Then, the same expression for the soft repulsive potential can be
used as for the case of spherical beads using minimal separation
$d(\mathrm{\bf q}_{ij})$ in place of the center-center distance [44]. However,
it turned out, that the soft repulsive spherocylinders form LC phases only in
the regime of high aspect ratio [45, 46]. By adding the attractive part to the
interactive potential, which represents the dispersion forces, one strengthens
the mesogenity of soft spherocylinders and is able to obtain ordered LC phases
at much moderate aspect ratios [47]. To emphasize strong attraction between
polar $c$ chromophores and a solvent, we opted to use such interaction
potential with the attractive contribution for this case, too.

All the cases of the pair interaction potential between $i$th and $j$th beads,
discussed above, can be written in a single dimensionless form

$V^{\mathrm{SAP}}[d^{\prime}(\mathrm{\bf
q}_{ij})]=\left\\{\begin{array}[]{ll}U\big{\\{}[1-d^{\prime}(\mathrm{\bf
q}_{ij})]^{2}-\epsilon^{\prime}(\mathrm{\bf q}_{ij})\big{\\}},&0\leq
d^{\prime}(\mathrm{\bf q}_{ij})<1\vspace{2mm}\\\
U\big{\\{}[1-d^{\prime}(\mathrm{\bf
q}_{ij})]^{2}-\epsilon^{\prime}(\mathrm{\bf q}_{ij})\big{.}&\\\ \hskip
20.00003pt\big{.}-\frac{1}{4\epsilon^{\prime}(\mathrm{\bf
q}_{ij})}[1-d^{\prime}(\mathrm{\bf q}_{ij})]^{4}\big{\\}},&1\leq
d^{\prime}(\mathrm{\bf q}_{ij})\leq d^{\prime}_{c}\vspace{2mm}\\\
0,&d^{\prime}(\mathrm{\bf q}_{ij})>d^{\prime}_{c}\end{array}\right.$ (2)

where $U$ defines repulsion strength, $d^{\prime}(\mathrm{\bf
q}_{ij})=d(\mathrm{\bf q}_{ij})/\sigma_{ij}$ is the dimensionless minimal
separation between two beads, and $\sigma_{ij}=(\sigma_{i}+\sigma_{j})/2$ is
the length scaling factor for the $\\{i,j\\}$ interacting pair.

$\epsilon^{\prime}(\mathrm{\bf
q}_{ij})=\Bigg{\\{}4\Big{[}U^{\prime}_{a}-5\epsilon^{\prime}_{1}P_{2}(\hat{\mathrm{\bf
e}}_{i}\cdot\hat{\mathrm{\bf
e}}_{j})-5\epsilon^{\prime}_{2}\Big{(}P_{2}(\hat{\mathrm{\bf
r}}_{ij}\cdot\hat{\mathrm{\bf e}}_{i})+P_{2}(\hat{\mathrm{\bf
r}}_{ij}\cdot\hat{\mathrm{\bf e}}_{j})\Big{)}\Big{]}\Bigg{\\}}^{-1},$ (3)

is a dimensionless well depth of this potential, obtained from the condition,
that both the expression (2) and its first derivative on
$d^{\prime}(\mathrm{\bf q}_{ij})$ turn to zero when $d^{\prime}(\mathrm{\bf
q}_{ij})=d^{\prime}_{c}$, where
$d^{\prime}_{c}=1+\sqrt{2\epsilon^{\prime}(\mathrm{\bf q}_{ij})}$ is the
cutoff separation of the potential. $\hat{\mathrm{\bf r}}_{ij}=\mathrm{\bf
r}_{ij}/r_{ij}$ is a unit vector connecting the centers of two beads,
$U^{\prime}_{a}$, $\epsilon^{\prime}_{1}$ and $\epsilon^{\prime}_{2}$ are
dimensionless parameters that define the shape of the interaction potential
depending on mutual orientation of beads [47], $P_{2}(x)=(3x^{2}-1)/2$ is the
second Legendre polynomial.

Figure 2: Illustration of the potential
$V^{\mathrm{SAP}}[d^{\prime}(\mathrm{\bf q}_{ij})]$ shape for the case of two
spherical beads at $U=70\textrm{ j}$ and at a range of effective dimensionless
well depths $\epsilon^{\prime}(\mathrm{\bf q}_{ij})$ (respective cutoffs
$d^{\prime}_{c}$ are shown via vertical dashed lines). For the case of two
spherical beads, $d^{\prime}(\mathrm{\bf q}_{ij})$ reduces to the
dimensionless separation $r^{\prime}_{ij}$ between their centers. When
$\epsilon^{\prime}(\mathrm{\bf q}_{ij})\to 0$, the potential
$V^{\mathrm{SAP}}[d^{\prime}(\mathrm{\bf q}_{ij})]$ reduces to its purely
repulsive counterpart $V^{\mathrm{SRP}}[d^{\prime}(\mathrm{\bf q}_{ij})]$, Eq.
(4).

The effective well depth $\epsilon^{\prime}(\mathrm{\bf q}_{ij})$ influences
both the shape of the attractive part and, via $d^{\prime}_{c}$, its range.
When the parameters, contained in the expression for
$\epsilon^{\prime}(\mathrm{\bf q}_{ij})$, are such, that it asymptotically
reaches zero, the cutoff $d^{\prime}_{c}$ approaches $1$ and the interval for
the second line in Eq. (2) shrinks to zero. As the result, in this limit, one
retrieves the soft repulsive quadratic potential

$V^{\mathrm{SRP}}[d^{\prime}(\mathrm{\bf
q}_{ij})]=\left\\{\begin{array}[]{ll}U\left[1-d^{\prime}(\mathrm{\bf
q}_{ij})\right]^{2},&0\leq d^{\prime}(\mathrm{\bf q}_{ij})\leq
1\vspace{2mm}\\\ 0,&d^{\prime}(\mathrm{\bf q}_{ij})>1\end{array}\right.,$ (4)

that is used typically in the dissipative particle dynamics simulations [42].
This limit is illustrated in Fig. 2 for the case of two spherical beads and
the repulsive energy strength of $U=70\textrm{ j}$.

