Quantum phase transitions and Fermi surface reconstructionqpt.physics.harvard.edu/talks/sces11.pdfQuantum phase transitions and Fermi surface reconstruction HARVARD Talk online: sachdev.physics.harvard.edu

Quantum phase transitionsand

Fermi surface reconstruction

HARVARD

Talk online: sachdev.physics.harvard.edu

Wednesday, August 31, 2011

Max Metlitski

HARVARD

Matthias Punk Erez Berg


1. Fate of the Fermi surface: reconstruction or not ? Experimental motivations from cuprates and pnictides

2. Conventional theory and its breakdown in two spatial dimensions

3. Fermi surface reconstruction: onset of unconventional superconductivity

4. Fermi surface reconstruction without symmetry breaking: metals with “topological” order

5. The nematic transition: field theory of antipodal Fermi surface points








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, August 31, 2011

Fermi surfaces translated by K = (π,π).Wednesday, August 31, 2011

Fermi surface reconstructioninto electron and hole pockets in

antiferromagnetic phase with ��ϕ� �= 0



Quantum phase transition with Fermi surface reconstruction

��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


Hole-doped

Electron-doped

Electron-doped cuprate superconductors


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


TSDW Tc

T0

2.0

0

!"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

Temperature-doping phase diagram of the iron pnictides:

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


Quantum phase transition with Ising-nematic order

Fermi surface with full square lattice symmetry

x

y

No Fermi surface reconstruction


Spontaneous elongation along x direction:Ising order parameter φ > 0.

x

y

Quantum phase transition with Ising-nematic orderNo Fermi surface reconstruction


x

y

Spontaneous elongation along y direction:Ising order parameter φ < 0.

Quantum phase transition with Ising-nematic orderNo Fermi surface reconstruction


Ising-nematic order parameter

φ ∼�

d2k (cos kx − cos ky) c†kσckσ

Measures spontaneous breaking of square lattice

point-group symmetry of underlying Hamiltonian


rrc

Pomeranchuk instability as a function of coupling r

�φ� = 0�φ� �= 0

or

Quantum criticality of Ising-nematic ordering


Broken rotational symmetry in the pseudogap phase of a high-Tc superconductorR. Daou, J. Chang, David LeBoeuf, Olivier Cyr-Choiniere, Francis Laliberte, Nicolas Doiron-Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis TailleferNature, 463, 519 (2010).


http://arxiv.org/find/cond-mat/1/au:+Daou_R/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Daou_R/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Chang_J/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Chang_J/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+LeBoeuf_D/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+LeBoeuf_D/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1




http://arxiv.org/find/cond-mat/1/au:+Laliberte_F/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Laliberte_F/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1




http://arxiv.org/find/cond-mat/1/au:+Ramshaw_B/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Ramshaw_B/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Liang_R/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Liang_R/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Bonn_D/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Bonn_D/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Hardy_W/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Hardy_W/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Taillefer_L/0/1/0/all/0/1

http://arxiv.org/find/cond-mat/1/au:+Taillefer_L/0/1/0/all/0/1

steeply with decreasing temperature. This insulator-like behavior is cut off near TN for the lowestdoping levels but extends to much lower temper-atures for larger cobalt concentrations. In contrast,for currents flowing in the a direction, theresistivity behaves similarly to a normal metal,continuing to decrease with decreasing temper-ature over the entire temperature range, except fora small jump near TN. The superconducting tran-sition at the lowest temperatures causes both raand rb to drop to zero. The difference of the tem-perature derivatives of rb and ra (normalized bythe room temperature value), shown in Fig. 4B,reveals a strong correlation with the orthorhombicdistortion.

The composition dependence of the anisotro-py in the in-plane resistivity (Fig. 4, A and B) isin stark contrast to that of the structural anisot-ropy that develops below TS. The orthorhombicdistortion, characterized by the ratio of in-planelattice constants, has a maximum value for x = 0at low temperature of (a! b)/(a + b) = 0.36% anddecreases monotonically with increasing cobaltconcentration (25), whereas the resistivity anisot-ropy is a nonmonotonic function of doping, ex-hibiting a maximum near the beginning of thesuperconducting dome. These contrasting behav-iors suggest that the itinerant electrons do notpassively follow the lattice distortion. Rather, itappears that the material suffers an underlyingelectronic phase transition that profoundly affectsthe low energy excitations near the Fermi leveland in a manner that is much less apparent in theresponse of the crystal lattice.

