# Time evolution of the local density of states of strongly correlated
fermions coupled to a nanoprobe

T. Blum  Fachbereich Physik, Philipps-Universität Marburg, 35032 Marburg,
Germany  R.M. Noack  Fachbereich Physik, Philipps-Universität Marburg, 35032
Marburg, Germany  S.R. Manmana  Institut für Theoretische Physik, Georg-
August-Universität Göttingen, 37077 Göttingen, Germany

###### Abstract

We study the time evolution of a one-dimensional system of strongly correlated
electrons (a ‘sample’) that is suddenly coupled to a smaller, initially empty
system (a ‘nanoprobe’), which can subsequently move along the system. Our
purpose here is to study the role of interactions in this model system when it
is far from equilibrium. We therefore take both the sample and the nanoprobe
to be described by a Hubbard model with on-site repulsive interactions and
nearest-neighbor hopping. We compute the behavior of the local particle
density and the local density of states (LDOS) as a function of time using
time-dependent matrix product states at quarter and at half filling, fillings
at which the chain realizes a Luttinger liquid or a Mott insulator,
respectively. This allows us to study in detail the oscillation of the
particles between the sample and the nanoprobe. While, for noninteracting
systems, the LDOS is time-independent, in the presence of interactions, the
backflow of electrons to the sample will lead to nontrivial dynamics in the
LDOS. In particular, studying the time-dependent LDOS allows us to study how
the Mott gap closes locally and how this melting of the Mott insulator
propagates through the system in time after such a local perturbation—a
behavior that we envisage can be investigated in future experiments on
ultrashort time scales or on optical lattices using microscopy setups.

##  I Introduction

One very common and interesting scenario in physics occurs when two systems
are coupled to one another starting at a particular point in time and then
interact with one another in a time-dependent way. The salient physical
question is how the two systems then evolve in time. When the two systems are
quantum many-body systems, particularly interesting behavior can be expected.

The study of such nonequilibrium correlated systems is inspired by fundamental
issues such as the nature of thermalization in closed quantum systems [1, 2,
3, 4], the behavior found in controlled experimental investigations in cold
atomic gases on optical lattices [5, 6], and by the study of material
properties after a strong photo excitation, typically in the context of pump-
probe experiments [7, 8, 9, 10]. These advanced time-resolved experimental
techniques make it possible to study a variety of nonequilibrium phenomena,
such as the formation of transient order, light-induced phase transitions, or
hidden states on ultra-short time scales (pico- or femtoseconds) [11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. One such experimental approach is
time- and angular resolved photo emission spectroscopy (trARPES) [25], which
provides direct insights into nonequilibrium properties of spectral functions,
e.g., the existence of Floquet states [26, 27, 28, 29, 30, 31, 32, 33, 34],
which have been proposed as a means of engineering certain material properties
such as topological states in periodically driven systems [35, 36, 31].

Typically, trARPES measurements are made in momentum space; in order to gain
insights into the behavior of observables in real space, one would thus need
to Fourier-transform the data [37]. For locally restricted excitations (e.g.,
ones induced by the tip of a scanning tunneling microscope (STM) [38, 39]), it
can be important to investigate the evolution of spectral properties directly
in position space. While STM experiments typically deal with transport
properties in the linear-response regime and hence remain close to equilibrium
(see, e.g., Ref. 40 for a recent review article), it is also interesting to
consider such situations far away from equilibrium. Here we treat this case,
computing the time evolution of the local density of states (LDOS) when
strongly perturbing a strongly correlated system only locally. We envisage
that this quantity will be experimentally accessible in the future, also on
the ultrashort time scales treated by us here.

Inspired by these considerations, we ask the question of what happens when a
small, initially empty, test system (in the following referred to as a
‘nanoprobe’ or ‘probe’) is brought near a sample hosting a strongly correlated
state of matter. We assume a strong coupling between the sample and the
nanoprobe, which also allows for electrons to flow from the probe back to the
sample. This will induce a dynamics that is far from the linear-response
regime, so that an equilibrium description does not apply. One important
aspect is to what extent measures from linear response theory are useful to
describe such a situation. We quantify this by considering in detail the
properties of the LDOS computed from time-dependent single particle
propagators. The behavior of the LDOS is particularly interesting for strongly
correlated electrons because the time-evolved LDOS can exhibit nontrivial
features induced by significant electron-electron interaction: whereas the
LDOS remains time-independent after the sudden coupling for noninteracting
systems (as discussed later in the paper), this is no longer true for strongly
interacting systems, and the behavior of the LDOS can change significantly in
the course of time.

Here we study these aspects on a ‘standard’ model for strongly correlated
physics, namely, the Hubbard model in one spatial dimension [41, 42]. At
finite repulsive interaction strength, this model exhibits Mott-insulating
behavior [43] at half filling and is a Luttinger liquid [44] otherwise. Our
setup allows us to study and compare the time evolution of the LDOS in the two
qualitatively different phases by tuning the initial filling.

The remainder of the paper is organized as follows: In Sec. II, we describe
the model, the quantities that we study, and the methods that we use. In
particular, in Sec. II.1, we describe the setup of our model system and the
time-dependent coupling between sample and nanoprobe. In Sec. II.2, we discuss
the observables we compute, in particular the time-dependent LDOS. Section
II.3 describes our matrix-product-state (MPS) approach to simulating the
dynamics of the system as well as the exact solution for the $U=0$ case. In
Sec. III we present our results for both the stationary and the moving
nanoprobe. Finally, we discuss and summarize our findings in Sec. IV. In
addition, App. A describes estimates of the accuracy of our calculations by
comparing to the exact results at $U=0$, and App. B contains additional
results for a higher nanoprobe velocity, $v=1$.

##  II Model, Observables, and Methods

###  II.1 Model

……$U$$v$$t_{h}$$t_{h}$$t^{\prime}_{h}$ Figure 1:  Schematic depiction of the
setup. Blue sites are in the sample and red sites in the nanoprobe. The
tunneling strength $t_{h}$ and the on-site Hubbard $U$ have each the same
values in the sample and the probe, respectively. The tunneling strength
$t_{h}^{\prime}$ between sample and nanoprobe is finite only at times $t>0$.
The nanoprobe can move to the right with a constant speed $v\geq 0$ relative
to the sample.

We study the one-dimensional Hubbard model [41, 42], which, at time $t=0$, is
suddenly coupled to a set of initially empty interacting sites, which
represent the nanoprobe, as depicted in Fig. 1. This system is modeled by the
Hamiltonian

$\begin{split}H=&-t_{h}\sum_{k\in\left\\{\text{sample},\text{probe}\right\\}}\sum_{\sigma}\left(c^{\dagger}_{k,\sigma}c_{k+1,\sigma}^{\phantom{\dagger}}+c^{\dagger}_{k+1,\sigma}c_{k,\sigma}^{\phantom{\dagger}}\right)+\,U\sum_{k\in\left\\{\text{sample},\text{probe}\right\\}}n_{k,\uparrow}n_{k,\downarrow}\\\
&-t_{h}^{\prime}\sum_{i\in\text{sample}}\sum_{j\in\text{probe}}\sum_{\sigma}w(t,i,j)\left(c^{\dagger}_{i,\sigma}c_{j,\sigma}^{\phantom{\dagger}}+c^{\dagger}_{j,\sigma}c_{i,\sigma}^{\phantom{\dagger}}\right)\,,\end{split}$
(1)

with

$\begin{split}w(t,i,j)=\Theta(t)\left[\Theta\left(vt-i+j+\frac{L-l}{2}+1\right)-\Theta\left(vt-i+j+\frac{L-l}{2}-1\right)\right]\,,\\\
\end{split}$ (2)

where $\Theta(x)$ is the Heaviside function, $t$ the time, and we take the
lattice constant to be 1. We designate the number of sites in the sample $L$,
the number of sites in the nanoprobe $l$ with index $i\in\\{1,...,L\\}$ in the
sample and $j\in\\{1,...,l\\}$ in the probe and work in the canonical ensemble
with a fixed number of particles $N$. In Hamiltonian (1), we assume open
boundary conditions (OBC). The function $w(t,i,j)$ causes each site of the
nanoprobe to be coupled to the two nearest sites in the sample only for all
times $t>0$. The operator $c_{i,\sigma}^{(\dagger)}$ represents the
annihilation (creation) operator for an electron with spin $\sigma$ on lattice
site $i$, and
$n_{i,\sigma}=c^{\dagger}_{i,\sigma}c^{\phantom{\dagger}}_{i,\sigma}$ is the
electron density for spin $\sigma$ at site $i$. Here $t_{h}$ is the hopping
amplitude inside the sample and inside the nanoprobe; in the following, we
work in units in which $t_{h}\equiv 1$ and take $\hbar\equiv 1$. The coupling
$t_{h}^{\prime}$ is the hopping amplitude between sample and probe. The
parameter $U\geq 0$ denotes the strength of the repulsive on-site interaction
between the electrons, which we assume to be the same in the sample and in the
probe. If not stated explicitly otherwise, we take $U=4$, a value coresponding
to the bandwidth of the sample at $U=0$. In the following, we will also study
the situation in which the nanoprobe moves with a constant speed $v$ over the
sample. For speeds $v>0$, the hoppings between the nanoprobe and the sample
are adjusted in time according to the function $w(t,i,j)$ which leads to a
sequence of quantum quenches.

