The onset of antiferromagnetism in metalsqpt.physics.harvard.edu/talks/lorentz12_4.pdf · antiferromagnetism in metals sachdev.physics.harvard.edu HARVARD Lorentz Lecture, Leiden

The onset of antiferromagnetism in metals

HARVARDsachdev.physics.harvard.edu

Lorentz Lecture, Leiden June 4, 2012

Subir Sachdev

Wednesday, June 6, 2012

Max Metlitski

HARVARDMatthias Punk Erez Berg

Eun Gook Moon


The cuprate superconductors


Ground state has long-range Néel order

Square lattice antiferromagnet

H =�

�ij�

Jij�Si · �Sj

Order parameter is a single vector field �ϕ = ηi�Si

ηi = ±1 on two sublattices

��ϕ� �= 0 in Neel state.


Hole-doped

Electron-doped


Hole-doped

Electron-doped

Electron-doped cuprate superconductors


Hole-doped

Electron-doped

Resistivity∼ ρ0 +ATn

Electron-doped cuprate superconductors

Figure prepared by K. Jin and and R. L. Greenebased on N. P. Fournier, P. Armitage, and

R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010).

StrangeMetal


Ishida, Nakai, and HosonoarXiv:0906.2045v1

Iron pnictides: a new class of high temperature superconductors


TSDW Tc

T0

2.0

0

!"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF

Resistivity∼ ρ0 +ATα

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B 81, 184519 (2010)

StrangeMetal


Lower Tc superconductivity in the heavy fermion compounds

G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223

2

Generally, the ground state of a Ce heavy-fermion sys-tem is determined by the competition of the indirect Ru-derman Kittel Kasuya Yosida (RKKY) interaction whichprovokes magnetic order of localized moments mediatedby the light conduction electrons and the Kondo interac-tion. This last local mechanism causes a paramagneticground state due the screening of the local moment ofthe Ce ion by the conduction electrons. Both interac-tions depend critically on the hybridization of the 4felectrons with the conduction electrons. High pressure isan ideal tool to tune the hybridization and the positionof the 4f level with respect to the Fermi level. Thereforehigh pressure experiments are ideal to study the criti-cal region where both interactions are of the same orderand compete. To understand the quantum phase transi-tion from the antiferromagnetic (AF) state to the para-magnetic (PM) state is actually one of the fundamen-tal questions in solid state physics. Di!erent theoreticalapproaches exist to model the magnetic quantum phasetransition such as spin-fluctuation theory of an itinerantmagnet [10–12], or a new so-called ’local’ quantum criti-cal scenario [13, 14]. Another e"cient source to preventlong range antiferromagnetic order is given by the va-lence fluctuations between the trivalent and the tetrava-lent configuration of the cerium ions [15].

The interesting point is that in these strongly corre-lated electron systems the same electrons (or renormal-ized quasiparticles) are responsible for both, magnetismand superconductivity. The above mentioned Ce-115family is an ideal model system, as it allows to studyboth, the quantum critical behavior and the interplay ofthe magnetic order with a superconducting state. Espe-cially, as we will be shown below, unexpected observa-tions will be found, if a magnetic field is applied in thecritical pressure region.

PRESSURE-TEMPERATURE PHASE DIAGRAM

In this article we concentrate on the compoundCeRhIn5. At ambient pressure the RKKY interactionis dominant in CeRhIn5 and magnetic order appears atTN = 3.8 K. However, the ordered magnetic moment ofµ = 0.59µB at 1.9 K is reduced of about 30% in com-parison to that of Ce ion in a crystal field doublet with-out Kondo e!ect [17]. Compared to other heavy fermioncompounds at p = 0 the enhancement of the Sommerfeldcoe"cient of the specific heat (! = 52 mJ mol!1K!2) [18]and the cylotron masses of electrons on the extremal or-bits of the Fermi surface is rather moderate [19, 20]. Thetopologies of the Fermi surfaces of CeRhIn5 are cylin-drical and almost identical to that of LaRhIn5 which isthe non 4f isostructural reference compound. From thisit can be concluded that the 4f electrons in CeRhIn5

are localized and do not contribute to the Fermi volume[19, 20].

By application of pressure, the system can be tunedthrough a quantum phase transition. The Neel temper-

FIG. 2. Pressure–temperature phase diagram of CeRhIn5 atzero magnetic field determined from specific heat measure-ments with antiferromagnetic (AF, blue) and superconduct-ing phases (SC, yellow). When Tc < TN a coexistence phaseAF+SC exist. When Tc > TN the antiferromagnetic order isabruptly suppressed. The blue square indicate the transitionfrom SC to AF+SC after Ref. 16.

ature shows a smooth maximum around 0.8 GPa andis monotonously suppressed for higher pressures. How-ever, CeRhIn5 is also a superconductor in a large pres-sure region from about 1.3 to 5 GPa. It has been shownthat when the superconducting transition temperatureTc > TN the antiferromagnetic order is rapidly sup-pressed (see figure 2) and vanishes at a lower pressurethan that expected from a linear extrapolation to T = 0.Thus the pressure where Tc = TN defines a first criticalpressure p!

c and clearly just above p!c anitferromagnetism

collapses. The intuitive picture is that the opening of asuperconducting gap on large parts of the Fermi surfaceabove p!

c impedes the formation of long range magneticorder. A coexisting phase AF+SC in zero magnetic fieldseems only be formed if on cooling first the magnetic or-der is established. We will discuss below the microscopicevidence of an homogeneous AF+SC phase.

