One-dimensional Fermi polaron after a kick:
two-sided singularity of the momentum distribution,
Bragg reflection and other exact results

O. Gamayun1*, O. Lychkovskiy

1London Institute for Mathematical Sciences,

Royal Institution, 21 Albemarle St, London W1S 4BS, UK

*<EMAIL_ADDRESS>
## Abstract

A mobile impurity particle immersed in a quantum fluid forms a polaron – a
quasiparticle consisting of the impurity and a local disturbance of the fluid
around it. We ask what happens to a one-dimensional polaron after a kick, i.e.
an abrupt application of a force that instantly delivers a finite impulse to
the impurity. In the framework of an integrable model describing an impurity
in a one-dimensional gas of fermions or hard-core bosons, we calculate the
distribution of the polaron momentum established when the post-kick relaxation
is over. A remarkable feature of this distribution is a two-sided power-law
singularity that can correspond to one of two processes. In the first process,
the whole impulse is transferred to the polaron, without creating phonon-like
excitations of the fluid. In the second process, the impulse is shared between
the polaron and the center-of-mass motion of the fluid, again without creating
any fluid excitations. The latter process is, in fact, a Bragg reflection at
the edge of the emergent Brillouin zone. We carefully analyze the conditions
for each of the two cases and derive the asymptotic form of the distribution
in the vicinity of the singularity.

###### Contents

  1. 1 Introduction
  2. 2 McGuire model and its Bethe ansatz solution
    1. 2.1 The model
    2. 2.2 Bethe ansatz
    3. 2.3 Spectrum and a Bethe equation in the thermodynamic limit
    4. 2.4 Fermi polaron
  3. 3 Polaron after a kick
    1. 3.1 Distribution over polaron rapidities after a general quantum quench
    2. 3.2 Preparing the initial state by kicking the impurity
    3. 3.3 Polaron momentum distribution after the kick
    4. 3.4 Two-sided power-law singularity of the distribution
  4. 4 Discussion and outlook
  5. A Kick as a quantum quench
  6. B Momentum distribution: derivation
    1. B.1 Preliminary considerations
    2. B.2 Matrix element
    3. B.3 From sum over eigenstates to sum over independent integers
    4. B.4 Handling $\left(\det D\right)^{2}$
    5. B.5 Factorized representation of $\delta(\Lambda-\Lambda_{{\bf n},M})$
    6. B.6 Combining the pieces
  7. C Asymptotic analysis of the singularity
    1. C.1 Asymptotics of the Fredholm determinant
    2. C.2 Singularity of the momentum distribution

##  1 Introduction

The behavior of an impurity particle propagating in a host media is a
paradigmatic problem in physics. To address this problem, a concept of polaron
was introduced by Landau [1] and Pekar [2] at the down of the quantum theory
of condensed matter. A polaron is a quasiparticle consisting of the impurity
along with the local disturbance of the host media cased by the interaction
between the impurity and host particles. Polaron properties, such as mass or
dispersion relation, can be quite different from that of the bare impurity
[3]. The polaron framework can be universally applied to virtually any
combination of impurity particle and host media [4, 5, 6, 7].

Current experimental advance in the ultracold atomic gases allows one to
create and control two-component mixtures with large concentration imbalance,
thus offering a novel, extremely flexible platform for studying physics of
polarons [8, 9, 10, 11, 12, 13, 14, 15]. Importantly, such experiments can be
performed in the reduced spatial geometries, where effects of interactions are
more pronounced.

One-dimensional polarons are particularly remarkable. Their distinctive
feature is the ability to move perpetually at zero temperature – an effect
reminiscent but not identical to superfluidity [16, 17, 18]. This effect is
universal and stems from the non-trivial spectral edge of any one-dimensional
fluid [19]. It implies that the polaron momentum is a bona-fide quantum number
(at least at zero temperature).

A steadily increasing deal of attention is being attracted by non-equilibrium
aspects of the polaron formation and dynamics [20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].
Valuable insights into the nonequilibrium polaron physics [16, 22, 33] come
from studies of a one-dimensional integrable model introduced in 1965 by
McGuire [47, 48]. Importantly, integrability facilitates non-perturbative
analytical investigation of the strong correlation effects hardly available
otherwise.

In this paper, we consider the effect of kicking the polaron in the McGuire
model at zero temperature. In the context of cold atom experiments, the kick
can be realized by photon scattering or absorption, or by moving optical
tweezers [49]. On a formal level, the kick corresponds to applying an external
force to the impurity within a short time interval, so that the impurity
acquires a finite impulse. The kick is followed by a relaxation, when the
acquired momentum is shared between the polaron and the excitations of the
fluid created by the kick. The relaxation is effectively over when all created
fluid excitations have detached and moved away from the polaron, see Fig. 1.
We calculate and analyze the probability distribution of the polaron momentum
in the thus established (quasi-)steady state.

The rest of the paper is structured as follows. In the next section we provide
a concise but self-contained description of the McGuire model and its Bethe
ansatz solution, with the focus on the polaron properties. This section is
based on the prior literature [47, 50, 51, 52]. Original results are presented
in Section 3. There we provide the polaron momentum distribution and analyze
its singularity structure. The last section discusses the implications of the
results in a broader context. Derivations and proofs can be found in the
Appendix.

##  2 McGuire model and its Bethe ansatz solution

Figure 1: A cartoon of the out-of-equilibrium protocol under study. A polaron,
initially in equilibrium, is kicked by an instant application of a force to
the impurity. The kick changes the polaron momentum and creates excitations of
the fluid. We describe the steady state of the polaron after the relaxation is
over and all fluid excitations have broken apart.

###  2.1 The model

We use a simple and yet non-trivial integrable model of a polaron. This model
was introduced by McGuire who obtained its Bethe-ansatz solution in 1965 [47,
48]. Later the McGuire model turned out to represent a specific sector of a
more general Yang-Gaudin model [53, 54]. Much later it was realized that the
Bethe eigenstates of the McGuire model can be represented as Slater-like
determinants [55, 50, 16]. This discovery opened an avenue for a flurry of
exact results [56, 50, 16, 22, 57, 51, 33, 52, 58, 59]. In the present section
we review the McGuire model and its solution. The exposition mostly follows
ref. [51].

McGuire model describes a single impurity particle immersed into a one-
dimensional gas of $N$ spinless fermions (or, equivalently [19], hard-core
bosons), the mass of the impurity and a fermion being the same. Fermions do
not interact one with another but interact with the impurity. We work in the
first quantization, where the Hamiltonian of the model reads

$\begin{array}[]{c}\displaystyle H=\frac{1}{2}P_{\rm
imp}^{2}+\frac{1}{2}\sum_{j=1}^{N}P_{j}^{2}+\frac{2\,p_{\rm
F}}{\alpha}\sum_{j=1}^{N}\delta(x_{j}-x_{\rm imp}).\end{array}$ (1)

Here, $x_{j}$ ($P_{j}$) is the coordinate (momentum) of the $j$’th fermion,
$j=1,\dots,N,$ and $x_{\rm imp}$ ($P_{\rm imp}$) is that of the impurity.
Translation invariance is imposed by introducing periodic boundary conditions
with the circumference $L$. Any wave function should be periodic in any
coordinate with the period $L$ and antisymmetric in fermionic coordinates. The
number of fermions, $N$, is assumed to be odd. The Fermi momentum is defined
as $p_{\rm F}=\pi(N-1)/L$. We will be interested in the thermodynamic limit of
$N,L\rightarrow\infty$ with $p_{\rm F}$ being fixed. Only the case of
repulsion will be considered here, which corresponds to a positive interaction
strength $(2\,pF)/\alpha$.

The total momentum $P_{\rm tot}=P_{\rm imp}+\sum_{j=1}^{N}P_{j}$ is an
integral of motion of the model. Its eigenvalues (denoted by the same symbol
$P_{\rm tot}$) are quantized with the momentum quantum $2\pi/L$,

$P_{\rm tot}=\frac{2\pi}{L}M,$ (2)

where $M$ is an integer.

###  2.2 Bethe ansatz

The eigenstates of the McGuire model are labelled by $N+2$ integers. One of
them is $M$, and others are organized in an ordered set ${\bf
n}=\\{n_{1},n_{2},\dots n_{N+1}\\}$, $n_{1}<n_{2}<\dots<n_{N+1}$ satisfying
the constraint

$\sum_{l=1}^{N+1}n_{l}\in[M,M+N].$ (3)

An eigenstate $|{\bf n},M\rangle$ is given by

$|{\bf n},M\rangle={\mathcal{N}}\,e^{iP_{\rm tot}x_{\rm
imp}}\left|\begin{array}[]{cccc}e^{ik_{1}y_{1}}&e^{ik_{2}y_{1}}&...&e^{ik_{N+1}y_{1}}\\\
e^{ik_{1}y_{2}}&e^{ik_{2}y_{2}}&...&e^{ik_{N+1}y_{2}}\\\ .&.&.&.\\\ .&.&.&.\\\
e^{ik_{1}y_{N}}&e^{ik_{2}y_{N}}&...&e^{ik_{N+1}y_{N}}\\\\[5.0pt]
e^{-i\delta_{1}}\sin\delta_{1}&e^{-i\delta_{2}}\sin\delta_{2}&...&e^{-i\delta_{N+1}}\sin\delta_{N+1}\end{array}\right|,$
(4)

where $\cal N$ is a normalization constant, $y_{j}\equiv(x_{j}-x_{\rm
imp})\,{\rm mod}\,L$ is the position of the $j$’th fermion relative to the
position of the impurity, and $(N+1)$ pseudomomenta $k_{l}$ are fixed by
integers $n_{l}$ up to phase shifts $\delta_{l}\in[0,\pi)$,

$k_{l}=\frac{2\pi}{L}\left(n_{l}-\frac{\delta_{l}}{\pi}\right),\qquad
l=1,2,\dots,N+1.$ (5)

The phase shifts $\delta_{l}$ should be found from Bethe equations. We do not
need the complete set of Bethe equations since in the thermodynamic limit and
for eigenstates in the bottom of the spectrum (whose precise meaning is
discussed in the next subsection) the it reduces to a single equation on an
auxiliary but very important variable – polaron rapidity
$\Lambda\in(-\infty,+\infty)$. This is the equation (8) introduced in the next
subsection. It has a single root $\Lambda_{{\bf n},M}$ whenever the constraint
(3) is satisfied, and no roots otherwise. The $(N+1)$ phase shifts are
expressed through this root as

$\delta_{l}=\frac{\pi}{2}-\arctan\left(\Lambda_{{\bf
n},M}-\frac{2\pi}{L}\,\alpha\,n_{l}\right),$ (6)

up to corrections negligible in the thermodynamic limit.

