[a]B.F.L. Ward

# Overview of theoretical precision of the luminosity at future electron-
positron colliders

S. Jadach  W. Placzek  M. Skrzypek  S.A. Yost

###### Abstract

For both the FCC-ee and the ILC, to exploit properly the respective precision
physics program, the theoretical precision tag on the respective luminosity
will need to be improved from the analogs of the $0.054\%(0.061\%)$ results at
LEP at $M_{Z}$, where the former (latter) LEP result has (does not have) the
pairs correction. At the FCC-ee at $M_{Z}$ one needs improvement to $0.01\%$,
for example. We present an overview of the roads one may take to reach the
required $0.01\%$ precision tag at the FCC-ee and of what the corresponding
precision expectations would be for the FCC-ee350, ILC500, ILC1000, and
CLIC3000 setups.

BU-HEPP-22-06, Nov., 2022

##  1 Introduction

The importance of a precision MC event generator for the luminosity process at
the FCC-ee and other future colliders has been emphasized in Refs. [1, 2, 3].
Compared to the situation at the time of LEP, there has already been
substantial progress. In what follows, we give an overview of the expectations
for the theoretical precision of the luminosity at future electron-positron
colliders with the current situation in mind.

Specifically, we show in Table 1 a summary of the progress to date relative to
the time of LEP.

Type of correction / Error | 1999 | Update 2019
---|---|---
(a) Photonic ${\cal O}(L_{e}\alpha^{2})$ | 0.027% [4] | 0.027%
(b) Photonic ${\cal O}(L_{e}^{3}\alpha^{3})$ | 0.015% [5] | 0.015%
(c) Vacuum polariz. | 0.040% [6, 7] | 0.011% [8, 9]
(d) Light pairs | 0.030% [10] | 0.010% [11, 12]
(e) $Z$ and $s$-channel $\gamma$ exchange | 0.015% [13, 14] | 0.015%
(f) Up-down interference | 0.0014% [15] | 0.0014%
(g) Technical Precision | – | (0.027)%
Total | 0.061% [16] | 0.037%
Table 1: Summary of the total (physical+technical) theoretical uncertainty
with BHLUMI [17] for a typical calorimetric LEP luminosity detector within the
generic angular range of $18$–$52$ mrad. Total error is summed in quadrature.

We see that, in spite of there being no really dedicated effort, considerable
progress has been made.

##  2 Current Situation, Related to LEP and BHLUMI Upgrade

In Ref. [18] we have given the detailed description of the steps one would
take on the path to improving the theoretical precision of BHLUMI the desired
0.01% precision tag needed for future colliders like the FCC-ee on the $Z^{0}$
resonance. For completeness, we recall these steps in Table 2 and we refer the
reader to Ref. [18]

Type of correction / Error | Update 2018 | FCC-ee forecast
---|---|---
(a) Photonic $[{\cal O}\left(L_{e}\alpha^{2}\right)]\;{\cal O}\left(L_{e}^{2}\alpha^{3}\right)$ | 0.027% | $0.1\times 10^{-4}$
(b) Photonic $[{\cal O}\left(L_{e}^{3}\alpha^{3}\right)]\;{\cal O}\left(L_{e}^{4}\alpha^{4}\right)$ | 0.015% | $0.6\times 10^{-5}$
(c) Vacuum polariz. | 0.014% [19] | $0.6\times 10^{-4}$
(d) Light pairs | 0.010% [20, 21] | $0.5\times 10^{-4}$
(e) $Z$ and $s$-channel $\gamma$ exchange | 0.090% [22] | $0.1\times 10^{-4}$
(f) Up-down interference | 0.009% [15] | $0.1\times 10^{-4}$
(g) Technical Precision | (0.027)% | $0.1\times 10^{-4}$
Total | 0.097% | $1.0\times 10^{-4}$
Table 2: Anticipated total (physical+technical) theoretical uncertainty for a
FCC-ee luminosity calorimetric detector with the angular range being
$64$–$86\,$mrad (narrow), near the $Z$ peak. Description of photonic
corrections in square brackets is related to the 2nd column. The total error
is summed in quadrature.

for their detailed discussion accordingly. Here, we note that, with sufficient
research support111We have to be realistic that, given the far off start times
for the future electron colliders, such research support may also be far off.,
the path to the 0.01% precision tag for BHLUMI is indeed an opened one.

As we illustrated in the Introduction, progress on the theoretical effort to
improve the luminosity theory precision at electron colliders is occurring
even though there is no really dedicated effort in that direction. We note the
recent results in Refs. [23, 24] which feature the exact NNLO correction to
the Bhabha scattering (the luminosity process at electron colliders) with
next-to-soft stabilization. These results should viewed relative to the
results in Refs. [4, 16, 17, 25, 26] which realize the exact ${\cal
O}(\alpha^{2}L)$ corrections with the exact ${\cal O}(\alpha^{2}L^{2})$
correction done with amplitude-based resummation via MC event generator
methods. To be more precise, if we write the NNLO cross section as

$\sigma^{(2)}=(\frac{\alpha}{\pi})^{2}L^{2}\sigma^{(2)}_{2}+(\frac{\alpha}{\pi})^{2}L\sigma^{(2)}_{1}+(\frac{\alpha}{\pi})^{2}\sigma^{(2)}_{0},$
(1)

then the results in Refs. [23, 24] realize $\sigma^{(2)}_{i},\;i=0,\;1,\;2,$
while those in Refs. [26, 16, 25, 17, 4] realize $\sigma^{(2)}_{i},\;i=1,\;2,$
exactly. We note that the relevant big logarithm is $L=\ln(|t|/m_{e}^{2})$ in
an obvious notation and that the constant term $\sigma^{(2)}_{0}$ enters at
the level of $(\frac{\alpha}{\pi})^{2}\cong 5.4\times 10^{-6}$. From the
standpoint of progress on cross-checks for precision theory in the Bhabha
process, the authors in Refs.[23, 24] have made a comparison with the results
from BABAYAGA [27, 28, 29]: for the $\phi$ factory type setup of
$\sqrt{s}=1020$ MeV, with $E_{min}=408$ MeV,
$20^{\circ}<\theta_{\pm}<160^{\circ}$ ($\theta_{\pm}$ are the respective
$e^{\pm}$ cms scattering angles and $E_{min}$ is their minimum energy.), and
the accolinearity $\zeta$ cut $\zeta_{max}=10^{\circ}$, the agreement is at
the level $0.07\%$. This is two orders of magnitude larger than the level at
which $\sigma^{(2)}_{0}$ enters as noted above and suggests that one of the
two calcautions may have unknown technical errors, as both are supposed to be
exact at NNLO.

The semi-soft approximation used in Refs. [23, 24] using the notation therein
can be isolated via

$\lim_{\xi\to 0}\xi^{2}{\cal M}^{(\ell)}_{n+1}={\cal E}{\cal
M}^{(\ell)}_{n}+\xi\Delta{\cal M}^{(\ell)}_{n+1}+\ldots,$ (2)

where ${\cal M}^{(\ell)}_{n}$ is the $\ell-\text{loop}$ matrix element squared
amplitude with n final state particles, ${\cal E}$ is the respective YFS [30]
exponent and $\xi=2E_{\gamma}/\sqrt{s}$ the scaled photon energy. The semi-
soft approximation keeps the $1/\xi$ non-universal term determined by
$\Delta{\cal M}^{(\ell)}_{n+1}$ in eq.(2). This should be compared with the
corresponding treatment of the analogous effects in Refs. [26, 16, 25, 17, 4].
When the exact ${\cal O}(\alpha^{2}L)$ correction in latter references is
implemented in BHLUMI, the $(\frac{\alpha}{\pi})^{2}$ term will be missing
from $\bar{\beta}^{(2)}_{1U}$ and $\bar{\beta}^{(2)}_{1L}$ in the cross
section for the process $e^{+}(p_{1})+e^{-}(q_{1})\rightarrow
e^{+}(p_{2})+e^{-}(q_{2})+\gamma(k_{1})+...+\gamma(k_{n})+\gamma(k^{\prime}_{1})+...+\gamma(k^{\prime}_{n^{\prime}})$
as presented in Ref. [31] which we reproduce here for completeness:

$\displaystyle\sigma^{(r)}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{n^{\prime}=0}^{\infty}{1\over
n!}{1\over n^{\prime}!}\int{d^{3}p_{2}\over p_{2}^{0}}\int{d^{3}q_{2}\over
q_{2}^{0}}\;\prod_{j=1}^{n}\;\int\limits_{k_{j}\notin\Omega_{U}}{d^{3}k_{j}\over
k^{0}_{j}}\tilde{S}_{p}(k_{j})\;\prod_{l=1}^{n^{\prime}}\;\int\limits_{k^{\prime}_{l}\notin\Omega_{L}}{d^{3}{k^{\prime}}_{l}\over{k^{\prime}}^{0}_{l}}\tilde{S}_{q}(k^{\prime}_{l})$
$\displaystyle\delta^{(4)}\bigg{(}p_{1}-p_{2}+q_{1}-q_{2}-\sum_{j=1}^{n}k_{j}-\sum_{l=1}^{n^{\prime}}{k^{\prime}}_{l}\bigg{)}\;e^{Y_{p}(\Omega_{U})+Y_{q}(\Omega_{L})}$
$\displaystyle\Bigg{\\{}\bar{\beta}^{(r)}_{0}+\sum_{j=1}^{n}{\bar{\beta}^{(r)}_{1U}(k_{j})\over\tilde{S}_{p}(k_{j})}+\sum_{l=1}^{n^{\prime}}{\bar{\beta}^{(r)}_{1L}(k^{\prime}_{l})\over\tilde{S}_{q}(k^{\prime}_{l})}+\sum_{n\geq
j>k\geq 1}\;\
{\bar{\beta}^{(r)}_{2UU}(k_{j},k_{k})\over\tilde{S}_{p}(k_{j})\tilde{S}_{p}(k_{k})}$
$\displaystyle\qquad\qquad+\sum_{n^{\prime}\geq l>m\geq 1}\;\
{\bar{\beta}^{(r)}_{2LL}(k_{l},k_{m})\over\tilde{S}_{q}(k^{\prime}_{l})\tilde{S}_{q}(k^{\prime}_{m})}+\sum_{j=1}^{n}\sum_{l=1}^{n^{\prime}}\;{\bar{\beta}^{(r)}_{2UL}(k_{j},k^{\prime}_{l})\over\tilde{S}_{p}(k_{j})\tilde{S}_{q}(k^{\prime}_{l})}\Bigg{\\}}.$
(3)

where we refer the reader to Ref. [31] for the definitions of the respective
IR functions and constructs. There is no semi-soft approximation in the
approach in eq.(3).

##  3 Current Situation, Higher Energies

In Ref. [32] we have extended our analysis in Ref. [18] to include the
expectations for the luminosity theory error for higher energy future electron
colliders: FCCee350, ILC500, ILC1000, and CLIC3000, where the collider cms
energy in GeV is indicated by its subscript. We see in Table 3 that the key
variable is the geometric mean momentum transfer
$\bar{t}\equiv\sqrt{t_{min}t_{max}}$ for the respective acceptances for the
attendant cms energies using an obvious notation.

Generalizing [32] our methods first to the proposed 500 GeV ILC and
subsequently to the FCCee350, ILC1000, and CLIC3000, we get the analog of
Table 2 for the ILC500 which we show here in Table 4

as well as forecasts for the FCCee350, ILC1000, and CLIC3000 which we show
here in Table 5.

Of course, with appropriate resources, these results for the higher energies
could all be improved as it is indicated in Ref. [32].

We conclude with the following observations. The BHLUMI team, with appropriate
resources, can improve it to meet the physics requirements of the planned
future electron colliders. The far off nature of the colliders raises a
legitimate question on the timing of these resources. It is a question that
has yet to be answered.

## Acknowledgments

This work was partially supported by the National Science Centre, Poland Grant
No. 2019/34/E/
ST2/00457 and by The Citadel Foundation. We thank Dr. Patrick Janot for
helpful discussion. Two of us (SJ and BFLW) thank Prof. Gian Giudice for the
support and kind hospitality of the CERN TH Department.