At ambient pressure CeRhIn5 orders in an incommen-surate magnetic structure [21] with an ordering vectorof qic=(0.5, 0.5, ") and " = 0.297 that is a magneticstructure with a di!erent periodicity than the one of thelattice. Generally, an incommensurate magnetic struc-ture is not favorable for superconductivity with d wavesymmetry, which is realized in CeRhIn5 above p!

c [22].Neutron scattering experiments under high pressure donot give conclusive evidence of the structure under pres-sures up to 1.7 GPa which is the highest pressure studiedup to now [23–25]. The result is that at 1.7 GPa the in-commensurability has changed to " ! 0.4. The main dif-ficulty in these experiments with large sample volume isto ensure the pressure homogeneity. Near p!

c the controlof a perfect hydrostaticity is a key issue as the materialreacts quite opposite on uniaxial strain applied along thec and a axis.

From recent nuclear quadrupol resonance (NQR) data


Can quantum fluctuations near the onset of antiferromagnetism induce higher temperature superconductivity ?

Questions



Questions

How should such a theory be extended to apply to the hole-doped cuprates ?



Questions


What is the physics of the strange metal ?


1. Phenomenology of the onset of antiferromagnetism in a metal 2. Quantum field theory of the onset of antiferromagnetism in a metal 3. Quantum Monte Carlo without the sign problem

4. Fractionalization in metals, and the hole-doped cuprates

Outline




Outline


Fermi surface

Metal with “large” Fermi surface

Momenta with electronic

states empty

Momenta with electronic

states occupied


Fermi surface+antiferromagnetism

The electron spin polarization obeys�

�S(r, τ)�

= �ϕ(r, τ)eiK·r

where K is the ordering wavevector.

+



Metal with “large” Fermi surfaceWednesday, June 6, 2012

Fermi surfaces translated by K = (π,π).Wednesday, June 6, 2012

Electron and hole pockets inantiferromagnetic phase with ��ϕ� �= 0




��ϕ� = 0

Metal with electron and hole pockets

Increasing SDW order

��ϕ� �= 0

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Increasing interaction


N. P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002).

Photoemission in Nd2-xCexCuO4


Nd2−xCexCuO4

T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,

M. Lambacher, A. Erb, J. Wosnitza, and R. Gross,

Phys. Rev. Lett. 103, 157002 (2009).

Quantum oscillations


s




��ϕ� = 0



��ϕ� �= 0

sS. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995).

A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).Wednesday, June 6, 2012

FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Spin density wave (SDW)

Theory of quantum criticality in the cuprates

Underlying SDW ordering quantum critical pointin metal at x = xm


T*QuantumCritical


FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal


Relaxation and equilibration times ∼ �/kBT are robustproperties of strongly-coupled quantum criticality



T*QuantumCritical


FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal


Relaxation and equilibration times ∼ �/kBT are robustproperties of strongly-coupled quantum criticality



T*QuantumCriticalStrangeMetal ?


LargeFermi

surface

StrangeMetal


d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

SDW quantum critical point is unstable to d-wave superconductivityThis instability is stronger than that in the BCS theory

M. A. Metlitski andS. Sachdev,Physical ReviewB 82, 075128 (2010)


Fluctuating, paired Fermi

pockets


BaFe2(As1−x Px)2

K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science in press


BaFe2(As1−x Px)2



BaFe2(As1−x Px)2



BaFe2(As1−x Px)2

Notice shift between the position of the QCP in the superconductor, and the divergence in effective mass in the metal

measured at high magnetic fieldsWednesday, June 6, 2012

LargeFermi

surface

StrangeMetal




pairing fluctuations

SDW quantum critical point is unstable to d-wave superconductivityThis instability is stronger than that in the BCS theory

M. A. Metlitski andS. Sachdev,Physical ReviewB 82, 075128 (2010)



pockets


LargeFermi

surface

StrangeMetal




pairing fluctuations E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)


Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.


pockets


LargeFermi

surface

StrangeMetal




pairing fluctuations E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)




pockets



pocketsLargeFermi

surface

StrangeMetal






pockets E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)



pocketsLargeFermi

surface

StrangeMetal



T*


H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW


pairing fluctuationsLargeFermi

surface

StrangeMetal

d-waveSC

T

Tsdw


pockets

T*E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).


H

SC

M"Normal"


SDW(Small Fermi

pockets)

SC+SDW



surface

StrangeMetal

d-waveSC

T

Tsdw


pockets

T*

Neutron scatter-

ing experiments on

Nd2−xCexCuO4 show

that at low fields

xs = 0.14, while

quantum oscilla-

tions at high fields

show that xm = 0.165.

QuantumCritical

E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).


Nd2−xCexCuO4

T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,

M. Lambacher, A. Erb, J. Wosnitza, and R. Gross,

Phys. Rev. Lett. 103, 157002 (2009).

