theory of quantum matter: from quantum fields to …qpt.physics.harvard.edu/talks/salam14_3.pdf ·...
TRANSCRIPT
Theory of Quantum Matter: from Quantum Fields
to Strings
HARVARD
Salam Distinguished LecturesThe Abdus Salam International Center for Theoretical Physics
Trieste, ItalyJanuary 27-30, 2014
Subir Sachdev
Talk online: sachdev.physics.harvard.eduThursday, January 30, 14
1. The simplest models without quasiparticles
A. Superfluid-insulator transition
of ultracold bosons in an optical lattice
B. Conformal field theories in 2+1 dimensions and
the AdS/CFT correspondence
2. Metals without quasiparticles
A. Review of Fermi liquid theory
B. A “non-Fermi” liquid: the Ising-nematic
quantum critical point
C. Holography, entanglement, and strange metals
Outline
Thursday, January 30, 14
1. The simplest models without quasiparticles
A. Superfluid-insulator transition
of ultracold bosons in an optical lattice
B. Conformal field theories in 2+1 dimensions and
the AdS/CFT correspondence
2. Metals without quasiparticles
A. Review of Fermi liquid theory
B. A “non-Fermi” liquid: the Ising-nematic
quantum critical point
C. Holography, entanglement, and strange metals
Outline
Thursday, January 30, 14
Ishida, Nakai, and HosonoarXiv:0906.2045v1
Iron pnictides: a new class of high temperature superconductors
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
AF+nematic
Resistivity⇠ ⇢0 +AT↵
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)Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
Resistivity⇠ ⇢0 +AT↵
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)
Neel (AF) and “nematic” order
AF+nematic
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
Resistivity⇠ ⇢0 +AT↵
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)
Neel (AF) and “nematic” order
AF+nematic
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
AF
Resistivity⇠ ⇢0 +AT↵
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)
Neel (AF) and “nematic” order
AF+nematic
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
AF
Resistivity⇠ ⇢0 +AT↵
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)
Neel (AF) and “nematic” order
AF+nematic
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
Resistivity⇠ ⇢0 +AT↵
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)
SuperconductorBose condensate of pairs of electrons
AF+nematic
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
Resistivity⇠ ⇢0 +AT↵
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)
Ordinary metal(Fermi liquid)
AF+nematic
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
Resistivity⇠ ⇢0 +AT↵
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)
no quasiparticles,Landau-Boltzmann theory
does not apply
AF+nematic
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
Resistivity⇠ ⇢0 +AT↵
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
no quasiparticles,Landau-Boltzmann theory
does not apply
AF+nematic
Thursday, January 30, 14
LETTERdoi:10.1038/nature11178
Electronic nematicity above the structural andsuperconducting transition in BaFe2(As12xPx)2S. Kasahara1,2, H. J. Shi1, K. Hashimoto1{, S. Tonegawa1, Y. Mizukami1, T. Shibauchi1, K. Sugimoto3,4, T. Fukuda5,6,7, T. Terashima2,Andriy H. Nevidomskyy8 & Y. Matsuda1
Electronic nematicity, a unidirectional self-organized state thatbreaks the rotational symmetry of the underlying lattice1,2, hasbeen observed in the iron pnictide3–7 and copper oxide8–11 high-temperature superconductors. Whether nematicity plays anequally important role in these two systems is highly controversial.In iron pnictides, the nematicity has usually been associated withthe tetragonal-to-orthorhombic structural transition at temper-ature Ts. Although recent experiments3–7 have provided hints ofnematicity, they were performed either in the low-temperatureorthorhombic phase3,5 or in the tetragonal phase under uniaxialstrain4,6,7, both of which break the 906 rotational C4 symmetry.Therefore, the question remains open whether the nematicity canexist above Ts without an external driving force. Here we reportmagnetic torque measurements of the isovalent-doping systemBaFe2(As12xPx)2, showing that the nematicity develops well aboveTs and, moreover, persists to the non-magnetic superconductingregime, resulting in a phase diagram similar to the pseudogapphase diagram of the copper oxides8,12. By combining these resultswith synchrotron X-ray measurements, we identify two distincttemperatures—one at T*, signifying a true nematic transition,and the other at Ts (,T*), which we show not to be a true phasetransition, but rather what we refer to as a ‘meta-nematic trans-ition’, in analogy to the well-known meta-magnetic transition inthe theory of magnetism.
Magnetic torque measurements provide a stringent test of nematicityfor systems with tetragonal symmetry13. The torque t 5 m0VM 3 H is athermodynamic quantity, a differential of the free energy with respect toangular displacement. Here m0 is the permeability of vacuum, V is thesample volume, and M is the magnetization induced in the magneticfield H. When H is rotated within the tetragonal a–b plane (Fig. 1a, b), tis a periodic function of 2w, where w is the azimuthal angle measuredfrom the a axis:
t2w~12
m0H2V xaa{xbbð Þ sin 2w{2xab cos 2w½ $ ð1Þ
where the susceptibility tensor xij is defined by Mi 5SjxijHj. In a systemmaintaining tetragonal symmetry, t2w should be zero, because xaa 5 xbband xab 5 0. Finite values of t2w appear if a new electronic or magneticstate emerges that breaks the C4 tetragonal symmetry. In such a case,rotational symmetry breaking is revealed by xaa ? xbb and/or xab ? 0,depending on the direction of the nematicity.
BaFe2(As1–xPx)2 is a prototypical family of iron pnictides14–18, whosephase diagram is displayed in Fig. 1c. The temperature evolution of thetorque t(w) for the optimally doped compound (x 5 0.33) is depicted inthe upper panels of Fig. 1d. The two- and four-fold oscillations, t2w andt4w, obtained from the Fourier analysis are shown respectively in themiddle and lower panels of Fig. 1d. The distinct two-fold oscillationsappear at low temperatures, whereas they are absent at high temperatures
1Department of Physics, Kyoto University, Kyoto 606-8502, Japan. 2Research Center for Low Temperature and Materials Sciences, Kyoto University, Kyoto 606-8501, Japan. 3Research and UtilizationDivision, JASRI SPring-8, Sayo, Hyogo 679-5198, Japan. 4Structural Materials Science Laboratory, RIKEN SPring-8, Sayo, Hyogo 679-5148, Japan. 5Quantum Beam Science Directorate, JAEA SPring-8,Sayo, Hyogo 679-5148, Japan. 6Materials Dynamics Laboratory, RIKEN SPring-8, Sayo, Hyogo 679-5148, Japan. 7JST, Transformative Research-Project on Iron Pnictides (TRIP), Chiyoda, Tokyo 102-0075,Japan. 8Department of Physics and Astronomy, Rice University, 6100 Main Street, Houston, Texas 77005, USA. {Present address: Institute for Materials Research, Tohoku University, Sendai 980-8577,Japan.
