[a,b]M Battisti

# Overview of the Mini-EUSO $\mu s$ trigger logic performance

D. Barghini  A Belov  M Bertaina  F Bisconti  K Bolmgren  G Cambiè  F Capel  M
Casolino  T Ebisuzaki  F Fenu  C Fuglesang  A Golzio  P Gorodetzki  F Kajino
P Klimov  M Manfrin  L Marcelli  W Marszał  H. Miyamoto  E Parizot  P Picozza
L W Piotrowski  Z Plebaniak  G Prévôt  E Reali  M Ricci  N Sakaki  K Shinozaki
J Szabelski  Y Takizawa

# Overview of the Mini-EUSO $\mu s$ trigger logic performance

D. Barghini  A Belov  M Bertaina  F Bisconti  K Bolmgren  G Cambiè  F Capel  M
Casolino  T Ebisuzaki  F Fenu  C Fuglesang  A Golzio  P Gorodetzki  F Kajino
P Klimov  M Manfrin  L Marcelli  W Marszał  H. Miyamoto  E Parizot  P Picozza
L W Piotrowski  Z Plebaniak  G Prévôt  E Reali  M Ricci  N Sakaki  K Shinozaki
J Szabelski  Y Takizawa

###### Abstract

Mini-EUSO is the first detector of the JEM-EUSO program deployed on the ISS.
It is a wide field of view telescope currently operating from a nadir-facing
UV-transparent window on the ISS. It is based on an array of MAPMTs working in
photon counting mode with a $2.5~{}\mu s$ time resolution. Among the different
scientific objectives it searches for light signals with time duration
compatible to those expected from Extensive Air Showers (EAS) generated by
EECRs interacting in the atmosphere. Although the energy threshold for cosmic
ray showers is above $E>10^{21}eV$, due the constraints given by the size of
the UV-transparent window, the dedicated trigger logic has been capable of the
detection of other interesting classes of events, like elves and ground
flashers. An overview of the general performance of the trigger system is
provided, with a particular focus on the identification of classes of events
responsible for the triggers.

##  1 Detector description

###  1.1 Instrument overview

Mini-EUSO is a small telescope ($37\times 37\times 62~{}cm^{3}$) with an
optical system made of two Fresnel lenses of 25 cm diameter each. The focal
plane is made of 36 Multi-Anode Photomultiplier Tubes (MAPMTs Hamamatsu
Photonics R11265-M64), for a total of 2304 pixels arranged in a square matrix.
Four MAPMTs are grouped in a $2\times 2$ matrix called Elementary Cell (EC).
Each of the nine ECs shares a common high voltage power supply (HVPS) based on
a Cockroft-Walton circuit. The system has an internal safety circuit that
reduces the gain and the collection efficiency of the pixels in the EC unit in
case of high current drain due to bright light. This status is called cathode
2 mode. The focal plane is covered with a UV bandpass filter (2 mm of BG3
material), making the pixels mostly sensitive to $290~{}nm-430~{}nm$ band, in
the near UV. Mini-EUSO has a large field of view of roughly $42^{\circ}$, with
a pixel size on ground around $6.3\times 6.3~{}km^{2}$. The total footprint on
ground is slightly less than $350\times 350~{}km^{2}$ [1].

###  1.2 Data acquisition system

Mini-EUSO features a multi-level data acquisition system able to store events
in three different timescales at the same time, allowing the observation of
the same event with different time resolutions. The slowest timescale
($40.96~{}ms$) takes data continuously without a trigger system, and performs
a monitor of the UV emission of the Earth. A frame of $40.96~{}ms$ is called
D3 Gate Time Unite, or simply D3 GTU. The $320~{}\mu s$ timescale has its own
trigger system, and is mostly dedicated to atmospheric events. A $320~{}\mu s$
frame is called D2 GTU. The main focus will be on the $2.5~{}\mu s$ timescale
and its trigger system, used for the detection of very fast events. A
$2.5~{}\mu s$ frame is called D1 GTU. Hereinafter, GTU will always be referred
to D1 GTU, unless otherwise specified.

The L1 trigger code runs on a running buffer containing the $2.5~{}\mu s$ data
stream. When an event is triggered the system stores 128 D1 GTUs,
corresponding to $320~{}\mu s$ of data, 64 GTUs before and 64 GTUs after the
event. Mini-EUSO data acquisition system can store up to 4 consecutive events
within the same slot of 5.24 seconds. After the fourth triggered event the
system is in dead time and can not store any other data. The ability to save
data is restored at the start of the next slot of $5.24s$ [2]. The advantages
of this system are twofold. The first is the possibility to store consecutive
events without dead time, provided that there are still available slots. The
second advantage is that very long and very bright phenomena like, for
example, lightning strikes, that could last up to 1 second or more, do not
generate an enormous amount of triggers that would saturate the bandwidth.
Finally, it is possible to prevent given pixels from triggering through an
external command sent by the on-board CPU. This is usually done only for the
first orbit (from the switch-on moment until the first sunrise), when all the
border pixels and two entire MAPMTs are masked. The masking is needed because
some border pixels are more prone to crosstalk and other non-linear effect
that can generate fake triggers. An upgrade of the firmware, to store up to 8
triggers every 5.24 seconds, doubling the amount of data that it is possible
to store with $2.5~{}\mu s$ resolution, and to implement the pixel mask also
for other orbits is currently under test.

###  1.3 Trigger logic

Given the large pixel size on ground and the fast sampling rate of Mini-EUSO,
each pixel of Mini-EUSO has to be independent with its own threshold; a single
pixel above threshold is enough to issue a trigger. In fact, a signal moving
at the speed of light, like a EAS signal, takes slightly more than $20~{}\mu
s$ (8 GTUs) to cross 6.3 km, the field of view of a pixel. The logic therefore
looks for an excess of signal over 8 consecutive GTUs in the same pixel, where
the integration over 8 frames, namely $20~{}\mu s$, enhances the signal to
noise ratio. The 2304 thresholds are updated every $320~{}\mu s$, and set 16
$\sigma$ above the average value of the pixel over the previous slot of 128 D1
GTUs ($320~{}\mu s$). In Fig. 1 the scheme of the $2.5~{}\mu s$ trigger logic.

Figure 1: Mini-EUSO trigger logic. Every $320~{}\mu s$ the thresholds for each
pixel are updated, according to the formulas in the darker box. In parallel,
every GTU, the integral over the last 8 GTUs is compared with the threshold
for each pixel. If the sum is higher a trigger is issued

##  2 Analysis

###  2.1 Dataset

The following analysis is performed on a dataset of 37.4 hours of data, coming
from 31 different sessions; the excluded sessions were not performed in
nominal working conditions.

After each session a small portion of the data gathered are immediately
downlinked to ground ( $\sim$ 40 minutes of data, that corresponds to $\sim
20\%$) while the remaining is stored in USB pen drives and delivered by the
astronauts upon their return on ground. At the moment of writing more than
$50\%$ of data is still onboard the ISS. In the available dataset there are
more than $5\times 10^{4}$ L1 triggered events, an event being a collection of
128 frames with $2.5~{}\mu s$ time resolution.

###  2.2 D1 dead time

Figure 2 shows a map of the dead time of Mini-EUSO $2.5~{}\mu s$ trigger. In
blue-purple circles the triggers from the Atmosphere-Space Interactions
Monitor (ASIM), a detector installed outside the Columbus module of the ISS
and devoted to the study of atmospheric events, namely thunderstorms and
Transient-Luminous-Events (TLEs) [3]. The vast majority of its triggers are
classified as lightning strikes. From the map it is clear that Mini-EUSO does
not have higher dead time over cities and highly populated areas, since the
adaptive threshold is fast enough to prevent static light sources from
triggering. The higher concentration of dead time is usually due to
thunderstorms that are usually triggered 4 times in just a few milliseconds
and quickly saturate the 4 available slots.

Figure 2: Left: fraction of dead time. The globe has been divided into a
$1^{\circ}\times 1^{\circ}$ grid, the color representing the relative amount
of dead time in each cell. The average dead time is $\sim 25\%$ . Right:
triggers form ASIM detector, mostly induced by lightning strikes

As previously mentioned, during the first orbit the border pixels and two
MAPMTs are prevented from triggering to minimize the fake triggers due to non
linear electric effects. The trigger mask is not applied after the first
orbit, resulting in an increase of dead time, from an average of 11% in the
first orbits to 29% in all the others. The average dead time is $\sim 25\%$,
which represent an acceptable level for Mini-EUSO, given the detector’s
characteristics. Future space-based and balloon-based detectors (like EUSO-
SPB2 or POEMMA [4]) will work with a different logic for the data acquisition
system that will allow a lower dead time. In the meantime, the upgrade of the
firmware presented in 1.2 will significantly reduce this value for Mini-EUSO.

