Quantum Criticality of Metals Near the Pomeranchuk Instability

A non-Fermi liquid: Quantum criticality of metals near the Pomeranchuk instability

Subir Sachdev

sachdev.physics.harvard.edu HARVARD Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering y

Fermi surface with full square lattice symmetry

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering y

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

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering y

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

Tuesday, January 8, 13 Ising-nematic order parameter

2 d k (cos k cos k ) c† c ⇠ x y k k Z Measures spontaneous breaking of square lattice point-group symmetry of underlying Hamiltonian

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

or =0 =0 ⇥ ⇥ ⇤ r rc

Pomeranchuk instability as a function of coupling r

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering T Quantum critical TI-n

=0 =0 ⇥ ⇤ ⇥ r rc

Phase diagram as a function of T and r

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering T Quantum critical TI-n Classical d=2 Ising criticality

=0 =0 ⇥ ⇤ ⇥ r rc

Phase diagram as a function of T and r

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering T Quantum critical TI-n Classical d=2 Ising criticality

=0 =0 ⇥ ⇤ D=2+1 ⇥ Ising r criticalityrc ? Phase diagram as a function of T and r

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering T Quantum critical TI-n Classical d=2 Ising criticality

=0 =0 ⇥ ⇤ D=2+1 ⇥ Ising r criticalityrc ? Phase diagram as a function of T and r

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering T T ⇤ ? Quantum critical TI-n

=0 =0 ⇥ ⇤ ⇥ r rc

Phase diagram as a function of T and r

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering T T ⇤ ? StrangeQuantum Metalcritical ? TI-n

=0 =0 ⇥ ⇤ ⇥ r rc

Phase diagram as a function of T and r

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

Eective action for Ising order parameter

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

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

Eective action for Ising order parameter

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

Eective action for electrons:

Nf = d c† ⇤ c t c† c Sc i ⇥ i ij i i⇥ =1 i i

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering Coupling between Ising order and electrons

c = g d⌧ q (cos kx cos ky)c† ck q/2,↵ S k+q/2,↵ ↵=1 Z X Xk,q for spatially dependent

> 0 < 0 ⇥ ⇥

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

= d2rd⇥ (⌅ ⇤)2 + c2( ⇤)2 +( )⇤2 + u⇤4 S⇥ ⇤ c ⇤ ⇥ Nf = dc† (⇤ + ⇥ ) c Sc k ⇥ k k =1 k ⇥ Nf

c = g d⌧ q (cos kx cos ky)c† ck q/2,↵ S k+q/2,↵ ↵=1 Z X Xk,q

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

ﬂuctuation at wavevector ~q couples most eciently to fermions • near ~k . ± 0 Expand fermion kinetic energy at wavevectors about ~k0 and • boson () kinetic energy about ~q = 0. ±

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

ﬂuctuation at wavevector ~q couples most eciently to fermions • near ~k . ± 0 Expand fermion kinetic energy at wavevectors about ~k0 and • boson () kinetic energy about ~q = 0. ±

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

[ ,]= L ± 2 2 +† @⌧ i@x @y + + † @⌧ + i@x @y 1 2 +† + + † + (@y) 2g2

⇣ M. A. Metlitski and⌘ S. Sachdev, Phys. Rev. B 82, 075127 (2010) Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

2 2 = +† @⌧ i@x @y + + † @⌧ + i@x @y L 1 2 +† + + † + (@y) 2g2 ⇣ ⌘

One loop self-energy with Nf fermion ﬂavors: (a) d2k d⌦ 1 ⌃(~q, !)=Nf 4⇡2 2⇡ [ i(⌦ + !)+k + q +(k + q )2] i⌦ k + k2 Z x x y y x y N ! = f | | ⇥ ⇤ 4⇡ qy Landau-damping | | (b)

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

2 2 = +† @⌧ i@x @y + + † @⌧ + i@x @y L 1 2 +† + + † + (@y) (a) 2g2 ⇣ ⌘

(b) Electron self-energy at order 1/Nf :