We provide now the parameters for pair interaction between various
combinations of beads, that are based on their chemical origin. The $tt$ and
$cs$ pairs interact via attractive potential (2) with $U_{tt}=70\textrm{ j}$,
$U^{\prime}_{a}=21.43$, $\epsilon^{\prime}_{1}=1.714$ and
$\epsilon^{\prime}_{2}=-1.714$ (the “model A” in the nomenclature of Ref.
[47]). For the spherical $s$ bead we set $\hat{\mathrm{\bf e}}_{i}=0$. All the
rest pair interactions are soft repulsive (4), where
$U_{pp}=U_{ss}=U_{cc}=U_{ct}=70\textrm{ j}$ and $U_{ps}=U_{ts}=140\textrm{ j}$
to mimic a segregation between polar and non-polar beads. Most of these
parameters follow our previous studies [35, 36, 37, 38, 39].

The angular hole burning effect, discussed in section 1, can be modelled
either explicitly, via kinetic equations for the fractions of the $t$ and $c$
chromophores [48, 49], or implicitly, via effective field, that reorients the
$t$ chromophores prefferably perpendicularly to the polarization vector of
incident light [50, 51, 39]. As was shown in Ref. [48], explicit modelling
leads to a statistical effect of reorientation of the $t$ chromophores, and
this effect can be mapped onto the effective field with the strength,
dependent on the photoisomerization rates that appear in respective kinetic
equations. Here we choose the simpler approach of two, the implicit modelling
of the angular hole burning effect, by introducing the effective field via
respective energy term

$U^{\mathrm{ROT}}=F\sum_{i=1}^{n_{t}}P_{2}(\hat{\mathrm{\bf
e}}_{i}\cdot\hat{\mathrm{\bf p}}),$ (5)

where $F>0$ is the effective field strength, $n_{t}$ is the number of the $t$
chromophores at given time instance, and unit vectors $\hat{\mathrm{\bf
e}}_{i}$ and $\hat{\mathrm{\bf p}}$ provide the orientation of $i$th $t$
chromophore and of the polarization vector, respectively.

The chromophore is assumed to photoisomerise into the $c$ state under
ultraviolet (UV) light and to undergo a reverse photoisomerisation to the $t$
state under visible (Vis) one, following properties of $NO_{2}$ substituted
azobenzenes [52]. To simplify description, we consider two limit cases: (i) UV
light: all chromophores are in the $c$ state, and (ii) Vis light: all
chromophores are in the $t$ state. Simulation box is of a cuboid shape with
the following dimensions: $L_{x}=L_{y}=100\mathrm{\AA}$ and
$L_{z}=130\mathrm{\AA}$ with the periodic boundary conditions applied along
the $OX$ and $OY$ axes and with elastic reflective walls at both the $z=0$ and
$z=L_{z}$ planes. Polymer molecules are grafted to the bottom plane, $z=0$,
with the reduced grafting density
$\rho^{\prime}_{g}=N_{br}\sigma_{p}^{2}/(L_{x}L_{y})$, expressed via $N_{br}$,
the interval of $N_{br}=1-64$ is considered. Maximum reduced grafting density
is, therefore, equal to $0.135$. Simulation box interior is filled by a
solvent to keep the total density of about $0.5\mathrm{g}/\mathrm{cm}^{3}$.
Computer simulation is performed by molecular dynamics in the $NVT$ ensemble
with the time step of $20\mathrm{fs}$.

##  3 Photo-switchable liquid crystralline brush under ultraviolet light

We consider the idealised case, when illumination of the LCPB by the UV light
results in photoisomerization of all existent chromophores into the $c$ type.
Let us consider different energy terms that affect the structure of LCPB in a
solvent.

Each polymer molecule is expected to be of stretched bended conformation
because of implicit stiffness of their backbones. Side chain with terminal
chromophores are perpendicular to backbones according due to their strong
coupling to them. The molecules may rotate around their grafted beads as semi-
rigid objects subjected to the effect of excluded volume with respect to their
neighbours. Excluded volume effect is strengthened by solvation of the $c$
chromophores. These properties reduce the LCPB under UV light to the non-LC
brush, composed of semi-rigid side chain polymers. The bottom surface of the
simulation box has adsorbing properties for the molecules, because the polymer
$p$ beads have lesser repulsive neighbours near the surface than in a bulk.
The competition between the (i) adsorption of the polymers on the botom
surface, and (ii) their repulsion because of overcrowding will determine the
final state of the LCPB.

We found basically the same structure of the LCPB in a wide temperature range,
from $450K$ to $600K$, therefore, the case of the temperature $480K$ is
illustrated only. At small number $N_{br}$ of molecules in the LCPB, the
effect (i) prevails and we observe collapsed state, see Fig. 3, on the left.
With the increase of $N_{br}$, the molecules deadsorb from the bottom surface
and straighten up, along the $OZ$ axis, see Fig. 3, in the middle and on the
right.

Figure 3: Internal structure of the LCPB under UV illumination at various
number of grafted molecules $N_{br}$, indicated in the plots. Polymer $p$
beads are coloured in gray, the $c$ chromophores are yellow, solvent beads are
not shown. Temperature $T=480K$.

To examine the transition between the collapsed and stretched states of the
LCBR, we are evaluating its effective height

$\langle
h\rangle=\frac{\int_{0}^{L_{z}}z\rho(z)dz}{\int_{0}^{L_{z}}\rho(z)dz},$ (6)

where $\rho(z)$ is the density profile for the $p$ beads of its molecules,
obtained as the result of computer simulations. As one can see in Fig. 4, the
transition between the collapsed and straightened states of the LCPB under UV
light (rapid jump of $\langle h\rangle/L_{z}$) occurs already at $N_{br}=2$.

Figure 4: Reduced effective height $\langle h\rangle/L_{z}$ and orientational
order parameters $S_{z}$ and $S_{x}$ $N_{br}$ for the of the LCPB under UV
light as functions of the number of grafted molecules $N_{br}$ at the
temperature of $T=480K$.