###  II.2  Observables

####  II.2.1 Local particle density $\langle n_{i}\rangle$

In all cases, we compute the time evolution of the local particle density
$n_{i}(t)\equiv\langle\psi(t)|n_{i}|\psi(t)\rangle$. Due to the strong
tunneling between sample and nanoprobe, we expect the backflow of electrons to
lead to nontrivial behavior in this quantity. Furthermore, coupling the
nanoprobe to the sample is a local quench, so that we expect that a
perturbation will propagate through the sample, leading to a light-cone-like
signature in $n_{i}(t)$. Note that for the Mott insulator at half filling, the
flow of electrons from the sample to the nanoprobe will move the sample away
from half filling, so that it should then lose the properties of a Mott
insulator in the course of time. In particular, this should affect the Mott
gap, which is visible in the LDOS.

####  II.2.2 Time-dependent local density of states

We calculate the time evolution of the energy-resolved LDOS at position $i$,
$D_{\sigma}(t,\omega,i)$, as the Fourier transform of the retarded two-time
Green’s function (see, e.g., Ref. 45 for a similar approach to computing the
nonequilibrium single-particle spectral function),

$D_{\sigma}(t,\omega,i)=\text{Re}\left[\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Theta(\tau)\,\mathrm{d}\tau\,W(\tau)e^{i\omega\tau}\left\langle\left\\{c^{\phantom{\dagger}}_{i,\sigma}(t+\tau),c_{i,\sigma}^{\dagger}(t)\right\\}\right\rangle\right]\,,$
(3)

where the operator $c_{i,\sigma}^{(\dagger)}(t)$ is the annihilation
(creation) operator for an electron with spin $\sigma$ at position $i$ in the
Heisenberg picture evolved with the full, time-dependent Hamiltonian $H(t)$ at
time $t$, and $\\{\cdot,\cdot\\}$ denotes the usual anticommutator. Note that
alternative ways of computing momentum- and energy-resolved time-dependent
spectral functions that avoid carrying out a Fourier transform to frequency
space do exist; see, e.g., Ref. 46. Here, however, we study the local spectral
properties using the more direct definition of the LDOS based on linear
response theory, Eq. (3), and compare the results to the equilibrium
expectations, giving us a quantitative measure for how strongly our time-
dependent LDOS deviates from the linear-response regime.

As discussed further below, we can compute the two-time Green’s function up to
a restricted maximal time $\tau_{\text{max}}$, so that it is necessary to
regularize the integral either by introducing a damping term $e^{-\eta\tau}$
or to introduce a windowing function in order to avoid ringing effects caused
by a sudden cut-off of the data at $\tau=\tau_{\text{max}}$ [47]. We find here
that applying the Hann window, which is defined as

$W(\tau)=\Theta(\tau_{\text{max}}-|\tau|)\sin^{2}\left(\frac{\pi\tau}{2\tau_{\text{max}}}+\frac{\pi}{2}\right)\,,$

leads to the best results.

Note that, at equilibrium (linear response theory), the time evolution of
$c_{i,\sigma}^{(\dagger)}(t)$ is carried out using the Hamiltonian of the
unperturbed system, so that time-translation invariance can be used to reduce
the time dependence of the Green’s function to one time variable. Since time-
translation invariance is absent out of equilibrum (e.g., in the quench we
perfom here at time $t=0$), we need to treat the full time dependence of the
two-time Green’s function. Here we use a relative-time representation of the
retarded two-time Green’s function, i.e.,
$\left\langle\left\\{c_{x,\uparrow}^{\phantom{\dagger}}(t+\tau),c_{x,\uparrow}^{\dagger}(t)\right\\}\right\rangle$.
This has two helpful aspects: (i) we can interpret the variable $t$ as the
waiting time after switching on the coupling between the nanoprobe and the
sample and ask for the behavior of the LDOS at this waiting time. (ii) It
makes the numerical evaluation of this quantity easier. This is the case
because, due to the Heaviside function $\Theta(\tau)$, we only need to compute
the Green’s function for $\tau>0$. Rather than this relative-time
representation, one could instead use the so-called Wigner representation
(see, e.g., Ref. 48), for which, however, one would also need to compute
expectation values at further points in time, increasing the computational
cost. According to Ref. 48, the results in both representations do not differ
significantly, so that we choose to work in the relative-time representation
for the sake of efficiency.

Due to the limitations of the numerical approaches, we have evaluated the
integral numerically using a time step $d\tau=0.01$ up to times
$\tau_{\text{max}}=5$ (except that we take $\tau_{\text{max}}=20$ for the
$U=0$ case, which can be treated exactly). However, we discretize $\omega$ at
2000 equally spaced points between $\omega_{\text{min}}=-10$ and
$\omega_{\text{max}}=10$ by interpolation using zero-padding. Note that, at
equilibrium, the LDOS defined in this way can be expressed as a Lehmann
representation with positive weights, so that an interpretation as a spectral
weight is natural. However, this is no longer possible in the nonequilibrium
case, and negative weights can, in principle, appear [49]. As discussed
further below, we observe negative weights only at short times, at which the
system is strongly out of equilibrium, but, at later times, negative weights
seem to be absent, so that an interpretation of the results in terms of
spectral weights is possible. Similarly, in nonequilibrium situations, one
needs to be careful about identifying the occupied and unoccupied parts of the
LDOS: at equilibrium, one usually introduces the lesser and the greater parts
of the LDOS, $B_{\sigma}^{<}(\omega,i)$ and $B_{\sigma}^{>}(\omega,i)$ [50],
which indicate the populated and empty states on lattice site $i$,
respectively. Out of equilibrium, one can generalize these functions as
follows:

$B_{\sigma}^{<}(t,\omega,i)=\text{Re}\,\left[\frac{1}{\sqrt{2\pi}}\int_{0}^{\tau_{\text{max}}}\mathrm{d}\tau\,W(\tau)e^{i\omega\tau}\left\langle
c_{i,\sigma}^{\dagger}(t)c^{\phantom{\dagger}}_{i,\sigma}(t+\tau)\right\rangle\right]\,,$
(4)

$B_{\sigma}^{>}(t,\omega,i)=\text{Re}\,\left[\frac{1}{\sqrt{2\pi}}\int_{0}^{\tau_{\text{max}}}\mathrm{d}\tau\,W(\tau)e^{i\omega\tau}\left\langle
c^{\phantom{\dagger}}_{i,\sigma}(t+\tau)c_{i,\sigma}^{\dagger}(t)\right\rangle\right]\,.$
(5)

Note that, due to the lack of a Lehmann representation, both quantities can
become negative. Furthermore, linear response theory requires time-translation
invariance in order to interpret these quantities as occupied or empty parts
of the spectrum, respectively. Here, due to the nonequilibrium setup, we do
not have this symmetry when going to negative $\tau$, so that we can only
expect these quantities to represent the occupied (empty) part of the LDOS
approximately. We will come back to this interesting aspect in Sec. III.1.4,
where we discuss how to estimate the occupied states in
$D^{<}_{\sigma}(t,\omega,i)$ at lattice site $i$ by comparing to $\langle
n_{i,\sigma}\rangle(t)$.