###  2.3 Trigger performance

In order to evaluate the trigger performance, the $2.5~{}\mu s$ data have been
analyzed through an offline algorithm that mimics the online trigger logic
implemented in the FPGA. The values of the thresholds for each pixel are
estimated from the data, since Mini-EUSO does not store them. In particular,
the background for each pixel is assumed to be equal to the average over the
first 32 GTUs ($80~{}\mu s$) of each triggered packet. This is in general a
good estimation since the 8 consecutive GTUs that issued the trigger are saved
in GTUs 58-63 inside the 128 GTUs packet, therefore the first 32 GTUs contain
only background. When long lasting events are triggered in more than one
packet, the offline trigger code does not update the thresholds, using the
ones computed for the previous packet. The other option to estimate the
background value would be to use the value of each pixel registered with the
D3 timescale. The results of the two methods are quite similar in normal
conditions, in the following analysis only the first method has been used,
since it performs better when the background is changing quickly or in the
presence of a transient event, but with the downside of being more susceptible
to statistical fluctuations.

Given the results of this offline algorithm, it is possible to produce a first
categorization of the data, based upon the number of pixels over threshold in
the entire packet of 128 GTUs. Notably, the trigger signal is issued when the
first pixel is over threshold, all the others are irrelevant from the trigger
point of view, but the presence of other pixels over threshold may give
information on the event that produced the trigger.

Tables 2 and 2 show that almost half of the triggered packets have only one
pixel over threshold, as expected given the footprint on ground of a single
pixel. For $\sim 18\%$ of the triggered packets the algorithm was not able to
recognise any pixel over threshold; this is due to the statistical
fluctuations of the background over the first 32 GTUs of each packet, since an
average made over 32 GTUs have a statistical fluctuation which is roughly
double the one on 128 frames. To prove that, the same analysis has been
repeated, setting the threshold to 12 $\sigma$ above the background and
obtaining at least a pixel over threshold in $97\%$ of the packets. This
proves that in many of these packets there is at least one pixel with a signal
just slightly below the threshold computed by the offline code, but above the
real threshold used by the online trigger. The remaining packets probably
belong to some classes of events not always detected by the offline code, like
the triggers issued by the sun rising, in which the light intensity grows
constantly and quickly. These events might be difficult for the offline code
that sets the threshold based on the background $\sim 65~{}\mu s$ before the
triggered event, while the thresholds could be computed up to $320~{}\mu s$
before, when the background is significantly lower.

Pixels over threshold | Frequency
---|---
0 | 18.58 %
1 | 45.73 %
2 | 13.94 %
3 | 5.25 %
4 | 1.32 %
5 | 0.62 %
6 | 0.46 %
Table 1: The numeric values of the first seven bins in Table 2

Table 2: Frequency of triggered packets with a given number of pixels over
threshold. The peak around 256 pixels corresponds to an entire EC switching
back to nominal working conditions from cathode 2 mode. A smaller peak is also
present around pixel 512

The distribution of pixels over threshold shown in Fig.3 presents a few hot
spots, corresponding to border pixels more prone to generate fake triggers. On
some occasions these pixels are responsible for a huge increase in the dead
time, generating for example the orbit that in Fig.2 starts from New Zealand
and moves towards north-east, with an average dead time $\sim 90\%$. As
already mentioned, a procedure to mask those pixels is currently under test,
and will be implemented as soon as possible.

Figure 3: Left: the number of GTUs over threshold for each pixel, considering
only the packet with less than 20 pixels over threshold (46180 events, 88% of
the dataset). 12 border pixels (0.52% of the pixels) account for 13.2% of the
total amount of pixels over threshold. Right: the time position of the pixels
over threshold inside the packet of 128 GTUs, each point is the sum over 8
consecutive GTUs: the huge peak centered at GTU 57 shows that the algorithm
correctly recognises the events that issued the trigger, which are positioned
at the center of the packet. A few long lasting events produce the tail, while
almost no trigger are recognised before GTU 50

##  3 Categories of triggered events

Being an orbital detector, Mini-EUSO detects a wide range of different events.
A list of the most interesting or most common category of events is given in
this section.

  * •

Atmospheric events: lightning strikes are the brightest events detected by
Mini-EUSO. Lightnings are best observed by Mini-EUSO with the longer
timescale, but triggered also in the $2.5~{}\mu s$ time resolution. With this
fast time resolution it is possible to probe the initial part of the
development of a lighting, and the signal appears over a large area of the
focal plane where the brightness increases constantly, until the end of the
128 saved GTUs (Fig.4, a).

Figure 4: a): Lightcurve of a lightning event: after the trigger has been
issued ($\sim$ GTU 64) the light level increases constantly. b): The bright
disk of an elve. Mini-EUSO can observe the elve development as the ring
expands in the field of view. The left part of the focal plane is in cathode 2
mode, therefore the counts are usually 0. The bright spot is the center of the
elves, bright enough to be clearly visible even in cathode 2 mode. c): Two
frames from two different events: the light sources inside the blue circles
are static (cities) while the source inside the red circle is flashing and is
what caused the trigger.

  * •

Elves are horizontally expanding, fast donut-shaped light emissions at the
bottom ionosphere. The characteristic spatial dimension of elves are rings
extending to a horizontal radius of $\sim 500~{}km$. Mini-EUSO can provide
high-speed UV imaging of elves, complementary to informations gathered by
other instruments (i.e. ASIM, also on the ISS). So far 17 elves have been
detected (Fig.4, b). Other types of Transient Luminous Events (TLEs) are
present in Mini-EUSO data and are currently under study.

  * •

Ground flashers: several triggers come from ground sources and present a light
profile that lasts for tens on $\mu s$. These events are usually found near
airports, and are probably produced by warning lights (Fig.4, c).

  * •

EAS-like events: The TUS detector [5] has found several events with the shape
and characteristics that resemble the ones expected from EASs [6], the most
interesting being event TUS161003, over Minnesota [7]. In Mini-EUSO data there
are also several events with a lightcurve and time profile that matches the
bi-gaussian shape expected from an EAS, with a faster rising and a slower
decay, and the signal being confined in one or few neighbouring pixels. The
vast majority of these events are detected near the location of airports and
are triggered many times while moving in the focal plane, suggesting that
airport’s lights might be the sources of these events. In Fig. 5 a comparison
between TUS161003 and one Mini-EUSO event is presented. Through a comparison
with a simulation, the cosmic origin has been excluded also for the only event
with these characteristics found over the ocean and triggered only once
(Fig.6). The research of these events is still ongoing.

Figure 5: Blue: Lightcurve of TUS161003 event as seen by TUS. Even though the
event has probably an anthropogenic origin, it presents all the features of an
EAS signal. Red: An event detected by Mini-EUSO near lake Michigan. It
presents the same time profile of the event seen by TUS even though it is
$\sim$ 10 times brighter, assuming the detector efficiency quoted in [1]. It
is triggered four times by Mini-EUSO, it is therefore produced by a ground
source. The signal appears in an area near three small airports. Figure 6:
Left: Mini-EUSO event detected off the coast of Sri Lanka . Center and Right:
EAS simulated through ESAF [8] with different energy and zenith angle. The
simulation with $Z=50^{\circ}$ and energy $5\times 10^{21}~{}eV$ produce a
footprint on the focal plane similar to the event but the lightcurve is too
short, while the event at $Z=80^{\circ}$ and energy $2\times 10^{22}~{}eV$
correctly reproduces the lightcurve but has a different shape.

  * •

Direct cosmic rays are low energy cosmic rays directly impinging on the
detector. They are usually characterized by a signal that reaches the maximum
in only 1 GTU and then presents an exponential decay. Direct cosmic rays are
in general easy to recognise, and therefore do not represent a problem despite
that they represent a huge part of the $2.5~{}\mu s$ dataset. They can appear
with different shapes on the focal plane (a single bright pixel, a blob, a
track).

##  4 Conclusion

Mini-EUSO $2.5~{}\mu s$ trigger logic has proven to be working as expected.
The fast adaptive thresholds prevent bright but static sources from
triggering. The trigger logic and data acquisition system guarantee to keep
the average dead time $\sim~{}25\%$. This is obtained without a second level
trigger, and it is going to decrease once the firmware update presented in 1.2
will be installed. With the exception of a handful of pixels on the border,
the distribution of pixels over threshold is quite uniform and it is possible
to recognise different and interesting categories of events in the data.

## Acknowledgments

This work was supported by State Space Corporation ROSCOSMOS, by the Italian
Space Agency through the ASI INFN agreement n. 2020-26-HH.0 and contract n.
2016-1-U.0, by the French space agency CNES, National Science Centre in Poland
grant 2017/27/B/ST9/02162.

