the metallic states of the cuprates - harvard...

The metallic states of the cuprates:

Fermi liquids, fractionalized Fermi liquids, and non-Fermi liquids

School on Strongly Coupled Field Theories for Condensed Matter and Quantum Information Theory

International Institute of PhysicsNatal, Brazil

August 13-14, 2015

Subir Sachdev

HARVARD

Talk online: sachdev.physics.harvard.edu

YBa2Cu3O6+x

High temperature superconductors

CuO2 plane

Cu

O

“Undoped”Anti-

ferromagnet

FilledBand

“Undoped”Anti-

ferromagnet

Anti-ferromagnetwith p holesper square

Anti-ferromagnetwith p holesper square

But relative to the band

insulator, there are 1+ p holesper square, and

so a Fermi liquid has a

Fermi surface of size 1+ p

SM

FL

Figure: K. Fujita and J. C. Seamus Davis

YBa2Cu3O6+x

1. Conventional metal

Area enclosed by Fermi surface =1+p

M. Plate, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, Ruixing Liang, D. A. Bonn, W. N. Hardy,S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli, Phys. Rev. Lett. 95, 077001 (2005)

SM

FL

SM

FL

Pseudogap

2. Pseudogap metal

at low p

Kyle M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana,Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, Z.-X. Shen, Science 307, 901 (2005)

SM

FL

3. Strange metal

No quasiparticle excitations

SM

FL

1. Review of Fermi liquid theory Topological argument for the Luttinger theorem

2. Fractionalized Fermi liquid A Fermi liquid co-existing with topological order for the pseudogap metal

3. Quantum matter without quasiparticles(A) A mean-field model of a non-Fermi liquid, and

charged black holes(B) Field theory of a non-Fermi liquid (Ising-nematic

quantum critical point)(C) Theory of transport in strange metals: application to

the (less) strange metal in graphene

Fermisurface

kx

ky

Ordinary quantum matter: the Fermi liquid (FL)• Fermi surface separates empty

and occupied states in mo-

mentum space.

• Area enclosed by Fermi

surface = total density of

electrons (mod 2) = 1+p.

• Density of electrons can

be continuously varied at

zero temperature.

• Long-lived electron-like quasi-

particle excitations near

the Fermi surface: lifetime

of quasiparticles 1/T 2.

• Fermi surface separates empty


mentum space.






zero temperature.





Fermisurface

kx

ky

Ordinary quantum matter: the Fermi liquid (FL)

Fermisurface

kx

ky

Ordinary quantum matter: the Fermi liquid (FL)• Fermi surface separates empty


mentum space.






zero temperature.





Oshikawa’s non-perturbative proof of the Luttinger theorem

Lx

Ly

Φ

We take N particles, each with charge Q, on a Lx

Ly

lattice on a torus.

We pierce flux = hc/Q through a hole of the torus.

An exact computation shows that the change in crystal momentum of the

many-body state due to flux piercing is

Pxf

Pxi

=

2N

Lx

(mod 2) = 2Ly

(mod 2)

where = N/(Lx

Ly

) is the density.

M. Oshikawa, PRL 84, 3370 (2000)A. Paramekanti and A. Vishwanath,

PRB 70, 245118 (2004)

Oshikawa’s non-perturbative proof of the Luttinger theorem

Proof of

Pxf

Pxi

=

2N

Lx

(mod 2) = 2Ly

(mod 2).

The initial and final Hamiltonians are related by a gauge transformation

UG

Hf

U1G

= Hi

, UG

= exp

i2

Lx

X

i

xi

ni

!.

while the wavefunction evolves from | i

i to UT

| i

i, where UT

is the time evolutionoperator. We want to work in a fixed gauge in which the initial and final Hamilto-nians are the same: in this gauge, the final state is |

f

i = UG

UT

| i

i. Let Tx

bethe lattice translation operator. Then we can show that

Tx

| i

i = eiP

xi | i

i , Tx

| f

i = eiP

xf | f

i ,

using the easily established properties

Tx

UT

= UT

Tx

, Tx

UG

= exp

i2

N

Lx

UG

Tx

Oshikawa’s non-perturbative proof of the Luttinger theoremP

x

= 2Ly

(mod 2) , Py

= 2Lx

(mod 2)Now we compute the momentum balance assuming that the only low energy excita-

tions are quasiparticles near the Fermi surface, and these react like free particles to a

suciently slow flux insertion. Then we can write

Px

=

2

Lx

Lx

Ly

42VFS , P

y

=

2

Ly

Lx

Ly

42VFS

where VFS is the volume of the Fermi surface. Actually, the quasiparticles are only

defined near the Fermi surface, but by using Gauss’s Law on the momentum acquired

by quasiparticles near the Fermi surface, we can convert the answer to an integral over

the volume enclosed by the Fermi surface, as shown above.

Now we equate these values to those obtained above, and obtain

N Lx

Ly

VFS

42= L

x

mx

, N Lx

Ly

VFS

42= L

y

my

for some integers mx

, my

. By choosing Lx

, Ly

mutually prime integers we can now

show

=

N

Lx

Ly

=

VFS

42+m

for some integer m: this is Luttinger’s theorem.