Quantum oscillations


s


0 100 200 300 4005

10

20

50

100

200

500

Temperature (K)

Spin

cor

rela

tion

leng

th

/a

x=0.154x=0.150x=0.145x=0.134x=0.129x=0.106x=0.075x=0.038

E. M. Motoyama, G. Yu, I. M. Vishik, O. P. Vajk, P. K. Mang, and M. Greven,Nature 445, 186 (2007).

Nd2−xCexCuO4


H

SC

M"Normal"


SDW(Small Fermi

pockets)

SC+SDW



surface

StrangeMetal

d-waveSC

T

Tsdw


pockets

T*

Neutron scatter-

ing experiments on

Nd2−xCexCuO4 show

that at low fields

xs = 0.14, while

quantum oscilla-

tions at high fields

show that xm = 0.165.

QuantumCritical

E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).


Similar phase diagram for CeRhIn5

G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223.Tuson Park, F. Ronning, H. Q. Yuan, M. B. Salamon, R. Movshovich, J. L. Sarrao, and J. D. Thompson, Nature 440, 65 (2006)


H

SC

M"Normal"


SDW(Small Fermi

pockets)

SC+SDW



surface

StrangeMetal

d-waveSC

T

Tsdw


pockets

T*QuantumCritical

There is a muchlarger shift fromxm to xs in the

hole-dopedcuprates.


H

SC

M"Normal"


SDW(Small Fermi

pockets)

SC+SDW



surface

StrangeMetal

d-waveSC

T

Tsdw


pockets


QuantumCritical

Many experiments in hole-doped cuprates

have presented evidence for the

predicted green quantum phase

transition line from SC to SC+SDW

in a magnetic field


H

SC

M"Normal"


SDW(Small Fermi

pockets)

SC+SDW



surface

StrangeMetal

d-waveSC

T

Tsdw


pockets


QuantumCritical





in a magnetic field


H

SC

M"Normal"


SDW(Small Fermi

pockets)

SC+SDW



surface

StrangeMetal

d-waveSC

T

Tsdw


pockets


QuantumCritical





in a magnetic field


hole doping x

YBCO6.5

YBCO6.45YBCO6.35

SC

SC+

AFm

agne

tic fi

eld H

AF

D. Haug, V. Hinkov, Y. Sidis, P. Bourges, N. B. Christensen, A. Ivanov,

T. Keller, C. T. Lin, and B. Keimer, New J. Phys. 12, 105006 (2010)


H

SC

M"Normal"


SDW(Small Fermi

pockets)

SC+SDW



surface

StrangeMetal

d-waveSC

T

Tsdw


pockets

T*QuantumCritical

This opens a wideintermediate regimefor new physics:

bond-nematic order,T -breaking,

fractionalization andMott physics etc.




Outline




Outline



Quantum phase transition with Fermi surface reconstruction

��ϕ� = 0



��ϕ� �= 0

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Increasing interaction


Boson-fermion theory for both phases

S =

�d2rdτ [Lc + Lϕ + Lcϕ]

Lc = c†aε(−i∇)ca

Lϕ =1

2(∇ϕα)

2 +r

2ϕ2α +

u

4

�ϕ2α

�2

Lcϕ = λϕα eiK·r c†a σαab cb.


Boson-fermion theory for both phases

“Yukawa” coupling between fermions and

antiferromagnetic order:

λ2 ∼ U , the Hubbard repulsion

S =

�d2rdτ [Lc + Lϕ + Lcϕ]

Lc = c†aε(−i∇)ca

Lϕ =1

2(∇ϕα)

2 +r

2ϕ2α +

u

4

�ϕ2α

�2

Lcϕ = λϕα eiK·r c†a σαab cb.


• Integrate out Fermi surface quasiparticles and

obtain an effective theory for the order param-

eter �ϕ alone.

• This is dangerous, and will lead to non-local in

the �ϕ theory. Hertz focused on only the simplest

such non-local term.

• However, there are an infinite number of non-

local terms at higher order, and these lead to

a breakdown of the Hertz theory in two spatial

dimensions.

Hertz-Moriya-Millis theory

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).Wednesday, June 6, 2012



eter �ϕ alone.







dimensions.


Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).Wednesday, June 6, 2012


Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).



eter �ϕ alone.







dimensions.


Sung-Sik Lee, Phys. Rev. B 80, 165102 (2009) M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010)M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075128 (2010)

• In d = 2, we must work in local theoreswhich keeps both the order parameter andthe Fermi surface quasiparticles “alive”.

• The theories can be organized in a 1/N ex-pansion, where N is the number of fermion“flavors”.

• At subleading order, resummation of all“planar” graphics is required (at least): thistheory is even more complicated than QCD.












Fermi surfaces translated by K = (π,π).Wednesday, June 6, 2012

“Hot” spotsWednesday, June 6, 2012

Low energy theory for critical point near hot spotsWednesday, June 6, 2012

Low energy theory for critical point near hot spotsWednesday, June 6, 2012

v1 v2

ψ2 fermionsoccupied


Theory has fermions ψ1,2 (with Fermi velocities v1,2)and boson order parameter �ϕ,interacting with coupling λ

kx

ky


Pairing “glue” from antiferromagnetic fluctuationsWednesday, June 6, 2012

Unconventional pairing at and near hot spots

∆−∆

�c†kαc

†−kβ

�= εαβ∆(cos kx − cos ky)


At stronger coupling, different effects compete:

• Pairing glue becomes stronger.