This research has been supported by the Interdisciplinary Scientific and
Educational School of Moscow University "Fundamental and Applied Space
Research". The article has been prepared based on research materials carried
out in the space experiment "UV atmosphere"

## References

  * [1] S. Bacholle et al.; 2021 ApJS 253 36
  * [2] A. Belov et al.; Advances in Space Research,Volume 62, Issue 10, Pages 2966-2976
  * [3] Neubert, T., Østgaard, N., Reglero, V. et al.; Sci Rev 215, 26 (2019).
  * [4] POEMMA collaboration et al.; JCAP06(2021)007
  * [5] AIP Conference Proceedings 566, 57 (2001); https://doi.org/10.1063/1.1378622
  * [6] Klimov et al., PoS(ICRC2021), contribution #598
  * [7] B.A. Khrenov et al.; JCAP03(2020)033
  * [8] Berat et al., Astroparticle Physics Volume 33, doi.org/10.1016/j.astropartphys.2010.02.005

The JEM-EUSO Collaboration
(author-list as of July 1st, 2021)

G. Abdellaouiah, S. Abefq, J.H. Adams Jr.pd, D. Allardcb, G. Alonsomd, L.
Anchordoquipe, A. Anzaloneeh,ed, E. Arnoneek,el, K. Asanofe, R. Attallahac, H.
Attouiaa, M. Ave Pernasmc, M. Bagheriph, J. Balázla, M. Bakiriaa, D.
Barghiniel,ek, S. Bartocciei,ej, M. Battistiek,el, J. Bayerdd, B.
Beldjilaliah, T. Belenguermb, N. Belkhalfaaa, R. Bellottiea,eb, A.A. Belovkb,
K. Benmessaiaa, M. Bertainaek,el, P.F. Bertonepf, P.L. Biermanndb, F.
Biscontiel,ek, C. Blaksleyft, N. Blancoa, S. Blin-Bondilca,cb, P. Bobikla, M.
Bogomilovba, K. Bolmgrenna, E. Bozzoob, S. Brizpb, A. Brunoeh,ed, K.S.
Caballerohd, F. Cafagnaea, G. Cambiéei,ej, D. Campanaef, J-N. Capdeviellecb,
F. Capelde, A. Carameteja, L. Carameteja, P. Carlsonna, R. Carusoec,ed, M.
Casolinoft,ei, C. Cassardoek,el, A. Castellinaek,em, O. Catalanoeh,ed, A.
Cellinoek,em, K. Černýbb, M. Chikawafc, G. Chiritoija, M.J. Christlpf, R.
Colalilloef,eg, L. Contien,ei, G. Cottoek,el, H.J. Crawfordpa, R. Cremoniniel,
A. Creusotcb, A. de Castro Gónzalezpb, C. de la Tailleca, L. del Peralmc, A.
Diaz Damiancc, R. Diesingpb, P. Dinaucourtca, A. Djakonowia, T. Djemilac, A.
Ebersoldtdb, T. Ebisuzakift, J. Eserpb, F. Fenuek,el, S. Fernández-Gonzálezma,
S. Ferrareseek,el, G. Filippatospc, W.I. Finchpc C. Fornaroen,ei, M. Foukaab,
A. Franceschiee, S. Franchinimd, C. Fuglesangna, T. Fujiifg, M. Fukushimafe,
P. Galeottiek,el, E. García-Ortegama, D. Gardiolek,em, G.K. Garipovkb, E.
Gascónma, E. Gazdaph, J. Gencilb, A. Golzioek,el, C. González Alvaradomb, P.
Gorodetzkyft, A. Greenpc, F. Guarinoef,eg, C. Guépinpl, A. Guzmándd, Y.
Hachisuft, A. Haungsdb, J. Hernández Carreteromc, L. Hulettpc, D. Ikedafe, N.
Inouefn, S. Inoueft, F. Isgròef,eg, Y. Itowfk, T. Jammerdc, S. Jeonggb, E.
Jovenme, E.G. Juddpa, J. Jochumdc, F. Kajinoff, T. Kajinofi, S. Kalliaf, I.
Kanekoft, Y. Karadzhovba, M. Kasztelania, K. Katahiraft, K. Kawaift, Y.
Kawasakift, A. Kedadraaa, H. Khalesaa, B.A. Khrenovkb, Jeong-Sook Kimga, Soon-
Wook Kimga, M. Kleifgesdb, P.A. Klimovkb, D. Kolevba, I. Kreykenbohmda, J.F.
Krizmanicpf,pk, K. Królikia, V. Kungelpc, Y. Kuriharafs, A. Kusenkofr,pe, E.
Kuznetsovpd, H. Lahmaraa, F. Lakhdariag, J. Licandrome, L. López Campanoma, F.
López Martínezpb, S. Mackovjakla, M. Mahdiaa, D. Mandátbc, M. Manfrinek,el, L.
Marcelliei, J.L. Marcosma, W. Marszałia, Y. Martínme, O. Martinezhc, K.
Masefa, R. Matevba, J.N. Matthewspg, N. Mebarkiad, G. Medina-Tancoha, A.
Menshikovdb, A. Merinoma, M. Meseef,eg, J. Meseguermd, S.S. Meyerpb, J.
Mimouniad, H. Miyamotoek,el, Y. Mizumotofi, A. Monacoea,eb, J.A. Morales de
los Ríosmc, M. Mastafapd, S. Nagatakift, S. Naitamorab, T. Napolitanoee, J. M.
Nachtmanpi A. Neronovob,cb, K. Nomotofr, T. Nonakafe, T. Ogawaft, S. Ogiofl,
H. Ohmorift, A.V. Olintopb, Y. Onelpi G. Osteriaef, A.N. Otteph, A.
Pagliaroeh,ed, W. Painterdb, M.I. Panasyukkb, B. Panicoef, E. Parizotcb, I.H.
Parkgb, B. Pastircakla, T. Paulpe, M. Pechbb, I. Pérez-Grandemd, F.
Perfettoef, T. Peteroc, P. Picozzaei,ej,ft, S. Pindadomd, L.W. Piotrowskiib,
S. Pirainodd, Z. Plebaniakek,el,ia, A. Pollinioa, E.M. Popescuja, R.
Preveteef,eg, G. Prévôtcb, H. Prietomc, M. Przybylakia, G. Puehlhoferdd, M.
Putisla, P. Reardonpd, M.H.. Renopi, M. Reyesme, M. Ricciee, M.D. Rodríguez
Fríasmc, O.F. Romero Matamalaph, F. Rongaee, M.D. Sabaumb, G. Saccáec,ed, G.
Sáez Canomc, H. Sagawafe, Z. Sahnouneab, A. Saitofg, N. Sakakift, H.
Salazarhc, J.C. Sanchez Balanzarha, J.L. Sánchezma, A. Santangelodd, A. Sanz-
Andrésmd, M. Sanz Palominomb, O.A. Saprykinkc, F. Sarazinpc, M. Satofo, A.
Scagliolaea,eb, T. Schanzdd, H. Schielerdb, P. Schovánekbc, V. Scottief,eg, M.
Serrame, S.A. Sharakinkb, H.M. Shimizufj, K. Shinozakiia, J.F. Sorianope, A.
Sotgiuei,ej, I. Stanja, I. Strharskýla, N. Sugiyamafj, D. Supanitskyha, M.
Suzukifm, J. Szabelskiia, N. Tajimaft, T. Tajimaft, Y. Takahashifo, M.
Takedafe, Y. Takizawaft, M.C. Talaiac, Y. Tamedafp, C. Tenzerdd, S.B.
Thomaspg, O. Tibollahe, L.G. Tkachevka, T. Tomidafh, N. Toneft, S. Toscanoob,
M. Traïcheaa, Y. Tsunesadafl, K. Tsunoft, S. Turrizianift, Y. Uchihorifb, O.
Vaduvescume, J.F. Valdés-Galiciaha, P. Vallaniaek,em, L. Valoreef,eg, G.
Vankova-Kirilovaba, T. M. Venterspj, C. Vigoritoek,el, L. Villaseñorhb, B.
Vlcekmc, P. von Ballmooscc, M. Vrabellb, S. Wadaft, J. Watanabefi, J. Watts
Jr.pd, R. Weigand Muñozma, A. Weindldb, L. Wienckepc, M. Willeda, J. Wilmsda,
D. Winnpm T. Yamamotoff, J. Yanggb, H. Yanofm, I.V. Yashinkb, D. Yonetokufd,
S. Yoshidafa, R. Youngpf, I.S Zguraja, M.Yu. Zotovkb, A. Zuccaro Marchift

