# Local-in-time Conservative Binary Dynamics at Fourth Post-Minkowskian Order

Christoph Dlapa  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germany  Gregor Kälin  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Zhengwen Liu  Niels Bohr International
Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
School of Physics and Shing-Tung Yau Center, Southeast University, Nanjing
210018, China  Rafael A. Porto  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany

###### Abstract

Leveraging scattering information to describe binary systems in generic orbits
requires identifying local- and nonlocal-in-time tail effects. We report here
the derivation of the universal (non-spinning) local-in-time conservative
dynamics at fourth Post-Minkowskian order, i.e. ${\cal O}(G^{4})$. This is
achieved by computing the nonlocal-in-time contribution to the deflection
angle, and removing it from the full conservative value in [1, 2]. Unlike the
total result, the integration problem involves two scales—velocity and mass
ratio—and features multiple polylogarithms, complete elliptic and iterated
elliptic integrals, notably in the mass ratio. We reconstruct the local radial
action, center-of-mass momentum and Hamiltonian, as well as the exact
logarithmic-dependent part(s), all valid for generic orbits. We incorporate
the remaining nonlocal terms for elliptic-like motion to sixth Post-Newtonian
order. The combined Hamiltonian is in perfect agreement in the overlap with
the Post-Newtonian state of the art. The results presented here provide the
most accurate description of gravitationally-bound binaries harnessing
scattering data to date, readily applicable to waveform modelling.

††preprint: DESY 24-029

Introduction. Motivated by the impending era of high-precision gravitational-
wave (GW) astronomy with observatories such as LISA [3], the Einstein
Telescope [4] and the Cosmic Explorer [5], and the incredibly rich amount of
information expected from compact binary sources [6, 7, 8, 9, 10, 11], the
(long dormant [12]) Post-Minkowskian (PM) expansion in general
relativity—entailing a perturbative series in $G$ (Newton’s constant) but to
all orders in the relative velocity—has experienced a resurgence in recent
years, e.g. [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 1, 2, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].
This is, in part, thanks to the repurposing of modern integration techniques
from collider physics (see [42, 44] and references therein), which have led to
a plethora of new results. Notably, using worldline effective field theory
(EFT) methodologies [47, 48, 49, 50, 51], the rapidly evolving state of the
art includes the total relativistic impulse (yielding the scattering angle and
emitted GW flux) of non-spinning [1, 2] and spinning [39, 40] bodies to ${\cal
O}(G^{4})$, akin of a ‘three-loop’ calculation in particle physics, as well as
partial results in the conservative sector at 5PM [46].

The derivations in [1, 2], together with a (Firsov-type [52, 53]) resummation
scheme [21, 22], have led to an unprecedented agreement between analytic
results and numerical simulations [54, 55, 56], paving the way to more
accurate waveform models for hyperbolic encounters. However, due to nonlocal-
in-time effects [57, 58], unbound results cannot be used to describe generic
elliptic-like motion (away from the large-eccentricity limit [1]). As shown in
[23], the binding energy for quasi-circular orbits obtained from scattering
results (via the “boundary-to-bound” (B2B) analytically continuation [21, 22])
does not reproduce—other than logarithms—the known Post-Newtonian (PN) values
[57, 58, 59, 60, 61] (see also [62]). Hence, to fully harness the power of
scattering calculations, a separation between local- and nonlocal-in-time
effects was thus imperative. In this letter we report the derivation of the
universal (non-spinning) local-in-time conservative dynamics of binary systems
at ${\cal O}(G^{4})$. This is obtained via a direct computation of the
nonlocal-in-time contribution to the scattering angle. Following [23, 63], the
calculation entails an integral over the energy spectrum times the logarithm
of the center-of-mass GW frequency. To solve the integration problem, we
implement the methodology of differential equations, already used in [1, 2].
However, unlike the total impulse, which obeys a simple (power-law) mass
scaling [21, 16], isolating (non)local effects in a gauge-invariant fashion
entails dealing with two relevant scales: the velocity and mass ratio. Despite
the complexity of the two-scale problem, we find that it can be factorized
into solving two second-order Picard-Fuchs (PF) equations. The nonlocal part
of the angle features multiple polylogarithms (MPLs), complete elliptic
integrals, and integrations thereof. We find agreement in the overlap with the
6PN values in [63].

We derive the local-in-time contribution to the conservative scattering angle
by removing the (unbound) nonlocal terms from the total result in [1, 2]. The
local radial action follows directly via the B2B map [21, 22, 23]. Using the
relations in [21, 22], we reconstruct the universal local-in-time center-of-
mass momentum and Hamiltonian in isotropic gauge, together with the complete
logarithmic dependence, all applicable to generic motion. We also provide—for
all practical purposes—results expanded to 30 orders in the (symmetric) mass
ratio and all orders in the velocity (with an error beyond 30PN). To
incorporate the remaining (non-logarithmic) nonlocal part of the bound
dynamics, we adapt to our isotropic gauge the values obtained in [63] to 6PN
order. The combined Hamiltonian at ${\cal O}(G^{4})$ perfectly matches in the
overlap with the state of the art in PN theory [64, 63, 62]. The results
presented here can be directly inputed onto waveforms models for
gravitationally-bound eccentric orbits, potentially increasing their accuracy
by incorporating an infinite tower of (local-in-time) velocity corrections.

(Non)local-in-time tail effects. The scattering of the emitted radiation off
of the binary’s gravitational potential, or “tail effect”, enters in the 4PM
conservative dynamics both through local- as well as nonlocal-in-time
interactions [57, 58, 59, 60, 61]. Because of this, although an effectively
local description is possible to any order [1, 2], the coefficients of the
radial action (or Hamiltonian) depend on the type of motion, and therefore are
not related via analytic continuation for generic orbits. Our strategy is to
identify the local- and nonlocal-in-time parts of ${\cal S}_{r}$, the total
radial action. Due to the structure of tail effects [57, 58, 23], the
nonlocal-in-time tail terms can be shown to take the gauge-invariant form

${\cal S}^{(\rm
nloc)}_{r\,}=-\frac{GE}{2\pi}\int_{\omega}\frac{dE}{d\omega}\log\left(\frac{4\omega^{2}}{\mu^{2}}e^{2\gamma_{E}}\right)\,,$
(1)

where $\int_{\omega}\equiv\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}$, $E$
and $\frac{dE}{d\omega}$ are the total (incoming) energy and emitted GW
spectrum in the center-of-mass frame. The ‘renormalization scale’ $\mu$, which
cancels against a similar term in the local-in-time part [34, 1, 58], can be
arbitrarily chosen. The factor of $4e^{2\gamma_{E}}$ ( with $\gamma_{E}$
Euler’s constant) follows the PN conventions [57, 58]. An explicit derivation
of (1) in the context of the PM expansion can be found in [23], see also [63]
for a discussion in the PN regime. For unbound motion, the scattering angle is
given by $\frac{\chi}{2\pi}=-\partial_{j}{\cal I}_{r}$ , with ${\cal
I}_{r}\equiv\frac{{\cal S}_{r}}{GM^{2}\nu}$ and $j\equiv\frac{J}{GM^{2}\nu}$
the (reduced) radial action and angular momentum, and $M=m_{1}+m_{2}$,
$q=m_{2}/m_{1}$ ($m_{2}\leq m_{1}$), $\nu=m_{1}m_{2}/M^{2}$ the total mass,
mass ratio, and symmetric mass ratio, respectively. We split the PM
coefficients of the deflection angle in impact parameter space as,

$\frac{\chi}{2}=\sum_{n=1}\left(\chi_{b}^{(n)}+\chi_{b}^{(n)\log}\log\frac{\mu
b}{\Gamma}\right)\left(\frac{GM}{b}\right)^{n},$ (2)

where $\gamma\equiv u_{1}\cdot u_{2}$ (using the mostly negative metric
convention), $u_{1,2}$ the incoming velocities, and $\Gamma\equiv
E/M=\sqrt{1+2\nu(\gamma-1)}$. (The reader should keep in mind that logarithms
of the velocity may still appear in both coefficients.) In the remaining of
this letter we choose $\mu\equiv 1/GM$ for the renormalization scale.

Integrand construction. Because of the overall factor of $G$ in (1), it is
sufficient to construct the integrand to ${\cal O}(G^{3})$. Using the results
in [33], and multiplying by a factor of $(2e^{\gamma_{E}}k\cdot u_{\rm
com})^{2\tilde{\epsilon}}$, with $k=(\omega,{\bm{k}})$ the (on-shell) radiated
momentum, and $u_{\rm com}\equiv\frac{(m_{1}u_{1}+m_{2}u_{2})}{E}$ the center-
of-mass velocity, we readily derive a covariant version which, after
projecting on the center-of-mass frame, matches at ${\cal
O}(\tilde{\epsilon})$ the expression in (1). We find it convenient to
distinguish the $\tilde{\epsilon}$-expansion from the standard $D=4-2\epsilon$
that we use for dimensional regularization. To the families of (two-loop)
scalar integrals introduced in [33] for computing the total impulse, we add,

$\displaystyle I^{\pm\pm;T_{5}\dots T_{9}}_{\nu_{1}\dots\nu_{10}}$
$\displaystyle=\int_{\ell_{1},\ell_{2}}\frac{{\delta}^{(\nu_{1}-1)}(\ell_{1}\cdot
u_{1}){\delta}^{(\nu_{2}-1)}(\ell_{2}\cdot u_{2})}{(\pm\ell_{1}\cdot
u_{2})^{\nu_{3}}(\pm\ell_{2}\cdot u_{1})^{\nu_{4}}}$ (3)
$\displaystyle\qquad\qquad\times(k\cdot
u_{\textrm{com}})^{2\tilde{\epsilon}-\nu_{10}}\prod_{j=5}^{9}\frac{1}{D_{j,T_{j}}^{\nu_{j}}}\,,$

with a noninteger powered propagator, where we use the same notation as in
[33] (see also [42]). The radiative momentum is rewritten as
$k^{\alpha}=\ell^{\alpha}_{1}+\ell^{\alpha}_{2}-q^{\alpha}$, with
$q^{\alpha}\equiv(q^{0},{\bm{q}})$ the momentum transfer, obeying $q\cdot
u_{a}=0$ (not to be confused with the mass ratio). The choice of
$i0^{+}$-prescription for the square propagators, either retarded or advanced,
is encoded in $T_{j}\in\\{\textrm{ret},\textrm{adv}\\}$:

$\displaystyle D_{5,\textrm{ret/adv}}$ $\displaystyle=$
$\displaystyle(\ell_{1}^{0}\pm
i0)^{2}-{\bm{\ell}}_{1}^{2}\,,D_{6,\textrm{ret/adv}}=(\ell_{2}^{0}\pm
i0)^{2}-{\bm{\ell}}_{2}^{2}\,,$ $\displaystyle D_{7,\textrm{ret/adv}}$
$\displaystyle=$ $\displaystyle(\ell_{1}^{0}+\ell_{2}^{0}+q^{0}\pm
i0)^{2}-({\bm{\ell}}_{1}+{\bm{\ell}}_{2}-{\bm{q}})^{2}\,,$ $\displaystyle
D_{8,\textrm{ret/adv}}$ $\displaystyle=$ $\displaystyle(\ell_{1}^{0}-q^{0}\pm
i0)^{2}-({\bm{\ell}}_{1}-{\bm{q}})^{2}\,,$ $\displaystyle
D_{9,\textrm{ret/adv}}$ $\displaystyle=$ $\displaystyle(\ell_{2}^{0}-q^{0}\pm
i0)^{2}-({\bm{\ell}}_{2}-{\bm{q}})^{2}\,.$ (4)