###  2.3 Spectrum and a Bethe equation in the thermodynamic limit

Figure 2:  Schematic dependence of the polaron energy $\cal E$, velocity $v$
and rapidity $\Lambda$ on the polaron momentum $Q$. The units of the vertical
axis are arbitrary. $\cal E(Q)$ is shifted by a constant to ensure ${\cal
E}(0)=0$.

The total momentum $P_{\rm tot}$ and energy $E_{\rm tot}$ of the eigenstate
$|{\bf n},M\rangle$ can be expressed through the corresponding pseudomomenta,

$\displaystyle P_{\rm tot}=$ $\displaystyle\sum_{l=1}^{N+1}k_{l},$
$\displaystyle E_{\rm tot}=$
$\displaystyle\frac{1}{2}\sum_{l=1}^{N+1}k_{l}^{2}.$ (7)

In fact, the right hand sides (r.h.s.) of these equations give the expectation
values of operators $P_{\rm tot}$ and $H$, respectively, for a state of the
form (4) for arbitrary $k_{l}$, not necessarily the solutions of Bethe
equations. In the thermodynamic limit, one can obtain the sole relevant Bethe
equation on the rapidity by plugging expressions (5),(6) into the first
equation (7) and replacing the sum by the integral. The resulting equation on
$\Lambda$ reads

$Q(\Lambda)=\frac{2\pi}{L}\left(M-\sum_{l=1}^{N+1}\left(n_{l}-\frac{1}{2}\right)\right),$
(8)

where the function $Q(\Lambda)$ is defined as

$\frac{Q(\Lambda)}{p_{\rm
F}}=\frac{1}{\pi\alpha}\left(\left(\Lambda+\alpha\right)\arctan\left(\Lambda+\alpha\right)-\left(\Lambda-\alpha\right)\arctan\left(\Lambda-\alpha\right)+\frac{1}{2}\log\frac{1+(\alpha-\Lambda)^{2}}{1+(\alpha+\Lambda)^{2}}\right).$
(9)

As we discuss in the next subsection, $Q(\Lambda)$ is interpreted as the
polaron momentum. It is a monotonically increasing function of $\Lambda$ (see
Fig. 2), therefore the equation (8) has at most one root $\Lambda_{{\bf
n},M}$. Further, $Q(\Lambda)\in[-p_{\rm F},p_{\rm F}]$, which implies that a
root exists whenever the constraint (3) is satisfied.

For a given total momentum, one can define a (momentum-dependent) ground
state. Its energy is denoted by ${\cal E}(P_{\rm tot})$. The latter function
constitutes the lower edge of the many-body spectrum. In the thermodynamic
limit, it is periodic with the period $2p_{\rm F}$. The spectrum is therefore
divided into Brillouin zones of width $2p_{\rm F}$. The main Brillouin zone
corresponds to $P_{\rm tot}\in[-p_{\rm F},p_{\rm F}]$.

The set ${\bf n}$ of each ground state consists of consecutive integers. All
eigenstates within a Brillouin zone share the same set ${\bf n}$ but have
different $M$. The main Brillouin zone corresponds to the set ${\bf
n}=\\{-(N-1)/2,\dots,(N+1)/2\\}$. This implies $P_{\rm tot}=Q(\Lambda_{{\bf
n},M})$ for each ground state $|{\bf n},M\rangle$ from the main Brillouin
zone. Other Brillouin zones have total momentum $P_{\rm tot}=Q(\Lambda_{{\bf
n},M})+2\,m\,p_{\rm F}$, where an integer $m$ is the number of the zone. The
momentum shift $(2\,m\,p_{\rm F})$ corresponds to the center-of-mass motion of
the Fermi gas.

We say that an eigenstate is in the bottom of the spectrum whenever its energy
differs from ${\cal E}(P_{\rm tot})$ by a value that is $O(1)$ in the
thermodynamic limit. The set ${\bf n}$ for such an eigenstate differs from
that for a ground state by “particle-hole excitations” whose number is $o(N)$
in the thermodynamic limit.

###  2.4 Fermi polaron

The impurity along with the disturbance of the Fermi gas around it is commonly
known as Fermi polaron (another term “depleton” is also used in the one-
dimensional context [60, 61, 62]). In the McGuire model the polaron can be
characterized in a remarkably precise and rigorous way. Namely, it turns out
that, for any eigenstate in the bottom of the spectrum, any local property of
the impurity (e.g. its momentum distribution, static correlation function etc)
depends only on the corresponding polaron rapidity. In other words, if the
rapidities of two eigenstates $|{\bf n},M\rangle$ and $|{\bf
n}^{\prime},M^{\prime}\rangle$ are identical up to finite size corrections,
$\Lambda_{{\bf n}^{\prime},M^{\prime}}=\Lambda_{{\bf n},M}+O(1/N)$, then local
properties of the impurity in these two states are also identical up to finite
size corrections. Therefore one can consistently interpret an eigenstate
$|{\bf n},M\rangle$ as containing a polaron with the rapidity $\Lambda_{{\bf
n},M}$ and a certain number of Fermi sea excitations. The latter do not alter
local polaron properties since their density vanishes in the thermodynamic
limit.

Ground states is the main Brillouin zone contain only a polaron, without
additional excitations of the Fermi sea. For this reason the momentum
$Q(\Lambda)$ and the energy ${\cal E}(Q(\Lambda))={\cal E}(\Lambda)$ of such
ground states are interpreted as the polaron momentum and energy,
respectively. The polaron energy is explicitly given by

$\frac{{\cal E}(\Lambda)}{p_{\rm
F}^{2}}=\frac{1}{\pi\alpha}-\frac{1+\alpha^{2}-\Lambda^{2}}{2\pi\alpha^{2}}\big{(}\arctan\left(\Lambda+\alpha\right)+\arctan\left(\Lambda-\alpha\right)\big{)}+\frac{\Lambda}{2\pi\alpha^{2}}\,\log\frac{1+(\alpha-\Lambda)^{2}}{1+(\alpha+\Lambda)^{2}}.$
(10)

A distinct feature of a polaron in one dimension is that it can move
perpetually with a velocity below a critical one (no matter whether the model
is integrable or not) [16, 17, 18, 22]. The critical velocity does not exceed
the speed of sound in the medium hosting the polaron [17, 18]. The velocity
operator is defined through the Heisenberg equation as $i[H,x_{\rm imp}]$ and,
for the Hamiltonian (1), coincides with the impurity’s momentum operator
$P_{\rm imp}$.

In the McGuire model the impurity’s velocity $v(\Lambda)$ is expressed through
the polaron rapidity as

$\frac{v(\Lambda)}{v_{\rm F}}=\langle{\bf n},M|P_{\rm imp}|{\bf
n},M\rangle=\frac{\Lambda}{\alpha}+\frac{1}{2\alpha}\,\frac{\log\frac{1+(\alpha-\Lambda)^{2}}{1+(\alpha+\Lambda)^{2}}}{\arctan(\alpha-\Lambda)+\arctan(\alpha+\Lambda)},$
(11)

where $v_{\rm F}=p_{\rm F}$ is the Fermi velocity, see Fig. 2. One can verify
that the polaron velocity satisfies the usual relation for the group velocity,

$v(\Lambda)=\frac{\partial{\cal E}}{\partial\Lambda}\left(\frac{\partial
Q}{\partial\Lambda}\right)^{-1}=\frac{\partial{\cal E}}{\partial Q}.$ (12)

Explicit formula for the complete velocity distribution of an impurity in a
polaron eigenstate can be found in ref. [52].

Since $Q(\Lambda)$ is a monotonic function, it can be inverted,
$\Lambda=\Lambda(Q)$. As a consequence, the polaron can be unambiguously
labelled not only by its rapidity $\Lambda\in(-\infty,\infty)$ but also by its
momentum $Q\in[-p_{\rm F},p_{\rm F}]$.