90 K
0 180
35 K
0 180 360
65 K
0 180
75 K
0 180
b c
a
HSingle crystal
I (deg.) W
(a.u
.)cd
Paramagnetic(tetragonal)
(100)T
Antiferromagnetic(orthorhombic)
(010)T
Fe As/P
T > T* T < T*T *
Electronic nematic
Superconducting
0 0.2 0.4 0.60
100
200
x
T (K
)
W2I
(a.u
.) W
4I (a
.u.)
I (deg.) I (deg.) I (deg.)
V
c
a bM
HI
TorqueW = P0M × H≠ 0
a b
Figure 1 | Torque magnetometry and the doping–temperature phasediagram of BaFe2(As12xPx)2. a, b, Schematic representations of theexperimental configuration for torque measurements under in-plane fieldrotation. In a nematic state, domain formation with different preferreddirections in the a–b plane (‘twinning’) will occur. We used very small singlecrystals with typical size ,70mm 3 70mm3 30mm, in which a significantdifference in volume between the two types of domains enables the observationof uncompensated t2w signals. The equation given in the figure for t assumesunit volume; see text for details. A single-crystalline sample (brown block) ismounted on the piezo-resistive lever which is attached to the base (blue block)and forms an electrical bridge circuit (orange lines) with the neighbouringreference lever. A magnetic field H can be rotated relative to the sample, asillustrated by a blue arrow on a sphere. In this experiment, the field is preciselyapplied in the a–b plane. c, Phase diagram of BaFe2(As1–xPx)2. This system isclean and homogeneous14,16,17, as demonstrated by the quantum oscillationsobserved over a wide x range16. The antiferromagnetic transition at TN (filledcircles)15 coincides or is preceded by the structural transition at Ts (opentriangles)18. The superconducting dome extends over a doping range0.2 , x , 0.7 (open squares), with maximum Tc 5 31 K. Crosses indicate thenematic transition temperature T* determined by the torque and synchrotronX-ray diffraction measurements. The insets illustrate the tetragonal FeAs/Player. xab 5 0 above T* yielding an isotropic torque signal (green-shadedcircle), whereas xab ? 0 below T*, indicating the appearance of the nematicityalong the [110]T (Fe–Fe bond) direction, illustrated with the green-shadedellipse. d, The upper panels depict the temperature evolution of the raw torquet(w) at m0H 5 4 T for BaFe2(As0.67P0.33)2 (Tc 5 30 K). All torque curves arereversible with respect to the field rotation. t(w) can be decomposed as t(w) 5 t2w 1 t4w 1 t6w 1 ???, where t2nw 5 A2nw sin 2n(w 2 w0) has 2n-foldsymmetry with integer n. The middle and lower panels display the two- andfour-fold components obtained from Fourier analysis. The four-foldoscillations t4w (and higher-order terms) arise primarily from the nonlinearsusceptibilities13. a.u., arbitrary units.
3 8 2 | N A T U R E | V O L 4 8 6 | 2 1 J U N E 2 0 1 2
Macmillan Publishers Limited. All rights reserved©2012
BaFe2(As1�x
Px
)2
S. Kasahara, H.J. Shi, K. Hashimoto, S. Tonegawa, Y. Mizukami,T. Shibauchi, K. Sugimoto, T. Fukuda, T. Terashima, A.H. Nevidomskyy, and
Y. Matsuda, Nature 486, 382 (2012).
LETTERdoi:10.1038/nature11178
Electronic nematicity above the structural andsuperconducting transition in BaFe2(As12xPx)2S. Kasahara1,2, H. J. Shi1, K. Hashimoto1{, S. Tonegawa1, Y. Mizukami1, T. Shibauchi1, K. Sugimoto3,4, T. Fukuda5,6,7, T. Terashima2,Andriy H. Nevidomskyy8 & Y. Matsuda1
Electronic nematicity, a unidirectional self-organized state thatbreaks the rotational symmetry of the underlying lattice1,2, hasbeen observed in the iron pnictide3–7 and copper oxide8–11 high-temperature superconductors. Whether nematicity plays anequally important role in these two systems is highly controversial.In iron pnictides, the nematicity has usually been associated withthe tetragonal-to-orthorhombic structural transition at temper-ature Ts. Although recent experiments3–7 have provided hints ofnematicity, they were performed either in the low-temperatureorthorhombic phase3,5 or in the tetragonal phase under uniaxialstrain4,6,7, both of which break the 906 rotational C4 symmetry.Therefore, the question remains open whether the nematicity canexist above Ts without an external driving force. Here we reportmagnetic torque measurements of the isovalent-doping systemBaFe2(As12xPx)2, showing that the nematicity develops well aboveTs and, moreover, persists to the non-magnetic superconductingregime, resulting in a phase diagram similar to the pseudogapphase diagram of the copper oxides8,12. By combining these resultswith synchrotron X-ray measurements, we identify two distincttemperatures—one at T*, signifying a true nematic transition,and the other at Ts (,T*), which we show not to be a true phasetransition, but rather what we refer to as a ‘meta-nematic trans-ition’, in analogy to the well-known meta-magnetic transition inthe theory of magnetism.
Magnetic torque measurements provide a stringent test of nematicityfor systems with tetragonal symmetry13. The torque t 5 m0VM 3 H is athermodynamic quantity, a differential of the free energy with respect toangular displacement. Here m0 is the permeability of vacuum, V is thesample volume, and M is the magnetization induced in the magneticfield H. When H is rotated within the tetragonal a–b plane (Fig. 1a, b), tis a periodic function of 2w, where w is the azimuthal angle measuredfrom the a axis:
t2w~12
m0H2V xaa{xbbð Þ sin 2w{2xab cos 2w½ $ ð1Þ
where the susceptibility tensor xij is defined by Mi 5SjxijHj. In a systemmaintaining tetragonal symmetry, t2w should be zero, because xaa 5 xbband xab 5 0. Finite values of t2w appear if a new electronic or magneticstate emerges that breaks the C4 tetragonal symmetry. In such a case,rotational symmetry breaking is revealed by xaa ? xbb and/or xab ? 0,depending on the direction of the nematicity.
BaFe2(As1–xPx)2 is a prototypical family of iron pnictides14–18, whosephase diagram is displayed in Fig. 1c. The temperature evolution of thetorque t(w) for the optimally doped compound (x 5 0.33) is depicted inthe upper panels of Fig. 1d. The two- and four-fold oscillations, t2w andt4w, obtained from the Fourier analysis are shown respectively in themiddle and lower panels of Fig. 1d. The distinct two-fold oscillationsappear at low temperatures, whereas they are absent at high temperatures
1Department of Physics, Kyoto University, Kyoto 606-8502, Japan. 2Research Center for Low Temperature and Materials Sciences, Kyoto University, Kyoto 606-8501, Japan. 3Research and UtilizationDivision, JASRI SPring-8, Sayo, Hyogo 679-5198, Japan. 4Structural Materials Science Laboratory, RIKEN SPring-8, Sayo, Hyogo 679-5148, Japan. 5Quantum Beam Science Directorate, JAEA SPring-8,Sayo, Hyogo 679-5148, Japan. 6Materials Dynamics Laboratory, RIKEN SPring-8, Sayo, Hyogo 679-5148, Japan. 7JST, Transformative Research-Project on Iron Pnictides (TRIP), Chiyoda, Tokyo 102-0075,Japan. 8Department of Physics and Astronomy, Rice University, 6100 Main Street, Houston, Texas 77005, USA. {Present address: Institute for Materials Research, Tohoku University, Sendai 980-8577,Japan.
