Strange metals and black holes - Harvard...
Transcript of Strange metals and black holes - Harvard...
Strange metals and
black holes Subir Sachdev
Dynamics and Disorder in Quantum Many Body Systems Far from Equilibrium
Les Houches Summer School, August 19-21, 2019
HARVARD
Remarkable recent observation of‘Planckian’ strange metal transport in cuprates,pnictides, magic-angle graphene, andultracold atoms: the resistivity, ⇢, is
⇢ =m⇤
ne21
⌧
with a universal scattering rate
1
⌧⇡ kBT
~ ,
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REPORT◥
QUANTUM SIMULATION
Bad metallic transport in a cold atomFermi-Hubbard systemPeter T. Brown1, Debayan Mitra1, Elmer Guardado-Sanchez1, Reza Nourafkan2,Alexis Reymbaut2, Charles-David Hébert2, Simon Bergeron2, A.-M. S. Tremblay2,3,Jure Kokalj4,5, David A. Huse1, Peter Schauß1*, Waseem S. Bakr1†
Strong interactions in many-body quantum systems complicate the interpretation ofcharge transport in such materials. To shed light on this problem, we study transport in aclean quantum system: ultracold lithium-6 in a two-dimensional optical lattice, a testingground for strong interaction physics in the Fermi-Hubbard model. We determine thediffusion constant by measuring the relaxation of an imposed density modulation andmodeling its decay hydrodynamically. The diffusion constant is converted to a resistivity byusing the Nernst-Einstein relation. That resistivity exhibits a linear temperaturedependence and shows no evidence of saturation, two characteristic signatures of a badmetal. The techniques we developed in this study may be applied to measurements ofother transport quantities, including the optical conductivity and thermopower.
In conventional materials, charge is carriedby quasiparticles and conductivity is under-stood as a current of these charge carriersdeveloped in response to an external field.For the conductivity to be finite, the charge
carriers must be able to relax their momentumthrough scattering. The Boltzmann kinetic equa-tion in conjunction with Fermi liquid theoryprovides a detailed description of transport inconventionalmaterials, including two trademarksof resistivity. The first is the Fermi liquid predictionthat the temperature (T)–dependent resistivity r(T)should scale like T2 at low temperature (1). Thesecond is that the resistivity should not exceedamaximum value rmax, obtained from the Druderelation assuming the Mott-Ioffe-Regel (MIR)limit, which states that the mean free path of aquasiparticle cannot be less than the lattice spacing(2, 3). This resistivity bound itself is sometimesreferred to as the MIR limit.Strong interactions can, however, lead to a
breakdown of Fermi liquid theory. One signal ofthis breakdown is anomalous scaling of r withtemperature, including the linear scaling ob-served in the strange metal state of the cuprates(4) and other anomalous scalings in d- and f-electron materials (5). Another is the violation ofthe resistivity bound r < rmax, which is observed
in a wide variety of materials (6). Additionally,interactions may lead to a situation where themomentum relaxation rate alone does not de-termine the conductivity, in contrast to thesemiclassical Drude formula, generalizationsof which hold for a large class of systems calledcoherent metals (7). Approaches introduced tounderstand these anomalous behaviors includehidden Fermi liquids (8), marginal Fermi liquids(9), proximity to quantum critical points (10) andassociated holographic approaches (11), andmany numerical studies of model systems,most notably the Hubbard model (12) and thet − J model (13).Disentangling strong interaction physics from
other effects, such as impurities and electron-phonon coupling, is difficult in real materials.Cold atom systems are free of these complica-tions, but transport experiments are challengingbecause of the finite and isolated nature of thesesystems. Most fermionic charge transport ex-periments have focused on studying either massflow through optically structured mesoscopicdevices (14–17) or bulk transport in lattice sys-tems (18–22). In this study, we explored bulktransport in a Fermi-Hubbard system by study-ing charge diffusion, which is a microscopicprocess related to conductivity through theNernst-Einstein equation s = ccD, where D isthe diffusion constant and cc ¼ @n
@m
! "jT is the
compressibility. This requires only the assumptionof linear response and the absence of thermo-electric coupling (23) anddoes not rest on assump-tions concerning quasiparticles.We realized the two-dimensional Fermi-Hubbard
model by using a degenerate spin-balancedmix-ture of two hyperfine ground states of 6Li in anoptical lattice (24). Our lattice beams producea harmonic trapping potential, which leads to avarying atomic density in the trap. To obtain a
system with uniform density, we flatten ourtrapping potential over an elliptical region ofmean diameter 30 sites by using a repulsivepotential created with a spatial light modulator.We superimpose an additional sinusoidal po-tential that varies slowly along one direction ofthe lattice with a controllable wavelength (Fig. 1,A and B). By adiabatically loading the gas intothese potentials, we prepare a Hubbard systemin thermal equilibrium with a small-amplitude(typically 10%) sinusoidal density modulation.The average density in the region with the flat-tened potential is the same with and without thesinusoidal potential. Next, we suddenly turn offthe added sinusoidal potential and observe thedecay of the density pattern versus time (Fig. 1,C and D), always keeping the optical lattice atfixed intensity. We measure the density of asingle spin component, hn↑i, by using techniquesdescribed in (24), giving us access to the totaldensity through hni ¼ h2n↑i.Wework at average total densityhni ¼ 0:82ð2Þ.
This value is close to a conjectured quantumcritical point in the Hubbard model (25). Ourlattice depth is 6.9(2) ER, where ER is the latticerecoil energy and ER/h = 14.66 kHz, leading toa tunneling rate of t/h = 925(10) Hz. Here h isPlanck’s constant. We adjust the scatteringlength, as = 1070(10)ao, by working at a mag-netic bias field of 616.0(2) G, in the vicinity ofthe Feshbach resonance centered near 690 G.These parameters lead to an on-site interaction–to–tunneling ratio U/t = 7.4(8), which is in thestrong-interaction regime and close to the valuethatmaximizes antiferromagnetic correlations athalf-filling (26).We observe the decay of the initial sinusoidal
density pattern over a period of a few tunnelingtimes. The short time scale ensures that theobserved dynamics are not affected by the in-homogeneous density outside of the centralflattened region of the trap. To obtain betterstatistics, we apply the sinusoidal modulationalong one dimension and average along theother direction (Fig. 1, A and C). We fit theaverage modulation profile to a sinusoid, wherethe phase and frequency are fixed by the initialpattern (Fig. 2A). The time dependence of theamplitude of the sinusoid quantifies the decayof the density modulation (Fig. 2B). Our experi-mental technique is analogous to that ofHild et al.(27), who studied the decay of a sinusoidallymodulated spin pattern in a bosonic system.The decay of the sinusoidal density pattern
versus the wavelength l of the modulation be-comes consistent with diffusive transport at longwavelengths. In diffusive transport, the ampli-tude of a density pattern at wave vector k = 2p/lwill decay exponentially with time constant t =1/Dk2, where D is the diffusion constant. We ob-serve exponentially decaying amplitudes withdiffusive scaling for wavelengths longer than15 sites. However, the decay curves are flat atearly times, showing clear deviation from ex-ponential decay. For short wavelengths, we ob-serve deviations from diffusive behavior in theform of underdamped oscillations, which can be
RESEARCH
Brown et al., Science 363, 379–382 (2019) 25 January 2019 1 of 4
1Department of Physics, Princeton University, Princeton, NJ08544, USA. 2Département de Physique, Institut Quantique,and Regroupement Québécois sur les Matériaux de Pointe,Université de Sherbrooke, Sherbrooke, Québec J1K 2R1,Canada. 3Canadian Institute for Advanced Research, Toronto,Ontario M5G 1Z8, Canada. 4Faculty of Civil and GeodeticEngineering, University of Ljubljana, SI-1000 Ljubljana,Slovenia. 5Jožef Stefan Institute, Jamova 39, SI-1000Ljubljana, Slovenia.*Present address: Department of Physics, University of Virginia,Charlottesville, VA 22904, USA.†Corresponding author. Email: [email protected]
on February 13, 2019
http://science.sciencemag.org/
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nloaded from
Strange metal in magic-angle graphene with near Planckian dissipation
Yuan Cao,1, ⇤ Debanjan Chowdhury,1, ⇤ Daniel Rodan-Legrain,1 Oriol Rubies-Bigorda,1
Kenji Watanabe,2 Takashi Taniguchi,2 T. Senthil,1, † and Pablo Jarillo-Herrero1, †
1Department of Physics, Massachusetts Institute of Technology, Cambridge Massachusetts 02139, USA.
2National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan.