1. Review of Fermi liquid theory Topological argument for the Luttinger theorem

2. Fractionalized Fermi liquid A Fermi liquid co-existing with topological order for the pseudogap metal

3. Quantum matter without quasiparticles(A) A mean-field model of a non-Fermi liquid, and

charged black holes(B) Field theory of a non-Fermi liquid (Ising-nematic

quantum critical point)(C) Theory of transport in strange metals: application to

the (less) strange metal in graphene

Fractionalized Fermi liquid (FL*) = | |

Realizes a metal with a

Fermi surface of

area p co-existing

with “topological

order”

M. Punk, A. Allais, and S. S., PNAS 112, 9552 (2015)

A fermionic “dimer” describing a “bonding” orbital between two sites


Fermi surface of

area p co-existing

with “topological

order”Density of fermionic dimers = p;

density of holes relative to filled band = 1+ pM. Punk, A. Allais, and S. S., PNAS 112, 9552 (2015)

Fractionalized Fermi liquid (FL*) = | |


Fermi surface of

area p co-existing

with “topological

order”


Place pseudogap metal on a

torus;

Topological order

obtain “topological” states nearly

degenerate with the ground state:

number of dimers crossing

red line is (almost)

conserved modulo 2

Topological order Place

pseudogap metal on a

torus;

Topological order = | |





conserved modulo 2


torus;

Topological order = | |

to change dimer number parity

across red line, it is necessary to create a pair of unpaired spins and move them

around the sample.


torus;

= | | Topological order

to change dimer number parity

across red line, it is necessary to create a pair of unpaired spins and move them

around the sample.


torus;





conserved modulo 2

Topological order Place

pseudogap metal on a

torus;

Topological order: flux insertion

Upon inserting a

flux quantum =

h/e coupling onlyto " electrons (say),the states with an

even (odd) num-

ber of dimers cross-

ing the cut pick

up a factor of +1

(1). The hole

of the torus has

a “vison” after the

flux insertion.

N. E. Bonesteel, PRB 40, 8954 (1989).

Fractionalized Fermi liquid (FL*)

T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004)

Px

= 2Ly

(mod 2) , Py

= 2Lx

(mod 2)

Momentum balance for a Z2

fractionalized Fermi liquid:

Momentum acquired by quasiparticles near the Fermi surface:

(Px

)

quasiparticles

=

2

Lx

Lx

Ly

42

VFS

Momentum acquired by vison:

(Px

)

vison

= Ly

Using Px

= (Px

)

quasiparticles

+ (Px

)

vison

we obtain the Fermi

surface volume:

=

N

Lx

Ly

= 2

VFS

42

+2m+1 for some integer m. For the

cuprates, this implies a Fermi volume of density p and not 1 + p.

Quantum dimer model with bosonic and fermionic dimers

Connection to the

t-t0-t00-J model:

t1 = (t+ t0)/2t2 = (t t0)/2

t3 = (t+ t0+ t00)/4

,J V)

,t1 ,

,

t2

t3


Quantum dimer model with bosonic and fermionic dimers

Dispersion and quasiparticle residue of a single fermionic dimer for J = V = 1,

and hopping parameters obtained from the t-J model for the cuprates,

t1 = 1.05, t2 = 1.95 and t3 = 0.6, on a 8 8 lattice.


“Back side” of Fermi surface is suppressed for observables which change electron number in the square lattice

0

2

4

6

0p2p

0p

2p

Y. Qi and S. Sachdev,Phys. Rev. B 81, 115129 (2010)

M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

1. Conventional metal

Area enclosed by Fermi surface =1+p

M. Plate, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, Ruixing Liang, D. A. Bonn, W. N. Hardy,S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli, Phys. Rev. Lett. 95, 077001 (2005)

SM

FL

SM

FL

Pseudogap

2. Pseudogap metal

at low p


SM

FL

Evolution of the Hall Coefficient and the Peculiar Electronic Structureof the Cuprate Superconductors

Yoichi Ando,* Y. Kurita,† Seiki Komiya, S. Ono, and Kouji SegawaCentral Research Institute of Electric Power Industry, Komae, Tokyo 201-8511, Japan

(Received 3 July 2003; published 13 May 2004)

Although the Hall coefficient RH is an informative transport property of metals and semiconductors,its meaning in the cuprate superconductors has been ambiguous because of its unusual characteristics.Here we show that a systematic study of RH in La2!xSrxCuO4 single crystals over a wide doping rangeestablishes a qualitative understanding of its peculiar evolution, which turns out to reflect a two-component nature of the electronic structure caused by an unusual development of the Fermi surfacerecently uncovered by photoemission experiments.

DOI: 10.1103/PhysRevLett.92.197001 PACS numbers: 74.25.Fy, 74.25.Jb, 74.72.Dn

During the past 17 years after the high-Tc supercon-ductivity was discovered in cuprates, virtually all mea-surable properties of their ‘‘normal state,’’ the state in theabsence of superconductivity, have been studied to under-stand the stage for novel superconductivity. However,there is yet no established picture for even such basicproperties as the resistivity and the Hall coefficient [1],not to mention other more elaborate properties. The Hallcoefficient RH of conventional metals is independent oftemperature and signifies the Fermi surface (FS) topol-ogy and carrier density, but in cuprates RH shows strong,sometimes peaked, temperature dependences as well as acomplicated doping dependence. An advance in under-standing came when Chien,Wang, and Ong found [2] thatthe cotangent of the Hall angle, cot!H (which is the ra-tio of the in-plane resistivity !ab to the Hall resistivity!H), approximately shows a simple linear-in-T2 behavior,which suggests the existence of a quasiparticle-relaxationrate that changes as "T2. However, while it appears thatthe Hall problem in cuprates can be simplified whenanalyzed in terms of cot!H, it was argued by Ong andAnderson [3] that cot!H is after all a derived quantity,and the central anomaly resides in the directly measuredquantities !ab and RH.

In this Letter, we address the notoriously difficult prob-lem of the Hall effect with the recent knowledge on thephysics of lightly doped cuprates and the peculiar evolu-tion of the FS recently elucidated by the angle-resolvedphotoemission spectroscopy (ARPES) experiments [4,5].We first show that the behavior of RH and !ab in thelightly doped cuprates mimics rather well the behaviorof a conventional Fermi liquid, and discuss that thisbehavior signifies the physics on the ‘‘Fermi arc,’’ a smallportion of the FS near the Brillouin-zone diagonals. Wethen discuss that the peculiar hole-doping dependenceand the temperature dependence of RH reflect a gradualparticipation of the ‘‘flatband’’ near #"; 0$ of the Brillouinzone, which brings about a sort of two-band nature to thetransport. The measurements of RH and !ab using a stan-dard six-probe method are done on high-quality single

crystals of La2!xSrxCuO4 (LSCO) and YBa2Cu3Oy(YBCO), the details of which have been describedelsewhere [6].