## References

  * [1] A. Blondel et al., CERN Yellow Reports: Monographs, CERN-2020-003 (2020) 1905.05078.
  * [2] A. Blondel et al., CERN Yellow Rep. Monogr. 3 (2019) (2019) 1809.01830.
  * [3] A. Blondel and P. Janot, 1912.11871.
  * [4] S. Jadach et al., Acta Phys. Polon. B30 (1999) 1745 – 1750.
  * [5] S. Jadach and B. F. L. Ward, Phys. Lett. B389 (1996) 129–136.
  * [6] H. Burkhardt and B. Pietrzyk, Z. Phys. C 67 (1995) 585 – 602.
  * [7] S. Eidelman and F. Jegerlehner, Phys. Lett. B. 356 (1995) 398 – 403\.
  * [8] F. Jegerlehner, CERN Yellow Reports: Monographs, eds. A. Blondel et al., CERN-2020-003 (2020) 9, 1905.05078.
  * [9] P. Janot and S. Jadach, Phys. Lett. B 803 (2020) 135319, 1912.02067.
  * [10] S. Jadach, M. Skrzypek, and B. F. L. Ward, Phys. Rev. D 55 (1997) 1206 – 1215.
  * [11] G. Montagna et al., Nucl. Phys. B 547 (1999) 49 – 59.
  * [12] G. Montagna et al., Phys. Lett. B 459 (1999) 649 – 652.
  * [13] S. Jadach, W. Placzek, and B. Ward, Phys. Lett. B 353 (1995) 349 – 361.
  * [14] A. Arbuzov et al., Phys. Lett. B383 (1996) 238–242, hep-ph/9605239.
  * [15] S. Jadach, E. Richter-Wa̧s, B. F. L. Ward, and Z. Wa̧s, Phys. Lett. B253 (1991) 469 – 477.
  * [16] S. Jadach, M. Melles, B. F. L. Ward, and S. A. Yost, Phys. Lett. B450 (1999) 262–266.
  * [17] S. Jadach et al., Comput. Phys. Commun. 102 (1997) 229\.
  * [18] S. Jadach, W. Placzek, M. Skrzypek, B. F. L. Ward, and S. A. Yost, Phys. Lett. B 790 (2018) 314.
  * [19] F. Jegerlehner, talk in FCC-ee Mini-Workshop, Physics Behind Precision, https://indico.cern.ch/event/469561/contributions /1977974/attachments/1221704/1786449/ SMalphaFCCee16.pdf.
  * [20] G. Montagna et al., Nucl. Phys. B 547 (1999) 39 – 59, hep-ph/9811436.
  * [21] G. Montagna et al., Phys. Lett. B459 (1999) 649 – 652, hep-ph/9905235.
  * [22] S. Jadach, W. Placzek, and B. Ward, Phys. Lett. B 353 (1995) 349 – 361.
  * [23] P. Banerjee et al., Phys. Lett. B 820 (2021) 136547, 2106.07469.
  * [24] T. Engel et al., in Initial state QED radiation aspects of $e^{+}e^{-}$ colliders, eds. S. Frixione and E. Laenen (2022) 2203.12557.
  * [25] S. Jadach, E. Richter-Wa̧s, B. F. L. Ward, and Z. Wa̧s, Comput. Phys. Commun. 70 (1992) 305\.
  * [26] S. Jadach et al., Phys. Lett. B377 (1996) 168–176.
  * [27] G. Balossini, C. M. Carloni-Calame, G. Montagna, O. Nicrosini, and F. Piccinini, Nucl. Phys. B 758 (2006) 227 – 253, hep-ph/0607181.
  * [28] C. M. Carloni-Calame et al., J. High Energy Phys. 07 (2011) 126, hep-ph/1106.3178.
  * [29] C. M. Carloni-Calame et al., EPJ Web Conf. PHIPSI17 218 (2019) 07004.
  * [30] D. R. Yennie, S. Frautschi, and H. Suura, Ann. Phys. (NY) 13 (1961) 379 – 452.
  * [31] S. Jadach et al., Phys. Lett. B353 (1995) 362–372.
  * [32] S. Jadach et al., Eur. J. Phys. C 81 (2021) 1047.


# Chirped amplitude mode in photo-excited superconductors

Thomas Blommel  Department of Physics, University of Michigan, Michigan, USA
Jason Kaye  Center for Computational Quantum Physics, Flatiron Institute, 162
5th Avenue, New York, NY 10010, USA  Center for Computational Mathematics,
Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA  Yuta Murakami
Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
Emanuel Gull  Department of Physics, University of Michigan, Michigan, USA
Denis Golež  Jožef Stefan Institute, SI-1000, Ljubljana, Slovenia  Faculty of
Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia

###### Abstract

We show that the amplitude mode in superconductors exhibits chirped
oscillations under resonant excitation and that the chirping velocity
increases as we approach the critical excitation strength. The chirped
amplitude mode enables us to determine the local modification of the effective
potential even when the system is in a long-lived pre-thermal state. We then
show that this chirped amplitude mode is an experimentally observable quantity
since the photo-induced (super)-current in pump-probe experiments serves as an
efficient proxy for the dynamics of the order parameter, including the chirped
dynamics. Our result is based on the attractive Hubbard model using dynamical
mean-field theory within the symmetry-broken state after a resonant excitation
across the superconducting gap. Since the collective response takes place on
emergently long timescales, we extend the hierarchical low-rank compression
method for nonequilibrium Green’s functions to symmetry-broken states and show
that it serves as an efficient representation despite long-lived memory
kernels.

Introduction– Soon after the seminal paper on BCS superconductivity [1],
Anderson pointed out that the electromagnetic response of a superconductor
leads to a collective response [2, 3, 4]. This realization had a profound
impact on condensed matter physics, as well as particle physics, with the
prediction of the Higgs mode [5, 6]. It took almost two decades before such an
excitation was observed in 2H-NbSe2 with the coexistence of superconductor and
charge-density-wave [7, 8, 9, 10] phases, and another three decades for direct
observation using terahertz spectroscopy either in a pump-probe setup [11] or
with third-harmonics generation [12, 13].

This progress opened the field of Higgs spectroscopy, which has now been
applied to numerous superconducting materials, including superconductors with
running supercurrents [14, 15] and unconventional superconductors [16, 17].
Ever more precise measurements of collective mode response enables the study
of their long-time behavior after excitation both in solids [11] as well as
quantum simulators [18, 19, 20].

The theoretical description of the the time evolution of the Higgs mode is
difficult. Time-dependent Landau-Ginzburg theory assumes an ad-hoc rapid
modification of the effective potential and damping is considered on the
phenomenological level [21, 22]. Microscopic time-dependent BCS theory
predicts richer dynamics, including a power-law decay [23, 24] or solitonic
solution [25, 26, 27]. However, all these theoretical statements assume
dissipationless electronic dynamics, which is questionable in real materials
and opens up a question on the decay of collective modes upon the inclusion of
electronic scattering.

The main difficulty of the theoretical description of electronic scattering on
collective mode dynamics stems from the fact that collective response takes
place on emergently long time scales. This has limited previous studies to
dissipationless dynamics [23, 24, 25, 26, 27], or when fluctuations were
included, the description was limited to relatively short times with few
coherent oscillations and limited information about amplitude mode lifetime
[28, 29, 30, 31].

Only recently, progress in numerical solvers for nonequilibrium Green’s
functions [32], including the generalized Kadanoff-Baym ansatz [33, 34, 35,
36] and memory kernel truncation methods [37, 38], has allowed access to long
enough times to examine these questions. However, the application of these
techniques to symmetry-broken phases is much less understood due to the long-
lived power-law correlators acting as memory kernels.

In this work, we study the long-time evolution of collective modes by solving
the full Kadanoff-Baym equations (KBE), and show that the order parameter
displays an extremely slow approach to a thermal state. Within such a
nonthermal state, the amplitude mode exhibits chirping (dynamical decrease in
the frequency). The velocity of the chirping increases as we approach the
nonthermal critical excitation strength at which the order parameter vanishes
in the long time limit. We show that the effect is experimentally accessible
in tera-hertz pump-probe experiments by directly measuring the probe-induced
currents, which can be used to extract the dynamics of the order parameter
with a small imprint of the quasiparticle response.

Figure 1: Time evolution of the order parameter $\phi$ after photoexcitation
with pump amplitude $E_{0}$ in the (a) weak excitation regime $E_{0}<E_{0c}$,
in which amplitude mode oscillations are present, and (b) the strong
excitation regime $E_{0}\geq E_{0c}$, in which we observe superexponential
decay. The noise at small values in (b) is consistent with the truncation
tolerance $10^{-8}$ used in the HODLR compression scheme. (c) The system
exponentially approaches the expected thermal values (thick bars on right-hand
side of (a)). $E_{0}=0.08,0.085$ are integrated further in time to emphasize
the exponential decay.

We also highlight our use of the hierarchical off-diagonal low-rank (HODLR)
compression method for the numerical solution of the KBE [39]. The standard
$\mathcal{O}\left(N^{3}\right)$ scaling of the direct solution of the KBE with
the number of time steps $N$, resulting from memory terms, typically prevents
access to the time scales necessary to analyze the lifetime of collective
excitations [32, 28, 29, 30]. A new class of data compression-based methods,
including the HODLR approach, have recently been proposed to overcome this
bottleneck [40, 41, 42, 43, 44]. Viewing two-time Green’s functions as
matrices, the HODLR method decomposes these matrices into blocks, refined
towards the diagonal, which in many cases have been observed to be numerically
low-rank. A truncated singular value decomposition of these blocks is
systematically updated on-the-fly during time-stepping and used to compute
history integrals with controllable accuracy, yielding an
$\mathcal{O}\left(k^{2}N^{2}\log N\right)$ computational complexity and
$\mathcal{O}\left(kN\log N\right)$ memory complexity, for $k$ the maximum
block rank, without modifying the underlying KBE. Using this method, we
demonstrate a 100-fold decrease in the computational cost required to
propagate to the time scales studied compared with direct time stepping, with
50-200 times less memory.

Model and method– We study superconductivity within the attractive Hubbard
model

$H=-t_{0}\sum_{\langle
j,k\rangle\sigma}e^{iqA(t)}d^{\dagger}_{j\sigma}d_{k\sigma}-U\sum_{j}(n_{j\uparrow}-1/2)(n_{j\downarrow}-1/2),$
(1)

where $d_{i\sigma}$ is the annihilation operator at site $i$ and spin
$\sigma$, $n_{i\sigma}$ is the spin-dependent density operator and $q$ is the
charge. We set the energy to the hopping amplitude $t_{0}=1$, the Coulomb
attraction to $U=2$, and fix the occupation at half filling.

We solve the problem within time-dependent dynamical mean-field theory (DMFT)
on the Bethe lattice and explicitly break the superconducting symmetry. This
leads to matrix-valued Green’s functions
$G_{\alpha\beta}(t,t^{\prime})=-i\langle\mathcal{T}\psi_{\alpha}(t)\psi^{\dagger}_{\beta}(t^{\prime})\rangle,$
with the Nambu spinor $\psi=\\{d_{\uparrow},d^{\dagger}_{\downarrow}\\}$. For
the impurity solver, we use the self-consistent second-order perturbation
theory known as second Born, or GF2 [45, 46, 47]. Collective order parameter
dynamics take place on long time scales, and such long-time integration of the
KBE is made possible by using a compressed representation of the Green’s
function to reduce both the computational and memory complexity [39]. We refer
to the appendices for further implementation details.

We induce a dynamical perturbation by a short pump pulse introduced by a
Peierls substitution in Eq. 1. The electric field $E_{p}=-\partial_{t}A$ is
parameterized as

$E_{P}(t)=E_{0}\,\text{exp}\left(-\frac{(t-t_{c})^{2}}{\sigma^{2}}\right)\text{sin}(\omega(t-t_{c})),$
(2)

with pulse center $t_{c}$, pulse width $\sigma$, driving frequency $\omega$,
and amplitude $E_{0}$.

Figure 2: (a-d): Windowed Fourier transform $F[\phi](t_{w})$ of the order
parameter centered around $t_{w}$ with exponential background subtracted, for
pump amplitudes $E_{0}=0.02$, $0.06$, $0.08$, and $0.1$. The red line is a
linear fit to the maximum. (e) Slope of the maxima (chirping velocity), shown
by red lines, for different pump intensities (blue) with the best fit
(orange).

Results– For an interaction strength of $U=2$ the attractive Hubbard model in
equilibrium has a phase transition at inverse temperature
$1/T_{c}=\beta_{c}\approx 10.6$ within the GF2 approximation. Across this
transition, the system develops a nonzero superconducting order parameter
$\phi(t)=\sum_{k}\langle c_{k\downarrow}c_{-k\uparrow}\rangle$. We choose a
temperature $\beta=18$ deep in the ordered state, and find that the system has
a band gap of $\Delta\approx 0.65$.

To excite the amplitude mode, we pump the system with a pulse tuned to twice
the band gap ($t_{c}=16$, $\sigma=6.5$, $\omega=2\Delta$). The time evolution
of the order parameter is presented in Fig. 1. At weak excitation strength,
the order parameter decays at an extremely slow exponential rate [48, 49]. The
expected thermal states, marked by the horizontal lines, are not reached on
the time scale of our simulation; see also Fig. 1(c), in which the thermal
values are subtracted. For intermediate pump strengths, we observe the decay
of the order parameter through several orders of magnitude. On top of the
exponential relaxation, well-defined oscillations correspond to the amplitude
mode excitation [50, 10]. As the pump strength is increased, the initial
amplitude of these oscillations increases; however, they also have shorter
lifetimes. Increasing the pump amplitude further, we reach a critical point at
which these oscillations disappear. In this regime, the order parameter goes
to zero in the long-time limit, see Fig. 1(b), and transiently critical
behaviour of the order parameter takes place [48, 51, 52].

We observe that the frequency of the collective mode oscillations gradually
chirps to smaller values. To extract the chirping velocity, we first subtract
the background exponential decay from the data in Fig. 1, and then compute a
windowed Fourier transform
$\text{FT}[\phi](t_{w})=\int_{0}^{T_{max}}dt\,\phi(t)e^{-i\omega
t-(t-t_{w})^{6}/\sigma^{6}_{w}}$. We define the chirping velocity
$\dot{\omega}$ as the slope of the maximum of $\text{FT}[\phi](t_{w})$, which
is tracked in Fig. 2(a-d) for increasing pump amplitudes. Fig. 2(e) shows that
the chirping velocity follows a power-law scaling $\dot{\omega}\propto
E_{0}^{3.41}$ with the pump amplitude, slightly faster than the scaling
$j\propto E_{0}^{3}$ of the current due to photo-induced amplitude mode
oscillations [24, 50].

Whereas the dissipationless BCS dynamics lead to power-law decay of the
oscillation amplitude [23], the description based on the attractive Hubbard
model includes fluctuations, and exhibits exponential decay. This is
consistent with the effect of thermal noise on the dissipationless BCS
dynamics [25, 53]. The lifetime of the amplitude mode decays with increasing
pump strength (see Fig. 2(a-d)), and changes by more than an order of
magnitude from the weak to the strong excitation regime.

The chirping of the amplitude mode is the main result of this work. We
emphasize that observing such phenomena requires microscopic treatment of
fluctuations beyond the mean-field theory, in a full nonequilibrium setup, as
the resulting state is highly nonthermal. Moreover, the chirping phenomena is
not limited to superconductors, but should be observed in any system with
broken symmetry, like magnetic [54] or charge density waves [55, 56]. In
actual experimental setups, there is a trade-off in pump strength between the
chirping velocity and the lifetime of the amplitude mode, which depends on the
details of the pulse and the material studied.

Now, we interpret the microscopic dynamics within the time-dependent Landau-
Ginzburg theory. The chirping of the amplitude mode corresponds to a dynamical
reduction of the local curvature due to the feedback of the electronic
scattering on the collective order. While it would be tempting to interpret
the chirping as a heating effect within the Landau-Ginzburg theory, we
emphasize that the system is closed and in a prethermal regime, which the
temperature change cannot describe. In principle, one could construct a time-
dependent Landau-Ginzburg theory of chirping. However, it is difficult in
practice to extract the evolution of global structure of the Landau-Ginzburg
potential, even when microscopic dynamics are available. An important
consequence of our analysis is that the chirped frequency can act as a
microscopic measure of the dynamically modified local curvature.

