1/29

冷却原子気体における交流スピン伝導率
Yuta Sekino, Hiroyuki Tajima, & Shun Uchino, arXiv:2103.02418 (accepted by PRResearch)
Hiroyuki Tajima, Yuta Sekino, & Shun Uchino, PRB 105, 064508 (2022)

Yuta Sekino
RIKEN iTHEMS

関野 裕太 （理研CPR, 理研iTHEMS）

共同研究者：
田島裕之（東大理） 内野瞬（原研先端研）
熱場の量子論2022 @京大基研

20th Sep, 2022

2/29

Outline of this talk
1. Introduction
2. Optical spin conductivity• Proposal of measurement
3. Theoretical studies• Formalism• Fermi super uids
• Tomonaga-Luttinger liquidfl4. Summary

3/29

Outline of this talk
1. Introduction
2. Optical spin conductivity• Proposal of measurement
3. Theoretical studies• Formalism• Fermi super uids
• Tomonaga-Luttinger liquidfl4. Summary

Ultracold atoms
Very pure & highly controllable atomic gases
Ideal research platform for quantum many-body phenomena

From Schauss Group, University of Virginia (https://ultracold.phys.virginia.edu/public̲html/)

4/29

5/29

High controllability
1. Quantum statistics & spin degrees of freedom
7Li, 23Na,
39K,…
Bose atom:
REVIEW
ARTICLE

Hyper ne statesa

SITE-RESOLVED IMAGING OF ULTRACOLD...

2. Spatial geometryof gas(a)
3D, 2D, 1D, Lattice, …

RTICLE

Lattice beam

T1

1D
b

Cubic latticecomplete control of the experimentalist. For example,
the geometry of the trapping potentials can be changed
by interfering laser beams under a different angle, thus
making even more complex lattice confiRPgurations19,
such as Kagomé lattices20. The depth of such opticalpotentials can even be varied dynamically during
T2
R1 by simply increasing or
an experimental sequence
T3
decreasing the intensityof the laser light, thus turning
experimental investigations of the time dynamics offundmental phase transitions into a reality.
T1x
Each periodic potential formed by a single
standing wave has the formSpincomplete control of the experimentalist. For example,
QUANTUMthe geometry of the PRX
trapping
potentials can2,be020344
changed (2021)
by interfering laser beams under a different angle, thus
making even more complex lattice configurations19,(b) 20. The depth of such opticalsuch as Kagomé lattices
0.3
potentials can even be varied dynamically during
an experimental sequence by simply increasing or
decreasing the intensity of the laser light, thus turning
experimental investigations of the time dynamics of 0.2
fundmental phase transitions into a reality.
Each periodic potential formed by a single
standing wave has the form
0.1
Optical density

Fermi atom: 6Li, 40K,…

Vlat(x) = V0sin2(kLx),

Triangular lattice0.0

fi

Signal (arb. units)

where kL = 2π/λL is the wave vector of the laser(c)the optical standing wave and V0
light used to form
represents the lattice potential depth,
usually given
_
2 2
in
units
of
the
recoilenergyE=hk
R2R
L /2m (m being
the mass of a single neutral atom), which is a naturalenergy scale for neutral atoms in periodic light fields.
Note that by choosing to interfere two laser beams
at an angle less than 180°, one can form periodiczpotentials with a larger period.
The motion of a single particle in such periodic
T
T2
potentials
is described in terms of Bloch waves
y+with crystal momentum q. However, an additionalT
T3
harmonic confinement arises due to the gaussianVlat(x) = V0sin2(kLx),
profile of the laser beams (see Fig. 2). Although this
FIG. 1. Triangular-lattice quantum gas microscope. (a) Sketch of triangularlatticeconfi
and nement
Raman
imagingbeamsPRX
and their
Yang
etissidebandal.(Virginia),
harmonic
usuallyweak(typicallythe vacuum
chamber.
The
stainlesssteel
octagonis equippedwithoscillation
an outer copper
coil pair for the MOT= 2π/λwave
vectorof thelaser
where
kbyL superimposing
Nat.
Phys.
(2005)
Figure Bloch,1alignment
Optical latticerelative
potentialsto
formedtwoor threeorthogonal
standing
waves.chamber
L is thearound10–200 Hz
frequencies)
light
to to
form
theofopticalstanding
V0 propagate
field
andlattice,
innerthecoil
pairforFeshbachfield.
The
dipoletrapandbeams
in thex-ytheplane
andare approximatelyaligned with
Quantum(2021)
a, For a 2D
opticalatoms
are usedconfithened
an arraytightly confining
1Dwavepotential
tubes.
comparedwith
confi
nementof theatoms onrepresents
the lattice
depth,
usually
given beam throughaxistheR1.
Thelattice
triangularlatticeis formed
by recycling
thelatticethe recessed
top andbottomwindows,
b, In the the3D case,opticalcan be approximatedby potential
a 3D simple
cubic
array
of tightly
each latticesite (typicallyaround
10–40kHz), itleaving just
_
2 2
in
units
of
the
recoilenergy
E
=
h
k
/2m
(m
being
confiningenoughharmonicspace
oscillatorat each lattice
forpotentials
the objectiveat thesite.
top window.
and third focusgenerally
are createdby to
1:1animaging
systems, which
are not shown.
R The secondL
leads
inhomogeneousenvironment
the
mass
of
a
singleneutralatom),
whichisanaturalThree orange arrows (T1, T2, and T3) indicate the direction of the three beams
cross
at thethe atoms where thefor thewhichtrappedatoms.
Oneposition
must beofcareful,6/29

High controllability
1. Quantum statistics & spin degrees of freedom
Bose atom: 7Li, 23Na, 39K,…

Hyper ne states

Fermi atom: 6Li, 40K,…

2. Spatial geometry of gas

3. Interaction between atoms

3D, 2D, 1D, Lattice, …

Feshbach resonance, Lattice depth, …

PHYSICA L R EVIEW LET T ERS(1)! a79 $;

fih=2", n9 is the number=2 state, and a59 (a79 ) iss between atoms in the

40K

week ending
13 JUNE 2003

3000

scattering length (ao)and the measured line

m the mean-field energy
ween j9=2; !9=2i andhbach resonance. The
density-dependent fre-

Spin

2000
1000
0
-1000
-2000
-3000

215

220
225
B (gauss)

230

Regal
& Jinversus(JILA),
PRL(2003)
FIG. 3. Scattering
lengthmagnetic
field near the7/29

Major research directions
Ideal platform to study quantum many-body phenomena
1. Novel quantum phenomena

2. Quantum computation

3. Analog quantum simulation:
cold-atomic systems equivalent/similar to other interesting systems

Superconductors

Fermi atoms
in super uid phase

Neutron super uid inneutron stars© HENNING DALHOFF/SCIENCE PHOTO LIBRARY
flfl

https://www.riken.jp/en/news̲pubs/research̲news/rr/20200207̲2/index.html

GD?:9K5: B767:B4 L?@5<9 ;<5C:5B67:B 3:9: ?9:?69:C @5 B:I:96A

a lower phase space density of the molecular gas, we
B7:?B4 *6B:9 ;<<A@5M 65C :I6?<967@I: ;<<A@5M 3:9: 8B:C 7< ?9<C8;:
observe an essentially unchanged condensate fraction. B<C@8= ;<5C:5B67:B @5 7G: !" ! ! 1 B767: @5 6 ;A<I:9A:6E =6M5:7@;
796?.+4 #G: ;<5C:5B67:B 3:9: 7G:5 7965BE:99:C @57< 65 <?7@;6AFinally, we note that, as in our previous measurements
C@?<A: 796? ;<5B@B7@5M <E 6 B@5MA: E<;8B:C @5E969:C A6B:9 J:6=N4
performed in the BEC limit, the measured condensate"E7:9 7G: B?@5 ?9:?6967@<5O 6 J@6B K:AC #P 65C 6 K:AC M96C@:57 #! 3:9:
fraction always remains well below one [16]. As part 6??A@:Cof
E<9 6 I69@6JA: 6=<857 <E 7@=: Q6B A<5M 6B +P BRO C89@5M 3G@;G
7G: 67<=B 9:A6>:C 7<369CB 7G:@9 :F8@A@J9@8= C@B79@J87@<5 QS@M4 .R4our probing procedure the magnetic field is set well below
the Feshbach resonance where the molecule lifetime is#G: ?9<KA:B @5 S@M4 + 3:9: <J76@5:C E9<= I:97@;6A ;87B 7G9<8MG
6JB<9?7@<5 @=6M:B4 #G:D ?9<I@C: ;A:69 :I@C:5;: <E 657@E:99<=6MH
only on the order of milliseconds [15,40]. This results in5:7@;
a @57:96;7@<54 Q#G: <??<B@7: ;6B: @B ?9:C@;7:C E<9 7G: 2T%J " ! 1
measured loss of 50% of the molecules and may also
B?@5 =8A7@?A:7.N4R #G: B?@5 B798;789: @B ;<5B@B7:57 3@7G 7G: ;<99:H
B?<5C@5M B?@5HC<=6@5 C@6M96= @5 S@M4 164 )I:9A6??@5M !" ! " 1
reduce the measured condensate fraction.
;A<8CB 6B <JB:9I:C 69: @5;<=?67@JA: 3@7G 7G: 6BB8=?7@<5 <E
In conclusion, we have introduced a method for probE:99<=6M5:7@; @57:96;7@<54
ing the momentum distribution of fermionic atom pairs#G: B79:5M7G $ ! "UP " .P# VW <E 7G: 657@E:99<=6M5:7@; @57:96;Hand employed this technique to observe fermionic con7@<5 36B :B7@=67:C JD C:7:9=@5@5M %JO 7G: A<;67@<5 <E 7G:!" ! P 7< 7G: !" ! " 1 J<85C69DO 65C JD ?A<77@5M & ! !#!%Jdensates near a Feshbach resonance. By projecting the
I:9B8B 7G: F86C967@; X::=65 BG@E7 ' ! '(#.P QS@M4 NR4 #G: ;<5B7657B !
system onto a molecular gas, we map out condensation 65Cof '( 69: C:K5:C @5 7G: KM89: A:M:5C4 Y@7G ) !