aa Centre for Development of Advanced Technologies (CDTA), Algiers, Algeria
ab Dep. Astronomy, Centre Res. Astronomy, Astrophysics and Geophysics (CRAAG),
Algiers, Algeria
ac LPR at Dept. of Physics, Faculty of Sciences, University Badji Mokhtar,
Annaba, Algeria
ad Lab. of Math. and Sub-Atomic Phys. (LPMPS), Univ. Constantine I,
Constantine, Algeria
af Department of Physics, Faculty of Sciences, University of M’sila, M’sila,
Algeria
ag Research Unit on Optics and Photonics, UROP-CDTA, Sétif, Algeria
ah Telecom Lab., Faculty of Technology, University Abou Bekr Belkaid, Tlemcen,
Algeria
ba St. Kliment Ohridski University of Sofia, Bulgaria
bb Joint Laboratory of Optics, Faculty of Science, Palacký University,
Olomouc, Czech Republic
bc Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic
ca Omega, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France
cb Université de Paris, CNRS, AstroParticule et Cosmologie, F-75013 Paris,
France
cc IRAP, Université de Toulouse, CNRS, Toulouse, France
da ECAP, University of Erlangen-Nuremberg, Germany
db Karlsruhe Institute of Technology (KIT), Germany
dc Experimental Physics Institute, Kepler Center, University of Tübingen,
Germany
dd Institute for Astronomy and Astrophysics, Kepler Center, University of
Tübingen, Germany
de Technical University of Munich, Munich, Germany
ea Istituto Nazionale di Fisica Nucleare - Sezione di Bari, Italy
eb Universita’ degli Studi di Bari Aldo Moro and INFN - Sezione di Bari, Italy
ec Dipartimento di Fisica e Astronomia "Ettore Majorana", Universita’ di
Catania, Italy
ed Istituto Nazionale di Fisica Nucleare - Sezione di Catania, Italy
ee Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati,
Italy
ef Istituto Nazionale di Fisica Nucleare - Sezione di Napoli, Italy
eg Universita’ di Napoli Federico II - Dipartimento di Fisica "Ettore
Pancini", Italy
eh INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo, Italy
ei Istituto Nazionale di Fisica Nucleare - Sezione di Roma Tor Vergata, Italy
ej Universita’ di Roma Tor Vergata - Dipartimento di Fisica, Roma, Italy
ek Istituto Nazionale di Fisica Nucleare - Sezione di Torino, Italy
el Dipartimento di Fisica, Universita’ di Torino, Italy
em Osservatorio Astrofisico di Torino, Istituto Nazionale di Astrofisica,
Italy
en Uninettuno University, Rome, Italy
fa Chiba University, Chiba, Japan
fb National Institutes for Quantum and Radiological Science and Technology
(QST), Chiba, Japan
fc Kindai University, Higashi-Osaka, Japan
fd Kanazawa University, Kanazawa, Japan
fe Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Japan
ff Konan University, Kobe, Japan
fg Kyoto University, Kyoto, Japan
fh Shinshu University, Nagano, Japan
fi National Astronomical Observatory, Mitaka, Japan
fj Nagoya University, Nagoya, Japan
fk Institute for Space-Earth Environmental Research, Nagoya University,
Nagoya, Japan
fl Graduate School of Science, Osaka City University, Japan
fm Institute of Space and Astronautical Science/JAXA, Sagamihara, Japan
fn Saitama University, Saitama, Japan
fo Hokkaido University, Sapporo, Japan
fp Osaka Electro-Communication University, Neyagawa, Japan
fq Nihon University Chiyoda, Tokyo, Japan
fr University of Tokyo, Tokyo, Japan
fs High Energy Accelerator Research Organization (KEK), Tsukuba, Japan
ft RIKEN, Wako, Japan
ga Korea Astronomy and Space Science Institute (KASI), Daejeon, Republic of
Korea
gb Sungkyunkwan University, Seoul, Republic of Korea
ha Universidad Nacional Autónoma de México (UNAM), Mexico
hb Universidad Michoacana de San Nicolas de Hidalgo (UMSNH), Morelia, Mexico
hc Benemérita Universidad Autónoma de Puebla (BUAP), Mexico
hd Universidad Autónoma de Chiapas (UNACH), Chiapas, Mexico
he Centro Mesoamericano de Física Teórica (MCTP), Mexico
ia National Centre for Nuclear Research, Lodz, Poland
ib Faculty of Physics, University of Warsaw, Poland
ja Institute of Space Science ISS, Magurele, Romania
ka Joint Institute for Nuclear Research, Dubna, Russia
kb Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Russia
kc Space Regatta Consortium, Korolev, Russia
la Institute of Experimental Physics, Kosice, Slovakia
lb Technical University Kosice (TUKE), Kosice, Slovakia
ma Universidad de León (ULE), León, Spain
mb Instituto Nacional de Técnica Aeroespacial (INTA), Madrid, Spain
mc Universidad de Alcalá (UAH), Madrid, Spain
md Universidad Politécnia de madrid (UPM), Madrid, Spain
me Instituto de Astrofísica de Canarias (IAC), Tenerife, Spain
na KTH Royal Institute of Technology, Stockholm, Sweden
oa Swiss Center for Electronics and Microtechnology (CSEM), Neuchâtel,
Switzerland
ob ISDC Data Centre for Astrophysics, Versoix, Switzerland
oc Institute for Atmospheric and Climate Science, ETH Zürich, Switzerland
pa Space Science Laboratory, University of California, Berkeley, CA, USA
pb University of Chicago, IL, USA
pc Colorado School of Mines, Golden, CO, USA
pd University of Alabama in Huntsville, Huntsville, AL; USA
pe Lehman College, City University of New York (CUNY), NY, USA
pf NASA Marshall Space Flight Center, Huntsville, AL, USA
pg University of Utah, Salt Lake City, UT, USA
ph Georgia Institute of Technology, USA
pi University of Iowa, Iowa City, IA, USA
pj NASA Goddard Space Flight Center, Greenbelt, MD, USA
pk Center for Space Science & Technology, University of Maryland, Baltimore
County, Baltimore, MD, USA
pl Department of Astronomy, University of Maryland, College Park, MD, USA
pm Fairfield University, Fairfield, CT, USA


# Field induced conducting state in Mott insulator

X. Z. Zhang  College of Physics and Materials Science, Tianjin Normal
University, Tianjin 300387, China  Z. Song<EMAIL_ADDRESS>School of
Physics, Nankai University, Tianjin 300071, China

###### Abstract

Electron–electron repulsion, on the one hand, can result in bound pair, which
has heavy effective mass. On the other hand, it is also the cause of Mott
insulator. We study the effect of a staggered magnetic field on a Hubbard
model. We find that a bound pair with large energy bandwidth can be formed
under the resonant staggered field, being the half of Hubbard repulsion
strength. Accordingly, the system exhibits following dynamical behaviors: (i)
When an electric field is applied, fast bound pair Bloch oscillation occurs,
while a single electron is frozen. (ii) When a quenching resonant field is
applied to an initial antiferromagnetic Mott insulating state, the final state
becomes doublon conducting state manifested by the non-zero $\eta$ correlator
and large charge fluctuation. Our finding indicates that the cooperation of
electron-electron correlation and modulated external field can induce novel
quench dynamics.

##  I Introduction

The Hubbard model represents a minimal starting point for modeling strongly
correlated systems Hubbard and Flowers (1964); Fisher et al. (1989); Georges
et al. (1996); Sachdev (2011); Keimer et al. (2015). It has relevance for a
broad range of correlation phenomena, including high-temperature
superconductivity Lee et al. (2006), metal-insulator transition Fazekas
(1999), quantum criticality Sachdev (2011), to interacting topological states
of matter Wen (2007). Over the past decades, the Hubbard model lies at the
heart of investigating the cold atom quantum simulations since these strongly
correlated systems are faithful representations of the bosonic or fermionic
Hubbard models Jaksch and Zoller (2005); Bloch et al. (2008, 2012). Recent
developments in quantum simulations of the Hubbard model suggest that these
cold atom systems are particularly suitable for studying quantum many-body
dynamics Giamarchi et al. (2016), which is less refereed in the content of
condensed matter systems. A fascinating subject is quantum quench dynamics in
the strongly correlated system. It studies how a many-body system is prepared
initially in a certain initial state, and evolves unitarily in time following
the sudden change of the parameters to a final Hamiltonian Calabrese and Cardy
(2006). Dynamics induced by a quantum quench has become an active topic of
research because it poses many fundamental questions that can also be studied
by current generation experiments.