##  3 Polaron after a kick

###  3.1 Distribution over polaron rapidities after a general quantum quench

A quantum quench initializes a system in an out-of-equilibrium state $|{\rm
in}\rangle$. The quench is followed by relaxation and, eventually, by
establishing a post-quench equilibrium state. The long-time expectation value
${\cal O}_{\infty}$ of an observable $\cal O$ can be obtained by averaging the
observable over the diagonal ensemble [63]. Specifically, employing our
notations for eigenstates, one obtains

${\cal O}_{\infty}=\sum_{|{\bf n},M\rangle}\big{|}\langle{\bf n},M|{\rm
in}\rangle\big{|}^{2}\,\langle{\bf n},M|{\cal O}|{\bf n},M\rangle.$ (13)

The particular quench studied in the present paper excites only eigenstates in
the bottom of the spectrum. As discussed above, for such states the diagonal
matrix element of any polaron observable depends only on the polaron momentum,
$\langle{\bf n},M|{\cal O}|{\bf n},M\rangle={\cal O}(\Lambda_{{\bf n},M})$.
Therefore, it makes sense to rewrite eq. (13) as

${\cal O}_{\infty}=\int_{-\infty}^{\infty}dQ\,\Gamma(\Lambda)\,{\cal
O}(\Lambda),$ (14)

where

$\Gamma(\Lambda)=\sum_{|{\bf n},M\rangle}\big{|}\langle{\bf n},M|{\rm
in}\rangle\big{|}^{2}\,\delta(\Lambda-\Lambda_{{\bf n},M})$ (15)

is the probability distribution of the equilibrium post-quench state over
polaron rapidities.

Eq. (14) allows for a transparent physical interpretation. The post-quench
equilibrium state features a polaron of rapidity $\Lambda$ with the
probability $\Gamma(\Lambda)$. In addition, this state contains Fermi sea
excitation that, however, break off far apart from the polaron (since their
velocity always exceeds that of the polaron [17, 18, 64]) and thus have no
effect on the observables related to the impurity.

###  3.2 Preparing the initial state by kicking the impurity

In the present paper we consider a specific way to prepare the initial out-of-
equilibrium state. It consists of applying a large force $F$ to the impurity
over a small time interval $\tau$. We consider the limit of
$F\rightarrow\infty$, $\tau\rightarrow 0$ with the delivered impulse $\Delta
P=F\tau$ fixed. This may be viewed as an instant kick applied to the impurity.

We assume that prior to the kick the system is in an eigenstate $|{\bf
n}^{0},M^{0}\rangle$ with the polaron rapidity $\Lambda_{0}\equiv\Lambda_{{\bf
n}^{0},M^{0}}$ and momentum $Q_{0}=Q(\Lambda_{0})$. This eigenstate is assumed
to belong to the bottom of the spectrum. Naively, kicking the impurity at time
$t=0$ can be described by adding the term $-F\,x_{\rm imp}\,\delta(t/\tau)$ to
the Hamiltonian (1). Then the out-of-equilibrium state immediately after the
kick reads

$|{\rm in}\rangle=e^{i\,\Delta P\,x_{\rm imp}}|{\bf n}^{0},M^{0}\rangle.$ (16)

The above simple consideration is not rigorous since the linear potential
breaks the translation invariance and is incompatible with periodic boundary
conditions. Nevertheless, eq. (16) remains correct, provided $\Delta P$ is an
integer of momentum quanta $2\pi/L$ which we assume in what follows. We
justify eq. (16) in Appendix A by employing a more elaborate (although
somewhat cumbersome) argument.

###  3.3 Polaron momentum distribution after the kick

Figure 3:  Left: Color plots of $\Gamma_{Q_{0}\rightarrow Q}$ for various
impulses $\Delta P$. Bragg reflection is clearly seen for smaller values of
$\Delta P$. The bright green color indicates the position of the singularity.
Top right: The range of the initial polaron momentum $Q_{0}$ and impulse
$\Delta P$ where the post-kick polaron momentum distribution
$\Gamma_{Q_{0}\rightarrow Q}$ has a two-sided power law singularity. Points
indicate values of $Q_{0}$ and $\Delta P$ chosen for the bottom right plot.
Dashed lines mark the boundaries of the Brillouin zone. Bottom right: Three
typical shapes of $\Gamma_{Q_{0}\rightarrow Q}$ as a function of $Q$ (with
$Q_{0}=0$ and values $\Delta P$ specified in the plot). For $\Delta
P=0.3p_{\rm F}$ the kick acts within the main Brillouin zone and
$\Gamma_{Q_{0}\rightarrow Q}$ features a sharp singularity at $Q=Q_{0}+\Delta
P$. For $\Delta P=1.1p_{\rm F}$ the Bragg reflection from the Brillouin zone
boundary occurs, the singularity at $Q=Q_{0}+\Delta P-2p_{\rm F}$ is
accompanied by a broad maximum on the opposite side of the Brillouin zone. For
$\Delta P=1.7p_{\rm F}$ the singularity is absent. The coupling constant is
$\alpha=2$.

Here we present the result for the polaron rapidity distribution established
as a result of relaxation after the kick. We choose to employ a more detailed
notation $\Gamma_{\Lambda_{0}\rightarrow\Lambda}$ for this distribution. It
has the same meaning as $\Gamma(\Lambda)$ in eq. (14) but explicitly contains
the pre-kick rapidity $\Lambda_{0}$. The new notation highlights its physical
meaning: $\Gamma_{\Lambda_{0}\rightarrow\Lambda}$ is the density of the
probability that the polaron with the initial rapidity $\Lambda_{0}$ will
acquire the rapidity $\Lambda$ after the kick with the impulse $\Delta P$.

The explicit expression for the rapidity distribution is the first main result
of the present paper. It reads

$\Gamma_{\Lambda_{0}\rightarrow\Lambda}=\frac{\Delta
P^{2}}{(\Lambda-\Lambda_{0})^{2}\,\,Q^{\prime}(\Lambda_{0})}{\rm
Re}\int\limits_{0}^{\infty}\frac{dx}{\pi}e^{-i\,\Delta P\,x}\det(1+\hat{K}).$
(17)

Here $\det(1+\hat{K})$ is a Fredholm determinant, and the operator $\hat{K}$
acts on $L^{2}[-p_{\rm F},p_{\rm F}]$ and has an integrable kernel

$\begin{array}[]{c}\displaystyle
K(q,q^{\prime})=(\Lambda_{0}-\Lambda)^{2}\,\,\frac{e^{-ix(q+q^{\prime})/2}}{\pi\sqrt{(\alpha
q/p_{\rm F}-\Lambda_{0})^{2}+1}\,\,\sqrt{(\alpha q^{\prime}/p_{\rm
F}-\Lambda_{0})^{2}+1}}\,\frac{e(q)-e(q^{\prime})}{q-q^{\prime}}\,p_{\rm
F},\end{array}$ (18) $\begin{array}[]{c}\displaystyle
e(q)=\frac{e^{ixq}-e^{ix\Lambda p_{\rm F}/\alpha}e^{-x\,p_{\rm
F}/\alpha}}{\alpha\,q/p_{\rm
F}-\Lambda-i}-\frac{e^{ixq}}{\Lambda_{0}-\Lambda}.\end{array}$ (19)

Note that $\hat{K}$ depends on $x$, $\Lambda_{0}$, $\Lambda$ and $\alpha$ as
parameters. Note also a shorthand notation $Q^{\prime}(\Lambda)=\partial
Q/\partial\Lambda$ introduced in eq.(17).

One can view the Fredholm determinant as a thermodynamic limit of a finite
determinant with matrix elements $\delta_{ij}+(2\pi/L)K(q_{i},q_{j})$, where
$q_{i},q_{j}\in[-p_{\rm F},p_{\rm F}]$ are quantized momenta with momentum
quantum $2\pi/L$. More details on properties of Fredholm determinants and an
algorithm for their effective numerical computation can be found in ref. [65].
The derivation of eq. (17) is presented in Appendix B.

Since the polaron momentum $Q$ is a more intuitive quantity compared to the
polaron rapidity $\Lambda$, we also introduce the polaron momentum
distribution $\Gamma_{Q_{0}\rightarrow Q}$. Its physical meaning is as
follows: $\Gamma_{Q_{0}\rightarrow Q}$ is the density of the probability that
the polaron with the initial momentum $Q_{0}$ will acquire the momentum $Q$
after the kick with the impulse $\Delta P$. The two distributions are related
as

$\Gamma_{Q_{0}\rightarrow
Q}=\Gamma_{\Lambda_{0}\rightarrow\Lambda}/Q^{\prime}(\Lambda)$ (20)

Plots of the momentum distribution for various impulses and initial polaron
momenta is shown in Fig. 3. The distribution can feature two related effects,
the Bragg reflection from the edge of the Brillouin zone and power-law
singularities. Both of them are discussed in detail in the next subsection.

The polaron momentum can be hard to measure in the experiment since it is
shared between the impurity and the accompanying disturbance of the Fermi sea.
The polaron rapidity also does not have a direct operational meaning. In
contrast, the polaron velocity coincides with the velocity of the impurity and
thus can be readily measured. One can straightforwardly obtain the velocity
distribution $\Gamma_{v_{0}\rightarrow v}$ from the rapidity distribution (17)
and expression (11) for $v(\Lambda)$, keeping in mind that the latter map is
two-to-one and thus folding should be employed.

To compute the average polaron velocity in the post-quench equilibrium state,
one should convolve $v(\Lambda)$ with $\Gamma_{\Lambda_{0}\rightarrow\Lambda}$
according to eq. (14). In practice, the integration in eq. (14) is performed
(analogously to those in ref. [33]) over a deformed contour in the complex
plane. The result is shown in Fig. 4. It is compared with an analogous result
for a different quench protocol described in ref. [33] – injection of the
impurity with the momentum $\Delta P$ into initially undisturbed Fermi sea.
Although these two protocols, the injection and the kick, are very similar at
the first sight, they result in quite different values of the equilibrium
impurity velocity, particularly for larger interactions strength.

Figure 4: Steady state polaron velocity established after the kick as a
function of the impulse (solid lines). The initial polaron momentum is
$Q_{0}=0$. Different curves correspond to different coupling constants
$2\pi/\alpha=3,\,6,\,10$ (from top to bottom). For comparison, the results for
a different quench protocol are shown (dashed lines) [33], where a bare
impurity is injected in the initially undisturbed fluid with the momentum
$\Delta P$.