9<8MG 6M9::=:57 3@7G 6 7G:<9:7@;6A ;6A;8A67@<5 <E *. ! *P !
"U!U " P!U#*0 Q9:E4 .NR4 #G: 657@E:99<=6M5:7@; @57:96;7@<5 :5:9MD;<99:B?<5CB 7< .4U 5Z @5 <89 ;<5C:5B67:B4 L7@AAO 7G: =6M5:7<B767@;
QE:99<=6M5:7@;R @57:96;7@<5 J:73::5 7G: 67<=@; =6M5:7@; =<=:57B@B 6J<87 7:5 7@=:B 3:6[:94 Y: 5<7: 7G67 7G: <?7@;6AAD 796??:C
B6=?A:B @5 3G@;G 7G: C<=6@5B 3:9: <JB:9I:C 3:9: 67 6 7:=?:96789:<E 7G: <9C:9 <E 1PP 5ZO E69 A69M:9 7G65 7G: 657@E:99<=6M5:7@; :5:9MD4#G:aE<9=67@<5 <E B?@5 C<=6@5B <;;89B <5AD @5 6 0<B:\&@5B7:@5b;<5C:5B67:4
S@M89: +; BG<3B 6 ?9<KA: <E 7G: C:5B@7D C@B79@J87@<5 E<9 6 ;A<8C 67
StrangeM96C@:57
metal Q#! $ . =] ;= ! 1 R4 !<#P ! .P =] 65C 6A=<B7H;65;:AA:C!" ! P 9:M@<5 ;65 J: @C:57@K:C4 #G: ;A<8C 36B ?9:?69:C 3@7G 6
B=6AA 7<76A 65M8A69 =<=:578=4 ^8: 7< 7G: 6A=<B7HW:9< M96C@:57
65C 7G: 5<5HW:9< 65M8A69 =<=:578=O 7G: ;A<8C ;<99:B?<5CB 7< 6?<@57 @5 7G: BG6C:C 9:M@<5 @5 S@M4 16O 967G:9 7G65 6 I:97@;6A A@5: 3@7G
5< <EEB:7 6B C@B;8BB:C J:E<9: 3@7G K5@7: M96C@:57B 65C W:9< 65M8A69=<=:578=4 #G: C@EE:9:57 3@C7GB <E 7G: ?9<KA:B 69: ?9<J6JAD ;68B:C
JD 9:B@C86A K:AC @5G<=<M:5@7@:B4 S@M89: +; C:=<5B7967:B 7G: ;<=H
?A:7: =@B;@J@A@7D <E 7G: !" ! " 1 ;<=?<5:57B4
d-waveS<9 6 G<=<M:5<8B 73<H;<=?<5:57
BDB7:=
superconductor"!!!!!!!!7G: ;9@7:9@<5 E<9=@B;@J@A@7D Q@==@B;@J@A@7DR @B1NO1UO12 *6J $ " %# *6 *J O 3G:5 7G: =:65

8/29

Various many-body states with spin

m

0.15

Mott

Temperature

0.8 s

1 s

2 s

4 s

a

7 s

0.10
0

0.2

∆B (gauss)

0.4

0.15
0.6

0.8

0.20

T/TF

qu
id

0.0

-1
b
Optical density

-0.2

Fe
rm
i li

↑+↓

0

+1

0
0.05
-0.4

0.5

0
-1
0 s

0.5 s

2 s

3 s

5 s

0.5

m F = +1

Spin removal

mF = 0m F = -1

0.0↑cPotential

N0 /N

0
-0.05

Antiferromagnetic
phase

Doping

+1

0.10
0.05

antiferromagnet

c

F

e
as

Condensate (BEC)ph".!, " P!U# # 1P1N ;= ! + O 7G: C@EE:9:5;: J:73::5 7G: B;677:9@5M

Heisenbergp(BCS-BEC crossover)ga

Spinor Bose-Einstein

do
eu
Ps

Fermi super uid

8

0.5!"#$%& ' !"#$%&'"( ") *#"+(,-.&%&/ .0'( ,"$%'(.1 23."#0&'"( '$%*/. ")3%44'.&'5%446 /70%(,'(* .0'("# 5"(,/(.%&/. .8"9 3"&8 &8/ .0%&'%4 %(, 860/#:(/,'.&#'3+&'"(.1 2#3'&#%#6 0"0+4%&'"(. ") &8/ &8#// 860/#:(/ .&%&/. 9/#/ 0#/0%#/,
FIG. 4 (color online). Transition to condensation as a func1 | ")Probing
antiferromagnetism
Hubbard model
with
a
surr
1 2& Figure% 3'%.
:/4,!@A BC &8/+.'(* #1)1 &#%(.'&'"(.
;<%(,%+=>/(/#.9//0.?
Regal
et
al.,
(JILA)
PRL(2004)
Stenger
et al., (MIT)Nature(1998)
Mazurenko inet theal.,(Harvard)
Nature(2017)
tion of both !B and T=TF . The data for this phase diagram
0.0! A &" ! !
$D microscope.
,'))/# '()#"$ ! ! ! D &" ! ! A %(, )#"$ !quantum
gasa, Schematic of the two-dimensional Hubbard
ima
were collected with the same procedure as shown in the inset &#%(.'&'"(.
in-200We explore the
0 trajectories 200)#/E+/(56 36 !A1F GHI ,+/ &" &8/ E+%,#%&'5 >//$%(.8')&C diagram,%(, &8/6 5"+4,
3/
650
phase
includingpredicted
phases.
Fig. 2 with thold " 2 ms. The area between the dashed lines
Position(µm),#'J/( ./0%#%&/461 K8/ '$%*/. ") 54"+,. 9'&8 J%#'"+.traced,9/44 &'$/.
'(
&8/
&#%0
.8"9
raw
by the red arrows for a Hubbard model with U/t = 7.2(2). The
around !B ! 0 reflects the uncertainty in the Feshbach reso&8/ /J"4+&'"( &" &8/ .%$/ /E+'4'3#'+$ )"# 5"(,/(.%&/. 0#/0%#/, '( /'&8/# % 0+#/exp
strongest antiferromagnetic order is observed at the starred point.
nance location from the width of the feature in Fig. 1. The!"#$%& ( G'.5'34/%(, '$$'.5'34/ .0'( ,"$%'(.1 27'%4 5"4+$( ,/(.'&6 0#":4/. ")! ! A .&%&/ ;+00/# #"9? "# '( /E+%446 0"0+4%&/, ! ! " D .&%&/. ;4"9/# #"9?1
6b, Experimental set-up..0'("#We trap
Li atoms in a two-dimensional squarecorr
L/&9//( M %(, DM . ,9/44 &'$/C &8/ ,'.&#'3+&'"( ,', ("& 58%(*/ .'*(':5%(&46C
L"./=W'(.&/'( 5"(,/(.%&/. %#/ .8"9(C "3&%'(/, )#"$ &'$/-")-R'*8&
surface and contour plots are obtained using a Renka-Cline
opticallattice.K8/
We3'%.
use thecombinedtheopticallattice
D 5"$0"(/(&. rem
9/#/%4&8"+*8 &8/ ,/(.'&6 ,/5#/%./, ,+/ &" &8#//-3",6#/5"$3'(%&'"(1%3."#0&'"('$%*/.potential%. '( !'*1 N1of
K8/
0#":4/.
") &8/! ! "andinterpolation of approximately 200 distinct data points (#)
theanticonfinementthatisgeneratedbythedigitalmicromirror
device
ima
!
NA
$BC
%(,
&8/
:/4,
*#%,'/(&
9%.
!A.8')&/,&"+(,"&8/
S&/#(=B/#4%58./0%#%&'"(1
2&
4"9
3'%.
:/4,.
;)?C
&8/
!
:/4,
,+#'(*
&8/
,9/44
&'$/
9%.
#
[36]. One measure of when the gas becomes strongly interacting#! ! DD $B 5$ 1 K8/./ '$%*/. 9/#/ &%O/( %)&/# &8/
5"$0"(/(&
9%. .O'#&/,"( 3"&8
36 ! ! " D 5"$0"(/(&.
"0&'5%4 &#%0(DMD)
to 9%.
trap.+,,/(46the atoms
in a central
sampleΩ .',/.
of homogeneousdensity,9'&8 .'*(':5%(&" F is the Fermi momentumis the criterion jkF aj > 1, where hk.9'&58/, "))C %(, &8/ %&"$. 9/#/ %44"9/, &" /70%(,1 P+/ &" &8/ 4%#*/ %.0/5& #%&'"! ! & D %,$'7&+#/.C &8+. ,/$"(.&#%&'(* &8/ %(&')/##"$%*(/&'5 '(&/#%5&'"( ;%4."[20,21,37,38]. For these data, j!Bj < 0:6 corresponds ;&60'5%446
toJ'.'34/ '( !'*1 N?1 2& 8'*8/# :/4,. ;*?C &8/ ! ! " D 5"$0"(/(&. %#/ 0+.8/, %0%#&
QA?C &8/ /70%(.'"( 9%. %4$".& 0+#/46 '( &8/ #%,'%4 ,'#/5&'"(.1 244 &8/
we find spin correlations
across the entire disk that alternate in sign U/t
jkF aj > 1.
)+#&8/# 36 % 4%#*/# ! ! A 5"$0"(/(&C %(, &8/ ! ! & D %,$'7&+#/. J%('.81 K8/6$/%(-:/4, /(/#*6 9%. #/4/%./, %)&/# $D $.C %)&/# 98'58 &8/ %&"$. /70%(,/, %.
@

"

"

"

"

"

"

"

A

"

!D

"

"