Notably, the resistivity anisotropy is evidentfor temperatures well above TS. Even though thecrystal symmetry is tetragonal in this temperaturerange, the uniaxial stress applied by the cantileverbreaks the fourfold symmetry in the basal plane.The external symmetry-breaking field induces anonzero order parameter above the critical tem-

perature, smoothing the divergent behavior asso-ciated with the critical point observed underambient conditions (26). As the temperature is re-duced toward TS from above, the resistivity anisot-ropy for the strained samples increases rapidly,indicating that the nematic susceptibility divergesat the critical temperature and providing com-pelling evidence that the resistivity anisotropyobserved above TS reflects fluctuations associ-ated with the phase transition that occurs at TS inthe zero stress limit (more discussion can befound in the supporting online material). Thereis no evidence in either thermodynamic or trans-port properties for a third phase transition aboveTS (27).

Co-doped BaFe2As2 was chosen for this ini-tial study because the structural and magnetictransition are clearly separated in temperature andthe material can be readily tuned in a controlledand reproducible fashion. However, given therather generic phase diagram found in this familyof compounds (28), it is possible that the ob-served anisotropy is quite general to underdopediron pnictides. For example, recent scanning tun-neling microscopy measurements also reveal alarge electronic anisotropy ofCo-dopedCaFe2As2at low temperatures (29).

In comparison to other electron nematicsystems, such as Sr3Ru2O7 (19) and the quantumHall system (18), the phase transition of ironpnictides occurs at zero field and at a temperaturerange between 50 and 150 K, providing an ac-cessible platform for spectroscopy and thermo-dynamic measurements. Although the pseudo-gapphase in underdoped cuprates exists for similarconditions to these, the iron arsenides have awell-defined phase boundary associated with thenematic phase transition. In this sense, the ironarsenides present a cleaner system in which toinvestigate the physical origin of nematic orderand the consequences for superconductivity.

References and Notes1. Y. Kamihara, T. Watanabe, M. Hirano, H. Hosono,

J. Am. Chem. Soc. 130, 3296 (2008).2. M. Rotter, M. Tegel, D. Johrendt, Phys. Rev. B 78,

020503 (2008).3. S. E. Sebastian et al., J. Phys. Condens. Matter 20,

422203 (2008).4. J. G. Analytis et al., Phys. Rev. B 80, 064507 (2009).5. Z. Ren et al., Chin. Phys. Lett. 25, 2215 (2008).6. C. de la Cruz et al., Nature 453, 899 (2008).7. C. Lester et al., Phys. Rev. B 79, 144523 (2009).8. D. K. Pratt et al., Phys. Rev. Lett. 103, 087001 (2009).9. C. Fang, H. Yao, W.-F. Tsai, J. Hu, S. A. Kivelson,

Phys. Rev. B 77, 224509 (2008).10. T. Yildirim, Phys. Rev. Lett. 102, 037003 (2009).11. C. Xu, M. Muller, S. Sachdev, Phys. Rev. B 78, 020501

(2008).12. M. Johannes, I. Mazin, Phys. Rev. B 79, 220510 (2009).13. F. Krüger, S. Kumar, J. Zaanen, J. van den Brink,

Phys. Rev. B 79, 054504 (2009).14. C.-C. Chen, B. Moritz, J. van den Brink, T. P. Devereaux,

R. R. P. Singh, Phys. Rev. B 80, 180418 (2009).15. W. Lv, J. Wu, P. Phillips, Phys. Rev. B 80, 224506

(2009).16. C.-C. Lee, W.-G. Yin, W. Ku, Phys. Rev. Lett. 103, 267001

(2009).17. Nematic order refers to a spontaneously broken rotational

symmetry without the breaking of translational symmetry.Systems for which the crystal lattice appears to preservethe original rotational symmetry (as found in the quantumHall effect and Sr3Ru2O7) might be dubbed “electronnematic,” whereas systems for which the crystal latticealso breaks rotational symmetry [for example, YBa2Cu3O7!dand Ba(Fe1!xCo x)2As2] might be more appropriately named“system nematic.” In both cases, a strong electronicanisotropy is observed that is much more pronounced thanconsideration of the crystal lattice alone would otherwisesuggest. We refer to this general phenomenon as “nematicorder” in the rest of the text. For a recent review, see (30).