####  II.2.3 Nonequilibrium occupation energy

At equilibrium, the LDOS is obtained as

$\displaystyle D^{\rm eq}_{\sigma}(\omega,i)$ $\displaystyle=$
$\displaystyle\text{Re}\left[\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Theta(\tau)\,\mathrm{d}\tau\,W(\tau)e^{i\omega\tau}\right.$
(6)
$\displaystyle\left.\left\langle\left\\{c^{\phantom{\dagger}}_{i,\sigma}(\tau)^{H_{0}},c_{i,\sigma}^{\dagger}(0)^{H_{0}}\right\\}\right\rangle\right]\,,$

where $H_{0}$ indicates that the time evolution of the operators is carried
out using the unperturbed Hamiltonian, and, due to time-translation
invariance, only one time variable needs to be treated. Integrating the
equilibrium LDOS $D^{\rm eq}_{\sigma}(\omega,i)$ in energy up to the Fermi
level $E_{F}$ will yield the local particle density $\langle
n_{i,\sigma}\rangle$, so that the LDOS below $E_{F}$ can be interpreted as the
occupied part. The Fermi energy $E_{F}$ is hence the value of the energy of
the highest populated state in the system. Similarly, at equilibrium, one can
introduce

$\displaystyle B^{<,\rm eq}_{\sigma}(\omega,i)$ $\displaystyle=$
$\displaystyle\text{Re}\left[\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\,\mathrm{d}\tau\,W(\tau)e^{i\omega\tau}\right.$
(7) $\displaystyle\left.\left\langle
c^{\dagger}_{i,\sigma}(0)^{H_{0}}c_{i,\sigma}^{\phantom{\dagger}}(\tau)^{H_{0}}\right\rangle\right]$

as the lesser part of the LDOS. Note that $B_{\sigma}^{<,\rm eq}(\omega,i)$
does not have weights for energies higher than $E_{F}$, so that
$\int_{-\infty}^{\infty}d\omega\,B_{\sigma}^{<,\rm eq}(\omega,i)=\langle
n_{i,\sigma}\rangle$. Hence, $B_{\sigma}^{<,\rm eq}(\omega,i)$ represents the
occupied part of the LDOS. Out of equilibrium, however, one needs to be more
careful. A simple way to identify the populated states is to assume that only
the lowest-energy parts of the nonequilibrium LDOS $D_{\sigma}(t,\omega,i)$
are occupied. Integrating $D_{\sigma}(t,\omega,i)$ at site $i$ at fixed
waiting time $t$ up to a certain energy, which we call $E_{\rm occ}$, should
give the same result as $\langle n_{i,\sigma}\rangle(t)$. The value $E_{\rm
occ}(t)$ can then be interpreted as the nonequilibrium generalization of
$E_{F}$. We define $E_{\rm occ}$ implicitly via

$\frac{1}{C}\int_{-\infty}^{E_{\rm
occ}(t,i)}\mathrm{d}\omega\,D_{\sigma}(t,\omega,i)\,=\langle
n_{i,\sigma}\rangle(t)\,,$ (8)

with the normalization
$C=\int_{-\infty}^{\infty}\mathrm{d}\omega\,D_{\sigma}(t,\omega,i)$. In the
following, spin-flip symmetry is present; thus, for simplicity, we only
discuss the $\sigma=\,\uparrow$ component of both the LDOS and the particle
density.

Furthermore, we can compare the expectation value for the number of particles
as obtained from the energy integral $\int
d\omega\,B^{<}_{\sigma}(t,\omega,i)$ at fixed $t$ and $i$ with the directly
computed expectation value $\langle n_{\sigma,i}\rangle(t)$, which, in
equilibrium, is identical, as discussed above. Differences from this
expectation value can be considered to be a measure of how far away from
equilibrium the system is. As discussed further below, significant deviations
are obtained at short times in particular, indicating strong nonequilibrium
behavior, while, at later times, both quantities agree to within a few
percent. The time evolution of $E_{\rm occ}(t)$ also indicates how particles
are redistributed in the course of time, in particular, for the cases in which
the LDOS is changing in time due to the strong interactions in the system.

###  II.3  MPS-based methods

####  II.3.1 Initial state preparation and time evolution

All calculations for the interacting system have been performed at $U=4$ using
a density matrix renormalization group (DMRG) algorithm [51, 52] within the
MPS framework [53, 54], which is available in the SymMps toolkit [55].

The general procedure is as follows. For $U=4$, we calculate the ground state
on a system with $L=50$ sample lattice sites and $l=5$ nanoprobe sites for the
quarter- and half-filled samples, with $N=26$ and $N=50$ particles,
respectively. (We take $N=26$ rather than $N=25$ particles for quarter filling
so that the number of spin-up and spin-down particles is equal.) Initially,
the nanoprobe is not yet coupled to the system, i.e., $t_{h}^{\prime}=0$. To
ensure a state with zero occupation at time $t=0$ in the nanoprobe, we add a
small repulsive electrostatic potential $\sum\limits_{j\in\,{\rm
probe}}\mu_{j}(n_{j,\uparrow}+n_{j,\downarrow})$ on these sites. We then
perform a ground-state search with a maximum bond dimension of $\chi=2500$ and
a maximum discarded weight of $\delta=10^{-14}$ with a total of 50 sweeps.
This allows us to approximate the ground state of the system with an absolute
error in the ground state energy of $3.0\times 10^{-6}$ in the quarter-filled
case, compared to the exact value known from Bethe Ansatz [56, 42]. The
absolute error in the ground-state energy in the half-filled case is even
smaller, $1.7\times 10^{-7}$.

At time $t=0$, the electrostatic potential in the nanoprobe is then set to
zero, and it is immediately coupled to the system with the full hopping
strength, $t_{h}^{\prime}=t_{h}$. We compute the subsequent time evolution of
the total system using the time-dependent variational principle (TDVP) in its
two-site version using MPS [57, 58]. This allows us to treat systems with
arbitrary coupling ranges and to adjust the bond dimension in the course of
the time evolution, keeping track of the growth of entanglement with time. It
is known that the TDVP can have substantial problems with product initial
states. Alternatively, one could use the matrix product operator (MPO) WII
time-evolution algorithm [59, 58], which does not suffer from such problems.
Here we have tested both algorithms, comparing with exact diagonalization for
systems with 3 particles, and we find that the errors of the TDVP for the
given setup are an order of magnitude smaller than those of the MPO WII
algorithm. The MPO WII has a maximum absolute error of $\approx 4\times
10^{-4}$ in $n_{i}(t)$ for this test case, whereas the TDVP algorithm reaches
a maximum error of $\approx 5\times 10^{-5}$. In addition, in App. A, we
compare the time evolution of the half-filled case at $U=0$ as obtained from
MPS with that of the exact solution (see Sec. II.3.3). As shown there, the
absolute errors grow in time. For $v=0$, we find that the absolute errors in
$\langle n_{i,\sigma}\rangle(t)$ are $\lesssim 0.05$ at later times, and the
absolute errors in the LDOS are $\lesssim 0.015$, while for the moving probe
the errors in the LDOS are even smaller, although the absolute errors in
$\langle n_{i,\sigma}\rangle(t)$ can be as high as $\sim 0.1$ at isolated
points. Hence, we believe that our results are sufficiently accurate to
extract the essential physical behavior.

We set the threshold for the discarded weight during the time evolution to
$\delta=10^{-10}$ and the maximum bond dimension to $\chi_{\rm max}=2500$. The
time step size is $\delta t=0.01$ for all results presented. As discussed in
App. A, the resulting discarded weight at later times is rather large, almost
$10^{-4}$; however, due to the high computational cost, we do not further
increase the bond dimension or carry out calculations to longer times.

We perform the time evolution so that, for all waiting times, the integration
time is $\tau_{\text{max}}=5$ after the waiting time (i.e., for waiting time
$t=5$, we perform the simulations up to times $t+\tau_{\text{max}}=10$). In
this way, all the presented results at different waiting times have the same
resolution in $\omega$.

For a finite speed $v>0$ of the probe, the Hamiltonian is explicitly time-
dependent. In principle, more accurate results can be obtained by applying
more elaborate discretization schemes on the time variable, such as
commutator-free exponential time (CFET) propagators [60]. However, in our
case, due to the structure of the function $w(t,i,j)$ in the Hamiltonian (1),
the time-dependence of $H$ is not continuous, but comes in steps at points in
time determined by the value of $v$. The time evolution for $v>0$ is,
therefore, realized as a sequence of quantum quenches, for which the usual
methods (here the TDVP) perform well.

####  II.3.2 Swap operator

For the case of the moving nanoprobe, i.e., $v>0$, the hopping terms to the
sample become longer range with time, which leads to larger entanglement
between the two parts of the system and hence makes the MPS description less
accurate. In our case, we have only nearest-neighbor and next-nearest-neighbor
hopping at time $t=0$; it is useful to keep the hopping as short-range as
possible in the course of the time evolution. We address this by introducing
the swap operator (see Ref. 61 for the usage of swap operators for time-
evolution algorithms),

$P_{i}=\prod_{\sigma=\uparrow,\downarrow}\left[1-\left(c_{i,\sigma}^{\dagger}-c_{i+1,\sigma}^{\dagger}\right)\left(c^{\phantom{\dagger}}_{i,\sigma}-c_{i+1,\sigma}^{\phantom{\dagger}}\right)\right]\,,$
(9)

which is applied at the points in time when the probe, according to the
function $w(t,i,j)$, Eq. (2), couples to new sites in the sample. In this way,
the site labeling is adapted so that the hopping terms for all times treated
are either nearest-neighbor or next-nearest-neighbor terms, thus minimizing
the entanglement growth in the course of time.