A typical feature of quench dynamics is that the considered many-body systems
are usually far from equilibrium, which can induce many intriguing phenomena
in condensed-matter materials Tokura (2006); Yonemitsu and Nasu (2008); Aoki
et al. (2014); Giannetti et al. (2016). One striking example of recent
experimental observations is the light-induced superconducting-like properties
in some high-$T_{c}$ cuprates Fausti et al. (2011); Hu et al. (2014); Kaiser
et al. (2014); Nicoletti et al. (2014) and alkali-doped fullerides Mitrano et
al. (2016); Cantaluppi et al. (2018), which have stimulated many theoretical
investigations Sentef et al. (2016); Patel and Eberlein (2016); Knap et al.
(2016); Kennes et al. (2017); Sentef (2017); Babadi et al. (2017); Murakami et
al. (2017); Wang et al. (2018). In addition, the pump-probe experiments
realized in cold-atom setups have been also triggered an intensive theoretical
investigation of out-of-equilibrium dynamics in quantum many-body systems (see
e.g. the reviews Polkovnikov et al. (2011); Calabrese et al. (2016); Vasseur
and Moore (2016); Gogolin and Eisert (2016); Tomita et al. (2017)). They have
been successfully used for understanding the mechanisms of relaxation
Cazalilla (2006); Barthel and Schollwöck (2008); Cramer et al. (2008);
Calabrese et al. (2011); Mossel and Caux (2012); Collura et al. (2013);
Fagotti and Essler (2013); Kormos et al. (2014); Piroli and Calabrese (2017),
showing that local-in-space observables relax at large times. Such novel
dynamic phenomena are ultimately attributed to the cooperation between the
interaction and the external field.

This paper aims to study the effect of the staggered external field on a
Fermi-Hubbard model in the deep Mott insulating phase. We show that the quench
resonant field can build a bridge for the transport of the particles
demonstrated by the large charge fluctuation and enhancement of the doublon-
doublon correlation. Specifically, an initial anti-ferromagnetic insulating
state becomes a conducting state under the action of the post-quench
Hamiltonian. The underlying mechanism, which can be elucidated by the two-
particle system, is that: The combination between the on-site interaction and
the presence of resonant staggered field culminates in the formation of a
bound pair (BP) with large energy bandwidth. However, the bandwidth of a
single particle is shrunk. Correspondingly, the system exhibit two distinct
dynamic behaviors when the dc electric field is applied. For the scattering
states, the particles are frozen in the initial place. As for the BPs, they
show breathing behavior populating a large interval that is determined by the
bandwidth of BP. This is in stark difference with the case of the staggered
field free, wherein the BP is frozen at the starting point. Hence, the large-
scale migration of BPs induced by the external field provides an insight into
the dynamics transition of the system from the Mott insulating to conducting
phase. It is hoped that these results can motivate further studies of both
fundamental aspects and potential applications of quench interacting many-body
systems.

Figure 1: Schematic illustration of the structure of the Hubbard model
concerned in this work, which supports the transition from insulating to
conducting states induced by the competition between on-site repulsion and
staggered magnetic field. It consists of two sub-lattices $A$ (empty) and $B$
(grey). The electrons are subjected by opposite magnetic fields which are
denoted by orange and green shaded regions, respectively. We present a
possible electron filling configuration to illustrate the underlying mechanism
of the transition. The corresponding on-site energy is denoted under each
sits. (i) In the case of $V=0$ and large $U$, the hopping processes denoted by
red arrows are allowed, while green is forbidden. The half-filled ground state
is Mott insulating state. (ii) When the resonant magnetic field is switched
on, the hopping processes denoted by green arrows are allowed, while red is
forbidden, supporting conducting state even at half filling.

The rest of the paper is structured as follows: In Sec. II, we introduce the
Hubbard model subjected to a staggered field. The exact two-particle
equivalent Hamiltonian and effective Hamiltonian about the BP solution are
investigated based on the symmetry of the system. Sec. III is devoted to
investigate the dynamics demonstrated by the BO. Physical insight into the
many-body system with a resonant staggered field is provided by the so-called
breathing oscillation of BP. In Sec. IV, the transition from the Mott
insulating to doublon conducting phase is investigated through quench dynamics
by examining the charge fluctuation and $\eta$ correlator. We conclude and
discuss our results in Sec. V.

##  II Model

###  II.1 the Hubbard model with staggered potential

We consider a many-body quantum system $H=H_{0}+H_{s}$ and show the basic
property determined by the interplay between the interaction and external
field. The system Hamiltonian given by the Hubbard model with bipartite
lattice

$H_{0}=-\sum_{i\in A,j\in
B}^{N}\sum_{\sigma=\uparrow,\downarrow}J_{ij}c_{i,\sigma}^{\dagger}c_{j,\sigma}+\text{H.c.}+U\sum_{j\in
A\cup B}n_{j,\uparrow}n_{j,\downarrow},$ (1)

is subjected to a staggered external field

$H_{s}=\frac{V}{2}\sum_{j\in
A}\left(n_{j,\uparrow}-n_{j,\downarrow}\right)-\frac{V}{2}\sum_{j\in
B}\left(n_{j,\uparrow}-n_{j,\downarrow}\right).$ (2)

where the operator $c_{j,\sigma}$ is the annihilation operator of a
spin-$\sigma$ fermion at site $j$, satisfying the usual fermion
anticommutation relations $\\{c_{i,\sigma}^{\dagger},$
$c_{j,\sigma^{\prime}}\\}=\delta_{i,j}\delta_{\sigma,\sigma^{\prime}}$ and
$\\{c_{i,\sigma},$ $c_{j,\sigma^{\prime}}\\}=0$; the system can be divided
into two disjoint sets $A$ and $B$ and the hopping matrix elements $J_{ij}$
are required to be real and satisfy $J_{ij}=J_{ji}$; the number operators are
$n_{j,\sigma}=c_{j,\sigma}^{\dagger}c_{j,\sigma}$, while $U$ is the on-site
energy denoting the repulsive particle-particle interaction; the term $V$
describes spin-dependent staggered potentials. For convenience and clarity,
the number of the sublattice sites and the filled-particles are denoted by $N$
and $M$, respectively. The considered system is illustrated in Fig. 1. Here,
we want to stress that the staggered magnetic field can alter significantly
the mobility of the particles. A simple example is employed to elucidate the
underlying mechanism in Fig.
LABEL:fig_illustration._It_is_shown_that_the_external_staggered_field_provides_an_energy_shell_that_allows_particle_pairs_to_move_through_more_channels,_which_may_facilitate_the_enhancement_of_the_conductivity._In_the_subsequent_sections,_we_will_gradually_dissect_the_influence_of_external_magnetic_field_on_the_energy_spectrum_and_corresponding_dynamics._To_this_end,_we_first_focus_on_the_symmetry_of_the_system.
In the absence of the staggered field $V=0$, the system Hamiltonian $H$ has
two sets of SU(2) symmetry. The first of these relates to spinful particles
which can be characterized by the generators

$\displaystyle s^{+}$ $\displaystyle=$
$\displaystyle\left(s^{-}\right)^{\dagger}=\sum_{j\in A\cup B}s_{j}^{+},$ (3)
$\displaystyle s^{z}$ $\displaystyle=$ $\displaystyle\sum_{j\in A\cup
B}s_{j}^{z},$ (4)

where the local operators $s_{j}^{+}=c_{j,\uparrow}^{\dagger}c_{j,\downarrow}$
and $s_{j}^{z}=\left(n_{j,\uparrow}-n_{j,\downarrow}\right)/2$ obey the Lie
algebra, i.e., $[s_{j}^{+},$ $s_{j}^{-}]=2s_{j}^{z}$, and $[s_{j}^{z},$
$s_{j}^{\pm}]=\pm s_{j}^{\pm}$. The second, often referred to as $\eta$
symmetry, relates to spinless quasiparticles. The corresponding generators can
be given as

$\displaystyle\eta^{+}$ $\displaystyle=$
$\displaystyle\left(\eta^{-}\right)^{\dagger}=\sum_{j\in A\cup
B}\eta_{j}^{+},$ (5) $\displaystyle\eta^{z}$ $\displaystyle=$
$\displaystyle\sum_{j\in A\cup B}\eta_{j}^{z},$ (6)

with $\eta_{j}^{+}=\lambda c_{j,\uparrow}^{\dagger}c_{j,\downarrow}^{\dagger}$
and $\eta_{j}^{z}=\left(n_{j,\uparrow}+n_{j,\downarrow}-1\right)/2$ satisfying
commutation relation, i.e., $[\eta_{j}^{+},$ $\eta_{j}^{-}]=2\eta_{j}^{z}$,
and $[\eta_{j}^{z},$ $\eta_{j}^{\pm}]=\pm\eta_{j}^{\pm}$. Here we assume a
bipartite lattice and $\lambda=1$ for $j\in\left\\{A\right\\}$ and $-1$ for
$j\in\left\\{B\right\\}$. Evidently, the presence of the staggered field
spoils such SU(2) symmetries but remains $s^{z}$ symmetry, that is $[s^{z},$
$H]=0$. Hence, the system can be divided into different invariant subspaces in
terms of the quantum number of $s^{z}$.