18. M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer,K. W. West, Phys. Rev. Lett. 82, 394 (1999).

19. R. A. Borzi et al., Science 315, 214 (2007).20. Y. Ando, K. Segawa, S. Komiya, A. N. Lavrov, Phys. Rev. Lett.

88, 137005 (2002).21. V. Hinkov et al., Science 319, 597 (2008).22. R. Daou et al., Nature 463, 519 (2010).23. M. A. Tanatar et al., Phys. Rev. B 79, 180508 (2009).24. J.-H. Chu et al., Phys. Rev. B 81, 214502 (2010).25. R. Prozorov et al., Phys. Rev. B 80, 174517 (2009).26. L. D. Landau, E. M. Lifshitz, Statistical Physics: Part 1

(Butterworth-Heinemann, Oxford, ed. 3, 1980).27. J.-H. Chu, J. G. Analytis, C. Kucharczyk, I. R. Fisher,

Phys. Rev. B 79, 014506 (2009).28. For the Fe arsenides and oxy arsenides typified by

BaFe2As2 and LaFeAsO, the structural transition isfrom tetragonal to orthorhombic symmetry. The closelyrelated Fe chalcogenides, typified by FeTe, also suffera structural distortion, but in this case from tetragonalto monoclinic (31).

29. T. M. Chuang et al., Science 327, 181 (2010).30. E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein,

A. P. Mackenzie, http://arxiv.org/abs/0910.416631. S. Li et al., Phys. Rev. B 79, 054053 (2009).32. The authors thank C. D. Batista, C.-C. Chen,

T. P. Devereaux, S. A. Kivelson, A. P. Mackenzie,R. D. McDonald, S. C. Riggs, D. J. Scalapino, andZ.-X. Shen for helpful discussions. This work is supportedby the U.S. Department of Energy (DOE), Office of BasicEnergy Sciences, under contract DE-AC02-76SF00515.Use of the APS is supported by the DOE, Office ofScience, under contract DE-AC02-06CH11357.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/329/5993/824/DC1Materials and MethodsFigs. S1 to S4References

5 April 2010; accepted 12 July 201010.1126/science.1190482

Fig. 4. (A) Evolution of the in-plane resistivity anisotropy as a function of temperature and doping,expressed in terms of the resistivity ratio rb /ra. Structural, magnetic and superconducting criticaltemperatures, determined following (27), are shown as circles, squares, and triangles, respectively.Notably, the resistivity ratio deviates from unity at a considerably higher temperature than TS, indicatingthat nematic fluctuations extend far above the phase boundary. (B) The difference in the temperaturederivative of ra and rb: d(ra ! rb)/dT , where ra = ra/ra(300K) and rb = rb /rb(300K), as a function oftemperature and doping. The resistivity has been normalized by its room temperature value to avoiduncertainty due to geometric factors. Regions of highest intensity are those regions where rb appears tobe insulator-like (drb /dT < 0) while ra remains metallic (dra /dT > 0). This behavior is clearly correlatedwith the nematic phase between the structural and magnetic transitions.

13 AUGUST 2010 VOL 329 SCIENCE www.sciencemag.org826

REPORTS

on

Augu

st 2

9, 2

011

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

In-plane resistivity anisotropy in Ba(Fe1-xCox)2As2

Jiun-Haw Chu, J. G. Analytis, K. De Greve, P. L. McMahon, Z. Islam, Y. Yamamoto, and I. R. Fisher, Science 329, 824 (2010)














• 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, August 31, 2011



eter �ϕ alone.







dimensions.


Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).Wednesday, August 31, 2011


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.














��ϕ� = 0



��ϕ� �= 0




• Steglich: Evidence for two types of quantum criti-cal points between the SDW metal and the heavyfermion metal (both Fermi liquid phases) ind = 3: one described by SDW theory, and the otherby Kondo breakdown in Kondo lattice model(Q Si, P. Coleman)

• Our theory: Because the phases on either side arequalitatively identical, there is only one theory, whetherone uses the SDW or Kondo lattice models. In d = 2,this theory is strongly coupled. In d = 3, we can showthat the weak coupling (as in Hertz-Millis-Moriya)theory is stable; however there may also be anotherfixed point at strong coupling.


• Steglich: Evidence for two types of quantum criti-cal points between the SDW metal and the heavyfermion metal (both Fermi liquid phases) ind = 3: one described by SDW theory, and the otherby Kondo breakdown in Kondo lattice model(Q Si, P. Coleman)

• Our theory: Because the phases on either side arequalitatively identical, there is only one theory, whetherone uses the SDW or Kondo lattice models. In d = 2,this theory is strongly coupled. In d = 3, we can showthat the weak coupling (as in Hertz-Millis-Moriya)theory is stable; however there may also be anotherfixed point at strong coupling.
