####  II.3.3 Solution for $U=0$

In the noninteracting case, $U=0$, we can solve the system exactly by
diagonalizing the Hubbard Hamiltonian (1) for a single particle. In doing
this, we add an additional chemical potential of $\mu=10^{8}$ to the sites of
the nanoprobe so that the wave-function amplitude is vanishingly small there.
This ensures that the initial many-body state has zero particle density on the
nanoprobe before the coupling between nanoprobe and sample is turned on at
$t=0$. This then yields the single-particle wave functions $\phi_{i,n}(0)$,
where $i$ denotes the lattice site and $n$ refers to the $n$-th excited state
for a single particle, with $n=1$ being the single-particle ground state. The
time-dependent state $|\psi_{i}(t)\rangle$ is then easily obtained by applying
the time-evolution operator $e^{-iHt}$:

$\displaystyle|\psi_{i}(t)\rangle$ $\displaystyle=$
$\displaystyle\left(\sum_{n=1}^{N_{\uparrow}}\phi_{i,n}(t)c_{n,\uparrow}^{\dagger}+\sum_{n=1}^{N_{\downarrow}}\phi_{i,n}(t)c_{n,\downarrow}^{\dagger}\right)|0\rangle,$

where $\phi_{i,n}(t)=e^{-iHt}\phi_{i,n}(0)$ are the time-evolved single-
particle states. The time-dependent LDOS $D_{\sigma}(t,\omega,i)$ can then be
calculated via the transformation
$c_{i,\uparrow}(t)=\sum_{n^{\prime}}\phi_{i,n^{\prime}}(t)\,c_{n^{\prime},\uparrow}\,.$
Note that $n$ refers to the single-particle states for the decoupled system;
$n^{\prime}$, however, refers to the single-particle eigenstates after
coupling the nanoprobe to the Hubbard chain. Inserting this expression into
Eq. (3) leads to a simple expression for the two-time Green’s function:

$\displaystyle\left\langle\left\\{c^{\phantom{\dagger}}_{i,\sigma}(t+\tau),c_{i,\sigma}^{\dagger}(t)\right\\}\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{n^{\prime}}\phi^{\phantom{\ast}}_{i,n^{\prime}}(t+\tau)\phi_{i,n^{\prime}}^{\ast}(t)\,.$
(10)

This expression is independent of the number of particles in the system, as
particles and holes lead to the same contribution for the LDOS in the $U=0$
case. Furthermore, at $v=0$, i.e., for a time-independent Hamiltonian, the
LDOS of the noninteracting system is time-independent because the dependence
on $t$ in Eq. (10) vanishes for $e^{-iH(t+\tau)}\,e^{iHt}=e^{-iH\tau}$. Note
that Eq. (10) can be calculated accurately to arbitrarily long times, so that,
in contrast to the MPS results at finite $U$, we can obtain much better
resolution in $\omega$.

##  III Results

In this section, we present representative results for the time evolution of
our setup. We organize the discussion by first considering a stationary
nanoprobe, i.e., one that does not move over the sample, $v=0$, then going on
to a relatively slow nanoprobe with $v=0.55$. We have also carried out
calculations at a higher velocity, $v=1$, which are presented in App. B; we
will also describe the salient aspects of the behavior relative to that of
$v=0.55$ in the main text. For all values of $v$, we first discuss the
behavior of the noninteracting case, $U=0$, then present results for the
interacting case, taking the Hubbard interaction to have intermediate
strength, $U=4$. For all cases, we take a system with $L=50$ sample sites with
$l=5$ initially empty probe sites and treat two values of the band filling,
quarter filling with $N=26$ particles and half filling, with $N=50$ particles.
We present the results for the time evolution of the local particle density in
the sample and in the nanoprobe as well as the LDOS at three fixed positions:
in the sample far away from the nanoprobe, site $i_{s}=13$, in the sample
directly under the nanoprobe, site $i_{s}=26$, and in the center of the
nanoprobe, site $i_{p}=3$. All results are obtained for samples with $L=50$
lattice sites and nanoprobes with $l=5$ lattice sites.

###  III.1 Resting nanoprobe: $v=0$

We first consider the case in which the nanoprobe does not move, i.e., $v=0$.
As soon as the tunneling between the system and the nanoprobe is turned on at
time $t=0$, the system evolves nontrivially in time as it undergoes a quantum
quench. As the tunneling will also be turned on at $t=0$ for the cases of a
moving nanoprobe, the $v=0$ behavior will provide a basis for the
interpretation of the moving cases. Since the nanoprobe initially holds no
particles, particles will begin tunneling from the system to the nanoprobe at
$t=0$.

####  III.1.1 Noninteracting case $U=0$

We start with the noninteracting case, $U=0$. In Fig. 2(a), we display the
expectation value of the local particle density $n_{i}(t)$ for the quarter-
filled system as a color density plot plotted as a function of position and
time. As can be seen in the plot, at time $t=0$, the nanoprobe is completely
empty of particles. After the tunneling to the system is turned on ($t>0$),
the particles begin to tunnel back and forth, leading to oscillations in time
in the local particle density in both the nanoprobe and in the region of the
system in its vicinity. These oscillations are somewhat spatially
inhomogeneous both in the system and in the nanoprobe, with the strongest
fluctuations of particle density occurring near the center of the nanoprobe.
As time progresses, the time dependence in local particle density spreads out
in time. In particular, a “light cone” that spreads out across the sample at
constant speed $c\approx 1.6$, which we have estimated by roughly fitting the
wave front, is evident. In Figs. 2(b)–(d), we display the LDOS
$D_{\uparrow}(t,\omega,x)$ for this case at sites $i_{s}=13$ and $i_{s}=26$ in
the sample and $i_{p}=3$ in the nanoprobe. Sites $i_{s}=26$ and $i_{p}=3$ are
directly next to each other and become directly linked by hopping terms as
soon as the coupling between probe and sample is turned on. For $U=0$ and a
speed of $v=0$, the LDOS is time-independent for all sites after the coupling
is turned on. This is to be expected because the Hamiltonian is time-
independent for $t>0$ in this case, and there is no scattering between the
electrons; see also the discussion in Sec. II.3.3. Thus, we display the LDOS
for time $t=0$ only. For $i_{s}=13$, Fig. 2(b), we observe peaks at
$\omega\approx-2$ and $\omega\approx 2$, with an oscillating and non-vanishing
LDOS in between the peaks. At sample site $i_{s}=26$ and probe site $i_{p}=3$,
which are adjacent to one another, we can see that the LDOS for both sites
have peaks that are largely at the same $\omega$ positions, with only the
amplitudes differing somewhat. In contrast to the LDOS for $i_{s}=13$, there
is a major peak at $\omega\approx-4$ for these two sites. Regarding the
occupation energies, we find that $E_{\rm occ}(t)$ is essentially time-
independent on lattice site $i_{s}=13$, indicating no (or only very little)
redistribution of particles on the time scale considered. This is to be
expected, as the light-cone-like perturbation in $n_{i}(t)$ reaches this
position only at later times, $t>5$, which are not shown. At site $i_{s}=26$,
$E_{\rm occ}(t)$ is initially close to the equlibrium Fermi energy at site
$i_{s}=13$. After the nanoprobe is coupled to the sample, $E_{\rm occ}(t)$
begins to oscillate, almost reaching the left edge of the LDOS (i.e., the
lattice site is nearly completely depleted) at time $t=4$. The occupation
energy at site $i_{p}=3$ inside the nanoprobe is not defined at time $t=0$
because the system is empty. However, after the probe is coupled to the
sample, $E_{\rm occ}(t)$ begins to oscillate, similarly to the value on site
$i_{s}=26$, which is directly coupled to this lattice site of the nanoprobe.
The oscillations of $E_{\rm occ}(t)$ on these two sites have approximately
opposite phase, indicating strong tunneling of particles between the two
lattice sites. This is to be expected, since at $U=0$ there is no scattering
between the particles, which could lead to equilibration of local observables
on short time scales.

Figure 2:  Exact results for the nanoprobe model with $L=50$ sample sites and
$l=5$ nanoprobe sites for the system initially at quarter filling, i.e.,
$N=26$, with $U=0$ and nanoprobe velocity $v=0$. Here (a) depicts the
expectation value of the local particle density, $\langle{n}_{i}\rangle$, as a
color density plot as a function of lattice site $i$ and time $t$, and (b),
(c), and (d) plot the LDOS for the lattice sites $i_{s}=13$, $i_{s}=26$, and
$i_{p}=3$, respectively, as a function of the frequency $\omega$. Note that
the LDOS is rigorously time-independent for $U=0$ when $v=0$. The occupation
energy $E_{\rm occ}(t)$ is shown as an inset for each of the three lattice
sites as a function of $t$ and is also shown at five indicated times as
vertical lines in the LDOS.