In slightly hole-doped LSCO and YBCO, which areusually considered to be antiferromagnetic insulators, itwas demonstrated [6] that the charge transport shows asurprisingly metallic behavior with a hole mobility com-parable to that of optimally doped superconductors atmoderate temperatures. Detailed ARPES measurementswere subsequently performed on lightly doped LSCO [5]and YBCO [7], which revealed that only patches of FS,called ‘‘Fermi arcs’’ [8], are observed at the zone diago-nals, where quasiparticlelike peaks were detected in har-mony with the transport results. The ARPES resultsindicate that, for some reason, a significant fraction ofthe large FS (that is observed in optimally doped cuprates[9]) is destroyed and the remaining small portion isresponsible for the metallic transport. Thus, by lookingat the transport properties of the lightly doped cuprates,one can gain insight into the physics of the Fermi arc,whose origin is currently under debate [10].

Figures 1(a) and 1(b) show an example of the behaviorof !ab#T$ and RH#T$ in a lightly doped cuprate: Here thedata are for YBCO with y % 6:30 (hole doping of about3% per Cu [6]), which is an antiferromagnet with the Neeltemperature of 230 K [11]. As we have reported previ-ously [6,12], RH is virtually T independent (as in conven-tional metals) at moderate temperatures in the lightlydoped samples; as a result, !ab and cot!H have thesame T dependence [see Figs. 1(c) and 1(d)], which, in-triguingly, is most consistent with "T2 and not with "T.This implies that the relaxation rate of the ‘‘quasipar-ticles’’ on the Fermi arc changes as "T2, which inciden-tally is the same as the behavior of conventional Fermiliquids. Note that the low-temperature upturn in !ab andRH is due to localization effects [13], which just obscurethe intrinsic low-temperature behavior of the system.

We found that the T dependence of !ab is consistentwith "T2 not only in lightly doped YBCO but also inlightly doped LSCO, as shown in Fig. 2(a) for x % 0:02;

P H Y S I C A L R E V I E W L E T T E R S week ending14 MAY 2004VOLUME 92, NUMBER 19

197001-1 0031-9007=04=92(19)=197001(4)$22.50 © 2004 The American Physical Society 197001-1

however, it should be noted that the temperature rangefor the T2 law is a bit narrow [Fig. 2(a) shows the dataup to 250 K] and !ab!T" deviates downwardly from the T2

dependence at high temperatures. As is shown for x #0:08 [Fig. 2(b)], the deviation tends to start from lowertemperature at larger x. One possible way to interpret thisdeviation is to ascribe it to an increase in the density ofstates at the Fermi energy EF with increasing T, whichhappens, for example, when a gap is filled in with T. Infact, the band structure of LSCO elucidated by ARPES[4,5] suggests such a possibility: In lightly doped LSCO aflatband [located near !"; 0" of the Brillouin zone] liesbelow EF and this band gradually moves up to EF withincreasing doping; therefore, if thermal activation causessome holes to reside on the flatband and to contribute tothe conductivity, the high-temperature deviation from the

T2 behavior and its doping dependence can be understood,at least qualitatively.

The systematics of RH!T" and cot!H!T" of LSCO fora very wide range of doping (x # 0:02–0:25) is shownin Fig. 3. Let us first compare the robustness of the T2

behavior in cot!H to that in !ab: One can see in Fig. 3(b)that cot!H for both x # 0:02 and 0.08, where the behaviorof !ab was discussed, shows no high-temperature devia-tion from the T2 behavior up to 300 K. This contrastbetween !ab!T" and cot!H!T" regarding the robustnessof the T2 law is qualitatively understandable if both !aband RH reflect a change in the effective carrier densityneff due to the temperature-dependent participation of theflatband, since such a change tends to be canceled incot!H; remember, H cot!H is equal to the inverse Hallmobility and, thus, would normally be free from a changein the carrier density. This observation suggests that therelative simplicity in the behavior of cot!H comes fromits lack of a direct dependence on neff , while both !ab andRH depend directly on neff .

Thus, our data are most consistent with an emerg-ing picture that a Fermi-liquid-like transport resultsfrom the quasiparticles on the Fermi arcs in lightlyhole-doped cuprates, and the rest of the FS starts tocontribute to the transport at higher doping and/or tem-perature. It should be noted, however, that there cannot bea real ‘‘Fermi liquid’’ on the Fermi arcs, because the largemagnitude of !ab in the lightly doped cuprates wouldindicate that the mean-free path of the electrons at EF

0

2

4

6

8

0 1 2 3 4 5 6

ρ ab (

mΩ

cm)

T2 (104 K2)

La1.98Sr0.02CuO4

x = 0.02(a)

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5 6

ρ ab (

mΩ

cm)

T2 (104 K2)

x = 0.08(b)

La1.92Sr0.08CuO4

FIG. 2 (color online). Validity of the T2 law in !ab!T" ofLSCO single crystals. At a lightly doped composition x #0:02 (a), the T2 law (shown by a solid line) holds for the tem-perature range of 130–230 K, while at x # 0:08 (b) the rangeis 60–160 K.

1

10

0 50 100 150 200 250 300

RH (

10-3

cm3 /C

)

Temperature (K)

La2-x

SrxCuO

4

x = 0.02

0.03

0.040.050.07

0.08

0.11

0.13

0.14

0.17

0.18

0.21

(a)

0.25

0.23

50

100

150

200

250

300

0 2 4 6 8

cot Θ

H a

t 10

T

T2 (104 K2)

x = 0.02

0.030.070.08

(b)

0

100

200

300

400

500

0 2 4 6 8

cot Θ

H a

t 10

T

T2 (104 K2)

(c)

x = 0.21

0.18

0.17

0.140.130.11

FIG. 3 (color online). Hall response in LSCO single crystalsfor a wide range of doping. (a) Variation of the T dependenceof RH for x#0:02–0:25, all measured on high-quality singlecrystals; the x values are determined by the inductivelycoupled-plasma atomic-emission-spectroscopy analyses andare accurate within $5%. (b),(c) Plots of cot!H vs T2 forrepresentative x values; for selected data, solid lines emphasizethe T2 law in cot!H!T", which holds well for 0:02 % x % 0:14.