(Dated: January 15, 2019)
Recent experiments on magic angle twisted bilayer graphene have discovered correlated insulating
behavior and superconductivity at a fractional filling of an isolated narrow band. In this paper we show
that magic angle bilayer graphene exhibits another hallmark of strongly correlated systems — a broad
regime of T�linear resistivity above a small, density dependent, crossover temperature— for a range of
fillings near the correlated insulator. We also extract a transport “scattering rate”, which satisfies a near
Planckian form that is universally related to the ratio of (kBT/~). Our results establish magic angle bilayer
graphene as a highly tunable platform to investigate strange metal behavior, which could shed light on
this mysterious ubiquitous phase of correlated matter.
A panoply of strongly correlated materials have metallicparent states that display properties at odds with the expecta-tions in a conventional Fermi liquid and are marked by the ab-sence of coherent quasiparticle excitations. Some well knownfamilies of materials, such as the ruthenates [1, 2], cobal-tates [3, 4] and a subset of the iron-based superconductors [5],show non-Fermi liquid (NFL) behavior over a broad inter-mediate range of temperatures, with a crossover to a conven-tional Fermi liquid below an emergent low-energy scale, Tcoh,at which coherent electronic quasiparticles emerge as well-defined excitations. Even more striking examples of NFLbehavior are observed in the hole-doped cuprates [6, 7] andcertain quantum critical heavy-fermion compounds [8] wherethe incoherent features appear to survive down to the lowestmeasurable temperatures, i.e. Tcoh ! 0, when superconduc-tivity is suppressed externally. In the incoherent regime, all ofthese materials in spite of being microscopically distinct havea resistivity, ⇢(T ) ⇠ T , and exhibit a number of anomalousfeatures [8–11] clearly indicating the absence of sharp elec-tronic quasiparticles. In strongly interacting non-quasiparticlesystems, it has been conjectured [12] that transport “scatteringrates” (�) satisfy a universal ‘Planckian’ bound � . O(kBT/~)at a temperature T . It is however worth noting that in a NFLthere is in general no clear definition for �; the scattering ratesdefined through di↵erent measurements, such as dc and opti-cal conductivity, need not be identical [13]. One of the mostsurprising aspects of incoherent transport in these systems, inspite of these subtleties, is that � extracted from the dc resis-tivity (through a procedure specified in Ref. [14]) appears tosatisfy a universal form � = CkBT/~with C a number of order1 [14, 15].
Recent experiments [16, 17] have reported the discoveryof a correlation driven insulator at fractional fillings (with re-spect to a fully filled isolated band) in magic-angle bilayergraphene (MABLG). In MABLG, the relative rotation be-tween two sheets of graphene generates a moire pattern (Fig.1a) with a periodicity that is much larger than the underly-ing interatomic distances in graphene. The theoretically esti-mated electronic bandwidth, W, is strongly renormalized near
these small magic-angles [18–20]; then the strength of the typ-ical Coulomb interactions, U, becomes at least comparable to(if not greater than) the bandwidth, U & W. Investigatingthe properties of MABLG as a function of temperature andcarrier density with unprecedented tunability in a controlledsetup can lead to new insights into the nature of electronictransport in other low-dimensional strongly correlated metal-lic systems.
For our experiments, we have fabricated multiple high-quality encapsulated MABLG devices (see Fig. 1a) usingthe ‘tear and stack’ technique [21, 22]. As reported earlier[16, 22], we obtain band-insulators with a large gap nearn ⇡ ±ns, where n is the carrier density tuned externally byapplying a gate voltage and ns corresponds to four electronsper moire unit cell. On the other hand, correlation driven in-sulators with much smaller gap-scales [16, 23] appear nearn ⇡ ±ns/2; we denote these fillings as ⌫ = ±2 from nowon (the fully filled band corresponds to ⌫ = +4). Dop-ing away from these correlated insulators by an additionalamount, �, with holes (⌫ = ±2 � �) and electrons (⌫ = ±2 + �)leads to superconductivity (SC) [17, 23]. The superconduct-ing transition temperature (Tc) measured relative to the Fermi-energy ("F), as inferred from low temperature quantum oscil-lations measurements [17], is high, with the largest value of(Tc/"F) ⇠ 0.07 � 0.08, indicating strong coupling supercon-ductivity [17]. A schematic ⌫�T phase-diagram for MABLGis shown in Fig.1b.
In this work, we investigate the transport phenomenologyof the metallic states in MABLG as a function of increasingtemperature over a large range of externally tuned fillings. Wehave analyzed the temperature dependence of the longitudinalDC resistivity, ⇢(T ), for a number of devices (MA1 - MA6)over a wide range of fillings, the results of which appear tobe qualitatively similar across devices. In order to highlightthe universal aspects of the behavior, both within and acrossdi↵erent samples, we show the temperature dependent tracesof ⇢(T ) for a range of di↵erent ⌫ in two devices MA1 andMA4 in Fig.1c and Fig.1d respectively. The range of ⌫ chosenfor this purpose is shown as a color-bar in Fig.1b; see caption
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Planckian dissipation and scale invariance in a quantum-critical disordered pnictide
Yasuyuki Nakajima,1, 2 Tristin Metz,2 Christopher Eckberg,2 Kevin Kirshenbaum,2 Alex Hughes,2
Renxiong Wang,2 Limin Wang,2 Shanta R. Saha,2 I-Lin Liu,2, 3, 4 Nicholas P. Butch,2, 4
Zhonghao Liu,5, 6 Sergey V. Borisenko,5 Peter Y. Zavalij,7 and Johnpierre Paglione2, 8
1Department of Physics, University of Central Florida, Orlando, Florida 328162Center for Nanophysics and Advanced Materials, Department of Physics,
University of Maryland, College Park, Maryland 207423Chemical Physics Department, University of Maryland, College Park, Maryland 20742
4NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 208995IFW-Dresden, Helmholtzstraße 20, 01171 Dresden, Germany
6Shanghai Institute of Microsystem and Information Technology,Chinese Academy of Sciences, Shanghai 200050, China
7Department of Chemistry, University of Maryland, College Park, Maryland 207428The Canadian Institute for Advanced Research, Toronto, ON M5G 1Z8, Canada
(Dated: February 5, 2019)
Quantum-mechanical fluctuations between competing phases at T = 0 induce exotic finite-temperature collective excitations that are not described by the standard Landau Fermi liquidframework [1–4]. These excitations exhibit anomalous temperature dependences, or non-Fermi liq-uid behavior, in the transport and thermodynamic properties [5] in the vicinity of a quantumcritical point, and are often intimately linked to the appearance of unconventional Cooper pairingas observed in strongly correlated systems including the high-Tc cuprate and iron pnictide supercon-ductors [6, 7]. The presence of superconductivity, however, precludes direct access to the quantumcritical point, and makes it difficult to assess the role of quantum-critical fluctuations in shapinganomalous finite-temperature physical properties, such as Planckian dissipation !/τp = kBT [8–10].Here we report temperature-field scale invariance of non-Fermi liquid thermodynamic, transportand Hall quantities in a non-superconducting iron-pnictide, Ba(Fe1/3Co1/3Ni1/3)2As2, indicative ofquantum criticality at zero temperature and zero applied magnetic field. Beyond a linear in temper-ature resistivity, the hallmark signature of strong quasiparticle scattering, we find a more universalPlanck-limited scattering rate that obeys a scaling relation between temperature and applied mag-netic fields down to the lowest energy scales. Together with the emergence of hole-like carriers closeto the zero-temperature and zero-field limit, the scale invariance, isotropic field response and lackof applied pressure sensitivity point to the realization of a novel quantum fluid predicted by theholographic correspondence [11] and born out of a unique quantum critical system that does notdrive a pairing instability.
Non-Fermi liquid (NFL) behavior ubiquitously ap-pears in iron-based high-temperature superconductorswith a novel type of superconducting pairing symme-try driven by interband repulsion [7, 12]. The putativepairing mechanism is thought to be associated with thetemperature-doping phase diagram, bearing striking re-semblance to cuprate and heavy-fermion superconductors[13, 14]. In iron-based superconductors, the supercon-ducting phase appears to be centered around the point ofsuppression of antiferromagnetic (AFM) and orthorhom-bic structural order [12]. Close to the boundary betweenAFM order and superconductivity, the exotic metallicregime emerges in the normal state. In addition to theAFM order, the presence of an electronic nematic phaseabove the structural transition complicates the under-standing of the SC and NFL behavior [15–18]. More-over, the robust superconducting phase prohibits inves-tigations of zero-temperature limit normal state physicalproperties associated with the quantum critical (QC) in-stability due to the extremely high upper critical fields.