90 K
0 180
35 K
0 180 360
65 K
0 180
75 K
0 180
b c
a
HSingle crystal
I (deg.)
W (a
.u.)c
d
Paramagnetic(tetragonal)
(100)T
Antiferromagnetic(orthorhombic)
(010)T
Fe As/P
T > T* T < T*T *
Electronic nematic
Superconducting
0 0.2 0.4 0.60
100
200
x
T (K
)
W2I
(a.u
.) W
4I (a
.u.)
I (deg.) I (deg.) I (deg.)
V
c
a bM
HI
TorqueW = P0M × H≠ 0
a b
Figure 1 | Torque magnetometry and the doping–temperature phasediagram of BaFe2(As12xPx)2. a, b, Schematic representations of theexperimental configuration for torque measurements under in-plane fieldrotation. In a nematic state, domain formation with different preferreddirections in the a–b plane (‘twinning’) will occur. We used very small singlecrystals with typical size ,70mm 3 70mm3 30mm, in which a significantdifference in volume between the two types of domains enables the observationof uncompensated t2w signals. The equation given in the figure for t assumesunit volume; see text for details. A single-crystalline sample (brown block) ismounted on the piezo-resistive lever which is attached to the base (blue block)and forms an electrical bridge circuit (orange lines) with the neighbouringreference lever. A magnetic field H can be rotated relative to the sample, asillustrated by a blue arrow on a sphere. In this experiment, the field is preciselyapplied in the a–b plane. c, Phase diagram of BaFe2(As1–xPx)2. This system isclean and homogeneous14,16,17, as demonstrated by the quantum oscillationsobserved over a wide x range16. The antiferromagnetic transition at TN (filledcircles)15 coincides or is preceded by the structural transition at Ts (opentriangles)18. The superconducting dome extends over a doping range0.2 , x , 0.7 (open squares), with maximum Tc 5 31 K. Crosses indicate thenematic transition temperature T* determined by the torque and synchrotronX-ray diffraction measurements. The insets illustrate the tetragonal FeAs/Player. xab 5 0 above T* yielding an isotropic torque signal (green-shadedcircle), whereas xab ? 0 below T*, indicating the appearance of the nematicityalong the [110]T (Fe–Fe bond) direction, illustrated with the green-shadedellipse. d, The upper panels depict the temperature evolution of the raw torquet(w) at m0H 5 4 T for BaFe2(As0.67P0.33)2 (Tc 5 30 K). All torque curves arereversible with respect to the field rotation. t(w) can be decomposed as t(w) 5 t2w 1 t4w 1 t6w 1 ???, where t2nw 5 A2nw sin 2n(w 2 w0) has 2n-foldsymmetry with integer n. The middle and lower panels display the two- andfour-fold components obtained from Fourier analysis. The four-foldoscillations t4w (and higher-order terms) arise primarily from the nonlinearsusceptibilities13. a.u., arbitrary units.
3 8 2 | N A T U R E | V O L 4 8 6 | 2 1 J U N E 2 0 1 2
Macmillan Publishers Limited. All rights reserved©2012
Thursday, January 30, 14
YBa2Cu3O6+x
High temperature superconductors
Thursday, January 30, 14
� �
Pseudogap
Thursday, January 30, 14
� �
Pseudogap
Thursday, January 30, 14
� �
StrangeMetal
Pseudogap
Thursday, January 30, 14
LETTERS
PUBLISHED ONLINE: 20 MAY 2012 | DOI: 10.1038/NPHYS2321
Visualization of the emergence of the pseudogapstate and the evolution to superconductivity in alightly hole-doped Mott insulatorY. Kohsaka1*, T. Hanaguri2, M. Azuma3, M. Takano4, J. C. Davis5,6,7,8 and H. Takagi1,2,9
Superconductivity emerges from the cuprate antiferromag-netic Mott state with hole doping. The resulting electronicstructure1 is not understood, although changes in the state ofoxygen atoms seem paramount2–5. Hole doping first destroysthe Mott state, yielding a weak insulator6,7 where electronslocalize only at low temperatures without a full energy gap.At higher doping levels, the ‘pseudogap’, a weakly conductingstate with an anisotropic energy gap and intra-unit-cell break-ing of 90� rotational (C4v) symmetry, appears3,4,8–10. However,a direct visualization of the emergence of these phenomenawith increasing hole density has never been achieved. Here wereport atomic-scale imaging of electronic structure evolutionfrom the weak insulator through the emergence of the pseu-dogap to the superconducting state in Ca2� x
Nax
CuO2Cl2. Thespectral signature of the pseudogap emerges at the lowestdoping level from aweakly insulating but C4v-symmetricmatrixexhibiting a distinct spectral shape. At slightly higher holedensity, nanoscale regions exhibiting pseudogap spectra and180� rotational (C2v) symmetry form unidirectional clusterswithin the C4v-symmetric matrix. Thus, hole doping proceedsby the appearance of nanoscale clusters of localized holeswithin which the broken-symmetry pseudogap state is stabi-lized. A fundamentally two-component electronic structure11then exists in Ca2� x
Nax
CuO2Cl2 until the C2v-symmetric clus-ters touch at higher doping levels, and the long-range super-conductivity appears.
To visualize at the atomic scale how the pseudogap andsuperconducting states are formed sequentially from the weakinsulator state, we performed spectroscopic imaging scanningtunnelling microscopy (SI-STM) studies on Ca2�x
Nax
CuO2Cl2(0.06 x 0.12; see also the Methods sections). The crystalstructure is simple tetragonal (I4/mmm) and thereby advantageousbecause the CuO2 planes are unbuckled and free from orthorhom-bic distortion. More importantly Ca2CuO2Cl2 can be doped fromthe Mott insulator to the superconductor by introduction of Naatoms. Figure 1c,d shows differential conductance images mea-sured using SI-STM of bulk-insulating x = 0.06 and x = 0.08samples taken in the field of views of the topographic imagesin Fig. 1a,b. The wavy, bright, arcs in Fig. 1c,d have never beenobserved in superconducting samples (x > 0.08) but appear onlyin such quasi-insulating samples (x 0.08). They are created by
1Inorganic Complex Electron Systems Research Team, RIKEN Advanced Science Institute, Wako, Saitama 351-0198, Japan, 2Magnetic MaterialsLaboratory, RIKEN Advanced Science Institute, Wako, Saitama 351-0198, Japan, 3Materials and Structures Lab., Tokyo Institute of Technology, Yokohama,Kanagawa 226-8503, Japan, 4Institute for Integrated Cell-Material Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan, 5LASSP, Department ofPhysics, Cornell University, Ithaca, New York 14853, USA, 6CMPMS Department, Brookhaven National Laboratory, Upton, New York 11973, USA, 7Schoolof Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, UK, 8Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca,New York 14853, USA, 9Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan. *e-mail: [email protected].
spectral peaks in differential conductance spectra whose energy isdependent on location (Fig. 1f). Consequently, the wavy arcs shrinkwith increasing bias voltages and finally disappear. This behaviour,due to tip-induced impurity charging12–14, is characteristic of poorelectronic screening in a weakly insulating state.