"

even up to the largest5"+4,
distanceof d =)"#|d|E+%,#%&'5= 10, >//$%(
as expected
for
aHI1
state
pera
("& 3/ #/."4J/,/(/#*'/. $% NA
K8/ %(&')/##"040403-3
A %(, &8/ ! !exp
"D$%*(/&'5'(&/#%5&'"(
4/%,. &"'$$'.5'3'4'&6") &8/the! !
") R'*8& &" 6'/4, % S&/#(-B/#4%58 ./0%#%&'"( ") &8/ 54"+,C&#%(.4%&/, &8/
with$/#/46
antiferromagnetic
long-rangeorder.
We
determinetempera""fl)#// 0%#&'54/.1 K8+.C % $%*(/&'5 :/4, *#%,'/(&C 98'58 9%. %004'/, %)&/# M $. &'$/""

Spin dynamics with cold atoms

9/29

Ideal
grounds to study spin dynamics
RESEARCH experimental
LETTER
Spin-resolved
manipulation & detectiona2_
C sd dzv
z d~0

1.3 mmin the case of strongly overdamped motion realized here. Fitting an
exponential with decay time t to the displacement gives the spin dragcoefficient of the trapped system as C sd ~v2z t. In the deeply degenerate
regime,2.
the relationshipbetween the measured
and the microscopic
1. Spin di usion w/o latticeSpin impurity
on latticespin drag coefficient might be affected by a weak enhancement of the23
effective
massDOI:
andthe10.1038/NPHYS2561
attractive interaction energy between the
NATURE
NATUREPHYSICSPHYSICS
10.1038/NPHYS2561
DOI:
10,22,24
clouds.
The spin drag coefficient is found to be greatest on resonance, and thus
a conductionab Supplementarybc
spin
is slowest on resonance (seeInformabtion). On resonance, Csd in a homogeneous system must be given by afunction of the reduced temperature T/TF times the Fermi rate EF/B. At
high temperatures, we expect the spin drag coefficient to obey a universal
scaling C sd !nsv! EBF ðT=TF Þ{1=2 . In Fig. 2 we show the spin dragcoefficient as a function of T/TF; Csd is normalized by EF/B, where EF
and TF are the local values at the centre of total mass. We observe T 21/2
{1=2
y
EF
scalingy for T/T. At lower
F . 2, finding C sd ~0:16ð1Þ B ðT=TF Þtemperatures, we observe a crossover from classical to non-classicalx drag coefficient reaches a maximum of approxibehaviour asxthe spin
mately 0.1EF/B near the Fermi 0temperature.
We interpret this saturation1 ms 1 ms
ms
0 ms
Time (1 ms per frame)
0.6
0.6
0.2
0.2
of the spin drag coefficient as a consequenceof0.2Fermi statistics andcd↓↓↓↑↓↓↓
4,5
unitarity , as s and v approach values determined by the Fermi wave0.3
0.2
vectorkF0.3. The spin drag coefficient is inversely proportional to the spin
0.2
0.3
0.1
0.1
0.1
0.1
conductivity, which describes the spin current
response
to an external
0.0
0.0
spin-dependent force. Near the Fermi temperature, the maximum spin
Sommeretal.,(MIT)Nature(2011)
Fukuhara
ettoal.,(MPI)
Nat.
Phys. (2013)
–0.1
drag
corresponds
a minimum
0.0 spin0.0conductivity of the
0.0 coefficient
0.0
0.0
–0.2
¬10
¬10
¬5
¬50
05
10
5
10 ¬10
¬10
¬5
¬50
05
10
5
10 ¬10–0.2
order of kF/B. This is the slowest spin conduction possible in three
Positionsites)(lattice sites)
Position (lattice
Positionsites)(lattice sites)–0.3
dimensionsPositionin the(latticeabsenceof localization.
0.0 0.4
040
80 120 160
400
800
At low temperatures,the spin drag coefficient decreases withdd
1.3 mmTime (ms)

V/kB (μK)

22Probability

Probability

Probability

Probability

Position (mm)Position (mm)Probability

↓

ff

Space

↑

Spin dynamics with cold atoms

10/29

Ideal
grounds to study spin dynamics
RESEARCH experimental
LETTER
Spin-resolved
manipulation & detectiona2_
C sd dzv
z d~0

1.3 mmin the case of strongly overdamped motion realized here. Fitting an
exponential with decay time t to the displacement gives the spin dragcoefficient of the trapped system as C sd ~v2z t. In the deeply degenerate
regime,2.
the relationshipbetween the measured
and the microscopic
1. Spin di usion w/o latticeSpin impurity
on latticespin drag coefficient might be affected by a weak enhancement of the23
effective
massDOI:
andthe10.1038/NPHYS2561
attractive interaction energy between the
NATURE
NATUREPHYSICSPHYSICS
10.1038/NPHYS2561
DOI:
10,22,24
clouds.
The spin drag coefficient is found to be greatest on resonance, and thus
a conductionab Supplementarybc
spin
is slowest on resonance (seeInformab 3. AC spin transport (Our proposal)tion). On resonance, Csd in a homogeneous system must be given by aof the reduced
temperature
EF/B. At
F times the Fermi rate
YS, Tajima,function& Uchino,
accepted
byT/T
PRResearch(2022)
high temperatures, we expect the spin drag coefficient to obey a universal
scaling C sd !nsv! EBF ðT=TF Þ{1=2 . In Fig. 2 we show the spin dragcoefficient as a function of T/TF; Csd is normalized by EF/B, where EF
and TF are the local values at the centre of total mass. We observe T 21/2
{1=2
y
EF
S,x
scalingy for T/T. At lower
F . 2, finding C sd ~0:16ð1Þ B ðT=TF Þtemperatures, we observe a crossover from classical to non-classicalx drag coefficient reaches a maximum of approxibehaviour asxthe spin
mately 0.1EF/B near the Fermi 0temperature.
We interpret this saturation1 ms 1 ms
ms
0 ms
Time (1 ms per frame)
0.6
0.6
0
0.2
0.2
of the spin drag coefficient as a consequenceof0.2Fermi statistics andcd↓↓↓↑↓↓↓
4,5
unitarity , as s and v approach values determined by the Fermi wave0.3
0.2
vectorkF0.3. The spin drag coefficient is inversely proportional to the spin
0.2
0.3
0.1
0.1
0.1
0.1
conductivity, which describes the spin current
response
to an external
0.0
0.0
x
spin-dependent force. Near the Fermi temperature,
the maximum spin
Sommeretal.,(MIT)Nature(2011)
Fukuhara
etal.,(MPI)
Nat.
Phys. (2013)
–0.1
drag
coefficientcorresponds to a minimum
0.0 spin0.0conductivity of the
Drivenbymagnetic
oroptical0.0elds
0.0
0.0
0
–0.2
¬10
¬10
¬5
¬50
05
10
5
10 ¬10
¬10
¬5
¬50
05
10
5
10 ¬10–0.2
order of kF/B. This is the slowest spin conduction possible in three
Positionsites)(lattice sites)
Position (lattice
Positionsites)(lattice sites)–0.3
dimensionsPositionin the(latticeabsenceof localization.
0.0 0.4
040
80 120 160
400
800
At low temperatures,the spin drag coefficient decreases withdd
V/kB (μK)

fi

f

t

x

Time (ms)

ff

+f

2

2

Probability

Position (mm)Position (mm)-f

J

Probability↓(b)Probability(a)Probability

JS,x

x

Probability

1.3 mm↑11/29Todayʼs topic
Optical (AC) spin conductivity JS,x

(b)

x

(a)

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˜S,↵ (!)/f˜ (!)
(!)
=
J
↵,

-f

+f

JS,x

0

fx

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0t

YS, Tajima, & Uchino, accepted by PRResearch (2022)<latexit sha1_base64="xnoy+BxfulVJhh96bIO/AAz/p0Q=">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</latexit>

Signi cance of

(S)
↵ (!)

YS, Tajima, & Uchino, accepted by PRResearch (2022)
Tajima, YS, & Uchino, PRB (2022)

1. Elusive in solid-state systems
2. Powerful probe for quantum many-body states
BCS-BEC crossover, Tomonaga-Luttinger liquid(TLL), Spinor BEC,…

fi

3. Widely applicable probe for clean systems

PH YSICAL REVI EWLETTERS

6 SEPTEMBER 1993

(meV)solids
2.
Opticalconductivity
for
20
40
60
80
20
30

(meV)

I

I

I

I

I

I

I

2oo — (')

I

I

[

]

YBa2cu3OQ
7P
PH YSICAL

0

I

I

I

I

I

]

YBa2Cu30~

REVI EW LETTERS

P95 K—II]

SEPTEMBER
6 895
1993
K

110
Powerful probe for exotic electron systems————— 70
10(meV)
S (meV)SEPTEMBER
6
1993
20
40
60
80
20
30
Superconductor,
Pseudogap phase,
100—Non-Fermi liquid, Dirac fermions, …
2oo — (')
YBa2Cu30~
YBa2cu3OQ
150 K

K
K
K

—

I

I

I

I

I

I

I

I

I

I

I

[

]

7P
(meV)
P95 K—
40
60

20

)

I

I

I

I

I

I

I

I

]

895 K
150 K

80
I

I

]

I

]

I

110 K
—————— 70
K

Homes et al., PRL
10 K (1993)I]

YBa2Cu30~

100—

895 K

l

I
I

I
I

]

I

I

I

I

I

I

I

I

I

]

I

I

I

]

150 K

110 K
—————— 70
K

5

10 K

I

I

I

I

I

I

200
ENCY (cm ')

I

I

300
Tc=63K

I

I
I

I

I

I

I
I

.
.

I
I

]

I

I

I

I

I

I

I

I

I

]

I

I

I

]

0 ~2,

YBaqCu306. yo along the c axis
and below T, (63 K). At room
00reflectance200
metallic
displays a 300l
QUENCY (cm ')]weak upturnin the reflectancemetallic
but thethematebehavior,yo along
of YBaqCu306.
c axisBelowthe(63
condensation0). and
T„T,
ve
below
K). At room
encyreflectancemode is displays
clearly shownhea metallic
by
I

40 g~

l

I
I

I

I

I

I

I

I

I

I

]

5

I

I

I

I

I

I

]

I

I

FIG. 2.