0

1

2

3

4

0 100 200 300

ρ ab (

mΩ

cm)

Temperature (K)

YBa2Cu3O6.30

(a)

0

10

20

30

0 100 200 300

RH (

10-3

cm

3 /C

)

Temperature (K)

YBa2Cu3O6.30

(b)

0

1

2

3

4

0 2 4 6 8

ρ ab (

mΩ

cm)

T2 (104 K2)

(c)

Resistivity

0

50

100

150

200

0 2 4 6 8

cot Q

H a

t 10

T

T2 (104 K2)

(d)

Hall Angle

FIG. 1 (color online). Temperature dependences of transportproperties of lightly hole-doped YBa2Cu3O6:30. At this insu-lating composition, !ab!T" shows positive curvatures (a) andRH is virtually T independent at moderate temperatures (b). Itturns out that !ab!T" well obeys the T2 law (c), which alsoholds for cot!H [ & !ab=RH, (d)], suggesting that a Fermi-liquid-like T2 scattering rate governs the transport on theFermi arcs.

P H Y S I C A L R E V I E W L E T T E R S week ending14 MAY 2004VOLUME 92, NUMBER 19

197001-2 197001-2

PRL 92, 197001 (2004)

T-independent Hall effect in a magnetic field of fermions of

charge +e and density p

Electrical and optical evidence for Fermi surface of long-lived quasiparticles of density p

PNAS 110, 5774 (2013)

Spectroscopic evidence for Fermi liquid-like energyand temperature dependence of the relaxationrate in the pseudogap phase of the cupratesSeyed Iman Mirzaeia, Damien Strickera, Jason N. Hancocka,b, Christophe Berthoda, Antoine Georgesa,c,d,Erik van Heumena,e, Mun K. Chanf, Xudong Zhaof,g, Yuan Lih, Martin Grevenf, Neven Bari!si"cf,i,j,and Dirk van der Marela,1

aDépartement de Physique de la Matière Condensée, Université de Genève, 1211 Geneva, Switzerland; bDepartment of Physics and the Institute of MaterialsScience, Storrs, CT 06119; cCentre de Physique Théorique, École Polytechnique, Centre National de la Recherche Scientifique, 91128 Palaiseau, France; dCollègede France, 75005 Paris, France; eVan der Waals-Zeeman Instituut, Universiteit van Amsterdam,1098 XH Amsterdam, The Netherlands; fSchool of Physics andAstronomy, University of Minnesota, Minneapolis, MN 55455; gState Key Lab of Inorganic Synthesis and Preparative Chemistry, College of Chemistry, JilinUniversity, Changchun 130012, China; hInternational Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China; iInstitute ofPhysics, 10000 Zagreb, Croatia; and jService de Physique de l’Etat Condensé, Commissariat à l’Energie Atomique, Direction des Sciences de la Matière(DSM)-Institut Rayonnement Matière de Saclay (IRAMIS), 91198 Gif-sur-Yvette, France

Edited by David Pines, University of California, Davis, CA, and approved February 25, 2013 (received for review October 29, 2012)

Cuprate high-Tc superconductors exhibit enigmatic behavior in thenonsuperconducting state. For carrier concentrations near “opti-mal doping” (with respect to the highest Tcs) the transport andspectroscopic properties are unlike those of a Landau–Fermi liquid.On the Mott-insulating side of the optimal carrier concentration,which corresponds to underdoping, a pseudogap removes quasi-particle spectral weight from parts of the Fermi surface and causesa breakup of the Fermi surface into disconnected nodal and anti-nodal sectors. Here, we show that the near-nodal excitations ofunderdoped cuprates obey Fermi liquid behavior. The lifetime τ(ω,T) of a quasi-particle depends on its energy ω as well as on thetemperature T. For a Fermi liquid, 1/τ(ω, T) is expected to collapseon a universal function proportional to (h#ω)2 + (pπkBT)

2. Magneto-transport experiments, which probe the properties in the limit ω =0, have provided indications for the presence of a T2 dependenceof the dc (ω = 0) resistivity of different cuprate materials. How-ever, Fermi liquid behavior is very much about the energy depen-dence of the lifetime, and this can only be addressed byspectroscopic techniques. Our optical experiments confirm theaforementioned universal ω- and T dependence of 1/τ(ω, T), withp∼ 1.5. Our data thus provide a piece of evidence in favor of a Fermiliquid-like scenario of the pseudogap phase of the cuprates.

optical spectroscopy | superconductivity | mass renormalization |self energy

The compound HgBa2CuO4+δ (Hg1201) is the single-layercuprate that exhibits the highest Tc (97 K). We therefore

measured the optical conductivity of strongly underdoped singlecrystals of Hg1201 ðTc = 67 KÞ. Here we are interested in theoptical conductivity of the CuO2 layers. We therefore express theoptical conductivity as a 2D sheet conductance GðωÞ= dcσðωÞ,where dc is the interlayer spacing. The real part of the sheetconductance normalized by the conduction quantum G0 = 2e2=his shown in Fig. 1. As seen in the figure, a gap-like suppressionbelow 140 meV is clearly observable for temperatures below Tcand remains visible in the normal state up to ∼250 K. This is aclear optical signature of the pseudogap. We also observe thezero-energy mode due to the free charge carrier response, whichprogressively narrows upon lowering the temperature. In mate-rials where the charge carrier relaxation is dominated by impu-rity scattering, the width of this “Drude” peak corresponds to therelaxation rate of the charge carriers. Relaxation processesarising from interactions have the effect of replacing the constant(frequency-independent) relaxation rate with a frequency-dependent one. The general expression for the optical conduc-tivity of interacting electrons is then