###  3.4 Two-sided power-law singularity of the distribution

The singularity in the distribution $\Gamma_{Q_{0}\rightarrow Q}$ is present
for the range of parameters depicted in the top right panel of Fig. 3 and
analytically specified below in eq. (25). It can be obtained from the
asymptotics of the Fredholm determinant, see Appendix C. The result reads

$\Gamma^{\rm sing}_{Q_{0}\to Q}=\begin{dcases}\frac{C_{+}}{(Q-Q_{\rm
sing})^{1-(\xi_{m}^{-})^{2}-(\xi_{m}^{+})^{2}}},&Q>Q_{\rm
sing},\\\\[11.99998pt] \frac{C_{-}}{(Q_{\rm
sing}-Q)^{1-(\xi_{m}^{-})^{2}-(\xi_{m}^{+})^{2}}},&Q<Q_{\rm
sing}.\end{dcases}$ (21)

The position of the singularity $Q_{\rm sing}$ is determined by the value of
$(Q_{0}+\Delta P)$, namely,

$Q_{\rm sing}=Q_{0}+\Delta P-2\,m\,p_{\rm F},$ (22)

where the integer $m$ is chosen such that

$Q_{\rm sing}\in[-p_{\rm F},p_{\rm F}].$ (23)

The exponent of the singularity is constructed from

$\xi_{m}^{\pm}=\frac{1}{\pi}\,\Big{(}\arctan(\Lambda_{0}\mp\alpha)-\arctan(\Lambda\mp\alpha)\Big{)}-m,$
(24)

where $\Lambda=\Lambda(Q)$ and $\Lambda_{0}=\Lambda(Q_{0})$. The constants
$C_{\pm}$ are given in Appendix C, see eq. (52). The true distribution
$\Gamma_{Q_{0}\to Q}$ is close to $\Gamma^{\rm sing}_{Q_{0}\to Q}$ for polaron
momenta in the vicinity of $Q_{\rm sing}$.

From the mathematical point of view, the singularity (21) is similar in origin
to the threshold X-ray singularity [66]. While there is no any threshold here,
the amplitudes $C_{\pm}$ of the left and right parts of the singularity
differ, therefore it resembles two threshold singularities glued together. We
adopt the term two-sided for such type of singularity.

The condition for the existence of the singularity is that the exponent in the
asymptotics (21) is positive,

$1-(\xi_{m}^{-})^{2}-(\xi_{m}^{+})^{2}>0.$ (25)

In general, this is a complicated nonlinear condition, see Fig. 3 for
illustration. However, one simple fact about it can be easily obtained: the
condition is never satisfied when $|m|\geq 2$. Therefore, in fact only $m=0$
or $m=\pm 1$ are allowed. Let us discuss these two cases separately.

When $m=0$, the kick acts within the main Brillouin zone, i.e. $Q_{0}+\Delta
P\in[-p_{\rm F},p_{\rm F}]$. In this case the singularity corresponds to the
process where the whole impulse of the kick is transferred to the polaron,
with no Fermi sea excitations created. The smaller is the impulse $\Delta P$,
the more probability weight is concentrated in the vicinity of the
singularity, see Fig. 3. In the limit of vanishing $\Delta P$, the singularity
turns into the delta-function embracing all the weight, i.e.

$\Gamma_{Q_{0}\to Q}\xrightarrow{\Delta P\to 0}\delta(Q-Q_{0}),$ (26)

see Appendix C for the proof.

When $m=\pm 1$, the kick drives the system through the boundary between two
Brillouin zones. In this case the Bragg reflection occurs: The impulse of the
kick is shared between the polaron and the center-of-mass motion of the Fermi
sea, the latter acquiring the momentum $\pm 2p_{\rm F}$. Again, no Fermi sea
excitations are created at $Q=Q_{\rm sing}$. One can see from Fig. 3 that in
this case the singularity carries a relatively low share of probability
weight, with the majority weight being carried by a broad peak on the opposite
side of the Brillouin zone.

##  4 Discussion and outlook

We have calculated the polaron momentum distribution established after a kick.
The kick can be thought as a limiting case of a more general external driving
with the force $F$ applied to the impurity for the time interval $\tau$, the
acquired impulse being given by $\Delta P_{\tau}=F\,\tau$. In one dimension,
such driving in general can lead to Bragg reflection from the boundary of the
emergent Brillouin zone of the fluid, an effect first theoretically predicted
for adiabatic driving [20, 61, 60] and subsequently observed for a finite
driving force in an experiment with ultracold atoms [11]. We are able to
rigorously determine the conditions for Bragg reflection in our setting. In
particular, in contrast to the conventional Bragg reflection from crystals,
here such reflection is operational only between neighbouring Brillouin zones,
with the only available momentum change equal to $\pm 2p_{\rm F}$.

In the adiabatic limit of fixed $\Delta P$ and $F=\Delta P/\tau\rightarrow 0$
the polaron was predicted to experience Bloch-like oscillations [20, 61, 60]
with the polaron momentum ${Q\simeq Ft\mod(2p_{\rm F})}$ (see also [67, 68,
69, 22, 18]). Adiabatic driving can be emulated by a periodic sequence of
small kicks. Whenever the interval between the kicks exceeds the polaron
relaxation time, our result for the polaron momentum distribution can be
applied after each kick. Since in the limit of the small impulse the
distribution approaches the delta-function, see eq. (26), the above simple
picture of Bloch-like oscillations is restored. This is consistent with the
fact that Bloch-like oscillations are particularly robust for polarons that
are heavier than the host particles [22, 18, 70, 71] (the effective mass of
the polaron was calculated in [47]; it always exceeds the mass of the host
fermion).

The two-sided power-law singularities that show up in the polaron momentum
distribution correspond to processes where no host medium excitations are
created, analogously to such genuinely solid state effects like X-ray
singularities [66] or Mössbauer effect [72]. This highlights the peculiarity
of one-dimensional fluids that originates from geometrically-enhanced quantum
correlations.

An interesting question for further exploration is to what extent the
qualitative picture established here survives the breakdown of the
integrability. We note in this respect that the results obtained in integrable
systems are often quite robust and do not change quantitatively away from the
integrable point [22, 73]. Other intriguing directions of further research
include the polaron dynamics at a finite temperature and the dynamics of
attractive polarons.

##  Appendix A Kick as a quantum quench

In order to rigorously define the kick in the translation-invariant system,
one introduces a more general Hamiltonian

$\begin{array}[]{c}\displaystyle H_{\Phi}=\frac{1}{2}\left(P_{\rm
imp}-\Phi\right)^{2}+\frac{1}{2}\sum_{j=1}^{N}P_{j}^{2}+\frac{2p_{\rm
F}}{\alpha}\,\sum_{j=1}^{N}\delta(x_{j}-x_{\rm imp}),\end{array}$ (27)

depending on the parameter $\Phi$. This Hamiltonian is solvable by Bethe
ansatz for arbitrary value of $\Phi$ (not necessarily an integer of the
momentum quantum), with eigenstates $|{\bf n},M\rangle_{\Phi}$ of the form (4)
and the equation on the rapidity that can be obtained from eq. (8) by
substituting $(2\pi/L)M=P_{\rm tot}\rightarrow P_{\rm tot}-\Phi$. If the value
of $\Phi$ is an integer of momentum quanta $2\pi/L$, the eigenstates $|{\bf
n},M\rangle_{\Phi}$ are related to the eigenstates $|{\bf n},M\rangle=|{\bf
n},M\rangle_{\Phi=0}$ in a simple way,

$|{\bf n},M\rangle_{\Phi}=e^{i\Phi\,x_{\rm imp}}\,|{\bf n},M\rangle,$ (28)

which can be verified directly by applying $H_{\Phi}$ to both sides of the
above relation.

The “kick” protocol of impurity preparation described in Section 3.2
corresponds to preparing the system in an eigenstate of the Hamiltonian
$H_{\Phi}$ with $\Phi=\Delta P$ and subsequently quenching the value of $\Phi$
to zero. In view of the relation (28), such quantum quench results in the
state (16).

##  Appendix B Momentum distribution: derivation

###  B.1 Preliminary considerations

To lighten the notations we employ the convention

$p_{\rm F}=1$ (29)

throughout the rest of the Appendix. One can always restore $p_{\rm F}$ by
dimensionality.

In this section we outline the derivation of the rapidity distribution defined
as

$\Gamma(\Lambda)=\sum_{|{\bf n},M\rangle}\big{|}\langle{\bf n},M|\,e^{i\Delta
P\,x_{\rm imp}}|{\bf
n}^{0},M^{0}\rangle\big{|}^{2}\,\delta(\Lambda-\Lambda_{{\bf n},M}),$ (30)

cf. eqs. (15),(16). We use the technique employed earlier to calculate the
Green’s function of the impurity [57, 51, 74].

It turns convenient to introduce functions $k(n,\Lambda)$ and
$\delta(n,\Lambda)$,

$k(n,\Lambda)=\frac{2\pi}{L}\left(n-\frac{\delta(n,\Lambda)}{\pi}\right),\qquad\delta(n,\Lambda)=\frac{\pi}{2}-\arctan\left(\Lambda-\frac{2\pi}{L}\,\alpha\,n\right).$
(31)

They can be used to specify solutions of Bethe equations. Indeed,
$k(n_{l},\Lambda_{{\bf n},M})=k_{l}$ and $\delta(n_{l},\Lambda_{{\bf
n},M})=\delta_{l}$ are, respectively, the $l$’th Bethe pseudomomentum and
phase of the eigenstate $|{\bf n},M\rangle$, cf. eqs. (5),(6).