Using integration-by-parts (IBP) reduction techniques implemented in the
packages LiteRed [65] and FiniteFlow [66], we find 17 master integrals
contributing to the radiation region (where the $k$ momentum goes on-shell),
which isolates the contribution to the energy loss from the total impulse. It
is possible to select integrals such that we can take $\nu_{1}=\nu_{2}=1$,
$\nu_{10}=0$. The final set, specified by $\nu_{3\cdots 9}$, becomes

$\displaystyle(-1,0,0,0,1,1,1)\,,$ $\displaystyle(-1,0,0,0,1,1,2)\,,$
$\displaystyle(-1,0,0,0,1,2,1)\,,$ (5) $\displaystyle(-1,0,0,0,2,1,1)\,,$
$\displaystyle(0,-1,0,0,1,1,1)\,,$ $\displaystyle(0,-1,0,0,1,1,2)\,,$
$\displaystyle(-1,0,0,1,1,0,1)\,,$ $\displaystyle(0,-1,0,1,1,0,1)\,,$
$\displaystyle(-1,0,0,1,1,1,1)\,,$ $\displaystyle(-1,0,1,0,1,1,0)\,,$
$\displaystyle(0,-1,1,0,1,1,0)\,,$ $\displaystyle(-1,0,1,0,1,1,1)\,,$
$\displaystyle(-1,0,1,1,1,1,1)\,,$ $\displaystyle(-1,0,1,1,1,1,2)\,,$
$\displaystyle(-1,0,1,1,2,1,1)\,,$ $\displaystyle(0,1,1,1,1,0,0)\,,$
$\displaystyle(1,0,0,0,1,1,1)\,,$

modulo different choices of $i0^{+}$-prescriptions ($T_{5\cdots 9}$) and signs
in front of linear propagators.

Integration. To solve for the master integrals, we derive differential
equations in $x$ and the mass ratio, $q$, where $x$ is given by
$\gamma=\frac{1}{2}\left(x+\frac{1}{x}\right)$. We then adopt the strategy of
an $\epsilon$\- (and $\tilde{\epsilon}$-)regular basis [67], such that we can
set $\epsilon=0$, and consider the expansion of the integrand, differential
equations, and boundary constants, only to $\mathcal{O}(\tilde{\epsilon})$.
The latter are determined via a small-$q$ expansion, together with the
techniques described in [42] (adapted to the new factors of
$\tilde{\epsilon}$). From this setting, it is then straightforward to find a
solution of the differential equations through iterated integration.

For the parts containing MPLs, and similarly to the $x$ variable, it is useful
to rationalize the square root of the energy $\frac{E}{m_{1}}=\sqrt{1+2\gamma
q+q^{2}}=\sqrt{(q+x)\left(q+\frac{1}{x}\right)}$, by introducing a new
variable, $y$, defined through
${q^{-1}=-\gamma-\frac{v_{\infty}}{2}\left(y+\frac{1}{y}\right)}$, with
$v_{\infty}\equiv\sqrt{\gamma^{2}-1}$. Hence, we find the traditional harmonic
polylogarithms with letters $\\{x,1+x,1-x\\}$ [26, 33], as well as MPLs which
depend on the velocity and mass ratio via the new letters:
${\\{y,1+y,1-y,y-\frac{1+x}{1-x},y-\frac{1-x}{1+x},1+2\frac{1-x}{1+x}y+y^{2}\\}}$.
In addition to MPLs, the solution to the differential equations depend on
another set of functions, through an _a priori_ irreducible fourth-order PF
equation, already at ${\cal O}(\tilde{\epsilon}^{0})$. However, a Baikov
representation [68, 69] of the maximal cut suggests a simpler Calabi-Yau two-
fold as the relevant geometry. Indeed, in terms of the variables
$(qx,\frac{q}{x})$, the differential equations can be solved, in the first and
subsequently the second variable, via two equivalent second-order PF equations
(per variable). The solution can then be written in terms of products of
$\mathrm{K}$’s (such as the $f_{1}$ in (6) below) as well as the leading three
derivatives w.r.t. the mass ratio. As in previous PM computations, e.g. [34,
1, 2], $\mathrm{K}(z)=\int_{0}^{1}\frac{dt}{\sqrt{(1-t^{2})(1-zt^{2})}}$, is
the complete elliptic integral of the first kind.

After the leading order solution is known, it is then straightforward to
obtain the $\mathcal{O}(\tilde{\epsilon})$ part. We find that it can be
written in terms of (at most) two-fold iterated integrals, with elliptic
kernels depending on the mass ratio, $q$, as the integration variable. The
full set is given by:

$\displaystyle f_{1}$
$\displaystyle=\frac{\mathrm{K}(-qx)\mathrm{K}\left(1+\frac{q}{x}\right)-\mathrm{K}\left(-\frac{q}{x}\right)\mathrm{K}(1+qx)}{\pi},\qquad$
$\displaystyle f_{2}$ $\displaystyle=\frac{f_{1}}{q}\,,\quad\quad
f_{3}=\partial_{x}f_{1}\,,\quad\quad f_{4}=\frac{\partial_{x}f_{1}}{q}\,,$ (6)
$\displaystyle f_{5}$
$\displaystyle=\left[\frac{1-x^{2}}{x}(1+q\,\partial_{q})-\frac{1-q^{2}}{q}x\,\partial_{x}\right]\frac{f_{1}}{\sqrt{(q+x)\left(q+\frac{1}{x}\right)}}.$

Remarkably, while individually this is not the case, the combination of
complete elliptic integrals in $f_{1}$ has a simple power-series expansion in
the PN limit ($x\to 1$). Furthermore, the $f_{i}$’s are real, and have (at
most) simple poles in $q$. Let us point out, however, that a simplified
version of the iterated integrals may still be possible. In particular, upon
assigning to $\mathrm{K}(z)$ a transcendental weight one, we notice that the
iterated integrals would have up to weight four, in contrast to the MPL part
with maximum weight two. Hence, we expect that either the naïve assignment is
incorrect or an even simpler form exists. We leave this open for future work.

Scattering angle. After solving for the master integrals and plugging them
back into the integrand, we arrive at the radial action, and from there to the
nonlocal-in-time contribution to the deflection angle, $\chi_{b(\rm
nloc)}^{(4)}$ and $\chi_{b(\rm nloc)}^{(4)\log}$, at 4PM order. As anticipated
in [34], the logarithmic part takes on a simple closed form,

$\displaystyle\frac{1}{\pi\Gamma}\chi_{b(\rm
nloc)}^{(4)\log}=-2\nu\chi_{2\epsilon}(\gamma)=$ (7)
$\displaystyle\frac{-2\nu}{{\left(\gamma^{2}-1\right)^{2}}}\left(h_{5}+h_{9}\log\left(\frac{\gamma+1}{2}\right)+\frac{h_{10}\operatorname{arccosh}(\gamma)}{\sqrt{\gamma^{2}-1}}\right)\,,$

with $\chi_{2\epsilon}$ introduced in [34], and the $h_{5,9,10}$ are
polynomials depending only on $\gamma$, which also enter the non-logarithmic
part (see below). The latter, on the other hand, also involves a set of
iterated integrals in the mass ratio. Despite its complexity, it is
straightforward to construct a “self-force” (SF) expanded version, for which
we find the generic form, valid to any $n$SF order,

$\displaystyle\frac{1}{\pi\Gamma}\chi_{b(\rm nloc)}^{(4)(n\rm SF)}$
$\displaystyle=\frac{\nu}{\left(\gamma^{2}-1\right)^{2}}\left\\{h_{1}+\frac{\pi^{2}h_{2}}{\sqrt{\gamma^{2}-1}}+h_{3}\log\left(\frac{\gamma+1}{2}\right)+\frac{h_{4}\operatorname{arccosh}(\gamma)}{\sqrt{\gamma^{2}-1}}+h_{5}\log\left(\frac{\gamma-1}{8}\right)+h_{6}\log^{2}\left(\frac{\gamma+1}{2}\right)\right.$
(8)
$\displaystyle\quad+h_{7}\operatorname{arccosh}(\gamma)^{2}+\frac{h_{8}\log(2)\operatorname{arccosh}(\gamma)}{\sqrt{\gamma^{2}-1}}+h_{9}\log\left(\frac{\gamma-1}{8}\right)\log\left(\frac{\gamma+1}{2}\right)$
$\displaystyle\quad\left.+\frac{h_{10}\log\left(\frac{\gamma^{2}-1}{16}\right)\operatorname{arccosh}(\gamma)}{\sqrt{\gamma^{2}-1}}+h_{11}\text{Li}_{2}\left(\frac{\gamma-1}{\gamma+1}\right)+\frac{h_{12}\left[\operatorname{arccosh}(\gamma)^{2}+4\text{Li}_{2}\left(\sqrt{\gamma^{2}-1}-\gamma\right)\right]}{\sqrt{\gamma^{2}-1}}\right\\}\,.$

Due to the structure of the full solution, except for the $h_{1}$, $h_{3}$ and
$h_{4}$ carrying information from the (iterated) elliptic sector ($h_{1,4}$)
and the new letters in the MPLs depending on the mass ratio ($h_{3}$), the
remaining $h_{i}$’s are SF-exact. We find the $n$SF coefficients may be split
as

$h_{i}=h_{i}^{(0)}(\gamma)+\sqrt{1-4\nu}\,h_{i}^{(1)}(\gamma)+\Delta
h_{i}(\gamma,\nu)\,,$ (9)

where the $h_{i}^{(0)}(\gamma)$, $h_{i}^{(1)}(\gamma)$, are polynomials in
$\gamma$ only. The $\Delta h_{i}(\gamma,\nu)$ vanish except when $i=1,3,4$,
for which they become polynomials both in $\gamma$ and $\nu$, up to ${\cal
O}(\nu^{n})$. We provide in an ancillary file their values up to $n=30$. The
30SF result (with an error beyond 30PN) is in perfect agreement in the overlap
with the 6PN values in [63].Let us emphasize that the definition of nonlocal-
in-time in [64, 63] includes not only the expression in (1) ($W_{1}$ in [63]),
but also an extra contribution ($W_{2}$ in [63]). Due to the local-in-time
(and gauge-dependent) nature of $W_{2}$, we do not add it to (1). Therefore,
(8) agrees (in the overlapping realm of validity) with the scattering angle
obtained from the $W_{1}$-only terms in Eq. (3.14) of [63].After subtracting
from the total conservative angle in [1, 2], we arrive at the local-in-time
counterpart,111Although amenable to a conservative-like description of the
relative dynamics, we keep the other (time-symmetric) radiation-reaction
corrections, i.e. “2rad” in [2], in the dissipative part.