While AFM spin fluctuations are widely believed toprovide the pairing glue in the iron-pnictides, other mag-
netic interactions are prevalent in closely related ma-terials, such as the cobalt-based oxypnictides LaCoOX(X=P, As) [19], which exhibit ferromagnetic (FM) or-ders, and Co-based intermetallic arsenides with coexist-ing FM and AFM spin correlations [20–22]. For instance,a strongly enhanced Wilson ratio RW of ∼ 7-10 at 2 K[23] and violation of the Koringa law [20–22] suggestproximity to a FM instability in BaCo2As2. BaNi2As2,on the other hand, seems to be devoid of magnetic or-der [24] and rather hosts other ordering instabilities inboth structure and charge [25]. Confirmed by extensivestudy, Fe, Co, and Ni have the same 2+ oxidation statein the tetragonal ThCr2Si2 structure, thus adding one delectron- (hole-) contribution by Ni (Fe) substitution forCo in BaCo2As2 [26–29], and thereby modifying the elec-tronic structure subtly but significantly enough to tune inand out of different ground states and correlation types.Utilizing this balance, counter-doping a system to achievethe same nominal d electron count as BaCo2As2 can re-alize a unique route to the same nearly FM system whiledisrupting any specific spin correlation in the system.
Here, we utilize this approach to stabilize a novel
arXiv:1902.01034v1 [cond-mat.str-el] 4 Feb 2019
Planck
ian
dissip
atio
nand
scale
invaria
ncein
aquantu
m-critic
aldiso
rdered
pnictid
e
Yasu
yuki
Naka
jima, 1
,2Tristin
Metz, 2
Christop
her
Eckb
erg, 2Kevin
Kirsh
enbau
m, 2
Alex
Hugh
es, 2
Renxion
gWan
g, 2Lim
inWan
g, 2Shanta
R.Sah
a, 2I-L
inLiu, 2,3,4
Nich
olasP.Butch
, 2,4
Zhon
ghao
Liu, 5,6
Sergey
V.Borisen
ko, 5Peter
Y.Zavalij, 7
andJoh
npierre
Paglion
e2,8
1Depa
rtmen
tofPhysics,
University
ofCen
tralFlorid
a,Orla
ndo,Florid
a32816
2Cen
terforNanophysics
andAdvanced
Materia
ls,Depa
rtmen
tofPhysics,
University
ofMaryland,College
Park,
Maryland
20742
3Chem
icalPhysics
Depa
rtmen
t,University
ofMaryland,College
Park,
Maryland20742
4NIST
Cen
terforNeu
tronResea
rch,Natio
nalInstitu
teofStandardsandTech
nology
,Gaith
ersburg,
Maryland20899
5IFW
-Dresd
en,Helm
holtzstra
ße
20,01171
Dresd
en,Germ
any
6ShanghaiInstitu
teofMicro
system
and
Inform
atio
nTech
nology
,Chinese
Academ
yofScien
ces,Shanghai200050,China
7Depa
rtmen
tofChem
istry,University
ofMaryland,College
Park,
Maryland20742
8TheCanadian
Institu
teforAdvanced
Resea
rch,Toronto,ON
M5G
1Z8,Canada
(Dated
:Feb
ruary
5,2019)
Quan
tum-m
echan
icalfluctu
ationsbetw
eencom
petin
gphases
atT
=0
induce
exotic
finite-
temperatu
recollective
excitation
sthat
arenot
describ
edby
thestan
dard
Lan
dau
Ferm
iliq
uid
framew
ork[1–4].
These
excitation
sex
hibit
anom
aloustem
peratu
redep
enden
ces,or
non
-Ferm
iliq
-uid
beh
avior,
inthetran
sport
and
therm
odynam
icprop
erties[5]
inthevicin
ityof
aquan
tum
criticalpoin
t,an
dare
oftenintim
atelylin
kedto
theap
pearan
ceof
uncon
vention
alCoop
erpairin
gas
observed
instron
glycorrelated
system
sinclu
dingthehigh
-Tccu
prate
andiron
pnictid
esupercon
-ductors
[6,7].
Thepresen
ceof
supercon
ductiv
ity,how
ever,preclu
des
direct
accessto
thequan
tum
criticalpoin
t,an
dmakes
itdiffi
cult
toassess
therole
ofquan
tum-critical
fluctu
ationsin
shap
ing
anom
alousfinite-tem
peratu
rephysical
prop
erties,such
asPlan
ckian
dissip
ation!/τ
p=
kBT
[8–10].Here
werep
orttem
peratu
re-field
scaleinvarian
ceof
non
-Ferm
iliq
uid
therm
odynam
ic,tran
sport
andHall
quan
titiesin
anon
-supercon
ductin
giron
-pnictid
e,Ba(F
e1/3 C
o1/3 N
i1/3 )
2 As2 ,
indicative
ofquan
tum
criticalityat
zerotem
peratu
rean
dzero
applied
magn
eticfield
.Beyon
dalin
earin
temper-
ature
resistivity,
thehallm
arksign
ature
ofstron
gquasip
articlescatterin
g,wefindamore
universal
Plan
ck-lim
itedscatterin
grate
that
obey
sascalin
grelation
betw
eentem
peratu
rean
dap
plied
mag-
netic
field
sdow
nto
thelow
esten
ergyscales.
Togeth
erwith
theem
ergence
ofhole-like
carriersclose
tothezero-tem
peratu
rean
dzero-fi
eldlim
it,thescale
invarian
ce,isotrop
icfield
respon
sean
dlack
ofap
plied
pressu
resen
sitivity
poin
tto
therealization
ofanovel
quan
tum
fluid
pred
ictedbythe
holograp
hic
correspon
den
ce[11]
andborn
outof
auniquequan
tum
criticalsystem
that
does
not
drive
apairin
ginstab
ility.
Non
-Ferm
iliqu
id(N
FL)
behavior
ubiqu
itously
ap-
pears
iniron
-based
high
-temperatu
resupercon
ductors
with
anovel
typeof
supercon
ductin
gpairin
gsym
me-
trydriven
byinterb
andrep
ulsion
[7,12].
Theputative
pairin
gmech
anism
isthou
ghtto
beassociated
with
the
temperatu
re-dop
ingphase
diagram
,bearin
gstrikin
gre-
semblan
ceto
cuprate
andheavy-ferm
ionsupercon
ductors
[13,14].
Iniron
-based
supercon
ductors,
thesupercon
-ductin
gphase
appears
tobecentered
aroundthepoint
ofsuppression
ofantiferrom
agnetic
(AFM)an
dorth
orhom
-bic
structu
ralord
er[12].
Close
tothebou
ndary
betw
eenAFM
order
and
supercon
ductivity,
theexotic
metallic
regimeem
ergesin
thenorm
alstate.
Inad
dition
tothe
AFM
order,
thepresen
ceof
anelectron
icnem
aticphase
above
thestru
ctural
transition
complicates
theunder-
standingof
theSC
and
NFL
behavior
[15–18].More-
over,therob
ust
supercon
ductin
gphase
proh
ibits
inves-tigation
sof
zero-temperatu
relim
itnorm
alstate
physical
prop
ertiesassociated
with
thequ
antum
critical(Q
C)in-
stability
dueto
theextrem
elyhigh
upper
criticalfield
s.
While
AFM
spin
fluctu
ationsare
widely
believed
toprovid
ethepairin
gglu
ein
theiron
-pnictid
es,oth
ermag-
netic
interactionsare
prevalent
inclosely
relatedma-
terials,such
asthecob
alt-based
oxypnictid
esLaC
oOX
(X=P,As)
[19],which
exhibit
ferromagn
etic(F
M)or-
ders,
andCo-b
asedinterm
etallicarsen
ides
with
coexist-ingFM
andAFM
spin
correlations[20–22].
For
instan
ce,astron
glyenhan
cedW
ilsonratio
RW
of∼
7-10at
2K
[23]an
dviolation
oftheKorin
galaw
[20–22]suggest
proxim
ityto
aFM
instab
ilityin
BaC
o2 A
s2 .