• There is stronger fermion-boson

scattering, and fermionic quasi-

particles lose their integrity.

• Other instabilities can appear

e.g. to charge density waves/stripeorder.


















BCS theory

1 + λe-ph log�ωD

ω

�

Electron-phononcoupling

Debyefrequency


M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010)

Enhancement of pairing susceptibility by interactions

Antiferromagnetic critical point

1 +sin θ

2πlog2

�EF

ω

�





1 +sin θ

2πlog2

�EF

ω

�





1 +sin θ

2πlog2

�EF

ω

�



Fermienergy


(see also Ar. Abanov, A. V. Chubukov, and A. M. Finkel'stein, Europhys. Lett. 54, 488 (2001))


1 +sin θ

2πlog2

�EF

ω

�

θ is the angle between Fermi lines.Independent of interaction strength

U in 2 dimensions.


θ





1 +sin θ

2πlog2

�EF

ω

�

• Universal log2 singularity arises from Fermi lines;singularity at hot spots is weaker.

• Interference between BCS and quantum-critical logs.

• Momentum dependence of self-energy is crucial.

• Not suppressed by 1/N factor in 1/N expansion.




Outline




Outline


v1 v2




kx

ky


v1 v2


kx

ky

To faithfully realize low energy theory in quantum Monte Carlo,we need a UV completion in which Fermi lines don’t end

and all weights are positive.


K

Hot spots in a single band model

QMC for the onset of antiferromagnetism


K

Hot spots in a two band model


E. Berg, M. Metlitski, and

S. Sachdev, arXiv:1206.0742


K






K


No sign problem in fermion determinant Monte Carlo !

Determinant is positive because of Kramer’sdegeneracy, and no additional symmetries are needed; holds for

arbitrary band structure and band filling, provided K only connects hot spots in distinct bands






Electrons with dispersion εkinteracting with fluctuations of the

antiferromagnetic order parameter �ϕ.

Z =

�DcαD�ϕ exp (−S)

S =

�dτ

�

k

c†kα

�∂

∂τ− εk

�ckα

+

�dτd2x

�1

2(∇x�ϕ)

2 +r

2�ϕ2 + . . .

�

− λ

�dτ

�

i

�ϕi · (−1)xic†iα�σαβciβ


Electrons with dispersions ε(x)k and ε(y)kinteracting with fluctuations of the


Z =

�Dc(x)α Dc(y)α D�ϕ exp (−S)

S =

�dτ

�

k

c(x)†kα

�∂

∂τ− ε(x)k

�c(x)kα

+

�dτ

�

k

c(y)†kα

�∂

∂τ− ε(y)k

�c(y)kα

+

�dτd2x

�1

2(∇x�ϕ)

2+

r

2�ϕ2

+ . . .

�

− λ

�dτ

�

i

�ϕi · (−1)xic(x)†iα �σαβc

(y)iβ +H.c.





Electrons with dispersions ε(x)k and ε(y)kinteracting with fluctuations of the


Z =

�Dc(x)α Dc(y)α D�ϕ exp (−S)

S =

�dτ

�

k

c(x)†kα

�∂

∂τ− ε(x)k

�c(x)kα

+

�dτ

�

k

c(y)†kα

�∂

∂τ− ε(y)k

�c(y)kα

+

�dτd2x

�1

2(∇x�ϕ)

2+

r

2�ϕ2

+ . . .

�

− λ

�dτ

�

i

�ϕi · (−1)xic(x)†iα �σαβc

(y)iβ +H.c.


No sign problem !




K






!1

!0.5

0

0.5

1k y/! K

a)


Center Brillouin zone at (π,π,)




!1 0 1!1

!0.5

0

0.5

1

kx/!

k y/!

!1 0 1!1

!0.5

0

0.5

1

k y/! kx/!

K

a) b)


Move one of the Fermi surface by (π,π,)




!1

!0.5

0

0.5

1k y/!

a)


Now hot spots are at Fermi surface intersections




v1 v2


kx

ky

To faithfully realize low energy theory in quantum Monte Carlo,we need a UV completion in which Fermi lines don’t end

and all weights are positive.


!1

!0.5

0

0.5

1k y/!

a)


Now hot spots are at Fermi surface intersections




!1 0 1!1

!0.5

0

0.5

1

kx/!

k y/!

!1 0 1!1

!0.5

0

0.5

1k y/!

kx/!

K

a) b)


Expected Fermi surfaces in the AFM ordered phase




E. Berg, M. Metlitski, and S. Sachdev, arXiv:1206.0742


!1 0 1!1

!0.5

0

0.5

1r = !0.5

kx/!

k y/!

!1 0 1!1

!0.5

0

0.5

1

kx/!

r = 0

!1 0 1!1

!0.5

0

0.5

1

kx/!

r = 0.5

0.5

1

1.5

Electron occupation number nk

as a function of the tuning parameter r



!0.5 0 0.5 1

0

0.1

0.2

0.3

0.4

rBi

nder

cum

ulan

t!0.5 0 0.5 10

0.2

0.4

0.6

r

! "/(L

2 #)

L=8L=10L=12L=14

a) b)

AF susceptibility, χϕ, and Binder cumulantas a function of the tuning parameter r




!2 !1 0 1 2 3!2

0

2

4

6

8

10 x 10!4

r

P ±(xmax

)L = 10

L = 14

L = 12

rc

!