Targeting application of the LCPB as an UPL aligning surface for low molecular
weight LCs, two conditions are relevant: (a) the chromophores are
preferentially perpendicular to the $OZ$ axis, and (b) their orientation is
uniaxial in the $XY$ plane. Let us assume the $OX$ axis as a reference one for
the condition (b). Then, these conditions are quantified via orientational
order parameters $S_{z}$ and $S_{x}$, evaluated with respect to these two axes

$S_{z}=\langle P_{2}(\hat{\mathrm{\bf e}}_{i}\cdot\hat{\mathrm{\bf
k}})\rangle,~{}~{}S_{x}=\langle P_{2}(\hat{\mathrm{\bf
e}}_{i}\cdot\hat{\mathrm{\bf i}})\rangle,$ (7)

where $\hat{\mathrm{\bf e}}_{i}$ defines the orientation of $i$th chromophore,
whereas $\hat{\mathrm{\bf k}}$ and $\hat{\mathrm{\bf i}}$ are unit vectors
oriented along the $OZ$ and $OX$ axes. The averaging is performed over all
chromophores within the LCPB.

At $N_{br}>2$, the $S_{z}$ order parameter is very close to $-0.5$, as follows
from Fig. 4, indicating hight level of perpendicularity of chromophores to the
$OZ$ axis. This is achieved because of two factors: stretchening of backbones
along this axis (excluded volume driven “dense brush” regime) and strong
coupling of side chains to the backbone. This opens up the possibility of
manipulating the orientation order of chromophores by stretching or coiling
the backbones, as remarked in the experimental works [20, 12, 53]. However,
the order parameter $S_{x}\leq 0.1$ for all $N_{br}$ being examined,
indicating the isotropy of chromophores in the $XY$ plane. On a top of that,
the $c$ chromophores are non-mesogenic. Therefore, the LCPB under UV light may
be used for inducing random planar orientation for low molecular weight LC in
bulk. In the next section we will analyse the same set of characteristics for
the LCBR under Vis light, where chromophores are strongly mesogenic ones of
the $t$ type.

##  4 Photo-switchable liquid crystralline brush under visible light

We consider here another idealised case, when illumination of the LCPB by the
Vis light results in a photoisomerization of all existent chromophores into
the $t$ type ones. The switch from the $c$ to $t$ type changes some of energy
contributions. Namely, in contrary to the pair of two $c$ chromophores, their
mesogenic $t$ counterparts are attracted to each other via the potential (2)
with the parameters provided in section 2. This has two consequences: the
first is formation of the intermolecular physical crosslinks that turns the
LCPB into a macromolecular network [54, 55]. The effect is also enforced by
strong repulsion between the nonpolar $t$ chromophores and polar solvent $s$
beads, where we remind the choice for the repulsion strength
$U_{ts}=140\textrm{ j}$, being made in section 2. The second consequence is
the possibility of formation of the globally ordered LC phase, and it is
related directly to the application of the LCPB as the aligning surface for
low molecular weight LC.

Figure 5: Internal structure of the LCPB under Vis illumination at various
number of grafted molecules $N_{br}$, indicated in the plots. Polymer $p$
beads are coloured in gray, the $t$ chromophores are in blue, solvent beads
are not shown. Self-assembly at $T=490K$.

We will try different ways by which such LCPB can be assembled into an UPL LC
phase, starting form the attempts of a spontaneous self-assembly. Following
our experience with the mesogens of the same type [35, 36], such runs are
performed at $T=490K$, just below the phase transition to the isotropic phase.
The results for the internal structure of the LCPB at various number of
grafted molecules $N_{br}$ are shown in Fig. 5. Similarly to the case of UV
light, discussed in section 3, one observes collapsed state at small $N_{br}$
(see, left frame in Fig. 5) and the stretched state at medium and large
$N_{br}$ (see, middle and right frames in Fig. 5).

Figure 6: Reduced effective height $\langle h\rangle/L_{z}$ and orientational
order parameters $S_{z}$ and $S_{x}$ for the LCPB under Vis light as functions
of the number of grafted molecules $N_{br}$. Self-assembly at $T=490K$.

The results of a quantitative analysis, via characteristics $\langle
h\rangle$, $S_{z}$, and $S_{x}$, is illustrated in Fig. 6. Similarly to the
case of UV light (Fig. 4), one observes the transition between the collapsed
and stretched states of the brush, but it is less sharp than under UV light
and occurs at higher $N_{br}\approx 10-16$. Enhacement of the collapsed state
at $N_{br}=4-9$ should be attributed to both strong attraction between the $t$
chromophores (resulting in formation of their clusters), and adsortption of
these clusters on the bottom surface because of their strong repulsion with a
solvent in a bulk, see left frame in Fig. 5.

At $N_{br}>16$, the order parameter $S_{z}\approx-0.5$, this indicates again a
high degree of perpendicularity of chromophores to the $OZ$ axis, but the
order parameter $S_{x}$ fluctuates between $-0.2$ and $0.4$ suggesting a
dynamically changed polydomain LC structure of the chromophores subsystem.
This conclusion is supported by the snapshot given in the rigth frame in Fig.
5, where one can see clusters with a broad distribution of their local nematic
directors. Simulations of duration $200\mathrm{ns}$ were performed at each
number $N_{br}$ of brush molecules, but an UPL order within the $XY$ plane was
observed in none of these cases. The attempts were repeated at the
temperatures $T=490K$ and $500K$ with the same result. This indicates that the
model LCPB does not self-assemble into the phase with the UPL ordering and
produces the globally isotropic planar phase of the LCPB instead, similar to
the one observed under UV light.