Gðω;TÞ= iπKZω+Mðω;TÞ

G0: [1]

The spectral weight K corresponds to minus the kinetic en-ergy if the frequency integration of the experimental data isrestricted to intraband transitions. The effect of electron–electroninteractions and coupling to collective modes is described bythe memory function Mðω;TÞ=M1ðω;TÞ+ iM2ðω;TÞ, whereZ−1M2ðω;TÞ= 1=τðω;TÞ represents the dynamical (or optical) re-laxation rate in the case of a Fermi liquid.The zero frequency limit of the optical conductivity of Fig. 1

corroborates the recently reported temperature dependence of thedc resistivity (1) as shown in Fig. 2. Because K is practically tem-perature independent in the normal state (2), the low-temperatureT2 dependence of the resistivity is due to the quadratic tempera-ture variation of M2ð0;TÞ= Z=τð0;TÞ. The infrared data confirmthat Hg1201 exhibits the lowest residual resistance among thecuprates and a change to a linear temperature dependence aboveT* associated with the sudden closing of a pseudogap (3, 4). Fig. 2Bshows this as a clear departure from the T2 curve at ∼5 × 104 K2.The dc transport data, owing to the higher precision, allow forHg1201 crystals of the same composition and doping to identifyT*∼ 350 K as the temperature above which the resistivity hasa linear temperature dependence, and T**∼ 220 K as the tem-perature below which the temperature dependence is purely qua-dratic. Finally, superconducting fluctuations become noticeableat T′∼ 85 K.The doping dependences of K and of the coherent spectral

weight, defined as K*=K/(1+M1(ω,T)Zω)jω=0, are summarizedin Fig. 3 for a number of hole-doped cuprates. The theoreticalvalues of K based on the band parameters obtained from localdensity approximation (LDA) ab initio calculations are abouta factor of 2 larger than the measured values, which is due tostrong correlation predicted by the Hubbard model for U=t≥ 4(6). K decreases when the hole doping decreases, but does notextrapolate to zero for zero doping in accordance with the

Author contributions: S.I.M., D.S., N.B., and D.v.d.M. designed research; S.I.M., D.S., J.N.H.,C.B., A.G., E.v.H., M.K.C., X.Z., Y.L., M.G., N.B., and D.v.d.M. performed research; S.I.M.,D.S., J.N.H., C.B., A.G., E.v.H., M.K.C., X.Z., Y.L., M.G., N.B., and D.v.d.M. analyzed data;and S.I.M., D.S., C.B., A.G., M.G., N.B., and D.v.d.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: [email protected].

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1218846110/-/DCSupplemental.

5774–5778 | PNAS | April 9, 2013 | vol. 110 | no. 15 www.pnas.org/cgi/doi/10.1073/pnas.1218846110

χ″ðΩÞ, where spin (charge) refers to electron–hole pairs carrying(no) net spin. χ″ðΩÞ can be strongly renormalized, but theproperty that χ″ðΩÞ∝Ω in the limit Ω→ 0 is generic for Fermiliquids (24). Integration of the susceptibility multiplied with theinteraction vertex I2χ″ðΩÞ over all possible decay channels fromzero to ξ leads us to conclude that indeed M2 ∝ ξ2, as reportedexperimentally in the present article. In this description thecross-over ξ0 corresponds to the energy where I2χ″ðΩÞ is trun-cated, leading to a leveling off of M2 for ξ> ξ0. The strongtemperature dependence of M1ðω;TÞ is also a natural conse-quence of this description; it was shown in ref. 36 that, in leadingorders of temperature, χ″ðΩÞ of a correlated Fermi liquiddecreases as a function of temperature.In summary, we have shown from optical spectroscopy meas-

urements that the ungapped near-nodal excitations of underdopedcuprate superconductors obey Fermi liquid behavior when mate-rials with reduced amount of disorder are considered. This ob-servation, which is at variance with some established paradigms,provides leads toward understanding of themetallic state and high-temperature superconductivity in these materials.

Materials and MethodsSample Preparation. Single crystals were grown using a flux method, char-acterized, and heat treated to the desired doping level as described in refs. 37and 38. The conductivity data in Fig. 1 are of a sample which has an onsetcritical temperature of 67 K and a transition width of 2 K. The crystal surfaceis oriented along the a–b plane with a dimension of about 1.51 × 1.22 mm2.Hg1201 samples are hygroscopic. Therefore, the last stage of the prepara-tion of the sample surface is done under a continuous flow of nitrogen,upon which the sample is transferred to a high-vacuum chamber (10−7 mbar)

within a few minutes. Before each measurement the surface is carefullychecked for any evidence of oxidation.

Comparison with dc Resistivity. Transport measurements have been per-formed using the four-terminal method. Due to the irregular shape of thecleaved samples the absolute value of the dc resistivity can only be de-termined with about 20% accuracy. However, we obtained very high relativeaccuracy of the temperature dependence of the dc resistivity, as seen fromidentical temperature dependences of samples of the same composition anddoping, regardless of having significantly different dimensions and shapes.An independent check of the dc resistivity was obtained from the ω= 0 limitof the experimental infrared optical conductivity (Fig. 2). The dc resistivityhad to be scaled by a factor of 0.66 to match the optical data, most likely dueto the aforementioned influence of the irregular shape of the crystals on theabsolute value of the measured dc resistances. The excellent match of thetwo temperature dependences demonstrates the high quality of both dcresistivity and optical conductivity data.

ACKNOWLEDGMENTS. We thank A. Chubukov, A. J. Leggett, T. Giamarchi,T. M. Rice, and T. Timusk for discussions and communications. This work wassupported by the Swiss National Science Foundation (SNSF) through Grant200020-140761 and the National Center of Competence in Research, Materialswith Novel Electronic Properties. The crystal growth and characterization workwas supported by the US Department of Energy, Office of Basic Energy Scien-ces. X.Z. acknowledges support from the National Natural Science Foundation,China; N.B. acknowledges support from the European Commission underthe [Intra-European Fellowships (IEF)] Marie Curie Fellowship Programme;and E.v.H. acknowledges support through the Vernieuwingsimpuls (VENI) pro-gram funded by the Netherlands Organisation for Scientific Research.