$\chi_{b(\rm loc)}^{(4)}=\chi_{b\rm(tot)}^{(4)\rm cons}-\chi_{b(\rm
nloc)}^{(4)}\,,\quad\chi_{b(\rm loc)}^{(4)\log}=-\chi_{b(\rm
nloc)}^{(4)\log}\,,$ (10)

where we used the fact that the $\log\frac{\mu b}{\Gamma}$ cancels out in the
total value. The result in (10) can now be used to describe generic bound
orbits, as we discuss next.

Local-in-time conservative dynamics. Following the B2B dictionary, the local-
in-time (reduced) bound radial action takes the form [21]

$i^{4\rm PM}_{r(\rm loc)}=\frac{2v_{\infty}^{4}}{3(\Gamma
j)^{3}}\left(\frac{\chi_{b(\rm loc)}^{(4)}}{\pi\Gamma}+\frac{\chi_{b(\rm
loc)}^{(4)\log}}{2\pi\Gamma}\log\frac{j^{2}}{v_{\infty}^{2}}\right)\,.$ (11)

Using the expressions in [21, 22], and a dimensionally rescaled distance
$\hat{r}=r/(GM)$, we can also reconstruct the center-of-mass momentum (notice
we use different conventions w.r.t. [21, 22]),

${\hat{\bm{p}}}^{2}=\frac{v_{\infty}^{2}}{\Gamma^{2}}\left(1+\sum_{n=1}\frac{1}{\hat{r}^{n}}\left(f_{n}+f_{n}^{\log}\log\hat{r}\right)\right)\,,$
(12)

with $\hat{\bm{p}}\equiv{\bm{p}}/(M\nu)$, and Hamiltonian, $\hat{H}\equiv
H/(M\nu)$,

$\hat{H}=\hat{E}+\sum_{i=1}\frac{1}{\hat{r}^{i}}\left(\hat{c}_{i}+\hat{c}_{i}^{\log}\log\hat{r}\right)\,,$
(13)

where $\hat{E}=\sum_{a}\hat{E}_{a}$,
$\hat{E}_{a}=\sqrt{\hat{\bm{p}}^{2}+\left(\frac{m_{a}}{M\nu}\right)^{2}}$.The
coefficients $\big{(}\hat{c}_{4(\rm loc)},\hat{c}^{\log}_{4\rm(loc)}\big{)}$
are displayed in ancillary files.

Universal logarithms. Nonlocal-in-time tail effects also contribute with a
$\log\hat{r}$ term in the bound dynamics. Performing a small-eccentricity
expansion of (1), and using Kepler’s law
($\log\Omega=-\frac{3}{2}\log\hat{r}+\cdots$, with $\Omega$ the 1PM orbital
frequency), we find

$\frac{\hat{c}_{4(\rm
nloc)}^{\log}}{\hat{r}^{4}}=-\frac{3}{2}\frac{\hat{c}_{4(\rm
loc)}^{\log}}{\hat{r}^{4}}=-3G\frac{\Gamma}{\nu}\left.\frac{dE}{dt}\right|_{\rm
3PM}\,,$ (14)

where
($\xi\equiv\frac{\hat{E}_{1}\hat{E}_{2}}{(\hat{E}_{1}+\hat{E}_{2})^{2}},\gamma=\nu(\hat{E}_{1}\hat{E}_{2}+\hat{\bm{p}}^{2})$)

$G\left.\frac{dE}{dt}\right|_{\rm
3PM}(\hat{r},\hat{\bm{p}}^{2})=-\frac{4\nu^{3}}{3{\hat{r}}^{4}}\frac{\gamma^{2}-1}{\Gamma^{3}\xi}\chi_{2\epsilon}(\gamma)\,,$
(15)

is the energy flux at 3PM order [34, 23]. Similarly,

$\displaystyle f_{4(\rm nloc)}^{\log}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}f_{4(\rm
loc)}^{\log}=-8\Gamma\nu\,\chi_{2\epsilon}\,,$ (16)

consistently with (7). Hence, adding both terms,

$\displaystyle\hat{H}^{{\rm ell(\log)}}_{\rm 4PM}$ $\displaystyle=$
$\displaystyle\frac{4\nu^{2}}{3\hat{r}^{4}}\frac{(\gamma^{2}-1)}{\Gamma^{2}\xi}\chi_{2\epsilon}\log\hat{r}\,,$
(17)

and likewise,

$\displaystyle\left(\hat{\bm{p}}^{2}\right)^{{\rm ell(\log)}}_{\rm 4PM}$
$\displaystyle=$ $\displaystyle-\frac{8\nu
v_{\infty}^{2}}{3\Gamma\hat{r}^{4}}\,\chi_{2\epsilon}\log\hat{r}\,,$ (18)

for the full logarithmic dependence of the bound Hamiltonian and center-of-
mass momentum at 4PM.

Towards the complete bound dynamics. Putting together the local-in-time
coefficient plus exact logarithmic part, the total bound Hamiltonian up to 4PM
order may be written as

$\displaystyle\hat{H}^{\rm ell}_{\rm 4PM}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{i=4}\frac{\hat{c}_{i\rm(loc)}}{\hat{r}^{i}}+\sum_{i=1}^{i=4}\frac{\hat{c}_{i\rm(nloc)}}{\hat{r}^{i}}$
$\displaystyle+$
$\displaystyle\frac{4\nu^{2}}{3\hat{r}^{4}}\frac{(\gamma^{2}-1)}{\Gamma^{2}\xi}\chi_{2\epsilon}\log\left(\frac{\hat{r}}{e^{2\gamma_{E}}}\right)\,,$

where we have absorbed the factor of $e^{2\gamma_{E}}$ that arises from (1)
into the logarithm. The $\hat{c}_{1|2|3(\rm loc)}$ are the known local-in-time
PM coefficients up to 3PM order [18, 25, 20, 26], and $\hat{c}_{4(\rm loc)}$
is reported here for the first time. To complete the knowledge of the bound
dynamics, we are still missing the
[non-$\log\left(\frac{\hat{r}}{e^{2\gamma_{E}}}\right)$] nonlocal-in-time
contributions, $\hat{c}_{i\rm(nloc)}$, which depend on the trajectory. These
are more difficult to compute in a PM scheme, since they are often needed in
the opposite limit of quasi-circular orbits, thus entering at all PM orders!
Yet, they can be readily obtained within the PN approximation by evaluating
the radial action in (1) in a small-eccentricity expansion. Adapting the
($W_{1}$-only) results in [63] to the isotropic gauge, we quote their values
in the supplemental material to 6PN and eight order in the eccentricity. The
combined Hamiltonian in (Local-in-time Conservative Binary Dynamics at Fourth
Post-Minkowskian Order) is in perfect agreement to ${\cal
O}(G^{4}\hat{\bm{p}}^{6})$ with the $\hat{H}^{\rm ell}_{\rm 6PN(4PM)}$ derived
in [62] using the state of the art in PN theory, while at the same time it
incorporates all-order-in-velocity corrections. Ready-to-use expressions for
the full results and 30SF-approximate are collected in ancillary files.

Conclusions. Novel integration techniques in combination with EFT
methodologies have been extremely successful in reaching the very state of the
art in our understanding of scattering dynamics in general relativity,
including conservative and dissipative effects [42, 46]. However, as
illustrated in [23, 62], although local-in-time and logarithms are universal,
the full hyperbolic results fail to describe quasi-circular binaries. This is
due to the presence of orbit-dependent (non-logarithmic) nonlocal-in-time
effects, which preclude a smooth analytic continuation via the B2B map [21,
22]. Hence, up until now, we were lacking a direct correspondence to generic
bound motion, notably for the conservative sector.We have computed the
nonlocal-in-time contribution to the deflection angle, and removed it from the
total conservative value in [1, 2], thus yielding the local-in-time
counterpart. We then derived the radial action, center-of-mass (isotropic-
gauge) momentum and Hamiltonian, as well as the total logarithmic-dependent
part(s), all applicable to generic motion. Upon adapting the (non-logarithmic)
nonlocal-in-time effects for elliptic-like orbits computed in the PN expansion
[63], the combined total Hamiltonian becomes the most accurate description of
gravitationally-bound binary systems obtained from PN/PM data to date, readily
applicable to waveform modelling. Studies assessing the implications of our
results towards constructing high-precision GW templates, as well as the
derivation of a PM version of nonlocal-in-time effects for bound orbits, are
underway.

Acknowledgements. We thank Massimiliano M. Riva for useful discussions. We
thank Sebastian Pögel and Yu Jiao Zhu for suggesting the Calabi-Yau two-fold
as the relevant geometry and Matthias Wilhelm for useful discussions on the
integration problem. We are grateful to Donato Bini and Mohammed Khalil for
their help adapting the nonlocal-in-time (elliptic-like) terms from $W_{1}$ in
[63] into the isotropic gauge. The work of CD, GK, and RAP is supported by the
ERC-CoG Precision Gravity: From the LHC to LISA provided by the European
Research Council (ERC) under the European Union’s H2020 research and
innovation programme (grant No. 817791). ZL is supported partially by DFF
grant 1026-00077B, the Carlsberg foundation, and the European Union’s Horizon
2020 research and innovation program under the Marie Sklodowska-Curie grant
agreement No. 847523 ‘INTERACTIONS’.

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##  Appendix A Supplemental Material

An accurate hybrid description of elliptic-like orbits can be constructed by
incorporating into the Hamiltonian in (Local-in-time Conservative Binary
Dynamics at Fourth Post-Minkowskian Order) the
[non-$\log\left(\frac{\hat{r}}{e^{2\gamma_{E}}}\right)$] nonlocal-in-time
($W_{1}$-only) contributions computed in PN theory to 6PN and ${\cal
O}(e^{8})$ in the small-eccentricity expansion in [63], yielding

$\displaystyle\hat{H}^{\rm ell}(\hat{r},{\bm{p}}^{2},\nu)$ $\displaystyle=$
$\displaystyle\hat{E}+\sum_{i=1}^{i=4}\frac{\hat{c}_{i\rm(loc)}(\hat{\bm{p}}^{2},\nu)}{\hat{r}^{i}}+\frac{4\nu^{2}}{3\hat{r}^{4}}\frac{(\gamma^{2}-1)}{\Gamma^{2}\xi}\chi_{2\epsilon}(\gamma)\log\left(\frac{\hat{r}}{e^{2\gamma_{E}}}\right)$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{i=4}\frac{1}{\hat{r}^{i}}\left\\{\hat{c}^{{\rm
6PN}(e^{8})}_{i\rm(nloc)}(\hat{\bm{p}}^{2},\nu)+{\cal
O}\big{(}\hat{\bm{p}}^{2(8-i)}\big{)}\right\\}+\frac{1}{\hat{r}^{5}}\left(\hat{c}^{{\rm
4PN}(e^{8})}_{5(\rm
loc+nloc)}(\nu)-\frac{22\nu}{15}\log\left(\frac{\hat{r}}{e^{2\gamma_{E}}}\right)\right)+{\cal
O}\left(\frac{\hat{\bm{p}}^{2}}{\hat{r}^{5}}\right)\,,$

where we have included also the known (static) 4PN correction at ${\cal
O}(G^{5})$,