###  II.2 bound pair solution

Figure 2: Schematic of the structures of equivalent Hamiltonians for the one-
dimensional two-particle Hubbard model with the staggered field. (a) In the
invariant subspace with center momentum $k$, the equivalent Hamiltonian
$H_{k}$ describes a $4$-leg ladder with $k$-dependent hopping. The on-site
interaction can be treated as the on-site potential that only exists in the
$j=0$ unit cell. The staggered field appears periodically within each unit
cell. (b) The purple box describes the effective Hamiltonian about the BP
approximate solution. In the near-resonant condition and large $U$ limit,
$H_{\mathrm{eff}}$ characterizes the first-order perturbation.

To gain physical insight into subsequent many-body dynamics, we first
investigate the two-particle invariant subspaces. When $V=0$, the 1D
homogeneous system can harbor the bound pair (BP) solution with the form of
$\varepsilon_{\mathrm{bp}}^{k}=\sqrt{U^{2}+16J^{2}\cos^{2}\frac{k}{2}}$
Winkler et al. (2006); Jin and Song (2011); Zhang et al. (2016); Zhang and
Song (2021). The corresponding bandwidth is $8J^{2}/U$ as $J\ll U$. This
becomes very complicated when staggered field presents. For clarity and
simplicity, we suppose that the Hamiltonian $H$ still describes a 1D
homogeneous ring system. The translation symmetry combing with $s^{z}$
symmetry allows us to construct the basis of invariant subspace $s^{z}=0$ as
follow

$\left(\begin{array}[]{c}\left|j,1,k\right\rangle\\\
\left|j,2,k\right\rangle\\\ \left|j,3,k\right\rangle\\\
\left|j,4,k\right\rangle\end{array}\right)=\sum_{l}\frac{1}{\sqrt{N}}e^{ik(l+j)}\left(\begin{array}[]{c}\alpha_{l,\uparrow}^{\dagger}\alpha_{l+j,\downarrow}^{\dagger}|\mathrm{Vac}\rangle\\\
\alpha_{l,\uparrow}^{\dagger}\beta_{l+j,\downarrow}^{\dagger}|\mathrm{Vac}\rangle\\\
\beta_{l,\uparrow}^{\dagger}\beta_{l+j,\downarrow}^{\dagger}|\mathrm{Vac}\rangle\\\
\beta_{l,\uparrow}^{\dagger}\alpha_{l+j,\downarrow}^{\dagger}|\mathrm{Vac}\rangle\end{array}\right),$
(7)

with the notation $\alpha_{l,\sigma}^{\dagger}=c_{l,\sigma}^{\dagger}$
($l\in\left\\{A\right\\}$) and
$\beta_{l,\sigma}^{\dagger}=c_{l,\sigma}^{\dagger}$
($l\in\left\\{B\right\\}$). Here $k=2n\pi/N$ is the momentum vector indexing
the subspace and $j=0...N-1$ describes the relative distance between the two
particles. The other two invariant subspaces $s^{z}=\pm 1$ are not considered
since the bound states only reside in the subspace with $s^{z}=0$. In such a
two-particle Hilbert space, the Hamiltonian $H$ can be written as
$H=\sum_{k}H_{k}$ where

$\displaystyle H_{k}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}h_{j,k}-J\sum_{j=0}^{N-1}\sum_{m}\left|j,m,k\right\rangle\left\langle
j,m+1,k\right|+\text{H.c.}$ (8)
$\displaystyle+\sum_{j=0}^{N-1}\sum_{m}\mu_{m}\left|j,m,k\right\rangle\left\langle
j,m,k\right|+$ $\displaystyle+U(\left|0,1,k\right\rangle\left\langle
0,1,k\right|+\left|0,3,k\right\rangle\left\langle 0,3,k\right|),$

and

$\displaystyle h_{j,k}$ $\displaystyle=$
$\displaystyle\lambda_{1}\left|j,1,k\right\rangle\left\langle
j+1,4,k\right|+\lambda_{2}\left|j,2,k\right\rangle\left\langle j+1,1,k\right|$
(9) $\displaystyle+\lambda_{3}\left|j,2,k\right\rangle\left\langle
j+1,3,k\right|$ $\displaystyle+\lambda_{4}\left|j,3,k\right\rangle\left\langle
j+1,4,k\right|+\text{H.c.}$

with $\lambda_{1,2,3,4}=-J\left(1,\text{ }e^{ik},\text{ }1,\text{
}e^{ik}\right)$ and $\mu_{1,2,3,4}=\left(0,\text{ }V,\text{ }0,\text{
}-V\right)$. Note that $\left|N,m,k\right\rangle=\left|0,m,k\right\rangle$.
The expression of $H_{k}$ in Eq. (8) has a clear physical meaning: $j$, $m$,
$k$ denotes the site state for the $j$-th site on the $m$-leg of a $4$-leg
ladder system with the modulated inter-cell coupling. The structure of $H_{k}$
is schematically illustrated in Fig. 2(a). The BP state mainly piles up at the
$0$-th site due to the near resonance condition of $\left|\Delta\right|/J\ll
1$, where $\Delta=U-V$. Hence, the corresponding approximate solution can be
obtained within such unitcell. The involved four kets are denoted by purple
box in Fig. 2(b). In the basis of $\left\\{\left|0,1,k\right\rangle\text{,
}\left|0,2,k\right\rangle\text{, }\left|0,3,k\right\rangle\text{,
}\left|N-1,2,k\right\rangle\right\\}$, the matrix form of effective BP
Hamiltonian is given as

$H_{\mathrm{eff}}=-J\left(\begin{array}[]{cccc}-\Delta/J&1&0&e^{-ik}\\\
1&0&1&0\\\ 0&1&-\Delta/J&1\\\ e^{ik}&0&1&0\end{array}\right)+VI_{4},$ (10)

Taking the linear transformation,

$\left(\begin{array}[]{c}\left|1,k\right\rangle_{\mathrm{bp}}\\\
\left|2,k\right\rangle_{\mathrm{bp}}\\\
\left|3,k\right\rangle_{\mathrm{bp}}\\\
\left|4,k\right\rangle_{\mathrm{bp}}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}-1&0&1&0\\\
0&-1&0&1\\\ 1&0&1&0\\\
0&1&0&1\end{array}\right)\left(\begin{array}[]{c}\left|0,1,k\right\rangle\\\
\left|0,2,k\right\rangle\\\ \left|0,3,k\right\rangle\\\
\left|N-1,2,k\right\rangle\end{array}\right),$ (11)

the effective BP Hamiltonian can be rewritten as

$H_{\mathrm{eff}}=\left(\begin{array}[]{cc}h_{1,\mathrm{eff}}&0\\\
0&h_{2,\mathrm{eff}}\end{array}\right)+VI_{4},$ (12)

with

$\displaystyle h_{1,\mathrm{eff}}$ $\displaystyle=$
$\displaystyle-J\left(\begin{array}[]{cc}\Delta/J&2i\sin\left(k/4\right)\\\
-2i\sin\left(k/4\right)&0\end{array}\right),$ (15) $\displaystyle
h_{2,\mathrm{eff}}$ $\displaystyle=$
$\displaystyle-J\left(\begin{array}[]{cc}\Delta/J&-2\cos\left(k/4\right)\\\
-2\cos\left(k/4\right)&0\end{array}\right).$ (18)

Evidently, the effective $H_{eff}$ can be divided into two invariant
subspaces, each of which is described by a $k$-dependent two-level system. The
corresponding eigen energies are

$\displaystyle\varepsilon_{1,\sigma}^{k}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{2}+\sigma\sqrt{(\frac{\Delta}{2})^{2}+4J^{2}\sin^{2}\frac{k}{4}},$
(19) $\displaystyle\varepsilon_{2,\sigma}^{k}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{2}+\sigma\sqrt{(\frac{\Delta}{2})^{2}+4J^{2}\cos^{2}\frac{k}{4}},$
(20)

with $\sigma=\pm$. For each subsystem, there exists an energy gap $\Delta$
between the two bands. When the resonance condition ($\Delta=0$) is achieved,
not only the upper and lower energy bands are connected with each other
($\varepsilon_{j,+}^{0}=\varepsilon_{j,-}^{0}$, $j=1,$ $2$), but also four
bands form a single BP band with the width $4J$. This evidence suggests that
the bound pair velocity is altered from the order of $J^{2}/U$ to $J$. To
check this point, we compare the result obtained by Eqs. (19)-(20) with that
of the exact diagonalization method in Fig. 3. It demonstrates that two kinds
of results accord with each other in the large $U$ limit. This characteristic
plays the key role to under the following Bloch oscillation (BO) dynamics.