Note that eq. (9) for $Q(\Lambda)$ is in fact obtained as the thermodynamic
limit of

$Q(\Lambda)=\frac{2\pi}{L}\sum_{n=-(N-1)/2}^{(N+1)/2}\left(\frac{1}{2}-\frac{1}{\pi}\delta(n,\Lambda)\right).$
(32)

This equation along with eq. (31) imply the identity

$\begin{array}[]{c}\displaystyle\sum\limits_{l=1}^{N+1}\frac{\partial
k}{\partial\Lambda}(n_{l},\Lambda)=Q^{\prime}(\Lambda)+O(1/N),\qquad
Q^{\prime}(\Lambda)\equiv\frac{\partial
Q(\Lambda)}{\partial\Lambda},\end{array}$ (33)

valid for an arbitrary state $|{\bf n},M\rangle$ from the bottom of the
spectrum.

The importance of the quantity $\partial k/\partial\Lambda$ stems from the
identity

$\begin{array}[]{c}\displaystyle\frac{\partial
k}{\partial\Lambda}(n_{l},\Lambda_{{\bf
n},M})=\frac{2}{L}\,(\sin\delta_{l})^{2}+O(1/{N^{2}})\end{array}$ (34)

that follows from the complete set of Bethe equations. Thanks to this
identity, $\partial k/\partial\Lambda$ enters matrix elements between
eigenstates (4).

###  B.2 Matrix element

The matrix element entering eq. (30) is obtained from the determinant
representation (4) of the eigenfunction [50]. It reads

$\displaystyle\big{|}\langle{\bf n},M|\,e^{i\Delta P\,x_{\rm imp}}|{\bf
n}^{0},M^{0}\rangle\big{|}^{2}=$ $\displaystyle\delta_{M,M^{0}+\Delta
M}\left(\frac{\Delta P}{\Lambda_{{\bf
n},M}-\Lambda_{0}}\right)^{2}\,\frac{1}{Q^{\prime}(\Lambda_{0})\,Q^{\prime}(\Lambda)}\,\left(\det
D\right)^{2}$ $\displaystyle\times\,(\Lambda_{{\bf
n},M}-\Lambda_{0})^{2N+2}\,\,\prod\limits_{l=1}^{N+1}\frac{\partial
k}{\partial\Lambda}(n_{l},\Lambda)\,\,\prod\limits_{l=1}^{N+1}\frac{\partial
k}{\partial\Lambda}(n_{l}^{0},\Lambda_{0}).$ (35)

Here $D$ is $(N+1)\times(N+1)$ Cauchy matrix constructed from the two sets of
pseudomomenta corresponding to eigenstates $|{\bf n}^{0},M^{0}\rangle$ and
$|{\bf n},M\rangle$,

$D=\left\|\frac{1}{k(n_{l},\Lambda_{{\bf
n},M})-k(n^{0}_{l^{\prime}},\Lambda_{0})}\right\|_{l,l^{\prime}=1,2,\dots,(N+1)}.$
(36)

Due to momentum conservation the matrix element is nonzero for a single value
of $M$ given by

$M=M^{0}+\Delta M,\qquad\Delta M\equiv\frac{L}{2\pi}\Delta P.$ (37)

It should be reminded that $\Lambda_{0}\equiv\Lambda_{{\bf n}^{0},M^{0}}$.

###  B.3 From sum over eigenstates to sum over independent integers

The next step is to replace the summation over eigenstates in eq. (30) by the
summation over independent integers $n_{l}$, $l=1,2,\dots$. This is done as
follows:

$\sum_{|{\bf
n},M\rangle}~{}\longrightarrow~{}\frac{1}{(N+1)!}\sum_{n_{1}}\sum_{n_{2}}...\sum_{n_{N+1}}\theta_{M}({\bf
n}).$ (38)

Here each $n_{l}$ runs over all integers, $M$ is fixed according to eq. (37),
the prefactor $1/(N+1)!$ accounts for permutations within the set ${\bf n}$,
the function $\theta_{M}({\bf n})$ equals $1$ provided the constraint (3) is
satisfied and $0$ otherwise, and terms where at least two integers are equal
vanish automatically thanks to the determinant $\det D$ in the matrix element
(B.2).

###  B.4 Handling $\left(\det D\right)^{2}$

The Cauchy-Binet theorem allows one to convert $(N+1)$ sums over $\left(\det
D\right)^{2}$ to a single determinant,

$\frac{1}{(N+1)!}\sum_{n_{1}}\sum_{n_{2}}...\sum_{n_{N+1}}\delta(\Lambda-\Lambda_{{\bf
n},M})\,\left(\det D\right)^{2}\,\prod_{l=1}^{N+1}f(n_{l},\Lambda_{{\bf
n},M})=\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta(\Lambda-\Lambda_{{\bf
n},M})\,\det\left\|\sum_{n=-\infty}^{\infty}\frac{f(n,\Lambda)}{\big{(}k(n,\Lambda)-k(n^{0}_{l},\Lambda_{0})\big{)}\big{(}k(n,\Lambda)-k(n^{0}_{l^{\prime}},\Lambda_{0})\big{)}}\right\|_{l,l^{\prime}=1,2,\dots,(N+1)},$
(39)

where $f(n,\Lambda)$ is an arbitrary function.

###  B.5 Factorized representation of $\delta(\Lambda-\Lambda_{{\bf n},M})$

Since the Bethe equation (8) has a single solution $\Lambda=\Lambda_{{\bf
n},M}$ provided ${\bf n}$ and $M$ satisfy the constraint (3), and no solutions
otherwise, one can rewrite $\theta_{M}({\bf n})\,\delta(\Lambda-\Lambda_{{\bf
n},M})$ as follows:

$\theta_{M}({\bf n})\,\delta(\Lambda-\Lambda_{{\bf
n},M})=\int\limits_{-\infty}^{+\infty}\frac{dx}{2\pi}\,Q^{\prime}(\Lambda)\,e^{ix\left(\sum\limits_{l=1}^{N+1}\left(k(n_{l},\Lambda)-k(n_{l}^{0},\Lambda_{0})\right)-\Delta
P)\right)}.$ (40)

Here the momentum conservation and the identity (33) has been employed. The
exponential in the integrand can be factorized, which will prove useful in
what follows.

###  B.6 Combining the pieces

We combine eqs. (B.2), (33), (38), (39) and (40) to obtain

$\Gamma=\frac{\Delta
P^{2}}{(\Lambda-\Lambda_{0})^{2}\,Q^{\prime}(\Lambda_{0})}\,{\rm
Re}\int\limits_{0}^{\infty}\frac{dx}{\pi}e^{-i\,\Delta P\,x}\det\mathcal{A}$
(41)

where the $(N+1)\times(N+1)$ matrix $\mathcal{A}$ has matrix elements

$\displaystyle\mathcal{A}_{ll^{\prime}}=$
$\displaystyle(\Lambda-\Lambda_{0})^{2}\,\sqrt{\frac{\partial
k}{\partial\Lambda}(n^{0}_{l},\Lambda_{0})}\,\,\sqrt{\frac{\partial
k}{\partial\Lambda}(n^{0}_{l^{\prime}},\Lambda_{0}\big{)}}\,e^{-ix\big{(}k(n^{0}_{l},\Lambda_{0})+k(n^{0}_{l^{\prime}},\Lambda_{0})\big{)}/2}$
$\displaystyle\times\sum\limits_{n=-\infty}^{\infty}\frac{\partial
k}{\partial\Lambda}(n,\Lambda)\,\,\frac{e^{ix\,k(n,\Lambda)}}{\big{(}k(n,\Lambda)-k(n^{0}_{l},\Lambda_{0})\big{)}\,\big{(}k(n,\Lambda)-k(n^{0}_{l^{\prime}},\Lambda_{0}))}$
(42)

For $l\neq l^{\prime}$ the second line of the above equation can be
reorganized as

$\frac{e\big{(}k(n^{0}_{l},\Lambda_{0})\big{)}-e\big{(}k(n^{0}_{l^{\prime}},\Lambda_{0})\big{)}}{k(n^{0}_{l},\Lambda_{0})-k(n^{0}_{l^{\prime}},\Lambda_{0})}$
(43)

with

$e\big{(}k(n^{0}_{l},\Lambda_{0})\big{)}=\sum\limits_{n=-\infty}^{\infty}\frac{\partial
k}{\partial\Lambda}(n,\Lambda)\frac{e^{ix\,k(n,\Lambda)}}{k(n,\Lambda)-k(n^{0}_{l},\Lambda_{0})}$
(44)

This function has a nice thermodynamic limit that can be obtained by
presenting the sum as a contour integral, a technique described in refs. [51,
74]. The result is given by eq. (19). For diagonal entries, $l=l^{\prime}$,
eq. (43) still can be used if interpreted in the l’Hôpital sense, with extra
care required to keep both the O(1) and O(1/N) terms:

$\mathcal{A}_{ll}=1+(\Lambda-\Lambda_{0})^{2}\,e^{-ixk}\frac{\partial
k}{\partial\Lambda}\partial_{k}e(k)\Big{|}_{k=k(n^{0}_{l},\Lambda_{0})}+O(1/N^{2}).$
(45)

Combining eqs. (41),(B.6),(43) and (19) one arrives at eq. (17).

##  Appendix C Asymptotic analysis of the singularity

###  C.1 Asymptotics of the Fredholm determinant

Figure 5: Comparison of the exact Fredholm determinant (18) (blue) and its
asymptotic expression (46) (magenta) for $\Lambda_{0}=0,$ $\Lambda=10$,
$\alpha=3$.