A wide variety of spectral shapes originating from electricheterogeneity were found in these samples. A typical example ofthe spectra is, as spectrum number 1 in Fig. 1e, the V-shapedpseudogap (⇠0.2 eV) spectrum with a small dip (⇠20meV) nearthe Fermi energy. This is indistinguishable from those found instrongly underdoped cuprate superconductors3, and establishesthat the pseudogap state appears locally at the nanoscale within theweak insulator. Besides the V-shaped pseudogap spectra in someareas, we find a new class of spectra that is predominant elsewherein the insulating samples. As for example spectrum number 2 inFig. 1e, such spectra are extremely asymmetric about the Fermienergy, U-shaped (concave in minus a few hundred millivolts) andexhibit no clear pseudogap. The growing asymmetry is stronglyindicative of approaching the Mott insulating state15,16 whereasthe non-zero conductance in the unoccupied state is distinctfrom the Mott insulating state17. The approach for spectroscopicexamination of the emergence of the pseudogap from the weakinsulator is therefore transformation from the U-shaped insulatingspectra to the V-shaped pseudogap spectra as a function oflocation and doping.
Figure 2a represents the transformation between these two typesof spectrum. The V-shaped pseudogap becomes larger and broader,and eventually is smoothly connected to the U-shaped insulatingspectra. To quantify this variation, we focus on positive biases wherethe edge of the pseudogap is clear. We fit the following functionto each spectrum18,
f (E)= c0Re
E+ i� (E)p(E+ i� (E))2 ��2
�+ c1E+ c2 (1)
where E is the energy, � is the broadening term, � is the energygap and c
i
(i = 0,1,2) are fitting constants. Use of equation (1)is merely for accurate quantitative parameterization of the gapmaximum and does not imply any particular electronic state.We use � (E) = ↵E as ref. 18 (↵ is a proportional constant) butmomentum-independent� for simplicity of fitting procedures (see
534 NATURE PHYSICS | VOL 8 | JULY 2012 | www.nature.com/naturephysics
Y. Kohsaka, T. Hanaguri, M. Azuma, M. Takano, J. C. Davis, and H. TakagiNature Physics, 8, 534 (2012).
8
6
4
2
0
dI/dV
(a. u
.)
2001000V (mV)
line 1
2001000V (mV)
line 2
120
100
80
∆ (m
V)
Cu O Cu O Cu O Cu O Cu O Cu O Cu O CuLateral position along line 1
O O O O O O O OLateral position along line 2
line 1 line 2
line 1 line 2
O Cu
1 2
Fig. 3
b
e f
2 nm
∆, Ca1.92Na0.08CuO2Cl2 ∆, Ca1.88Na0.12CuO2Cl2
R, Ca1.92Na0.08CuO2Cl2 R, Ca1.88Na0.12CuO2Cl2
0.50
0.80
0.65
300
100
200
(mV)300
100
200
(mV)
a
2 nm
2 nm2 nm 0.50
0.80
0.65
c
d50
040
030
0 500400300
500
400
300 500400300
2 nm
2 nm 0.90.5
1.40.8
XY
XY
a
b
Y XQy Qx
Sy Sx
XYQy Qx
Sy Sx
T ~ 0
T > Tc
c
Cu
Oy
Ox
Evidence for “nematic” order(i.e. breaking of 90� rotation symmetry) in Ca1.88Na0.12CuO2Cl2.
Thursday, January 30, 14
Evidence for “nematic” order(i.e. breaking of 90� rotation symmetry) in Ca1.88Na0.12CuO2Cl2.
LETTERS
PUBLISHED ONLINE: 20 MAY 2012 | DOI: 10.1038/NPHYS2321
Visualization of the emergence of the pseudogapstate and the evolution to superconductivity in alightly hole-doped Mott insulatorY. Kohsaka1*, T. Hanaguri2, M. Azuma3, M. Takano4, J. C. Davis5,6,7,8 and H. Takagi1,2,9
Superconductivity emerges from the cuprate antiferromag-netic Mott state with hole doping. The resulting electronicstructure1 is not understood, although changes in the state ofoxygen atoms seem paramount2–5. Hole doping first destroysthe Mott state, yielding a weak insulator6,7 where electronslocalize only at low temperatures without a full energy gap.At higher doping levels, the ‘pseudogap’, a weakly conductingstate with an anisotropic energy gap and intra-unit-cell break-ing of 90� rotational (C4v) symmetry, appears3,4,8–10. However,a direct visualization of the emergence of these phenomenawith increasing hole density has never been achieved. Here wereport atomic-scale imaging of electronic structure evolutionfrom the weak insulator through the emergence of the pseu-dogap to the superconducting state in Ca2� x
Nax
CuO2Cl2. Thespectral signature of the pseudogap emerges at the lowestdoping level from aweakly insulating but C4v-symmetricmatrixexhibiting a distinct spectral shape. At slightly higher holedensity, nanoscale regions exhibiting pseudogap spectra and180� rotational (C2v) symmetry form unidirectional clusterswithin the C4v-symmetric matrix. Thus, hole doping proceedsby the appearance of nanoscale clusters of localized holeswithin which the broken-symmetry pseudogap state is stabi-lized. A fundamentally two-component electronic structure11then exists in Ca2� x
Nax
CuO2Cl2 until the C2v-symmetric clus-ters touch at higher doping levels, and the long-range super-conductivity appears.
To visualize at the atomic scale how the pseudogap andsuperconducting states are formed sequentially from the weakinsulator state, we performed spectroscopic imaging scanningtunnelling microscopy (SI-STM) studies on Ca2�x
Nax
CuO2Cl2(0.06 x 0.12; see also the Methods sections). The crystalstructure is simple tetragonal (I4/mmm) and thereby advantageousbecause the CuO2 planes are unbuckled and free from orthorhom-bic distortion. More importantly Ca2CuO2Cl2 can be doped fromthe Mott insulator to the superconductor by introduction of Naatoms. Figure 1c,d shows differential conductance images mea-sured using SI-STM of bulk-insulating x = 0.06 and x = 0.08samples taken in the field of views of the topographic imagesin Fig. 1a,b. The wavy, bright, arcs in Fig. 1c,d have never beenobserved in superconducting samples (x > 0.08) but appear onlyin such quasi-insulating samples (x 0.08). They are created by
1Inorganic Complex Electron Systems Research Team, RIKEN Advanced Science Institute, Wako, Saitama 351-0198, Japan, 2Magnetic MaterialsLaboratory, RIKEN Advanced Science Institute, Wako, Saitama 351-0198, Japan, 3Materials and Structures Lab., Tokyo Institute of Technology, Yokohama,Kanagawa 226-8503, Japan, 4Institute for Integrated Cell-Material Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan, 5LASSP, Department ofPhysics, Cornell University, Ithaca, New York 14853, USA, 6CMPMS Department, Brookhaven National Laboratory, Upton, New York 11973, USA, 7Schoolof Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, UK, 8Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca,New York 14853, USA, 9Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan. *e-mail: [email protected].
spectral peaks in differential conductance spectra whose energy isdependent on location (Fig. 1f). Consequently, the wavy arcs shrinkwith increasing bias voltages and finally disappear. This behaviour,due to tip-induced impurity charging12–14, is characteristic of poorelectronic screening in a weakly insulating state.