2 2 = +† @⌧ i@x @y + + † @⌧ + i@x @y L 1 2 +† + + † + (@y) (a) 2g2 ⇣ ⌘

(b) Electron self-energy at order 1/Nf :

2 2 = +† @⌧ i@x @y + + † @⌧ + i@x @y L 1 2 +† + + † + (@y) 2g2 ⇣ ⌘ Schematic form of and fermion Green’s functions in d dimensions

1/Nf 1 D(~q, !)= ,Gf (~q, !)= 2 d/3 2 ! qx + q isgn(!) ! /Nf q + | | ? | | ? q | ?|

2 1/zb In the boson case, q ! with zb =3/2. ? ⇠ 2 1/zf In the fermion case, qx q ! with zf =3/d. ⇠ ? ⇠

Note zf 2 Fermions have higher energy than bosons, and perturbation theory) in g is OK. Strongly-coupled theory in d = 2.

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

2 2 = +† @⌧ i@x @y + + † @⌧ + i@x @y L 1 2 +† + + † + (@y) 2g2 ⇣ ⌘ Schematic form of and fermion Green’s functions in d =2

1/N 1 D(~q, !)= f ,G(~q, !)= ! f q + q2 isgn(!) ! 2/3/N q2 + | | x y | | f y q | y| 2 1/z In both cases qx qy ! ,withz =3/2. Note that the ⇠ 1 ⇠ bare term ! in G is irrelevant. ⇠ f Strongly-coupled theory without quasiparticles.

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

2 2 = +† @⌧ i@x @y + + † @⌧ + i@x @y L 1 2 +† + + † + (@y) 2g2 ⇣ ⌘ Simple scaling argument for z =3/2.

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

2 2 scaling = +† i@x @y + + † +i@x @y L 2 g +† + † +( @y) ⇣ ⌘ Simple scaling argument for z =3/2.

Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering

2 2 scaling = +† i@x @y + + † +i@x @y L 2 g +† + † +( @y) ⇣ ⌘ Simple scaling argument for z =3/2.

Under the rescaling x x/s, y y/s1/2, and ⌧ ⌧/sz,we ﬁnd invariance provided! ! !

s(2z+1)/4 ! s(2z+1)/4 ! (3 2z)/4 g gs ! So the action is invariant provided z =3/2.

Tuesday, January 8, 13 Quantum critical metals near the onset of antiferromagnetism: superconductivity and other instabilities

Subir Sachdev

sachdev.physics.harvard.edu HARVARD Tuesday, January 8, 13 Max Metlitski Erez Berg

HARVARD Tuesday, January 8, 13 Resistivity BaFe2(As1 x Px)2 ⇢ + AT n ⇠ 0

K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).

Tuesday, January 8, 13 BaFe2(As1 x Px)2

Tuesday, January 8, 13 2 Lower Tc superconductivity in the heavy fermion compounds