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LETTERShttps://doi.org/10.1038/s41567-018-0334-2
1Institut Quantique, Département de Physique & RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada. 2SPEC, CEA, CNRS-UMR 3680, Université Paris-Saclay, Gif sur Yvette Cedex, France. 3Laboratoire National des Champs Magnétiques Intenses (CNRS, EMFL, INSA, UJF, UPS), Toulouse, France. 4AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, Poland. 5Laboratoire de Physique des Solides, Université Paris-Sud, Université Paris-Saclay, CNRS UMR 8502, Orsay, France. 6Canadian Institute for Advanced Research, Toronto, Ontario, Canada. *e-mail: [email protected]; [email protected]
The perfectly linear temperature dependence of the electrical resistivity observed as T!→ !0 in a variety of metals close to a quantum critical point1–4 is a major puzzle of condensed-mat-ter physics5. Here we show that T-linear resistivity as T!→ !0 is a generic property of cuprates, associated with a universal scattering rate. We measured the low-temperature resistivity of the bilayer cuprate Bi2Sr2CaCu2O8+δ and found that it exhib-its a T-linear dependence with the same slope as in the single-layer cuprates Bi2Sr2CuO6+δ (ref.!6), La1.6−xNd0.4SrxCuO4 (ref.!7) and La2−xSrxCuO4 (ref.!8), despite their very different Fermi surfaces and structural, superconducting and magnetic prop-erties. We then show that the T-linear coefficient (per CuO2 plane), A1
□, is given by the universal relation A1□TF!= !h/2e2,
where e is the electron charge, h is the Planck constant and TF is the Fermi temperature. This relation, obtained by assum-ing that the scattering rate 1/τ of charge carriers reaches the Planckian limit9,10, whereby ħ/τ!= !kBT, works not only for hole-doped cuprates6–8,11,12 but also for electron-doped cuprates13,14, despite the different nature of their quantum critical point and strength of their electron correlations.
In conventional metals, the electrical resistivity ρ(T) normally varies as T2 in the limit T → 0, where electron–electron scattering dominates, in accordance with Fermi-liquid theory. However, close to a quantum critical point (QCP) where a phase of antiferromag-netic order ends, ρ(T) ~ Tn, with n < 2.0. Most striking is the obser-vation of a perfectly linear T dependence ρ(T) = ρ0 + A1T as T → 0 in several very different materials, when tuned to their magnetic QCP; for example, the quasi-one-dimensional (1D) organic conductor (TMTSF)2PF6 (ref. 4), the quasi-2D ruthenate Sr3Ru2O7 (ref. 3) and the 3D heavy-fermion metal CeCu6 (ref. 1). This T-linear resistiv-ity as T → 0 has emerged as one of the major puzzles in the physics of metals5, and while several theoretical scenarios have been pro-posed15, no compelling explanation has been found.
In cuprates, a perfect T-linear resistivity as T → 0 has been observed (once superconductivity is suppressed by a magnetic field) in two closely related electron-doped materials, Pr2−xCexCuO4±δ (PCCO)2,16,17 and La2−xCexCuO4 (LCCO)13,14, and in three hole-doped materials: Bi2Sr2CuO6+δ (ref. 6), La2−xSrxCuO4 (LSCO)8 and La1.6−xNd0.4SrxCuO4 (Nd-LSCO)7,11,12. On the electron-doped side, T-linear resistivity is seen just above the QCP16 where antiferromag-netic order ends18 as a function of x, and as such it may not come as a surprise. On the hole-doped side, however, the doping values
where ρ(T) = ρ0 + A1T as T → 0 are very far from the QCP where long-range antiferromagnetic order ends (pN ~ 0.02); for example, at p = 0.24 in Nd-LSCO (Fig. 1a) and in the range p = 0.21–0.26 in LSCO (Fig. 1b). Instead, these values are close to the critical dop-ing where the pseudogap phase ends (that is, at p* = 0.23 ± 0.01 in Nd-LSCO (ref. 11) and at p* ~ 0.18–0.19 in LSCO (ref. 8)), where the role of antiferromagnetic spin fluctuations is not clear. In Bi2201, p* is farther still (see Supplementary Section 10).
To make progress, several questions must be answered. Is T-linear resistivity as T → 0 in hole-doped cuprates limited to single-layer materials with low Tc, or is it generic? Why is ρ(T) = ρ0 + A1T as T → 0 seen in LSCO over an anomalously wide doping range8? Is there a common mechanism linking cuprates to the other metals where ρ ~ T as T → 0?
To establish the universal character of T-linear resistivity in cuprates, we have turned to Bi2Sr2CaCu2O8+δ (Bi2212). While Nd-LSCO and LSCO have essentially the same single electron-like diamond-shaped Fermi surface at p > p* (refs 19,20), Bi2212 has a very different Fermi surface, consisting of two sheets, one of which is also diamond-like at p > 0.22, but the other is much more circular21 (see Supplementary Section 1). Moreover, the structural, magnetic and superconducting properties of Bi2212 are very different to those of Nd-LSCO and LSCO: a stronger 2D character, a larger gap to spin excitations, no spin-density-wave order above p ~ 0.1 and a much higher superconducting Tc.
We measured the resistivity of Bi2212 at p = 0.23 by sup-pressing superconductivity with a magnetic field of up to 58 T. At p = 0.23, the system is just above its pseudogap critical point (p* = 0.22 (ref. 22); see Supplementary Section 2). Our data are shown in Fig. 2. The raw data at H = 55 T reveal a perfectly linear T dependence of ρ(T) down to the lowest accessible temperature (Fig. 1a). Correcting for the magnetoresistance (see Methods and Supplementary Section 3), as was done for LSCO (ref. 8), we find that the T-linear dependence of ρ(T) seen in Bi2212 at H = 0 from T ~ 120 K down to Tc simply continues to low temperature, with the same slope A1 = 0.62 ± 0.06 μ Ω cm K−1 (Fig. 2b). Measured per CuO2 plane, this gives A1
□ ≡ A1/d = 8.0 ± 0.9 Ω K−1, where d is the (aver-age) separation between CuO2 planes. Remarkably, this is the same value, within error bars, as measured in Nd-LSCO at p = 0.24, where A1
□ = 7.4 ± 0.8 Ω K−1 (see Table 1).The observation of T-linear resistivity in those two cuprates
shows that it is robust against changes in the shape, topology and
Universal T-linear resistivity and Planckian dissipation in overdoped cupratesA. Legros1,2, S. Benhabib3, W. Tabis3,4, F. Laliberté" "1, M. Dion1, M. Lizaire1, B. Vignolle3, D. Vignolles" "3, H. Raffy5, Z. Z. Li5, P. Auban-Senzier5, N. Doiron-Leyraud1, P. Fournier1,6, D. Colson2, L. Taillefer" "1,6* and C. Proust" "3,6*
NATURE PHYSICS | VOL 15 | FEBRUARY 2019 | 142–147 | www.nature.com/naturephysics142
LETTERShttps://doi.org/10.1038/s41567-018-0334-2
1Institut Quantique, Département de Physique & RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada. 2SPEC, CEA, CNRS-UMR 3680, Université Paris-Saclay, Gif sur Yvette Cedex, France. 3Laboratoire National des Champs Magnétiques Intenses (CNRS, EMFL, INSA, UJF, UPS), Toulouse, France. 4AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, Poland. 5Laboratoire de Physique des Solides, Université Paris-Sud, Université Paris-Saclay, CNRS UMR 8502, Orsay, France. 6Canadian Institute for Advanced Research, Toronto, Ontario, Canada. *e-mail: [email protected]; [email protected]
The perfectly linear temperature dependence of the electrical resistivity observed as T!→ !0 in a variety of metals close to a quantum critical point1–4 is a major puzzle of condensed-mat-ter physics5. Here we show that T-linear resistivity as T!→ !0 is a generic property of cuprates, associated with a universal scattering rate. We measured the low-temperature resistivity of the bilayer cuprate Bi2Sr2CaCu2O8+δ and found that it exhib-its a T-linear dependence with the same slope as in the single-layer cuprates Bi2Sr2CuO6+δ (ref.!6), La1.6−xNd0.4SrxCuO4 (ref.!7) and La2−xSrxCuO4 (ref.!8), despite their very different Fermi surfaces and structural, superconducting and magnetic prop-erties. We then show that the T-linear coefficient (per CuO2 plane), A1
□, is given by the universal relation A1□TF!= !h/2e2,
where e is the electron charge, h is the Planck constant and TF is the Fermi temperature. This relation, obtained by assum-ing that the scattering rate 1/τ of charge carriers reaches the Planckian limit9,10, whereby ħ/τ!= !kBT, works not only for hole-doped cuprates6–8,11,12 but also for electron-doped cuprates13,14, despite the different nature of their quantum critical point and strength of their electron correlations.
In conventional metals, the electrical resistivity ρ(T) normally varies as T2 in the limit T → 0, where electron–electron scattering dominates, in accordance with Fermi-liquid theory. However, close to a quantum critical point (QCP) where a phase of antiferromag-netic order ends, ρ(T) ~ Tn, with n < 2.0. Most striking is the obser-vation of a perfectly linear T dependence ρ(T) = ρ0 + A1T as T → 0 in several very different materials, when tuned to their magnetic QCP; for example, the quasi-one-dimensional (1D) organic conductor (TMTSF)2PF6 (ref. 4), the quasi-2D ruthenate Sr3Ru2O7 (ref. 3) and the 3D heavy-fermion metal CeCu6 (ref. 1). This T-linear resistiv-ity as T → 0 has emerged as one of the major puzzles in the physics of metals5, and while several theoretical scenarios have been pro-posed15, no compelling explanation has been found.