��ϕ� = 0



��ϕ� �= 0




Fermi surfaces translated by K = (π,π).Wednesday, August 31, 2011

“Hot” spotsWednesday, August 31, 2011

Low energy theory for critical point near hot spotsWednesday, August 31, 2011

Low energy theory for critical point near hot spotsWednesday, August 31, 2011

v1 v2

ψ2 fermionsoccupied

ψ1 fermionsoccupied

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

kx

ky


v1 v2

kx

ky

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

Critical point theory is strongly coupled in d = 2Results are independent of coupling λ


Unconventional pairing at and near hot spots

∆−∆


Unconventional pairing at and near hot spots

∆−∆

�c†kαc

†−kβ

�= εαβ∆(cos kx − cos ky)


BCS theory

1 + λe-ph log�ωD

ω

�

Cooperlogarithm

}Wednesday, August 31, 2011

BCS theory

1 + λe-ph log�ωD

ω

�

Electron-phononcoupling

Debyefrequency

ImpliesTc ∼ ωD exp (−1/λ)


Antiferromagnetic fluctuations: weak-coupling

1 +

�U

t

�2

log

�EF

ω

�

V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983)D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986)

K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010)

Enhancement of pairing susceptibility by interactions

Cooperlogarithm

}Wednesday, August 31, 2011

Fermienergy

Antiferromagnetic fluctuations: weak-coupling

1 +

�U

t

�2

log

�EF

ω

�

Applies in a Fermi liquidas repulsive interaction U → 0.

Implies

Tc ∼ EF exp�− (t/U)2

�


V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983)D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986)

K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010)


Spin density wave quantum critical point

1 +α

π(1 + α2)log2

�EF

ω

�





1 +α

π(1 + α2)log2

�EF

ω

�





1 +α

π(1 + α2)log2

�EF

ω

�





1 +α

π(1 + α2)log2

�EF

ω

�


Fermienergy




1 +α

π(1 + α2)log2

�EF

ω

�


Fermienergy

α = tan θ, where 2θ isthe angle between Fermi lines.

Independent of interaction strengthU in 2 dimensions.


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


2θ



Enhancement of pairing susceptibility by interactionsSpin density wave quantum critical point

1 +α

π(1 + α2)log2

�EF

ω

�

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

• 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.
















��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


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

SDW order

T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003)Wednesday, August 31, 2011

Y. Qi and S. Sachdev, Physical Review B 81, 115129 (2010)

FL* phase has Fermi pockets without long-range antiferromagnetism, along with emergent gauge excitations


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


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

Fractionalized Fermi liquid (FL*) phasewith no symmetry

breaking and “small” Fermi surface

��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)


only in the local critical regime.25Finally, based on the thermodynamic and transport data

down to !0.4 K and up to 140 kOe, we can construct ten-tative T-H phase diagrams for the two orientations of theapplied magnetic field "Fig. 10#. Both phase diagrams arevery similar. Initially increasing magnetic field drives firstthe lower and then the higher magnetic transitions to zero.With further increase in field signatures of the NFL behaviorappear in the temperature-dependent resistivity ($%&T) andheat capacity (Cmagn /T&!ln T) and at our highest appliedfield values FL-like low-temperature resistivity ($%&T2)"i.e., the coherence line25,26 on the T!H phase diagram# isobserved. Although the current lack of data below !0.4 Kimpairs our ability to fully delineate the critical field that

FIG. 9. The anisotropic field dependence of "a# the electronicspecific-heat coefficient '; "b# the resistivity coefficient A, and "c#the Kadowaki-Woods ratio A/'2. In "a# and "b# shifted data for H!care shown as * "see text#. The shift for "a# is !52 kOe and forfigure "b# it is !35 kOe.

FIG. 10. Tentative T-H phase diagram for "a# H!ab; "b# H!c .Long-range magnetic order "LRMO#, NFL, and FL regions aremarked on the phase diagram. Symbols: filled circles—from Cp"Hmeasurements, open circles—from %(T)"H , open triangles—from%(H)"T , and asterisks—temperatures below which $%&T2 in%(T)"H data "coherence temperature Tcoh). Dashed line—low tem-perature limit of our measurements, vertical line marks H"0.

MAGNETIC FIELD INDUCED NON-FERMI-LIQUID . . . PHYSICAL REVIEW B 69, 014415 "2004#

014415-7

Magnetic field induced non-Fermi-liquid behavior in YbAgGe single crystals

S. L. Bud’ko,1 E. Morosan,1,2 and P. C. Canfield1,21Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA

2Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA!Received 25 August 2003; published 23 January 2004"

Detailed anisotropic resistivity and heat-capacity measurements down to #0.4 K and up to 140 kOe arereported for a single crystalline YbAgGe. Based on these data YbAgGe, a member of the hexagonal RAgGeserie, can be classified as new, stoichiometric heavy-fermion compound with two magnetic ordering tempera-tures below 1 K and field-induced non-Fermi-liquid behavior above 45–70 kOe and 80–110 kOe for H!ab andH!c , respectively.