This can be explained by the known fact, that the LC phases formed by the LC
polymers are stabilized by chemical crosslinking of them [56, 57]. Similar
effect takes place in our model for the LCPB, where the physical
“crosslinking” occurs instead. When the initial distribution of chromophores
orientations is globally isotropic, then their photoisomerisation into the $t$
type, and consequent formation of physical links, stabilizes this isotropic
distribution. Therefore, to achieve a stable UPL orientation, it should be
induced by means of an external stimulus, and then stabilized via physical
links. This is the way an UPL orientation of chromophores is formed
experimentally, where role of aligning external stimulus is played by the
linearly polarized beam [16, 17, 18, 19, 20]. When such a beam propagates
along the $OZ$ axis, then its polarization vector $\hat{\mathrm{\bf p}}$ has a
fixed orientation within the $XY$ plane and can be chosen, without a loss of
generality, to coincide with the $OY$ axis. For this choice of
$\hat{\mathrm{\bf p}}$, the most energy favourable orientations for the $t$
chromophores will be located within the $XZ$ plane, see Eq. (5). But for the
case of LCPB, stretched along the $OZ$ axis (see Fig. 5), chromophores are
predominantly perpendicular to the $OZ$ axis (see, $S_{z}$ values in 6) due to
conformational energy terms. Therefore, application of the external field (5)
with $\hat{\mathrm{\bf p}}$ collinear with the $OY$ axis, leaves for this
model the only option for the chromophores to align predominantly along the
$OX$ axis.

Figure 7: LCPBs with a UPL orientation of chromophores induced via external
field (5) and then equilibrated at $T=450K$ for $100\mathrm{ns}$ with the
field switched off. The cases of $N_{br}=25$ and $64$ are shown.

The energy term (5) with the strength of $1\textrm{ j}$ and $\hat{\mathrm{\bf
p}}||OY$, mimicking the effect of linearly polarized light, was applied at
$T=490K$ for $10\mathrm{ns}$, resulting in formation of an UPL orientation of
chromophores within the LCPB for all cases with $N_{br}=4-64$. Then, the field
was switched off in each case and this configuration was equilibrated during
subsequent $100\mathrm{ns}$. As an illustration, we show the snapshots for the
configurations, obtained at $T=450K$ for two cases of $N_{br}=25$ and $64$,
see, Fig. 7.

Figure 8: Reduced brush height $\langle h\rangle/L_{z}$ and the order
parameters $S_{z}$ and $S_{x}$ of the LCPB under Vis light as functions of the
number of grafted molecules $N_{br}$. The result of equilibration of an UPL
phase, induced by an external stimulus, at three different temperatures
(indicated in the plots).

Let us consider three characteristics, $\langle h\rangle$, $S_{z}$ and
$S_{x}$, of the LCPB, equilibrated for various number $N_{br}$ of molecules
and at three different temperatures $T=450K$, $550K$ and $650K$, see, Fig. 8.
The observation (i) is: upon the increase of $N_{br}$, a UPL order
($S_{z}\approx-0.5$ and $S_{x}>0.3$) appears synchronously with the transition
of a brush into a straightened state. The observation (ii) is: with the raise
of temperature, this transition occurs at higher $N_{br}$ and became less
sharp for $S_{z}$ and $S_{x}$, eventually disappearing at the highest
temperature, $T=650K$, being shown. These observations reflect the role of
both the polymer and the chromophores subsystems in the “memory effect”, which
preserves a UPL order after the field is switched off. Namely, the role of the
polymer system is twofold: to ensure the conformations (backbones are
stretched and aligned along the $OZ$ axis and chromophores are kept
perpendicular to them) required for a planar order of chromophores; and to
stabilize an uniaxial order of chromophores, induced by external stimulus, by
a set of links. Both tasks require certain minimal number $N_{br}$ of
molecules. Now, the $t$ chromophores need to form orientationally ordered
phase. This is possible starting from certain threshold density of them
(again, one requires a some minimal number $N_{br}$ for that) and at the
temperature below their order-disorder transition. The combination of all
these factors explain both observations (i) and (ii) and can be summarized
that, to achieve a stable UPL orientation for the LCPB, one requires a brush
that is dense enough, and the temperature, where the LC order of chromophores
is strong enough. Particular intervals for both characteristics depend on the
parameters of the interaction potentials and their practical estimates for
real chemical substances require thorough parametrization procedures to be
employed.

Therefore, the model describes the mesoscopic mechanisms behind the
preparation and stability of an UPL orientation of the LCPB, that incorporates
the $t$ chromophores. By illuminating such brush by UV and Vis light, one can
switch its orientational order from isotropic to an UPL, aimed on inducing
required anchoring of low molecular crystals in a bulk.

##  5 Conclusions

Delevoping photo-switchable commanding surfaces, including the possibility of
triggering required anchoring of bulk liquid crystals, has found a lot of
experimental interest recently. Computer simulation of such systems faces
several difficulties. These are related to the relatively large length and
time scales of the problem, and turn rendering of these effects using
chemically-exact models problematic. Instead, one may attempt more physics-
oriented approach, when not the exact chemical details, but rather principal
features of the phenomena are taken into account. Such attempt is made in the
current study.

We build the mesoscopic model for a liquid crystalline polymer brush, which
incorporates chromophores. It reproduces such principal features of the real
systems as: side-chain molecular architecture, semi-rigidness of polymer
chains, polarity of chemical groups and of a solvent and photoisomerization of
the azobenzene chromophores. This allowed to capture essential features of
such brushes: transition from a collapsed to a straightened state with the
increase of grafting density, differences in behaviour of the brush under
ultraviolet and visible light, and the role of a temperature in its structure.

The main result of the study is the demonstration that the relatively simple
mesoscopic model of a complex macromolecular system is able to describe the
possibility to made a commanding surface, which can photo-switch between the
orientationally isotropic and uniaxial planar phase. Althought, such surfaces
have been already developed experimentally, their modelling provides a cheaper
and simpler alternative to look for their enhancements or possible changes of
their functionality. To this end we plan to perform further studies involving
special parametrizations of the model interaction parameters, aimed on closing
the gap between the models of this kind and real chemical architectures.

##  6 Acknowledgements

The authors acknowledge financial support via the grant program 6541230
(fundamental research) of the Nat. Acad. Sci. of Ukraine “Self-assembly and
formation of structures in soft matter: analytical approaches and computer
simulations”, grant reg. no. 0121U110784.