In cuprates, a perfect T-linear resistivity as T → 0 has been observed (once superconductivity is suppressed by a magnetic field) in two closely related electron-doped materials, Pr2−xCexCuO4±δ (PCCO)2,16,17 and La2−xCexCuO4 (LCCO)13,14, and in three hole-doped materials: Bi2Sr2CuO6+δ (ref. 6), La2−xSrxCuO4 (LSCO)8 and La1.6−xNd0.4SrxCuO4 (Nd-LSCO)7,11,12. On the electron-doped side, T-linear resistivity is seen just above the QCP16 where antiferromag-netic order ends18 as a function of x, and as such it may not come as a surprise. On the hole-doped side, however, the doping values
where ρ(T) = ρ0 + A1T as T → 0 are very far from the QCP where long-range antiferromagnetic order ends (pN ~ 0.02); for example, at p = 0.24 in Nd-LSCO (Fig. 1a) and in the range p = 0.21–0.26 in LSCO (Fig. 1b). Instead, these values are close to the critical dop-ing where the pseudogap phase ends (that is, at p* = 0.23 ± 0.01 in Nd-LSCO (ref. 11) and at p* ~ 0.18–0.19 in LSCO (ref. 8)), where the role of antiferromagnetic spin fluctuations is not clear. In Bi2201, p* is farther still (see Supplementary Section 10).
To make progress, several questions must be answered. Is T-linear resistivity as T → 0 in hole-doped cuprates limited to single-layer materials with low Tc, or is it generic? Why is ρ(T) = ρ0 + A1T as T → 0 seen in LSCO over an anomalously wide doping range8? Is there a common mechanism linking cuprates to the other metals where ρ ~ T as T → 0?
To establish the universal character of T-linear resistivity in cuprates, we have turned to Bi2Sr2CaCu2O8+δ (Bi2212). While Nd-LSCO and LSCO have essentially the same single electron-like diamond-shaped Fermi surface at p > p* (refs 19,20), Bi2212 has a very different Fermi surface, consisting of two sheets, one of which is also diamond-like at p > 0.22, but the other is much more circular21 (see Supplementary Section 1). Moreover, the structural, magnetic and superconducting properties of Bi2212 are very different to those of Nd-LSCO and LSCO: a stronger 2D character, a larger gap to spin excitations, no spin-density-wave order above p ~ 0.1 and a much higher superconducting Tc.
We measured the resistivity of Bi2212 at p = 0.23 by sup-pressing superconductivity with a magnetic field of up to 58 T. At p = 0.23, the system is just above its pseudogap critical point (p* = 0.22 (ref. 22); see Supplementary Section 2). Our data are shown in Fig. 2. The raw data at H = 55 T reveal a perfectly linear T dependence of ρ(T) down to the lowest accessible temperature (Fig. 1a). Correcting for the magnetoresistance (see Methods and Supplementary Section 3), as was done for LSCO (ref. 8), we find that the T-linear dependence of ρ(T) seen in Bi2212 at H = 0 from T ~ 120 K down to Tc simply continues to low temperature, with the same slope A1 = 0.62 ± 0.06 μ Ω cm K−1 (Fig. 2b). Measured per CuO2 plane, this gives A1
□ ≡ A1/d = 8.0 ± 0.9 Ω K−1, where d is the (aver-age) separation between CuO2 planes. Remarkably, this is the same value, within error bars, as measured in Nd-LSCO at p = 0.24, where A1
□ = 7.4 ± 0.8 Ω K−1 (see Table 1).The observation of T-linear resistivity in those two cuprates
shows that it is robust against changes in the shape, topology and
Universal T-linear resistivity and Planckian dissipation in overdoped cupratesA. Legros1,2, S. Benhabib3, W. Tabis3,4, F. Laliberté" "1, M. Dion1, M. Lizaire1, B. Vignolle3, D. Vignolles" "3, H. Raffy5, Z. Z. Li5, P. Auban-Senzier5, N. Doiron-Leyraud1, P. Fournier1,6, D. Colson2, L. Taillefer" "1,6* and C. Proust" "3,6*
NATURE PHYSICS | VOL 15 | FEBRUARY 2019 | 142–147 | www.nature.com/naturephysics142
Remarkable recent observation of ‘Planckian’ strange metal transportin cuprates, pnictides, magic-angle graphene, and ultracold atoms: theresistivity is associated with a universal scattering time ⇡ ~/(kBT ).
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REPORT◥
QUANTUM SIMULATION
Bad metallic transport in a cold atomFermi-Hubbard systemPeter T. Brown1, Debayan Mitra1, Elmer Guardado-Sanchez1, Reza Nourafkan2,Alexis Reymbaut2, Charles-David Hébert2, Simon Bergeron2, A.-M. S. Tremblay2,3,Jure Kokalj4,5, David A. Huse1, Peter Schauß1*, Waseem S. Bakr1†
Strong interactions in many-body quantum systems complicate the interpretation ofcharge transport in such materials. To shed light on this problem, we study transport in aclean quantum system: ultracold lithium-6 in a two-dimensional optical lattice, a testingground for strong interaction physics in the Fermi-Hubbard model. We determine thediffusion constant by measuring the relaxation of an imposed density modulation andmodeling its decay hydrodynamically. The diffusion constant is converted to a resistivity byusing the Nernst-Einstein relation. That resistivity exhibits a linear temperaturedependence and shows no evidence of saturation, two characteristic signatures of a badmetal. The techniques we developed in this study may be applied to measurements ofother transport quantities, including the optical conductivity and thermopower.
In conventional materials, charge is carriedby quasiparticles and conductivity is under-stood as a current of these charge carriersdeveloped in response to an external field.For the conductivity to be finite, the charge
carriers must be able to relax their momentumthrough scattering. The Boltzmann kinetic equa-tion in conjunction with Fermi liquid theoryprovides a detailed description of transport inconventionalmaterials, including two trademarksof resistivity. The first is the Fermi liquid predictionthat the temperature (T)–dependent resistivity r(T)should scale like T2 at low temperature (1). Thesecond is that the resistivity should not exceedamaximum value rmax, obtained from the Druderelation assuming the Mott-Ioffe-Regel (MIR)limit, which states that the mean free path of aquasiparticle cannot be less than the lattice spacing(2, 3). This resistivity bound itself is sometimesreferred to as the MIR limit.Strong interactions can, however, lead to a
breakdown of Fermi liquid theory. One signal ofthis breakdown is anomalous scaling of r withtemperature, including the linear scaling ob-served in the strange metal state of the cuprates(4) and other anomalous scalings in d- and f-electron materials (5). Another is the violation ofthe resistivity bound r < rmax, which is observed
in a wide variety of materials (6). Additionally,interactions may lead to a situation where themomentum relaxation rate alone does not de-termine the conductivity, in contrast to thesemiclassical Drude formula, generalizationsof which hold for a large class of systems calledcoherent metals (7). Approaches introduced tounderstand these anomalous behaviors includehidden Fermi liquids (8), marginal Fermi liquids(9), proximity to quantum critical points (10) andassociated holographic approaches (11), andmany numerical studies of model systems,most notably the Hubbard model (12) and thet − J model (13).Disentangling strong interaction physics from
other effects, such as impurities and electron-phonon coupling, is difficult in real materials.Cold atom systems are free of these complica-tions, but transport experiments are challengingbecause of the finite and isolated nature of thesesystems. Most fermionic charge transport ex-periments have focused on studying either massflow through optically structured mesoscopicdevices (14–17) or bulk transport in lattice sys-tems (18–22). In this study, we explored bulktransport in a Fermi-Hubbard system by study-ing charge diffusion, which is a microscopicprocess related to conductivity through theNernst-Einstein equation s = ccD, where D isthe diffusion constant and cc ¼ @n
@m
! "jT is the
compressibility. This requires only the assumptionof linear response and the absence of thermo-electric coupling (23) anddoes not rest on assump-tions concerning quasiparticles.We realized the two-dimensional Fermi-Hubbard
model by using a degenerate spin-balancedmix-ture of two hyperfine ground states of 6Li in anoptical lattice (24). Our lattice beams producea harmonic trapping potential, which leads to avarying atomic density in the trap. To obtain a
system with uniform density, we flatten ourtrapping potential over an elliptical region ofmean diameter 30 sites by using a repulsivepotential created with a spatial light modulator.We superimpose an additional sinusoidal po-tential that varies slowly along one direction ofthe lattice with a controllable wavelength (Fig. 1,A and B). By adiabatically loading the gas intothese potentials, we prepare a Hubbard systemin thermal equilibrium with a small-amplitude(typically 10%) sinusoidal density modulation.The average density in the region with the flat-tened potential is the same with and without thesinusoidal potential. Next, we suddenly turn offthe added sinusoidal potential and observe thedecay of the density pattern versus time (Fig. 1,C and D), always keeping the optical lattice atfixed intensity. We measure the density of asingle spin component, hn↑i, by using techniquesdescribed in (24), giving us access to the totaldensity through hni ¼ h2n↑i.Wework at average total densityhni ¼ 0:82ð2Þ.