####  III.1.2 Interacting case at quarter filling

We now turn on the Hubbard interaction to $U=4$, staying with quarter filling
and resting nanoprobe, $v=0$; see Fig. 3. As can be seen by comparing Fig.
3(a) with Fig. 2(a), the oscillation in local particle density between the
system and the nanoprobe is more damped than in the $U=0$ case. Again, a light
cone that spreads out over the system with the speed $c\approx 1.6$ is clearly
evident. Examining the LDOS for site $i_{s}=13$ (well to the left of the
nanoprobe), we obtain a double peak structure, and an additional separated
satellite peak at higher $\omega$. Again, as in Fig. 2(b), $E_{\rm occ}(t)$ is
essentially time-independent on this lattice site on the time scales shown. As
before, this is to be expected from the light-cone signal in $n_{i}(t)$, which
reaches this position only after the times shown.

For lattice sites $i_{s}=26$, Fig. 2(c), and $i_{p}=3$, Fig. 2(d), the overall
structure of the LDOS is similar at all times. However, in contrast to the
noninteracting case, the LDOS is time-dependent at both sites. This is due to
the interactions between the electrons. Interestingly, at the later times
treated by us, they seem to settle to a stationary value, and the change in
time is smaller than at the beginning of the evolution. Again, $E_{\rm
occ}(t)$ changes significantly in time on both sites and oscillates with the
period of the oscillations of the local densities on these sites. The phase of
the oscillation again shows a tendency to be opposite on both sites, but the
electron-electron interaction now induces a damping.

Note that, as discussed in Sec. II, the resolution in $\omega$ is
significantly lower than for the $U=0$ case, so that finer structures are not
resolvable. In addition, on general grounds, the interaction $U$ leads to
self-energy contributions, which tend to broaden out peaks in spectral
functions relative to the noninteracting case.

Figure 3:  MPS results for (a) the local particle density and the LDOS for (b)
$i_{s}=13$, (c) $i_{s}=26$, and (d) $i_{p}=3$, plotted as in Fig. 2, for
$U=4$, quarter filling, i.e., $N=26$, and $v=0$.

####  III.1.3 Interacting case at half filling

We now consider the effect of changing the initial filling of the system to
half filling. Figure 4 depicts the time evolution of the local particle
density for the noninteracting ($U=0$) half-filled system. In contrast to the
quarter-filled case, Fig. 2(a), we can see that the oscillatory behavior is
now spread across the whole nanoprobe rather than being concentrated in its
center. As discussed in Sec. II.3.3, the LDOS for the $U=0$ case is
independent of band filling. However, $E_{\rm occ}$ changes in time, as shown
in Fig. 2.

Figure 4: Exact results for the local particle density plotted as in Fig. 2
for $U=0$, half filling, i.e., $N=50$, and $v=0$. The occupation energy
$E_{\rm occ}(t)$ is shown as an inset for sites $i_{s}=26$ and $i_{p}=3$. The
LDOS in this case is not explicitly shown, as it is the same as in Fig. 2.

At finite interaction, $U=4$, Fig. 6(a) shows that the oscillations in the
local particle density between the sample and the nanoprobe are also present,
but are heavily damped. The large damping is not unexpected, as the system is
insulating for $U=4$ and half filling, inhibiting charge transport.

In the LDOS at $i_{s}=13$, Fig. 6(b), we observe two peaks, which, at the
beginning of the time evolution, have essentially the same height. The
splitting between the peaks is consistent with the size of the Mott-Hubbard
gap at this value of $U$, which, according to the Bethe Ansatz, is
$\Delta\approx 1.29$ [43]. However, note that the weight of the LDOS in Fig. 6
does not go to zero in the gap region. This is because of the limited
resolution of our calculations; see Sec. II. As in the previously discussed
cases, the LDOS and $E_{\rm occ}$ on this site initially show only very weak
time dependence, while at $t=5$ a clear change is visible. We associate this
onset of change with the arrival of the light cone, which induces a change of
the number of particles on this site, and hence a change of the LDOS and
$E_{\rm occ}$.

For $i_{s}=26$ and $i_{p}=3$, Figs. 6(c) and (d), we obtain a continuous LDOS
with a smaller peak at energies below $E_{\rm occ}$ and a higher peak at
energies larger than $E_{\rm occ}$. On both sites, the LDOS changes in time,
while, at later times, the changes become smaller. The overall structure of
the LDOS for both sites and at all times is again comparable, as in the case
of quarter filling. However, the gap is no longer present after the coupling
to the probe is turned on, indicating a melting of the Mott insulator in this
region.

We expect that this melting of the Mott insulator will propagate through the
system and that the gap in the LDOS will close with the arrival of the light
cone on the respective site. It remains an open question as to whether the
shape of the LDOS at longer times will approach the results on site $i_{s}=26$
also further away from the nanoprobe. Due to the high computational cost, we
do not address this question further in this paper.

From the insets of Figs. 6(b)-(d), it can be seen that $E_{\rm occ}(t)$
behaves similarly to the previously discussed cases, but that the damping is
much stronger. The behavior on sites $i_{s}=26$ and $i_{p}=3$ indicates that
the amplitudes of the oscillations are damped nearly completely to zero on the
time scales that we have been able to treat; in addition, the values of
$E_{\rm occ}(t)$ at time $t=5$ are quite similar, which corresponds to what we
would expect if equilibration takes place. Both aspects indicate that the LDOS
reaches equilibrium on both adjacent sites on the short time scales treated
here. This is interesting because, in typical materials, the hopping strength
is $t_{h}\sim 1$eV, which translates to time scales ${\cal O}(1\,\text{fs})$,
corresponding to one time unit in Fig. 6. Thus, equilibration after coupling a
Mott insulator to a nanoprobe seems to happen on ultrafast time scales of a
few fs. To further investigate the equilibration behavior of the Mott-
insulating case, longer times and additional Mott-insulating systems would
need to be treated, which, due to the high computational expense, is left as a
subject for future research.

####  III.1.4 Interpretation of the results as a nonequilibrium LDOS

In the preceeding discussion, we have tacitly interpreted our results as if we
were dealing with an equilibrium or near-equilibrium setup (i.e., have applied
linear response), so that the corresponding Fourier transform of the Green’s
function, Eq. (3), can be rewritten in terms of a Lehmann representation with
explicitly positive weights. Out of equilibrium, the weights cannot be proven
to be positive in general [49], so that, in principle, negative weights can
appear. This makes it difficult to interpret the corresponding results as a
time-dependent LDOS. However, as can be seen in Figs. 2-6, all of the weights
calculated here are positive, up to small artifacts, which may be due, at
least in part, to the way we compute the Fourier transform. Hence, for the
cases treated so far, it appears reasonable to interpret the results as time-
evolving LDOS.

This issue can be further investigated by studying the lesser and greater
parts of the LDOS as defined in Eqs. (4) and (5), respectively. In
equilibrium, the expectation is that $\int
d\omega\,B_{\sigma}^{<}(\omega,x)=\langle n_{x,\sigma}\rangle$. Figure 5 shows
results for the discrepancy between the two quantities in the out-of-
equilibrium case for both the noninteracting and the interacting, $U=4$, cases
at half filling with a resting nanoprobe. The agreement between $\int
d\omega\,B_{\sigma}^{<}(\omega,x)$ and the independently computed expectation
values $\langle n_{x,\sigma}(t)\rangle$ at later times is within a few
percent, thus substantiating our interpretation of $D_{\sigma}(\omega,x,t)$,
as defined in Eq. (3), as a good approximation to the time evolution of the
LDOS for the later times. Interestingly, for the interacting system, the
discrepancy is smaller than for the noninteracting system. At shorter times,
$t\approx 1$, however, the discrepancy can become as large as $\sim 30\%$.
This shows that here the system is in a strongly out-of-equilibrium regime in
which the interpretation of our results as a time-dependent LDOS must be
treated with caution. Note that, in all cases, the absolute value of the
discrepancy is very small; at times $t\approx 1$, however, the small particle
numbers lead to a large relative discrepancy. The discrepancy in the actual
values is larger at small times, and, again, is smaller for the interacting
system.

For the case of a moving probe, similar behavior is obtained (see section
III.2 and App. B). However, as we will see next, the computed LDOS can, in
addition, take on negative values, in particular, in the strongly out-of-
equilibrium regime at short times.

Figure 5:  Absolute and relative discrepancy between the local expectation
value of the local densities $\langle n_{\sigma,x}\rangle(t)$ and the integral
over the lesser spectral function $\int d\omega\,B_{\sigma}^{<}(\omega,x)$.
Left column: half filling, i.e., $N=50$, $U=0,\,v=0$. Right column: the same
at $U=4$. (Top row: relative difference, bottom row: absolute difference.) We
define the absolute difference here as $|\int
d\omega\,B_{\sigma}^{<}(\omega,x)-\langle n_{\sigma,x}\rangle(t)|$ and the
relative difference as $|\left(\int
d\omega\,B_{\sigma}^{<}(\omega,x)\right)/\langle n_{\sigma,x}\rangle(t)-1|$.