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††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Far-from-equilibrium universal critical behavior of quantum geometry

Shuohang Wu  Wilczek Quantum Center and Key Laboratory of Artificial
Structures and Quantum Control, School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai 200240, China  Zhengxin Guo  Wilczek Quantum Center
and Key Laboratory of Artificial Structures and Quantum Control, School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Zi Cai<EMAIL_ADDRESS>Wilczek Quantum Center and Key Laboratory of
Artificial Structures and Quantum Control, School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China  Shanghai Research
Center for Quantum Sciences, Shanghai 201315, China

###### Abstract

In this study, we show that the quantum critical point in the ground state of
quantum many-body systems, can also govern the universal dynamical behavior
when the systems are driven far from equilibrium, which can be captured by the
evolution of the quantum geometry of the systems. By investigating quantum
quench dynamics in a variety of quadratic fermionic models, we find that the
quantum volume of these systems typically grows linearly over time, with a
growth velocity demonstrating universal behavior: its first derivative over
the control parameter exhibits a discontinuity at the quantum critical point,
with an universal jump value that is independent of specific models, but is
crucially determined by the system dimension. This result reveals universal
dynamical properties of non-equilibrium quantum many-body systems.

Introduction – The quantum phase transition, driven by quantum fluctuations
rather than by thermal motion, features non-analytic behavior in the ground
state of a quantum many-body system. Although it is impossible to cool matter
to ground state at absolute zero, the impact of a quantum critical point
extends across a broad range of low-energy excited states. This creates a
V-shaped quantum criticality regime that fans out to finite temperatures from
the quantum phase transition point Sachdev (1999). Such influence brings the
quantum criticality from a concept of theoretical interest to a tangible
phenomenon that critically determines the finite-temperature properties of
quantum critical materials Coleman and Schofield (2005). To date, most studies
have concentrated on the thermal equilibrium properties of quantum critical
systems, while the non-equilibrium aspects of the quantum criticality are far
less understoodDalla Torre et al. (2010).

The physics of non-equilibrium systems is generally more complex and richer
than in equilibrium ones. Regarding quantum criticality, substantial efforts
have explored universal dynamical behaviors near quantum critical points.
Examples include universal relaxation at a quantum critical point Sachdev and
Young (1997) and ramping dynamics Zurek et al. (2005); Dziarmaga (2005);
Damski (2005) governed by the Kimble-Zurek mechanism Kibble (1976); Zurek
(1985). These universal phenomena, despite their non-equilibrium nature,
involve only low-lying excited states above the critical transition point.
Compared to these near-equilibrium dynamics fundamentally tied to the
underlying quantum critical points in the ground state, the role of quantum
criticality in non-equilibrium dynamics far from the ground state remains less
clear. Notably, recent progress has shown intrinsic non-equilibrium critical
scaling in the quantum quench dynamics of long-range interacting systems Titum
and Maghrebi (2020); De et al. (2023). In this study, we reveal that even in a
system as simple as quadratic free fermions, the quantum phase transition in
the ground state can lead to universal non-analytic behavior in out-of-
equilibrium dynamics far from the ground state, which can be captured by the
evolution of the quantum geometryResta (2011).

To measure distances between quantum states, the quantum geometry is
introduced, which contains information of both the phase and amplitude
distances. The former is characterized by the Berry curvature, and the latter
by the quantum metric Provost and G.Vallee (1980); Matsuura and Ryu (2010); Ma
et al. (2010). Compared to the Berry curvature which has been intensively
studied in the context of topological physicsBohm et al. (2003), the quantum
metric has been rarely investigated until very recently. It has been measured
experimentally in various synthetic quantum systems including cold atoms Yi et
al. (2023), nitrogen-vacancy centers Yu et al. (2019), and superconducting
circuits Zheng et al. (2022) It is now understood that the full quantum
geometry, including the quantum metric, influences diverse phenomena, from
flat-band superconductivity and superfluidity Peotta and Törmä (2015); Julku
et al. (2016); Peotta et al. (2023); Tian et al. (2023); de Jesús Espinosa-
Champo and Naumis (2023) to the anomalous Hall effect Gianfrate et al. (2020);
Wang et al. (2021); Gao et al. (2023), quantum Fisher information Braunstein
and Caves (1994); Zanardi et al. (2007); Hauke et al. (2016), electron-phonon
interactions Yu et al. (2023), and fractional Chern insulators Parameswaran et
al. (2013); Neupert et al. (2015); Mera and Ozawa (2021a, b) (for a review,
see Ref.Törmä (2023)). Here, we demonstrate the crucial role of quantum metric
in interpreting out-of-equilibrium phenomena in quantum systems, particularly
in critical dynamics far from the ground state.

Figure 1: (Color online) (a)Quantum volume $g(t)$ for the 1D Kitaev model in
the quantum quench dynamics starting from the same initial states with
$\lambda_{i}=0.2J$, but evolving under the Hamiltonian with different
$\lambda_{f}$. (b)The growth velocity $v$ as a function of $\lambda_{f}$. The
inset is its derivative with respect to $\lambda_{f}$ , which exhibits a
discontinuity with an universal jump value $2\pi$ at $\lambda_{f}=J$.
(c)$\frac{dv}{d\lambda_{f}}$ as a function of $\lambda_{f}$ starting from
initial states with different $\lambda_{i}$, which share the same jump value
of $2\pi$ at $\lambda_{f}=J$.