The singularities in the rapidity and momentum distributions stem from the
asymptotic behavior of ${\det}(1+\hat{K})$ for large $x$. The latter is given
by

$\det(1+\hat{K})\simeq\sum_{m=-1}^{1}\frac{C_{\Lambda_{0},\Lambda}[\xi_{m}]}{(2i)^{(\xi^{-}_{m})^{2}}(-2i)^{(\xi^{+}_{m})^{2}}}\,\,\frac{e^{i\,x\,(Q-Q_{0}+2m)}}{x^{(\xi^{-}_{m})^{2}+(\xi^{+}_{m})^{2}}},\qquad
x\gg 1/p_{\rm F},$ (46)

where $\xi_{m}$ is a function given by

$\xi_{m}(q)=\frac{1}{\pi}\big{(}\arctan(\Lambda_{0}-\alpha
q)-\arctan(\Lambda-\alpha q)\big{)}-m,$ (47)

$\xi_{m}^{\pm}$ equals $\xi_{m}(\pm 1)$ (which is consistent with the
definition (24) in the main text), the functional
$C_{\Lambda_{0},\Lambda}[\xi]$ reads

$C[\xi]=\left(G(1-\xi^{-})\,G(1+\xi^{+})\right)^{2}\,(2\pi)^{\xi^{-}-\xi^{+}}\,e^{-\tilde{C}_{\Lambda_{0},\Lambda}[\xi]},$
(48)

with the subscript $m$ in $\xi_{m}(q)$, $\xi_{m}^{\pm}$ omitted for brevity,
$G(a)$ is the Barnes function, and

$\displaystyle\tilde{C}_{\Lambda_{0},\Lambda}[\xi]=$
$\displaystyle\int\limits_{-1}^{1}dq\,\frac{\xi(q)\cot\big{(}\pi\xi(q)\big{)}}{1+(\Lambda-\alpha
q)^{2}}-\frac{2\alpha}{\Lambda_{0}-\Lambda}\int\limits_{-1}^{1}dq\,\xi(q)$
$\displaystyle~{}~{}~{}+\frac{1}{2}\int\limits_{-1}^{1}dq\int\limits_{-1}^{1}dq^{\prime}\left(\frac{\xi(q^{\prime})-\xi(q)}{q^{\prime}-q}\right)^{2}-\int\limits_{-1}^{1}dq\,\frac{\xi(-1)^{2}-\xi(q)^{2}}{1+q}-\int\limits_{-1}^{1}dq\,\frac{\xi(1)^{2}-\xi(q)^{2}}{1-q}.$
(49)

The asymptotics (46) is derived with the help of the standard soft mode
resummation technique [75, 76, 77, 78] (see section 5 in ref. [74] for a
pedagogical introduction).

It should be emphasized that we have restricted the summation over $m$ in eq.
(46) by three terms $m=0,\pm 1$. Only these terms can be relevant for the
singularity of the distribution, as discussed in Section 3.4.

Importantly,

$C_{\Lambda_{0},\Lambda}[\xi]=C_{\Lambda,\Lambda_{0}}[-\xi],$ (50)

thanks to the following property of the Barnes function:

$G(1-z)=\frac{G(1+z)}{(2\pi)^{z}}\exp\left(\pi\int\limits_{0}^{z}\,x\cot(\pi
x)\,dx.\right)$ (51)

We compare the asymptotics (46) to the exact Fredholm determinant (18) in Fig.
5. One can see an excellent agreement for sufficiently large $x$. High
frequency oscillations at small $x$ that are not captured by the asymptotics
(46) stem from the term $e^{ix\Lambda p_{\rm F}/\alpha}e^{-|x|p_{\rm
F}/\alpha}$ in eq. (19). They are visible only for large values of $\Lambda$.

###  C.2 Singularity of the momentum distribution

Plugging the asymptotics (46) into eq.(17) and performing the integration over
$x$, one obtains eq. (21) with

$C_{\pm}=\frac{\Delta
P^{2}}{(\Lambda-\Lambda_{0})^{2}\,\,Q^{\prime}(\Lambda_{0})Q^{\prime}(\Lambda)}\,\frac{\Gamma\big{(}1-(\xi_{m}^{-})^{2}-(\xi_{m}^{+})^{2}\big{)}}{\pi\,2^{(\xi_{m}^{-})^{2}+(\xi_{m}^{+})^{2}}}\,C_{\Lambda_{0},\Lambda}[\xi_{m}]\,\sin\big{(}\pi\,(\xi^{\mp}_{m})^{2}\big{)}.$
(52)

Here $\Gamma(a)$ is the gamma-function and $C_{\Lambda_{0},\Lambda}[\xi_{m}]$,
$\xi_{m}$, $\xi_{m}^{\pm}$ are defined in the previous subsection, with $m$
chosen to satisfy the condition (23). The latter rule implies that a single
term from the sum in eq. (46) contributes to the singularity. Note also that
the argument of the gamma-function is always in the interval $(0,1)$ thanks to
the condition (25).

The limit $\Delta P\to 0$ deserves a separate consideration. In this limit
$m=0$, the position of the singularity $Q_{\rm sing}$ approaches $Q_{0}$, and
$\xi_{0}(q),\xi_{0}^{\pm}\rightarrow 0$,
$C_{\Lambda_{0},\Lambda}[\xi_{0}]\rightarrow 1$. Therefore the Fourier
transform of the asymptotics (46) leads to the delta-function,

$\lim_{\Delta P\rightarrow 0}\Gamma^{\rm sing}_{Q_{0}\rightarrow
Q}=\frac{1}{\pi}{\rm Re}\frac{i}{(Q-Q_{0}+i0)}=\delta(Q-Q_{0}),$ (53)

where the latter equality is the Plemelj-Sokhotski formula. The normalization
condition implies that $\Gamma^{\rm sing}_{Q_{0}\rightarrow
Q}=\Gamma_{Q_{0}\rightarrow Q}$ in the limit $\Delta P\to 0$. This way one
arrives at eq. (26).

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# Characteristics and Correlation of Nonaffine Particle Displacements in the
Plastic Deformation of Athermal Amorphous Materials

Meenakshi L  Bhaskar Sen Gupta<EMAIL_ADDRESS>Department of
Physics, School of Advanced Sciences, Vellore Institute of Technology,
Vellore, Tamil Nadu - 632014, India

###### Abstract

When an amorphous solid is deformed homogeneously, the response exhibits
heterogeneous plastic instabilities with localized cooperative rearrangement
of cluster of particles. The heterogeneous behavior plays an important role in
deciding the mechanical properties of amorphous solids. In this paper, we
employ computer simulation to study the characteristics and the spatial
correlations of these clusters characterized by the non-affine displacements
in amorphous solids under simple shear deformation in the athermal quasistatic
limit. The clusters with large displacements are found to be homogeneously
distributed in space in the elastic regime, followed by a localization within
a system-spanning shear band after yielding. The distributions of the
displacement field exhibit power-law nature with exponents strongly dependent
on the deformation. The non-affine displacements show strong spatial
correlations which become long-ranged with increasing strain. From our
results, it is evident that the decay of the correlation functions is
exponential in nature in the elastic regime. The yielding transition is marked
by an abrupt change in the decay after which it is well described by power-
law. These results demonstrate a scale-free character of non-affine
correlations in the stead flow regime. These results are found to be robust
and independent of the strain window over which the total non-affine
displacement is calculated.

##  I Introduction

There has been a long-standing interest to understand the mechanical
instabilities of amorphous materials because of their paramount importance in
a wide range of fields, from fracture of metallic glasses in material
engineering metglassfrac1 ; metglassfrac2  to earthquake and landslide in
geology earthquake . Despite extensive research, the physics of mechanical
failure in amorphous materials is still poorly understood compared to its
crystalline counterparts. Unlike crystalline solids where the topological
defects or dislocations are considered as the basis of deformation and plastic
flow crystal , there is no long-range order in amorphous materials compared to
which defects can be defined. Therefore, understanding amorphous plasticity
requires an entirely different approach from crystalline plasticity.

The elementary process of amorphous plasticity involves the shear localization
under the influence of mechanical loading and the subsequent collective
rearrangement of a cluster of particles, often referred to as the shear
transformation zone (STZ) argon ; falk . Rearrangements are found to occur at
any strain ($\gamma$) value. For small $\gamma$, an amorphous sample shows
elastic behavior, and the stress ($\sigma$) increases linearly with $\gamma$.
However, this branch is punctuated by localized plastic instabilities, where
the irreversible structural rearrangements occur locally argon1 ; schall . The
typical size of such STZs is extended up to a few particle diameter. With
increasing $\gamma$, these plastic drops trigger more events nearby and this
correlated process spreads across the material in an avalanche-like manner,
and subsequently the material yields via system spanning shear band formation
Alexander ; Bonn ; 04VBB ; 04ML ; 05DA ; 06TLB ; 06ML ; 09LP ; 11RTV ; 06SLG ;
13KTG ; 13NSSMM .

Elucidating the influence on a plastic event by its predecessors spatially
would be a significant advance in understanding the unwanted mechanical
failure. Numerous experimental and computer simulation studies reported the
existence of correlations between the STZs. For example, using confocal
microscope in a shear colloidal glass it was observed that the global, as well
as local non-affine displacement fields show strong spatial correlations in
the plastic flow region chikkadi . The long-range correlations that exhibit
power-law decay originate from the elastic deformation field due to plastic
events. These findings were substantiated by numerical simulations on hard-
sphere glasses mandal ; varnik . In a similar spirit, spatial autocorrelation
in the non-affine deformation field of sheared bulk metallic glasses was
studied using computer simulations murali . The decay of the correlation
function was found to be exponential in nature. This was related to a length
scale associated with the size of the STZs in the pre-yield regime. In ref.
cubuk  results of experiments and simulations on a wide range of amorphous
materials were reported and the correlation was found to be decaying
exponentially. Finite rate deformation of Lennard Jones glasses at nonzero
temperature showed a transition from exponential to power-law decay of
correlation functions across the yielding transition nikolai . Therefore, it
is evident that the long debate over power-law vs exponential decay in the
correlations of non-affine displacement fields is not settled yet.