This value is close to a conjectured quantumcritical point in the Hubbard model (25). Ourlattice depth is 6.9(2) ER, where ER is the latticerecoil energy and ER/h = 14.66 kHz, leading toa tunneling rate of t/h = 925(10) Hz. Here h isPlanck’s constant. We adjust the scatteringlength, as = 1070(10)ao, by working at a mag-netic bias field of 616.0(2) G, in the vicinity ofthe Feshbach resonance centered near 690 G.These parameters lead to an on-site interaction–to–tunneling ratio U/t = 7.4(8), which is in thestrong-interaction regime and close to the valuethatmaximizes antiferromagnetic correlations athalf-filling (26).We observe the decay of the initial sinusoidal
density pattern over a period of a few tunnelingtimes. The short time scale ensures that theobserved dynamics are not affected by the in-homogeneous density outside of the centralflattened region of the trap. To obtain betterstatistics, we apply the sinusoidal modulationalong one dimension and average along theother direction (Fig. 1, A and C). We fit theaverage modulation profile to a sinusoid, wherethe phase and frequency are fixed by the initialpattern (Fig. 2A). The time dependence of theamplitude of the sinusoid quantifies the decayof the density modulation (Fig. 2B). Our experi-mental technique is analogous to that ofHild et al.(27), who studied the decay of a sinusoidallymodulated spin pattern in a bosonic system.The decay of the sinusoidal density pattern
versus the wavelength l of the modulation be-comes consistent with diffusive transport at longwavelengths. In diffusive transport, the ampli-tude of a density pattern at wave vector k = 2p/lwill decay exponentially with time constant t =1/Dk2, where D is the diffusion constant. We ob-serve exponentially decaying amplitudes withdiffusive scaling for wavelengths longer than15 sites. However, the decay curves are flat atearly times, showing clear deviation from ex-ponential decay. For short wavelengths, we ob-serve deviations from diffusive behavior in theform of underdamped oscillations, which can be
RESEARCH
Brown et al., Science 363, 379–382 (2019) 25 January 2019 1 of 4
1Department of Physics, Princeton University, Princeton, NJ08544, USA. 2Département de Physique, Institut Quantique,and Regroupement Québécois sur les Matériaux de Pointe,Université de Sherbrooke, Sherbrooke, Québec J1K 2R1,Canada. 3Canadian Institute for Advanced Research, Toronto,Ontario M5G 1Z8, Canada. 4Faculty of Civil and GeodeticEngineering, University of Ljubljana, SI-1000 Ljubljana,Slovenia. 5Jožef Stefan Institute, Jamova 39, SI-1000Ljubljana, Slovenia.*Present address: Department of Physics, University of Virginia,Charlottesville, VA 22904, USA.†Corresponding author. Email: [email protected]
on February 13, 2019
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excitations.Even
more
strikingexam
plesof
NFL
behaviorare
observedin
thehole-doped
cuprates[6,7]
andcertain
quantumcriticalheavy-ferm
ioncom
pounds[8]w
herethe
incoherentfeaturesappear
tosurvive
down
tothe
lowest
measurable
temperatures,i.e.
Tcoh!
0,when
superconduc-tivity
issuppressedexternally.In
theincoherentregim
e,allofthese
materials
inspite
ofbeingm
icroscopicallydistincthave
aresistivity,
⇢(T
)⇠
T,and
exhibitanum
berof
anomalous
features[8–11]
clearlyindicating
theabsence
ofsharp
elec-tronic
quasiparticles.Instrongly
interactingnon-quasiparticle
systems,ithasbeen
conjectured[12]thattransport“scattering
rates”(�)satisfy
auniversal‘Planckian’bound
�.
O(k
BT/~)
atatem
peratureT
.Itis
howeverw
orthnoting
thatina
NFL
thereisin
generalnocleardefinition
for�;the
scatteringrates
definedthrough
di↵erentmeasurem
ents,suchas
dcand
opti-calconductivity,need
notbeidentical[13].
One
ofthem
ostsurprising
aspectsofincoherenttransportin
thesesystem
s,inspite
ofthesesubtleties,is
that�
extractedfrom
thedc
resis-tivity
(througha
procedurespecified
inR
ef.[14])appears
tosatisfy
auniversalform
�=
Ck
BT/~
with
Ca
numberoforder
1[14,15].
Recent
experiments
[16,17]
havereported
thediscovery
ofacorrelation
driveninsulatoratfractionalfillings
(with
re-spect
toa
fullyfilled
isolatedband)
inm
agic-anglebilayer
graphene(M
AB
LG).
InM
AB
LG,
therelative
rotationbe-
tween
two
sheetsofgraphene
generatesa
moire
pattern(Fig.
1a)w
itha
periodicitythat
ism
uchlarger
thanthe
underly-ing
interatomic
distancesin
graphene.The
theoreticallyesti-
mated
electronicbandw
idth,W
,isstronglyrenorm
alizednear
thesesm
allmagic-angles[18–20];then
thestrength
ofthetyp-
icalCoulom
binteractions,
U,becom
esatleastcom
parableto
(ifnot
greaterthan)
thebandw
idth,U&
W.
Investigatingthe
propertiesof
MA
BLG
asa
functionof
temperature
andcarrier
densityw
ithunprecedented
tunabilityin
acontrolled
setupcan
leadto
newinsights
intothe
natureof
electronictransportin
otherlow-dim
ensionalstronglycorrelated
metal-
licsystem
s.For
ourexperim
ents,w
ehave
fabricatedm
ultiplehigh-
qualityencapsulated
MA
BLG
devices(see
Fig.1a)
usingthe
‘tearand
stack’technique
[21,22].A
sreported
earlier[16,
22],w
eobtain
band-insulatorsw
itha
largegap
nearn⇡±
ns ,w
heren
isthe
carrierdensity
tunedexternally
byapplying
agate
voltageand
ns
correspondsto
fourelectrons
permoire
unitcell.O
nthe
otherhand,correlationdriven
in-sulators
with
much
smaller
gap-scales[16,
23]appear
nearn⇡±
ns /2;
we
denotethese
fillingsas⌫=±
2from
nowon
(thefully
filledband
correspondsto⌫=+
4).D
op-ing
away
fromthese
correlatedinsulators
byan
additionalam
ount,�,with
holes(⌫=±
2��)and
electrons(⌫=±
2+�)
leadsto
superconductivity(SC
)[17,23].
Thesuperconduct-
ingtransition
temperature
(Tc )m
easuredrelative
tothe
Fermi-
energy("
F ),asinferred
fromlow
temperature
quantumoscil-
lationsm
easurements
[17],ishigh,w
iththe
largestvalueof
(Tc /"
F )⇠0.07�
0.08,indicatingstrong
couplingsupercon-
ductivity[17].A
schematic⌫�
Tphase-diagram
forMA
BLG
isshow
nin
Fig.1b.In
thisw
ork,we
investigatethe
transportphenomenology
ofthe
metallic
statesin
MA
BLG
asa
functionof
increasingtem
peratureovera
largerange
ofexternallytuned
fillings.We
haveanalyzed
thetem
peraturedependence
ofthelongitudinal
DC
resistivity,⇢(
T),for
anum
berof
devices(M
A1
-M
A6)
overa
wide
rangeof
fillings,theresults
ofw
hichappear
tobe
qualitativelysim
ilaracross
devices.In
orderto
highlightthe
universalaspectsof
thebehavior,both
within
andacross
di↵erentsamples,w
eshow
thetem
peraturedependenttraces
of⇢(T
)for
arange
ofdi↵erent
⌫in
two
devicesM
A1
andM
A4
inFig.1c
andFig.1d
respectively.Therange
of⌫
chosenforthis
purposeis
shown
asa
color-barinFig.1b;see
caption
arXiv:1901.03710v1 [cond-mat.str-el] 11 Jan 2019
18
Table 1 | Slope of T-linear resistivity and Planckian limit in seven materials.