P+

_

_P_

|

s/d pairing amplitudes P+/P−as a function of the tuning parameter r




!2 !1 0 1 2 3!2

0

2

4

6

8

10 x 10!4

r

P ±(xmax

)L = 10

L = 14

L = 12

rc

!

P+

_

_P_

|

Notice shift between the position of the QCP in the superconductor, and the position of maximum pairing



BaFe2(As1−x Px)2

Notice shift between the position of the QCP in the superconductor, and the divergence in effective mass in the metal

measured at high magnetic fieldsWednesday, June 6, 2012



Outline




Outline


Hole-doped

Electron-doped


Hole-doped

Electron-doped

?



Quantum phase transition with Fermi surface reconstruction

��ϕ� = 0



��ϕ� �= 0





��ϕ� �= 0 ��ϕ� = 0

Separating onset of SDW orderand Fermi surface reconstruction





��ϕ� �= 0 ��ϕ� = 0

T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003)

��ϕ� = 0


Electron and/or hole Fermi pockets form in “local” SDW order, but quantum fluctuations destroy long-range

SDW order





��ϕ� �= 0 ��ϕ� = 0

Fractionalized Fermi liquid (FL*) phasewith no symmetry

breaking and “small” Fermi surface

��ϕ� = 0



SDW order

T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003)Wednesday, June 6, 2012

Heavy Fermi liquid with “large” Fermi

surface of hydridized f and

c-conduction electrons

Magnetic order and the heavy Fermi liquid in the Kondo lattice

��ϕ� = 0

Magnetic Metal: f-electron moments

and c-conduction electron

Fermi surface

��ϕ� �= 0

f

c

f+c


Separating onset of SDW order and the heavy Fermi liquid in the Kondo lattice



Fermi surface

��ϕ� �= 0

f

c




��ϕ� = 0

f+c





Fermi surface

��ϕ� �= 0

f

c




��ϕ� = 0

f+c

c

f

Conduction electronFermi surface

andspin-liquid of f-electrons

��ϕ� = 0





Fermi surface

��ϕ� �= 0

f

c




��ϕ� = 0

f+c

c

f



��ϕ� = 0


f- sp

in:

loca

lized

delo

caliz

ed

Fractionalized Fermi liquid (FL*)

AFM Metal

large Fermi surface heavy Fermi liquid

YbAlB4

K

Q

Kc

Qc

QC1

QC2

YbRh2Si2Yb(Rh0.94Ir0.06)2Si2

YbAgGe

YbRh2(Si0.95Ge0.05)2

Yb(Rh0.93Co0.07)2Si2

QTC

Kondo screened paramagnet

spin

liqu

id

YbIr2Si2

CeCu2Si2

Experimental perpective on same phase diagrams of Kondo lattice

J. Custers, P. Gegenwart, C. Geibel, F. Steglich,

P. Coleman, and S. Paschen, Phys. Rev. Lett.

104, 186402 (2010)


finite B range (factor of 45 in T, factor of 2.3 in B consid-ering the expected crossover to a quadratic dependence ateven lower T in the AFM state at B< Bc1 ! 0:3 T, seeFig. 3) has not previously been observed in any HF com-pound [9–11]. In analogy with YbRh2Si2 [16], the resis-tivity !"B# isotherms have been examined. Clear crossoverbehavior is seen for B ? c and B k c which is character-ized by inflection points [16] denoted as Binfl in Figs. 1(b)and 1(c), respectively. It is clear from these figures that Binfl

increases with increasing T. Like Co- and Ir-substitutedYbRh2Si2 [9], the crossover behavior for the Ge-substituted compound investigated here is found to be al-most identical with the one of pure YbRh2Si2 [Fig. 1(b)].

This is further supported by another measure of thecrossover scale T$, the position Tmax of maxima in iso-B""T# curves [16], cf. Fig. 2. Like !"B#, also the ""T# datashow that, while TN is strongly suppressed upon substitut-ing YbRh2Si2 with Ge, T$ does not move (Fig. 2, inset).

Figure 3 summarizes all characteristic features ofYbRh2"Si0:95Ge0:05#2 in a T-B phase diagram. As indicatedby the shaded area, a finite range of NFL behavior at zero Tappears between the critical fields Bc1 and Bc2 for thesuppression of TN and T$.

In pure YbRh2Si2, the in-T linear resistivity extends tothe lowest accessible T (20 mK) at a single critical B, yet inYbRh2"Si0:95Ge0:05#2 this canonical behavior is violated,and instead, in-T linear resistivity extends to the lowest Tover a substantial B range. In isolation, this behavior mightbe dismissed as an anomaly. However, similar behavior hasrecently been observed also in other Yb-based HF com-pounds [9–11].

Conservatively, we might attribute these observations todisorder. In the Hertz-Millis theory, the in-T linear resis-tivity of HF systems is itself attributed to disorder [18,19].Furthermore, disorder is expected to smear a well-definedQCP into a region [20].