DOI: 10.1103/PhysRevB.69.014415 PACS number!s": 75.30.Mb, 75.30.Kz

I. INTRODUCTION

YbAgGe is the penultimate member of the hexagonalRAgGe series1 and was recently identified1–4 as a new Yb-based heavy-fermion compound. Magnetization measure-ments on YbAgGe down to 1.8 K !Ref. 1" show moderateanisotropy !at low temperatures $ab /$c%3) and a loss oflocal moment character below #20 K !Fig. 1" !also see Fig.33 and related discussion in Ref. 1". The in-plane M (H) atT!2 K shows a trend toward saturation whereas H!c field-dependent magnetization continues to be virtually linear be-low 140 kOe !Fig. 1 inset". Initial thermodynamic and trans-port measurements down to 0.4 K !Refs. 1–3" reveal twomagnetic transitions, a higher one at %1 K, and a lower one,with very sharp features in &(T) and Cp(T), at %0.65 K!Fig. 2". Given that the magnetic entropy inferred from Fig. 2is only #5% of R ln 2 at 1 K and only reaches R ln 2 by#25 K it seems likely that these transitions are associatedwith a small moment ordering. Based on these measurementsthe compound was anticipated to be close to the quantumcritical point. The linear component of Cp(T), ' , is#150 mJ/mol K2 between 12 K and 20 K. Cp(T)/T rises upto #1200 mJ/mol K2 for T#1 K but given the presence ofthe magnetic transitions below 1 K, it is difficult to unam-biguously evaluate the electronic specific heat. Grosslyspeaking, 150 mJ/mol K2"'"1 J/mol K2 leading to an es-timate 10 K "TK"100 K for the Kondo temperature, TK .Since the number of the Yb-based heavy-fermion com-

pounds is relatively small,5–7 any new member of the familyattracts attention.6,8 As an up-to-date example, YbRh2Si2, aheavy-fermion antiferromagnet,9 became a subject of inten-sive, rewarding exploration.10–12 The case of YbAgGe ap-pears to have the potential of being somewhat similar toYbRh2Si2: the relatively high value of ' and the proximityof the magnetic ordering temperature to T!0 suggest thatYbAgGe is close to a quantum critical point !QCP" andmakes it a good candidate for a study of the delicate balanceand competition between magnetically ordered and stronglycorrelated ground states under the influence of a number ofparameters such as pressure, chemical substitution, and/ormagnetic field.In this work we report on the magnetic-field-induced evo-

lution of the ground state of YbAgGe as seen in anisotropic

resistivity and specific-heat measurements up to 140 kOe.We show that on increase of the applied magnetic field, theprogression from small moment magnetic order to QCP withthe evidence of non-Fermi-liquid !NFL" behavior and, inhigher fields, to low-temperature FL state is observed.

II. EXPERIMENTAL

YbAgGe crystallizes in hexagonal ZrNiAl-typestructure.13,14 YbAgGe single crystals in the form of cleanhexagonal cross-section rods of several millimeters lengthand 0.3–0.8 mm2 cross section were grown from high-temperature ternary solutions rich in Ag and Ge. Their struc-ture and the absence of impurity phases were confirmed bypowder X-ray diffraction !see Ref. 1 for details of thesamples’ growth". Temperature- and field-dependent resistiv-ity &(H ,T) and heat capacity Cp(H ,T) were measured downto 0.4 K in an applied magnetic field up to 140 kOe in aQuantum Design PPMS-14 instrument with He-3 option. Forresistivity a standard ac four-probe resistance technique ( f!16 Hz, I!1–0.3 mA) was used. Pt leads were attached tothe sample with Epotek H20E silver epoxy so that the cur-

FIG. 1. Anisotropic temperature-dependent dc susceptibility and!inset" field-dependent magnetization of YbAgGe.

PHYSICAL REVIEW B 69, 014415 !2004"

0163-1829/2004/69!1"/014415!8"/$22.50 ©2004 The American Physical Society69 014415-1


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)


LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS1299

!