Figure 3: Comparison between the approximate BP solution and exact
diagonalization results. There are three typical cases: (a) $U/J=8$, $V/J=7$;
(b) $U/J=8$, $V/J=8$; (c) $U/J=8$, $V/J=9$; One can see that when a large $U$
limit takes, two such results accord well with each other. $H_{eff}$ can well
capture the information about the touching between the upper and lower BP
band. The black shaded region represents the scattering band, which can be
obtained by the single-particle solution.

##  III dynamics of two-particle system

It is well known that a particle in a periodic potential with an additional
constant force performs BO, which can be explained by the formation of a
Wannier-Stark ladder energy spectrum. BO is an ubiquitous phenomenon of
coherent wave transport that can be observed in a wide variety of physical
system Voisin et al. (1988); Feldmann et al. (1992); Ben Dahan et al. (1996);
Wilkinson et al. (1996); Morandotti et al. (1999); Pertsch et al. (1999);
Rosam et al. (2001); Morsch et al. (2001); Longhi (2008); Dreisow et al.
(2009); Zhang et al. (2017) and reflects the spectrum property of the system
in the absence of the weak dc electric field Hartmann et al. (2004); Breid et
al. (2006). In addition, a cluster of bound particles can also present the BO
which shares the same mechanism of the single-particle case Lin et al. (2014);
Zhang et al. (2016). In this section, we use the BO to investigate the
influence of a staggered field on the transport properties of the system.

Figure 4: Band structures for the systems with and without staggered field
$V$: (a1) $U/J=8$, $V/J=7$; (a2) $U/J=8$, $V=0$. The presence of $V$ results
in the two-band structure for the single-particle subspace so that the
scattering bands of (a1) consists of three parts with different energy scale.
Such the scattering band can be understood by the superposition of the
energies of two-free particles. The resonant staggered field expands the BP
band but suppresses the scattering band. The different band structures
determine the system dynamics in the presence of a dc electric field. The
panels (b1)-(b2) describes the BP dynamics, while the (c1)-(c2) exhibit the
time evolution of the two scattering-particle state
$\left|\Phi_{1}\left(0\right)\right\rangle$. It clearly shows that the
scattering particles are frozen at their initial locations and the BP can
widen and shrink periodically populating an interval $4J/F$ when dc electric
field is applied. This indicates that the BP band induced by staggered field
is $4J$, which again confirms our analysis.

###  III.1 single-electron solution: Rice-Mele model

We first study the 1D bipartite Hamiltonian in the single-particle invariant
subspace, wherein the interaction $U$ has no effect. On the basis of

$\displaystyle\left|\lambda,j,+\right\rangle$ $\displaystyle=$ $\displaystyle
c_{j,\uparrow}^{\dagger}\left|\mathrm{Vac}\right\rangle,$ (21)
$\displaystyle\left|\lambda,j,-\right\rangle$ $\displaystyle=$ $\displaystyle
c_{j,\downarrow}^{\dagger}\left|\mathrm{Vac}\right\rangle,$ (22)

where $\left|\mathrm{Vac}\right\rangle$ is the vacuum state of fermion
$c_{j,\sigma}$, and $\lambda=A$ ($B$) when $j\in A$ ($B$), the corresponding
Hamiltonian can be given as

$H=H_{0}+H_{s}+H_{e},$ (23)

with

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle-J\sum_{j,\sigma}\left(\left|A,j,\sigma\right\rangle\left\langle
B,j,\sigma\right|\right.$ (24)
$\displaystyle\left.+\left|B,j,\sigma\right\rangle\left\langle
A,j+1,\sigma\right|+\text{H.c.}\right),$ $\displaystyle H_{s}$
$\displaystyle=$
$\displaystyle-\frac{V}{2}\sum_{j,,\sigma}\left(\sigma\left|A,j,\sigma\right\rangle\left\langle
A,j,\sigma\right|\right.$ (25)
$\displaystyle\left.-\left|B,j,\sigma\right\rangle\left\langle
B,j,\sigma\right|\right),$ $\displaystyle H_{e}$ $\displaystyle=$
$\displaystyle
F\sum_{j,\sigma}\left(2j+1\right)\left|A,j,\sigma\right\rangle\left\langle
A,j,\sigma\right|$ (26)
$\displaystyle+2j\left|B,j,\sigma\right\rangle\left\langle B,j,\sigma\right|.$

Here $H_{e}$ describes the dc electric field. In the absence of $H_{d}$, the
Hamiltonian characterizes a Rice-Mele model with homogeneous hopping Rice and
Mele (1982). Performing the Fourier transformation,

$\left|\lambda,q,\sigma\right\rangle=\frac{1}{\sqrt{N}}\sum_{j}e^{iqj}\left|\lambda,j,\sigma\right\rangle,$
(27)

where $q=2l\pi/N$, $l\in\left[0,\text{ }N-1\right]$, the Hamiltonian $H$ can
be block diagonalized due to its translational symmetry, i.e.,
$H=\sum_{q,\sigma}H_{q,\sigma}$. In this new basis, the matrix form of $H_{q}$
is

$H_{q,\sigma}=-J\left(\begin{array}[]{cc}\sigma V/2J&1+e^{ik}\\\
1+e^{-ik}&-\sigma V/2J\end{array}\right),$ (28)

which admits the eigen spectrum as

$\epsilon_{q,\pm}^{\left(1\right)}=\pm\sqrt{(\frac{V}{2})^{2}+4J^{2}\cos^{2}\left(q/2\right)}.$
(29)

Evidently, the two-particle spectrum consists of scattering and bound bands.
The scattering band can be theoretically constructed through the Eq. (29). In
particular, the upper scattering band of Figs. 4(a1)-(a2) can be obtained by
the combination of two individual particles, i.e.,
$\epsilon_{k,\text{{u}}}^{\left(2\right)}=\epsilon_{q_{1},+}^{\left(1\right)}+\epsilon_{q_{2},+}^{\left(1\right)}$.
Analogously, the middle and lower scattering bands can be given as
$\epsilon_{k,\text{{m}}}^{\left(2\right)}=\epsilon_{q_{1},+}^{\left(1\right)}+\epsilon_{q_{2},-}^{\left(1\right)}$,
and
$\epsilon_{k,\text{{l}}}^{\left(2\right)}=\epsilon_{q_{1},-}^{\left(1\right)}+\epsilon_{q_{2},-}^{\left(1\right)}$,
respectively. When $J\ll V$, the single-particle bandwidth is in proportion to
$J^{2}/U$ which is similar to that of the bound pair spectrum as $V=0$. This
indicates that the presence of the staggered field suppresses the group
velocity of the single particle but enhances the bound pair group velocity. It
is conceivable that when $V$ is large enough, the existence of dc electric
field will not induce the large-scale oscillation of scattering particles in
coordinate space. In the following, we examine the dynamics of the system in
the single-particle and two-particle Hilbert spaces, respectively.

###  III.2 Bloch oscillation

In the preceding section, the effect of the staggered field on the band
structure is discussed. For simplicity, we only consider the dynamics when the
resonant field is applied. The Hamiltonian can be given as
$H_{\mathrm{BO}}=H+H_{e}$, where the dc electric field is
$H_{e}=F\sum_{\sigma,j\in A,B}jn_{j,\sigma}$. To show the difference between
the dynamics of the single-particle and two-particle cases, the initial states
are chosen as

$\displaystyle\left|\Phi_{1}\left(0\right)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}c_{N/2,\uparrow}^{\dagger}c_{3N/2,\downarrow}^{\dagger}\left|\mathrm{Vac}\right\rangle,$
(30) $\displaystyle\left|\Phi_{2}\left(0\right)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}c_{N,\uparrow}^{\dagger}c_{N,\downarrow}^{\dagger}\left|\mathrm{Vac}\right\rangle.$
(31)

These are two types of wave packets, which are initially located at two
separate sites and a single site, respectively. Evidently, the existence of
interaction $U$ only affects $\left|\Phi_{2}\left(0\right)\right\rangle$ but
does no effect on $\left|\Phi_{1}\left(0\right)\right\rangle$. For the initial
state $\left|\Phi_{1}\left(0\right)\right\rangle$, the two-particle dynamics
can be understood as the superposition of the motions of two independent
particles regardless of whether the staggered external field is switched on.
For the second type of initial state
$\left|\Phi_{2}\left(0\right)\right\rangle$, the interaction $U$ binds the two
particles together so that the two particles behave like a single composite
particle. The dc electric field forces the spectrum of the system to consists
of Wannier–Stark ladders Hartmann et al. (2004); Breid et al. (2006) and hence
drives the two initial states to undergo the BO, respectively. A typical
breathing behavior can be observed in Figs. 4(b1) and (c2). It shows that the
width of the wave packet oscillates strongly, while its position remains
constant. In general, the dynamics is confined to a spatial interval, the
width of which is determined by the ratio of the bandwidth of the considered
system with $H_{e}=0$ to the strength $F$. Specifically, the BP can be moved
on the lattice with effective hopping $2J^{2}/U$ when $V=0$ and large $U$
limit are supposed, whereas a single particle’s hopping is $J$. As such, the
breathing mode of two scattering particles
$\left|\Phi_{1}\left(t\right)\right\rangle$ extends over an interval $4J/F$,
and $\left|\Phi_{2}\left(t\right)\right\rangle$ is frozen at the initial site.
Note that the period of BO for BP is $\pi/2F$ rather than $\pi/F$ since the
external field felt by the BP is $2F$ when it migrates. In contrast, the
existence of the resonant field shrinks the bandwidth of the scattering band
to the order of $J^{2}/U$. On the other hand, it broadens the width of the
bound band to $4J$ instead of $J^{2}/U$ with frequency $F$. As a consequence,
the dynamics of the BP and scattering particles are interchanged as shown in
Figs. 4 (c1)-(c2). So far, we have demonstrated how the external resonance
field can improve the mobility of BP in the matter by changing the structure
of the energy spectrum through the example of two particles. This mechanism
could provide some physical insights to understand the subsequent many-body
dynamics.

