metals without quasiparticles - harvard...

Metals without quasiparticles

A. Review of Fermi liquid theory

B. A “non-Fermi” liquid: the Ising-nematic

quantum critical point

C. Fermi surfaces and gauge fields

L = f†↵

✓@⌧ � r2

2m� µ

◆f↵

+ u f†↵f

†�f�f↵

The Fermi liquid

�� kF !

Occupied states

Empty states

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↵


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↵


S[ ↵

] =

Zd

d�1y dx d⌧

h

†↵

�@

⌧

� i@

x

� @

2y

�

↵

+ u

†↵

†�

�

↵

i


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 !

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:


S[ ↵

] =

Zd

d�1y dx d⌧

h

†↵

�@

⌧

� i@

x

� @

2y

�

↵

+ u

†↵

†�

�

↵

i


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:

• 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


fermion density






The Fermi liquid

L = f†✓@⌧ � r2

2m� µ

◆f

+ 4 Fermi terms

⇥| q|�

�� kF !

Occupied states

Empty states

L = f†✓@⌧ � r2

2m� µ

◆f

+ 4 Fermi terms


fermion density






The Fermi liquid

⇥| q|�

�� kF !

Occupied states

Empty states

| 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

| 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

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)

⇥| 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 .

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

YBa2Cu3O6+x

High temperature superconductors

� �

Pseudogap

� �

Strange Metal

Pseudogap

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.

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

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:+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:+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:+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:+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:+Hardy_W/0/1/0/all/0/1

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

Quantum criticality of Ising-nematic ordering in a metal

x

yOccupied states

Empty states

A metal with a Fermi surfacewith full square lattice symmetry

x

y

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


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

x

yQuantum criticality of Ising-nematic ordering in a metal

Ising-nematic order parameter

� ⇠Z

d2k (cos kx

� cos ky

) c†k�

ck�

Measures spontaneous breaking of square latticepoint-group symmetry of underlying Hamiltonian

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

x

yQuantum criticality of Ising-nematic ordering in a metal

x

y

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


Pomeranchuk instability as a function of coupling �

��c

��⇥ = 0⇥�⇤ �= 0

or


T

Phase diagram as a function of T and �

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

TI-n


��c

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

TI-n Classicald=2 Isingcriticality



��c

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n


D=2+1 Ising

criticality ?


�

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n

D=2+1 Ising

criticality ?



�

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n

Strongly-coupled“non-Fermi liquid”

metal with no quasiparticles



�

T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n




Fermiliquid

Fermiliquid

Quantum critical


�

T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n



StrangeMetal


Fermiliquid

Fermiliquid


�

Pomeranchuk instability as a function of coupling �

��c

��⇥ = 0⇥�⇤ �= 0

or


E�ective action for Ising order parameter

S⇥ =⇤

d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� c)⇤2 + u⇤4

⇥


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�


��⇥ > 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 �


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�



• � fluctuation at wavevector ~q couples most e�ciently to fermions

near ±~k0, where ~q is tangent to the Fermi surface, and so

the initial and final fermion states, after scattering of �, arenearly degenerate.

• Expand fermion kinetic energy at wavevectors about ±~k0 andboson (�) kinetic energy about ~q = 0.

L[ ±,�] =

†+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

��⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)


(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

|


Computation of Landau dampingWe will only be interested in terms in ⌃

�

that are singular in q and !n

, and will drop

regular contributions from regions of high momentum and frequency. In this case, it

is permissible to reverse the conventional order of integrating over frequency first, and

to first integrate over kx

. It is a simple matter to perform the integration over kx

in

using the method residues to yield (working temporarily with d� 1 spatial dimensions

along the y direction)

⌃

�

(~q,!n

) =

Nf

2vF

Zdd�1k

y

(2⇡)d�1

Zd⌦

n

2⇡

sgn(⌦

n

+ !n

)� sgn(⌦

n

)⇣!n

+ ivF

qx

+ iq2y

/2 + i~qy

· ~ky

⌘

=

Nf

|!n

|2⇡v

F

Zdd�1k

y

(2⇡)d�1

1⇣!n

+ ivF

qx

+ iq2y

/2 + i~qy

· ~ky

⌘ .