7.2

7.3

7.4

7.5

7.6

(µ"

cm

)

0 0.02 0.04 0.06T (K)

0.08 0.10

0 0.5 1.0µ0H (T)

A (

µ" c

m K

¬2)

0

20

40

0 40 80µ0H (mT)

µ0H (mT)

T (

K)

0

0.1

6% lr

020

30

40

50

6070

80

2

1nT #

TN

TFL

Figure 4 | Signatures of Fermi-liquid and non-Fermi-liquid behaviour in

the resistivity of Yb(Rh0.94Ir0.06)2Si2. Temperature dependence of the

resistivity ρ(T) at selected magnetic fields. The lines indicate Fermi-liquid

behaviour, that is, fits to ρ(T)= ρ0+ATnwith the exponent n= 2 and ρ0

being the residual resistivity, for temperatures below TFL (marked by blue

arrows). Bottom inset: The field dependence of the coefficient

A= (ρ −ρ0)/Tnin the Fermi-liquid regime. The red line corresponds to

A(H)∝ (H−HAc)−1

yielding a critical field of µ0HAc

= 30(5)mT. Top

inset: Colour-coded representation of the resistivity exponent calculated as

n= dlog(ρ −ρ0)/dlogT. The energy scales TN, T�and TFL are reproduced

from Fig. 1, top panel. The red square on the abscissa depicts HAc, the critical

field of the divergence of the A coefficient (see bottom inset). The error

bars represent standard errors.

ref. 22 resembles our experimental observations. Also for YbAgGe,a finite field range was reported where the resistivity exhibitssimilar non-Fermi-liquid behaviour to the lowest temperatures23.However, the specific heat of YbAgGe shows a saturation ofC(T )/Tin this field range, discarding a spin-liquid ground state24. Incontrast, preliminarymeasurements on 6% Ir down to 0.06 K reveala strong divergence of C(T )/T with decreasing temperature in thefield range below 50mT (not shown), supporting our claimof a spinliquid. In addition, the susceptibility continues to increase towardsthe lowest temperatures (see Fig. 2a). The experimental evidenceof such a new, non-magnetic ground state is fascinating and willcertainlymotivate future experimental and theoretical studies.

Figure 5 shows the evolution of the two different QCPs as afunction of Ir/Co substitution. The following main results canbe deduced from this figure. (1) The antiferromagnetic state isstabilized through the application of positive chemical pressure,as expected. (2) The position of the suggested breakdown ofthe Kondo effect depends only weakly on chemical pressure—although the Kondo effect itself is known to be strongly pressuredependent. (3) As a consequence, for positive pressure, theantiferromagnetic QCP at HN is located in the regime with intactKondo screening (HN >H

�) where the SDW theory is expected tobe applicable in accordance with our observations. (4) For negativechemical pressure, on the other hand, HN is separated from H

�

towards lower fields with an intermediate spin-liquid-type groundstate emerging. Obviously, here, antiferromagnetic order and theFermi-liquid ground state are not connected by a single QCP,but are separated by a spin liquid, that is, a non-Fermi-liquidrange as previously observed for MnSi (ref. 25) and, perhaps, inβ-YbAlB4 (refs 26, 27).

0.06 0.04y x

0.02 0 0.060.040.02

‘SL’

0

0.1

0.2

µ 0H

(T)

Yb(Rh1¬yIr

y)2Si2 Yb(Rh1¬x

Cox)2Si2

T 0

FL

AF

H!

HN

Figure 5 | Experimental phase diagram in the zero temperature limit. The

zero-temperature phase diagram depicts the extrapolated critical fields of

the various energy scales. The red line represents the critical field of T�(H)

and the green line the antiferromagnetic (AF) critical field HN. The blue and

green regions mark the Fermi-liquid (FL) and magnetically ordered ground

state, respectively. The orange region situated between these two is a

phase that resembles a spin liquid (SL). Data for Yb(Rh1−xMx)2Si2 with

2.5% Ir and 3% Co have been included. The error bars represent

standard errors.

To conclude, the application of chemical pressure provides awider view on the global phase diagram of YbRh2Si2 by liftingthe coincidence of the multiple energy scales in the stoichiometriccompound. The results and their interpretation presented here posea formidable challenge for those theories describing the breakdownof the Kondo effect near an antiferromagnetic QCP in Kondolattice systems. It remains to be explored under which conditionsantiferromagnetic ordering and the Fermi-surface reconstructionmay eventually become separated as observed for YbRh2Si2 withIr substitution. Equally important, it needs to be understoodwhy in pure YbRh2Si2, the antiferromagnetic QCP coincides withthe Kondo breakdown.