However, various aspects speak against this conservativeview point. First, it is unlikely that the smearing of a QCPwill be ‘‘asymmetric’’. The position of the T$ line inYbRh2"Si1%xGex#2 and hence of the entrance into theLFL phase is not affected by going from x ! 0 to x !0:05 (see Refs. [15,16] for the phase diagram ofYbRh2Si2); the NFL region in YbRh2"Si0:95Ge0:05#2 thusspreads only to the left of T$. Second, the NFL power lawdependencies are identical for YbRh2"Si0:95Ge0:05#2 andYbRh2Si2 [12]. Thus, either both systems are disorderdominated or none. And finally, values for the normalizedlinear rise of resistivity !!=!0 are, with &4 forYbRh2"Si0:95Ge0:05#2 [21],&5 for early YbRh2Si2 samples[22], and & 20 for the new generation of ultrapureYbRh2Si2 (where ! ! !0 ' AT# with # ! 1( 0:2 holdsup to 20 K) [23], all beyond the maximum value of unityexpected within the Hertz-Millis type scenario for disor-dered systems [18]. !!=!0 values more compatible withthis scenario are observed for CeCu5:9Au0:1 (!!=!0 &0:5) [24] and YbAgGe (!!=!0 & 1) [10], values muchlarger than unity for CeCoIn5 (!!=!0 & 100 for I ? c)[25]. Of course, the significance of !!=!0 in estimatingthe role of disorder is questionable in systems such asYbRh2Si2 and YbRh2"Si0:95Ge0:05#2 where the Hertz-Millis theory fails [12,15,16].

FIG. 3 (color online). Phase diagram of YbRh2"Si0:95Ge0:05#2for B k c. Symbols represent Binfl (r) and the upper boundary ofLFL behavior (.). The dashed TLFL line is the polynomial fitshown in the inset of Fig. 1(a). Data points from measurementswith B ? c are included by multiplying B with the factor 11: )symbolizes Binfl, * displays Tmax from "ac"T#. The solid T$ lineis taken from the inset of Fig. 2. Hexagons represent T$ [16] (orTHall [15]) of YbRh2Si2. j marks TN observed by specific heat.The dotted TN line indicates the typical evolution of TN forYbRh2Si2, TN"B# ! TN"0#"1% B=Bc#0:36 [9], using the respec-tive parameters for YbRh2"Si0:95Ge0:05#2 (TN"0# ! 18 mK, Bc !11+ B?c

c & 0:3 T) [12]. The hatched area 0:3 T , B , 0:66 Tmarks the zero T NFL phase characterized by !!- T. The insetcompares the evolution of the resistivity exponent ", derivedfrom the dependence "!% !0# - T" (see also Ref. [12]), forYbRh2Si2 (top) and YbRh2"Si0:95Ge0:05#2 (bottom) in the same Band T range.

0 0.1 0.2 0.30

0.5

1

0.03 0.1 1 101

3

5

7

YbRh2(Si

1-xGe

x)

2

x = 0x = 0.05

Tmax , Binfl

,,

B ! c

T (

K)

B (T)

x = 0.05

B = 0.28T

0.22T

0.12T

0.1T

0.075T

0.065T

0.05T

Tmax

B ! c

"

(# 1

0-6m

3 m

ol$1

)

x = 0

YbRh2(Si

1-x Ge

x)2 ;

ac

T (K)

FIG. 2. Ac susceptibility of YbRh2"Si1%xGex#2 for x ! 0:05(d) and, for comparison, for x ! 0 (*). Inset: Positions of themaxima in iso-B "ac"T# curves (circles) and of the inflectionpoints [16] Binfl of !"B# isotherms (diamonds) of both samples.The power law "B% Bc#0:75 (solid line) with Bc ! 0:06 T is agood description of all data points.

PRL 104, 186402 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending7 MAY 2010

186402-2

J. Custers, P. Gegenwart, C. Geibel, F. Steglich, P. Coleman, and S. Paschen, Phys. Rev. Lett. 104, 186402 (2010)





��ϕ� �= 0 ��ϕ� = 0



��ϕ� = 0



SDW order


0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

k_x

k_y

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

k_x

k_y

Hole pocket of a Z2-FL* phase

in a single-band t-J model

M. Punk and S. Sachdev, Phys. Rev. B 85, 195123 (2012)Wednesday, June 6, 2012

are larger than the presumed doping levels, 0.11, 0.085, and<0:05, respectively, it is clear that the pocket size scalesqualitatively with the doping level as predicted theoreti-cally [6,10]. Interestingly, two fluid models of the pseudo-gap state do predict the observed discrepancy between thepocket size and carrier concentration or doping level [22].The finding of a finite nodal FS rather than a ‘‘nodal’’ pointat low T for the Tc ! 0 K sample is at variance withrecently reported findings under the same conditions[23]. The measured Fermi pockets are, however, ingood agreement with those predicted by the YRZansatz. In Fig. 2(b) we show the spectral function calcu-

lated at EF as a function of doping, where A" ~k; 0# !$"1=!# ImGYRZ" ~k; 0# and where GYRZ" ~k; 0# is Green’sfunction taken from Ref. [6]. The experimental observa-tions are remarkably well reproduced by this model withthe doping level as the only adjustable parameter.