0 g~
40
0 ~2,

100
200
T(K)
1

200

I

I

]

I

400

(cm

FREQUENCY

l

I

I

600

800

The optical conductivity 100of 200YBa2Cu306. 7p' obtained by aalong the c 0axis from -25 to 800 cm T(K)200 of the
400 reflectance
600 with800(a) the
Kramers-Kronig
analysis
FREQUENCY (cmatphonons
and 630 cm
150,
192,
286,
317, 557,
present andance.
heweak upturnin the reflectance
Opticalspin conductivity
would
also
be
a
usefulprobe
subtracted
to yieldthe electronic
2. TheNote that
opticalof YBa2Cu306.
conductivitybackground.
(b) FIG.
7p
of metallic behavior, but the mate' obtained
the
-25
the
axis
from
formation
of
c
along
to
cm
a
is
800
visiblewell aboveaclearly
pseudogap
by T
for
nontrivial
spin
dynamics
BelowthecondensationT„
g 0).
Kramers-Kronigof the reflectancearea represents
the spectral with
(63K). The shadedanalysis
theweight(a)oftheequency mode is clearly shown byphonons at 150, 192, 286, 317, 557, and 630 cm
present and
1

I

I

I

]

l

I

I

12/29

13/29Outline of this talk
1. Introduction
2. Optical spin conductivity• Proposal of measurement
3. Theoretical studies• Formalism• Fermi super uids
• Tomonaga-Luttinger liquidfl4. Summary

14/29Measurement scheme: set up

YS, Tajima, & Uchino, arXiv:2103.02418

Total Hamiltonian:<latexit sha1_base64="EpAcXf9OPlmcBXShLQCwgfjbOEA=">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</latexit>

H (t)<latexit sha1_base64="i1LtyJ1H14r6X4ZW9YKF5H7l3m0=">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</latexit>

H(t) = H + H (t)H<latexit sha1_base64="eNxE3cHpVQ0mXo/ajC/IBWnX+Lw=">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</latexit>

Cold atoms with spin (at least Sz) conserved
Spin: S=1/2, 1, 3/2, …
Zeeman eld, synthetic gauge eld, …
trap & lattice potentials, …
BCS-BEC crossover, Spinor BEC, ferromagnets/antiferromagnets, …

fi

fi(Extension to spin-nonconserving & nonequilibrium systems is possible)

How to induce AC spin current

+f

15/29

YS, Tajima, & Uchino, arXiv:2103.02418

Time-dependent force coupled to spin density Sz
Z<latexit sha1_base64="1WGptdntrxIisl4mNJtZ9uLcaPY=">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</latexit>

H (t) =

dr f (t)r Sz (r),<latexit sha1_base64="8jnSiW2N7O+0Eb9GlroaPcHxJ6Y=">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</latexit>( = x, y, z)<latexit sha1_base64="8utBSvbb2UeDzhKy9dzNPDHLk1U=">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</latexit>

V (r, t)
(b)

JS,x

0

r
<latexit sha1_base64="9n7YKWVDezP7A4hDYyS8+yuRIT8=">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</latexit>

fx
0

t

How to induce AC spin current

YS, Tajima, & Uchino, arXiv:2103.02418

Time-dependent force coupled to spin density Sz
Z<latexit sha1_base64="1WGptdntrxIisl4mNJtZ9uLcaPY=">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</latexit>

H (t) =(b)

dr f (t)r Sz (r),JS,x<latexit sha1_base64="8jnSiW2N7O+0Eb9GlroaPcHxJ6Y=">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</latexit>( = x, y, z)

Single-frequency driving force toward β = x, y, z
<latexit sha1_base64="t6MkkbC/VHGHmDNXEEA+EhwIGbs=">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</latexit>f (t) = F cos(!0 t)

0

fx

t

0

1. Magnetic eld gradientMedley et al., (MIT) PRL (2011); Jotzu et al., (ETH) PRL (2015)

2. Optical Stern-Gerlach e ect

ff

Taie et.al., (Kyoto) PRL (2010)

fi

+f

16/29

17/29RESEARCH LETTER
How to extract spin conductivity

YS, Tajima, & Uchino, arXiv:2103.02418

adhJS (t)i =dt

1. Spin current:

dr rSz (r, t)d⌘ hXS (t)i
dt

(S)
˜
↵ (!)f (!)<latexit sha1_base64="SDPkQ6dbjTo15FXS1rmlZfI1wnA=">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</latexit>

2. Spin conductivity:⌧Z

1.3 mm<latexit sha1_base64="wYuMN1Xn+spea+u27bfu9aZxtmw=">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</latexit>hJ˜S,↵ (!)i =(Spin conservation)(Ohmʼs law in frequency space)<latexit sha1_base64="t6MkkbC/VHGHmDNXEEA+EhwIGbs=">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</latexit>f (t) = F cos(!0 t)

3. Driving force:bh XS,↵ (t)i
=
F

Im

(S)
↵ (!0 )

!0

1.3 mm<latexit sha1_base64="J63eHYEIYQqWAujKMhKSYAD+ldg=">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</latexit>cos(!0 t) +Re!0<latexit sha1_base64="dMhbi6kShDCo7U/9TZUWhnRMag8=">AAAGTXiclZQ9b9NAGMefBtyW8tIUFiSWiqgoVaXogipACFAFC+2UOqSNVAfr7F5dK37T2QkpVkYWNiYGJpAYEB+AFcTCxsTQj4AYC2KpEM9dTBu3TRzOSu7u8e//3N/3ZgSOHUaE7I7lTp1Wxicmz0ydPXf+wnR+5uJa6Le4yWqm7/i8btCQObbHapEdOawecEZdw2HrRvOBeL/eZjy0fe9RtBOwhkstz96yTRphSM8vaKFtuVSPNeoE21QzWES7j+Nidb5b1HyXWfRur9LJvJ4vkBKRZfZ4o5w0CpCUij+TewYabIIPJrTABQYeRNh2gEKIzwaUgUCAsQbEGOPYsuV7Bl2YQm0LKYYExWgT/y3sbSRRD/siZyjVJo7i4I+jchbmyDfyjuyRL+Q9+U72B+aKZQ7hZQdro6dlgT79/HL1d6bKxTqC7UPVUM8RbMEt6dVG74GMiK8we/r205d71dvqXHyNvCE/0P9rsks+4xd47Z/m21Wmvhrix0AvYsYE4WHkiZwPVzr0cAU0WEBKrIVQCA1PZvhk+o6kHdm3+taDS8cnKe5JBf8PRRMVQiO8x9jrymewp06K72TyQYoP+vjBnnhKwzPHWEnxKyONUUlpKpljqClezeTrKb6ewceJ101kxHlsyIjYmR15Vv+dqviAiqGAXFZOS/JiDrLy9pOj5TZQ25I7bHjmQy6dd/DaqPI0aQcnmydzzkbQLkv2qHZZnkzU4u1ZPnpXHm+sXS+Vb5QWVxcLS/eTe3QSrsBVKKL/m7AED3HH1HDsF/ABPsIn5avyS9lX/vTQ3FiiuQSpMj7xF0gYXD0=</latexit>

Measurement of(S)↵ (!0 )hXS,↵ (t)i
<latexit sha1_base64="ywd/+dDRpO+4QRquk5zO6MSfihc=">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</latexit>

sin(!0 t)(S)↵ (! = !0 )Time (1 ms per frame)e.g. Experiment on spin di usion (w/o f (t) )hX# (t)i

ff

Sommer et al.,(MIT) Nature (2011)

Position (mm)hXS (t)i = hX" (t)i
<latexit sha1_base64="93BcrHS2h909qXMsdLgT987sdAY=">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</latexit>

d

0.3
0.2
0.1
0.0
–0.1
–0.2
–0.3

hX" (t)i<latexit sha1_base64="zCK041EnGQdOknoUjPg/QnNlvA0=">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</latexit>hX# (t)i<latexit sha1_base64="7WXNB/Wo0YT9YnE7zhhiE6SWZkQ=">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</latexit>

0

40

80

120 160
Time (ms)

400

800

Position (mm)c<latexit sha1_base64="vuwRLFZH3xsH2Pgf79gnF1M57MU=">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</latexit>

0.2
0.0
–0.2
0.0 0.4
V/kB (μK)

18/29Outline of this talk
1. Introduction
2. Optical spin conductivity• Proposal of measurement
3. Theoretical studies• Formalism• Fermi super uids
• Tomonaga-Luttinger liquidfl4. Summary

Theoretical studies
Our works on optical spin conductivity in homogeneous gases
YS, Tajima, & Uchino, arXiv:2103.02418
Tajima, YS, & Uchino, PRB (2022)

1. S=1/2 super uid Fermi gas with spin gap
2. 1D p-wave Fermi super uid with topological phase transition
3. Tomonaga-Luttinger liquidflfl

4. S=1 polar BEC with gapped or gapless spin modes

19/29

20/29General relations
Enss & Haussmann PRL (2012)

Enss, Euro. Phys. J. Special Topics (2013)<latexit sha1_base64="cpfA5TflU+YP/XIj8PQFON+dXps=">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</latexit>

YS, Tajima, & Uchino, arXiv:2103.02418(~ = kB = 1)

1. Kubo formula<latexit sha1_base64="RJPq7LSA81nX/WnKYHEPv5IO00k=">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</latexit>(S)↵ (!) =i!+↵

X s 2 Nszmsz

Magnetic quantum #:

+

↵

(!)

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Spin-current response func.:↵,

1

2. f-sum rule
Z 1

2 {x, y, z}

S, S + 1, · · · , S
Z 1
i! + tidt e✓(t)h[JS,↵ (t), JS, (0)]ieq↵ (!) =

N sz
<latexit sha1_base64="4a2XB196uMLSxZJphKSuN1auQ54=">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</latexit><latexit sha1_base64="FAQwtE/yTDW1/Dw+fJOYLtGMZ+k=">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</latexit>

d!
Re
1 ⇡

(S)
↵ (!) =↵

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Particle # in the sz channel:z!