MethodsSingle crystals were grown from In flux, analogous to the stoichiometric samplesdescribed earlier12. The In flux was subsequently removed in hydrochloric acid.The presented results prove the absence of residual In. X-ray diffraction confirmsthe single crystallinity. All low-temperature measurements were carried out withthe magnetic field aligned perpendicular to the crystallographic c axis, H ⊥ c .The a.c.-susceptibility measurements were carried out at low frequencies witha modulation field amplitude of 4 µT down to 0.02 K. As no imaginary signalwas detected, the real part χ � is a direct measure of the field derivative of themagnetization. The temperature-dependent susceptibility χ �(T ) was measured inselected static magnetic fields. The isothermal susceptibility χ �(H ) was measuredas a function of a field applied in addition to the modulation field. The electricalresistivity ρ was monitored by a standard four-point lock-in technique at lowfrequencies down to 0.02 K. An extremely small out-of-phase signal of lessthan 1% proves the high quality of the spot-welded contacts. With the helpof low-temperature transformers, a very high sensitivity of better than 0.1 nVwas realized. In all samples, the resistivity was measured perpendicular to thecrystallographic c axis, and the magnetic field was applied parallel to the current.The magnetic field dependence of the magnetization M (H ) was isothermallymeasured in a high-resolution Faraday magnetometer down to 0.05 K (ref. 28).Background contributions from the sample platform and the torque exerted onthe sample have been subtracted. The magnetization was analysed in the formM̃ =M +χ �

H by fitting

�H

0A2 −(A2 −A1)/(1+ (H �/H0)p) dH � (1)

to the data from which the crossover field H0 was obtained14. M̃ is preferredfor the analysis as it enables a more precise fitting compared with M itself,although the conclusions drawn from M (H ) are identical (see ref. 14 and itssupporting online material).

468 NATURE PHYSICS | VOL 5 | JULY 2009 | www.nature.com/naturephysics

LETTERSPUBLISHED ONLINE: 31 MAY 2009 | DOI: 10.1038/NPHYS1299

Detaching the antiferromagnetic quantum criticalpoint from the Fermi-surface reconstructionin YbRh2Si2S. Friedemann1*, T. Westerkamp1, M. Brando1, N. Oeschler1, S. Wirth1, P. Gegenwart1,2, C. Krellner1,C. Geibel1 and F. Steglich1*

A continuous phase transition driven to zero temperature by

a non-thermal parameter, such as pressure, terminates in a

quantum critical point (QCP). At present, two main theoretical

approaches are available for antiferromagnetic QCPs in heavy-

fermion systems. The conventional one is the quantum

generalization of finite-temperature phase transitions, which

reproduces the physical properties in many cases1–5. More

recent unconventional models incorporate a breakdown of the

Kondo effect, giving rise to a Fermi-surface reconstruction6–8

—

YbRh2Si2 is a prototype of this category5,9–11

. In YbRh2Si2,

the antiferromagnetic transition temperature merges with the

Kondo breakdown at the QCP. Here, we study the evolution of

the quantum criticality in YbRh2Si2 under chemical pressure.

Surprisingly, for positive pressure we find the signature

of the Kondo breakdown within the magnetically ordered

phase, whereas negative pressure induces their separation,

leaving an intermediate spin-liquid-type ground state over an

extended range. This behaviour suggests a new quantumphase

arising from the interplay of the Kondo breakdown and the

antiferromagnetic QCP.

In heavy-fermion systems, the Kondo effect leads to theformation of composite quasiparticles of the f and conduction-electron states with largely renormalized masses forming a LandauFermi-liquid ground state in the paramagnetic regime well belowthe Kondo temperature TK. These quasiparticles are assumed tostay intact at the quantum critical point (QCP) in the conventionalmodels in which magnetic order arises through a spin-density-wave (SDW) instability. However, the observation of magneticcorrelations in CeCu5.9Au0.1 being of local character11 prompted aseries of theoretical descriptions that discard this basic assumption.Rather, they focus on the breakdown of the Kondo effect, whichcauses the f states to become localized and decoupled from theconduction-band states at the QCP where one expects the Fermisurface to be reconstructed7. Consequently, a new energy scaleT

� is predicted reflecting the finite-temperature T crossover ofthe Fermi-surface volume. This picture has been scrutinized intetragonal YbRh2Si2 (TK ≈ 25K; ref. 12), a stoichiometric andvery clean heavy-fermion metal that seems to be ideally suitedfor this kind of study9,12: antiferromagnetic order sets in at a verylow temperature TN = 0.07K and can easily be suppressed by asmall magnetic field of µ0HN = 60mT (H ⊥ c , with c being themagnetically hard axis). Hall-effect experiments13 have detecteda rapid change of the Hall coefficient along a line T

�(H ) thatconverges with HN, the width of the Hall crossover extrapolating

1Max-Planck-Institut für Chemische Physik fester Stoffe, D-01187 Dresden, Germany,