Figure 6:  MPS results for (a) the local particle density and the LDOS for (b)
$i_{s}=13$, (c) $i_{s}=26$, and (d) $i_{p}=3$, plotted as in Fig. 2, for
$U=4$, half filling, i.e., $N=50$, and $v=0$.

###  III.2 Moving nanoprobe: $v>0$

Having explored the behavior when coupling a stationary nanoprobe to the
sample at time $t=0$, we now treat cases in which the nanoprobe is moving at
constant speed. As in the stationary case, we also turn on the coupling
between sample and a nanoprobe at the center of the sample at time $t=0$;
subsequently, the nanoprobe moves to the right relative to the sample with
speed $v$.

A natural scale for the velocities in the system is the velocity of
propagation of disturbances in the local particle density, i.e., the speed
associated with the previously discussed light cone, $c\sim 2$. We treat two
different velocities, $v=0.55$ ($\approx c/4$) and $v=1$ ($\approx c/2$). Here
we present the findings for $v=0.55$ in detail and complement the discussion
by highlighting the similarities and differences with the case of the higher
velocity of $v=1$, for which we present additional results in App. B.

In the following, we will use the resting case, $v=0$, as a reference,
highlighting the similarities and differences of the behavior in the two
moving cases. For all velocities $v>0$, we find that there are stronger wave
fronts to the left of the right-moving nanoprobe and weaker ones to its right,
see, e.g., Fig. 7(a) for $v=0.55$, $U=0$, and quarter filling. Note that the
slope of the wave fronts is still consistent with a constant speed $c\approx
1.6$ for the light cone. The effect of the movement of the probe is that
particles “leak out” of the nanoprobe in the wake of its motion. The speed of
$v=0.55$ is an interesting case because, for this speed, we find that these
“wake” effects are particularly pronounced, in contrast to the $v=1$ case (see
App. B). For $v=0.55$, the motion of the probe is spatiotemporally
commensurate with the oscillations of the particle density, which leads to
constructive interference to the left of the moving probe. Therefore, the wave
fronts are very pronounced for this particular velocity, as can be seen in
Fig. 7(a). Strong oscillations in the local particle density between the
nanoprobe and the system are still present, but, for the moving nanoprobe,
they are mostly concentrated at its trailing edge rather than its center as in
the stationary case.

The LDOS for this case is displayed in Figs. 7(b)–(d), for the same three
sites as before, $i_{s}=13$, $i_{s}=26$, and $i_{p}=3$. Note that the LDOS is
now, in general, time-dependent, also for $U=0$. As at $v=0$, the LDOS at site
$i_{s}=13$ is essentially time-independent due to the fact that the nanoprobe
is moving away from this site so that, at the times treated, the wake caused
by the moving nanoprobe has not yet reached it. At site $i_{s}=26$, Fig. 7(c),
the evolution of the LDOS differs significantly from that of the resting case,
Fig. 2(c). One marked feature is that the results can now take on
substantially negative values, in particular, for times $t<3$. However, at
later times, the discrepancy between $\langle n_{i,\sigma}\rangle$ and $\int
d\omega\,B^{<}_{\sigma}(\omega)$ is only a few percent, a behavior similar to
that of the case of the stationary nanoprobe. Note that $i_{s}=26$ is a
stationary point in the center of the sample and therefore becomes decoupled
from the nanoprobe at time $\approx 6.4$, which is shorter than the time
interval over which we perform the Fourier transform from $t$ to $\omega$. For
the faster moving case, $v=1$, (see App. B), site $i_{s}=26$ already decouples
from the nanoprobe at times $t\approx 3.5$, so that the LDOS, Fig. 14(b),
takes on, approximately, the decoupled equilibrium form, i.e., that of Fig.
2(b), for $t\geq 4$.

The LDOS at $i_{p}=3$, Fig. 7(d), is essentially time-independent, despite the
fast motion of the nanoprobe, a behavior also found in the $v=1$ case; see
App. B. The structure and the positions of the peaks are roughly the same as
for the resting nanoprobe, Fig. 3(d). Regarding $E_{\rm occ}(t)$, we see a
similar behavior to that of the resting case for $i_{s}=13$ and $i_{p}=3$.
However, $E_{\rm occ}(t)$ appears to increase with time for $i_{s}=26$, unlike
in the resting case.

Figure 7: Exact results for (a) the local particle density and the LDOS for
(b) $i_{s}=13$, (c) $i_{s}=26$, and (d) $i_{p}=3$, plotted as in Fig. 2, for
$U=0$, quarter filling, i.e., $N=26$, and $v=0.55$.

Turning on the interaction to $U=4$ in the quarter-filled case with $v=0.55$,
Fig. 8, we see that $\langle n_{i}(t)\rangle$ has similar features to that of
the $U=0$ case, Fig. 7(a). Note, however, that now the oscillation is
significantly more damped. As before, the LDOS at site $i_{s}=13$, Fig. 8(b),
is approximately time-independent since the light cone has not reached this
position on the time scales treated by us. Comparing the LDOS on site
$i_{s}=26$ to the resting case, $v=0$, we observe that the overall picture is
comparable, but that there are now more changes in time in the relevant energy
region; notably, at $t=3$, a dip evolves between two peaks at
$\omega\approx-2.5$ and $\omega\approx 1$. However, this dip disappears at
later times. It is notable that—in contrast to the noninteracting case—the
LDOS takes on (up to minimal effects) only positive values for all values of
$\omega$ shown. As discussed in Section III.1.4, the discrepancy of the
occupations is only a few percent, so that, again, interpreting the spectrum
as a quasi-equilibrium LDOS seems to be justified. For $i_{p}=3$, Fig. 8(d),
there is only weak time dependence. The results are very similar to the ones
for $v=0$, with only minor qualitative differences. This is interesting, as it
indicates that the motion of the nanoprobe seems to not significantly affect
the time evolution of the LDOS on its sites. This similarity of the LDOS on
site $i_{p}=3$ to the $v=0$ case remains present when the nanoprobe velocity
is higher, $v=1$; see App. B.

The time-dependent behavior of $E_{\rm occ}(t)$ for $v=0.55$, depicted in the
insets of Figs. 8(b)-(d), is very similar to that of the resting case, Figs.
3(b)-(d), However, for $i_{s}=26$, $E_{\rm occ}(t)$ appears to settle to a
stationary value within the simulation time, in contrast to the $v=0$ case.
This difference in behavior is due to the fact that the probe moves away from
this position so that local observables equilibrate faster than when the probe
is resting. At site $i_{p}=3$, however, $E_{\rm occ}(t)$ keeps changing in
time, as might be expected because the nanoprobe continues to move over the
sample.

Figure 8:  MPS results for (a) the local particle density and the LDOS for (b)
$i_{s}=13$, (c) $i_{s}=26$, and (d) $i_{p}=3$, plotted as in Fig. 2, for
$U=4$, quarter filling, i.e., $N=26$, and $v=0.55$.

We continue examining probe velocity $v=0.55$, but now increase the initial
band filling to half filling, first taking $U=0$. Figure 9 displays the $U=0$
local particle density. As can be seen by comparing with Fig. 7(a), the wave
fronts to the left of the nanoprobe are even more pronounced than in the
quarter-filled case. Furthermore, there is an even stronger build-up of local
particle density inside the probe. At larger times, the local particle density
rapidly changes over to being concentrated at the leading edge of the
nanoprobe. Note that the build-up of local density is even stronger in the
$v=1$ case, with it being concentrated at the trailing edge of the probe for
times up to $t\approx 7$; see App. B.

Figure 9:  Exact results for the local particle density, plotted as in Fig. 2,
for $U=0$, half filling, i.e., $N=50$, and $v=0.55$. The occupation energy
$E_{\rm occ}(t)$ is shown as an inset for sites $i_{s}=26$ and $i_{p}=3$. The
LDOS in this case is not explicitly shown, as it is the same as in Fig. 7.

We now consider the interacting case, $U=4$, Fig. 10, remaining at half
filling and $v=0.55$. As in the half-filled, interacting, $v=0$ case, Fig.
6(a), we see once again that the oscillation between the nanoprobe and the
system is heavily damped. We notice that there is a lower particle density
towards the left of the probe inside the light cone, consistent with the probe
moving to the right. For higher probe velocity, $v=1$ (see App. B), the local
particle density behaves similarly, except that the oscillations in particle
density in the sample are less strong, as the probe moves more quickly over
the sample.