Material n (1027 m-3)
m*(m0)
A1 / d (! / K)
h / (2e2 TF)(! / K)
⍺
Bi2212 p = 0.23 6.8 8.4 ± 1.6 8.0 ± 0.9 7.4 ± 1.4 1.1 ± 0.3
Bi2201 p ~ 0.4 3.5 7 ± 1.5 8 ± 2 8 ± 2 1.0 ± 0.4
LSCO p = 0.26 7.8 9.8 ± 1.7 8.2 ± 1.0 8.9 ± 1.8 0.9 ± 0.3
Nd-LSCO p = 0.24 7.9 12 ± 4 7.4 ± 0.8 10.6 ± 3.7 0.7 ± 0.4
PCCO x = 0.17 8.8 2.4 ± 0.1 1.7 ± 0.3 2.1 ± 0.1 0.8 ± 0.2
LCCO x = 0.15 9.0 3.0 ± 0.3 3.0 ± 0.45 2.6 ± 0.3 1.2 ± 0.3
TMTSF P = 11 kbar 1.4 1.15 ± 0.2 2.8 ± 0.3 2.8 ± 0.4 1.0 ± 0.3
Table 1 | Slope of T-linear resistivity vs Planckian limit in seven materials.
Comparison of the measured slope of the T-linear resistivity in the T = 0 limit,
A1 , with the value predicted by the Planckian limit (Eq. 1; penultimate column),
for four hole-doped cuprates (Bi2212, Bi2201, LSCO and Nd-LSCO), two
electron-doped cuprates (PCCO and LCCO) and the organic conductor
(TMTSF)2PF6 , as discussed in the text (and Supplementary Information).
The ratio α of the experimental value, A1☐ = A1 / d, over the predicted value,
is given in the last column. Although A1☐ varies by a factor 5, the ratio m* / n
(~1/TF) is seen to vary by the same amount, so that α = 1.0 in all cases,
within error bars.
18
Table 1 | Slope of T-linear resistivity and Planckian limit in seven materials.
Material n (1027 m-3)
m*(m0)
A1 / d (! / K)
h / (2e2 TF)(! / K)
⍺
Bi2212 p = 0.23 6.8 8.4 ± 1.6 8.0 ± 0.9 7.4 ± 1.4 1.1 ± 0.3
Bi2201 p ~ 0.4 3.5 7 ± 1.5 8 ± 2 8 ± 2 1.0 ± 0.4
LSCO p = 0.26 7.8 9.8 ± 1.7 8.2 ± 1.0 8.9 ± 1.8 0.9 ± 0.3
Nd-LSCO p = 0.24 7.9 12 ± 4 7.4 ± 0.8 10.6 ± 3.7 0.7 ± 0.4
PCCO x = 0.17 8.8 2.4 ± 0.1 1.7 ± 0.3 2.1 ± 0.1 0.8 ± 0.2
LCCO x = 0.15 9.0 3.0 ± 0.3 3.0 ± 0.45 2.6 ± 0.3 1.2 ± 0.3
TMTSF P = 11 kbar 1.4 1.15 ± 0.2 2.8 ± 0.3 2.8 ± 0.4 1.0 ± 0.3
Table 1 | Slope of T-linear resistivity vs Planckian limit in seven materials.
Comparison of the measured slope of the T-linear resistivity in the T = 0 limit,
A1 , with the value predicted by the Planckian limit (Eq. 1; penultimate column),
for four hole-doped cuprates (Bi2212, Bi2201, LSCO and Nd-LSCO), two
electron-doped cuprates (PCCO and LCCO) and the organic conductor
(TMTSF)2PF6 , as discussed in the text (and Supplementary Information).
The ratio α of the experimental value, A1☐ = A1 / d, over the predicted value,
is given in the last column. Although A1☐ varies by a factor 5, the ratio m* / n
(~1/TF) is seen to vary by the same amount, so that α = 1.0 in all cases,
within error bars.
A. Legros, S. Benhabib, W. Tabis, F. Laliberté, M. Dion, M. Lizaire, B. Vignolle, D. Vignolles, H. Raffy, Z. Z. Li, P. Auban-Senzier, N. Doiron-Leyraud, P. Fournier, D. Colson, L. Taillefer, and C. Proust, Nature Physics 15, 142 (2019)
1
⌧= ↵
kBT
~<latexit sha1_base64="Q0pGo+lkjPvE7fTn3atRPbcYr7A=">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</latexit>
Horizon radius R =2GM
c2
Objects so dense that light is gravitationally bound to them.
Black Holes
In Einstein’s theory, the region inside the black hole horizon is disconnected from
the rest of the universe.
LIGOSeptember 14, 2015
• The ring-down is predicted by General Relativity to happen in a
time8⇡GM
c3⇠ 8 milliseconds. Curiously this happens to equal
~kBTH
: so the ring down can also be viewed as the approach of a
quantum system to thermal equilibrium at the fastest possible rate.!
Black holes
• Black holes have an entropy anda temperature, TH = ~c3/(8⇡GMkB).
• The entropy is proportional totheir surface area.
• They relax to thermal equilib-rium in a Planckian time⇠ ~/(kBTH).
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Black holes
Holography:Quantum black holes “look like” quantum many-particle systems
without quasiparticle excitations, residing “on” the surface of the black hole
J. Maldacena, IJTP 38, 1113 (1999); S.S. Gubser, I.R. Klebanov, and A.M. Polyakov Phys. Lett. B 428, 105 (1998); E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)
• Black holes have an entropy anda temperature, TH = ~c3/(8⇡GMkB).
• The entropy is proportional totheir surface area.
• They relax to thermal equilib-rium in a Planckian time⇠ ~/(kBTH).
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1. Quantum matter with quasiparticles: random matrix model
2. Quantum matter without quasiparticles: the complex SYK model
3. Fluctuations, and the Schwarzian
4. Models of strange metals
5. Einstein-Maxwell theory of charged black holes in AdS space
1. Quantum matter with quasiparticles: random matrix model
2. Quantum matter without quasiparticles: the complex SYK model
3. Fluctuations, and the Schwarzian
4. Models of strange metals
5. Einstein-Maxwell theory of charged black holes in AdS space
• Note: The electron liquid in one dimension and the fractional
quantum Hall state both have quasiparticles; however, the quasi-
particles do not have the same quantum numbers as an electron.
What are quasiparticles ?
• Quasiparticles are additive excitations:The low-lying excitations of the many-body systemcan be identified as a set {n↵} of quasiparticles withenergy "↵
E =P
↵ n↵"↵ +P
↵,� F↵�n↵n� + . . .
In a lattice system ofN sites, this parameterizes the energyof ⇠ e↵N states in terms of poly(N) numbers.
Ordinary metals and quasiparticles• Quasiparticles eventually collide with each other. Such collisions even-
tually leads to thermal equilibration in a chaotic quantum state, but theequilibration takes a long time. In a Fermi liquid, this time diverges as
⌧eq ⇠ ~E3F
U2(kBT )2, as T ! 0,
where U is the strength of interactions, and EF is the Fermi energy.
• Similarly, a quasiparticle model implies a resistivity
⇢ =m⇤
ne21
⌧⇠ U2T 2 with ⌧ ⇠ ⌧eq
• These times are much longer than the‘Planckian time’ ~/(kBT ), which we willfind in systems without quasiparticle excitations.
⌧ ⇠ ⌧eq � ~kBT
, as T ! 0.<latexit sha1_base64="9/VubhQoZ2AU9raguSa8JnhCjTk=">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</latexit>
Ordinary metals and quasiparticles• Quasiparticles eventually collide with each other. Such collisions even-
tually leads to thermal equilibration in a chaotic quantum state, but theequilibration takes a long time. In a Fermi liquid, this time diverges as
⌧eq ⇠ ~E3F
U2(kBT )2, as T ! 0,
where U is the strength of interactions, and EF is the Fermi energy.
• Similarly, a quasiparticle model implies a resistivity
⇢ =m⇤
ne21
⌧⇠ U2T 2 with ⌧ ⇠ ⌧eq
• These times are much longer than the‘Planckian time’ ~/(kBT ), which we willfind in systems without quasiparticle excitations.
⌧ ⇠ ⌧eq � ~kBT
, as T ! 0.<latexit sha1_base64="9/VubhQoZ2AU9raguSa8JnhCjTk=">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</latexit>
Fermions occupying the eigenstates of a N x N random matrix
tij are independent random variables with tij = 0 and |tij |2 = t2
A simple model of a metal with quasiparticles
H =1
(N)1/2
NX
i,j=1
tijc†i cj � µ
X
i
c†i ci
cicj + cjci = 0 , cic†j + c
†jci = �ij
1
N
X
i
c†i ci = Q
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A simple model of a metal with quasiparticles
⇢(!) =� 1
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Feynman graph expansion in tij.., and graph-by-graph average,yields exact equations in the large N limit:
G(⌧) ⌘ �T⌧
Dci(⌧)c
†i (0)
E
G(i!) =1
i! + µ� ⌃(i!), ⌃(⌧) = t2G(⌧)
G(⌧ = 0�) = Q.