We now integrate along the component of

~ky

parallel to the direction of ~qy

to obtain

⌃

�

(~q,!n

) =

Nf

|!n

|2⇡v

F

|qy

|

Zdd�2k

y

(2⇡)d�2

=

Nf

|!n

|2⇡v

F

|qy

|⇤d�2

Note that in d = 2 the last non-universal factor is not present, and the result for ⌃

�

is

universal with ⇤

d�2= 1. Also, in our action v

F

= 1 and = 2.

(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.


Computation of fermion self energy

The fermion self energy can be evaluated by methods similar to those used for the

computation of Landau damping. Integrating over qx

, with d � 1 spatial dimensions

along the y direction, we find (with � = 1/(2⇡vF

))

⌃(k,⌦n

) = i1

vF

Zdd�1q

y

(2⇡)d�1

Zd!

n

2⇡

sgn(!n

+ ⌦

n

)|qy

||q

y

|3 + �|!n

|

= i1

⇡vF

�sgn(⌦

n

)

Zdd�1q

y

(2⇡)d�1|q

y

| ln✓|q

y

|3 + �|⌦n

||q

y

|3

◆.

Evaluation of the qy

integral yields ⌃ ⇠ sgn(⌦)|⌦|d/3 near d = 2. In the physically

important case of d = 2, the qy

integral evaluates to

⌃(k,⌦n

) =

1

⇡vF

�1/3p3

sgn(⌦

n

)|⌦n

|2/3 , d = 2.

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.


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.


Simple scaling argument for z = 3/2.

L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2



L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

X X



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.


⇥| q|�

�� kF !

FL Fermi liquid




⇥ ⇥ |q|z, and z = 1.






• 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




⇥ ⇥ |q|z, and z = 1.






FL Fermi liquid

n(k)

kkF

• Fermi surface

with kdF ⇠ Q.



• S ⇠ T (d�✓)/z

with ✓ = d� 1.


M. A. Metlitski and S. Sachdev,Phys. Rev. B 82, 075127 (2010)

⇥| q|�

�� kF ! �� kF !⇥| q

|�




⇥ ⇥ |q|z, and z = 1.






FL Fermi liquid

NFLNematic

QCP

• Fermi surface

with kdF ⇠ Q.



• S ⇠ T (d�✓)/z

with ✓ = d� 1.


⇥| q|�

�� kF !⇥| q

|�

�� kF !




⇥ ⇥ |q|z, and z = 1.






FL Fermi liquid

NFLNematic

QCP

Logarithmic violation of “area law”: SE = CE kFP ln(kFP )


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)







B

A P

⇥| q|�

�� kF !⇥| q

|�

�� kF !




⇥ ⇥ |q|z, and z = 1.






FL Fermi liquid

NFLNematic

QCP

• Fermi surface

with kdF ⇠ Q.



• S ⇠ T (d�✓)/z

with ✓ = d� 1.


T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n



StrangeMetal


Fermiliquid

Fermiliquid


�

A Fermi surface of ck↵ fermions coupled to a gauge field aµ =

(a⌧ ,a):

Sc =

NfX

↵=1

X

k

Zd⌧c†k↵ (@⌧ � ia⌧ + "k�a) ck↵

Sa =

Zddrd⌧

1

2g2(✏µ⌫�@⌫a�)

2

�

The a⌧ component couples to the fermion density, and the den-

sity fluctuations screen the long-range “Coulomb” interactions. So

we are only in interested in the transverse component of the gauge

field, which couples to the fermion current. We work in the Coulomb

gauge q · a(q) = 0.

Fermi surfaces and gauge fields

• ~a fluctuation at wavevector ~q couples most e�ciently to fermions near

±~k0, where ~q is tangent to the Fermi surface, and so the initial and

final fermion states, after scattering of �, are nearly degenerate. In the

Coulomb gauge, we can write ~a = (�, 0), with x, y axes chosen as above.

• Expand fermion kinetic energy at wavevectors about ±~k0 and boson (�)

kinetic energy about ~q = 0.



L[ ±,�] =

†+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

��⇣ †+ + � †

� �

⌘+

1

2g2(@

y

�)2




L[ ±,�] =

†+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

��⇣ †+ + � †

� �

⌘+

1

2g2(@

y

�)2

Only difference from the nematic case is the

change in this sign !All previous analysis of

the breakdown of quasiparticles applies

here too.

metals without quasiparticles - harvard...

Documents