$\displaystyle\hat{c}^{{\rm 4PN}(e^{8})}_{5(\rm loc+nloc)}$ $\displaystyle=$
$\displaystyle-\frac{1}{16}+\nu^{2}\left(\frac{287\pi^{2}}{128}-\frac{1849}{48}\right)-\frac{5\nu^{3}}{8}$
$\displaystyle+$
$\displaystyle\nu\left(\frac{76645\pi^{2}}{12288}-\frac{334259}{720}+\frac{5421492}{5}\log(2)+\frac{631071}{20}\log(3)-\frac{1953125}{4}\log(5)\right)\,,$

and

$\displaystyle\chi_{2\epsilon}(\gamma)$ $\displaystyle=$
$\displaystyle-\frac{210\gamma^{6}-552\gamma^{5}+339\gamma^{4}-912\gamma^{3}+3148\gamma^{2}-3336\gamma+1151}{32(\gamma^{2}-1)^{2}}$
$\displaystyle+$
$\displaystyle\frac{3\left(35\gamma^{4}+60\gamma^{3}-150\gamma^{2}+76\gamma-5\right)}{16(\gamma^{2}-1)}\log\left(\frac{\gamma+1}{2}\right)-\frac{3\gamma\left(2\gamma^{2}-3\right)\left(35\gamma^{4}-30\gamma^{2}+11\right)}{32\left(\gamma^{2}-1\right)^{2}}\frac{\operatorname{arccosh}(\gamma)}{\sqrt{\gamma^{2}-1}}$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\nu(\hat{E}_{1}\hat{E}_{2}+\hat{\bm{p}}^{2})=\nu\hat{\bm{p}}^{2}+\sqrt{1+{\bm{p}}^{2}+\nu\hat{\bm{p}}^{2}(\nu\hat{\bm{p}}^{2}-2)}\,,\quad\Gamma^{2}\xi=\nu^{2}\hat{E}_{1}\hat{E}_{2}=\nu\sqrt{1+\hat{\bm{p}}^{2}+\nu\hat{\bm{p}}^{2}(\nu\hat{\bm{p}}^{2}-2)}\,.$

For the remaining terms, translating from the results in [63] to our isotropic
gauge, one finds

$\displaystyle\hat{c}^{{\rm 6PN}(e^{8})}_{1\rm(nloc)}(\hat{{\bm{p}}}^{2},\nu)$
$\displaystyle=\hat{{\bm{p}}}^{8}\nu\left[-\frac{17886}{175}+\frac{10834496\log(2)}{45}+\frac{6591861\log(3)}{700}-\frac{27734375\log(5)}{252}\right]$
(23)
$\displaystyle\quad+\hat{{\bm{p}}}^{10}\begin{multlined}\left[\nu^{2}\left(-\frac{17886}{35}+\frac{10834496\log(2)}{9}+\frac{6591861\log(3)}{140}-\frac{138671875\log(5)}{252}\right)\right.\\\
\left.+\nu\left(\frac{8943}{175}-\frac{5417248\log(2)}{45}-\frac{6591861\log(3)}{1400}+\frac{27734375\log(5)}{504}\right)\right]\end{multlined}\left[\nu^{2}\left(-\frac{17886}{35}+\frac{10834496\log(2)}{9}+\frac{6591861\log(3)}{140}-\frac{138671875\log(5)}{252}\right)\right.\\\
\left.+\nu\left(\frac{8943}{175}-\frac{5417248\log(2)}{45}-\frac{6591861\log(3)}{1400}+\frac{27734375\log(5)}{504}\right)\right]$
(26)
$\displaystyle\quad+\hat{{\bm{p}}}^{12}\begin{multlined}\left[\nu^{3}\left(-\frac{53658}{35}+\frac{10834496\log(2)}{3}+\frac{19775583\log(3)}{140}-\frac{138671875\log(5)}{84}\right)\right.\\\
+\nu^{2}\left(\frac{80487}{175}-\frac{5417248\log(2)}{5}-\frac{59326749\log(3)}{1400}+\frac{27734375\log(5)}{56}\right)\\\
\left.+\nu\left(-\frac{26829}{700}+\frac{1354312\log(2)}{15}+\frac{19775583\log(3)}{5600}-\frac{27734375\log(5)}{672}\right)\right]\,,\end{multlined}\left[\nu^{3}\left(-\frac{53658}{35}+\frac{10834496\log(2)}{3}+\frac{19775583\log(3)}{140}-\frac{138671875\log(5)}{84}\right)\right.\\\
+\nu^{2}\left(\frac{80487}{175}-\frac{5417248\log(2)}{5}-\frac{59326749\log(3)}{1400}+\frac{27734375\log(5)}{56}\right)\\\
\left.+\nu\left(-\frac{26829}{700}+\frac{1354312\log(2)}{15}+\frac{19775583\log(3)}{5600}-\frac{27734375\log(5)}{672}\right)\right]\,,$
(30) $\displaystyle\hat{c}^{{\rm
6PN}(e^{8})}_{2\rm(nloc)}(\hat{{\bm{p}}}^{2},\nu)$
$\displaystyle=\hat{{\bm{p}}}^{6}\nu\left[\frac{578843}{1050}-\frac{21212984\log(2)}{15}-\frac{282031389\log(3)}{5600}+\frac{1296484375\log(5)}{2016}\right]$
(31)
$\displaystyle\quad+\hat{{\bm{p}}}^{8}\begin{multlined}\left[\nu^{2}\left(\frac{48747851}{13440}-\frac{2125906693\log(2)}{378}+\frac{242787134673\log(3)}{286720}+\frac{3671798828125\log(5)}{1548288}\right.\right.\\\
\left.-\frac{96889010407\log(7)}{221184}\right)+\nu\left(\frac{81402259}{53760}-\frac{249145033\log(2)}{60}-\frac{103980982797\log(3)}{229376}\right.\\\
\left.\left.+\frac{12166079921875\log(5)}{6193152}+\frac{96889010407\log(7)}{884736}\right)\right]\end{multlined}\left[\nu^{2}\left(\frac{48747851}{13440}-\frac{2125906693\log(2)}{378}+\frac{242787134673\log(3)}{286720}+\frac{3671798828125\log(5)}{1548288}\right.\right.\\\
\left.-\frac{96889010407\log(7)}{221184}\right)+\nu\left(\frac{81402259}{53760}-\frac{249145033\log(2)}{60}-\frac{103980982797\log(3)}{229376}\right.\\\
\left.\left.+\frac{12166079921875\log(5)}{6193152}+\frac{96889010407\log(7)}{884736}\right)\right]$
(35)
$\displaystyle\quad+\hat{{\bm{p}}}^{10}\begin{multlined}\left[\nu^{3}\left(\frac{1764244493}{134400}-\frac{50772177511\log(2)}{3780}+\frac{15140243287719\log(3)}{2867200}+\frac{15746212109375\log(5)}{3096576}\right.\right.\\\
\left.-\frac{1065779114477\log(7)}{442368}\right)+\nu^{2}\left(\frac{922629733}{107520}-\frac{188966394467\log(2)}{7560}-\frac{7125985899279\log(3)}{2293760}\right.\\\
\left.+\frac{147239183828125\log(5)}{12386304}+\frac{484445052035\log(7)}{589824}\right)+\nu\left(-\frac{28327409}{35840}+\frac{154094423\log(2)}{72}\right.\\\
\left.\left.+\frac{527065116993\log(3)}{2293760}-\frac{12533579921875\log(5)}{12386304}-\frac{96889010407\log(7)}{1769472}\right)\right]\,,\end{multlined}\left[\nu^{3}\left(\frac{1764244493}{134400}-\frac{50772177511\log(2)}{3780}+\frac{15140243287719\log(3)}{2867200}+\frac{15746212109375\log(5)}{3096576}\right.\right.\\\
\left.-\frac{1065779114477\log(7)}{442368}\right)+\nu^{2}\left(\frac{922629733}{107520}-\frac{188966394467\log(2)}{7560}-\frac{7125985899279\log(3)}{2293760}\right.\\\
\left.+\frac{147239183828125\log(5)}{12386304}+\frac{484445052035\log(7)}{589824}\right)+\nu\left(-\frac{28327409}{35840}+\frac{154094423\log(2)}{72}\right.\\\
\left.\left.+\frac{527065116993\log(3)}{2293760}-\frac{12533579921875\log(5)}{12386304}-\frac{96889010407\log(7)}{1769472}\right)\right]\,,$
(40) $\displaystyle\hat{c}^{{\rm
6PN}(e^{8})}_{3\rm(nloc)}(\hat{{\bm{p}}}^{2},\nu)$
$\displaystyle=\hat{{\bm{p}}}^{4}\nu\left[-\frac{175727}{150}+\frac{233388968\log(2)}{75}+\frac{16351713\log(3)}{160}-\frac{405859375\log(5)}{288}\right]$
(41)
$\displaystyle\quad+\hat{{\bm{p}}}^{6}\begin{multlined}\left[\nu^{2}\left(-\frac{45488833}{4480}+\frac{105895904239\log(2)}{13230}-\frac{10808816520303\log(3)}{2007040}-\frac{28119126171875\log(5)}{10838016}\right.\right.\\\
\left.+\frac{2588320706587\log(7)}{1105920}\right)+\nu\left(-\frac{4548912331}{627200}+\frac{2922687496621\log(2)}{132300}+\frac{13469503195629\log(3)}{5734400}\right.\\\
\left.\left.-\frac{150369012359375\log(5)}{14450688}-\frac{2588320706587\log(7)}{4423680}\right)\right]\end{multlined}\left[\nu^{2}\left(-\frac{45488833}{4480}+\frac{105895904239\log(2)}{13230}-\frac{10808816520303\log(3)}{2007040}-\frac{28119126171875\log(5)}{10838016}\right.\right.\\\
\left.+\frac{2588320706587\log(7)}{1105920}\right)+\nu\left(-\frac{4548912331}{627200}+\frac{2922687496621\log(2)}{132300}+\frac{13469503195629\log(3)}{5734400}\right.\\\
\left.\left.-\frac{150369012359375\log(5)}{14450688}-\frac{2588320706587\log(7)}{4423680}\right)\right]$
(45)
$\displaystyle\quad+\hat{{\bm{p}}}^{8}\begin{multlined}\left[\nu^{3}\left(-\frac{1108386017}{24192}+\frac{38790406370519\log(2)}{1786050}-\frac{244047465883413\log(3)}{10035200}\right.\right.\\\
\left.+\frac{1009279694921875\log(5)}{877879296}+\frac{453841966033589\log(7)}{89579520}\right)+\nu^{2}\left(-\frac{361592268899}{5644800}\right.\\\
\left.+\frac{116606471572979\log(2)}{1071630}-\frac{448065058976289\log(3)}{40140800}-\frac{181279182489765625\log(5)}{3511517184}\right.\\\
\left.+\frac{3680972377512689\log(7)}{358318080}\right)+\nu\left(-\frac{3438069458633}{304819200}+\frac{369057536315537\log(2)}{9185400}\right.\\\
\left.\left.+\frac{607401830370627\log(3)}{80281600}-\frac{44240036362654375\log(5)}{2341011456}-\frac{1267373911442149\log(7)}{429981696}\right)\right]\,,\end{multlined}\left[\nu^{3}\left(-\frac{1108386017}{24192}+\frac{38790406370519\log(2)}{1786050}-\frac{244047465883413\log(3)}{10035200}\right.\right.\\\
\left.+\frac{1009279694921875\log(5)}{877879296}+\frac{453841966033589\log(7)}{89579520}\right)+\nu^{2}\left(-\frac{361592268899}{5644800}\right.\\\
\left.+\frac{116606471572979\log(2)}{1071630}-\frac{448065058976289\log(3)}{40140800}-\frac{181279182489765625\log(5)}{3511517184}\right.\\\
\left.+\frac{3680972377512689\log(7)}{358318080}\right)+\nu\left(-\frac{3438069458633}{304819200}+\frac{369057536315537\log(2)}{9185400}\right.\\\
\left.\left.+\frac{607401830370627\log(3)}{80281600}-\frac{44240036362654375\log(5)}{2341011456}-\frac{1267373911442149\log(7)}{429981696}\right)\right]\,,$
(51) $\displaystyle\hat{c}^{{\rm
6PN}(e^{8})}_{4\rm(nloc)}(\hat{{\bm{p}}}^{2},\nu)$
$\displaystyle=\hat{{\bm{p}}}^{2}\nu\left[\frac{27636}{25}-\frac{680106004\log(2)}{225}-\frac{37122381\log(3)}{400}+\frac{196484375\log(5)}{144}\right]$
(52)
$\displaystyle\quad+\hat{{\bm{p}}}^{4}\begin{multlined}\left[\nu^{2}\left(\frac{23663203}{1680}-\frac{9735062548\log(2)}{33075}+\frac{14337306321183\log(3)}{1254400}-\frac{2701666015625\log(5)}{1354752}\right.\right.\\\
\left.-\frac{650540498447\log(7)}{138240}\right)+\nu\left(\frac{116416141}{9408}-\frac{526259559517\log(2)}{13230}\right.\\\
\left.\left.-\frac{641012819877\log(3)}{143360}+\frac{33874913921875\log(5)}{1806336}+\frac{650540498447\log(7)}{552960}\right)\right]\end{multlined}\left[\nu^{2}\left(\frac{23663203}{1680}-\frac{9735062548\log(2)}{33075}+\frac{14337306321183\log(3)}{1254400}-\frac{2701666015625\log(5)}{1354752}\right.\right.\\\
\left.-\frac{650540498447\log(7)}{138240}\right)+\nu\left(\frac{116416141}{9408}-\frac{526259559517\log(2)}{13230}\right.\\\
\left.\left.-\frac{641012819877\log(3)}{143360}+\frac{33874913921875\log(5)}{1806336}+\frac{650540498447\log(7)}{552960}\right)\right]$
(56)
$\displaystyle\quad+\hat{{\bm{p}}}^{6}\begin{multlined}\left[\nu^{3}\left(\frac{10075896877}{120960}-\frac{2348423027149\log(2)}{51030}+\frac{8674336284777\log(3)}{286720}\right.\right.\\\
\left.-\frac{2232609748046875\log(5)}{125411328}+\frac{250707235071713\log(7)}{17915904}\right)+\nu^{2}\left(\frac{60245673427}{376320}\right.\\\
\left.-\frac{126132398166437\log(2)}{1071630}+\frac{763693932388383\log(3)}{8028160}+\frac{204623745011171875\log(5)}{3511517184}\right.\\\
\left.-\frac{4304025048065071\log(7)}{71663616}\right)+\nu\left(\frac{102339488389}{2177280}-\frac{616925145960877\log(2)}{3214890}\right.\\\
\left.\left.-\frac{144912376553769\log(3)}{4014080}+\frac{52541416380715625\log(5)}{585252864}+\frac{1554400159532395\log(7)}{107495424}\right)\right]\,.\end{multlined}\left[\nu^{3}\left(\frac{10075896877}{120960}-\frac{2348423027149\log(2)}{51030}+\frac{8674336284777\log(3)}{286720}\right.\right.\\\
\left.-\frac{2232609748046875\log(5)}{125411328}+\frac{250707235071713\log(7)}{17915904}\right)+\nu^{2}\left(\frac{60245673427}{376320}\right.\\\
\left.-\frac{126132398166437\log(2)}{1071630}+\frac{763693932388383\log(3)}{8028160}+\frac{204623745011171875\log(5)}{3511517184}\right.\\\
\left.-\frac{4304025048065071\log(7)}{71663616}\right)+\nu\left(\frac{102339488389}{2177280}-\frac{616925145960877\log(2)}{3214890}\right.\\\
\left.\left.-\frac{144912376553769\log(3)}{4014080}+\frac{52541416380715625\log(5)}{585252864}+\frac{1554400159532395\log(7)}{107495424}\right)\right]\,.$
(62)