G(!) can be determined by solving a quadratic equation.<latexit sha1_base64="TvyYmNq1c/d62y589QXYkoEDoqw=">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</latexit><latexit sha1_base64="TvyYmNq1c/d62y589QXYkoEDoqw=">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</latexit><latexit sha1_base64="TvyYmNq1c/d62y589QXYkoEDoqw=">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</latexit><latexit sha1_base64="TvyYmNq1c/d62y589QXYkoEDoqw=">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</latexit>
A simple model of a metal with quasiparticles
"↵ levelspacing ⇠ 1/N
⇢(!) =� 1
⇡ ImG(!)<latexit sha1_base64="3hkVLGGiMUm+AUx/4pxVADH9CSY=">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</latexit><latexit sha1_base64="3hkVLGGiMUm+AUx/4pxVADH9CSY=">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</latexit><latexit sha1_base64="3hkVLGGiMUm+AUx/4pxVADH9CSY=">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</latexit><latexit sha1_base64="3hkVLGGiMUm+AUx/4pxVADH9CSY=">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</latexit>
Let "↵ be the eigenvalues of the matrix tij/pN .
The fermions will occupy the lowest NQ eigen-values, upto the Fermi energy EF . The single-particle density of states is⇢(!) = (1/N)
P↵ �(! � "↵), and ⇢0 ⌘ ⇢(! = 0).
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A simple model of a metal with quasiparticles
The grand potential ⌦(T ) at low T is (from the Sommerfeld expansion)
⌦(T )� E0 = N
✓�⇡2
6⇢0T
2 +O(T 4)
◆+ . . .
where ⇢0 ⌘ ⇢(0) is the single particle density of states at the Fermi level.We can also define the many body density of states, D(E), via
Z = e�⌦(T )/T =
Z 1
�1dED(E)e�E/T
The inversion from ⌦(T ) to D(E) has to performed with care (it need not commutewith the 1/N expansion), and we obtain
D(E) ⇠ exp
⇡
r2N⇢0(E � E0)
3
!, E > E0 ,
1
N⌧ ⇢0(E � E0) ⌧ N
andD(E) = 0 for E < E0. This is related to the asymptotic growth of the partitionsof an integer, p(n) ⇠ exp(⇡
p2n/3). Near the lower bound, there are large sample-
to-sample fluctuations due to variations in the lowest quasiparticle energies.<latexit 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A simple model of a metal with quasiparticles
Quasiparticleexcitations withspacing ⇠ 1/N
There are 2N manybody levels with energy
E =NX
↵=1
n↵"↵,
where n↵ = 0, 1. Shownare all values of E for asingle cluster of size
N = 12. The "↵ have alevel spacing ⇠ 1/N .
Many-bodylevel spacing
⇠ 2�N⇠ Nt<latexit sha1_base64="TLjbKtS021mdKXlJRfXvFeutNo8=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=</latexit><latexit sha1_base64="TLjbKtS021mdKXlJRfXvFeutNo8=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=</latexit><latexit sha1_base64="TLjbKtS021mdKXlJRfXvFeutNo8=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=</latexit><latexit sha1_base64="TLjbKtS021mdKXlJRfXvFeutNo8=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=</latexit>
Fermi liquid state: Two-body interactions lead to a scattering timeof quasiparticle excitations from in (random) single-particle eigen-states which diverges as ⇠ T�2 at the Fermi level.
A simple model of a metal with quasiparticlesNow add weak interactions
H =1
(N)1/2
NX
i,j=1
tijc†i cj � µ
X
i
c†i ci +
1
(2N)3/2
NX
i,j,k,`=1
Uij;k` c†i c
†jckc`
Uij;k` are independent random variables with Uij;k` = 0 and |Uij;k`|2 = U2. We
compute the lifetime of a quasiparticle, ⌧↵, in an exact eigenstate ↵(i) of the
free particle Hamitonian with energy "↵. By Fermi’s Golden rule, for "↵ at the
Fermi energy
1
⌧↵= ⇡U
2⇢30
Zd"�d"�d"�f("�)(1� f("�))(1� f("�))�("↵ + "� � "� � "�)
=⇡3U
2⇢30
4T
2
where ⇢0 is the density of states at the Fermi energy, and f(✏) = 1/(e✏/T
+1) is
the Fermi function.<latexit sha1_base64="P2XD/1G9epCNe0KQg6JmiM9I8aY=">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</latexit>
1. Quantum matter with quasiparticles: random matrix model
2. Quantum matter without quasiparticles: the complex SYK model
3. Fluctuations, and the Schwarzian
4. Models of strange metals
5. Einstein-Maxwell theory of charged black holes in AdS space
A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)
S. Sachdev and J. Ye, PRL 70, 3339 (1993)
(See also: the “2-Body Random Ensemble” in nuclear physics; did not obtain the large N limit;T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, and S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981))
U↵�;�� are independent random variables with U↵�;�� = 0 and |U↵�;��|2 = U2
N ! 1 yields critical strange metal.<latexit sha1_base64="yQteaZaLoEYlztplY68/SkIu+Hc=">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 UfkMXlKYvKc7JPX5IAMCPPeeR+8j94nP/M/+1/8r+tU39twHpI/zP/2C0qR6Mw=</latexit>
H =1
(2N)3/2
NX
↵,�,�,�=1
U↵�;�� c†↵c
†�c�c� + ✏
X
↵
c†↵c↵
c↵c� + c�c↵ = 0 , c↵c†� + c
†�c↵ = �↵�
Q =1
N
X
↵
c†↵c↵
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The complex SYK model
U U⌃ =<latexit sha1_base64="pk3ZmbMDJBp861UaFugr9daoJp8=">AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==</latexit><latexit sha1_base64="pk3ZmbMDJBp861UaFugr9daoJp8=">AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==</latexit><latexit sha1_base64="pk3ZmbMDJBp861UaFugr9daoJp8=">AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==</latexit><latexit sha1_base64="pk3ZmbMDJBp861UaFugr9daoJp8=">AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==</latexit>
G<latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit>
G<latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit>
G<latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit>
The complex SYK model
S. Sachdev and J. Ye, PRL 70, 3339 (1993)
Feynman graph expansion in U↵�;��, and graph-by-graph average, yieldsexact equations in the large N limit:
G(i!) =1
i! � ✏� ⌃(i!), ⌃(⌧) = �U2G2(⌧)G(�⌧)
G(⌧ = 0�) = Q.<latexit sha1_base64="S4bYhQ+/icXvh6gGGPmW8BXeedU=">AAADanicdVLbbtNAELUbLiVceuEJ9WVEQ9VKSWSHmlYgpAokyhNqBWkrddNovJ44q67XxrsujSx/G9/BB/CG4J0n1knKTbDSSkdn5syZnZ0wk0Ibz/vkLjSuXb9xc/FW8/adu/eWlldWj3Ra5Jz6PJVpfhKiJikU9Y0wkk6ynDAJJR2H5y/r+PEF5Vqk6p2ZZDRIMFZiJDgaSw1X3BOmUqEiUqb5iiYqQQVxjtkY6DJDVetAKGj1hyVDmY2RhWTwGYsxSZBFJA1WrTagimayTjjpzPRobTGmNkwEyUjbesgN0Pti6qzrqmZMIDGPCVpvWiBFIszTpjWIhSpRilhVzf1NwdKEYtyC58BGOfLSr0oBMxI6wCjTQtouLXwr4gThp6ICZt0i2GjPwVUCM1jU9ToA/bMe7Ns75/Y3OzNgp6KKJKQcGLNN1KQVeGedLdioO0nQjDnK8rDq/sptMlLRVefD5XWv6+0E20988Lq+5+0GgQXBdhAEPfC73vSsO/NzMFz+zqKUF4n9Ci5R61Pfy8ygxNwILqlqskJThvzcDvXUQoUJ6XZ0ITI9hYNyug0VPLLBCEZpbq8yMGV/F5eYaD1JQptZv0L/HavJf8VOCzPaHZRCZYUhxWdGo0KCSaFeLYhETtzIiQXIc2HbBj5G+2XGLmCTabLbqWIzLpmhS/NBRNan3O4GQlV2WFcTgf+Do17Xf9ztHfbW917Mx7borDkPnU3Hd3acPee1c+D0He5+dD+7X91vC18aq40HjbVZ6oI719x3/jiN1g9qfhJv</latexit>