X s 2 Ns
z

sz

m

z

.

21/29Outline of this talk
1. Introduction
2. Optical spin conductivity• Proposal of measurement
3. Theoretical studies• Formalism• Fermi super uids
• Tomonaga-Luttinger liquidfl4. Summary

22/29Super uid Fermi gas
H=

Z

2

dx 4

X

=",#

†

✓

r2
2m

µ

◆

g

†
"

†
#

#

3

5
"

g > 0 : S-wave attraction<latexit sha1_base64="arGaIYaN3t+C/SH157kA4n9af4A=">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</latexit>

week ending
13 JUNE 2003

PHYSICA L R EVIEW LET T ERSLUME 90, N UMBER 23

ximum transfer is reduced and the measured line
pe is wider.
Both of these effects arise from the mean-field energy
to strong interactions between j9=2; !9=2i and2; !5=2i atoms at the Feshbach resonance. The
an-field energy produces a density-dependent frency shift given by2h"!! " n9 #a59 ! a79 $;
m

(1)

3000

scattering length (ao)<latexit sha1_base64="tqyKCk5qlQ9+u2S/0YoWvjbHiNE=">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</latexit>

2000
1000
0
-1000
-2000
-3000

215

220
225
B (gauss)

230

fi

fl

ere m is the atom mass, h" " h=2", n9 is the number
Regal & Jin (JILA), PRL(2003)
sity of atoms in the mf " !9=2 state, and a59 (a79 ) is
FIG. 3. Scattering length versus magnetic field near theStrengthofattraction
scattering length for collisions between atoms in the
Feshbach resonance peak. The scattering length a59 was ob" !9=2 and mf " !5=2 (mf " !7=2) states. Heretained from measurements of the mean-field energy taken for
Randeria & Taylor (2014)
have ignored a nonresonant interaction term proporT=TF " 0:4 and two different densities: np " 1:8 * 1014 cm!3al to the population difference between the mf "(circles) and np " 5:8 * 1013 cm!3 (squares). We estimate a
Eagles (1969); Leggett (1980)
BCS-Leggett
eld theory=2 and mf " !5=2
states; this term is mean0 for a perfectsystematic uncertainty in a59 of '50% due to uncertainty in23/29BCS-BEC crossover
Chemical potential @ T=0
Μ#ΕF

µ/✏F

<latexit sha1_base64="SLCOp2SxDWQidf627NBJ//t9wxI=">AAACgnichVHLLgRBFD3aezxmsCCxEZMRERnVCCIWQiKWzBgkRibdrYaKfqW7ehImfsAPWFghFl5fYeMHLHyCWI7ExsKdnk4Ewa1U1bmn7rl1b5XumsKXjD3VKfUNjU3NLa2xtvaOzniiq3vddwLP4DnDMR1vU9d8bgqb56SQJt90Pa5Zusk39P3F6vlGiXu+cOw1eeDybUvbtUVRGJokqpDozVvBWJ67vjDJzVua3POs8tJRIZFkaRbawE+gRiCJyFacxDXy2IEDAwEscNiQhE1o8GlsQQWDS9w2ysR5hER4znGEGGkDiuIUoRG7T+sueVsRa5NfzemHaoNuMWl6pBxAij2yK1ZhD+yGPbP3X3OVwxzVWg5o12ta7hbix33Zt39VFu0SexglpfWp/bNyiSJmwooFdeCGTLUXo6YvHZ5UsrOZVHmInbMX6uKMPbF76sMuvRqXqzxzihh9g/r90X+C9fG0OpWeXJ1Mzi9EH9KCfgximF59GvNYxgpy4b0XuMWd0qCMKKoyUQtV6iJND76YMvcB56qT+A==</latexit>

BCS state

BEC of diatomic molecules!kF a"!1

1
0
!1
!2
!3
!4 T = 0!5!2 !1<latexit sha1_base64="+QQbRah7CncnJ+WPIOVfOxX7CcY=">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</latexit>µ ' ✏F<latexit sha1_base64="DPe5kI9bPKMGlA4UNfGcdcvVxuY=">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</latexit>

Ek,F =

p

1/kF a

Ek,F
("k

µ)2 +

("k = k2 /2m)<latexit sha1_base64="MZ5g+22dQNSjWU9PtX27ym0/Kqc=">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</latexit>

1

2

µ!

<latexit sha1_base64="bTPqUgN9nCef2Ca6/NN9CRdGKos=">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</latexit>

Energy dispersion<latexit sha1_base64="R5fAbCHUCR1ZtSHkhMdbfkq9xXE=">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</latexit>

0<latexit sha1_base64="5kBTuWO5YDrgq8vyuPLIhgl1hLI=">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</latexit>

Strength of attraction

Ek,F

µ>0
<latexit sha1_base64="UzAOh2BJe8YgSh0kvnpMvJJP4/E=">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</latexit>2µ<0<latexit sha1_base64="OZO2hoj+LxhnF4x9n5SGOvLG5Cc=">AAACbnichVE9SwNBEH05v+NXVBBBxGCIpJAwEVERC9HGUo2JgorcnZu4eF/cXQIa/AM2lipWChbqz7DxD1j4E8RGiGBj4eRyICrqLLv79u282ZkdzTGk5xM9RpSGxqbmlta2aHtHZ1d3rKc379klVxc53TZsd11TPWFIS+R86Rti3XGFamqGWNP2Fmr3a2XhetK2Vv19R2yZatGSBamrPlO5TbM0S9uxBKUpsPhPkAlBAqEt2bEbbGIHNnSUYELAgs/YgAqPxwYyIDjMbaHCnMtIBvcCh4iytsRegj1UZvd4LfJpI2QtPtdieoFa51cMni4r40jSA11Tle7plp7o/ddYlSBGLZd93rW6Vjjb3UcD2bd/VSbvPnYxxkrzU/tn5j4KmA4yllyBEzC1WvS6vnxwUs3OrCQro3RJz1zFBT3SHddhlV/1q2Wxco4otyHz/dN/gvx4OjOZnlieSMzNhw1pxSBGkOJfn8IcFrGEHL8rcYxTnEVelH5lSBmuuyqRUNOHL6akPgAevox3</latexit>

Egap

Egap

0

Emolecule /2µ<latexit sha1_base64="4i7EkUaOOiZXvIolYSciitL2+QI=">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</latexit>"k<latexit sha1_base64="J8kG72sjmfwvb4VYsBEkMJ7k/Yc=">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</latexit>

0"k<latexit sha1_base64="J8kG72sjmfwvb4VYsBEkMJ7k/Yc=">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</latexit>24/29Result for a Fermi super uid
0.01(kFa)-1 = -1
=0
=1

0.008
0.006

<latexit sha1_base64="kT1/iCUMOX5akmokqWxfNA8QBm8=">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</latexit>

Re

0.002
0

0

(S)
xx (!) /X

kx2 (|!|

2Ek,F )

k

1. Spin is insulated for small ω

0.004

2. Behaviors for ω → 2Egap + 0
1

2

3

4

/EF

5

6

7

μ > 0 [(kFa)-1= -1,0] → coherence peak8

μ < 0 [(kFa)-1= 1]

ω/2

Egap

ω/2

0

→ decay

Ek,F

Ek,F

µ
<latexit sha1_base64="4i7EkUaOOiZXvIolYSciitL2+QI=">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</latexit>"k<latexit sha1_base64="J8kG72sjmfwvb4VYsBEkMJ7k/Yc=">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</latexit>

Egap

ω/2

ω/2

0

"k
<latexit sha1_base64="J8kG72sjmfwvb4VYsBEkMJ7k/Yc=">AAAFmniclVTdThNBFD5UVxB/AL0x0QtigzExaaaGoFFiCN7YeAOFAglbm91lqJPO7k5mt6W44QV8AU28gsQL42N44wt4wSMYLzHxxgvPnK7QBdrV2bR75uz3ffPNmR9XSRHFjB2OFC5ctC6Njl0ev3L12vWJyakba1HY1h6veaEM9YbrRFyKgNdiEUu+oTR3fFfydbf13Hxf73AdiTBYjXcVr/tOMxDbwnNiTL2yO47mKhIyDBp2qzFZZCVGbfpsUE6DIqRtKZwqPAUbtiAED9rgA4cAYowlOBDhswllYKAwV4cEcxojQd857ME4ctuI4ohwMNvC/yb2NtNsgH2jGRHbw1Ek/jQyp2GGfWOf2BH7yj6z7+z3QK2ENIyXXXy7PS5XjYm3t1Z+5bJ8fMfw+oQ11HMM2/CYvAr0rihjZuH1+J03745WnlRnknvsgP1A//vskH3BGQSdn97HZV79MMSPi15MxQwiwMwO1cMnhwGugA0PEGXWwjAMR6cVPh89T2hJ/WbfemhyfB7jGTH0fzBayDAc4z3B3h49gz2xDJ7l4rsZfDcXrzJ41YcfPAed4eicMZJUawsxZu/XKWN2QZfOxd8dnByjEigiLk+zSXjjMU+3H/lv2i5y27Saw5VPcFndwbWr0s61j0+RphpWSSOPWyHsaW6FTgFy8aYqn76XzgZrD0vludLs8mxxYTG9s8bgNtyF++j/ESzAC1iCGo6t4T3sw4F1x1q0KtbLHrQwknJuQqZZq38AYqwwiA==</latexit>(kFa)-1= -1,0(kFa)-1= 1
flRe[ xx(S) ( )]/(mvF )YS, Tajima, & Uchino, arXiv:2103.02418

25/29Topological Fermi super uid

Tajima, YS, & Uchino, PRB (2022)