Examining the LDOS, first at site $i_{s}=13$, Fig. 10(b), we can again see the
peaks at $\omega\approx 0$ and $\omega\approx 4$ for $i_{s}=13$ and a gap at
$\omega\approx 2$, similar to the $v=0$ case, Fig. 6(b). The movement of the
probe causes the peak at $\omega\approx 4$ to shrink slightly for later times
$t$, a feature also found at higher probe velocity $v=1$ (see App. B). In the
LDOS at site $i_{s}=26$, Fig. 10(c), one can see that the Mott gap in the LDOS
again disappears due to coupling to the empty nanoprobe. The LDOS features
peaks at $\omega\approx-2.5$ and $\omega\approx 2$, and only changes slightly
with time. At the latest time that we have reached, $t=5$, the LDOS does show
small changes, building up spectral weight between the two peaks. This is most
likely due to the nanoprobe becoming decoupled from site $i_{s}=26$ at
$t\approx 6.4$, which is within the time interval over which the Fourier
transform is carried out. This effect is not yet noticeable at $t=4$, probably
because the Hann window that we apply weights times closer to the lower bound
of the integral more strongly. At the initial time, $t=0$, the LDOS at probe
site $i_{p}=3$, Fig. 10(d), exhibits peaks at $\omega\approx-4$ and
$\omega\approx 2$. For $t\geq 1$, both peaks diminish slightly, and the LDOS
is shifted a small amount towards larger $\omega$ values. The LDOS then
appears to settle in to this slightly altered shape for all times $t\geq 1$.
Interestingly, this form closely resembles that of the $i_{s}=26$ LDOS, cf.
Fig. 10(c), which seems reasonable because the two sites are directly
connected via a hopping term.

Examining the behavior of $E_{\rm occ}(t)$ [insets of Figs. 10(b), (c), and
(d)], it appears to be heavily damped in time, similarly as in the stationary
($v=0$) half-filled system, Figs. 6(b), (c), and (d). For $i_{s}=26$ and
$i_{p}=3$, $E_{\rm occ}(t)$ seems to settle to approximately the same value,
which is slightly larger than zero.

Figure 10:  MPS results for (a) the local particle density and the LDOS for
(b) $i_{s}=13$, (c) $i_{s}=26$, and (d) $i_{p}=3$, plotted as in Fig. 2, for
$U=4$, half filling, i.e., $N=50$, and $v=0.55$.

##  IV Discussion and Summary

In this work, we have investigated the coupling of an empty nanoprobe to a
one-dimensional system of correlated electrons for two different fillings,
half filling and quarter filling, treating the case of a stationary nanoprobe
as well as that of a moving one. We have taken both the sample and the
nanoprobe to have intermediate local Coulomb interaction strength, $U=4$, and
have contrasted the behavior with that of the noninteracting, $U=0$, case. In
all scenarios that we have studied, the coupling between the sample and the
nanoprobe, consisting of nearest-neighbor hopping terms between proximate
lattice sites of the sample and corresponding sites of the nanoprobe, is
suddenly turned on. Using an MPS formulation of the DMRG and treating the time
dependence using an TDVP scheme, we have studied the time-dependent behavior
of the local particle density $\langle n_{i}\rangle(t)$ and of the LDOS
$D_{\sigma}(t,\omega,x)$, as defined by Eq. (3).

As a reference system, we take the noninteracting case with a stationary
nanoprobe. Turning on the hopping between the sample and the empty nanoprobe
at $t=0$ induces long-lived oscillations in the local particle density
$\langle n_{i}\rangle(t)$ in the sites in the probe and in the sample in the
vicinity of the nanoprobe. As can be shown rigorously, the LDOS for this
system is time-independent. However, the occupation energy $E_{\rm occ}(t)$,
defined by Eq. (8), which is a measure of the local chemical potential,
oscillates in time, with opposing phases of oscillation on the site on the
nanoprobe and the proximate site on the sample due to the strong tunneling
between them, reflecting the oscillations in the local particle density.

For systems with finite interaction strength, $U=4$, oscillations in the local
particle density are still present, but become more strongly damped,
especially at half filling. At both quarter and half filling, the LDOS is no
longer time-independent, but does evolve towards a stationary form at longer
times; this holds for both stationary and moving nanoprobes. The oscillations
in $E_{\rm occ}(t)$ are, in general, still present, but are more strongly
damped than for $U=0$, indicating that the system seems to already attain
local equilibrium on the short time scales that we have been able to treat
here. This damping is somwhat stronger for the half-filled case than for the
quarter-filled one, reflecting the reduced charge mobility in the initially
Mott insulating phase, consistent with the stronger damping in the local
particle density.

For the initially half-filled sample, the gap in the LDOS in the sample site
situated at the center of the nanoprobe closes immediately upon suddenly
coupling the sample and probe, indicating the local breakdown of the Mott
insulator. However, for sample sites further away from the nanoprobe, some
time is required before the perturbation induced by coupling the two
subsystems reaches the observation position. In this way, one observes a
“melting” of the Mott insulator that propagates through the system with time.
For the quarter-filled sample, the LDOS is clearly metallic far from the
nanoprobe, where it remains stationary, in the sample under the nanoprobe,
where there is moderate change with time, and in the nanoprobe, where there
are only small changes with time once the connection between sample and probe
is turned on.

For a moving nanoprobe, the local environment in the sample, in particular,
the local particle density, changes as the nanoprobe moves over it. Note,
however, that changes in the sample propagate outwards with a velocity given
by the light cone, which is significantly larger than the speed of movement of
the nanoprobe and can be clearly seen in the time and space dependence of the
local particle density. The primary effect of the movement of the nanoprobe is
to reduce backflow effects into the sample, as the back-tunneling from the
probe to the sample is spread over different sample sites as the probe moves.
The behavior of the LDOS is similar to that of the case of a resting
nanoprobe. Interestingly, on the sites of the moving probe, the LDOS behaves
essentially as if the probe were at rest. On the sites in the sample, the
behavior of the LDOS is similar to that in the stationary case, up to times at
which the site initially under the nanoprobe is no longer under the influence
of the nanoprobe because it moves far enough away.

These features of the cases with a moving nanoprobe remain qualitatively the
same when the speed of the nanoprobe is increased. The exceptions are that for
the slower nanoprobe velocity, $v=0.55$, there are interference effects in the
local particle density due to commensuration of spatial and temporal
oscillations of the local particle density that are not present at the higher
nanoprobe speed, $v=1$. In addition, the decoupling of the nanoprobe from
sample sites initially under the nanoprobe occurs on shorter time scales for
the fast nanoprobe velocity, so that quasi-equilibration of the LDOS occurs
sooner on such sites.

We remark that the nonequilibrium LDOS that we have computed [Eq. (3)] is not
restricted to be strictly positive semi-definite. Negative values are
incompatible with the interpretion as spectral weights that one is used to for
a equilibrium LDOS. However, we, in fact, find that negative weights only
occur in very few cases (most markedly, in the $U=0$ case with a moving
nanoprobe), and, for these cases, only at short times. In these cases, we also
find that there is a discrepancy between the local particle density directly
calculated as $\langle n_{i,\sigma}\rangle(t)$ and that obtained from $\int
d\omega\,B^{<}_{\sigma}(\omega)$. We characterize this anomalous short-time
behavior as being strongly nonequilibrium. Thus, the general picture is that
the discrepancies in the local particle density and the appearance of negative
values of the LDOS disappear at later times (especially in the interacting
system), which is consistent with reaching an equilibrium state, so that a
description in terms of standard linear response theory is applicable. It is
interesting to see that this happens on the very short time scales treated by
us; for a typical material, the hopping strength $t_{h}\sim 0.5-1\,\text{eV}$.
Since the units of our time scale are $\hbar/t_{h}$, this implies that our
calculations typically reach $\sim 5-10\,\text{fs}$, which lies in the
ultrafast regime of even highly time-resolved experimental techniques such as
pump-probe experiments. Developing such time-resolved local spectroscopy with
STM is an ongoing challenge [62]. However, one can envisage that similar
scenarios to the ones proposed here could be studied using single-site
microscopes in experiments on optical lattices, which realize the Hubbard
model [63, 64, 65, 66, 67]. In either case, it will be interesting to study
how the states developing on the ultrashort time scales treated by us here
affect the behavior at the later times accessible to these experiments.

It would therefore be interesting to also study the time evolution of the LDOS
and other dynamical quantities for other strongly correlated systems. For
example, investigating a similar scenario to the one treated here in
correlated charge density wave (CDW) insulators could allow one to study if
and how the melting of a CDW state propagates through the system after a local
perturbation. Since topological phases are either protected by local
symmetries (e.g., symmetry-protected topological phases in 1D), or by long-
range entanglement (in particular, in 2D systems), it would be interesting to
study the interplay of the local perturbation with these protection mechanisms
by studying the time evolution of the LDOS.