Figure 3: Current induced by (a) Gaussian probe pulse $j_{g}$ with parameters
$E_{0}=0.1E^{\text{Pump}}_{0}$, $t_{c}=t^{\text{Pump}}_{c}+\tau$, $\sigma=1$,
and (b) monocycle probe pulse $j_{s}$ with parameters
$E_{0}=0.1E^{\text{Pump}}_{0}$, $t_{c}=t^{\text{Pump}}_{c}+\tau$, $\sigma=1$,
$\omega=1$. (c) Comparison of the order parameter oscillations with subtracted
background $\tilde{\phi}(t)$ with two separate quantities: the maximum current
response $\text{max}(j_{g})$ induced by Gaussian pulse, and the integral
$\hat{j}_{s}(\omega=0)$ over the current, for pump amplitudes $E_{0}=0.08$ (c)
and $E_{0}=0.04$ (d).

We next discuss how this physics could be observed experimentally. The
amplitude mode does not couple linearly to light; however, it can be observed
in nonlinear processes, such as pump-probe setups [11] or the third-harmonics
response [12, 50, 24]. Our numerical tests agree with previous observations
that it is difficult to extract the amplitude mode signal from the optical
response due to the overlap with the quasiparticle continuum [30]. However, we
observe that the photo-induced current serves as an excellent proxy for the
dynamics of the order parameter. While in principle the optical conductivity
and photo-induced current carry similar information, we found that in practice
it is much easier to extract coherent oscillations from the time-dependent
current, compared with spectral analysis of the optical conductivity.

We evaluate the probe-induced current by directly simulating an additional
probe pulse at a time $\tau$ after the pump pulse, and measuring the induced
current [57, 58, 59]. This approach is theoretically convenient as it includes
vertex corrections to the response function, which represents the contribution
from the collective modes [60, 61]. We propose two probe protocols and show
that the current response in both cases could be connected with the collective
dynamics of the amplitude mode. In the first setup, we model the probe as a
small Gaussian pulse, with amplitude one-tenth that of the pump pulse. We
measure the current response to the probe, $j_{g}(t)$, shown in Fig. 3(a),
where each line corresponds to the same probe applied at different values of
$\tau$. Figs. 3(c,d) show the first local minima versus the delay time $\tau$.
We see that the current tracks the oscillations of the order parameter for
both pump strengths considered, giving information about both the chirped
frequency and the lifetime of the amplitude mode.

Since measuring the fast current response might be experimentally difficult,
we show that the time-integrated current also tracks the order parameter. For
this case, we focus on a monocycle pulse parameterized by Eq. 2, with
parameters $\sigma=1$, $t_{c}=16+\tau$, $\omega=1$, and an amplitude one-tenth
that of the pump pulse. Here, since the electric field integrates to zero, the
current decays to zero after some time. Fig. 3(b) shows the current for this
experimental setup. The green data points in Figs. 3(c,d) correspond to the
integral of the current, and also closely track the order parameter.

Based on these results, we propose that the direct measurement of the photo-
induced current is a convenient way to measure amplitude mode response. The
photo-induced current and chirping can be measured in the pump-probe setup
using either free-space electro-optic sampling, like in Ref. [11], or
transmission line experiments, which were recently used to detect
superconducting nonlinear transport [62], the light-induced anomalous quantum
Hall effect due to circularly polarized light [63], and ultrafast resistive
switching in 1T-TaS2 [64].

Figure 4: (a) Wall clock time and (b) compression factor with increasing
number of timesteps for HODLR propagation scheme versus direct time stepping,
for various pump intensities $I$ starting from the equilibrium state at
$\beta=18$.

Since our analysis was enabled by the HODLR compression technique for the
numerical solution of the KBE [39], we consider the compressibility of the
dynamics of symmetry-broken states, and the performance of the scheme compared
with the direct solution of KBE. In Fig. 4(a), we observe approximately
quadratic scaling of the computational cost with the propagation time over a
range of pump intensities and equilibrium state temperatures. We reach 32000
time steps in roughly one day on a single-core of a workstation using 0.5 GB
of memory. By contrast, we extrapolate that cubic-scaling direct time stepping
using the NESSi code [32] would have taken 129 days with 137 GB of memory.
Fig. 4(b) shows a factor $50$-$200$ memory compression of the Green’s function
and self-energy compared with direct storage on a two-time grid. The Appendix
considers the compressibility in more detail.

We have shown that the collective amplitude mode displays chirped dynamics
after the above gap excitation, and have analyzed a direct experimental
signature of the response based on the photo-induced current within the pump-
probe setup. Future applications include the nonlinear response of
unconventional superconductors [16, 17], and extensions beyond dynamical mean-
field theory to study spatial fluctuations [65] and the build-up of the
fluctuating order in systems with competing orders [66, 67]. Methodologically,
this work represents the first application of the HODLR compression technique
for the KBE to symmetry-broken states, and demonstrates that it can access
long enough time scales for the practical study of a broad field of photo-
induced phase transitions [68, 69, 70, 71].

###### Acknowledgements.

DG acknowledges the support of the projects J1-2463, N1-0318, MN-0016-106, and
P1-0044 program of the Slovenian Research Agency. TB was funded by the
Department of Energy via Grant No. DE-SC0020347 until Aug. 2023. TB as of Aug.
2023 and EG were supported by the U.S. Department of Energy, Office of
Science, Office of Advanced Scientific Computing Research and Office of Basic
Energy Sciences, Scientific Discovery through Advanced Computing (SciDAC)
program under Award Number(s) DE-SC0022088. YM is supported by Grant-in-Aid
for Scientific Research from JSPS, KAKENHI Grant Nos. JP20K14412, JP21H05017
and JST CREST Grant No. JPMJCR1901. The Flatiron Institute is a division of
the Simons Foundation.

## References

  * Bardeen  _et al._ [1957] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
  * Anderson [1958a] P. W. Anderson, Phys. Rev. 110, 827 (1958a).
  * Anderson [1958b] P. W. Anderson, Phys. Rev. 112, 1900 (1958b).
  * Anderson [1963] P. W. Anderson, Phys. Rev. 130, 439 (1963).
  * Higgs [1964a] P. Higgs, Physics Letters 12, 132–133 (1964a).
  * Higgs [1964b] P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964b).
  * Sooryakumar and Klein [1980] R. Sooryakumar and M. V. Klein, Phys. Rev. Lett. 45, 660 (1980).
  * Littlewood and Varma [1981] P. B. Littlewood and C. M. Varma, Phys. Rev. Lett. 47, 811 (1981).
  * Littlewood and Varma [1982] P. B. Littlewood and C. M. Varma, Phys. Rev. B 26, 4883 (1982).
  * Pekker and Varma [2015] D. Pekker and C. Varma, Annual Review of Condensed Matter Physics 6, 269 (2015), https://doi.org/10.1146/annurev-conmatphys-031214-014350 .
  * Matsunaga  _et al._ [2013] R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang, and R. Shimano, Phys. Rev. Lett. 111, 057002 (2013).
  * Matsunaga  _et al._ [2014] R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka, K. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki, and R. Shimano, Science 345, 1145 (2014).
  * Matsunaga  _et al._ [2017] R. Matsunaga, N. Tsuji, K. Makise, H. Terai, H. Aoki, and R. Shimano, Phys. Rev. B 96, 020505 (2017).
  * Moor  _et al._ [2017] A. Moor, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett. 118, 047001 (2017).
  * Nakamura  _et al._ [2019] S. Nakamura, Y. Iida, Y. Murotani, R. Matsunaga, H. Terai, and R. Shimano, Phys. Rev. Lett. 122, 257001 (2019).
  * Katsumi  _et al._ [2018] K. Katsumi, N. Tsuji, Y. I. Hamada, R. Matsunaga, J. Schneeloch, R. D. Zhong, G. D. Gu, H. Aoki, Y. Gallais, and R. Shimano, Phys. Rev. Lett. 120, 117001 (2018).
  * Chu  _et al._ [2020] H. Chu, M.-J. Kim, K. Katsumi, S. Kovalev, R. D. Dawson, L. Schwarz, N. Yoshikawa, G. Kim, D. Putzky, Z. Z. Li, _et al._ , Nature communications 11, 1793 (2020).
  * Young  _et al._ [2024] D. J. Young, A. Chu, E. Y. Song, D. Barberena, D. Wellnitz, Z. Niu, V. M. Schäfer, R. J. Lewis-Swan, A. M. Rey, and J. K. Thompson, Nature 625, 679–684 (2024).
  * Lewis-Swan  _et al._ [2021] R. J. Lewis-Swan, D. Barberena, J. R. K. Cline, D. J. Young, J. K. Thompson, and A. M. Rey, Phys. Rev. Lett. 126, 173601 (2021).
  * Behrle  _et al._ [2018] A. Behrle, T. Harrison, J. Kombe, K. Gao, M. Link, J.-S. Bernier, C. Kollath, and M. Köhl, Nature Physics 14, 781 (2018).
  * Aranson and Kramer [2002] I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 (2002).
  * Zong  _et al._ [2019a] A. Zong, P. E. Dolgirev, A. Kogar, E. Ergeçen, M. B. Yilmaz, Y.-Q. Bie, T. Rohwer, I.-C. Tung, J. Straquadine, X. Wang, Y. Yang, X. Shen, R. Li, J. Yang, S. Park, M. C. Hoffmann, B. K. Ofori-Okai, M. E. Kozina, H. Wen, X. Wang, I. R. Fisher, P. Jarillo-Herrero, and N. Gedik, Phys. Rev. Lett. 123, 097601 (2019a).
  * Volkov and Kogan [1973] A. F. Volkov and S. M. Kogan, Zh. Eksp. Theor. Fiz. 65, 2038 (1973).
  * Tsuji and Aoki [2015] N. Tsuji and H. Aoki, Phys. Rev. B 92, 064508 (2015).
  * Barankov  _et al._ [2004] R. A. Barankov, L. S. Levitov, and B. Z. Spivak, Phys. Rev. Lett. 93, 160401 (2004).
  * Yuzbashyan  _et al._ [2006] E. A. Yuzbashyan, O. Tsyplyatyev, and B. L. Altshuler, Phys. Rev. Lett. 96, 097005 (2006).
  * Barankov and Levitov [2006] R. A. Barankov and L. S. Levitov, Phys. Rev. Lett. 96, 230403 (2006).
  * Kemper  _et al._ [2017] A. F. Kemper, M. A. Sentef, B. Moritz, T. P. Devereaux, and J. K. Freericks, Annalen der Physik 529, 1600235 (2017).
  * Nosarzewski  _et al._ [2017] B. Nosarzewski, B. Moritz, J. K. Freericks, A. F. Kemper, and T. P. Devereaux, Phys. Rev. B 96, 184518 (2017).
  * Kumar and Kemper [2019] A. Kumar and A. F. Kemper, Phys. Rev. B 100, 174515 (2019).
  * Murakami  _et al._ [2016a] Y. Murakami, P. Werner, N. Tsuji, and H. Aoki, Phys. Rev. B 94, 115126 (2016a).
  * Schüler  _et al._ [2020] M. Schüler, D. Golez, Y. Murakami, N. Bittner, A. Herrmann, H. U. Strand, P. Werner, and M. Eckstein, Computer Physics Communications 257, 107484 (2020).
  * Lipavský  _et al._ [1986] P. Lipavský, V. Špička, and B. Velický, Phys. Rev. B 34, 6933 (1986).
  * Tuovinen  _et al._ [2020] R. Tuovinen, D. Golež, M. Eckstein, and M. A. Sentef, Phys. Rev. B 102, 115157 (2020).
  * Schlünzen  _et al._ [2017] N. Schlünzen, J.-P. Joost, F. Heidrich-Meisner, and M. Bonitz, Phys. Rev. B 95, 165139 (2017).
  * Pavlyukh  _et al._ [2022] Y. Pavlyukh, E. Perfetto, D. Karlsson, R. van Leeuwen, and G. Stefanucci, Physical Review B 105, 125134 (2022).
  * Schüler  _et al._ [2018] M. Schüler, M. Eckstein, and P. Werner, Phys. Rev. B 97, 245129 (2018).
  * Stahl  _et al._ [2022] C. Stahl, N. Dasari, J. Li, A. Picano, P. Werner, and M. Eckstein, Physical Review B 105, 115146 (2022).
  * Kaye and Golež [2021] J. Kaye and D. Golež, SciPost Phys. 10, 091 (2021).
  * Yin  _et al._ [2022] J. Yin, Y. hao Chan, F. H. da Jornada, D. Y. Qiu, S. G. Louie, and C. Yang, J. Comput. Sci. 64, 101843 (2022).
  * Reeves  _et al._ [2023] C. C. Reeves, J. Yin, Y. Zhu, K. Z. Ibrahim, C. Yang, and V. c. v. Vlček, Phys. Rev. B 107, 075107 (2023).
  * Shinaoka  _et al._ [2023] H. Shinaoka, M. Wallerberger, Y. Murakami, K. Nogaki, R. Sakurai, P. Werner, and A. Kauch, Phys. Rev. X 13, 021015 (2023).
  * Murray  _et al._ [2023] M. Murray, H. Shinaoka, and P. Werner, arXiv preprint arXiv:2312.03809 (2023).
  * Meirinhos  _et al._ [2022] F. Meirinhos, M. Kajan, J. Kroha, and T. Bode, SciPost Phys. Core 5, 030 (2022).
  * van Leeuwen and Dahlen [2006] R. van Leeuwen and N. E. Dahlen, An introduction to nonequilibrium green functions (2006).
  * Dong  _et al._ [2022] X. Dong, E. Gull, and H. U. R. Strand, Phys. Rev. B 106, 125153 (2022).
  * Rusakov and Zgid [2016] A. A. Rusakov and D. Zgid, The Journal of Chemical Physics 144, 054106 (2016), https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.4940900/14800495/054106_1_online.pdf .
  * Tsuji  _et al._ [2013] N. Tsuji, M. Eckstein, and P. Werner, Phys. Rev. Lett. 110, 136404 (2013).
  * Werner  _et al._ [2012] P. Werner, N. Tsuji, and M. Eckstein, Phys. Rev. B 86, 205101 (2012).
  * Shimano and Tsuji [2020] R. Shimano and N. Tsuji, Annual Review of Condensed Matter Physics 11, 103 (2020), https://doi.org/10.1146/annurev-conmatphys-031119-050813 .
  * Golež  _et al._ [2016] D. Golež, P. Werner, and M. Eckstein, Phys. Rev. B 94, 035121 (2016).
  * Picano and Eckstein [2021] A. Picano and M. Eckstein, Phys. Rev. B 103, 165118 (2021).
  * Murakami  _et al._ [2016b] Y. Murakami, P. Werner, N. Tsuji, and H. Aoki, Phys. Rev. B 94, 115126 (2016b).
  * Hong  _et al._ [2017] T. Hong, M. Matsumoto, Y. Qiu, W. Chen, T. R. Gentile, S. Watson, F. F. Awwadi, M. M. Turnbull, S. E. Dissanayake, H. Agrawal, _et al._ , Nature Physics 13, 638 (2017).
  * Demsar  _et al._ [1999] J. Demsar, K. Biljaković, and D. Mihailovic, Phys. Rev. Lett. 83, 800 (1999).
  * Yusupov  _et al._ [2010] R. Yusupov, T. Mertelj, V. V. Kabanov, S. Brazovskii, P. Kusar, J.-H. Chu, I. R. Fisher, and D. Mihailovic, Nature Physics 6, 681 (2010).
  * Lenarčič  _et al._ [2014] Z. Lenarčič, D. Golež, J. Bonča, and P. Prelovšek, Phys. Rev. B 89, 125123 (2014).
  * Shao  _et al._ [2016] C. Shao, T. Tohyama, H.-G. Luo, and H. Lu, Phys. Rev. B 93, 195144 (2016).
  * Eckstein and Kollar [2008] M. Eckstein and M. Kollar, Phys. Rev. B 78, 205119 (2008).
  * Murakami  _et al._ [2016c] Y. Murakami, P. Werner, N. Tsuji, and H. Aoki, Phys. Rev. B 93, 094509 (2016c).
  * Golež  _et al._ [2020] D. Golež, Z. Sun, Y. Murakami, A. Georges, and A. J. Millis, Phys. Rev. Lett. 125, 257601 (2020).
  * Wang  _et al._ [2023] E. Wang, J. Adelinia, M. Chavez-Cervantes, T. Matsuyama, M. Fechner, M. Buzzi, G. Meier, and A. Cavalleri, Nature Communications 14, 7233 (2023).
  * McIver  _et al._ [2020] J. W. McIver, B. Schulte, F.-U. Stein, T. Matsuyama, G. Jotzu, G. Meier, and A. Cavalleri, Nature Physics 16, 38 (2020).
  * Venturini  _et al._ [2022] R. Venturini, A. Mraz, I. Vaskivskyi, Y. Vaskivskyi, D. Svetin, T. Mertelj, L. Pavlovič, J. Cheng, G. Chen, P. Amarasinghe, _et al._ , Applied Physics Letters 120 (2022).
  * Stahl and Eckstein [2021] C. Stahl and M. Eckstein, Phys. Rev. B 103, 035116 (2021).
  * Sun and Millis [2020] Z. Sun and A. J. Millis, Phys. Rev. X 10, 021028 (2020).
  * Zong  _et al._ [2019b] A. Zong, A. Kogar, Y.-Q. Bie, T. Rohwer, C. Lee, E. Baldini, E. Ergeçen, M. B. Yilmaz, B. Freelon, E. J. Sie, H. Zhou, J. Straquadine, P. Walmsley, P. E. Dolgirev, A. V. Rozhkov, I. R. Fisher, P. Jarillo-Herrero, B. V. Fine, and N. Gedik, Nature Physics 15, 27 (2019b).
  * Nasu [2004] K. Nasu,  _Photoinduced Phase Transitions_ (WORLD SCIENTIFIC, 2004).
  * Murakami  _et al._ [2023] Y. Murakami, D. Golež, M. Eckstein, and P. Werner, Photo-induced nonequilibrium states in mott insulators (2023), arXiv:2310.05201 [cond-mat.str-el] .
  * de la Torre  _et al._ [2021] A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W. McIver, and M. A. Sentef, Rev. Mod. Phys. 93, 041002 (2021).
  * Giannetti  _et al._ [2016] C. Giannetti, M. Capone, D. Fausti, M. Fabrizio, F. Parmigiani, and D. Mihailovic, Advances in Physics 65, 58 (2016), https://doi.org/10.1080/00018732.2016.1194044 .
  * Kaye  _et al._ [2022a] J. Kaye, K. Chen, and O. Parcollet, Phys. Rev. B 105, 235115 (2022a).
  * Kaye  _et al._ [2022b] J. Kaye, K. Chen, and H. U. R. Strand, Comput. Phys. Commun. 280, 108458 (2022b).
  * Kaye and Strand [2023] J. Kaye and H. U. R. Strand, Adv. Comput. Math. 49, 10.1007/s10444-023-10067-7 (2023).