Upon PN expanding the total Hamiltonian in (A), we find agreement with the
expression for $\hat{H}^{\rm ell}_{\rm 6PN(4PM)}$ in [62], in the overlapping
realm of validity.

Let us conclude with a few remarks on the (non)local-in-time dynamics in (A).
In addition to $\gamma=u_{1}\cdot u_{2}$, the nonlocality in time introduces a
new (mass-dependent) scale in the scattering problem, i.e. $u_{1(2)}\cdot
u_{\rm com}$, yielding an intricate (non-algebraic) mass dependence for the
nonlocal part (for fixed $\gamma$). Given the mass structure of the total
result—guaranteed by the Lorentz invariance of the (unbound) radial
action—this mass dependence transitions unscathed into the local-in-time
(universal) contribution in (A). One may then wonder, similarly to the
hyperbolic case, whether the inclusion of a PM version of the $\hat{c}_{i(\rm
nloc)}$’s for elliptic-like motion would also lead to a simpler structure for
the total bound dynamics. After all, that is expected to be the case—through
the B2B analytic continuation—in the large-eccentricity limit [23].222Notice
that $\gamma=1+\frac{\Gamma^{2}-1}{2\nu}$, which introduces rather nontrivial
mass dependence for fixed (binding) energy. Remarkably, when fully valid, the
B2B map guarantees that adiabatic invariants for the bound problem can be
written in terms of $\gamma$ and the angular momentum (up to conversion
factors) [21, 22]. However, we would expect this to be manifest only in the
full solution,333Modulo some subtleties, it is in principle possible to find
an analytic continuation between the full (local+nonlocal) hyperbolic and
elliptic integrands for the radial actions [23], yielding e.g. the scattering
angle and periastron advance, which would imply that (although not directly
B2B-related for generic motion) the latter may still inherit the structure
from the former. and not for a given (PN/PM) truncation in the (opposite)
regime of small eccentricities. For instance, due to the differences in the
region of integration and energy spectrum (continuum vs. discrete), while
nonlocal-in-time tail terms for hyperbolic-like motion first arise at ${\cal
O}(G^{4})$, the above $\hat{c}_{i(\rm nloc)}$ coefficients for elliptic-like
orbits enter at all orders in $G$ (starting at 4PN order). We will return to
these issues elsewhere.


# Parameterized Algorithms for Editing to Uniform Cluster Graph

Ajinkya Gaikwad Indian Institute of Science Education and Research, Pune,
India 11email: ajinkya.gaikwad@students.iiserpune.ac.in;11email:
<EMAIL_ADDRESS>Hitendra Kumar Indian Institute of Science Education
and Research, Pune, India 11email:
ajinkya.gaikwad@students.iiserpune.ac.in;11email<EMAIL_ADDRESS>Soumen Maity Indian Institute of Science Education and Research, Pune, India
11email: ajinkya.gaikwad@students.iiserpune.ac.in;11email:
<EMAIL_ADDRESS>
###### Abstract

Given a graph $G=(V,E)$ and an integer $k\in\mathbb{N}$, we investigate the
2-Eigenvalue Vertex Deletion (2-EVD) problem. The objective is to remove at
most $k$ vertices such that the adjacency matrix of the resulting graph has at
most two eigenvalues. It is established that the adjacency matrix of a graph
has at most two eigenvalues if and only if the graph is a collection of equal-
sized cliques. Thus, the 2-Eigenvalue Vertex Deletion amounts to removing a
set of at most $k$ vertices to transform the graph into a collection of equal-
sized cliques. The 2-Eigenvalue Edge Editing (2-EEE), 2-Eigenvalue Edge
Deletion (2-EED) and 2-Eigenvalue Edge Addition (2-EEA) problems are defined
analogously. We present a kernel of size $\mathcal{O}(k^{3})$ for $2$-EVD,
along with an FPT algorithm with a running time of $\mathcal{O}^{*}(2^{k})$.
For the problem $2$-EEE, we provide a kernel of size $\mathcal{O}(k^{2})$.
Additionally, we present linear kernels of size $5k$ and $6k$ for $2$-EEA and
$2$-EED respectively. For the $2$-EED, we also construct an algorithm with
running time $\mathcal{O}^{*}(1.47^{k})$ . These results address open
questions posed by Misra et al. (ISAAC 2023) regarding the complexity of these
problems when parameterized by the solution size.

###### Keywords:

Parameterized Complexity FPT Eigenvalue Uniform Cluster Graphs

##  1 Introduction

Numerous algorithmic challenges involving graphs can be framed as tasks of
modifying a graph to meet specific criteria. Notably, over the last three
decades, these graph modification tasks have been a significant source of
inspiration for innovating new methodologies in parameterized algorithms and
complexity theory. This paper delves into a particular graph modification
problem known as Uniform Cluster Vertex Deletion, along with several related
variations of the same problem. Previously explored in
[misra_et_al:LIPIcs.ISAAC.2023.53], the complexity of reducing the count of
distinct eigenvalues of a graph by either removing vertices or editing edges
was examined. Notably, it can be observed that an adjacency matrix of graph
possesses at most two distinct eigenvalues if and only if it comprises of
disjoint unions of equally sized cliques. Our notation aligns with that of the
aforementioned paper. Let $G=(V,E)$ be a simple undirected graph, where $V$
denotes the vertex set of $G$ and $E$ denotes the edge set of $G$. We use $n$
to denote $|V|$. For $u\in V$, we define $N(u)=\\{v\in V:(u,v)\in E\\}$ and
$N[u]=N(u)\cup\\{u\\}$. The _degree_ of $u\in V$ is $|N(u)|$ and denoted by
$d_{G}(u)$. A _clique_ $C$ in an undirected graph $G=(V,E)$ is a subset of the
vertices $C\subseteq V$ such that every two distinct vertices are adjacent. A
_cluster graph_ is a graph where every component is a clique. Observe that a
graph is a cluster graph if and only if it does not have an induced $P_{3}$,
that is, an induced path on three vertices. Let $U\subseteq V$ be a subset of
vertices of $G$ and $F\subseteq\binom{V}{2}$ be a subset of pairs of vertices
of $G$. The subgraph induced by $U\subseteq V$ is denoted by $G[U]$. We define
$G-U=G[V\setminus U]$, $G-F=(V,E\setminus F)$, $G+F=(V,E\cup F)$ and
$G\triangle F=(V,E\triangle F)$. Here, $E\triangle F$ is the symmetric
difference between $E$ and $F$. If $U=\\{u\\}$ or $F=\\{e\\}$ then we simply
write $G-u$, $G-e$ and $G+e$ for $G-U$, $G-F$ and $G+F$, respectively. It is
known that the spectrum of the adjacency matrix of a graph can be computed in
polynomial time. The following lemma is crucial to our discussions:

###### Lemma 1

[Doob1970OnCC, GOLDBERG2020163] The adjacency matrix of a graph $G$ has at
most two distinct eigenvalues if and only if $G$ is a disjoint union of equal
sized cliques.