Model and method – We examine a general quadratic fermionic model with
translational symmetry, whose Hamiltonian in momentum space is expressed as:

$H=\sum_{k}C^{\dagger}_{k}\hat{\mathcal{H}}_{k}C_{k}$ (1)

where $\hat{\mathcal{H}}_{k}=\vec{h}_{k}\cdot\vec{\sigma}$ represents a
$2\times 2$ matrix. Here,
$\vec{\sigma}=(\hat{\sigma}^{x},\hat{\sigma}^{y},\hat{\sigma}^{z})$ denotes
the Pauli matrices, and $\vec{h}_{k}=(h_{k}^{x},h_{k}^{y},h_{k}^{z})$ is a
three-dimensional vector. The annihilation and creation fermionic operators,
$C_{k}$ and $C^{\dagger}_{k}$ respectively, can be represented either as a
two-component fermion $C_{k}=[c_{\alpha,k},c_{\beta,k}]^{\text{T}}$, with the
indices $\alpha,\beta$ signifying the degrees of freedom (such as sublattice,
orbital, or spin), or through the Nambu representation as
$C_{k}=[c_{k},c_{-k}^{\dagger}]^{\text{T}}$, which describes a fermionic model
with pairing terms. The energy spectrum of the system is given by
$\epsilon_{k}^{\pm}=\pm\lvert d_{k}\rvert$, leading to an energy gap defined
as $\Delta_{k}=2\lvert d_{k}\rvert$. Typically, a quantum phase transition in
such a quadratic fermionic model is characterized by the closure of this gap
and a non-analytic behavior of $\Delta_{k}$ at one or more specific momenta
$k_{c}$.

The non-analytic behavior of $\Delta_{k}$ at the quantum critical point
affects not only static properties such as the ground state energy but also
has striking implications for the dynamics far from the ground state. To
elucidate this aspect, we examine the quantum quench dynamics where the
system, initially in the ground state of a Hamiltonian $H_{i}$, evolves
following an abrupt parameter change to a final Hamiltonian $H_{f}$. The
quench dynamics of such an integrable system has been extensively studied
Barthel and Schollwöck (2008); Calabrese et al. (2011); Mitra (2018); however,
we reexamine this problem from a novel perspective of quantum geometry. The
Hamiltonian in Eq.(1) is viewed as a set of independent $k$-modes, each is a
two level system evolving as a spin precession at a frequency $\Delta_{k}$.
Therefore, the singularity in $\Delta_{k}$ critically affects the system’s
dynamics, as manifested in the evolution of the quantum geometry.

Considering two $k$-modes with the initial states $|u^{0}_{k}\rangle$ and
$|u^{0}_{k+dk}\rangle$ that are close to each other, the difference in their
precession frequencies will separate these two states after a sufficiently
long time. The distance between the two states can be captured by the Fubini-
Study metric $\mathbb{B}(k)$, which is a rank-2 tensor for 2D systems:

$\mathbb{B}_{ij}(\mathbf{k})=\langle\partial_{i}u_{\mathbf{k}}|\partial_{j}u_{\mathbf{k}}\rangle-\langle\partial_{i}u_{\mathbf{k}}|u_{\mathbf{k}}\rangle\langle
u_{\mathbf{k}}|\partial_{j}u_{\mathbf{k}}\rangle,$ (2)

where $\partial_{i}=\frac{\partial}{\partial k_{i}}$ with $i=x,y$, and
$|u_{\mathbf{k}}\rangle$ is the wavefunction at momentum $\mathbf{k}$. The
imaginary part of $\mathbb{B}_{ij}(k)$ is the well-established Berry
curvature, which characterizes the phase differences of quantum states under a
small change of $\mathbf{k}$. The real part of $\mathbb{B}_{ij}(k)$ is the
quantum metric, which measures the orthogonality of two vectors and is
proportional to the amplitude distance of quantum states that are sufficiently
close to each other. To measure the average distance between the quantum
states at adjacent momenta, we introduce the quantum volume to characterize
the roughness of the manifold composed of the state vectors in the Brillouin
zone (BZ). The quantum volume $g$ is defined as

$g_{2D}=\int d\mathbf{k}\sqrt{\det\rm{Re}[\mathbb{B}_{ij}(\mathbf{k})]}$ (3)

where the integral is over the 1st BZ. For a 1D system, Eq.(2) reduces to a
rank-1 tensor (a number) as

$\mathbb{B}(k)=1-\langle\dot{u}_{k}|u_{k}\rangle|\langle
u_{k}|\dot{u}_{k}\rangle$ (4)

where $\dot{u}_{k}=du_{k}/dk$, and the quantum volume $g_{1D}=\int
dk\sqrt{\mathbb{B}(k)}$.

In quantum quench dynamics, we observe that the quantum volume increases
linearly with time, $g(t)\sim v[\lambda_{f}]t$, where the growth velocity
$v[\lambda_{f}]$ is dependent on the control parameter $\lambda_{f}$ in the
final Hamiltonian $H_{f}$. If $H_{f}$ undergo a quantum phase transition at
$\lambda_{f}=\lambda_{c}$, our key result is that the derivative of
$v[\lambda_{f}]$ demonstrates a universal discontinuity at the quantum
critical point:

$\frac{dv}{d\lambda_{f}}\bigg{|}_{\lambda_{f}=\lambda_{c}^{+}}-\frac{dv}{d\lambda_{f}}\bigg{|}_{\lambda_{f}=\lambda_{c}^{-}}=\mathcal{N}C_{d}$
(5)

where $\mathcal{N}$ is the number of the gap closing points in the 1st BZ, and
$C_{d}$ represents a dimensionless universal constant (the lattice constant is
set to be unit), independent on most system details including initial states
prior to the quench, the form of Hamiltonian $H_{f}$, and its energy spectrum
near the critical point. However, it is fundamentally determined by the system
dimension $d$.

$\displaystyle C_{d}=\left\\{\begin{array}[]{cl}2\pi,&d=1\\\ 8,&d=2\\\
\end{array}\right.$ (8)

Subsequently, we elucidate this phenomenon through specific examples in 1D and
2D systems.

1D models – The first model under consideration is the 1D Kitaev chain Kitaev
(2001), characterized by a Hamiltonian represented by Eq.(1) in momentum
space. Here, $C_{k}=[c_{k},c_{-k}^{\dagger}]$ denotes the spinless fermionic
operator in the Nambu representation, and the Hamiltonian vector is given by
Kitaev (2001)

$\vec{h}(k)=[0,i\Delta\sin k,-J\cos k-\lambda(t)],$ (9)

where $J$ and $\Delta$ represent the hopping and pairing amplitudes of
fermions on adjacent sites, respectively. For simplicity, we assume
$\Delta=J$. The term $\lambda(t)$, signifying the chemical potential, acts as
a time-dependent control parameter in this context. Notably, this model
facilitates a topological quantum phase transition in its ground state at the
critical threshold $\lambda_{c}=J$, where the gap closes at $k_{c}=\pi$, and
near the critical point, the energy spectrum approximates to $\epsilon_{k}\sim
J|k-k_{c}|$.