Generally, the ground state of a Ce heavy-fermion system is determined by the competition of the indirect Ru- derman Kittel Kasuya Yosida (RKKY) interaction which provokes magnetic order of localized moments mediated by the light conduction electrons and the Kondo interaction. This last local mechanism causes a paramagnetic ground state due the screening of the local moment of the Ce ion by the conduction electrons. Both interactions depend critically on the hybridization of the 4f electrons with the conduction electrons. High pressure is an ideal tool to tune the hybridization and the position of the 4f level with respect to the Fermi level. Therefore high pressure experiments are ideal to study the critical region where both interactions are of the same order G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223. and compete. To understand the quantum phase transi- 5 TusonFIG. Park, 2.F. Ronning, Pressure–temperature H. Q. Yuan, M. B. Salamon, phase R. diagram Movshovich, of CeRhIn at tion from the antiferromagnetic (AF) state to the para-J. L. Sarrao,zero and magnetic J. D. Thompson, field determined Nature 440, from65 (2006) specific heat measure- magnetic (PM) state is actually one of the fundamen- ments with antiferromagnetic (AF, blue) and superconduct- Tuesday, January 8, 13 tal questions in solid state physics. Different theoretical ing phases (SC, yellow). When Tc TN the antiferromagnetic order is transition such as spin-fluctuation theory of an itinerant abruptly suppressed. The blue square indicate the transition magnet [10–12], or a new so-called ’local’ quantum criti- from SC to AF+SC after Ref. 16. cal scenario [13, 14]. Another efficient source to prevent long range antiferromagnetic order is given by the valence fluctuations between the trivalent and the tetrava- ature shows a smooth maximum around 0.8 GPa and lent configuration of the cerium ions [15]. is monotonously suppressed for higher pressures. How- The interesting point is that in these strongly corre- ever, CeRhIn5 is also a superconductor in a large pres- lated electron systems the same electrons (or renormal- sure region from about 1.3 to 5 GPa. It has been shown ized quasiparticles) are responsible for both, magnetism that when the superconducting transition temperature and superconductivity. The above mentioned Ce-115 Tc >TN the antiferromagnetic order is rapidly sup- family is an ideal model system, as it allows to study pressed (see figure 2) and vanishes at a lower pressure both, the quantum critical behavior and the interplay of than that expected from a linear extrapolation to T =0. the magnetic order with a superconducting state. Espe- Thus the pressure where Tc = TN defines a first critical ! ! cially, as we will be shown below, unexpected observa- pressure pc and clearly just above pc anitferromagnetism tions will be found, if a magnetic field is applied in the collapses. The intuitive picture is that the opening of a critical pressure region. superconducting gap on large parts of the Fermi surface ! above pc impedes the formation of long range magnetic order. A coexisting phase AF+SC in zero magnetic field PRESSURE-TEMPERATURE PHASE DIAGRAM seems only be formed if on cooling first the magnetic order is established. We will discuss below the microscopic In this article we concentrate on the compound evidence of an homogeneous AF+SC phase. CeRhIn5.AtambientpressuretheRKKYinteraction At ambient pressure CeRhIn5 orders in an incommen- is dominant in CeRhIn5 and magnetic order appears at surate magnetic structure [21] with an ordering vector TN =3.8K.However,theorderedmagneticmomentof of qic=(0.5, 0.5, δ)andδ =0.297 that is a magnetic µ =0.59µB at 1.9 K is reduced of about 30% in com- structure with a different periodicity than the one of the parison to that of Ce ion in a crystal field doublet with- lattice. Generally, an incommensurate magnetic struc- out Kondo effect [17]. Compared to other heavy fermion ture is not favorable for superconductivity with d wave ! compounds at p =0theenhancementoftheSommerfeld symmetry, which is realized in CeRhIn5 above pc [22]. coefficient of the specific heat (γ =52mJmol−1K−2)[18] Neutron scattering experiments under high pressure do and the cylotron masses of electrons on the extremal or- not give conclusive evidence of the structure under pres- bits of the Fermi surface is rather moderate [19, 20]. The sures up to 1.7 GPa which is the highest pressure studied topologies of the Fermi surfaces of CeRhIn5 are cylin- up to now [23–25]. The result is that at 1.7 GPa the in- drical and almost identical to that of LaRhIn5 which is commensurability has changed to δ ≈ 0.4. The main dif- the non 4f isostructural reference compound. From this ficulty in these experiments with large sample volume is ! it can be concluded that the 4f electrons in CeRhIn5 to ensure the pressure homogeneity. Near pc the control are localized and do not contribute to the Fermi volume of a perfect hydrostaticity is a key issue as the material [19, 20]. reacts quite opposite on uniaxial strain applied along the By application of pressure, the system can be tuned c and a axis. through a quantum phase transition. The Néel temper- From recent nuclear quadrupol resonance (NQR) data

YBCO a 150

100 ( T )