Turning to the question of whether the pocket areas aretemperature dependent, we show in Fig. 3(a) the observedFermi arc for the Tc ! 45 K sample measured at threedifferent temperatures: 60, 90, and 140 K, all in the normalstate but well below T%. The measured FS crossings inthe figure are determined by the same method used inFigs. 1 and 2 rather than from the spectral weight at theFermi level. In Fig. 3(b) we show the measured arc lengthas a function of temperature. It is clear that any changewithtemperature is minimal and certainly not consistent with anincrease by more than a factor of 2 between the data takenat 140 and 60 K as would be expected by a T=T% scaling ofthe arc length [21]. The discrepancy arises because pre-vious experiments have not fully determined whether ornot a band actually crosses the Fermi level.

The picture of the low energy excitations of the normalstate emerging from the present study is of a nodal FScharacterized by a Fermi ‘‘pocket’’ that, at temperaturesabove Tc, shows a minimal temperature dependence and anarea proportional only to the doping level. We now turn ourattention to the antinodal pseudogap itself.

Several theories of the pseudogap phase propose theformation of preformed singlet pairs above Tc in the anti-nodal region of the Brillouin zone [24]. The YRZ spinliquid based on the RVB picture is one such model as itrecognizes the formation of resonating pairs of spin sin-glets along the copper-oxygen bonds of the square latticeas the lowest energy configuration. Figures 4(a)–4(d) showa series of spectral plots along the straight sector of theLDA FS in the antinodal region at a temperature of 140 Kfor the Tc ! 65 K sample at the locations indicated inFig. 4(e). Figure 4(f) shows intensity cuts through theseplots along the horizontal lines indicated in Figs. 4(a)–4(d).It is evident that a symmetric gap exists at all points alongthis line. The particle-hole symmetry in binding energyobserved here is in marked contrast to the particle-holesymmetry breaking predicted in the presence of densitywave order and is a necessary condition for the formationof Cooper pairs. Thus the present observations add supportto the hypothesis that the normal state is characterized bypair states forming along the copper-oxygen bonds and isconsistent with earlier studies.The combination of Figs. 2 and 4 points to a more

complete picture of the low energy excitations in the nor-mal state of the underdoped cuprates. For Tc < T < T%, aFermi pocket exists in the nodal region with an area pro-portional to the doping level. One does not need to invokediscontinuous Fermi arcs to describe the FS of underdopedBi2212, and Luttinger’s sum rule, properly understood, isseen to still approximately stand. However, as is evident inthe inset of Fig. 2(a), the area of the hole pockets wouldappear to be larger than assumed doping level at the higherdoping levels. This may reflect the presence of electronpockets at the higher doping level or it may reflect thepresence of a bilayer splitting, even though the latter isnot observed in the present study. We note that the splittingwill be smaller in the underdoped region and in the nodalregion. Although not verified in the present study, one

Tc65K@140K Tc45K@140K Tc0K@30K

(0,0) (!,0)

(0,!) (!,!)

(a)

kx

ky

(0,0) (!,0)

(0,!) (!,!)

(b) x = 0.03x = 0.12x = 0.14

kx

ky

0.2

0.1

0.0

x AR

PE

S

0.100.00 xn

FIG. 2 (color online). (a) The pseudopockets determined forthree different doping levels. The black data correspond to theTc ! 65 K sample, the blue data correspond to the Tc ! 45 Ksample, and the red data correspond to the nonsuperconductingTc ! 0 K sample. The area of the pockets xARPES scales with thenominal of doping level xn, as shown in the inset. (b) The Fermipockets derived from YRZ ansatz with different doping level.

(!,0)(0,0)

(0,!)140K 90K 60K

(!,!)

"arc

kx

ky

40

30

20

10

0

" arc (

°)

200150100500Temperature (K)

(a) (b)

FIG. 3 (color online). (a) The Fermi surface crossings deter-mined for the Tc ! 45 K sample at three different temperatures.The triangles indicate measurements at a sample temperature of140 K, the circles measurements at 90 K, and the diamondsmeasurements at 60 K. (b) The measured arc lengths in (a)plotted as a function of temperature. We note that rather thancycling the temperatures on the same sample, the data in (a) aremeasured on different samples cut from the same crystal.

PRL 107, 047003 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending22 JULY 2011

047003-3

Reconstructed Fermi Surface of Underdoped Bi2Sr2CaCu2O8!! Cuprate Superconductors

H.-B. Yang,1 J. D. Rameau,1 Z.-H. Pan,1 G. D. Gu,1 P. D. Johnson,1 H. Claus,2 D.G. Hinks,2 and T. E. Kidd3

1Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA2Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

3Physics Department, University of Northern Iowa, Cedar Falls, Iowa 50614, USA(Received 6 August 2010; published 20 July 2011)

The Fermi surface topologies of underdoped samples of the high-Tc superconductor Bi2Sr2CaCu2O8!!

have been measured with angle resolved photoemission. By examining thermally excited states above the

Fermi level, we show that the observed Fermi surfaces in the pseudogap phase are actually components of

fully enclosed hole pockets. The spectral weight of these pockets is vanishingly small at the magnetic zone

boundary, creating the illusion of Fermi ‘‘arcs.’’ The area of the pockets as measured in this study is

consistent with the doping level, and hence carrier density, of the samples measured. Furthermore, the

shape and area of the pockets is well reproduced by phenomenological models of the pseudogap phase

as a spin liquid.