A wide variety of spectral shapes originating from electricheterogeneity were found in these samples. A typical example ofthe spectra is, as spectrum number 1 in Fig. 1e, the V-shapedpseudogap (⇠0.2 eV) spectrum with a small dip (⇠20meV) nearthe Fermi energy. This is indistinguishable from those found instrongly underdoped cuprate superconductors3, and establishesthat the pseudogap state appears locally at the nanoscale within theweak insulator. Besides the V-shaped pseudogap spectra in someareas, we find a new class of spectra that is predominant elsewherein the insulating samples. As for example spectrum number 2 inFig. 1e, such spectra are extremely asymmetric about the Fermienergy, U-shaped (concave in minus a few hundred millivolts) andexhibit no clear pseudogap. The growing asymmetry is stronglyindicative of approaching the Mott insulating state15,16 whereasthe non-zero conductance in the unoccupied state is distinctfrom the Mott insulating state17. The approach for spectroscopicexamination of the emergence of the pseudogap from the weakinsulator is therefore transformation from the U-shaped insulatingspectra to the V-shaped pseudogap spectra as a function oflocation and doping.
Figure 2a represents the transformation between these two typesof spectrum. The V-shaped pseudogap becomes larger and broader,and eventually is smoothly connected to the U-shaped insulatingspectra. To quantify this variation, we focus on positive biases wherethe edge of the pseudogap is clear. We fit the following functionto each spectrum18,
f (E)= c0Re
E+ i� (E)p(E+ i� (E))2 ��2
�+ c1E+ c2 (1)
where E is the energy, � is the broadening term, � is the energygap and c
i
(i = 0,1,2) are fitting constants. Use of equation (1)is merely for accurate quantitative parameterization of the gapmaximum and does not imply any particular electronic state.We use � (E) = ↵E as ref. 18 (↵ is a proportional constant) butmomentum-independent� for simplicity of fitting procedures (see
534 NATURE PHYSICS | VOL 8 | JULY 2012 | www.nature.com/naturephysics
Y. Kohsaka, T. Hanaguri, M. Azuma, M. Takano, J. C. Davis, and H. TakagiNature Physics, 8, 534 (2012).
O(1)
O(2) O(4)
O(3)
Cu
O’(1)
O’(2) O’(3)
O’(4)
f
g h
e
+
-
0
+
-
0
+
-
0
+
-
0
2 nm 2 nm
2 nm2 nm
Qxx(R), Ca1.88Na0.12CuO2Cl2 Qxy(R), Ca1.88Na0.12CuO2Cl2
Qxy(∆), Ca1.88Na0.12CuO2Cl2Qxx(∆), Ca1.88Na0.12CuO2Cl2
Qxx(R), Ca1.92Na0.08CuO2Cl2 Qxy(R), Ca1.92Na0.08CuO2Cl2
Qxy(∆), Ca1.92Na0.08CuO2Cl2
+
-
0
+
-
0
+
-
0
+
-
0
Fig. 4b
c d
2 nm
Qxx(∆), Ca1.92Na0.08CuO2Cl2a
2 nm
2 nm2 nm
500
400
300 500400300
500
400
300 500400300
2 nm
2 nm 0.90.5
1.40.8
XY
XY
a
b
Y XQy Qx
Sy Sx
XYQy Qx
Sy Sx
T ~ 0
T > Tc
c
Cu
Oy
Ox
Thursday, January 30, 14
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).
Thursday, January 30, 14
Quantum criticality of Ising-nematic ordering in a metal
x
yOccupied states
Empty states
A metal with a Fermi surfacewith full square lattice symmetry
Thursday, January 30, 14
x
y
Spontaneous elongation along y direction:Ising order parameter � < 0.
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
Spontaneous elongation along x direction:Ising order parameter � > 0.
x
yQuantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
Ising-nematic order parameter
� ⇠Z
d2k (cos kx
� cos ky
) c†k�
ck�
Measures spontaneous breaking of square latticepoint-group symmetry of underlying Hamiltonian
Thursday, January 30, 14
Spontaneous elongation along x direction:Ising order parameter � > 0.
x
yQuantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
x
y
Spontaneous elongation along y direction:Ising order parameter � < 0.
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
Pomeranchuk instability as a function of coupling �
��c
��⇥ = 0⇥�⇤ �= 0
or
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
T
Phase diagram as a function of T and �
��⇥ = 0
Quantumcritical
⇥�⇤ �= 0
TI-n
Quantum criticality of Ising-nematic ordering in a metal
��c
Thursday, January 30, 14
T
��⇥ = 0
Quantumcritical
⇥�⇤ �= 0
TI-n Classicald=2 Isingcriticality
Quantum criticality of Ising-nematic ordering in a metal
Phase diagram as a function of T and �
��c
Thursday, January 30, 14
T
��⇥ = 0
Quantumcritical
⇥�⇤ �= 0
rc
TI-n
Quantum criticality of Ising-nematic ordering in a metal
D=2+1 Ising
criticality ?
Phase diagram as a function of T and �
�
Thursday, January 30, 14
T
��⇥ = 0
Quantumcritical
⇥�⇤ �= 0
rc
TI-n
D=2+1 Ising
criticality ?