2I. Physik. Institut, Georg-August-Universität Göttingen, D-37077

Göttingen, Germany. *e-mail: [email protected]; [email protected].

to zero for T → 0. This change was considered evidence foran abrupt change of the Fermi-surface volume indicating acorrespondence between T

�(H ) and the Kondo-breakdown energyscale. Subsequent thermodynamic and transport investigationsconfirmed T

�(H ) to be a new energy scale14. The a.c. susceptibilityχ �(T ) turned out to exhibit a pronounced maximum at T �, whichcan be particularly well distinguished from the sharp signature atTN. The magnetoresistance exhibits a step-like crossover similar tothe Hall coefficient. In fact, recent calculations for a Kondo latticepredict such a feature for both transport properties15. Furthermore,the magnetization shows a smeared kink at T �(H ) between twoalmost linear regimes with different slopes14. The anomaly atT �(H )in all isothermal measurements can be described either by the samecrossover function proposed for theHall effect or its integral versionresembling a smeared kink. The maximum in χ �(T ) is a naturalconsequence of the magnetization kink being smeared and shiftedto higher fields as the temperature is raised.

The Fermi-surface reconstruction may also occur away from theantiferromagnetic QCP as observed in CeIn3 and CeRh1−xCoxIn5(refs 16, 17). It is therefore very important to understandthe interplay between the phenomena assigned to the Kondobreakdown and the magnetic order. We address this issue byinvestigating YbRh2Si2 under positive and negative chemicalpressure, which was realized by partial isoelectronic substitutionof smaller Co or larger Ir for Rh, respectively (the analogy ofchemical and external pressure is discussed in SupplementaryInformation A). In Yb systems, pressure yields a stabilizationof magnetism, in particular an increase of TN (ref. 18). Onthe other hand, negative pressure, corresponding to a latticeexpansion, reduces TN.

The T–H phase diagrams of Yb(Rh0.94Ir0.06)2Si2 andYb(Rh0.93Co0.07)2Si2 (labelled 6% Ir and 7% Co in the following)are compared with that of YbRh2Si2 in Fig. 1. This set emphasizesthe evolution of the various energy scales. First, the magneticone follows the expected pressure dependence: for 6% Ir, TN isdepressed below 0.02 K, whereas in the case of 7% Co, TN isenhanced to 0.41 K. Second, the energy scale T �(H ) virtually doesnot change its position in the T–H phase diagram. Consequently,T

�(H ) is separated from TN(H ) in 6% Ir, whereas they intersect in7% Co. Finally, at fields above the respective critical fields H � (for6% Ir) andHN (for 7% Co) at which T

�(H ) and TN(H ) vanish, theFermi-liquid phase forms below TFL(H ).

These findings were mainly deduced from the temperature-dependent a.c. susceptibility χ �(T ) shown in Fig. 2. For 6% Ir,

NATURE PHYSICS | VOL 5 | JULY 2009 | www.nature.com/naturephysics 465

Nature Physics 5, 465 (2009)


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.














rrc

Pomeranchuk instability as a function of coupling r

�φ� = 0�φ� �= 0

or



• Critical point is described by an infinite set of 2+1dimensional field theories, one for each direction q̂.Each theory has a finite Fermi surface curvature,with quasiparticle dispersion = vF qx + q2y/(2m

∗).


M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010)Wednesday, August 31, 2011


∗).




∗).

Strong coupling problem:Infinite number of 2+1

dimensional field theoriesat Ising-nematic

quantum critical point



Conclusions

All quantum phase transitions of metalsin two spatial dimensions

involving symmetry breakingare strongly-coupled

andand very different from the “Stoner” mean field theory


Conclusions

There is an instability of universal strength to

unconventional superconductivity near the onset of antiferromagnetism

in a two-dimensional metal


Conclusions

The strong-coupling physics may also apply in three spatial dimensions,

but the “Stoner” mean field theory

is locally stable.


Conclusions

There can be an intermediate non-Fermi liquid phase

between the two Fermi liquids:the antiferromagnetic metal with “small” Fermi surfaces

and the metal with “large” Fermi surfaces


Conclusions

This non-Fermi liquid phase has neutral S=1/2 excitations,

and “topological” gauge excitations,which account for

deficits in the Luttinger count of the volume enclosed by the Fermi surfaces


Documents

Quantum phase transitions and Fermi surface reconstructionqpt.physics.harvard.edu/talks/sces11.pdfQuantum phase transitions and Fermi surface reconstruction HARVARD Talk online: sachdev.physics.harvard.edu