##  Appendix A Implementation details

The attractive Hubbard Hamiltonian in Eq. (1) can be rewritten as

$\displaystyle H$ $\displaystyle=-t_{0}\sum_{\langle j,k\rangle\sigma}\sigma
e^{iqA(t)\sigma}\psi^{\dagger}_{j\sigma}\psi_{k\sigma}$ (3)
$\displaystyle+U\sum_{i}\psi^{\dagger}_{i\downarrow}\psi_{i\downarrow}\psi^{\dagger}_{i\uparrow}\psi_{i\uparrow}-\frac{U}{2}\sum_{i\sigma}\psi^{\dagger}_{i\sigma}\psi_{i\sigma},$

where we have used the Nambu spinors

$\displaystyle\psi_{i}=\begin{pmatrix}d_{i\uparrow}\\\
d^{\dagger}_{i\downarrow}\end{pmatrix}.$ (4)

We again use the Nambu spinors to define the anomalous Green’s function

$G(t,t^{\prime})=\langle\psi(t)\psi^{\dagger}(t^{\prime})\rangle=\begin{pmatrix}\langle
d_{\uparrow}d^{\dagger}_{\uparrow}\rangle(t,t^{\prime})&&\langle
d_{\uparrow}d_{\downarrow}\rangle(t,t^{\prime})\\\ \langle
d^{\dagger}_{\downarrow}d^{\dagger}_{\uparrow}\rangle(t,t^{\prime})&&\langle
d^{\dagger}_{\downarrow}d_{\downarrow}\rangle(t,t^{\prime})\end{pmatrix},$ (5)

which is the quantity we compute by solving the KBE.

We solve this system within the DMFT approximation on a Bethe lattice, giving
the hybridization function

$\displaystyle\Delta(t,t^{\prime})=\Delta_{R}(t,t^{\prime})+\Delta_{L}(t,t^{\prime})$
(6)
$\displaystyle\Delta_{R}(t,t^{\prime})=\frac{1}{2}\bar{t}_{0}(t)\sigma_{z}G(t,t^{\prime})\sigma_{z}\bar{t}_{0}^{*}(t^{\prime})$
(7)
$\displaystyle\Delta_{L}(t,t^{\prime})=\frac{1}{2}\bar{t}_{0}^{*}(t)\sigma_{z}G(t,t^{\prime})\sigma_{z}\bar{t}_{0}(t^{\prime}),$
(8)

with hopping matrix elements given by

$\displaystyle\bar{t}_{0}$ $\displaystyle=\begin{pmatrix}e^{i\phi(t)}&&0\\\
0&&e^{-i\phi(t)}\end{pmatrix},$ (9) $\displaystyle\phi(t)$
$\displaystyle=\int_{0}^{t}E(s)ds$ (10)

due to the Peierls substitution.

The impurity problem is solved using the fully self-consistent second Born
approximation. In this approximation, the Fock term is given by

$\Sigma^{F}_{ij}(t)=iU(t)G^{<}_{ij}(t,t^{-})\delta_{i\bar{j}},$ (11)

and the Hartree term cancels the chemical potential since we are at half-
filling. There are two second-order diagrams:

$\displaystyle\Sigma^{B}_{ij}(t,t^{\prime})$
$\displaystyle=U(t)U(t^{\prime})G_{ij}(t,t^{\prime})G_{\bar{i}\bar{j}}(t,t^{\prime})G_{ij}(t^{\prime},t),$
(12) $\displaystyle\Sigma^{E}_{ij}(t,t^{\prime})$
$\displaystyle=-U(t)U(t^{\prime})G_{\bar{i}j}(t,t^{\prime})G_{i\bar{j}}(t,t^{\prime})G_{\bar{i}\bar{j}}(t^{\prime},t).$
(13)

The hybridization $\Delta[G](t,t^{\prime})$ and the self-energy
$\Sigma[G](t,t^{\prime})$ depend on the Green’s function, and we maintain
self-consistency using fixed-point iteration terminating when the maximum
difference of the Green’s function at subsequent iterates is less than
$10^{-10}$.

We use the fifth-order backward differentiation formula multistep method to
discretize the KBE. Gregory quadrature is used to discretize memory integrals
to fifth-order accuracy. The discrete Lehmann representation [72], implemented
in libdlr [73], is used to discretize the imaginary time variables appearing
in the vertical leg of the Keldysh contour, as in Ref. [74]. We have confirmed
that the compression of the Green’s function using the HODLR scheme preserves
conservation laws, namely the density and energy, to within the SVD truncation
tolerance used in the scheme, which we set to $10^{-8}$.

In the pump-probe experiments, we measure the current, given by

$\displaystyle j(t)$
$\displaystyle=\text{Im}(\text{Tr}[\sigma_{z}(\Gamma_{L}(t)-\Gamma_{R}(t))])$
(14) $\displaystyle\Gamma_{R/L}(t)$
$\displaystyle=-i[G\ast\Delta_{L/R}]^{<}(t,t),$ (15)

with $\ast$ the convolution operator on the three-legged Keldysh contour.

##  Appendix B Numerical ranks in HODLR scheme

Figure 5: (a) Rank of the truncated SVD of the largest block in a HODLR
decompostion of Green’s function at timestep $T$. Results for several
different pump strengths are presented. $\beta=6$ is a result in the
equilibrium disordered state. (b) Total memory required to store Green’s
function and self-energy for the presented pump strengths.

Our time propagation scheme compresses the Green’s function using a HODLR
decomposition [39]. A truncated SVD is performed on each block of the
decomposition, and updated on-the-fly, with singular values below $10^{-10}$
discarded. The numerical ranks of these blocks (within this precision)
determine the compressibility of the Green’s functions, and the performance of
the scheme.

Fig. 5(a) shows the growth of the maximum block rank $k$ with the number of
time steps. At inverse temperature $\beta=6$, the system is in a disordered
state, and the Green’s functions decay exponentially on their off-diagonal.
This leads to a rapid saturation of the ranks, giving an
$\mathcal{O}\left(N^{2}\log N\right)$ computational complexity of the
algorithm, and an $\mathcal{O}\left(N\log N\right)$ memory complexity, with
$N$ the number of time steps. The behavior is similar for the simulations in
which a low-temperature state is excited with a strong pump (i.e.
$E_{0}=0.15$), destroying the superconducting state. At low temperature, in
the ordered state, although the blocks are still numerically low rank with
$k\ll N$, they grow as $k\propto\sqrt{N}$, leading to an
$\mathcal{O}\left(N^{3}\log N\right)$ computational complexity and
$\mathcal{O}\left(N^{3/2}\log N\right)$ memory complexity. Nevertheless, while
the memory usage, shown in Fig. 5(b), reflects the expected scaling, the wall
clock time shown in Fig. 4 reflects approximately
$\mathcal{O}\left(N^{2}\right)$ scaling, suggesting that lower-scaling steps
of the algorithm dominate throughout the time scale of this simulation.


# Asymptotics of the conformal modulus of a nonsymmetric unbounded doubly-
connected domain under stretching

Giang V. Nguyen  Kazan Federal University, Kremlyovskaya str. 35, Tatarstan,
420008, Russian Federation<EMAIL_ADDRESS>and  Semen R. Nasyrov
Kazan Federal University, Kremlyovskaya str. 35, Tatarstan, 420008, Russian
Federation<EMAIL_ADDRESS>
###### Abstract.

We describe the asymptotic behavior of the conformal modulus of an unbounded
doubly-connected domain, non-symmetric with respect to the coordinate axes,
when stretched in the direction of the abscissa axis with coefficient
$H\to+\infty$. Therefore, we give a partial answer to a problem, suggested by
M. Vourinen, for the case of an arbitrary unbounded domain.

###### Key words and phrases:

Conformal modulus, doubly-connected domain, quadrilateral, quasiconformal
mapping, convergence of domains to a kernel.

##  1\. Introduction

###  1.1. The conformal modulus of a doubly-connected domain

Let $D$ be a doubly-connected planar domain with nondegenerate boundary
components, then $D$ is conformally equivalent to an annulus $\\{r<|z|<R\\}$
(see, e.g., [6, Chap. V, § 1], [17, Sect. 3]), and its conformal modulus
$m(D)$ is the number defined as follows:

$m(D):=\dfrac{1}{2\pi}\log\dfrac{R}{r}\,.$

Another approach to define the conformal modulus is in using the concept of
extremal length of curve family (see, e.g., [1, Chap. 1, Sect. D], [17, Chap.
3, Sect. 11]). Specifically,

$m(D)=\lambda(\Gamma)$

where $\lambda(\Gamma)$ is the extremal length of the family $\Gamma$ of
curves in the domain $D$ which join the boundary components of $D$. Moreover,

$m(D)={1}/{\lambda(\Gamma^{\prime})}$

where $\Gamma^{\prime}$ is the family of curves in $D$ separating its boundary
components.