Fermi atoms in quasi 1D<latexit sha1_base64="kPT/V1MbYNGPfIV1Cw8UZUIo1qg=">AAAFyHiclVTLbtNAFL0NGEp5NIUNEpuIqBVSq2iCqhaVhyrYtKzalKSV6hDZztRY8UtjO02wsmHJD7BgA0gsEJ+BhNjBhkU/AbEsEhsW3Lkxbdw2cRkr8Z3rc86cufPQfdsKQsb2xnJnzirnzo9fmLh46fKVyfzU1VrgRcLgVcOzPbGlawG3LZdXQyu0+ZYvuOboNt/UW4/k9802F4HluU/Crs/rjma61o5laCGmGvmllZn7ahA5jbg1pwaW6Wg9tWNhr2cMpJ6qTc0sDGZma3ONfJGVGLXC8aCcBEVI2po3lbsLKjTBAwMicICDCyHGNmgQ4LMNZWDgY64OMeYERhZ959CDCeRGiOKI0DDbwn8Te9tJ1sW+1AyIbeAoNv4EMgswzb6zD2yffWEf2Q/2Z6hWTBrSSxffep/L/cbky+sbvzNZDr5DeHbIGuk5hB24Q14t9O5TRs7C6PPbz1/tbyxVpuMZ9o79RP9v2R77hDNw27+M9+u88nqEHx29yIpJhIuZXaqHQw5dXAEVZhEl10IyJEckFT4ZfY/QNvXNgfUQ5PgkxgNiiP9gtJAhOdJ7jL0ePcM9sRSeZeI7KXwnE++n8P4AfvgcRIojMsaIE60mYuTer1NG7oIOnYt/Ozg+QMVQRFyWpkl46TFLdxB5Om0duRGt5mjlQ1xad3jtKrRz1YNTJKiGFdLI4q4S9ih3lU4BcvGmKh+9l44Htdul8kJpfn2+uPwwubPG4QbchFvofxGWYQXWoIpjv4HP8BW+KY8VX9lVun1obizhXINUU178Bd7WQpE=</latexit>H=X⇠k c†k,SCHNYDER et al.

ck, + V,<latexit sha1_base64="/t4wM4HwMP8sFpvoCvE1pGrwfPU=">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</latexit>⇠k = k 2 /(2m)

PHYSICALµTABLE I. Ten symmetry classes of single-particle Hamiltonians classified in terms ofabsence of time-reversal symmetry !TRS" and particle-hole symmetry !PHS", as well as“chiral”" symmetry !SLS" !Refs. 37 and 38". In the table, the absence of symmetries is denpresence of these symmetries is denoted by either “+1” or “−1,” depending on whether
operator implementing the symmetry at the level of the single-particle Hamiltonian squares!see text". #The index !1 equals "c in Eq. !1b"; here #Triplet
for TRS and PHS, respc = + 1 and −1
paringfirst six entries of the table !which can be realized in nonsuperconducting systems", TRS= + 1
is an integer #called TRS !even" in the text$ and TRS= −1 when it is a half-integer #cal
0 +q/2,"
k0 +q/2,# spink
the text$. For the last four entries, the superconductor “Bogoliubov–dez Gennes” !BdG" symC, DIII, and CI, the Hamiltonian preserves SU!2" spin-1/2 rotation symmetry when PHS=
!singlet" in the text$, while it does not preserve SU!2" when PHS= + 1 #called PHS !triplet"
last three columns list all topologically non-trivial quantum ground states as a function of sym
spatial dimension. The symbols Z and Z2 indicate whether the space of quantum ground sta
into topological sectors labeled by an integer or a Z2 quantity, respectively.

k,

P-wave Feshbach resonance in the ↑-↓ channel:<latexit sha1_base64="N52QypMN+LgSfGmIh0fm/sUYx5A=">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</latexit>

V =

U

X

kk 0 c†k+q/2," c† k+q/2,# cc

k,k0 ,qBdG Hamiltonian
X †
HMF =
k HBdG (k)<latexit sha1_base64="sWJqqB/0Z/wXFIOG3sFyRDY7snA=">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</latexit><latexit sha1_base64="bjPz8I9Hvt49qtb2yW0PN6eNSuM=">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</latexit>

k =

k

k

<latexit sha1_base64="wZh246XS+7/s697jSm9BYZrbv4k=">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</latexit>

HBdG (k) =· R(k) =

✓

(k) +

x

ck,"
c† k,#z ⇠k ,<latexit sha1_base64="FwhQcFzYaChcPQaFftUAm/I1u9k=">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</latexit>(k) = kD◆

(D > 0)⌫2Z

S = 1, S = 0<latexit sha1_base64="eXCUgOw4X3ffixIe/jxCpmdLzOM=">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</latexit>

PHS

SLS

d=1

Standard!Wigner-Dyson"

A !unitary"
AI !orthogonal"
AII !symplectic"

0
+1
−1

0
0
0

0
0
0

-

Chiral!sublattice"

AIII !chiral unitary"
BDI !chiral orthogonal"
CII !chiral symplectic"

0
+1
−1

0
+1
−1

1
1
1

Z
Z
Z

BdG

D
C
DIII
CI

0
0
−1
+1

+1
−1
+1
−1

0
0
1
1

Z2
Z2
-

<latexit sha1_base64="39sgPjiG6me3RcUZC3KHeu7de8A=">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</latexit>

Schnyder,Ryu, Furusaki(2008)
we shallPRB
be mostly
concerned
der and interactions
on the Z2 topologicalinsulator have &beenLudwig,
flClass BDI with winding #TRS

26/29Spectrum of spin conductivity

Tajima, YS, & Uchino, PRB (2022)

2
ν=1
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2 -1.5 -1 -0.5 0 0.5

1/(kFa)Closing of the spectral gap

0.04

ν=0

1

1.5

Re[ (S)( )]mEF/kF

Egap/EF

Topological phase transition

2

1/(kFa) = 1=0=1= 1.27
= 1.6

0.03
0.02
0.01
0

0

4

(!) /

X

<latexit sha1_base64="KA4H6XGLiZNV9qWlCivGOUbT7hI=">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</latexit>

Re

(S)

6

2

/EF

k

k 2 (|!|

8

10

2Ek )

27/29

Tomonaga-Luttinger liquid
REVIEW ARTICLEa

One-dimensional systems with spin-charge separation
Described by 4 parameters vc, vs, Kc, Ks
PHYSICAL REVIEW LETTERS 125, 190401 (2020)

compthe geby intmakinsuch apoten
an exp
decre
exper
fundm
E
stand

V (x)

lat
Based on the charge excitation spectrum (9), the interactionbonly modifies the effective mass with the Fermi point kF
where
%
c
light
u
replaced by kc ¼ m vc =ℏ [24]. As a consequence, it will
repre
move the resonance position from ω ¼ vF q to ω ≈ vc q inin un
the excitation spectrum. Here we observe that for T → 0,
the m
c
s
energ
DSF Sðq; ωÞ ≠ 0 only for ω− ≤ ω ≤ ωþ, where ω$ ¼
2%
Note
vc jqj $ ðℏq =2m Þ captures the dispersion (9). Taking
Senaratne et al. (Rice), to beat an
6
the setting for a gas of spin-balanced Li with particle
potenpublished in Science (2022)
number N ¼ 60, several different values of interTh
poten
action strength at temperature T ¼ 120 nk, tube length
withs F [24,48], we
L ¼ 20 μm, and q ¼ 1.47 μm−1 ≈ 0.15k
harm
demonstrate in Fig. 4(a) the Bragg spectrum as a function
profilharmof Bragg frequency. The peak frequency of the DSF
Figure 1 Optical lattice potentials formed by superimposing two or three orthogonal standing waves.
aroun
signal is plotted in Fig. 4(b) as a function of γ, from whicha, For a 2D optical lattice, the atoms are confined to an array of tightly confining 1D potential tubes.
compwe can read off the peak velocity defined as the ratio ofb, In the 3D case, the optical lattice can be approximated by a 3D simple cubic array of tightly
each
Fig.
4. potentials
Dispersion
wa
confining harmonic
oscillatorat each latticeof
site.spin and charge density gener
peakfrequency
and
q.
As
Fig.
4(b)
demonstrates,
this
peak
2
Fig. 3. Spin-charge separation. Peaks of measured Bragg
1/e axial width of outcoupled atoms following a Braggpfor thvelocity
is solely
determinedby the charge
velocity,spectrafor charge(red triangles)
andsoundspin (bluecircles) conand 150 µs time-of-flight for (A) charge (dc , red triangles)
there
whereasthe effective
mass from
affects0 to
the 500
width
the frequency
figurations
for a ranging
a0 . ofPeak
deriv
(B)
spin
(d
,
blue
circles)
excitations,
witharanging
fro
s
a
DSF.
Ourdetermined
results on charge
and its dependence
to thevaluesvia fitsvelocity
of a parabolic
function to the datato 500 a0 . The widths are the Gaussian fits to thetrapppos
of points
the interaction
are consistent
withvalue,
the and error
above 50% strengthof the maximum
measuredoutcoupled signal at !p . Error bars are standard errors
Oexperimental
measurement
anderrors
analysisbars are statisticalstandardof thereported
relevantinfit parametermined by bootstrapping for at least 20 independentoptica
im
Ref.
[24].
A moredetailed study
will
presented
theis given by
techn
ters.
The
corresponding
speed
of be
sound
vp = !inp /q[30]. The horizontal axis gives the Lieb-Liniger paramete
the banear
future[44].
the right axis. The upper horizontal axis gives the interaction
calculated for a median tube occupancy of 30 atoms [27].
A goo⇤strength in terms of the Lieb-Liniger parameter , evaluatedblue dashed line shows an estimation for ds derived fromis theatthecenter
of
a
tube
with
occupancy
30
atoms.
Lines
show
band
finitelifetime
of
the
spin
bosons.
(C-F)
Representative
s
(a)
(b)
mom
b
the calculated values for !p for the charge- and the spin-mode
ples of column-density (⇢c ) images of the atom cloud after
can b(dash-dotted red and dashed blue, respectively). Symbols for
Bragg pulse and time-of-flight. (C) Charge mode withapoten
a = 0 and 100 a0 have been slightly displaced from one anand (D) a = 500 a0 . (E,F) Spin mode with a = 0 andof theother for clarity.SNon-monotonicity in the charge-mode data
Then,
500 a0 , respectively. Each frame corresponds to 40 µm
onto aand theory at low interaction is due to small residual di↵erµm.
Brilloences in the number profiles prepared at di↵erent interaction
equal
(c)
(d)
strengths. Non-monotonicity
in the spin-mode theory is likely
energ
mom
due to neglecting the e↵ects of band curvature, a q 3 correcACKNOWLEDGMENTSx
the firtion.