We refer to [marekcygan, Downey] for details on parameterized complexity. Our
goal is to present the parameterized complexity landscape of the following
problems:
$2$-Eigenvalue Vertex Deletion (2-EVD)
Input: An undirected graph $G=(V,E)$, and a positive integer $k$. Question: Is
there a subset $S\subseteq V$ with $|S|\leq k$ such that $G-S$ is a collection
of equal sized cliques?

$2$-Eigenvalue Edge Editing (2-EEE)
Input: An undirected graph $G=(V,E)$, and a positive integer $k$. Question: Is
there a subset $F\subseteq\binom{V(G)}{2}$ with $|F|\leq k$ such that
$G\triangle F=(V,E\Delta F)$ is a collection of equal sized cliques?

$2$-Eigenvalue Edge Deletion (2-EED)
Input: An undirected graph $G=(V,E)$, and a positive integer $k$. Question: Is
there a subset $F\subseteq E$ with $|F|\leq k$ such that $G-F=(V,E\setminus
F)$ is a collection of equal sized cliques?

$2$-Eigenvalue Edge Addition (2-EEA)
Input: An undirected graph $G=(V,E)$, and a positive integer $k$. Question: Is
there a subset $F\subseteq\binom{V(G)}{2}$ with $|F|\leq k$ such that
$G+F=(V,E\cup F)$ is a collection of equal sized cliques?

###  1.1 Our results

In this paper, we study 2-EVD, 2-EEE, 2-EED, and 2-EEA from the parameterized
complexity point of view. Clearly, one can study these problems for any fixed
number of distinct eigenvalues. Using some simple yet powerful structural
observations and non-trivial ideas, we have provided combinatorial algorithms
for all considered variants of the problem. We give a kernel of size
$\mathcal{O}(k^{3})$ for $2$-EVD. Additionally, we provide an FPT algorithm
that runs in time $\mathcal{O}^{*}(2^{k})$, which improves upon the FPT
algorithm given in [misra_et_al:LIPIcs.ISAAC.2023.53] based on the standard
branching method, which has a running time of $\mathcal{O}^{*}(3^{k})$. For
the problem $2$-EEE, we provide a kernel of size $\mathcal{O}(k^{2})$.
Furthermore, we provide linear kernels of sizes $6k$ and $5k$ for $2$-EED and
$2$-EEA, respectively, improving upon the quadratic kernel given in
[misra_et_al:LIPIcs.ISAAC.2023.53] for 2-EEA. We also present an FPT algorithm
running in time $\mathcal{O}^{*}(1.47^{k})$ for 2-EED, further improving the
algorithm from [misra_et_al:LIPIcs.ISAAC.2023.53]. Through this work, we
resolve three open questions listed by Misra et al.
[misra_et_al:LIPIcs.ISAAC.2023.53] concerning the complexity of these problems
parameterized by solution size. This study provides a clear picture of the
parameterized complexity of the studied problems.

Uniform cluster |  FPT algorithm |  Kernel
---|---|---
Vertex deletion |  Theorem (LABEL:FPTevd) $\mathcal{O}^{*}(2^{k})$ |  Theorem (2.1) $\mathcal{O}(k^{3})$
Edge editing |  - |  Theorem (LABEL:EEE) $\mathcal{O}(k^{2})$
Edge deletion |  Theorem (LABEL:EEDFPT) $\mathcal{O}^{*}(1.47^{k})$ |  Theorem (LABEL:EED) $6k$
Edge Addition |  - |  Theorem (LABEL:EEA) $5k$

###  1.2 Review of previous work

As mentioned in [misra_et_al:LIPIcs.ISAAC.2023.53], the problem of modifying
the graph to reduce the count of distinct eigenvalues to $r$ of the
corresponding adjacency matrix was first brought up in [MEESUM201670]. In
[misra_et_al:LIPIcs.ISAAC.2023.53], Misra et al. considered classical and
parameterized complexity of this problem. The paper studied mainly four
possible operations which are vertex deletion, edge deletion, edge addition
and edge editing. For the special case of $r=2$, the vertex deletion variant
was shown to be NP-complete even on triangle-free and $3d$-regular graphs for
any $d\geq 2$, and also NP-complete on $d$-regular graphs for any $d\geq 8$.
Moreover, the edge deletion, addition, and editing variants were proved to be
NP-complete for $r=2$ case. Furthermore, for any fixed $r\geq 3$, they showed
that $r$-EVD is NP-complete on bipartite graphs. Also, the $2$-EEA was shown
to be NP-complete when the input is either a cluster graph, a forest, or a
collection of cycles. Apart from studying the classical complexity, the
authors also studied and provided numerous results in the realm of
parameterized complexity. In particular they gave FPT algorithms for the
$2$-EVD and $2$-EED problems with running time $\mathcal{O}^{*}(3^{k})$ and
$\mathcal{O}^{*}(2^{k})$, respectively. We have improved both of these
algorithms. The paper also presented a quadratic kernel for $2$-EEA.

The clustering problems hold high importance in the complexity theory as they
can model multiple scenarios. One more way to view this problem is to modify
graph in such a way that the components of the resulting graph have smaller
diameter. This is called as $s$-Club Cluster Vertex Deletion problem where the
input is a graph with two integers $s\geq 2$ and $k\geq 1$ and the goal is to
decide whether it is possible to remove at most $k$ vertices from $G$ such
that each connected component of the resulting graph has diameter at most $s$.

We note that Madathil and Meeks [madathil2024parameterized] recently explored
a generalization of this problem, termed Balanced Cluster Editing, in which,
given a graph $G$, and two integers $0\leq\eta\leq n$ and $k$, we are
permitted to “edit” up to $k$ edges. This editing should result in a cluster
graph where the size difference between any two connected components of the
resulting cluster graph does not exceed $\eta$. One can see that when
$\eta=0$, this problem reduces to the Uniform Cluster Editing problem. The
authors presented polynomial kernels for Balanced Cluster Completion, Balanced
Cluster Deletion, and Balanced Cluster Editing. While they provided polynomial
kernels for the general cases, we have achieved better bounds in all
considered variants when $\eta=0$. We obtained our results simultaneously with
and independent from those by Madathil and Meeks [madathil2024parameterized].

##  2 Kernelization and FPT algorithm for 2-EVD parameterized by solution size

In this section, we begin by proving the following theorem.

###### Theorem 2.1

2-EVD parameterized by solution size admits a kernel of size
$\mathcal{O}(k^{3})$.

A cluster graph is a graph in which every connected component is a clique.
Note that a graph is a cluster graph if and only if it does not contain an
induced $P_{3}$. This observation serves as a cornerstone for our
kernelization technique. The first step of the kernelization process is to
compute a maximal set $\mathcal{P}_{3}$ of vertex-disjoint induced $P_{3}$s in
$G$. This can be accomplished using the greedy algorithm shown below:

Algorithm 1

1:A graph $G=(V,E)$.

2:A maximal set $\mathcal{P}_{3}$ of vertex-disjoint induced $P_{3}$ in $G$.

3:$\mathcal{P}_{3}=\emptyset$

4:while $G$ has an induced $P_{3}$ do

5: Identify an induced $P_{3}=(u,v,w)$ in $G$

6: $\mathcal{P}_{3}=\mathcal{P}_{3}\cup\\{(u,v,w)\\}$

7: $G=G-\\{u,v,w\\}$

8:end while

9:return $\mathcal{P}_{3}$

At the end, if $|\mathcal{P}_{3}|>k$, we have a no-instance. So we assume that
$|\mathcal{P}_{3}|\leq k$, and let $S$ be the vertices of $\mathcal{P}_{3}$.
We have $|S|\leq 3k$. Let us denote the set of cliques of $G-S$ by
$\mathcal{C}$. We have the following simple rules:

###### Reduction EVD 1

If there exists $s\in S$ with neighbours in at least $k+2$ cliques of
$\mathcal{C}=G-S$, then delete $s$ (and its incident edges from $G$) and
decrement the parameter $k$ by 1. The new instance is $(G-s,k-1)$

###### Lemma 2

Reduction Rule EVD 1 is safe.

###### Proof

Let $X$ be a uniform cluster vertex deletion set of $G$ of size at most $k$.
For the sake of contradiction, let us assume that $s\in S$, which has
neighbours in at least $k+2$ cliques of $\mathcal{C}=G-S$, is not in $X$.
Therefore, $s$ must be in a clique of $G-X$. That clique can have vertices
from at most one clique, say $C_{i}$ in $\mathcal{C}$. Since $s$ has
neighbours in at least $k+2$ cliques, we would need to include neighbours of
$s$ in at least $k+1$ cliques other than $C_{i}$ in the solution $X$. This
contradicts the condition that the solution size $|X|$ is at most $k$. Hence
if $(G,k)$ is a yes-instance (meaning there exists a uniform cluster vertex
deletion set of size at most $k$), then $s$ must be in $X$. ∎

Next, we partition the set of cliques in $\mathcal{C}$ based on whether they
have neighbours in $S$ or not. We define:

$\mathcal{C}_{0}=\\{C\in\mathcal{C}:\text{no vertex in }C\text{ has a
neighbour in }S\\}$

and

$\mathcal{C}_{1}=\\{C\in\mathcal{C}:\text{some vertex in }C\text{ has a
neighbour in }S\\}.$

Observe that the exhaustive application of reduction rule EVD 1 completely
removes the vertices of $S$ that have neighbours in at least $k+2$ cliques of
$\mathcal{C}_{1}$. In other words, after the exhaustive application of
reduction rule EVD 1, each vertex of $S$ can have neighbours in at most $k+1$
cliques of $\mathcal{C}_{1}$. Since $|S|\leq 3k$ and each vertex of $S$ can
have neighbours in at most $k+1$ cliques of $\mathcal{C}_{1}$, there are at
most $3k(k+1)$ cliques in $\mathcal{C}_{1}$. Therefore, we have

${|\mathcal{C}_{1}|\leq 3k(k+1)}.$

An $r$-clique is a clique of size $r$. Now, we have the following simple rule.

###### Reduction EVD 2

If for some $r\in\\{1,2,\ldots,n\\}$ there are more than $k+1$ $r$-cliques in
$\mathcal{C}_{0}$, then remove all but $k+1$ $r$-cliques from $G$.

###### Lemma 3

Reduction rule EVD 2 is safe.


# Ab-initio study of ultrafast charge dynamics in graphene

Q. Z. Li1, P. Elliott1, J. K. Dewhurst2, S. Sharma1 and S. Shallcross1 1 Max-
Born-Institute for Non-linear Optics and and Short Pulse Spectroscopy, Max-
Born Strasse 2A, 12489 Berlin, Germany  2 Max-Planck-Institut fur
Mikrostrukturphysik Weinberg 2, D-06120 Halle, Germany.