##  IV Quench dynamics at half filling

In this section, we turn to investigate the many-body quench dynamics. Before
quench, the Hamiltonian of the considered system is $H_{0}$, which is a
standard Hubbard Hamiltonian. When $J\ll U$, the system is in the Mott
insulating phase such that there exists an excitation gap in the spectrum. In
the following, we focus on the system at half filling. In this condition, each
site contains only one particle. The on-site interaction prohibits the
movement of the particles. With the spirit of the two-particle case, one can
imagine that the resonant staggered field provides a bridge for the mobility
of the particles since the basis of
{$c_{j,\uparrow}^{\dagger}c_{j,\downarrow}^{\dagger}\left|\mathrm{Vac}\right\rangle$,
$c_{j,\uparrow}^{\dagger}c_{j+1,\downarrow}^{\dagger}\left|\mathrm{Vac}\right\rangle$,
$c_{j+1,\uparrow}^{\dagger}c_{j+1,\downarrow}^{\dagger}\left|\mathrm{Vac}\right\rangle$,…}
stays at the same energy shell ($U$). Therefore, the conductivity of the
system is enhanced.

Figure 5: Time evolutions of correlator $C_{r}\left(t\right)$ and charge
fluctuation $\Delta n_{j}\left(t\right)$ for different staggered fields $V$
(units of $J$) with a given on-site interaction $U/J=50$. The system is
initialized in the ground state of $H_{0}$, which is $6$-site Hubbard model at
half filling and $S_{z}=0$. Evidently, the resonant field can induce the
oscillation behaviors of both quantities. It can be served as a dynamic
signature to identify the transition from the Mott insulating to conducting
state.

###  IV.1 Transition from Mott insulating to conducting state

To see how does the staggered field affect the conductivity, we propose a
quench scheme in the following. The scheme is that: The system is initialized
in the ground state of $H_{0}$ and then evolved with a post-quench Hamiltonian
$H=H_{0}+H_{s}$. To measure the conductivity of the system, two physical
quantities are introduced. The first one is the doublon-doublon correlation

$C_{i,j}=\langle\eta_{i}^{+}\eta_{j}^{-}\rangle\text{ }\forall\text{ }i\text{,
}j\text{, }i\neq j,$ (32)

the constant number of which denotes that the corresponding state has off-
diagonal long-range order (ODLRO). It also provides a possible definition of
superconductivity, as a finite value of this quantity can be shown to imply
both the Meissner effect and flux quantization Yang (1962). In general, the
$\eta$-pairing state of $H_{0}$ possesses ODLRO. Unfortunately, such state is
the excited state rather than the ground state of $H_{0}$. In addition, the
presence of $H_{s}$ spoils the $\eta$ symmetry so that such superconducting
properties of the system may not exist. However, the correlator $C_{i,j}$ can
reflect the conductivity of the system to some extent. The other quantity used
to examine the conductivity is the charge fluctuation $\Delta n_{j}$ defined
by

$\Delta n_{j}=\sqrt{\langle(n_{j})^{2}\rangle-\langle n_{j}\rangle^{2}}.$ (33)

It is well known that when the on-site repulsive interaction between fermions
is large enough in the Mott phase, the number fluctuation would become
energetically unfavorable, forcing the system into a number state and
exhibiting a vanishing particle charge fluctuation $\Delta n_{j}$. Beyond the
Mott insulator regime, the fermions are delocalized with the nonvanishing
charge fluctuation. For the half-filled free fermion system, straightforward
algebras show that $\Delta n_{j}=\sqrt{2}/2$, which can be served as a
benchmark to examine the following result.

Figure 6: Average $\overline{C}_{r}$ and $\overline{\Delta n_{j}}$ as
functions of $V$ for $6$-site Hubbard model at half filling and $S_{z}=0$. The
system parameters are $U/J=50$, and $t_{f}=15/J$. It indicates that as $V$
approaches $U$, a peak will appear demonstrating our prediction in the text.

Again, we focus on $1$D lattice realizations due to their numerical
tractability. We perform numerical calculation where a $6$-site Hubbard model
at half filling is considered. Taking the ground state of $H_{0}$ as the
initial state $\left|\Phi\left(0\right)\right\rangle$, one can observe the
variations of the correlator
$C_{r}\left(t\right)=\langle\Phi\left(t\right)|\eta_{i}^{+}\eta_{i+r}^{-}\left|\Phi\left(t\right)\right\rangle$
and $\Delta
n_{j}\left(t\right)=\sqrt{\langle\Phi\left(t\right)|n_{j}^{2}\left|\Phi\left(t\right)\right\rangle-\left(\langle\Phi\left(t\right)|n_{j}\left|\Phi\left(t\right)\right\rangle\right)^{2}}$
as time, where
$\left|\Phi\left(t\right)\right\rangle=e^{-iHt}\left|\Phi\left(0\right)\right\rangle$.
Note that the ground state $\left|\Phi\left(0\right)\right\rangle$ of $1$D
Hubbard model is the Mott insulator with strong antiferromagnetic
correlations. Fig. 5 shows that the value of $V$ which we quench to determines
the center value of two such oscillated quantities. When the resonant field is
applied, i.e. $V=U$, the maximum center values of $C_{r}\left(t\right)$ and
$\Delta n_{j}\left(t\right)$ can be reached, which is in agreement with our
previous analysis. To determine the effect of $V$, we introduce the average
correlator and charge fluctuation in the time interval $\left[0,\text{
}t_{f}\right]$, defined as follows

$\displaystyle\overline{C}_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{t_{f}}\int_{0}^{t_{f}}C_{r}\left(t\right)\text{d}t\text{,}$
(34) $\displaystyle\overline{\Delta n_{j}}$ $\displaystyle=$
$\displaystyle\frac{1}{t_{f}}\int_{0}^{t_{f}}\Delta
n_{j}\left(t\right)\text{d}t\text{,}$ (35)

where $t_{f}$ represents the cutoff quench time. Average $\overline{C}_{r}$
and $\overline{\Delta n_{j}}$ as functions of parameter $V$ for different $r$
values are plotted in Fig. 6. It can be shown that the peaks appear in both
$\overline{C}_{r}$ and $\overline{\Delta n_{3}}$ when resonant filed is taken.
Comparing to the free fermion case, the increase of such two quantities
suggests that the system experiences a transition from Mott insulating to
conducting phase. This scheme presents an alternative approach to control the
conductivity of the system from the dynamical prospective.

##  V Summary

In summary, we investigate how does the interplay between the external field
and on-site interaction $U$ alter the conductivity of the system. When the
resonant field is applied, the two-particle case shows that there can exist an
energy shell of $U$, which allows the movement of BP with the hopping energy
scale of $J$ rather than $J^{2}/U$. This phenomenon can be extended to the
dilute system. However, for the case of a single particle, the movement of the
particle is limited due to the existence of the staggered field, which is
reflected by the fact that the bandwidth of the single-particle spectrum is
suppressed from $J$ to $J^{2}/U$ order. These facts can be examined by the BO.
Specifically, when two particles are bound together, they show large-scale BO
in coordinate space. On the contrary, when the initial two particles are far
away, the dc electric field suppresses the two particles at the initial
position. This unique two-particle property paves the way to understand the
many-body quench dynamics. For the system at half filling, the cooperation of
resonant staggered field and on-site interaction forces the initial
antiferromagnetic Mott insulating state to become a steady conducting state.
Our finding provides a new perspective to understand the influence of external
fields on strongly correlated systems and suggests a new dynamical mechanism
to control the conductivity.

###### Acknowledgements.

We acknowledge the support of the National Natural Science Foundation of China
(Grants No. 11975166, and No. 11874225).

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