Conformal moduli of doubly-connected domains are also closely related to the
conformal moduli of quadrilaterals (see, e.g., [1, 17, 16]). A quadrilateral
$(Q;z_{1},z_{2},z_{3},z_{4})$ is a Jordan domain $Q$ with four fixed points
(vertices) $z_{k}$, $1\leq k\leq 4$, on its boundary; the increasing of the
index $k$ corresponds to the positive bypass of the boundary $\partial Q$.

For a given quadrilateral $\mbox{\boldmath$Q$}=(Q;z_{1},z_{2},z_{3},z_{4})$,
there exists a conformal mapping of $Q$ onto a quadrilateral
$(\Pi;0,1,1+hi,hi)$, where $\Pi=[0,1]\times[0,h]$, $h>0$, keeping the
vertices. By definition, the conformal modulus $m(\mbox{\boldmath$Q$})$ of $Q$
is equal to $h$. The conformal modulus $m(\mbox{\boldmath$Q$})$ coincides with
the extremal length of the family of curves connecting in $Q$ the boundary
arcs $z_{1}z_{2}$ and $z_{3}z_{4}$; it is also the inverse to the extremal
length of the family of curves in $Q$ joining the arcs $z_{2}z_{3}$ and
$z_{4}z_{1}$. The quadrilateral
$\mbox{\boldmath$Q$}^{*}=(Q;z_{2},z_{3},z_{4},z_{1})$ is called conjugate to
$Q$. It is clear that
$m(\mbox{\boldmath$Q$}^{*})=m^{-1}(\mbox{\boldmath$Q$})$. If
$\mbox{\boldmath$Q$}^{c}=(Q^{c};z_{4},z_{3},z_{2},z_{1})$, where
$Q^{c}=\overline{\mathbb{C}}\setminus Q$, then the conformal modulus of
$\mbox{\boldmath$Q$}^{c}$ is called the exterior modulus (see, e.g., [20]).

Over the years, there have been obtained many results on the conformal moduli
of doubly-connected domains, as well as on moduli of quadrilaterals (see,
e.g., [2, 3, 4, 7, 8, 9, 17, 11, 12, 13, 14, 15, 20]).

Conformal moduli of doubly-connected domains and quadrilaterals are
conformally invariant. They are also quasiinvariant under quasiconformal
mappings (see, e.g., [1, Chap. 2, Sect. A], [17, Chap. 3, Sect. 22]). If $f$
is an $H$-quasiconformal mapping, then

$\dfrac{1}{H}m(\Omega)\leq m(f(\Omega))\leq Hm(\Omega)$

where $\Omega$ is a doubly-connected domain. The same is valid for conformal
moduli of quadrilaterals.

Prof. M. Vuorinen posed the problem of investigating the distortion of the
conformal moduli of doubly-connected domains under the action of the
$H$-quasiconformal mappings,

(1) $f_{H}:x+iy\mapsto Hx+iy\quad(H>1);$

that is the stretching along the abscissa axis with coefficient $H$. In
particular, it is interesting to study its asymptotic behavior as
$H\to+\infty$ (see, e.g., [13]).

In [2], [3], [13], [14] the results on solving the Vuorinen problem were
obtained for the cases where $D$ is a rectangular frame, a symmetric or non-
symmetric bounded doubly-connected domain or unbounded doubly-connected
domains, symmetric with respect to the coordinate axes. Here we study the
problem by Vuorinen for the case of an unbounded doubly-connected domain of
sufficiently arbitrary form, non-symmetric with respect to the coordinate
axes.

###  1.2. The main result

Let $\mathfrak{S}$ be the set of all domains $\Omega$ which satisfy the
following conditions:

(i) $\Omega$ is an unbounded doubly-connected domain;

(ii) the boundary of $\Omega$ consists of the graphs of continuous functions:
functions $g_{1}$, $f_{1}$ are given on a segment $[a,b]$ and $f_{1}(x)\leq
g_{1}(x)$, $x\in[a,b]$; the functions $g_{2}$, $f_{2}$ are given on a segment
$[c,d]\subset[a,b]$ and $g_{2}(x)\leq f_{2}(x)<f_{1}(x)$, $x\in[c,d]$;
moreover, $f_{1}(a)=g_{1}(a),f_{1}(b)=g_{1}(b)$;
$f_{2}(c)=g_{2}(c),f_{2}(d)=g_{2}(d)$ (see. Fig. 1).

Figure 1. A domain $\Omega$ from the set $\mathfrak{S}$.

We will describe the behavior of the conformal modulus of the domain
$\Omega_{H}$ obtained from $\Omega\in\mathfrak{S}$ by the stretching $f_{H}$,
given by (1), as $H\to+\infty$.

######  Theorem 1.1.

If $\Omega\in\mathfrak{S}$ and $\Omega_{H}=f_{H}(\Omega)$, then

(2) $m(\Omega_{H})\sim\dfrac{1}{\gamma H}\,,\quad H\to\infty,\quad\mbox{\rm
where }\ \gamma=\int\limits_{c}^{d}\dfrac{dx}{f_{1}(x)-f_{2}(x)}\,.$

Moreover,

(3) $m(\Omega_{H})\leq\dfrac{1}{\gamma H}\quad\mbox{\rm for every}\quad H>0.$

##  2\. Local versions of the Radó convergence theorem

In this section we will obtain a local version of the Radó convergence theorem
and its generalization for a sequence of doubly-connected domains with
degenerate moduli (see, e.g., [3, Theorem 3]); these results will be used in
the proof of Theorem 1.1.

As it is well known, every conformal mapping $f$ between Jordan domains can be
extended, by continuity, up to a homeomorphism of the closures of the domains
(see, e.g., [6, Chap. 2, Sect. 3]); for simplicity of notation, we will also
denote this homeomorphism by $f$.

Firstly, we recall the classical Radó’s theorem on uniform convergence of
conformal mappings. Let $E$ denote the unit disk $\\{|z|<1\\}$ (see, e.g., [6,
Chap. 2, Sect. 5]).

######  Theorem 2.1 (Radó).

Let $D_{n}$ be a sequence of Jordan domains, bounded by curves $\Lambda_{n}$,
which converges to a kernel with respect to a point $z_{0}$, and let this
kernel be a Jordan domain $D$ bounded by a curve $\Lambda$. Then a sequence of
continuous functions $f_{n}:\overline{E}\to\overline{D}_{n}$, mapping
conformally ${E}$ onto $D_{n}$ and normalized by the conditions
$f_{n}(0)=z_{0}$, $f_{n}^{\prime}(0)>0$, converges uniformly to the continuous
function $f\,:\,\overline{E}\to\overline{D}$, $f(0)=z_{0},f^{\prime}(0)>0$
mapping conformally $E$ onto $D$ if and only if there exist homeomorphisms
between the curves $\Lambda_{n}$ and $\Lambda$ such that, for any
$\varepsilon>0$, there is an $N$ such that the distance between the respective
points of the curves $\Lambda_{n}$ and $\Lambda$ is smaller than $\varepsilon$
whenever $n\geq N$.

######  Remark 2.1.

It is easy to prove that, under the assumptions of Theorem 2.1,
$f^{-1}_{n}\rightrightarrows f^{-1}$ on every compact $K$ such that
$K\subset\overline{D}_{n}$ for sufficiently large $n$. In particular,
$f^{-1}_{n}\rightrightarrows f^{-1}$ on
$\bigcap_{n=1}^{\infty}\overline{D}_{n}$.

First we will establish a local version of Theorem 2.1.

Figure 2. The sequence of domains $D_{n}$ and its kernel.

######  Theorem 2.2.

Assume that $D_{n}$ is a sequence of Jordan domains, bounded by curves
$\Gamma_{n}$, which converges to a kernel with respect to a point $z_{0}$, and
this kernel is a Jordan domain $D$ bounded by a curve $\Gamma$. Let
$f_{n}:\overline{E}\to\overline{D}_{n}$ be a sequence of continuous functions,
mapping conformally ${E}$ onto $D_{n}$ and normalized by the conditions
$f_{n}(0)=z_{0}$, $f_{n}^{\prime}(0)>0,$ and $f:\overline{E}\to\overline{D}$
be a continuous function mapping conformally ${E}$ onto $D$ and satisfying the
conditions $f(0)=z_{0}$, $f^{\prime}(0)>0$. Let there exist a sequence of
subarcs $\gamma_{n}\subset\Gamma_{n}$ with endpoints $a_{n}$, $b_{n}$, a
sequence of continuous mappings $\psi_{n}:\gamma_{n}\to\Gamma$ and a point
$w_{0}\in\Gamma$ such that:

$(i)$ for every $n\geq 1$, $\psi_{n}$ is a homeomorphism of
$\gamma_{n}\setminus\\{a_{n},b_{n}\\}$ onto $\Gamma\setminus\\{w_{0}\\}$;

$(ii)$ for every $n\geq 1$, $\psi_{n}(a_{n})=\psi_{n}(b_{n})=w_{0}$;

$(iii)$ for every $\varepsilon>0$ there exists a number $N\geq 0$ such that

$\left|\psi_{n}(z)-z\right|<\varepsilon\quad\forall n\geq N\ \forall
z\in\gamma_{n};$

$(iv)$ there exists a sequence of crosscuts $\alpha_{n}$ of domains $D_{n}$
with endpoints $a_{n}$ and $b_{n}$, diameters of which tend to $0$ as
$n\to\infty$.

Then $f_{n}$ converges locally uniformly to $f$ on
$\overline{E}\setminus\\{\zeta_{0}\\}$, $\zeta_{0}=f^{-1}(w_{0})$, and
$f^{-1}_{n}$ converges uniformly to $f^{-1}$ on every compact subset of
$\bigcap_{n=1}^{\infty}\overline{D}_{n}\setminus\\{w_{0}\\}$.

###### Proof.

I) It is clear that the sequence of Jordan domains $D^{\prime}_{n}$, bounded
with $\gamma_{n}\cup\alpha_{n}$, converges to $D$ as to a kernel with respect
to $z_{0}$ (Fig. 2). Without loss of generality, we can assume that $z_{0}\in
D_{n}$, $n\geq 1$. Consider a sequence of conformal mappings

$\widetilde{f}_{n}:E\to D^{\prime}_{n}$

with normalization $\widetilde{f}_{n}(0)=z_{0}$,
$\widetilde{f}_{n}^{\prime}(0)>0$.

######  Lemma 2.1.

The sequence $\widetilde{f}_{n}$ converges uniformly to $f$ on the closed unit
disk $\overline{E}$.

###### Proof.

Using the conditions of the theorem, we can easily show that there exist
homeomorphisms

$\widetilde{\psi}_{n}:\alpha_{n}\cup\gamma_{n}\to\Gamma,$

satisfying the Radó condition, i.e. such that for any $\varepsilon>0$ there
exists $N\geq 0$ such that

$|\widetilde{\psi}_{n}(z)-z|<\varepsilon,\quad z\in\alpha_{n}\cup\gamma_{n},$

whenever $n\geq N$. By Theorem 2.1, we conclude that $\widetilde{f}_{n}$
converges uniformly to $f$ on $\overline{E}$. ∎

II) Here we will show that the preimages of the arcs $\alpha_{n}$ under the
mappings $\widetilde{f}_{n}$ are arcs on the unit circle which are contracted
to a point as $n\to\infty$.

According to the principle of the boundary correspondence under conformal
mappings, every $\beta_{n}:=\widetilde{f}_{n}^{-1}(\alpha_{n})$ is an arc on
the unit circle, with the end points
$\zeta_{n1}^{a}:=\widetilde{f}_{n}^{-1}(a_{n})$ and
$\zeta_{n1}^{b}:=\widetilde{f}_{n}^{-1}(b_{n})$. Similarly, the points
$\zeta^{a}_{n2}:=f^{-1}_{n}(a_{n})$ and $\zeta^{b}_{n2}:=f^{-1}_{n}(b_{n})$
are also on the boundaries of the domains $f^{-1}_{n}(D^{\prime}_{n})$.

######  Lemma 2.2.

The diameters of $\beta_{n}$ tend to $0$ as $n\to\infty$. Moreover, these arcs
are contracted to the point $\zeta_{0}$, as $n\to\infty$.

###### Proof.

First we recall that by (iv) the diameters of the arcs $\alpha_{n}$ tend to 0,
and by (ii) and (iii) their endpoints $a_{n}$ and $b_{n}$ converge to $w_{0}$.
Therefore, $\alpha_{n}$ are contracted to the point $w_{0}$, as $n\to\infty$.
Assume that $\mbox{\rm diam}(\beta_{n})\not\to 0$, as $n\to\infty$. Then there
exists a subsequence $\beta_{n_{k}}$ of the sequence $\beta_{n}$ such that the
$\beta_{n_{k}}$ tends to some nondegenerate subarc $\beta$ of the unit circle.
By Lemma 2.1, we have $\widetilde{f}_{n_{k}}\rightrightarrows f$ on
$\overline{E}$, therefore, $f\equiv w_{0}$ on $\beta$. By the boundary
uniqueness theorem (see, e.g., [6, Chap. 2, Sect. 3, Lemma 3]) this implies
that $f\equiv w_{0}$ on the unit disk. The contradiction proves that
$\mbox{\rm diam}(\beta_{n})\to 0$, as $n\to\infty$.

If $\beta_{n}$ are not contracted to the point $\zeta_{0}$, then there exist a
subsequence $\beta_{n_{k}}$ of $\beta_{n}$ converging to some
$\zeta_{1}\neq\zeta_{0}$. But $\widetilde{f}_{n_{k}}$ converges uniformly to
$f$ on the closed unit disk. For every sequence
$\zeta_{n_{k}}\in\beta_{n_{k}}$ we have $\zeta_{n_{k}}\to\zeta_{1}$ and
$f_{n_{k}}(\zeta_{n_{k}})\to w_{0}$, $k\to\infty$, since
$f_{n_{k}}(\zeta_{n_{k}})\in\alpha_{n_{k}}$. But, because of uniform
convergence, $f(\zeta_{1})=\lim_{k\to\infty}f_{n_{k}}(\zeta_{n_{k}})=w_{0}$,
therefore, $\zeta_{1}=\zeta_{0}$. The contradiction proves the lemma. ∎

III) Consider the arcs $\sigma_{n}:=f_{n}^{-1}(\alpha_{n})$. Denote by
$\widetilde{D}_{n}$ the extended complex plane with the excluded arc
$\beta_{n}$. Let $\widetilde{\sigma}_{n}$ be the arc symmetric to $\sigma_{n}$
with respect to the unit circle. Let $\widetilde{G}_{n}$ be the unbounded
domain in the extended complex plane with the boundary
$\sigma_{n}\cup\widetilde{\sigma}_{n}$.