DOI: 10.1103/PhysRevLett.107.047003 PACS numbers: 74.25.Jb, 71.18.+y, 74.72.Kf, 79.60."i

Understanding the pseudogap regime in the high-Tc

superconducting cuprates is thought to be key to under-standing the high-Tc phenomenon in general [1]. An im-portant component of that understanding will be thedetermination of the nature of the low lying normal stateelectronic excitations that evolve into the superconductingstate. It is therefore critically important to know the exactnature of the Fermi surface (FS). Photoemission studies ofthe pseudogap regime reveal gaps in the spectral function indirections corresponding to the copper-oxygen bonds and aFS that seemingly consists of disconnected arcs falling onthe FS defined within the framework of a weakly interact-ing Fermi liquid [2]. A number of different theories haveattempted to explain these phenomena in terms of compet-ing orders whereby the full FS undergoes a reconstructionreflecting the competition [3,4]. An alternative approachrecognizes that the superconducting cuprates evolve withdoping from a Mott insulating state with no low energycharge excitations to a new state exhibiting propertiescharacteristic of both insulators and strongly correlatedmetals.

Several theories have been proposed to describe thecuprates from the latter perspective [5–7]. One such ap-proach is represented by the so-called Yang-Rice-Zhang(YRZ) ansatz [6], which, based on the doped resonantvalence bond (RVB) spin liquid concept [8], has beenshown to successfully explain a range of experimentalobservations in the underdoped regime [9–12]. The modelis characterized by two phenomena, a pseudogap thatdiffers in origin from the superconducting gap and holepockets that satisfy the Luttinger sum rule for a FS definedby both the poles and zeros of Green’s function at thechemical potential [13]. The pockets manifest themselvesalong part of the FS as an ‘‘arc’’ possessing finite spectralweight corresponding to the poles of Green’s function as ina conventional metal. The remaining ‘‘ghost’’ component

of the FS is defined by the zeros of Green’s function andtherefore possesses no spectral weight to be directly ob-served. Importantly, the zeros of Green’s function at thechemical potential coincide with the magnetic zone bound-ary associated with the underlying antiferromagnetic(AFM) order of the Mott insulating state and thereforerestrict the pockets to lying on only one side of this line.The model further predicts that the arc and ghost portionsof the FS are smoothly connected into pockets. Severaltheoretical studies indicate that within this framework thepockets have an area that scales with the doping [6,10].Recent photoemission studies have indeed provided someindication that the pseudogap regime is characterized byhole pockets centered in the nodal direction [14,15].Furthermore, the possibility that FS in the underdopedmaterials consists of a pocket structure is at the heart ofthe interpretation of recent studies that identified quantumoscillations in these materials [16]. In the present study, wedemonstrate for the first time that the FS of the underdopedcuprates in the normal state is characterized by hole pock-ets with an area proportional to the doping level.The photoemission studies reported in this Letter were

carried out on underdoped cuprate samples, both Ca dopedand oxygen deficient. The Ca-rich crystal was grown froma rod with Bi2:1Sr1:4Ca1:5Cu2O8!! composition using anarc-image furnace with a flowing 20% O2-Ar gas mixture.The maximum Tc was 80 K. The sample was then annealedat 700 #C giving a 45 K Tc with a transition width of 2 K.The oxygen-deficient Bi2Sr2CaCu2O8!! (Bi2212) crystalswere produced by annealing optimally doped Bi2212 crys-tals, at 450 #C to 650 #C for 3–15 days. The spectra shownin this Letter were all recorded on beam line U13UB at theNSLS using a Scienta SES2002 electron spectrometer.Each spectrum was recorded in the pulse-counting modewith an energy and angular resolution of 15 meVand 0.1#,respectively.

PRL 107, 047003 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending22 JULY 2011

0031-9007=11=107(4)=047003(4) 047003-1 ! 2011 American Physical Society


Characteristics of FL* phase

• Fermi surface volume does not count

all electrons.

• Such a phase must have neutral S = 1/2 ex-

citations (”spinons”), and collective spinless

gauge excitations (“topological” order).

• These topological excitations are needed to

account for the deficit in the Fermi surface

volume, in M. Oshikawa’s proof of the

Luttinger theorem.




all electrons.







Luttinger theorem.




all electrons.







Luttinger theorem.



Questions





Questions and Answers

Yes; convincing evidence from field theory and sign-problem free quantum Monte Carlo









The QCP shift from the metal to the superconductor is large. New physics (charge order, fractionalization...) is likely present in the intermediate regime







The QCP shift from the metal to the superconductor is large. New physics (charge order, fractionalization...) is likely present in the intermediate regime

Strongly-coupled quantum criticality of Fermi surface change in a metal


Thank you !


Documents

The onset of antiferromagnetism in metalsqpt.physics.harvard.edu/talks/lorentz12_4.pdf · antiferromagnetism in metals sachdev.physics.harvard.edu HARVARD Lorentz Lecture, Leiden