The main objective of our work is to gain insight into the characteristics of
the non-affine displacement field and the nature of their spatial correlations
in the elastic and steady-state plastic region. We, therefore, focus on the
microscopic analysis of the non-affine displacement field of particles in a
simple amorphous solid under athermal quasistatic shear (AQS) using computer
simulation. The principle utility of the AQS algorithm is that it enables us
to probe the shear-induced structural changes of the system in the absence of
thermal fluctuations Lacks1999_JCP ; Barrat2002_PRB ; Lemaitre2004_PRL1 ;
Lemaitre2004_PRL2 ; Lemaitre2006_PRE ; Procaccia2009_PRE . Our results show
that in the elastic branch, the STZs are homogeneously distributed in space
across the system followed by the system spanning shear band formation post
yielding. This is reflected in the distinct change of the decaying nature of
the correlation function of the non-affine displacement. The spatial extent of
the displacement field is examined by computing the distribution function
which shows power-law decay, with an exponent dependent on the deformation.
Finally, we revisit the riddle of exponential vs power-law decay of the
correlations between STZs in the pre and post-yield regime. The robustness of
these results is tested by computing the non-affine displacements as a
function of accumulated deformation as well as various strain windows.

Our paper is structured as follows: a detailed description of the sample
preparation using molecular dynamics simulations and the athermal quasistatic
shear protocol is given in the next section. In section III we discuss the
methods to calculate the nonaffine displacement field and present results and
analysis of their characteristics and spatial correlation functions. Finally,
a conclusion is presented in Sec. IV.

##  II Numerical Simulation Details

To prepare amorphous samples we resort to molecular dynamics (MD) simulations
using the well studied Kob-Anderson binary mixture model Kob . The system
comprises of two types of particles labeled as $A$ and $B$ with their number
ration $80:20$. We simulate $N=32400$ particles confined in a cubic box of
volume $V$. For simplicity, the mass $m$ of both types of particles is taken
to be the same and equal to unity. The particles interact via a pairwise
Lennard-Jones potential given by

$\displaystyle U_{\alpha\beta}(r)$ $\displaystyle=$ $\displaystyle
4\epsilon_{\alpha\beta}\Big{[}\Big{(}\frac{\sigma_{\alpha\beta}}{r}\Big{)}^{12}-\Big{(}\frac{\sigma_{\alpha\beta}}{r}\Big{)}^{6}\Big{]},r\leq
r_{cut}$ (1) $\displaystyle=$ $\displaystyle
0\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}r\textgreater
r_{cut},$

where $\alpha,\beta\in\rm{A,B}$. The units of various quantities in our
simulation are as follows: length is expressed in the unit of $\sigma_{AA}$,
energy in the unit of $\epsilon_{AA}$, time in the unit of
$(m\sigma_{AA}^{2}/48\epsilon_{AA})^{1/2}$ and temperature in the unit of
$\epsilon_{AA}/k_{\rm B}$. Here $k_{\rm B}$ is the Boltzmann constant which is
unity. The parameters $\sigma_{\alpha\beta}$ and $\epsilon_{\alpha\beta}$ are
chosen as follows: $\sigma_{AA}=1.0,\sigma_{BB}=0.8,\sigma_{AB}=0.88$ and
$\epsilon_{AA}=1.0,\epsilon_{BB}=0.5,\epsilon_{AB}=1.5$. The asymmetric choice
of the parameters introduces frustration in to the system which inhibits
crystallization below $T_{g}$ and forms stable glass. The cutoff $r_{cut}=2.5$
is used to reduce computational efforts.

MD simulations are employed by integrating Newton’s equation of motion using
velocity Verlet algorithm Verlet  with a discretization time step $\Delta
t=0.005$ in Lennard-Jones units. We work in the $NVT$ ensemble at a constant
density $\rho=N/V=1.2$ with periodic boundary conditions in all directions.
The temperature of the system is controlled by the Berendsen thermostat
Berendsen . To prepare the glass samples, the binary mixture is first
equilibrated in the liquid state at temperature $T=1.0$. The glass transition
temperature for the KA model at this density is $T_{g}=0.435$ Kob . After
equilibration, the system is cooled linearly with a rate $10^{-5}$ to a very
low temperature $T=0.001$ which is sufficiently low to eliminate any
appreciable thermal effects. Finally, the system is brought to the minimum
energy state using conjugate gradient energy minimization algorithm where the
temperature is formally $T=0$. For statistical averaging 100 independent
glassy samples are generated by repeating the whole process with different
initial realizations.

To explore plastic failure the glassy samples are subjected to simple shear
deformation using AQS protocol with the limit $T\rightarrow 0$ and
$\dot{\gamma}\rightarrow 0$, where $\dot{\gamma}$ is the strain rate 06ML .
The AQS algorithm includes two iterating steps: (a) a freshly quenched glassy
sample is deformed by applying an affine simple shear transformation to each
particle $i$ of the system as

$r_{ix}\rightarrow r_{ix}+r_{iz}\delta\gamma,~{}~{}~{}r_{iy}\rightarrow
r_{iy},~{}~{}~{}r_{iz}\rightarrow r_{iz}$ (2)

using the Lees-Edwards boundary conditions lees . We choose sufficiently small
strain increment $\delta\gamma=10^{-4}$. (b) After every affine transformation
step, the potential energy of the deformed system is minimized using conjugate
gradient algorithm under the constraints imposed by the boundary condition. By
repeating steps (a) and (b) we can reach up to arbitrarily large strain
values. The AQS method ensures that the system is in mechanical equilibrium
after every differential strain increase and follows strain-induced changes of
the potential energy surface. Therefore, the change of particles position does
not map homogeneously with the macroscopic strain but evolves as
$\textbf{r}_{i}=\textbf{r}_{i}^{\prime}+\textbf{u}_{i}$, where
$\textbf{u}_{i}$ represents the nonaffine displacement of the $i$th particle.
Under the influence of increasing strain, the energy minimum changes its shape
continuously until it develops zero-curvature along a certain direction. Any
further infinitesimal strain increment causes the system to fall to a new
minimum with a large non-affine displacement of particles, identified as
plastic event. During AQS deformation the particle configurations are saved at
regular intervals of strain values for further analysis.

##  III Results

Figure 1: Typical stress $(\sigma)$ vs strain $(\gamma)$ curve for an
individual glassy sample of $N=32400$ particles under athermal quasistatic
simple shear deformation. Inset shows the same curve zoomed over a small
strain window in the elastic regime.

In Fig. 1 we show the typical stress vs strain curve obtained from the AQS
simulation. The system yields at around $\gamma\approx 8\%$ and after that
steady-state plastic flow sets in. The mechanical response comprises a series
of plastic events involving the rearrangement of cluster of particles that
produces long-range displacement fields. To investigate the heterogeneous
response, we resort to the microscopic analysis involving spatial
configurations of atoms with large relative non-affine displacements
associated with the deformation events. More precisely, we compute the non-
affine displacements of the local shear transformations in deformed solids
originally proposed by Falk and Langer falk  with some important modifications
involving the strain $\gamma$, defined as

$\begin{split}D^{2}(\gamma,\Delta\gamma)=&\frac{1}{N_{i}}\sum\limits_{j=1}^{N_{i}}\\{\bm{\mathrm{r}}_{j}(\gamma)-\bm{\mathrm{r}}_{i}(\gamma)-\\\
&\bm{\mathrm{J}}_{i}[\bm{\mathrm{r}}_{j}(\gamma-\Delta\gamma)-\bm{\mathrm{r}}_{i}(\gamma-\Delta\gamma)]\\}^{2}\end{split}$
(3)

Here $\bm{\mathrm{r}}_{i}(\gamma)$ is the position vector of the $i$th
particle at the strain value $\gamma$, $\bm{\mathrm{J}}_{i}$ is the
transformation matrix that maps the $i$th particle and its first nearest
neighbor at strain $\gamma$ and $\gamma-\Delta\gamma$ respectively via an
affine deformation. This quantity proposed in ref. falk  is extensively used
for the spatio-temporal analysis of non-affine displacements in shear-driven
amorphous solids varnik ; cubuk ; jana ; chikkadi ; ding ; Nikolai1 ; Nikolai2
; vijay1 . The choice of $\Delta\gamma$ in Eq. 3 plays an important role in
the resolution of different plastic events and the nature of $D^{2}$ field. In
this work, we present results for two different choices of the reference frame
with respect to which the $D^{2}$ field is computed; (a) the undeformed atomic
configuration which corresponds to $\Delta\gamma=\gamma$. In this case, the
accumulation of all the plastic events that occurred during the total
deformation $\gamma$ will contribute to the $D^{2}$ field calculation. (b) The
reference frame is at a fixed distance $\Delta\gamma$ from the applied strain
$\gamma$ and thus maintains a constant strain window. Here the plastic events
occurring within the deformation interval $\gamma-\Delta\gamma$ to $\gamma$
are considered.

The representative snapshots for the deformed glassy states in the elastic and
steady state regime are shown in Fig. 2. We compute the displacement field
$D^{2}(\gamma,\Delta\gamma)$ for these configurations from Eq. 3 for two
different choices of reference frames: $\Delta\gamma=\gamma$ and
$\Delta\gamma=1\%$.