######  Lemma 2.3.

The sequences of domains $\widetilde{D}_{n}$ and $\widetilde{G}_{n}$ converge
to the domain $\overline{\mathbb{C}}\setminus\\{\zeta_{0}\\}$, as to a kernel,
as $n\to\infty$. Moreover, for every $\delta>0$ the sequences
$f_{n}^{-1}\circ\widetilde{f}_{n}$ and $\widetilde{f}_{n}^{-1}\circ{f}_{n}$
converge locally uniformly to the identity mapping on the set
$\overline{E}\setminus\\{\zeta_{0}\\}$.

###### Proof.

Since, by Lemma 2.2, the arcs $\beta_{n}$ are contracted into the point
$\zeta_{0}$, as $n\to+\infty$, we see that $\widetilde{D}_{n}$ converges to
$\overline{\mathbb{C}}\setminus\\{\zeta_{0}\\}$.

Denote $g_{n}:=f_{n}^{-1}\circ\widetilde{f}_{n}$. By the symmetry principle
[10, Chap. X, Sect. 1], we can extend the mapping $g_{n}$ through the arc of
the unit disk, complement to $\beta_{n}$, in $\widetilde{D}_{n}$. The extended
mapping, which we also, for simplicity of notation, denote by $g_{n}$, maps
$\widetilde{D}_{n}$ onto $\widetilde{G}_{n}$. On the other hand, the family of
the functions $g_{n}$ is a normal family in
$\overline{\mathbb{C}}\setminus\\{\zeta_{0}\\}$. Actually, $g_{n}$ is bounded
in $E$, therefore, in $E$ it forms a normal family. This follows that in
$\overline{\mathbb{C}}\setminus\overline{E}$ it is also normal. Thus, we only
need to prove that $g_{n}$ is normal in a small neighborhood of every point
$\omega\in\partial E$ distinct from $\zeta_{0}$. Let us choose a neighborhood
$U$ of $\omega$ not containing the points $\zeta_{0}$, $0$ and $\infty$. Then
in $U$ every $g_{n}$ does not take the values $0$, $\infty$ and some
$e^{i\phi_{n}}\in\partial E$. Consequently, $e^{-i\phi_{n}}g_{n}$ does not
take three fixed values $0$, $1$ and $\infty$. By Montel’s theorem (see, e.g.,
[6, Chap. 2, Sect. 7], [18, Chap. 7, Sect. 2], [19, Chap. 2, Sect. 2]),
$e^{-i\phi_{n}}g_{n}$ is a normal family in $U$ and there exists a subsequence
$e^{-i\phi_{n_{k}}}g_{n_{k}}$ converging locally uniformly in $U$ to some
limit function $h$. But $\partial E$ is a compact set, therefore, passing, if
necessary, to a subsequence, we can assume that the sequence $e^{i\phi_{n}}$
converges to some $e^{i\phi_{0}}\in\partial E$ in $U$. Therefore, $g_{n_{k}}$
converges to $e^{i\phi_{0}}h$. This means that the family $g_{n}$ is normal.

Consider any subsequence $g_{n_{k}}$ converging locally uniformly to some $g$.
Since $g_{n_{k}}(0)=0$, we see that $g(0)=0$. If $g\equiv\mbox{\rm const}$,
then $g\equiv 0$, but from the symmetry of $g_{n}$ we see that if
$g_{n_{k}}(z)\to 0$, then $g_{n_{k}}(1/\overline{z})\to\infty$. The
controversy proves that $g\not\equiv\mbox{\rm const}$. Then $g$ is a univalent
function in $\overline{\mathbb{C}}\setminus\\{\zeta_{0}\\}$ as a nonconstant
limit of univalent functions (see, e.g., [6, Chap. 1, Sect. 2]), and $g(0)=0$,
$g^{\prime}(0)>0$. Moreover, $g(1/\overline{z})=1/\overline{g(z)}$. This
conditions imply that $g(z)\equiv z$,
$z\in\overline{\mathbb{C}}\setminus\\{\zeta_{0}\\}$. Therefore, by the
generalized Carathéodory convergence theorem (see, e.g., [6, Chap. 5, Sect.
5]), $\widetilde{G}_{n}$ converges to
$\overline{\mathbb{C}}\setminus\\{\zeta_{0}\\}$ as $n\to\infty$. By the same
theorem, we deduce that the sequence of inverse functions
$\widetilde{f}_{n}^{-1}\circ{f}_{n}$ converges locally uniformly in
$\overline{\mathbb{C}}\setminus\\{\zeta_{0}\\}$ to the identity mapping. ∎

Now we will complete the proof of Theorem 2.2.

From Lemmas 2.1 and 2.3 we obtain that
$f_{n}=\widetilde{f}_{n}\circ(\widetilde{f}_{n}^{-1}\circ{f}_{n})$ converges
locally uniformly to $f$ in $\overline{E}\setminus\\{\zeta_{0}\\}$. Now we
will prove that $f^{-1}_{n}$ converges uniformly to $f^{-1}$ on every compact
subset $K$ of $\bigcap_{n=1}^{\infty}\overline{D}_{n}\setminus\\{w_{0}\\}$.
Assume the opposite. Then there exist $\varepsilon>0$ and a subsequence of
points $z_{n_{k}}\in K$ such that

(4) $|f^{-1}_{n_{k}}(z_{n_{k}})-f^{-1}(z_{n_{k}})|\geq\varepsilon,\quad k\geq
1.$

Denote $\zeta_{n_{k}}=f^{-1}(z_{n_{k}})$. Since $\zeta_{n_{k}}\in\overline{E}$
and $\overline{E}$ is compact, we may assume that $\zeta_{n_{k}}$ converges to
some $\zeta^{*}\in\overline{E}$. Then $z_{n_{k}}=f(\zeta_{n_{k}})\to
f(\zeta^{*})=:z^{*}$. It is cleat that $z^{*}\in K$, therefore, $z^{*}\neq
w_{0}$. Consequently, $\zeta^{*}\neq\zeta_{0}$. From the uniform convergence
of $f^{-1}_{n_{k}}$ to $f^{-1}$ on $K$ it follows that

$f^{-1}_{n_{k}}(z_{n_{k}})=f^{-1}_{n_{k}}\circ f(\zeta_{n_{k}})\to\zeta^{*}.$

Thus, we have $f^{-1}_{n_{k}}(z_{n_{k}})\to f^{-1}(z^{*})=\zeta^{*}$, as
$k\to\infty$, but this contradicts to (4).

Therefore, Theorem 2.2 is fully verified. ∎

Now we will state a theorem which is a local version of Radó type theorem
proven in [3, Theorem 3]. In the cited theorem the case of convergence of
univalent functions defined on a sequence of doubly-connected domains
converging to a simply-connected domain $G$ is considered. Here we investigate
the case where the uniform convergence is violated at some point.

Let $\mathcal{D}_{n}$ be a sequence of non-concentric annuli, each of which is
bounded by the unit circle $\\{|\zeta|=1\\}$ and the circle
$\\{|\zeta+ir_{n}/2|=r_{n}/2\\}$, where $0<r_{n}<1$ and
$\lim\limits_{n\to\infty}r_{n}=1$. Then, $\mathcal{D}_{n}$ converges to the
circular lune $\mathcal{D}$ bounded by the unit circle and the circle
$\\{|\zeta+i/2|=1/2\\}$.

Let $\mathcal{G}_{n}$ be a sequence of doubly-connected domains in
$\mathbb{C}$, each of which is bounded by Jordan curves $\Gamma_{n1}$ and
$\Gamma_{n2}$, and $\Gamma_{n2}$ is the exterior component of
$\partial\mathcal{G}_{n}$. Let there exist a sequence of crosscuts
$\alpha_{n}$ of domains $\mathcal{G}_{n}$ with endpoints $a_{n}$, $b_{n}$,
such that $\lim\limits_{n\to\infty}a_{n}=\lim\limits_{n\to\infty}b_{n}=w_{0}$
for some $w_{0}\in\mathbb{C}$ and the diameters of $\alpha_{n}$ tend to $0$,
as $n\to\infty$. For every $n$ we denote by $\gamma_{n}$ a subarc of
$\Gamma_{n2}$ with endpoints $a_{n},b_{n}$ such that
$\gamma_{n}\cup\alpha_{n}$ is the boundary of the Jordan domain containing
$\Gamma_{n1}$. Let the sequence $\mathcal{G}_{n}$ converge to a simply-
connected domain $\mathcal{G}$, the boundary of which consists of two Jordan
curves $\Gamma_{1}$ and $\Gamma_{2}$, and
$\Gamma_{1}\cap\Gamma_{2}=\\{w_{0}\\}$ (Fig. 3).

Figure 3. The sequence of mappings $f_{n}$ and its limit.

######  Theorem 2.3.

Let $f_{n}$ be a sequence of functions mapping conformally $\mathcal{D}_{n}$
onto $\mathcal{G}_{n}$, and $f_{n}(0)=z_{n0}\in\Gamma_{n1}$. Let the function
$f$ map $\mathcal{D}$ onto $\mathcal{G}$, and $f(0)=z_{0}\in\Gamma_{1}$, where
$z_{0}\neq w_{0}$. Assume that $\lim\limits_{n\to\infty}z_{n0}=z_{0}$, and
there exist homeomorphisms

$\psi_{n1}:\Gamma_{1}\to\Gamma_{n1},\,\,\,\psi_{n2}:\Gamma_{2}\to\Gamma_{n2}$

such that

(5) $\forall\varepsilon>0\ \ \exists N:\quad\forall n\geq
N\quad|\psi_{n1}(t)-t|<\varepsilon,\
t\in\Gamma_{1};\qquad|\psi_{n2}(s)-s|<\varepsilon,\ s\in\Gamma_{2}.$

Then, $f_{n}$ converges locally uniformly to $f$ on
$\overline{\mathcal{D}}\setminus\\{-i\\}$ and $f^{-1}_{n}$ converges uniformly
to $f^{-1}$ on every compact subset of
$\bigcap_{n=1}^{\infty}\overline{\mathcal{G}}_{n}\setminus\\{w_{0}\\}$.

To prove the theorem, we need to establish some auxiliary results.

Denote
$\alpha_{n}^{\prime}:=f_{n}^{-1}(\alpha_{n}),a_{n}^{\prime}:=f_{n}^{-1}(a_{n}),b_{n}^{\prime}:=f_{n}^{-1}(b_{n})$.
According to the principle of the boundary correspondence under conformal
mappings, $\alpha_{n}^{\prime}$ is a crosscut in $\mathcal{D}_{n}$, with
endpoints $a_{n}^{\prime},b_{n}^{\prime}$ on the unit circle.

######  Lemma 2.4.

The crosscuts $\alpha^{\prime}_{n}$ are contracted to the point $(-i)$, as
$n\to\infty$.

The proof of Lemma 2.4 is similar to that of Lemma 2.2.

Let $\mathcal{D}^{\prime}_{n}$ be the sequence of doubly-connected domains,
each of which is bounded by the circle $\\{|\zeta+ir_{n}/2|=r_{n}/2\\}$ and
$\alpha^{\prime}_{n}\cup\alpha^{\prime\prime}_{n}$, where
$\alpha^{\prime\prime}_{n}$ is the arc of the unit circle $\\{|\zeta|=1\\}$
with endpoints $a^{\prime}_{n}$, $b^{\prime}_{n}$ such that the curve
$\alpha^{\prime}_{n}\cup\alpha^{\prime\prime}_{n}$ contains the circle
$\\{|\zeta+ir_{n}/2|=r_{n}/2\\}$ in its interior (Fig. 4).

Figure 4. The sequence of mappings $f_{n}$ on subdomains
$\mathcal{D}^{\prime}_{n}$.

Let $\widetilde{\mathcal{G}}_{n}$ be a part of $\mathcal{G}_{n}$, bounded by
$\gamma_{n}\cup\alpha_{n}$ and $\Gamma_{n1}$ (Fig. 5). From Lemma 2.4 and the
conditions on $\alpha_{n}$, we easily deduce the following lemma.

######  Lemma 2.5.

The sequences of domains $\mathcal{D}_{n}^{\prime}$ and
$\widetilde{\mathcal{G}}_{n}$ converge to the domains $\mathcal{D}$ and
$\mathcal{G}$, respectively, as $n\to\infty$.

Denote by $\widetilde{f}_{n}$ a sequence of functions mapping conformally
$\widetilde{\mathcal{D}}_{n}$ onto $\widetilde{\mathcal{G}}_{n}$ such that
$\widetilde{f}_{n}(0)=z_{n0}\in\Gamma_{n1}$; here
$\widetilde{\mathcal{D}}_{n}$ are nonconcentric annuli bounded by the unit
circle and the circles
$\\{|\zeta+i\widetilde{r}_{n}/2|=\widetilde{r}_{n}/2\\}$, where
$0<\widetilde{r}_{n}<1$ and $\lim\limits_{n\to\infty}\widetilde{r}_{n}=1$. The
arc $\widetilde{\alpha}_{n}^{\prime}:=\widetilde{f}^{-1}_{n}(\alpha_{n})$ is a
subarc of the unit circle with endpoints
$\widetilde{a}_{n}^{\prime}:=\widetilde{f}_{n}^{-1}(a_{n}),\widetilde{b}_{n}^{\prime}:=\widetilde{f}_{n}^{-1}(b_{n})$.
Applying the same arguments as in the proof of Lemma 2.2, we obtain that the
arcs $\widetilde{\alpha}^{\prime}_{n}$ are contracted to the point $(-i)$, as
$n\to\infty$.