Figure 2: (Color online) (a)Quantum volume $g(t)$ for the 2D Qi-Wu-Zhang model
in the quantum quench dynamics starting from the same initial states with
$\lambda_{i}=-0.5J$, but evolving under the Hamiltonian with different
$\lambda_{f}$. (b) and (c) $\frac{dv}{d\lambda_{f}}$ as a function of
$\lambda_{f}$ around the different quantum critical points at (b)
$\lambda_{f}=-2J$ and (c) $\lambda_{f}=0$. The number of the gap closing
points in the 1st BZ is $\mathcal{N}=1$ ($\mathcal{N}=2$) for the critical
point at $\lambda_{f}=-2J$ ($\lambda_{f}=0$).

We examine the quench dynamics of the model, initiating from the ground state
of the Hamiltonian with a control parameter $\lambda=\lambda_{i}$. This
parameter is abruptly altered to $\lambda_{f}$, and we study the subsequent
evolution under the new Hamiltonian. The dynamics of $g_{1D}(t)$, originating
from the same initial state but evolving under various $\lambda_{f}$, is
depicted in Fig. 1 (a), revealing a linear increase in $g(t)$ with
superimposed oscillations. The velocity of growth $v$, as a function of
$\lambda_{f}$, is presented in Fig. 1 (b), displaying a distinct cusp at the
quantum critical point $\lambda_{f}=J$. This cusp signifies a discontinuity in
the first-order derivative of $v(\lambda_{f})$ at $\lambda_{f}=\lambda_{c}$,
as illustrated in the inset of Fig. 1 (b). Noteworthy is Fig. 1 (c), which
indicates that the magnitude of the discontinuity $C_{1}=2\pi$ the value is
independent on the initial states parameter $\lambda_{i}$.

To analytically derive $C_{1}$, we consider $\lambda_{i}=0$ for simplicity
(the cases with general initial states are discussed in the supplementary
material(SM)Sup ). The initial state of the $k$-th mode is
$|u^{0}_{k}\rangle=[\sin\frac{k}{2},i\cos\frac{k}{2}]^{T}$. At time $t$, its
wavefunction evolves to
$|u_{k}(t)\rangle=e^{-i\hat{\mathcal{H}}_{k}t}|u^{0}_{k}\rangle$. The
$\mathbb{B}(k,t)$, defined in Eq. (4), is given by:

$\mathbb{B}(k,t)=A_{k}(t)+B_{k}(t)t+R_{k}t^{2}$ (10)

where the time-dependent coefficients $A_{k}(t)$ and $B_{k}(t)$ are functions
of $\cos\epsilon_{k}t$ and $\sin\epsilon_{k}t$, thus oscillate with frequency
$\epsilon_{k}=\sqrt{\lambda_{f}^{2}+J^{2}+2\lambda_{f}J\cos k}$ Sup . The
quadratic term in Eq. (10), proportional to $t^{2}$, governs the long-time
dynamics, with $R_{k}=\frac{\lambda_{f}^{4}J^{2}\sin^{4}k}{\epsilon_{k}^{4}}$.
Following its definition in 1D, the quantum volume linearly increases over
time for sufficiently long periods as $g(t)\sim vt$, where the velocity
$v=\sum_{k}\sqrt{R_{k}}$. Calculating the momentum integral yields
$v(\lambda_{f})=\frac{\lambda_{f}^{2}\pi}{J}$ for
$\lambda_{f}\in[0,\lambda_{c}]$, and $v(\lambda_{f})=J\pi$ otherwise. Thus,
$C_{1}=2\pi$ is readily verified from its definition in Eq.(5).

For initial states characterized by $\lambda_{i}\neq 0$, the analytic
expression for $v(\lambda_{f})$ becomes more intricate. Nevertheless, the
relationship $C_{1}=2\pi$ remains valid as demonstrated in Ref.Sup ,
indicating its independence from the choice of the initial state. More
notably, the universal value of $C_{1}$ transcends specific models, provided
they feature a gap-closing point within their energy spectrum. Examples
include the Su-Schrieffer-Heeger model Su et al. (1979) and the free-fermion
model with staggered chemical potential (refer to Ref.Sup ).

This universal discontinuity stems from the fact the jump at
$\lambda_{f}=\lambda_{c}$ is attributed to the singularity in
$\epsilon_{k}(\lambda_{c})$ at $k=k_{c}$, consequently, only $k$-modes close
to $k_{c}$ are important in determining $C_{d}$, while other k-modes only
contributes to the analytic part of $v[\lambda_{c}]$. This observation
simplifies the analysis to these relevant $k$-modes, yielding an effective
model described by the Hamiltonian vector

$\vec{h}(k)=[Jk^{n},0,\lambda],\quad k\in[-\Lambda,\Lambda],$ (11)

where $\Lambda$ represents the ultraviolet cutoff in momentum space. Here, $n$
is an integer, with $n=1$ characterizing the Dirac fermion with mass
$\lambda$, corresponding to a low-energy effective depiction of the previously
discussed models. The energy spectrum gap,
$\epsilon_{k}=\sqrt{\lambda^{2}+J^{2}k^{2n}}$, vanishes at $k_{c}=0$ and
$\lambda=0$, where $\epsilon_{k}\sim|k|^{n}$. Subsequently, we will
demonstrate the independence of $C_{1}$ from this spectral function. For
brevity, we consider a simple initial state as the ground state of Eq.(11)
with $\lambda_{i}=\infty$, while the analysis of other initial states is
elaborated in the SM Sup . The growth rate is given by

$v(\lambda_{f})=\int^{\Lambda}_{-\Lambda}dk\frac{nJ^{3}|k|^{3n-1}}{\lambda_{f}^{2}+J^{2}k^{2n}}=2\Lambda^{n}J-2\lambda_{f}\arctan\frac{\Lambda^{n}J}{\lambda_{f}}$
(12)

From Eq.(12), differentiation with respect to $\lambda_{f}$ yields the jump
value at $\lambda_{f}=0$ as $C_{1}=2\pi$. The independence of $C_{1}$ from the
value of $n$ indicates that this universal jump is also insensitive to the
spectrum function close to the quantum critical point.