1. Bari!si"c N, et al. (2012) Universal sheet resistance of the cuprate superconductors. ar-Xiv:1207.1504.

2. Norman M, Chubukov A, van Heumen E, Kuzmenko AB, van der Marel D (2007) Opticalintegral in the cuprates and the question of sum-rule violation. Phys Rev B 76(22):220509.

3. Bucher B, Steiner P, Karpinski J, Kaldis E, Wachter P (1993) Influence of the spin gapon the normal state transport in YBa2Cu4O8. Phys Rev Lett 70(13):2012–2015.

4. Ando Y, Komiya S, Segawa K, Ono S, Kurita Y (2004) Electronic phase diagram ofhigh-Tc cuprate superconductors from a mapping of the in-plane resistivitycurvature. Phys Rev Lett 93(26 Pt 1):267001.

5. van Heumen E, et al. (2009) Optical determination of the relation between theelectron-boson coupling function and the critical temperature in high-Tc cuprates.Phys Rev B 79(18):184512.

6. Rozenberg MJ, et al. (1995) Optical conductivity in Mott-Hubbard systems. Phys RevLett 75(1):105–108.

7. Comanac A, de’ Medici L, Capone M, Millis AJ (2008) Optical conductivity and thecorrelation strength of high-temperature copper-oxide superconductors. Nat Phys4(4):287–290.

8. Uchida S, et al. (1991) Optical spectra of La2-xSrxCuO4: Effect of carrier doping onthe electronic structure of the CuO2 plane. Phys Rev B Condens Matter 43(10):7942–7954.

9. Padilla WJ, et al. (2005) Constant effective mass across the phase diagram of high-Tccuprates. Phys Rev B 72(6):060511.

10. Brinkman WF, Rice TM (1970) Application of Gutzwiller’s variational method to themetal-insulator transition. Phys Rev B 2(10):4302–4304.

11. Fournier D, et al. (2010) Loss of nodal quasiparticle integrity in underdopedYBa2Cu3O6+x. Nat Phys 6(11):905–911.

12. Kanigel A, et al. (2006) Evolution of the pseudogap from Fermi arcs to the nodalliquid. Nat Phys 2(7):447–451.

0 10000 20000 30000 400000

100

200

300

400

0 2000 4000 60000

100

200

( )2 (meV2)

M2(ω)(meV)

( ω)2 (meV2)

M2(ω)(meV)

Fig. 5. Dynamical relaxation rate of underdoped Hg1201. Imaginary part ofthe memory function of underdoped Hg1021 ðTc = 67 KÞ for temperaturesbetween 10 and 390 K in 20-K steps as a function of ω2. Thick lines are usedto highlight the 70-, 280-, and 390-K data. (Inset) Zoom of the low-ω rangeshowing a linear fit; temperatures are from 70 to 270 K in 20-K steps.

0 1 2 3 40 1 2 30

100

200

300

400

500

0 1 2 3

Bi2201x = 0.1Tc= 10 K

C

ξ2 (104 meV2)

250 K210 K180 K140 K100 K60 K20 K

Hg1201x=0.1Tc= 67 K

A

M2(ω)(meV)

320 K280 K240 K200 K160 K120 K80 K

295 K244 K200 K171 K147 K126 K100 K67 K

BY123x = 0.1Tc = 57 K

Fig. 6. Collapse of the frequency and temperature dependence of the re-laxation rate of underdoped cuprate materials. Normal state M2ðω; TÞ as afunction of ξ2 ≡ ðZωÞ2 + ðpπkBTÞ2 with p=1:5. (A) Hg1201 ðx ≅ 0:1; Tc = 67 KÞ.(B) Y123 ðx ≅ 0:1; Tc = 57 KÞ, spectra by Hwang et al. (19) (digitized data ofFig. 6 represented here as a function of ξ2). (C) Bi2201 ðx ≅ 0:1; Tc = 10 KÞ;data of van Heumen et al. (20) represented here as a function of ξ2. The datadisplayed in A and C are in 10-K intervals with color coding indicated fortemperatures in 40-K steps. In between these steps the color evolves grad-ually as a function of temperature. In B the color coding is given for alltemperatures displayed.

Mirzaei et al. PNAS | April 9, 2013 | vol. 110 | no. 15 | 5777

PHYS

ICS


xx

1

(i! + 1/)

with1

!2 + T 2

In-Plane Magnetoresistance Obeys Kohler’s Rule in the Pseudogap Phaseof Cuprate Superconductors

M. K. Chan,1,* M. J. Veit,1 C. J. Dorow,1,† Y. Ge,1 Y. Li,1 W. Tabis,1,2 Y. Tang,1 X. Zhao,1,3

N. Barišić,1,4,5,‡ and M. Greven1,§1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA2AGH University of Science and Technology, Faculty of Physics and Applied Computer Science,

Al. A. Mickiewicza 30, 30-059 Krakow, Poland3State Key Lab of Inorganic Synthesis and Preparative Chemistry, College of Chemistry, Jilin University,

Changchun 130012, China4Service de Physique de l’Etat Condensé, CEA-DSM-IRAMIS, F 91198 Gif-sur-Yvette, France

5Institute of Solid State Physics, Vienna University of Technology, 1040 Vienna, Austria(Received 19 February 2014; published 21 October 2014)

We report in-plane resistivity (ρ) and transverse magnetoresistance (MR) measurements for underdopedHgBa2CuO4þδ (Hg1201). Contrary to the long-standing view that Kohler’s rule is strongly violated inunderdoped cuprates, we find that it is in fact satisfied in the pseudogap phase of Hg1201. The transverse MRshows a quadratic field dependence, δρ=ρ0 ¼ aH2, with aðTÞ ∝ T−4. In combination with the observedρ ∝ T2 dependence, this is consistent with a single Fermi-liquid quasiparticle scattering rate. We show thatthis behavior is typically masked in cuprates with lower structural symmetry or strong disorder effects.