YBCO a c2

150 H 50

100 ( T )

p1 p2 c2

H 0 0.0 0.1 0.2 0.3 Hole doping, p 50 100 YBCO b p1 p2 0 H = 0 0.0 0.1 0.2 0.3 75 15 T Hole doping, p 30 T

100

YBCO b ( K ) 50 c HHole-doped = 0 cuprates T 75 15 T 30 T 25 G. Grissonnanche et al., preprint

( K ) 50 c

T 0 0.0 0.1 0.2 0.3 25 Hole doping, p Tuesday, January 8, 13

0 0.0 0.1 0.2 0.3 Hole doping, p Tuesday, January 8, 13 Temperature ( K ) ρa ( !Ω cm ) 150 100 100 200 300 400 50 0 0 0.00 order Spin 0.06 G. Grissonnanche etal., preprint Charge order 0.12 T Hole doping max T 0 0.18 0.24 YBCO T c b a 0.30

100 0 25 50 75

c c ( % ) % ( T ∆ / T / Figure 4: Fermi surface reconstruction from order that is biaxial within each bilayer and stag- gered perpendicular to the bilayers. (A) A schematic of biaxial order yielding inequivalent sites (depicted by symbols of varying color and size) within the bilayers (considering an example where δ1 δ2 0.25) resulting in a body-centered tetragonal superstructure. (B) Reconstruc- ≈ ≈ tion of the Brillouin zone, with one instance of the pocket location indicated at the ` X' point in relation to the original Fermi surface (purple) and nodes in the superconducting wave function. (C) A three-dimensional rendition ofS. Sebastian the concentrically et al., preprint arranged Fermi surfaces resulting from bilayer splitting exhibiting a twofold screw warping.

Tuesday, January 8, 13 of the form we experimentallyFigure 1: observe Fermi surface occurs warping at the in X relation points to ofthe the crystal new structure. Brillouin (A and zone,C) Primitive as and body-centred tetragonal crystal structures. (B and D) Examples of fundamental and two-fold demonstrated in Fig. 1D.screw The Fermi warped surface Fermi surface pockets cylinders must re therefore ecting the be Brillouin located zone at the symmetry X point corresponding of to A and C respectively. In (D), two adjoining Brillouin zones are shown to illustrate the origin of the new body-centered tetragonalthe twofold Brillouin screw warping zone. for In a Fermiorder surface to locate section the centred Fermi at surface the X point. pockets in the original Brillouin zone,conductors we note [17, that 18, the 19] X (see point Fig. of 1B)). the Materials new reduced with a Brillouin body-centered zone tetragonal coincides structure (see with the nodes in the superconductingFig. 1C), by contrast, wavefunction exhibit a unique in the twofold original screw Brillouin symmetry zone at the (shown Brillouin by zone corner dotted lines). We are thus` X' able (see to Fig. locate 1D). the Examples reconstructed of layered Fermi materials surface with Fermi pocket surface as originating warpings exhibiting this symmetry include the ruthenates [7], the overdoped Tl-based cuprates [9] and pnictide super- from the nodal regions of the original Brillouin zone. For example, the biaxial reconstruction conductors of the ` 122' composition [8]. scheme discussed in Ref. [16] would yield the nodal Fermi surface shown in Fig. 4B. The stag- The symmetry of the Fermi surface warping is determined by tting to the angular-dependence gered unidirectional modelof the discussed quantum in oscillation Ref. [35] frequencies would also and amplitudes result in a in body-centred tilted magnetic tetragonal elds [7, 9]. In the case transformation, the Fermi surface resulting from such a model would be of interest to explore. 3 Our nding of a twofold screw symmetry rules out an origin of the pockets from non-