It is easy to see that the sequence $\widetilde{\mathcal{D}}_{n}$ tends to
$\mathcal{D}$ as $n\to\infty$.

Let the conditions of Theorem 2.3 hold. Then it is evident that there exist
homeomorphisms $\widetilde{\psi}_{n1}:\Gamma_{1}\to\Gamma_{n1}$ and
$\widetilde{\psi}_{n2}:\Gamma_{2}\to\gamma_{n}\cup\alpha_{n}$ such that

(6) $\forall\varepsilon>0\ \exists N:\quad\forall n\geq
N\quad|\widetilde{\psi}_{n1}(t)-t|<\varepsilon,\
t\in\Gamma_{1};\quad|\widetilde{\psi}_{n2}(s)-s|<\varepsilon,\
s\in\Gamma_{2}.$

From this fact and the generalized Radó theorem [2, Sect. 4, Theorem. 6] we
obtain

######  Lemma 2.6.

The sequence $\widetilde{f}_{n}$ converges uniformly to $f$ in
$\overline{\mathcal{D}}$.

Figure 5. The sequence of mappings $\widetilde{f}_{n}$.

Denote by ${\mathcal{D}}^{\prime}_{n1}$ and $\widetilde{\mathcal{D}}_{n1}$ the
domains which obtained from the domains $\mathcal{D}^{\prime}_{n}$ and
$\widetilde{\mathcal{D}}_{n}$ by gluing them with their reflections with
respect to the arcs on the unit circle $|\zeta|=1$ with endpoints
$a^{\prime}_{n},b^{\prime}_{n}$ and
$\widetilde{a}^{\prime}_{n},\widetilde{b}^{\prime}_{n}$, complement to
$\alpha_{n}$ and $\alpha^{\prime}_{n}$ respectively. It is clear that the
sequences ${\mathcal{D}}^{\prime}_{n1}$ and $\widetilde{\mathcal{D}}_{n1}$
tend to the domain $\mathcal{D}_{0}=\\{|z+1/2|>1/2,\,\text{Im}z>-1\\}$ as
$n\to\infty$.

Every function $f_{n}^{-1}\circ\widetilde{f}_{n}$ maps
$\widetilde{\mathcal{D}}_{n}$ onto $\mathcal{D}^{\prime}_{n}$. According to
the Riemann-Schwarz symmetry principle, the functions
$f_{n}^{-1}\circ\widetilde{f}_{n}$ can be extended to the domains
$\widetilde{\mathcal{D}}_{n1}$; for convenience, we will also denote the
extended functions by $f_{n}^{-1}\circ\widetilde{f}_{n}$; each of them maps
$\widetilde{\mathcal{D}}_{n1}$ onto $\widetilde{\mathcal{D}}^{\prime}_{n1}$.

Applying the same arguments as in the proof of Lemma 2.3 we obtain

######  Lemma 2.7.

The sequence of functions $f_{n}^{-1}\circ\widetilde{f}_{n}$ converges locally
uniformly to the identity mapping on $\mathcal{D}_{0}\setminus\\{-i\\}$.

The same statement also holds for the inverse functions
$\widetilde{f}_{n}^{-1}\circ f_{n}$. Then, with the use of Lemma 2.6, we
obtain that $f_{n}=\widetilde{f}_{n}\circ(\widetilde{f}_{n}^{-1}\circ{f}_{n})$
converges locally uniformly to $f$ in $\mathcal{D}_{0}\setminus\\{-i\\}$. The
uniform convergence of $f_{n}^{-1}$ to $f^{-1}$ can be proved just in the
proof of similar result from Theorem 2.2. This proves Theorem 2.3.

######  Remark 2.2.

Further we can apply Theorem 2.3 not only for sequences of conformal mappings
but also for one parameter families of such mappings. We can also apply this
theorem for the case of unbounded domains; then we understand the convergence
in the spherical metric.

##  3\. Proofs of Theorem 1.1

First we will show that (3) holds.

Consider four the points
$A(c,f_{1}(c)),B(c,f_{2}(c)),C(d,f_{2}(d)),D(d,f_{1}(d))$ on $\partial\Omega$.
The vertical segments $AB$ and $CD$ separate $\Omega$ into two domains
$\mathcal{Q}$ and $\mathcal{P}$. Let $\mathcal{Q}$ denote the bounded domain
and $\mathcal{P}$ be unbounded, i.e.

$\mathcal{Q}:=\\{(x,y)\in\mathbb{C}\,:\,c\leq x\leq d,f_{2}(x)\leq y\leq
f_{1}(x)\\},$

and $\mathcal{P}=\Omega\setminus\overline{\mathcal{Q}}$.

Consider the quadrilaterals

$\mbox{\boldmath$Q$}:=\left(\mathcal{Q};A,B,C,D\right),\quad\mbox{\boldmath$P$}:=\left(\mathcal{P};D,C,B,A\right)$

and their images

$\mbox{\boldmath$Q$}_{H}:=f_{H}(\mbox{\boldmath$Q$})=\left(\mathcal{Q}_{H};A_{H},B_{H},C_{H},D_{H}\right),\quad\mbox{\boldmath$P$}_{H}:=f_{H}(\mbox{\boldmath$P$})=\left(\mathcal{P}_{H};D_{H},C_{H},B_{H},A_{H}\right)$

under the mapping $f_{H}$. By the Grötzsch inequality (see, e.g., [6, Chap. 4,
Sect. 6], [17, Chap. 3, Sect. 7]), we have

(7) $\displaystyle m^{-1}(\Omega_{H})\geq
m(\mbox{\boldmath$Q$}_{H})+m(\mbox{\boldmath$P$}_{H}).$

From (7) we deduce that $m^{-1}(\Omega_{H})\geq m(\mbox{\boldmath$Q$}_{H})$
and from [14, Theorem 4.2] it follows that

(8) $m(\mbox{\boldmath$Q$}_{H})\geq\gamma
H,\quad\gamma=\int\limits_{c}^{d}\dfrac{dx}{f_{1}\left({x}\right)-f_{2}\left({x}\right)}.$

Therefore, $m^{-1}(\Omega_{H})\geq\gamma H$ and this implies (3). In
particular, we have

(9) $m(\Omega_{H})\to 0,\quad\text{ as }\quad H\to\infty.$

Denote by $\varphi_{H}$ a conformal mapping of $\Omega_{H}$ onto a concentric
annulus

$\mathfrak{D}_{H}:=\\{1<|\zeta|<\rho_{H}\\},$

such that the point $A_{H}$ is mapped to some point on the circle
$\\{|\zeta|=1\\}$, say $\zeta=1$, and $\varphi_{H}(B_{H})$ is on the circle
$\\{|\zeta|=\rho\\}$. We have

$m(\Omega_{H})=m(\mathfrak{D}_{H})=\frac{1}{2\pi}\log\rho_{H},$

therefore, by (9), $\lim\limits_{H\to\infty}\rho_{H}=1$.

######  Lemma 3.1.

We have

(10) $\displaystyle m^{-1}(\Omega_{H})\sim
m(\mbox{\boldmath$P$}_{H})+m(\mbox{\boldmath$Q$}_{H}),\quad\mbox{\rm as }\quad
H\to+\infty.$

###### Proof.

To show this, we will establish that for every $\varepsilon>0$ there exists
$H_{0}$ such that for all $H\geq H_{0}$ the images of the segments
$A_{H}B_{H}$ and $C_{H}D_{H}$ under the map $\varphi_{H}$ lie in the
intersections of the annulus $\mathfrak{D}_{H}$ and circular sectors of
solution less than $\varepsilon$ centered at the origin.

First we consider the images of the segment $A_{H}B_{H}$. The modulus is a
conformal invariant, consequently, it is invariant under the shifts of the
domain. Therefore, instead of $\Omega$ we can consider the domain which is
obtained from $\Omega$ by translating along the real axis by the vector
$(-c,0)$. For convenience, we also denote the obtained domain by $\Omega$. In
this case, we really have $c=0$, therefore the points $A_{H}=A$, $B_{H}=B$ and
the segment $A_{H}B_{H}=AB$ do not depend on $H$.

Denote $r_{H}=\frac{2\rho_{H}}{1+\rho_{H}^{2}}$. It is clear that $r_{H}<1$
and $r_{H}\to 1$ as $H\to\infty$.

The Möbius transformation

$\psi_{H}(\zeta)=\rho_{H}\,\dfrac{\rho_{H}\zeta+i}{\zeta+i\rho_{H}}$

maps conformally the non-concentric annulus

$\mathfrak{G}_{H}:=\\{\zeta\in\mathbb{C}\,:\,|\zeta+ir_{H}/2|>r_{H}/2,\,|\zeta|<1\\}.$

onto the annulus $\mathfrak{D}_{H}$.

Consider two cases.

a) Let $a<c=0$. Denote

$\mathfrak{G}:=\\{\zeta\in\mathbb{C}\,:\,|\zeta+i/2|>1/2\\},\,|\zeta|<1\\},$
$\Omega=\\{(x,y):y<f_{1}(0)\\}\setminus\\{(x,f_{2}(0)):x\geq 0\\}.$

By the definition of the kernel convergence, the family of domains
$\Omega_{H}$ converges to $\Omega$ as to a kernel and also $\mathfrak{G}_{H}$
converges to $\mathfrak{G}$, as $H\to+\infty$.

Now consider the conformal mappings
$\mathcal{F}_{H}:=\psi_{H}^{-1}\circ\varphi_{H}$. Then, by the Carathéodory
convergence theorem [6, Chap. 5, Sect. 5], $\mathcal{F}_{H}$ converge locally
uniformly in $\Omega$ to some conformal mapping
$\mathcal{F}:\Omega\to\mathfrak{G}$, as $H\to+\infty$.

Now with the help of Theorem 2.3, we will show that the mappings
$\mathcal{F}_{H}$ converge uniformly to the conformal mapping
$\mathcal{F}:\Omega\to\mathfrak{G}$ on the segment $AB$. We should note that
the limiting domain $\Omega$ is not Jordan since it is a half plane cut along
a ray and formally we can not employ Theorem 2.3 directly to the sequence
$\mathcal{F}_{H}$ converging to $\mathcal{F}$. But we can apply to
$\Omega_{H}$ additional conformal mappings

$g_{H}(z)=\delta_{H}\sqrt{\frac{D_{H}(z-A)}{D_{H}-z}}+c_{H},$

with appropriate fixed branches of the square root, where $\delta_{H}\to 1$
and $c_{H}\to 0$, as $H\to\infty$. Then the domains $g_{H}(\Omega_{H})$
converge to a Jordan domain which is the image of $\Omega$ under the mapping
$g(z)=\sqrt{z-A}$. Obviously, $g_{H}$ converges to $g$ uniformly on $AB$. We
can choose $\delta_{H}$ and $c_{H}$ such that
$g_{H}(AB)\subset\overline{g(\Omega)}$. Let us fix a compact subset $K$ of
$g(\Omega)$ such that $g_{H}(AB)\subset\overline{g(\Omega)}$. By Theorem 2.3,
taking into account Remark 2.2, we obtain that $\mathcal{F}_{H}\circ
g_{H}^{-1}$ converge to $\mathcal{F}\circ g^{-1}$ on $K$. From this fact we
conclude that $\mathcal{F}_{H}$ converge to $\mathcal{F}$ on $AB$.

Denote $\gamma_{H}=\varphi_{H}(AB)$. Let us fix a neighborhood $U$ of the
curve $\Gamma=\mathcal{F}(AB)$ which lies outside of some neighborhood $V$ of
the point $(-i)$. Because of the uniform convergence, for sufficiently large
$H$ the curves $\Gamma_{H}=\mathcal{F}_{H}(AB)$ lie in $U$. Consequently, they
do not intersect $V$. We also note that for every $\delta>0$ the Möbius
transforms $\psi_{H}$ converge uniformly to the constant $1$ on the set
$|\zeta+i|\geq\delta$, as $H\to\infty$. This follows that for sufficiently
large $H$ the curves $\gamma_{H}=\psi_{H}(\Gamma_{H})$ lie in an arbitrarily
small neighborhood of the point $1$.

b) Let now $a=c$. This case is considered similarly to the case $a>c$; we
should note that now the kernel of the sequence $\Omega_{H}$ equals to
$\widetilde{\Omega}$ which is the complex plane with excluded rays
$\\{(x,f_{k}(0)):x\geq 0\\}$, $k=1$, $2$.

This means that for every $\varepsilon>0$ there exists $H_{0}$ and $\alpha$
such that for all $H\geq H_{0}$ the image of the segments $A_{H}B_{H}$ under
the map $\varphi_{H}$ lie in the sector $\alpha<\arg z<\alpha+\varepsilon$.
Similarly we prove that for every $\varepsilon>0$ there exists $H_{1}$ and
$\beta$ such that for all $H\geq H_{1}$ the image of the segments $C_{H}D_{H}$
under the map $\varphi_{H}$ lie in the sector $\beta<\arg
z<\beta+\varepsilon$. Without loss of generality we can assume that
$H_{0}=H_{1}$ and $\alpha=0<\varepsilon<\beta<\beta+\varepsilon<2\pi$.
Comparing the modules of $\varphi_{H}(\mbox{\boldmath$P$}_{H})$ and
$\varphi_{H}(\mbox{\boldmath$Q$}_{H})$ with the modules of quadrilaterals
lying in the annulus $\mathfrak{D}_{H}$ inside of the sectors bounded with the
rays $\arg z=0,\varepsilon,\beta,$ and $\beta+\varepsilon$, taking into
account the conformal invariance of moduli, we conclude that for $H\geq H_{0}$

$(\beta-\varepsilon)/\log\rho_{H}<m(\mbox{\boldmath$Q$}_{H})<(\beta+\varepsilon)/\log\rho_{H},$
$(2\pi-\beta-\varepsilon)/\log\rho_{H}<m(\mbox{\boldmath$P$}_{H})<(2\pi-\beta+\varepsilon)/\log\rho_{H},$

Adding the above inequalities, we obtain

(11)
$\displaystyle(2\pi-2\varepsilon)/\log\rho_{H}<m(\mbox{\boldmath$Q$}_{H})+m(\mbox{\boldmath$P$}_{H})<(2\pi+2\varepsilon)/\log\rho_{H},\quad\text{
for all }\quad H\geq H_{0}.$

Because, $\varepsilon>0$ can be chosen arbitrarily small, from (11) we get

$m(\mbox{\boldmath$Q$}_{H})+m(\mbox{\boldmath$P$}_{H})\sim
2\pi/\log\rho_{H}=m^{-1}(\Omega_{H}).$

The lemma is entirely proven. ∎

Now we will estimate the behavior of the conformal modulus of
$\mbox{\boldmath$P$}_{H}$ as $H\to+\infty$.