Quantum criticality of Ising-nematic ordering in a metal
Phase diagram as a function of T and �
�
Thursday, January 30, 14
T
��⇥ = 0
Quantumcritical
⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
Quantum criticality of Ising-nematic ordering in a metal
Phase diagram as a function of T and �
�
Thursday, January 30, 14
T
��⇥ = 0⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
Quantum criticality of Ising-nematic ordering in a metal
Quantumcritical
Phase diagram as a function of T and �
�
Thursday, January 30, 14
T
��⇥ = 0⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
Quantum criticality of Ising-nematic ordering in a metal
Fermiliquid
Fermiliquid
Quantumcritical
Phase diagram as a function of T and �
�
Thursday, January 30, 14
T
��⇥ = 0⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
Quantum criticality of Ising-nematic ordering in a metal
Fermiliquid
Fermiliquid
Phase diagram as a function of T and �
�
Thursday, January 30, 14
T
��⇥ = 0⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
StrangeMetal
Quantum criticality of Ising-nematic ordering in a metal
Fermiliquid
Fermiliquid
Phase diagram as a function of T and �
�
Thursday, January 30, 14
1. The simplest models without quasiparticles
A. Superfluid-insulator transition
of ultracold bosons in an optical lattice
B. Conformal field theories in 2+1 dimensions and
the AdS/CFT correspondence
2. Metals without quasiparticles
A. Review of Fermi liquid theory
B. A “non-Fermi” liquid: the Ising-nematic
quantum critical point
C. Holography, entanglement, and strange metals
Outline
Thursday, January 30, 14
1. The simplest models without quasiparticles
A. Superfluid-insulator transition
of ultracold bosons in an optical lattice
B. Conformal field theories in 2+1 dimensions and
the AdS/CFT correspondence
2. Metals without quasiparticles
A. Review of Fermi liquid theory
B. A “non-Fermi” liquid: the Ising-nematic
quantum critical point
C. Holography, entanglement, and strange metals
Outline
Thursday, January 30, 14
L = f†↵
✓@⌧ � r2
2m� µ
◆f↵
+ u f†↵f
†�f�f↵
The Fermi liquid
�� kF !
Occupied states
Empty states
Thursday, January 30, 14
The Fermi liquid: RG
FIG. 3: Alternative low energy formulation of Fermi liquid theory. We focus on an extended patch
of the Fermi surface, and expand in momenta about the point ~k0 on the Fermi surface. This yields
a theory of d-dimensional fermions in (7), with dispersion (14). The co-ordinate y represents the
d� 1 dimensions parallel to the Fermi surface.
(5) in one-dimensional. One benefit of (7) is now immediately evident: it has zero energy
excitations when
vFkx +
k
2y
2= 0, (8)
and so (8) defines the position of the Fermi surface, which is then part of the low energy
theory including its curvature. Note that (7) now includes an extended portion of the Fermi
surface; contrast that with (5), where the one-dimensional chiral fermion theory for each n̂
describes only a single point on the Fermi surface.
The gradient terms in (7) define a natural momentum space cuto↵, and associated scaling
limit. We will take such a limit at fixed ⇣, vF and . Notice that momenta in the x direction
scale as the square of the momenta in the y direction, and so we can choose v2Fk2x+
2k
4y < ⇤4.
Notice that as we reduce ⇤, we scale towards the single point ~k0 on the Fermi surface, as we
7
• Expand fermion kinetic energy at wavevectors about
~k0, bywriting f↵(~k0 + ~q) = ↵(~q)
L = f†↵
✓@⌧ � r2
2m� µ
◆f↵
+ u f†↵f
†�f�f↵
Thursday, January 30, 14
The Fermi liquid: RG
FIG. 3: Alternative low energy formulation of Fermi liquid theory. We focus on an extended patch
of the Fermi surface, and expand in momenta about the point ~k0 on the Fermi surface. This yields
a theory of d-dimensional fermions in (7), with dispersion (14). The co-ordinate y represents the
d� 1 dimensions parallel to the Fermi surface.
(5) in one-dimensional. One benefit of (7) is now immediately evident: it has zero energy
excitations when
vFkx +
k
2y
2= 0, (8)
and so (8) defines the position of the Fermi surface, which is then part of the low energy
theory including its curvature. Note that (7) now includes an extended portion of the Fermi
surface; contrast that with (5), where the one-dimensional chiral fermion theory for each n̂
describes only a single point on the Fermi surface.
The gradient terms in (7) define a natural momentum space cuto↵, and associated scaling
limit. We will take such a limit at fixed ⇣, vF and . Notice that momenta in the x direction
scale as the square of the momenta in the y direction, and so we can choose v2Fk2x+
2k
4y < ⇤4.
Notice that as we reduce ⇤, we scale towards the single point ~k0 on the Fermi surface, as we
7
L[ ↵
] = †↵
�@⌧
� i@x
� @2y
� ↵
+ u †↵
†�
�
↵
• Expand fermion kinetic energy at wavevectors about
~k0, bywriting f↵(~k0 + ~q) = ↵(~q)
L = f†↵
✓@⌧ � r2
2m� µ
◆f↵
+ u f†↵f
†�f�f↵
Thursday, January 30, 14
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
Thursday, January 30, 14
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
The kinetic energy is invariant under the rescaling x ! x/s,
y ! y/s
1/2, and ⌧ ! ⌧/s
z, provided z = 1 and
! s
(d+1)/4.
Then we find u ! us
(1�d)/2, and so we have the RG flow
du
d`
=
(1� d)
2
u
Interactions are irrelevant in d = 2 !
Thursday, January 30, 14
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
The kinetic energy is invariant under the rescaling x ! x/s,
y ! y/s
1/2, and ⌧ ! ⌧/s
z, provided z = 1 and
! s
(d+1)/4.
Then we find u ! us
(1�d)/2, and so we have the RG flow
du
d`
=
(1� d)
2
u
Interactions are irrelevant in d = 2 !
Thursday, January 30, 14
The fermion Green’s function to order u2has the form (upto logs)
G(~q,!) =A
! � qx
� q2y
+ ic!2
So the quasiparticle pole is sharp. And fermion momentum distribution
function n(~k) =Df†↵
(
~k)f↵
(
~k)Ehad the following form:
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
Thursday, January 30, 14
The Fermi liquid: RG
S[ ↵
] =
Zd
d�1y dx d⌧
h
†↵
�@
⌧
� i@
x
� @
2y
�
↵
+ u
†↵
†�
�
↵
i
n(k)
kkF
A
The fermion Green’s function to order u2has the form (upto logs)
G(~q,!) =A
! � qx
� q2y
+ ic!2
So the quasiparticle pole is sharp. And fermion momentum distribution
function n(~k) =Df†↵
(
~k)f↵
(
~k)Ehad the following form:
Thursday, January 30, 14
• Fermi wavevector obeys the Luttinger relation kdF ⇠ Q, the
fermion density
• Sharp particle and hole of excitations near the Fermi surface
with energy ! ⇠ |q|z, with dynamic exponent z = 1.
• The phase space density of fermions is e↵ectively one-dimensional,
so the entropy density S ⇠ T . It is useful to write this is as S ⇠T (d�✓)/z
, with violation of hyperscaling exponent ✓ = d� 1.
The Fermi liquid
L = f†✓@⌧ � r2
2m� µ
◆f
+ 4 Fermi terms�� kF !
Occupied states
Empty states
Thursday, January 30, 14
• Fermi wavevector obeys the Luttinger relation kdF ⇠ Q, the
fermion density
• Sharp particle and hole of excitations near the Fermi surface
with energy ! ⇠ |q|z, with dynamic exponent z = 1.
• The phase space density of fermions is e↵ectively one-dimensional,
so the entropy density S ⇠ T . It is useful to write this is as S ⇠T (d�✓)/z
, with violation of hyperscaling exponent ✓ = d� 1.