DOI: 10.1103/PhysRevLett.113.177005 PACS numbers: 74.72.Kf, 74.25.fc, 74.72.Gh

The unusual metallic “normal state” of the cuprates hasremained an enigma. Atypical observations at odds withFermi-liquid theory have been made particularly in the so-called strange-metal regime above the pseudogap (PG)temperature T% [inset of Fig. 1(b)] [1]. In this regime, thein-plane resistivity exhibits an anomalous extended lineartemperature dependence, ρ ∝ T [2], and the Hall effect isoften described as RH ∝ 1=T [3,4]. In order to account forthis anomolous behavior without abandoning a Fermi-liquid formalism, some descriptions have been formulatedbased on a scattering rate whose magnitude varies aroundthe in-plane Fermi surface, for example, due to anisotropicumklapp scattering or coupling to a bosonic mode [1] (e.g.,spin [5] or charge [6] fluctuations). Prominent non-Fermi-liquid prescriptions, such as the two-lifetime picture [7] andthe marginal-Fermi liquid [8], have also been put forth. Theformer implies charge-spin separation while the latter is asignature of a proximate quantum critical point.The transport behavior in the PG state (T < T%) has

furthermore been complicated not only because of theopening of the PG along portions of the Fermi surface, butalso due to possible superconducting (SC) [10], antiferro-magnetic [5,11] and charge-spin stripe fluctuations [12].Recent developments, however, suggest that T% marks a

phase transition [13] into a state with broken time-reversalsymmetry [14,15]. Additionally, the measurable extent ofSC fluctuations is likely limited to only a rather smalltemperature range (≈ 30 K) above Tc [16,17]. Thesestrong indications that the PG regime is indeed a distinctphase calls for a clear description of its intrinsic properties.In fact, a simple ρ ¼ A2T2 dependence was recently

reported for underdoped HgBa2CuO4þδ (Hg1201) [9]. Itwas also found that this Fermi-liquid-like behavior appearsbelow a characteristic temperature T%% [Tc < T%% < T%;inset of Fig. 1(b)] and that the coefficient A2 per CuO2

0 100 200 300 4000

0.4

0.8(a)

T*≈305K

T*≈ 280Kρ (m

Ω.c

m)

T (K)0 100 200 300 400

0

0.4

0.8(a)

T*≈305K

T*≈ 280Kρ (m

Ω.c

m)

T (K)

65 80−1

0

T (K)

χ (a

rb)

0

0.2

0.4(b)

T**≈ 200KT**≈ 180K

0 1502 3002

p

T(K

)

T2 (K2)

0

0.2

0.4(b)

T**≈ 200KT**≈ 180K

0 1502 3002

p

T(K

)

0.05 0.150

300

PG

SC

SM

T*T**

0

300

FIG. 1 (color online). (a) Temperature dependence of the in-plane resistivity for two Hg1201 samples. Dotted lines are linearfits to the high-temperature behavior. Inset: Magnetic suscep-tibility shows Tc ¼ 70& 1 and 80.5& 0.5 K for the two samples,HgUD70b (black) and HgUD81 (red). The Tc values are definedas the midpoint of the transition, and the uncertainties correspondto 90% of the transition width. (b) Resistivity plotted versus T2.Dotted lines are fits to ρ ¼ A2T2. There is some uncertainty in theconversion to units of ρ due to difficulties in measuring the exactcleaved sample dimensions [9]. For consistency, we haveassumed the same magnitude of ρ for the two Tc ¼ 70 Ksamples. Inset: Schematic temperature-hole doping phase dia-gram. The superconducting (SC), strange metal (SM), andpseudogap (PG) phases as well as the characteristic temperaturesT% and T%% are indicated. The circles represent the two dopinglevels of the present study.

PRL 113, 177005 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

24 OCTOBER 2014

0031-9007=14=113(17)=177005(6) 177005-1 © 2014 American Physical Society

In-Plane Magnetoresistance Obeys Kohler’s Rule in the Pseudogap Phaseof Cuprate Superconductors

M. K. Chan,1,* M. J. Veit,1 C. J. Dorow,1,† Y. Ge,1 Y. Li,1 W. Tabis,1,2 Y. Tang,1 X. Zhao,1,3

N. Barišić,1,4,5,‡ and M. Greven1,§1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA2AGH University of Science and Technology, Faculty of Physics and Applied Computer Science,

Al. A. Mickiewicza 30, 30-059 Krakow, Poland3State Key Lab of Inorganic Synthesis and Preparative Chemistry, College of Chemistry, Jilin University,

Changchun 130012, China4Service de Physique de l’Etat Condensé, CEA-DSM-IRAMIS, F 91198 Gif-sur-Yvette, France

5Institute of Solid State Physics, Vienna University of Technology, 1040 Vienna, Austria(Received 19 February 2014; published 21 October 2014)

We report in-plane resistivity (ρ) and transverse magnetoresistance (MR) measurements for underdopedHgBa2CuO4þδ (Hg1201). Contrary to the long-standing view that Kohler’s rule is strongly violated inunderdoped cuprates, we find that it is in fact satisfied in the pseudogap phase of Hg1201. The transverse MRshows a quadratic field dependence, δρ=ρ0 ¼ aH2, with aðTÞ ∝ T−4. In combination with the observedρ ∝ T2 dependence, this is consistent with a single Fermi-liquid quasiparticle scattering rate. We show thatthis behavior is typically masked in cuprates with lower structural symmetry or strong disorder effects.