######  Lemma 3.2.

We have

(12) $\displaystyle m(\mbox{\boldmath$P$}_{H})=O(\log H),\quad\mbox{\rm as
}\quad H\to+\infty.$

###### Proof.

As in the proof of Lemma 3.1, we shift $\Omega$ by the vector $(-c,0)$ and
keep for it and its subdomains the same notation.

Consider the quadrilateral $P$ (Fig. 1). Let us fix some real numbers
$\kappa,\ell$ and positive $\varepsilon>0$ such that
$f_{2}(c)<\kappa-\varepsilon<\kappa<f_{1}(c)$,
$f_{2}(d)<\ell-\varepsilon<\ell<f_{1}(d)$. If we replace the vertices of $P$
with the points $d+i\ell$, $d+i(\ell-\varepsilon)$, $c+i(\kappa-\varepsilon)$,
$c+i\kappa$, then the modulus of the obtained quadrilateral
$\mbox{\boldmath$P$}^{1}$ will be greater than the modulus of $P$. Now we
choose some rays $L_{1}$ and $L_{2}$ which come from the points $c+i\kappa$
and $d+i\ell$ and lie in $\mathcal{P}$. If we replace the side of the
quadrilateral $\mbox{\boldmath$P$}^{1}$ with end points $c+i\kappa$ and
$d+i\ell$ by the union of the rays $L_{1}$ and $L_{2}$, then we obtain a
quadrilateral $\mbox{\boldmath$P$}^{2}$, the modulus of which is greater than
$m(\mbox{\boldmath$P$}^{1})$. Further, for sufficiently large $M$ the segment
with enpoints $c+i(\kappa-M)$ and $d+i(\kappa-M)$ is lower than the graph of
$y=f_{2}(x)$. Replacing the boundary arc of the quadrilateral
$\mbox{\boldmath$P$}^{2}$ connecting the points $c+i\kappa$ and $d+i\ell$ and
containing the graph of $y=f_{2}(x)$ with the three-link polygonal line $L$
with vertices at the points $c+i\kappa$, $c+i(\kappa-M)$, $d+i(\ell-M)$ and
$d+i\ell$, we obtain a quadrilateral $\mbox{\boldmath$P$}^{3}$, the modulus of
which is greater than $m(\mbox{\boldmath$P$}^{2})$. We point out that the
boundary of the quadrilateral consists of the rays $L_{1}$, $L_{2}$ and the
polygonal line $L$; its vertices are $d+i\ell$, $d+i(\ell-\varepsilon)$,
$c+i(\kappa-\varepsilon)$, and $c+i\kappa$.

Denote by $\mbox{\boldmath$P$}^{3}_{H}$ the quadrilateral which is obtained
from $\mbox{\boldmath$P$}^{3}$ by the stretching with coefficient $H$. It is
clear that

(13) $m(\mbox{\boldmath$P$}_{H})\leq m(\mbox{\boldmath$P$}^{3}_{H}),$

therefore, to prove the lemma, we only need to prove that
$m(\mbox{\boldmath$P$}^{3}_{H})=O(\log H)$, $H\to+\infty$. Let the equations
of the straight lines containing $L_{1}$, the segment with endpoints
$c+i(\kappa-M)$, $d+i(\kappa-M)$ and $L_{2}$ have the form $y=a_{k}x+b_{k}$,
$k=1$, $2$, $3$. Then their images under the stretching are on the lines
$y=(a_{k}/H)x+b_{k}$, $1\leq k\leq 3$. Now we define the mapping

$\eta_{H}:x+iy\mapsto x+iv(x,y),$

where

(14) $\displaystyle
v(x,y)={\begin{cases}y-\frac{a_{1}}{H}\,x-b_{1},\quad\mbox{\rm if}\quad x\leq
Hc,\\\\[2.84544pt] y-\frac{a_{2}}{H}\,x-b_{2}-M,\quad\mbox{\rm if }\quad
Hc\leq x\leq Hd,\\\\[2.84544pt] y-\frac{a_{3}}{H}\,x-b_{3},\quad\mbox{\rm
if}\quad x\geq Hd.\end{cases}}$

It is a $K$-quasiconformal homeomorphism of the plane with

$K=K(H)=\frac{1+k}{1-k},\quad
k=\frac{\mathfrak{a}}{\sqrt{\mathfrak{a}^{2}+4H^{2}}},\quad\mathfrak{a}:=\max_{1\leq
k\leq 3}|a_{k}|.$

Since $K(H)\to 1$, $H\to+\infty$, there exists $K_{0}\geq 1$ such that
$K(H)\leq K_{0}$.

Denote by $\mbox{\boldmath$T$}_{H}$ the quadrilateral which is the image of
$\mbox{\boldmath$P$}^{3}_{H}$ under the mapping $\eta_{H}$. Then, because of
quasiinvariance of conformal modulus under quasiconformal mappings, we have

(15) $m(\mbox{\boldmath$P$}^{3}_{H})\leq K(H)m(\mbox{\boldmath$T$}_{H})\leq
K_{0}m(\mbox{\boldmath$T$}_{H}).$

Because of (13) and (15), to prove Lemma 3.2, we only need to establish that

(16) $m(\mbox{\boldmath$T$}_{H})=O(\log H),\quad\mbox{\rm as}\quad
H\to+\infty.$

Since conformal modulus is invariant under shifts, we can assume that $c=-d$.
We note that
$\mbox{\boldmath$T$}_{H}=(T_{H};dH,dH-i\varepsilon,-dH-i\varepsilon,-dH)$
where $T_{H}$ is obtained from the lower half plane by removing the rectangle
$[-dH,dH]\times[-M,0]$. The quadrilateral $\mbox{\boldmath$T$}_{H}$ is
symmetric with respect to the imaginary axis. Its modulus is twice times
greater than the modulus of its right half, i.e. the quadrilateral
$\mbox{\boldmath$T$}_{H}^{1}$ which is the fourth quarter of the plane with
the excluded rectangle $[0,dH]\times[-M,0]$, with vertices $dH$,
$dH-i\varepsilon$, $-iM$, and $\infty$. If we replace the side of
$\mbox{\boldmath$T$}_{H}^{1}$ which is the vertical ray, going down from the
point $-iM$ to infinity, with the ray going from this point to the left and
shift the obtained quadrilateral by the vector $-dH$ then we obtain the
quadrilateral $\mbox{\boldmath$U$}_{H}=(U;0,-i\varepsilon,-d_{H}-iM,\infty)$
the modulus of which is bigger than the modulus of
$\mbox{\boldmath$T$}_{H}^{1}$. Here $U$ is the domain the boundary of which is
the polygonal line consisting of the horizontal rays $\\{(x,-M),x\leq 0\\}$,
$\\{(x,0),x\geq 0\\}$ and the vertical segment $[-iM,0]$. We have

(17) $m(\mbox{\boldmath$T$}_{H})\leq 2m(\mbox{\boldmath$U$}_{H}).$

Now we need the following auxiliary result.

######  Lemma 3.3.

We have

$m(\mbox{\boldmath$U$}_{H})=O(\log H),\quad\mbox{\rm as }\quad H\to+\infty.$

We note that this lemma, in fact, is a generalization of [14, Sect. 4, Lemma
4.6].

###### Proof.

We can map conformally the lower half plane $\mathbb{L}$ onto $U$ by a
function $g$ such that $g(-1)=-iM$, $g(1)=0$, $g(\infty)=\infty$. The function
$g$ has the form

(18)
$g(\zeta)=\frac{M}{\pi}\int_{1}^{\zeta}\sqrt{\frac{\omega+1}{\omega-1}}\,d\omega$

where the branch of the square root is fixed such that it is positive for real
$\omega>1$. Denote by $(-R_{H})$ and $\xi$ the preimages of the points $(-dH-
iM)$ and $(-i\varepsilon)$ under the mapping $g(\zeta)$, $R_{H}>0$,
$-1<\xi<1$. From (18) it follows that

$g(\zeta)\sim\frac{M}{\pi}\,\zeta,\quad\mbox{\rm as }\quad\zeta\to\infty,$

therefore,

$R_{H}\sim\frac{M}{\pi}\,dH.$

Further, the conformal modulus of $\mbox{\boldmath$U$}_{H}$ is equal to the
conformal modulus of the quadrilateral
$\mbox{\boldmath$L$}_{H}=(\mathbb{L};\infty,-R_{H},\xi,0)$. In its turn, by
the Riemann-Schwarz symmetry principle, $m(\mbox{\boldmath$L$}_{H})$ is a half
of the modulus of the doubly-connected domain
$\mathcal{A}=\overline{\mathbb{C}}\setminus\left((-\infty,-R_{H}]\cup[\xi,1]\right)$,
which is an image of the Teichmüller ring under a linear transform. Thus, we
have (see, e.g., [1, Chap.3]; [5])

$2\pi m(\mathcal{A})\sim\log R_{H}\sim\log H,\quad\mbox{\rm as }\quad
H\to\infty.$

Since $m(\mbox{\boldmath$U$}_{H})\leq 2m(\mathcal{A})$, we proved the
statement of the lemma. ∎

From Lemma 3.3 and (17) it follows that (16) holds. This proves Lemma 3.2.

At last, combining (8), (10) and (12) we obtain (2). Therefore, Theorem 1.1 is
completely proved. ∎

##  4\. Discussion

There are still many interesting issues concerning investigation of the
behavior of conformal moduli of domains under their geometric transformations.
Our next task includes studying the behavior of the exterior conformal moduli
of arbitrary quadrilaterals when stretched in the direction of the abscissa
axis.

## FUNDING

The work of the second author is performed under the development program of
the Volga Region Mathematical Center (agreement no. 075-02-2022-882).

## References

  * [1] Ahlfors L. V., _Lectures on Quasiconformal Mappings, Vol. 38_ (American Mathematical Society, 2006).
  * [2] D. N. Dautova and S. R. Nasyrov, “Asymptotics of the modules of mirror symmetric doubly connected domains under stretching,” Mathematical Notes 103 (3), 537–549 (2018).
  * [3] D. N. Dautova and S. R. Nasyrov, “Asymptotics of conformal module of nonsymmetric doubly connected domain under unbounded stretching along the real axis,” Lobachevskii Journal of Mathematics 40 (9), 1268–1274 (2019).
  * [4] Dautova D., Nasyrov S. and Vuorinen M., “Conformal Modulus of the Exterior of Two Rectilinear Slits,” Computational Methods and Function Theory (Springer) 21 (1), 109–130 (2021).
  * [5] Glen D. Anderson, Mavina K. Vamanamurthy, Matti K. Vuorinen, _Conformal invariants, inequalities and quasiconformal maps_ (Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. J. Wiley, 1997).
  * [6] G. M. Goluzin, _Geometric Theory of Functions of a Complex Variable_ (American Mathematical Soc., Providence, RI, Translations of Mathematical Monographs, Vol. 26, 1969).
  * [7] Hakula H., Nasyrov S. and Vuorinen M., “Conformal moduli of symmetric circular quadrilaterals with cusps,” Electronic Transactions on Numerical Analysis (Kent State University) 54, 460–482 (2021).
  * [8] Hakula H., Rasila A., and Vuorinen M., “On moduli of rings and quadrilaterals: algorithms and experiments,” SIAM Journal on Scientific Computing 33 (1), 279–302 (2011).
  * [9] Hakula H., Rasila A., and Vuorinen M., “Computation of exterior moduli of quadrilaterals,” Electron. Trans. Numer. Anal. 40, 436–451 (2013).
  * [10] Lang S., _Complex analysis, Vol. 103_ (Springer Science & Business Media, 2003).
  * [11] Nasser Mohamed MS, “PlgCirMap: A MATLAB toolbox for computing conformal mappings from polygonal multiply connected domains onto circular domains,” SoftwareX 11, 100464 (2020).
  * [12] Nasser M., Rainio O., Rasila A., Vuorinen M., Wallace T., Yu H. and Zhang X., “Circular arc polygons, numerical conformal mappings, and moduli of quadrilaterals,” arXiv preprint arXiv:2107.11485 (2021).
  * [13] Nasyrov S. R., “Riemann–Schwarz reflection principle and asymptotics of modules of rectangular frames,” Computational Methods and Function Theory 15 (1), 59–74 (2015).
  * [14] Nasyrov S. R., Van Giang N., “Asymptotics of the conformal modulus of unbounded symmetric doubly-connected domain under stretching,” Lobachevskii Journal of Mathematics 42 (12), 2895–2904 (2021).
  * [15] Nasyrov S. R., Sugawa T. and Vuorinen M., “Moduli of quadrilaterals and quasiconformal reflection,” arXiv:2111.08304 (2021).
  * [16] Papamichael N., Stylianopoulos N., _Numerical Conformal Mapping: Domain Decomposition and the Mapping of Quadrilaterals_ (World Scientific, Singapore, 2010).
  * [17] R. Kühnau, “The conformal module of quadrilaterals and of rings,” in: _Handbook of Complex Analysis_ , Ed. by R. Kühnau (Elsevier, 2005), Vol. 2, pp. 99–129.
  * [18] Remmert R., _Classical topics in complex function theory, Vol. 172_ (Springer Science & Business Media, 2013).
  * [19] Schiff J. L., _Normal families_ (Springer Science & Business Media, 1993).
  * [20] Vuorinen M., Zhang X., “On exterior moduli of quadrilaterals and special functions,” Journal of Fixed Point Theory and Applications 13 (1), 215–230 (2013).