The Fermi liquid
L = f†✓@⌧ � r2
2m� µ
◆f
+ 4 Fermi terms
⇥| q|�
�� kF !
Occupied states
Empty states
Thursday, January 30, 14
L = f†✓@⌧ � r2
2m� µ
◆f
+ 4 Fermi terms
• Fermi wavevector obeys the Luttinger relation kdF ⇠ Q, the
fermion density
• Sharp particle and hole of excitations near the Fermi surface
with energy ! ⇠ |q|z, with dynamic exponent z = 1.
• The phase space density of fermions is e↵ectively one-dimensional,
so the entropy density S ⇠ T . It is useful to write this is as S ⇠T (d�✓)/z
, with violation of hyperscaling exponent ✓ = d� 1.
The Fermi liquid
⇥| q|�
�� kF !
Occupied states
Empty states
Thursday, January 30, 14
| i ) Ground state of entire system,
⇢ = | ih |
⇢A = TrB⇢ = density matrix of region A
Entanglement entropy SE = �Tr (⇢A ln ⇢A)
B
Entanglement entropy
A P
Thursday, January 30, 14
| i ) Ground state of entire system,
⇢ = | ih |
Take | i = 1p2(|"iA |#iB � |#iA |"iB)
Then ⇢A = TrB⇢ = density matrix of region A=
12 (|"iA h"|A + |#iA h#|A)
Entanglement entropy SE = �Tr (⇢A ln ⇢A)= ln 2
Entanglement entropy
Thursday, January 30, 14
Logarithmic violation of “area law”: SE =
1
12
(kFP ) ln(kFP )
for a circular Fermi surface with Fermi momentum kF ,where P is the perimeter of region A with an arbitrary smooth shape.
The prefactor 1/12 is universal: it is independent of the shape of the
entangling region, and of the strength of the interactions.
B
A P
Entanglement entropy of the Fermi liquid
D. Gioev and I. Klich, Physical Review Letters 96, 100503 (2006)B. Swingle, Physical Review Letters 105, 050502 (2010)
Thursday, January 30, 14
Logarithmic violation of “area law”: SE =
1
12
(kFP ) ln(kFP )
for a circular Fermi surface with Fermi momentum kF ,where P is the perimeter of region A with an arbitrary smooth shape.
The prefactor 1/12 is universal: it is independent of the shape of the
entangling region, and of the strength of the interactions.
B
A P
Entanglement entropy of the Fermi liquid
D. Gioev and I. Klich, Physical Review Letters 96, 100503 (2006)B. Swingle, Physical Review Letters 105, 050502 (2010)
Thursday, January 30, 14
Logarithmic violation of “area law”: SE =
1
12
(kFP ) ln(kFP )
for a circular Fermi surface with Fermi momentum kF ,where P is the perimeter of region A with an arbitrary smooth shape.
The prefactor 1/12 is universal: it is independent of the shape of the
entangling region, and of the strength of the interactions.
B
A P
Entanglement entropy of the Fermi liquid
D. Gioev and I. Klich, Physical Review Letters 96, 100503 (2006)B. Swingle, Physical Review Letters 105, 050502 (2010)
Thursday, January 30, 14
Logarithmic violation of “area law”: SE =
1
12
(kFP ) ln(kFP )
for a circular Fermi surface with Fermi momentum kF ,where P is the perimeter of region A with an arbitrary smooth shape.
The prefactor 1/12 is universal: it is independent of the shape of the
entangling region, and of the strength of the interactions.
B
A P
Entanglement entropy of the Fermi liquid
D. Gioev and I. Klich, Physical Review Letters 96, 100503 (2006)B. Swingle, Physical Review Letters 105, 050502 (2010)
Thursday, January 30, 14
⇥| q|�
�� kF !
FL Fermi liquid
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
Thursday, January 30, 14
1. The simplest models without quasiparticles
A. Superfluid-insulator transition
of ultracold bosons in an optical lattice
B. Conformal field theories in 2+1 dimensions and
the AdS/CFT correspondence
2. Metals without quasiparticles
A. Review of Fermi liquid theory
B. A “non-Fermi” liquid: the Ising-nematic
quantum critical point
C. Holography, entanglement, and strange metals
Outline
Thursday, January 30, 14
1. The simplest models without quasiparticles
A. Superfluid-insulator transition
of ultracold bosons in an optical lattice
B. Conformal field theories in 2+1 dimensions and
the AdS/CFT correspondence
2. Metals without quasiparticles
A. Review of Fermi liquid theory
B. A “non-Fermi” liquid: the Ising-nematic
quantum critical point
C. Holography, entanglement, and strange metals
Outline
Thursday, January 30, 14
Pomeranchuk instability as a function of coupling �
��c
��⇥ = 0⇥�⇤ �= 0
or
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
E�ective action for Ising order parameter
S⇥ =⇤
d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� �c)⇤2 + u⇤4
⇥
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
E�ective action for Ising order parameter
S⇥ =⇤
d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� �c)⇤2 + u⇤4
⇥
E�ective action for electrons:
Sc =⌃
d�
Nf⇧
�=1
�
⇤⇧
i
c†i�⇤⇥ ci� �⇧
i<j
tijc†i�ci�
⇥
⌅
⇥Nf⇧
�=1
⇧
k
⌃d�c†k� (⇤⇥ + ⇥k) ck�
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
��⇥ > 0 ��⇥ < 0
Coupling between Ising order and electrons
S�c
= � g
Zd⌧
NfX
↵=1
X
k,q
�q (cos kx� cos ky
)c†k+q/2,↵ck�q/2,↵
for spatially dependent �
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
S�c
= � g
Zd⌧
NfX
↵=1
X
k,q
�q (cos kx� cos ky
)c†k+q/2,↵ck�q/2,↵
S⇥ =⇤
d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� �c)⇤2 + u⇤4
⇥
Sc =Nf�
�=1
�
k
⇥d�c†k� (⇤⇥ + ⇥k) ck�
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
• � fluctuation at wavevector ~q couples most e�ciently to fermions
near ±~k0.
• Expand fermion kinetic energy at wavevectors about ±~k0 andboson (�) kinetic energy about ~q = 0.
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
• � fluctuation at wavevector ~q couples most e�ciently to fermions
near ±~k0.
• Expand fermion kinetic energy at wavevectors about ±~k0 andboson (�) kinetic energy about ~q = 0.
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
L[ ±,�] =
†+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
��⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
(a)
(b)Landau-damping
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
One loop � self-energy with Nf
fermion flavors:
⌃�
(~q,!) = Nf
Zd2k
4⇡2
d⌦
2⇡
1
[�i(⌦+ !) + kx
+ qx
+ (ky
+ qy
)2]⇥�i⌦� k
x
+ k2y
⇤
=N
f
4⇡
|!||q
y
|
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
(a)
(b)Electron self-energy at order 1/N
f
:
⌃(
~k,⌦) = � 1
Nf
Zd2q
4⇡2
d!