###### Abstract

Monolayer graphene provides an ideal material to explore one of the
fundamental light-field driven interference effects: Landau-Zener-Stückelberg
interference. However, direct observation of the resulting interference
patterns in momentum space has not proven possible, with Landau-Zener-
Stückelberg interference observed only indirectly through optically induced
residual currents. Here we show that the transient electron momentum density
(EMD), an object that can easily be obtained in experiment, provides an
excellent description of momentum resolved charge excitation. We employ state-
of-the-art time-dependent density function theory calculations, demonstrating
by direct comparison of EMD with conduction band occupancy, obtained from
projecting the time propagated wavefunction onto the ground state, that the
two quantities are in excellent agreement. For even the most intense laser
pulses we find that the electron dynamics to be almost completely dominated by
the $\pi$-band, with transitions to other bands strongly suppressed. Simple
model based tight-binding approaches can thus be expected to provide an
excellent description for the laser induced electron dynamics in graphene.

Intense laser light offers the possibility to control electrons in matter on
femtosecond time scales. Triumphs of this burgeoning field include tuning the
optically induced current in graphene via the carrier envelope phase of
lightSchiffrin2013 ; Heide2018 ; Higuchi2017 , attosecond control over
magnetic order in thin films of magnetic overlayersdewhurst_laser-induced_2018
; Siegrist2019 , and controlled valley excitation in the semi-conducting few
layer dichalcogenides by circularly polarized lightmak_control_2012 ;
zeng_valley_2012  to name only a few examples. The two band Dirac cone found
in graphene provides an ideal materials platform for studying one of the
canonical light-field driven interference effects: Landau-Zener-Stückelberg
(LZS) interferenceShevchenko2010 ; Nori16_lzs , which before its observation
in grapheneHiguchi2017  had only been observed in designed two state quantum
systemsDQD_current_lzs ; GaAs_DQD_current_lzs ; TiAu_qubit_current_lzs ;
Si_current_lzs ; SiSiGe_QDQ_charge_transfer_lzs . This effect occurs when an
oscillating electromagnetic field drives intraband oscillation through the
Bloch acceleration theorem
$\bm{\mathrm{k}}\to\bm{\mathrm{k}}+\bm{\mathrm{A}}(t)/c$ and in the region of
an avoided crossing interband transitions occur even when the band gap exceeds
the dominant pulse frequency, so-called Landau-Zener transitions. Upon
repeated passing of the avoided crossing multiple pathways exist to the
conduction band with consequent constructive and destructive interference of
electron states. This offers rich possibilities for controlling electron
dynamics by intense laser light, demonstrated by the recent observation of
control over optical currents underpinned by LZS interferenceHiguchi2017 , a
result anticipated theoretically in Ref. ishikawa_electronic_2013, .

The ubiquity of the avoided crossing band structure in 2d materials, found not
only in the Dirac cone of graphene but also in the the semi-conducting
monolayer dichalcogesidestmdc_bands , phosphorenephosphorene_bands ;
Nematollahi2018 , silicenesilicene_bands , and stanenestanene_bands , points
towards the importance of LSZ interferometry in controlling electron dynamics
in 2d materials. However, while interference physics can be easily probed
theoretically through the conduction band populationNematollahi2018 ;
Kelardeh2015 ; Kelardeh2016 , the experimental situation is more difficult,
with to date only indirect observations of LSZ physics in materials reported.
In this paper we show that the transient electron momentum density (EMD)
difference, defined as

$\Delta\rho(\bm{\mathrm{p}},t_{f})=\rho(\bm{\mathrm{p}},t_{f})-\rho(\bm{\mathrm{p}},t=0)$
(1)

with ${\bf p}$ momentum and $\rho(\bm{\mathrm{p}},t)$ the electron momentum
densitydugdale  before ($t=0$) and after ($t_{f}$) the pump laser pulse,
offers a tool for directly probing LZS interference effects. The EMD may be
measured experimentally via tomographic reconstruction using Compton
profilesSTBK95 ; DH96 ; SSWK96 ; KSHH03 ; HBHA07  and, in particular, for
layered materialsHN17 ; emd3d_Kurp1996 . Combining these techniques with
ultrafast X-ray sources will allow the transient EMD to be experimentally
measured. This suggests a way in which the LZS physics may be directly
observed in 2d materials, opening the way to correlate indirect LZS physics
such as induced currents with the fundamental momentum space interference
patterns.

For graphene, we demonstrate that the EMD facilitates both the real time
observation of the formation of LZS interference patterns, as well as the
elucidation of subtle features in the relation between pump pulse and
interference in momentum space.

In contrast to previous works that have employed simple single particle tight-
binding Hamiltonians to study the LZS effectNematollahi2018 ; Higuchi2017 ;
Kelardeh2014 ; Kelardeh2015 ; Kelardeh2016 ; fillion-gourdeau_time-domain_2016
; lefebvre_carrier-envelope_2018 ; gagnon_pulse_2018 , we will here deploy the
time dependent version of density functional theory (TD-DFT). To establish the
accuracy of the EMD as a record of LZS interference we compare it with the
excited electron distribution, $N_{\text{ex}}$, defined within TD-DFT
aselliott_ultrafast_2016 :

$N_{\text{ex}}(\bm{\mathrm{k}},t)=\sum_{i}^{occ}\sum_{j}^{unocc}\left|\langle\psi_{i\bm{\mathrm{k}}}(t)|\psi_{j\bm{\mathrm{k}}}(t=0)\rangle\right|^{2}$
(2)

where $\psi_{j\bm{\mathrm{k}}}(t)$ is the time-dependent Kohn-Sham orbital at
time $t$, and $\psi_{i\bm{\mathrm{k}}}(t=0)$ is the ground state orbital. In
all cases we find that the pattern of excitation in momentum space generated
by transient EMD and Nex is nearly identical in the first BZ.

Finally, we consider the role of the non-$\pi$-band states in the electron
dynamics in graphene. Remarkably, despite electron excitation through the
whole energy range of the $\pi$-band (up to 10 eV above the Fermi energy, an
energy range encompassing the $\sigma^{\ast}$ bands as well as several high
$l$ character bands), it turns out that there occur almost no transitions to
states outside the $\pi$-band manifold. We attribute this to the near
vanishing of the corresponding dipole matrix elements. Our calculations thus
suggest that even for very significant laser excitation tight-binding based
models will provide a good description of the electron dynamics.

Figure 1: Conduction band occupation as a function of $\bm{\mathrm{k}}$-vector
as determined directly by projection of the time-dependent state onto the
ground-state Kohn-Sham states (first column), see Eq. (2), and, second column,
the transient electron momentum density (tr-EMD) difference, see Eq. (1).
Evidently, both quantities in a consistent way capture the momentum space
intensity fringes generated by Landau-Zener-Stückelberg interference. The
third column displays the electric field ($\bm{\mathrm{E}}$-field) of the pump
laser pulse (blue lines), the $\bm{\mathrm{A}}$-field scaled such that it can
be plotted on the same axis (green lines), and the induced current density
(red lines). Pulses in (a)-(l) have a full width half maximum (FWHM) of 1.935
fs, a central frequency of 1.4 eV, and peak intensity of $5.43\times 10^{12}$
W/cm2, and carrier envelope phase as indicated in the panels. The remaining
three rows have FWHM 2.758 fs, CEP of $\pi/2$, and central frequencies and
intensities as indicated in the panels. In the first and second columns, the
white hexagons represent the boundary of the 1st BZ while the lines in the
right bottom corner represent the effective $\bm{\mathrm{k}}$-space trajectory
given by the Bloch acceleration theorem.

According to Runge-Gross theoremRG1984 , which extends the Hohenberg-Kohn
theorem into the time domain, with common initial states there will be a one
to one correspondence between the time-dependent external potentials and
densitiesCarsten_book ; TDDFTbook . Based on this theorem, a system of non-
interacting particles can be chosen such that the density of this non-
interacting system is equal to that of the interacting system for all times,
with the wave function of this non-interacting system represented by a Slater
determinant of single-particle orbitals. These time-dependent Kohn-Sham (KS)
orbitals are governed by the Schrödinger equation (for the spin degenerate
case):

$\displaystyle i\partial_{t}\psi_{j}({\bf
r},t)=\Bigg{[}\frac{1}{2}\left(-i{\nabla}+\frac{1}{c}{\bf A}_{\rm
ext}(t)\right)^{2}+v_{s}({\bf r},t)\Bigg{]}\psi_{j}({\bf r},t).$ (3)

In the above equation ${\bf A}_{\rm ext}(t)$ is the vector potential
representing the applied laser field, the effective potential $v_{s}({\bf
r},t)$ is given by $v_{s}({\bf r},t)=v_{\rm ext}({\bf r},t)+v_{\rm H}({\bf
r},t)+v_{\rm xc}({\bf r},t)$, where $v_{\rm ext}({\bf r},t)$ is the external
potential, $v_{\rm H}({\bf r},t)$ the Hartree potential, and $v_{\rm xc}({\bf
r},t)$ is the exchange-correlation (xc) potential. For the latter we have used
the adiabatic local density approximation. From the Fourier transform of the
Kohn-Sham states, $\psi_{i\bm{\mathrm{k}}}({\bf r})$, the electron momentum
density can be constructed as
$\rho(\bm{\mathrm{p}})=\sum_{i\bm{\mathrm{k}}}|\psi_{i\bm{\mathrm{k}}}(\bm{\mathrm{p}})|^{2}$.
This EMD constructed from KS states has been found to provide excellent
agreement with that obtained from Compton scatteringdugdale .

All calculations employ the state-of-the art all-electron full potential
linearized augmented plane wave (LAPW) methodsingh , as implemented in the ELK
codeelk . We have used a $30\times 30$ k-point set; for further details of the
implementation of TD-DFT within the LAPW basis we refer the reader to Refs.
kriegerjctc,  and dewhurst15, .

Figure 2: Landau-Zener-Stückelberg interference in the first Brillouin zone
(BZ) reflected by 2D transient electron momentum density (tr-EMD) at various
time steps during and after the pulse. A pulse of central frequency 1.3 eV,
intensity $1.0\times 10^{13}$ W/cm2, full width half maximum 2.758 fs, and
carrier envelope phase $\pi/2$ is employed, with the $\bm{\mathrm{A}}$-field
exhibited and laser induced current exhibited in panel (a). The points on the
$\bm{\mathrm{A}}$-field curve indicate the times at which the tr-EMD is
evaluated, shown in panels (b-f). In these panels the full evolution of the
Landau-Zener-Stückelberg (LZS) interference can be seen, including both early
time $k_{x}<0$ (left hand side of the vertical BZ boundary line) conduction
band excitation, intense excitation at the pulse peak, panel (d), before the
development of the LZS interference fringes on the falling shoulder of the
pulse, panel (e), and the full time $k_{x}>0$ LZS interference.

LZS interference probed by 2D tr-EMD: the patterns of excited charge in
momentum space that most directly characterise Landau-Zener-Stückelberg
interference are generally presented by plotting the conduction band
occupation over the first Brillouin zone. However this information, while easy
to obtain theoretically, is difficult to obtain experimentally. We thus look
at an alternative quantity, the change in electron momentum density due to the
laser pulse.

In Fig. 1 are displayed the $N_{\rm ex}$, EMD, and induced currents for a
diverse set of laser pulses exhibiting variation of several pulse parameters:
carrier envelope phase (the angular difference between the
$\bm{\mathrm{E}}$-field and pulse envelope maxima), polarization, intensity,
frequency, and full width half maximum (FWHM). The magnitude of the electric
field is of the order of 5 V/nm, placing these pulses in the strong non-
perturbative regime for graphene. As can be seen, in all cases $N_{\rm ex}$
and EMD convey consistent information concerning the excited charge,
establishing the latter as a reliable probe of momentum space excitation.