10 Outline

1. Weak coupling theory of SDW ordering, and d-wave superconductivity

2. Universal critical theory of SDW ordering

3. Emergent pseudospin symmetry, and quadrupolar density wave

4. Quantum Monte Carlo without the sign problem

Tuesday, January 8, 13 Outline

1. Weak coupling theory of SDW ordering, and d-wave superconductivity

2. Universal critical theory of SDW ordering

3. Emergent pseudospin symmetry, and quadrupolar density wave

4. Quantum Monte Carlo without the sign problem

Tuesday, January 8, 13 The Hubbard Model

1 1 H = tijci†↵cj↵ + U ni ni µ ci†↵ci↵ " 2 # 2 i

ni↵ = ci†↵ci↵

ci†↵cj + cjci†↵ = ij↵ ci↵cj + cjci↵ =0

Tuesday, January 8, 13 Fermi surface+antiferromagnetism

Metal with “large” Fermi surface

The electron spin polarization obeys

iK r S⌃(r, ) = ⌃⇥ (r, )e · ⇥ where K is the ordering wavevector.

Tuesday, January 8, 13 The Hubbard Model Decouple U term by a Hubbard-Stratanovich transformation

= d2rd⌧ [ + + ] S Lc L' Lc' Z = c† "( i )c Lc a r a

1 r u 2 = ( ' )2 + '2 + '2 L' 2 r ↵ 2 ↵ 4 ↵ iK r ↵ = ' e · c† c . Lc' ↵ a ab b “Yukawa” coupling between fermions and antiferromagnetic order: 2 U, the Hubbard repulsion ⇠ Tuesday, January 8, 13 Fermi surface+antiferromagnetism Increasing SDW order

~' =0 ~' =0 h i6 h i Metal with electron Metal with “large” and hole pockets Fermi surface Increasing interaction

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

T* QuantumStrange Fluctuating CriticalMetal Fermi Large pockets Fermi surface

Increasing SDW order

Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = xm Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

T* QuantumStrange Fluctuating CriticalMetal Fermi Large pockets Fermi surface

Increasing SDW order

Spin density wave (SDW)

Relaxation and equilibration times ~/kBT are robust properties of strongly-coupled quantum⇠ criticality

Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

T* QuantumStrangeStrange Fluctuating MetalCriticalMetal ? Fermi Large pockets Fermi surface

Increasing SDW order

Spin density wave (SDW)

Relaxation and equilibration times ~/kBT are robust properties of strongly-coupled quantum⇠ criticality

Tuesday, January 8, 13 Pairing by SDW ﬂuctuation exchange

We now allow the SDW ﬁeld ⌦⌅ to be dynamical, coupling to electrons as H = ⌦⌅ c† ⌦⇥ c . sdw q · k, ⇥ k+K+q,⇥ k,q,,⇥ Exchange of a ⌦⌅ quantum leads to the eective interaction 1 Hee = V⇥,⇤⌅(q)c† ck+q,⇥cp† ,⇤ cp q,⌅, 2 k, q p,⇤,⌅ k,,⇥ where the pairing interaction is ⇤ V (q)=⌦⇥ ⌦⇥ 0 , ⇥,⇤⌅ ⇥ · ⇤⌅ 2 +(q K)2 2 with ⇤0 the SDW susceptibility and the SDW correlation length.

Tuesday, January 8, 13 BCS Gap equation

In BCS theory, this interaction leads to the ‘gap equation’ for the pairing gap k ck c k . ⇤⌅ ⇥ ⇤⇧

3⇥0 p k = 2 2 +(p k K) 2 2 p 2 ⇤ + ⇤ ⇥ p p ⌅ Non-zero solutions of this equation require that and have opposite signs when p k K. k p ⇥

Tuesday, January 8, 13 Pairing “glue” from antiferromagnetic ﬂuctuations

V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010) Tuesday, January 8, 13 ck† c† k⇥ = ⇥(cos kx cos ky) D E

Unconventional pairing at and near hot spots

Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

T* QuantumStrange Fluctuating CriticalMetal Fermi Large pockets Fermi surface

Increasing SDW order

Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = xm Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