2⇡
1
[�i(! + ⌦) + kx
+ qx
+ (ky
+ qy
)
2]
"q2y
g2+
|!||q
y
|
#
= �i2p3N
f
✓g2
4⇡
◆2/3
sgn(⌦)|⌦|2/3
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
(a)
(b)Electron self-energy at order 1/N
f
:
⌃(
~k,⌦) = � 1
Nf
Zd2q
4⇡2
d!
2⇡
1
[�i(! + ⌦) + kx
+ qx
+ (ky
+ qy
)
2]
"q2y
g2+
|!||q
y
|
#
= �i2p3N
f
✓g2
4⇡
◆2/3
sgn(⌦)|⌦|2/3
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
⇠ |⌦|d/3 in dimension d.
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
Schematic form of � and fermion Green’s functions in d dimensions
D(~q,!) =1/N
f
q2? +|!||q?|
, Gf
(~q,!) =1
qx
+ q2? � isgn(!)|!|d/3/Nf
In the boson case, q2? ⇠ !1/zb with zb
= 3/2.In the fermion case, q
x
⇠ q2? ⇠ !1/zf with zf
= 3/d.
Note zf
< zb
for d > 2 ) Fermions have higher energy thanbosons, and perturbation theory in g is OK.Strongly-coupled theory in d = 2.
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
Schematic form of � and fermion Green’s functions in d = 2
D(~q,!) =1/N
f
q2y
+
|!||q
y
|
, Gf
(~q,!) =1
qx
+ q2y
� isgn(!)|!|2/3/Nf
In both cases qx
⇠ q2y
⇠ !1/z, with z = 3/2. Note that the
bare term ⇠ ! in G�1f
is irrelevant.
Strongly-coupled theory without quasiparticles.
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
Simple scaling argument for z = 3/2.
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
Simple scaling argument for z = 3/2.
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
X X
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
Simple scaling argument for z = 3/2.
L = †+
�@⌧
� i@x
� @2y
� + + †
��@⌧
+ i@x
� @2y
� �
� �⇣ †+ + + †
� �
⌘+
1
2g2(@
y
�)2
X X
Under the rescaling x ! x/s, y ! y/s
1/2, and ⌧ ! ⌧/s
z, we
find invariance provided
� ! � s
! s
(2z+1)/4
g ! g s
(3�2z)/4
So the action is invariant provided z = 3/2.
Quantum criticality of Ising-nematic ordering in a metal
Thursday, January 30, 14
⇥| q|�
�� kF !
FL Fermi liquid
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
Thursday, January 30, 14
• Fermi surface
with kdF ⇠ Q.
• Di↵use fermionic
excitations with z = 3/2to three loops.
• S ⇠ T (d�✓)/z
with ✓ = d� 1.
• SE ⇠ kd�1F P lnP .
⇥| q|�
�� kF ! �� kF !
NFLNematic
QCP
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
FL Fermi liquid
n(k)
kkF
Thursday, January 30, 14
• Fermi surface
with kdF ⇠ Q.
• Di↵use fermionic
excitations with z = 3/2to three loops.
• S ⇠ T (d�✓)/z
with ✓ = d� 1.
• SE ⇠ kd�1F P lnP .
M. A. Metlitski and S. Sachdev,Phys. Rev. B 82, 075127 (2010)
⇥| q|�
�� kF ! �� kF !⇥| q
|�
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
FL Fermi liquid
NFLNematic
QCP
Thursday, January 30, 14
• Fermi surface
with kdF ⇠ Q.
• Di↵use fermionic
excitations with z = 3/2to three loops.
• S ⇠ T (d�✓)/z
with ✓ = d� 1.
• SE ⇠ kd�1F P lnP .
⇥| q|�
�� kF !⇥| q
|�
�� kF !
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
FL Fermi liquid
NFLNematic
QCP
Thursday, January 30, 14
• Fermi surface
with kdF ⇠ Q.
• Di↵use fermionic
excitations with z = 3/2to three loops.
• S ⇠ T (d�✓)/z
with ✓ = d� 1.
• SE ⇠ kd�1F P lnP .
⇥| q|�
�� kF !⇥| q
|�
�� kF !
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
FL Fermi liquid
NFLNematic
QCP
Thursday, January 30, 14
Logarithmic violation of “area law”: SE = CE kFP ln(kFP )
for a circular Fermi surface with Fermi momentum kF ,where P is the perimeter of region A with an arbitrary smooth shape.
The prefactor CE is expected to be universal but 6= 1/12:independent of the shape of the entangling region, and dependent
only on IR features of the theory.
B
A P
Entanglement entropy of the non-Fermi liquid
B. Swingle, Physical Review Letters 105, 050502 (2010)Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)
Thursday, January 30, 14
Logarithmic violation of “area law”: SE = CE kFP ln(kFP )
for a circular Fermi surface with Fermi momentum kF ,where P is the perimeter of region A with an arbitrary smooth shape.
The prefactor CE is expected to be universal but 6= 1/12:independent of the shape of the entangling region, and dependent
only on IR features of the theory.
Entanglement entropy of the non-Fermi liquid
B. Swingle, Physical Review Letters 105, 050502 (2010)Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)
B
A P
Thursday, January 30, 14
Logarithmic violation of “area law”: SE = CE kFP ln(kFP )
for a circular Fermi surface with Fermi momentum kF ,where P is the perimeter of region A with an arbitrary smooth shape.
The prefactor CE is expected to be universal but 6= 1/12:independent of the shape of the entangling region, and dependent
only on IR features of the theory.
B
A P
Entanglement entropy of the non-Fermi liquid
B. Swingle, Physical Review Letters 105, 050502 (2010)Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)
Thursday, January 30, 14
⇥| q|�
�� kF !⇥| q
|�
�� kF !
• kdF ⇥ Q, the fermion density
• Sharp fermionic excitations
near Fermi surface with
⇥ ⇥ |q|z, and z = 1.
• Entropy density S ⇥ T (d��)/z
with violation of hyperscaling
exponent � = d� 1.
• Entanglement entropy
SE ⇥ kd�1F P lnP .
FL Fermi liquid
NFLNematic
QCP
• Fermi surface
with kdF ⇠ Q.
• Di↵use fermionic
excitations with z = 3/2to three loops.
• S ⇠ T (d�✓)/z
with ✓ = d� 1.
• SE ⇠ kd�1F P lnP .
Thursday, January 30, 14
T
��⇥ = 0⇥�⇤ �= 0
rc
TI-n
Strongly-coupled“non-Fermi liquid”
metal with no quasiparticles
StrangeMetal
Quantum criticality of Ising-nematic ordering in a metal
Fermiliquid
Fermiliquid
Phase diagram as a function of T and �
�
Thursday, January 30, 14
TSDW Tc
T0
2.0
0
α"
1.0 SDW
Superconductivity
BaFe2(As1-xPx)2
Resistivity⇠ ⇢0 +AT↵
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
no quasiparticles,Landau-Boltzmann theory
does not apply
AF+nematic
Thursday, January 30, 14