v

ChargeMeasurement of v & v

Spin

v

Heenergy
et al.,
PRL(2020)
FIG. 3. Exact low
excitation
spectrain charge (yellow
green) and spin (dark green) at γ ¼ c=n ¼ 5.03ðas ¼ 700a0 Þ
with the Fermi surface kF ¼ nπ, density n ¼ N=L ¼
3 × 106 ð1=mÞ, ΔE ¼ ℏω. The yellow green shows the particle-hole continuum excitation. The black solid lines indicate
the thresholds of particle-hole excitation which remarkablys the free fermion-like dispersion (9) with an effective
manifest
mass m% ≈ 1.27m at low energy. The black dashed line in the
charge excitation stands for the charge velocity vc. The dark green
shows the two-spinon excitation, where the black dashed lines
stand for the spin velocities vs near ΔK ¼ 0 and ℏkF , respectively. The two red dashed lines indicates the positions of
excitation momenta in charge and spin sectors for Fig. 4.
<latexit sha1_base64="RzwgudtfrmchqhY86YrN924pmwk=">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</latexit>

Re

(S)

(!) / ! 4K

(Memory function method)

YS, Tajima, & Uchino, arXiv:2103.02418

5

Vdip

These CORs reveal a coexistence of liquid and gaslike
states; for more details see Ref. [44].
Exact low-lying excitations and dynamic structure
factor.—Solving the TBA equations (2) and (3), we obtain
precisely the low-lying excitations in both spin and charge.
As shown in Fig. 3, the excitations in the two sectors are
separated from each other. The charge particle and hole
excitations at low energy are given exactly by

Vdip

K can be experimentally determined by spin conductivity at low frequency

cf. Charge conductivity

Giamarchi, PRB (1991); (1992)

with wstate s

We would like to thank T. L. Yang for his contribut

28/29Outline of this talk
1. Introduction
2. Optical spin conductivity• Proposal of measurement
3. Theoretical studies• Formalism• Fermi super uids
• Tomonaga-Luttinger liquidfl4. Summary

29/29

Summary of this talkOptical (AC) spin conductivity JS,x

(b)

x

(a)

Measurable in cold-atom experiments<latexit sha1_base64="JgUSW3q6FURMsXHjkX+fyMQD3a4=">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</latexit>-f(S)
˜S,↵ (!)/f˜ (!)
(!)
=
J
↵,

<latexit sha1_base64="xnoy+BxfulVJhh96bIO/AAz/p0Q=">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</latexit>

Signi cance of

(S)
↵ (!)

+f

JS,x

0

fx

<latexit sha1_base64="OlJihG2Cuy0/c10Fps33ho94IgA=">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</latexit>(↵,= x, y, z)

x

0

t

YS, Tajima, & Uchino, accepted by PRResearch (2022)
Tajima, YS, & Uchino, PRB (2022)

1. Elusive in solid-state systems
2. Powerful probe for quantum many-body states
BCS-BEC crossover, Tomonaga-Luttinger liquid(TLL), Spinor BEC,…

3. Widely applicable probe for clean systemsfiFuture perspective:

Pseudogap of the unitary Fermi gas ?

30/29Backup Slides

31/29Ultracold atoms
Very pure & highly controllable atomic gases
n = 1013 ̶1015 cm-3, T = 10-6 ̶10-8 K (μK̶nK)
Coldest in the Universe !!
(cf. O2 in a room: n = 1019 cm-3, T = 103 K)

From Schauss Group, University of Virginia (https://ultracold.phys.virginia.edu/public̲html/)

tures will further modify the ratio of trap-averaged to local transport
and therefore our spin susceptibility data agree down to this point withcoefficients.
the expected behaviour for a Fermi liquid.
Finally, the measured transport coefficients give for the first time
In conclusion, we have studied spin transport in strongly interacting
access to the temperature dependence of the spin susceptibility,!
"xs(T), Fermi gases. The spin diffusivity was found to attain a limiting value of
L n: {n;" , the about 6.3B/m, establishing the quantum limit of diffusion for strongly
in strongly interacting Fermi gases. Defined as xs ~ !
L m: {m;interacting Fermi gases. Away from resonance, the diffusivity increases.
spin susceptibility describes the spin response to an infinitesimal effective magnetic field or chemical potential difference m" 2 m# applied to
a 3.0
the gas, and is a crucial quantity that can discriminate between different states of matter10. In a magnetic field gradient, particles with
2.5
opposite spin are forced apart at a rate determined by the spin con2.0
ductivity ss, while diffusion acts to recombine them. The balance
R E PO R T
between the processesof diffusionand conduction
therefore
deterSommer
et al.,(MIT)Nature(2011);…R ES E A RC H | 1.5
Nicholos et al.,(MIT) Nature (2019); …
mines the resulting magnetization gradient, a connection expressed
1.0

32/37

Transport phenomena
Normalized χs, κ

1. Spin di usivity

30

response (44). By comparing the measured values ofc with calculations
0.5from the numerical linkedcluster expansion (NLCE) technique (48), we candetermine the entropy per particle S/kBN (where
kB is the Boltzmann constant). We find an entropy
0.3where13
10
per particle of S/kB N = 1.1 (±0.1), a regime
NLCE is expected to converge at half-filling over
T/TF
the range ofbt/U explored here (23, 49, 50).
The equilibrated samples with a spin density
gradient provide the
1.2starting point for subsequent
measurements. Because the initial spin density
gradient is small, it acts as a small perturbation
to the untilted scenario,
ensuring that we are
0.8
probing properties of the homogeneous system
in linear response. After the sample has been
prepared at a fixed
0.4value of t/U, the magnetic
gradient is suddenly switched off. Following thisquench, the system begins to relax back to equilibrium, where hS^ z; j i ¼ 0 everywhere. Figure 2, A
0.3
3
1
10
to F, shows the decay of the spin density gradient
T/TF
after the quench for t/U = 0.23 (±0.01). Thisrelaxation implies that a spin current JS must bepresent.
To obtain
JS fromsusceptibility
the measured spin
Figure4 | Spinon resonance.
a, Spincoefficientsusceptibility(xs, open
Fig. 3. Spin diffusionof the half-filled
Fermi-Hubbard system versus t/U. The
profiles, we define the spin density imbalance,experimentallymeasuredspin
diffusionnormalized
coefficient ħDS /taredcircles)andisothermal
compressibility(k,filledblue
circles),by2 ≡ DS /D0 at half-filling (black circles) versus theI (t), at time t after the quench asχs/n2κ

mDs/

100

2. DC spin conductivity

10

3
1
T/TF

3

10

Downloaded from https://www.science.org at Rikagaku Kenkyusho Library of Riken on May 14, 2022

0.3

Figure 3 | Spin diffusivity of a trapped Fermi gas. Shown is the spin
Hubbard parameters t/U, and a linear fit to data points with t/U < 0.09 (black dot-dashed line).The vertical
ideal Fermi gas at error
zerobars
temperature.
For temperatures below
diffusivity on resonance (Ds, normalized by B/m; filled circles) as a function of the values
X for anX
represent the 1s statistical error in the measurement; the horizontal error bars represent theITðtÞhS^ z; j ðtÞi $suppressedhS^ z; j ðtÞi relative
ð2Þ
tok, owing
toininteractions
betweenthe dimensionless temperature T/TF. At high temperatures, Ds obeys the
1s statisticalerror
the calibrated valueof t/U. The blue solid line represents isentropic results for
F,¼xs becomesLR/D0aobtained
from NLCE calculationsof thefor
real-time spin current-current correlation function for the
non-interacting
Fermi gas
opposite-spin
atoms.
Dashed line, xDsS of
universal T 3/2 behaviour (solid line). At low temperatures, Ds approaches aP
2 entropy per particle of 1.1kB. With a finite temporal cutoff of ~ħ/t for the
Hubbard model (44), with an
dividedby the value of n k obtained
from theb, Red overcircles,
xsand
constant value of 6.3(3)B/m for temperatures below about 0.5TF, establishingwherecomparison.
the left
L,R denotes summation
real-time correlation functions, the2 NLCE theory is expected to provide a lower bound to the true diffusivity.
halvesof the box.
the continuity
Andersonet.interactingal.,[Toronto
group]
PRL(2019)
et modelal.[EHT],Nature(2015), the ratioaof
xs to nfor
kthe
approaches
sameclouds.
AtUsing
temperatures
aboveForTFcomparison,
the quantum limit of spin diffusion
for strongly
Fermi gases.
Errorright
prediction
spin diffusionunity
coefficient ofKrinnerthe 2D Heisenberg
at highequation for the spin density, one can relate I(t)temperatures, kBT >> Jex, where T is the temperature, is shown (dashed red line) (8, 51, 52).
(dashedline).
Error
bars,
61s.e.
bars, 61s.e.
to the spin current JS at the center of the box

3. Optical (AC) conductivity

IEW LETTERS 122, 153602 (2019)

4. Quantized conductance
Inset: A close-up view of the spin diffusion coefficient at half-filling for t/U < 0.09, where it

(a)

ttice

( j = 0) via JS(t) = –(a/2)(d/dt)I(t)|t, where a isis expected to scale approximately linearly with t2/U.
the lattice spacing.
1 4 A P R I L 2 0 1 1 | VO L 4 7 2 | N AT U R E | 2 0 3
Figure 2G shows I (t) measured for several
Limited.
All rights reserved©2011 Macmillan Publishers valuesof t/U. For all values of t/U explored, I (t)decays to zero. We have verified that the effectsof lattice heating during this decay are negligible
relative to the experimental uncertainty in themeasurement (44). I (t) is then fitted to an exponential curve, and the spin current JS is obtained through the time derivative of the fit. To
connect JS with the spin transport coefficients,
we first examine the dependence of JS on thespin density gradient at the center of the box,
∇hS^ z; j¼0 i. By extracting both quantities for a fixedt/U at various times t, we have access to thedependence of JS on ∇hS^z; j¼0 i over a large rangeof values (Fig. 2H). We find that to within experimental error, JS is linearly proportional to∇hS^ z; j¼0 i. This implies that the spin dynamics are
diffusive, so that JS ¼ DS ∇hS^z; j¼0 i, where DS is
the spin diffusion coefficient. The diffusive natureof the dynamics is also independently probed by ameasurement of the power-law dependence ofthe decay time of I(t), at a fixed value of t/U,
on the system size L (44).
Fig. 4. Spin conductivity of the half-filled Fermi-Hubbard system versus t/U. The measured spin
Figure 3 shows the measured spin diffusion
conductivity at half-filling from the initial spin current in an applied magnetic gradient (red squares)
coefficient DS of the half-filled, homogeneous
Hubbard model as a function of t/U, in units
and from the measured spin diffusion coefficient using the Einstein relation sS = DSc (black circles).