2D models – Generalizing these results to 2D systems is straightforward. We
consider the Qi-Wu-Zhang (QWZ) model Qi et al. (2006) as an example, with its
Hamiltonian vector defined as:

$\vec{h}_{QWZ}(\mathbf{k})=[J\sin k_{x},J\sin k_{y},J(\cos k_{x}+\cos
k_{y})+\lambda]$ (13)

where $(k_{x},k_{y})\in(-\pi,\pi]$. This model exhibits topological quantum
phase transitions at $\lambda=0,\pm 2J$ separately. In the cases of
$\lambda=0$, there are two gap closing points in the 1st BZ $\mathcal{N}=2$:
$\mathbf{k}_{c}^{1}=(0,\pi)$ and $\mathbf{k}_{c}^{2}=(\pi,0)$, while for
$\lambda=2J(-2J)$, there is only one gap closing point $\mathcal{N}=1$ at
$\mathbf{k}_{c}=(0,0)$ ($\mathbf{k}_{c}=(\pi,\pi)$). Analogous to the 1D
scenario, we initialize the system in the ground state of the Hamiltonian.
(13) with $\lambda=\lambda_{i}$, then abruptly change the parameter to
$\lambda_{f}$ and examine the subsequent evolution of the quantum volume
$g(t)$. Similar to the 1D case, $g(t)$ demonstrates linear growth over time,
as depicted in Fig. 2 (a). From this, we derive the growth velocity
$v(\lambda_{f})$ and evaluate its derivative with respect to $\lambda_{f}$.
Fig. 2(b) and (c) indicate that the universal jump values at $\lambda_{f}=0$
and $\lambda_{f}=-2J$ satisfy Eq.(5) with $C_{2}=8$ and $\mathcal{N}=2(1)$ for
$\lambda_{f}=0(-2J)$.

To ascertain that $C_{2}$ remains invariant across different models, we
investigate a Boron-Nitride-like lattice, characterized by a honeycomb lattice
with staggered chemical potential on alternating sublattice sites. The
Hamiltonian vector for the free fermion model on such a lattice is given by:

$\displaystyle\vec{h}_{BN}(\mathbf{k})=[$ $\displaystyle
J(\cos{k_{x}}+2\cos{\frac{k_{x}}{2}}\cos{\frac{\sqrt{3}k_{y}}{2}}),$ (14)
$\displaystyle
J(\sin{k_{x}}+2\sin{\frac{k_{x}}{2}}\cos{\frac{\sqrt{3}k_{y}}{2}}),\lambda]$

Here, the staggered chemical potential $\lambda$ acts as the mass term for
Dirac fermions. The gap closes at $\lambda=0$ at the two Dirac points
$\mathbf{k}_{c}^{1}=(0,\frac{4\pi}{3\sqrt{3}})$;
$\mathbf{k}_{c}^{2}=(\frac{2\pi}{3},\frac{2\pi}{3\sqrt{3}})$ in the 1st BZ.
Both phases flanking the critical points are topologically trivial. Fig. 3
confirms that $C_{2}=8$ with $\mathcal{N}=2$ for this model, indicating that
the universal jump is not necessarily linked to a topological quantum critical
point, but is universal for general models with gap closing in its energy
spectrum.

Figure 3: (Color online) In the quantum quench dynamics of a fermionic model
in a BN like lattice, $\frac{dv}{d\lambda_{f}}$ as a function of $\lambda_{f}$
starting from initial states with different $\lambda_{i}$, which also exhibit
universal jump at $\lambda_{f}=0$.

The universal constant $C_{2}$ can also be understood through a low-energy
effective model characterized by the Hamiltonian vector:

$\vec{h}(\mathbf{k})=[k_{x},k_{y},\lambda],\quad|k|\leq\Lambda,$ (15)

and the energy spectrum
$\epsilon_{\mathbf{k}}(\lambda)=\sqrt{|\mathbf{k}|^{2}+\lambda^{2}}$. At the
ground state, this model undergoes a third-order quantum phase transition at
$\lambda=0$ Cai et al. (2008). Beginning with the initial state where
$\lambda_{i}=\infty$, the growth velocity can be calculated as:

$v(\lambda_{f})=\int^{\Lambda}_{0}kdk\frac{2k^{2}}{(\lambda_{f}^{2}+k^{2})^{\frac{3}{2}}}=\frac{2[|\lambda_{f}|-\epsilon_{k}(\lambda_{f})]^{2}}{\epsilon_{k}(\lambda_{f})}$
(16)

The singularity in $v(\lambda_{f})$ arises due to the $|\lambda_{f}|$ term in
Eq. (16). From its definition in Eq. (5), the discontinuity at $\lambda_{f}=0$
yields a jump value of $C_{2}=8$.

Conclusion and outlook – In conclusion, we demonstrate that quantum
criticality can govern the non-equilibrium dynamics far from the ground state
and induce universal dynamical behavior. Further developments include
extending these results to interacting fermionic models. Although defining a
quantum metric for this case is challenging, recent progress about interacting
systems with well-resolved quasi-particle poles Kruchkov and Ryu (2023) have
shed light on this issue. Another significant direction involves rigorously
proving the universal jump value and investigating its potential topological
or geometrical origins, which would deepen our understanding of this universal
phenomena.

Acknowledgments.— This work is supported by the National Key Research and
Development Program of China (Grant No. 2020YFA0309000), NSFC of China (Grant
No.12174251), Natural Science Foundation of Shanghai (Grant No.22ZR142830),
Shanghai Municipal Science and Technology Major Project (Grant
No.2019SHZDZX01).

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