Figure 2: Typical snapshots of the deformed glassy samples for (a)
$\Delta\gamma=\gamma$ at $\gamma=0.07$, (b) $\Delta\gamma=\gamma$ at
$\gamma=0.12$, (c) $\Delta\gamma=1\%$ at $\gamma=0.07$ (d) $\Delta\gamma=1\%$
at $\gamma=0.12$. The particles are color-coded according to their non-affine
displacements $D^{2}(\gamma,\Delta\gamma)$ (see text).

In Fig. 2 we color-code the particles according to their non-affine
displacement field. It can be clearly seen that the STZs are homogeneously
distributed in space in the elastic regime. The scenario changes completely as
the mobile particles localize in a band-type structure running across the
system and form shear band.

Figure 3: Spatial configurations of particles with large non-affine
displacements in Fig. 2 for (a) $D^{2}>0.05$, (b) $D^{2}>0.05$, (c)
$D^{2}>0.01$ and (d) $D^{2}>0.01$

The spatial distribution of clusters is further examined by plotting the
configurations of particles with large nonaffine displacements. This is shown
in Fig. 3. In the elastic regime, we observe the aggregation of particles into
disconnected clusters that are uniformly distributed in the sample. Post
yielding the clusters tend to merge and form shear band of the order of system
size.

This can be better understood by the probability distribution of the
displacement field which involves spatially averaged profiles of $D^{2}(z)$
computed over a small bin of thickness $\Delta z$. This is shown in Fig. 4 for
a chosen set of strain values with the undeformed state as the reference
frame.

Figure 4: The spatial distribution of the non-affine displacements $D^{2}(z)$
vs $z$ for $\Delta\gamma=\gamma$ at different strain values mentioned in the
inset. The $D^{2}(z)$ shown here corresponds to the same sample in Fig. 2.

For small deformation, the distribution curves are almost flat, indicating
that the clusters of mobile particles are homogeneously distributed across the
sample. The average value of $D^{2}(z)$ increases gradually with strain. The
yielding transition is marked by the appearance of a distinct peak in the
distribution, indicating the formation of shear band. The typical width of the
shear band is of several particle diameters. The width of the peak increases
gradually with strain implying the expansion of the shear band.

To understand the role of $\Delta\gamma$ on the nature of the nonaffine
displacements and their spatial extent, we compute the probability
distribution functions (PDFs) of $D^{2}(\gamma,\Delta\gamma)$ for the two
different choices of $\Delta\gamma$ mentioned above for a set of strain values
below and above yielding. This is shown in Fig. 5. With increasing strain
amplitude, the tail of the PDFs shifts towards larger
$D^{2}(\gamma,\Delta\gamma)$ values. In both cases, the PDFs exhibit a power-
law decay for all strain values. However the power-law exponent changes with
strain. In the case of undeformed reference frame ($\Delta\gamma=\gamma$), the
exponent varies from about -2.63 to about -4.8 for the range of deformations
chosen in our simulation. Incidentally, our result near yielding is very close
to the exponent -2.8 observed experimentally in the steady-state shear of
colloidal glasses chikkadi . When the second choice of reference frame
$\Delta\gamma=1\%$ is used to compute $D^{2}(\gamma,\Delta\gamma)$, the
exponent changes from about -2.4 to about -3.4 for the same set of strain
values. In both cases, the distribution functions become wider with strain in
the elastic regime. But post yielding, unlike the first case, for
$\Delta\gamma=1\%$, the PDFs start to converge above $\gamma=13\%$ with an
exponent -3.4. After this, the exponent becomes insensitive to higher strain
values. Therefore, we find the results are strongly dependent on the choice of
reference frame for computing the nonaffine displacements. Hence, we can
anticipate that the method to compute the $D^{2}(\gamma,\Delta\gamma)$ field
will play a major role in determining the correlation between STZs.

Figure 5: The probability distribution of the non-affine displacements for (a)
$\Delta\gamma=\gamma$ and (b) $\Delta\gamma=1\%$ for the strain values
mentioned in the inset. The dashed lines indicate the slope.

To quantify the geometry of plastic events over space, we compute the spatial
correlations of $D^{2}(\gamma,\Delta\gamma)$ field as follows:

$C_{D^{2}}(\Delta\textbf{r})=\frac{\langle
D^{2}(\textbf{r}+\Delta\textbf{r})D^{2}(\textbf{r})\rangle-\langle
D^{2}(\textbf{r})\rangle^{2}}{\langle D^{2}(\textbf{r})^{2}\rangle-\langle
D^{2}(\textbf{r})\rangle^{2}}$ (4)

Here the angular brackets represent an average of spatial coordinate r over
100 independent samples. In Fig. 6 we show the correlation results involving
the nonaffine measure given by Eq. 3 with respect to two different choices of
reference frames mentioned above at different strain values.

Figure 6: Spatial correlation function of the non-affine displacement field
for (a) $\Delta\gamma=\gamma$ and (b) $\Delta\gamma=1\%$ in the log-log scale
for the strain values mentioned on the right. The dashed liens represent the
slope -1.3 (see text). In figs. (c) and (d) we show the same data of (a) and
(b) respectively in the log-linear scale.

As expected, in both cases, the correlation becomes long ranged with
increasing strain. The relative change in $C_{D^{2}}(\Delta\textbf{r})$
abruptly becomes larger near the yielding transition ($\gamma\approx 8\%$)
indicating the system has developed a system-spanning displacement field. This
is consistent with the observation in the peak height in Fig. 4.

Our results show the existence of long-range spatial correlation in the
$D^{2}$ field. To characterize the nature of decay of
$C_{D^{2}}(\Delta\textbf{r})$ we show in Fig. 6 the correlation function in
the log-linear scale for the choices $\Delta\gamma=\gamma$ and
$\Delta\gamma=1\%$ at different strain values. In both cases, a constant slope
in the semi-log plot is visible for $\gamma<8\%$. These observations confirm
the exponential decay of $C_{D^{2}}(\Delta\textbf{r})$ in the elastic regime
and the existence of a characteristic length scale of plastic events. Post
yielding the decay can be well described by a power-law decay. These results
are found to be independent of the choice of the reference frame for
$D^{2}(\gamma,\Delta\gamma)$ calculations. At large strain, the correlation
functions gradually converge in the plastic flow regime. The convergence rate
appears to be faster for $\Delta\gamma=\gamma$ than for cumulative strain. The
power-law exponent close to the yielding transition is estimated to be -1.3.
We mention here that the same exponent was reported in the experimental study
of steadily sheared colloidal glasses vijay1 . This is shown in Fig. 6 with
dashed lines for reference. The power-law decay demonstrates the scale-free
character of the STZs in the stead flow regime. The deviation of the tail part
can be attributed to the finite size of the system. We, therefore, conclude
that the characteristic nature of the spatial correlations of non-affine
displacements is entirely different in the elastic and the plastic regime and
transforms from an exponential to a power-law decay at the yielding
transition. These results are found to be robust and insensitive to the choice
of $\Delta\gamma$.

The role of the choice of reference frame on the correlation functions is
further investigated. We compute $C_{D^{2}}(\Delta\textbf{r})$ with the
undeformed configuration ($\Delta\gamma=\gamma$) as the reference frame as
well as for three different strain windows $\Delta\gamma=0.1\%,0.5\%$ and
$1.0\%$. The results are shown in Fig. 7 for the elastic ($\gamma=5\%$) and
steady state ($\gamma=12\%$) regime.

Figure 7: Spatial correlation function of the non-affine displacement field
corresponding to the different choice of reference frame
$\Delta\gamma=\gamma,0.005,0.01$ and $0.02$ at $\gamma=0.05$ and $0.12$

We observe that before yielding the correlations overlap and therefore do not
depend on the choice of the reference frame. Also, the nature of decay can be
well described by exponential function over the entire range of distance. This
behavior completely changes post yielding and we see a strong dependence on
the reference frame. The correlation functions decay as power-law and become
long ranged with increasing strain window.

##  IV Summary and Conclusion

In summary, we have performed computer simulations on a modeled amorphous
solid to gain insight into the spatial distribution and structural connection
of plasticity by computing the non-affine displacements of all the particles.
The glassy samples were prepared by slow annihilation of the Kob Andersen
binary LJ model to a very low temperature below $T_{g}$ and then simple shear
was applied following AQS protocol. Different choices of reference frames
($\Delta\gamma$) to compute non-affine displacements were explored. The
simulation results offer manifold conclusions. It was observed that, in the
elastic regime, particles rearrangement with large non-affine displacements
happens in isolated clusters that are relatively homogeneously distributed in
space. Post-yielding it becomes inhomogeneous and localized by a system
spanning shear band formation. The displacement field shows power-law
distribution and the exponent strongly depends on the choice of
$\Delta\gamma$. Our results for cumulative strain are found to be similar to
sheared colloidal glass in the plastic flow regime. Nonaffine displacements
exhibit a strong spatial correlation in both the deformation regimes. One of
the important results obtained from our work is the nature of the correlation
function which changes from exponential to the power-law decay, and the
yielding transition plays a pivotal role in this. Therefore, the long-range
correlation function demonstrates a scale-free steady-state plastic regime.
These results are found to be robust and independent of the choice of
$\Delta\gamma$.

## Acknowledgment

B. Sen Gupta acknowledges Science and Engineering Research Board (SERB),
Department of Science and Technology (DST), Government of India (no.
SRG/2019/001923) for financial support. Meenakshi L. acknowledges VIT for
doctoral fellowship. B. Sen Gupta would like to thank Vijaykumar Chikkadi for
many useful discussions.

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