(c)

(d)

XDT2

(e)

ce

ff

(b)

(f)

33/29

1. AC spin transport
Hot topic in spintronics
Multilayer systems are considered

PRL 117, 076602 (2016)

PHYSICAL REVI

Ferromagnetic resonance in Multilayer systems<latexit sha1_base64="xnoy+BxfulVJhh96bIO/AAz/p0Q=">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</latexit>

Bulk transport property

(S)
↵ (!)

Li et al., PRL (2016)is elusive !!

AC spin transport within bulk is accessible with cold atoms

34/29

3. Probe for clean gases
Optical mass conductivity for cold atoms
Proposal: Tokuno & Giamarchi, PRL (2011)
Wu, Taylor, & Zaremba, EPR (2015)
Experiment: Anderson et. al.,[Toronto group] PRL (2019)PHYSICALREVIEW
LETTERS
153602 (2019)PHYSICAL REVIEWLETTERS122,
153602 122,
(2019)

b

0
-2

(e)

(µm)

(µm)

2

c

y lattice

XDT2

XDT2

2

(e)

XDT1
(c)

1

20 Hz
0 50 Hz60 Hz(a)x lattice

(b)

(b)

FIG. 2. Conductivity spectra. (a) R
FIG. 2. Conductivity
spectra. (a) andReal(b)andim
diagonal conductivity,
differediagonal
conductivity,and (b)difference
in realconductivities,
versusdrive frequency
20 Hz
conductivities,
versus driveasfrequency.
Here,
50 Hz
background. Lines show
fits Vto ¼E
60 Hz
backgroundas Fourier-limited. Lines show fits
to Eq.is (2).
(c) F
response
observed,

1
Optical lattice
is essential !!!

b

response

-2

c

(d)

ponse

0

displacement (µm)

2

(d)

(a)

y latticex lattice
XDT1
(b)
(c)

(b)

2

z latticez lattice(a))

35/29

3. Probe for clean gases

Generalized Kohnʼs theorem: Kohn (1961); Brey et al. (1989); Li et al. (1991)
Strong constraint on mass conductivity of clean gases
Homogeneous

Massconductivity<latexit sha1_base64="EcFZzKqMiEUeMJ0bS5raQMfX4rU=">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</latexit>

N
(!)
m

Harmonically trapped<latexit sha1_base64="X7R7OMnk9j1keWbL3AR/tga8vzw=">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</latexit>N(!
m!trap )On latticeSensitive toquantum states

Spin conductivity is never constrained andworks as probes w/o lattice36/29RESEARCH LETTER
How to extract spin conductivity

YS, Tajima, & Uchino, arXiv:2103.02418

adhJS (t)i =dt

1. Spin current:

dr rSz (r, t)d⌘ hXS (t)i
dt

(S)
˜
↵ (!)f (!)<latexit sha1_base64="SDPkQ6dbjTo15FXS1rmlZfI1wnA=">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</latexit>

2. Spin conductivity:⌧Z

1.3 mm<latexit sha1_base64="wYuMN1Xn+spea+u27bfu9aZxtmw=">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</latexit>hJ˜S,↵ (!)i =(Spin conservation)(Ohmʼs law in frequency space)<latexit sha1_base64="t6MkkbC/VHGHmDNXEEA+EhwIGbs=">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</latexit>f (t) = F cos(!0 t)

3. Driving force:bh XS,↵ (t)i
=
F

Im

(S)
↵ (!0 )

!0

1.3 mm<latexit sha1_base64="J63eHYEIYQqWAujKMhKSYAD+ldg=">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</latexit>cos(!0 t) +Re!0<latexit sha1_base64="dMhbi6kShDCo7U/9TZUWhnRMag8=">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</latexit>

Measurement of(S)↵ (!0 )hXS,↵ (t)i
<latexit sha1_base64="ywd/+dDRpO+4QRquk5zO6MSfihc=">AAAGNXiclZS/b9NAFMdfA4ZSfjSlCxJLRVRUBIouqKIVqqoKFtopTUhrqY4i272mVvxLZyckWBlZ+AcYGBBISFT8AfwBLPwDDB3YEBJiYCgSCwPvXkwbt00cbCW+e/583319d+8M37aCkLH9scyZs8q58+MXJi5eunxlMjt1dSPwmsLkFdOzPaEaesBty+WV0AptrvqC645h802j8VC+32xxEVie+zjs+Lzq6HXX2rFMPcRQLTutLam1qHxH021/V+/Ohbe05Vo2x/KMrpmTjULcyEF8Fb2pzDPQYBs8MKEJDnBwIcS2DToEeG9BARj4GKtChDGBLYvec+jCBGqbSHEkdIw28L+Ova046mJf5gxIbeIoNv4EKmdgln1me+yAfWLv2Xf2Z2CuiHJILx18Gj0t92uTz6+Vf6eqHHyGsHukGuo5hB1YJK8WevcpIr/C7OlbT18clO+XZqOb7A37gf5fs332Eb/Abf0y367z0sshfgz0ImdMEi5GntB8OOTQxRXQ4DZSci2kQmpEPMOn00tE29Sv962HIMenKZZJIf5D0UCF1EjvEfa6dA/21E7w7VTeT/B+Hz/Yk0hoROoYawl+baQxiglNMXWMUoIvpfJqgldT+Cj2uo2MrMcqReTObFOt/quq6JCKIIdcWs468XIO0vL2k6PlNlDbpB02PPMRl8w7eG1KVE3aYWWLeM75CNpVYo9rV6kyUYunZ+H4WXmysXE3X7iXn1+fz608iM/RcbgON2AO/S/ACjzCHVPBsTvwCt7BnvJB+aJ8Vb710MxYrJmGxKX8/AvqA1LU</latexit>

sin(!0 t)(S)↵ (! = !0 )Time (1 ms per frame)e.g. Experiment on spin di usion (w/o f (t) )hX# (t)i

ff

Sommer et al.,(MIT) Nature (2011)

Position (mm)hXS (t)i = hX" (t)i
<latexit sha1_base64="93BcrHS2h909qXMsdLgT987sdAY=">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</latexit>

d

0.3
0.2
0.1
0.0
–0.1
–0.2
–0.3

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0

40

80

120 160
Time (ms)

400

800

Position (mm)c<latexit sha1_base64="vuwRLFZH3xsH2Pgf79gnF1M57MU=">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</latexit>

0.2
0.0
–0.2
0.0 0.4
V/kB (μK)

37/29Topological phase transition

Tajima, YS, & Uchino, PRB (2022)
Topological phase transition

Topological phase transition at μ = 0

μ/ /EF
Egap
F

12
(a)
ν=1 μ ν=0
0.75
1.8
(iii)
1.6
0.5
(i) ν = 1 (w/ CP)
ν=0
1.4
0.25
1.2
01
0.8
μ mD2
-0.25
0.6 (ii) ν = 1 (w/o CP)
-0.5
0.4
-0.75
0.2
-1
10
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2(a)0.75 (b)
μ
2.5
F
(iii)
0.5 (i) ν = 1 (w/ CP)
ν=0
2
0.25

μ/F/ F F
Dk

1/(k a)

0
1.5

-0.25
1
-0.5
0.5
-0.75

(ii) ν = 1 (w/o CP)

kF / F

2
1.5

(b)

p
Quasiparticle energy Ek = ("k1/(kFa)1 1.5

2

Coherence Peak (CP) disappearsµ)2 + D2 k 2

Ek

Ek

Egap

Egap
0.5

⌫=0

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μ mD2

-10
-2 -1.5 -1 -0.5 0
2.5

!

⌫=1
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µ

mD

w/ CP

2

0
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mD2 < 0w/o CP38/29Spectrum of spin conductivity

Tajima, YS, & Uchino, PRB (2022)
Coherence peak (CP)disappears

Topological phasetransition point

Re[ (S)( )]mEF/kF

0.04
1

μ/ F

(a)
0.75

μ

0.5 (i) ν = 1 (w/ CP)
0.25

(iii)
ν=0

0

-0.25

μ mD2

-0.5

(ii) ν = 1 (w/o CP)

Dk
F//EF
Egap
F

-12
ν=1
1.8 (b)
2.5
1.6
1.4
2
1.2
1
1.5
0.8
1
0.6
0.4
0.5
0.2
0
0
-1.5 -1-0.5 00 0.5
0.5-2 -1.5-1 -0.5-2

1/(kFFa)
a)
1/(k

0.03
0.02

Topological phasetransition point

0.01
0

-0.75

1/(kFa) = 1=0=1= 1.27
= 1.6

0

4

2

ν=0

Re

8

/EF

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(S)

6

(!) /

X

10

k 2 (|!|

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1.5
1 1.5
1

2
2

Closing of the spectral gap of Re

(S)

(!)

Topological phase transition