DOI: 10.1103/PhysRevLett.113.177005 PACS numbers: 74.72.Kf, 74.25.fc, 74.72.Gh

The unusual metallic “normal state” of the cuprates hasremained an enigma. Atypical observations at odds withFermi-liquid theory have been made particularly in the so-called strange-metal regime above the pseudogap (PG)temperature T% [inset of Fig. 1(b)] [1]. In this regime, thein-plane resistivity exhibits an anomalous extended lineartemperature dependence, ρ ∝ T [2], and the Hall effect isoften described as RH ∝ 1=T [3,4]. In order to account forthis anomolous behavior without abandoning a Fermi-liquid formalism, some descriptions have been formulatedbased on a scattering rate whose magnitude varies aroundthe in-plane Fermi surface, for example, due to anisotropicumklapp scattering or coupling to a bosonic mode [1] (e.g.,spin [5] or charge [6] fluctuations). Prominent non-Fermi-liquid prescriptions, such as the two-lifetime picture [7] andthe marginal-Fermi liquid [8], have also been put forth. Theformer implies charge-spin separation while the latter is asignature of a proximate quantum critical point.The transport behavior in the PG state (T < T%) has

furthermore been complicated not only because of theopening of the PG along portions of the Fermi surface, butalso due to possible superconducting (SC) [10], antiferro-magnetic [5,11] and charge-spin stripe fluctuations [12].Recent developments, however, suggest that T% marks a

phase transition [13] into a state with broken time-reversalsymmetry [14,15]. Additionally, the measurable extent ofSC fluctuations is likely limited to only a rather smalltemperature range (≈ 30 K) above Tc [16,17]. Thesestrong indications that the PG regime is indeed a distinctphase calls for a clear description of its intrinsic properties.In fact, a simple ρ ¼ A2T2 dependence was recently

reported for underdoped HgBa2CuO4þδ (Hg1201) [9]. Itwas also found that this Fermi-liquid-like behavior appearsbelow a characteristic temperature T%% [Tc < T%% < T%;inset of Fig. 1(b)] and that the coefficient A2 per CuO2

0 100 200 300 4000

0.4

0.8(a)

T*≈305K

T*≈ 280Kρ (m

Ω.c

m)

T (K)0 100 200 300 400

0

0.4

0.8(a)

T*≈305K

T*≈ 280Kρ (m

Ω.c

m)

T (K)

65 80−1

0

T (K)

χ (a

rb)

0

0.2

0.4(b)

T**≈ 200KT**≈ 180K

0 1502 3002

p

T(K

)

T2 (K2)

0

0.2

0.4(b)

T**≈ 200KT**≈ 180K

0 1502 3002

p

T(K

)

0.05 0.150

300

PG

SC

SM

T*T**

0

300

FIG. 1 (color online). (a) Temperature dependence of the in-plane resistivity for two Hg1201 samples. Dotted lines are linearfits to the high-temperature behavior. Inset: Magnetic suscep-tibility shows Tc ¼ 70& 1 and 80.5& 0.5 K for the two samples,HgUD70b (black) and HgUD81 (red). The Tc values are definedas the midpoint of the transition, and the uncertainties correspondto 90% of the transition width. (b) Resistivity plotted versus T2.Dotted lines are fits to ρ ¼ A2T2. There is some uncertainty in theconversion to units of ρ due to difficulties in measuring the exactcleaved sample dimensions [9]. For consistency, we haveassumed the same magnitude of ρ for the two Tc ¼ 70 Ksamples. Inset: Schematic temperature-hole doping phase dia-gram. The superconducting (SC), strange metal (SM), andpseudogap (PG) phases as well as the characteristic temperaturesT% and T%% are indicated. The circles represent the two dopinglevels of the present study.

PRL 113, 177005 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

24 OCTOBER 2014

0031-9007=14=113(17)=177005(6) 177005-1 © 2014 American Physical Society

PRL 113, 177005 (2014)


xx

1

(1 + aH22 + . . .)

with1

T 2

Can we have a metal with no broken translational symmetry, and with long-lived electron-like quasiparticles on a Fermi surface of size p ?

Can we have a metal with no broken translational symmetry, and with long-lived electron-like quasiparticles on a Fermi surface of size p ?

Answer: Yes. There can be a Fermi surface of size p,

but it must be accompanied by topological order, in a

“fractionalized Fermi liquid”.

At T=0, such a metal must be separated from a Fermi liquid (with a Fermi surface of size 1+p) by a quantum phase transition

Pseudogap

2. Pseudogap metal

at low p


SM

FL

A new metal — a fractionalized

Fermi liquid (FL*) — with electron-like quasiparticles on a Fermi surface

of size p

SM

FL

Pseudogap

Y. Qi and S. Sachdev, Phys. Rev. B 81, 115129 (2010)M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

FL

Y. Kohsaka et al., Science 315, 1380 (2007)

Pseudogap

Density wave (DW) order at low T and p

SM

FL

Identified as a predicted “d-form

factor density wave”SM

M. A. Metlitski and S. Sachdev, PRB 82, 075128 (2010). S. Sachdev R. La Placa, PRL 111, 027202 (2013).

K. Fujita, M. H Hamidian, S. D. Edkins, Chung Koo Kim, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi,

H. Eisaki, S. Uchida, A. Allais, M. J. Lawler, E.-A. Kim, S. Sachdev, and J. C. Davis, PNAS 111, E3026 (2014)

Q = (/2, 0)Pseudogap

Q = (/2, 0)

SM

FL

Q = (/2, 0)

SM

FLX

Fermi surface of a fractionalized

Fermi liquid (FL*)

Y. Qi and S. Sachdev, Phys. Rev. B 81, 115129 (2010)

M. Punk and S. Sachdev, Phys. Rev. B 85, 195123 (2012), and to appear

Density wave instability of

FL* leads to the observed

wavevectorand form-factor

D. Chowdhury and S. Sachdev, Physical Review B 90, 245136 (2014).

SM FL*

Q = (/2, 0)

d-form factor

density wave

FL

The high T FL* can help explain the “d-form factor density wave” observed at low T

D. Chowdhury and S.S., Phys. Rev. B 90, 245136 (2014).

the metallic states of the cuprates - harvard...

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