###### Acknowledgements.

We thank Martin Wenderoth for helpful discussions. SRM acknowledges financial
support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - 217133147/SFB 1073, project B03 and by Philipps-Universität
Marburg through a visiting professorship from 2020-21.

##  Appendix A Numerical Accuracy

The truncation error of the time evolution has been found to be proportional
to the square root of the discarded weight: $\epsilon\approx\sqrt{\delta}$
[58]. In Fig. 11, we find that, for the half-filled system, the discarded
weight for later times reaches just under $\delta\approx 10^{-4}$, resulting
in an error of $\approx 10^{-2}$. This level of accuracy should be sufficient
to discern features of the LDOS that dominate its qualitative behavior.
Furthermore, as discussed in Sec. II.3, this level of error is also
overshadowed by the limited resolution of our calculation of the LDOS caused
by the fact that we are only able to carry out the integrals in Eqs. (4) and
(5) to $\tau_{\text{max}}=5$.

In Fig. 12(a), it can be seen that the error in the local particle density
$\langle n_{i}\rangle$ becomes appreciable at times after $t\approx 4$ in and
in the vicinity of the nanoprobe and continues to grow and spread out as time
progresses. The error is thus concentrated in regions in which the most change
in time takes place. The LDOS far away from the nanoprobe, $i_{s}=13$, Fig.
12(b), shows no significant error, whereas that under and in the nanoprobe,
Figs. 12(c)-(d) shows absolute errors that grow with time, becoming
significant at later times. Nevertheless, we consider them to be small enough
to ensure that the qualitative behavior of the LDOS, which is limited anyway
due to taking the maximum time in our Fourier integration, Eqs. (4) and (5),
to be $\tau_{\text{max}}=5$. The maximum absolute difference in the LDOS at
sites $i_{s}=26$ and $i_{p}=3$ for later times is about $0.014$. The largest
peaks that are resolved in the LDOS (for $\tau_{\text{max}}=5$) are
approximately $0.15$ in amplitude. This is an error of about 10% for the
largest peaks and only occurs at later times. Despite this appreciable error,
the accuracy should be sufficient to resolve the larger peaks in the LDOS.
When evaluating the presence and size of the smallest peaks, however, which
reach from approximately $0.015$ to $0.05$, one cannot necessarily make a
definite determination. Nevertheless, we emphasize that the errors at smaller
times and for lattice site $i_{s}=13$ are almost negligible and yield more
than sufficiently accurate results.

Figure 11: Maximum discarded weight $\delta$ during a TDVP time step as a
function of the evolution time $t$ for $U=4$, with $L=50$ sample sites and
$l=5$ nanoprobe sites and initial filling and probe speed as indicated in the
legend.

Figure 12:  Results of the TDVP time evolution, with $L=50$ sample sites and
$l=5$ nanoprobe sites, compared to exact diagonalization for $U=0$ and probe
velocity $v=0$ at half filling, i.e., $N=50$. Here (a) depicts the absolute
difference in the particle density $\langle n_{i}\rangle$, and (b)-(d) depict
the absolute difference in the LDOS at the indicated sites. Note that the
maximum integration time $\tau_{\text{max}}=5$ for both the exact
diagonalization and TDVP calculations in order to make the results comparable.

Figure 13:  Results of the TDVP time evolution compared to exact
diagonalization as in Fig. 12, except for probe velocity $v=1$.

##  Appendix B Results for $v=1$

In this appendix, we present results for a probe velocity of $v=1$ for the
same sets of remaining parameters as the results for $v=0.55$ presented in
Sec. III.2. Note that most of the features of the $v=1$ case are very similar
to those of the $v=0.55$ case; many of the important differences and
similarities have already been highlighted in Sec. III.2. In the following, we
will present $v=1$ results for both quarter and half filling for both $U=0$
and $U=4$ and briefly describe salient aspects.

We first treat the case of zero interaction strength, $U=0$, and quarter
filling. Comparing the local particle density, Fig. 14(a), with that for the
$v=0.55$ case, Fig. 7(a), we observe the same qualitative behavior, albeit
with more pronounced wave fronts to the left of the nanoprobe for $v=0.55$, as
discussed in Sec. III.2. Comparing the LDOS for sample site $i_{s}=26$, Fig.
14(b), to that of the $v=0.55$ case, Fig. 7(c), we can see that the right peak
in the LDOS is preserved throughout the whole time evolution. However, the
left peak in Fig. 7(b) is lost after the coupling of the probe to the system
is turned on at $t=0$. Note, however, that this peak is restored at $t\approx
3.5$ for $v=1$, as can be seen in Fig. 14(b). This is due to the fact that, at
this point in time, the probe has traveled far enough across the system that
site $i_{s}=26$ is decoupled from the probe.

We now turn to the case of intermediate interaction strength, $U=4$, for the
quarter-filled system. Comparing the local particle density, Fig. 15(a), with
that for the $v=0.55$ case, Fig. 8(a), we again see mostly the same behavior
except that there are more pronounced wave fronts to the left of the
nanoprobe. For the LDOS, comparing Figs. 15 (b)-(d), with Figs. 8 (b)-(d), we
can see that the LDOS at sites $i_{s}=13$ and $i_{p}=3$ are almost unaffected
by the change in speed of the probe. In Fig. 15 (c), we again observe that the
left peak in the LDOS is restored for times $t\geq 4$ for the probe speed of
$v=1$ due to decoupling of the nanoprobe from site $i_{s}=26$.

For the $U=0$ case in the half-filled system, comparing Fig. 16 with Fig. 9,
the behavior is qualitatively the same except for the more pronounced wave
fronts for $v=0.55$, also seen at quarter filling. One can see that there are
more particles collected in the nanoprobe over time for $v=1$; this is due to
the fact that the higher probe speed makes more particles available to tunnel
into the probe.

Finally, for interaction strength $U=4$ and half filling, comparing Fig. 17
with the $v=0.55$ case, Fig. 10, the local particle density, Fig. 17(a),
behaves similarly for the two probe speeds, but again, more particles are
collected in the probe over time for $v=1$. In the LDOS, the behavior for site
$i_{s}=13$, Fig. 17(b), is essentially unaffected by the probe speed, showing
a double-peak structure characteristic of a Mott insulator where there is only
a slight shift in the relative weight of the two peaks with time. For site
$i_{s}=26$, the behavior at $v=1$, Fig. 17(c), is similar to that for
$v=0.55$, Fig. 10(c), in that the two-peak structure characterizing the Mott
insulator immediately disappears at $t=0$, and two weaker peaks at small times
evolve into a structure characterized by one peak. In contrast to $v=0.55$,
however, a new two-peak structure with a stronger, broader, left peak, emerges
and remains stationary at the two longest times, $t=4.0$ and $t=5.0$. Within
the nanoprobe at $i_{p}=3$, the behavior for $v=1$, Fig. 17(d), and $v=0.55$,
Fig. 10(c), is virtually identical, showing a two-peak structure without a
discernible gap at small times washing out into a broad structure with
vestigial peaks at larger times.

Figure 14: Exact results for the nanoprobe model, with $L=50$ sample sites and
$l=5$ nanoprobe sites, for the system initially at quarter filling, i.e.,
$N=26$, with $U=0$ and nanoprobe velocity $v=1$. Here (a) depicts the
expectation value of the local particle density, $\langle{n}_{i}\rangle$ as a
color density plot as a function of lattice site $i$ and time $t$, and (b)
plots the LDOS at site $i_{s}=26$ as a function of the frequency $\omega$. The
occupation energy $E_{\rm occ}(t)$ at site $i_{s}=26$ is shown as a solid
line. The dotted line indicates the decoupling of the probe from lattice site
$i_{s}=26$ at $t=3.5$.

Figure 15:  MPS results for (a) the local particle density and the LDOS for
(b) $i_{s}=13$, (c) $i_{s}=26$, and (d) $i_{p}=3$, plotted as in Fig. 2, for
$U=4$, quarter filling, i.e., $N=26$, and $v=1$.  Figure 16:  Exact results
for the local particle density, plotted as in Fig. 2, for $U=0$, half filling,
i.e., $N=50$, and $v=1$. The occupation energy $E_{\rm occ}(t)$ is shown as an
inset for sites $i_{s}=26$ and $i_{p}=3$. The LDOS in this case is not
explicitly shown, as it is the same as in Fig. 14.

Figure 17:  MPS results for (a) the local particle density and the LDOS for
(b) $i_{s}=13$, (c) $i_{s}=26$, and (d) $i_{p}=3$, plotted as in Fig. 2, for
$U=4$, half filling, i.e., $N=50$, and $v=1$.

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