Before exploring the LZS interference physics of graphene revealed in Fig. 1
we first provide a theoretical basis to this observed coincidence in the
pattern of momentum space excitation between $N_{\rm ex}$ and EMD. The Kohn-
Sham electron momentum density is defined as

$\rho(\bm{\mathrm{p}},t)=\sum_{j\bm{\mathrm{k}}}f_{j\bm{\mathrm{k}}}|\psi_{j\bm{\mathrm{k}}}(\bm{\mathrm{p}},t)|^{2}$
(4)

where

$\psi_{j\bm{\mathrm{k}}}(\bm{\mathrm{p}},t)=\int d\bm{\mathrm{p}}\
e^{i\bm{\mathrm{p}}\cdot\bm{\mathrm{r}}}\psi_{j\bm{\mathrm{k}}}(\bm{\mathrm{r}},t)$
(5)

is the Fourier transform of the KS wavefunction $\psi_{j\bm{\mathrm{k}}}$ and
$f_{j\bm{\mathrm{k}}}$ the occupation. Upon expansion of the Bloch functions
in plane waves of the reciprocal lattice vectors, $\bm{\mathrm{G}}$,

$\psi_{j\bm{\mathrm{k}}}(\bm{\mathrm{r}},t)=\sum_{\bm{\mathrm{G}}}c^{\bm{\mathrm{k}}}_{j\bm{\mathrm{G}}}(t)e^{i(\bm{\mathrm{k}}+\bm{\mathrm{G}})\cdot\bm{\mathrm{r}}}$
(6)

and insertion into Eqs. (4) and (5), we find that the EMD can be expressed as

$\rho(\bm{\mathrm{p}},t)=\sum_{j\bm{\mathrm{k}}}f_{j\bm{\mathrm{k}}}\sum_{\bm{\mathrm{G}}}|c^{\bm{\mathrm{k}}}_{j\bm{\mathrm{G}}}(t)|^{2}\delta(\bm{\mathrm{p}}-\bm{\mathrm{k}}-\bm{\mathrm{G}}).$
(7)

The EMD will therefore only change with respect to the ground-state EMD (see
Eq. (1)) at points $\bm{\mathrm{p}}=\bm{\mathrm{k}}+\bm{\mathrm{G}}$ where the
coefficients $c^{\bm{\mathrm{k}}}_{j\bm{\mathrm{G}}}(t)$ change. In
particular, in almost all systems, this will include the $\bm{\mathrm{G}}=0$
point, i.e. the $\bm{\mathrm{k}}$ point itself within the first Brillouin
zone. As the coefficient, $c^{\bm{\mathrm{k}}}_{j\bm{\mathrm{G}}=0}(t_{f})$,
will change (w.r.t the GS value) at points in k-space where
$N_{\text{ex}}(\bm{\mathrm{k}},t_{f})$ is non-zero,
$\Delta\rho(\bm{\mathrm{k}},t_{f})$ must also then be non-zero. Hence, any
interference pattern seen in $N_{\text{ex}}(\bm{\mathrm{k}},t_{f})$ will also
be seen in $\Delta\rho(\bm{\mathrm{k}},t_{f})$. For 2 electron systems, it is
known that the EMD produced from the KS wavefunction can differ significantly
from the exact EMDRHGM09 ; EM11 , however in periodic systems, it was shown
that the KS-EMD gives excellent agreement with Compton Scattering
profilesdugdale .

For a carrier envelope phase (CEP) of $\phi=\pm\pi/2$ the maximum
$\bm{\mathrm{E}}$-field intensity, and hence the interband transition at the
avoided crossing, occurs at turning point of the path in momentum space
executed due to the $\bm{\mathrm{A}}$-field. As a result, the LZ transitions
generate excited conduction band charge at either the positive ($\phi=+\pi/2$)
or negative ($\phi=-\pi/2$) $k_{x}$ sides of the Dirac point. This can be seen
in rows (a-c) and (h-j) of Fig. 1. Note that positive and negative $k_{x}$,
measured from the Dirac point, corresponds to the left and right hand sides of
the vertical BZ boundary as seen in Fig. 1. In contrast, for $\phi=0$ the
maximum $\bm{\mathrm{E}}$-field intensity occurs at
$\bm{\mathrm{A}}=\bm{\mathrm{0}}$ resulting in a symmetric excitation about
the Dirac point, see row (c-e). In the past such asymmetric LZS interference
has been indirectly accessed by means the net current that results from the
asymmetric momentum space occupation for non-zero CEP, and to date this
represents the only observation in experiment of LSZ in a materialHiguchi2017
. This coherent current (current per unit cell) induced by the laser pulse is
displayed in the third column of Figs. 1 and 2, and corresponds well with that
seen in experiment. The experimentally accessible EMD, however, provides a
wealth of additional information, as we now describe.

By comparing rows (g-i) and (m-o) we observe almost identical residual
coherent current, and yet a very different momentum space LZS excitation as
revealed by the EMD. In particular, in row (m-o) we observe a sub-dominant
$k_{x}<0$ charge excitation absent in row (g-i) and reflecting multiple passes
of the avoided crossing due to the side peaks of the former pulse, see panel
(i). The presence of the main and side peaks in the pulse structure allows for
multiple pass $\bm{\mathrm{k}}$-space trajectories which, due to the pulse
envelope, consist of a series of passes of the avoided crossing from
trajectories of different length in momentum space. This yields both
asymmetric occupation and more complex interference patterns. Further
enhancement of these side peaks, see row (s-u) of Fig. 1 for which the CEP is
again $\phi=+\pi/2$, results in well developed interference fringes both for
positive and negative $k_{x}$, quite different to the right hand side only
momentum space occupation seen for the lower frequency $\phi=+\pi/2$ pulse
shown in panels (g-i). The EMD thus represents a much more sensitive probe of
the LZS effect, able to unveil subtitles of the interference physics lost in
the residual current.

A striking example of this richness of information provided by EMD versus the
residual current can be found in rows (p-r). Here it can be seen that
widespread excitation occurs throughout the BZ driven by an intense pulse the
$\bm{\mathrm{A}}$-field of which drives trajectories right across the BZ
(indicated by the lines in the $N_{\rm ex}$/EMD panels). A very complex and
asymmetric LZS interference pattern results from this intense excitation,
however the widespread occupation of momentum space drives an overall
cancellation of current carrying states and a vanishingly small residual
current.

Experimentally, the short time coherent current ultimately generates heating
and a diffusive residual current. This can therefore not provide a real time
probe of the development of LSZ physics. Transient EMD, on the other hand,
potentially provides a real time probe of the ultrashort time evolution of LZS
interference patterns in momentum space. In Fig. 2 we show the transient EMD
evaluated before, during, and after a pump pulse inducing both a coherent
current and LZS interference. One can observe an early time excitation due to
pulse side peaks, panels (b) and (c), followed by a dramatic excitation in
momentum space at the maximum of the main peak, panel (d). Only after this
peak has passed do the final interference fringes develop, panels (e) and (f).

Figure 3: Time dependent partial density of states (PDOS) projected onto the
$l=1$ spherical harmonics. Here the PDOS (in states/atom/eV) is shown both at
$t=0$ before the pulse, and at the end of the silulation after the pulse has
been applied. The pump pulse for panels (a) and (b) is polarized in the
$x$-direction, with intensities $10^{12}$ W/cm2 and $10^{14}$ W/cm2
respectively. As can be seen, even for almost complete excitation of the
$\pi$-band in which charge is excited from the $\pi$-band minima up to the
$\pi^{\ast}$-band maxima, there is no excitation into states of $p_{x}$ or
$p_{y}$ character. (c) Band structure of graphene showing the $\pi$ and
$\sigma$ band character. Negative and positive numbers indicate dominance by
$\pi$\- and $\sigma$-character respectively.

Dominance of the $\pi$-manifold in electron dynamics: the results for the
momentum resolved conduction band occupation shown in the previous sections,
correspond very closely to results obtained on the basis of model $\pi$-band
only tight-binding Hamiltonians. This raises the question of whether this is
due simply to the relatively low energies of the excited charge (in Fig 1 and
Fig. 2 the excited charge resides predominantly at the K point and the K-M-K
line) or whether, for a more general reason, the $\pi$-band will always
dominate ultrafast laser induced electron dynamics in graphene. To explore
this in Fig. 3 we display the partial density of states calculated before and
after the laser pulse. As can be seen, see Fig. 3a, for the pulse of intensity
$10^{12}$ W/cm2, the partial DOS after the pulse shows conduction band
occupation only up to 2.5 eV. At these energies, see Fig. 3c, only the
$\pi$-band is available for excited charge. Remarkably, when we consider a
very strong pulse of intensity $10^{14}$ W/cm2 the excited electrons are again
only of $p_{z}$ character, Fig. 3b, despite the fact that the laser pulse is
sufficiently strong to excite charge from the minima of the $\pi$-manifold up
to the maxima of the $\pi^{\ast}$-manifold. As may be noted from the band
structure, Fig. 3c, within this energy range exist many other bands that
would, in principle, be expected to be involved in the electron dynamics at
such high energies. Examination of the relevant dipole matrix elements reveals
that transitions from $\pi$ to $\sigma^{\ast}$ and $\pi^{\ast}$ to $\sigma$
are negligible for laser pulses with in-plane polarization. Thus even in the
highly non-perturbative regime transitions from the ground state to the
$\sigma^{\ast}$ manifold will be strongly suppressed. It might be argued that
the partial DOS, a projection within (touching) muffin tins, does not account
for excitation to delocalized bands of high $l$ character. Comparison of the
interstitial density of states before and after the pulse shows that there is
indeed an increase in interstitial charge at around 9 eV, possibly indicating
transitions from the $\pi^{\ast}$ manifold to delocalized bands (note the
intersections between $\pi^{\ast}$ and high $l$ character bands on the
M-$\Gamma$ line), however this is a rather small effect. It would thus appear
that the model $\pi$-band only tight-binding Hamiltonians provide an excellent
description of the electron dynamics even for very intense laser pulses.

To summarize we have investigated _ab-initio_ the laser induced electron
dynamics in monolayer graphene. This system provides a canonical example of a
material for which Landau-Zener-Stückelberg interferometry can be explored,
and we have shown that direct visualisation of the interference fringes in
momentum space is possible via the transient electron momentum density (EMD),
establishing transient EMD as an excellent experimental tool for exploring LZS
interference in 2d materials. Examination of the excited state partial density
of states reveals that the $\pi$-band manifold decisively dominates ultrafast
laser induced dynamics in graphene, justifying the deployment of the popular
Hückel tight-binding model. Whether this remains true for the complex few
layer graphene systems, for which such an approach is the only one that can
reasonably be envisioned, remains an open question.

##  I Acknowledgements

QZL would like to thank DFG for funding through TRR227 (project A04). SS would
like to thank DFG for funding through SH498/4-1 and PE acknowledges funding
from DFG Eigene Stelle project 2059421. The authors acknowledge the North-
German Supercomputing Alliance (HLRN) for providing HPC resources that have
contributed to the research results reported in this paper.

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