Strange Fluctuating,Small Fermi Metal pairedpockets Fermi with Large pairingpockets fluctuations Fermi surface M. A. Metlitski and S. Sachdev, Physical Review d-wave B 82, 075128 (2010) superconductor

Spin density wave (SDW) SDW quantum critical point is unstable to d-wave superconductivity This instability is stronger than that in the BCS theory

Tuesday, January 8, 13 BaFe2(As1 x Px)2

Notice shift between the position of the QCP in the superconductor, and the divergence in effective mass in the metal measured at high magnetic ﬁelds Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

Spin density wave (SDW) SDW quantum critical point is unstable to d-wave superconductivity This instability is stronger than that in the BCS theory

Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

Strange Fluctuating,Small Fermi Metal pairedpockets Fermi with Large pairingpockets fluctuations Fermi E. G. Moon and surface S. Sachdev, Phy. Rev. B 80, 035117 (2009) d-wave superconductor

Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = xs

Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

Strange Fluctuating,Small Fermi Metal pairedpockets Fermi with Large pairingpockets fluctuations Fermi E. G. Moon and surface S. Sachdev, Phy. Rev. B 80, 035117 (2009) d-wave superconductor

Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = xs

Tuesday, January 8, 13 Theory of quantum criticality in the cuprates

Strange Fluctuating,Fluctuating, Metal pairedpaired FermiFermi Large pocketspockets Fermi E. G. Moon and surface S. Sachdev, Phy. Rev. B 80, 035117 (2009) d-wave superconductor

Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = xs

Tuesday, January 8, 13 At stronger coupling, different effects compete:

Pairing glue becomes stronger. • There is stronger fermion-boson • scattering, and fermionic quasiparticles lose their integrity. Other instabilities can appear • e.g. to charge density waves/stripe order.

Tuesday, January 8, 13 At stronger coupling, different effects compete:

Tuesday, January 8, 13 Outline

1. Weak coupling theory of SDW ordering, and d-wave superconductivity

2. Universal critical theory of SDW ordering

3. Emergent pseudospin symmetry, and quadrupolar density wave

4. Quantum Monte Carlo without the sign problem

Tuesday, January 8, 13 Outline

1. Weak coupling theory of SDW ordering, and d-wave superconductivity

2. Universal critical theory of SDW ordering

3. Emergent pseudospin symmetry, and quadrupolar density wave

4. Quantum Monte Carlo without the sign problem

Tuesday, January 8, 13 Fermi surface+antiferromagnetism

Metal with “large” Fermi surface

Tuesday, January 8, 13 Fermi surface+antiferromagnetism

Fermi surfaces translated by K =(, ).

Tuesday, January 8, 13 Fermi surface+antiferromagnetism

“Hot” spots

Tuesday, January 8, 13 Fermi surface+antiferromagnetism

Electron and hole pockets in antiferromagnetic phase with ~' =0 h i6

Tuesday, January 8, 13 “Hot” spots

Tuesday, January 8, 13 Low energy theory for critical point near hot spots

Tuesday, January 8, 13 Theory has fermions 1,2 (with Fermi velocities v1,2) and boson order parameter ~' , interacting with coupling

v1 v2

1 fermions 2 fermions occupied occupied

Tuesday, January 8, 13 = † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵

v1 v2

1 fermions 2 fermions occupied occupied

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004). Tuesday, January 8, 13 = † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵

v1 v2

“Hot spot”

“Cold” Fermi surfaces

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004). Tuesday, January 8, 13 = † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵ 1 ⇣ s u Order parameter: = ( ~' )2 + (@ ~' )2 + ~' 2 + ~' 4 L' 2 rr 2 ⌧ 2 4 e

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004). Tuesday, January 8, 13 = † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵

v1 v2

Metal with “large” Fermi surface ~' =0 h i

"k1 < 0 "k2 < 0

Fermion dispersions: " = v k and " = v k k1 1 · k2 2 ·

Tuesday, January 8, 13 = † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵ ~' † ~ + † ~ · 1↵ ↵ 2 2↵ ↵ 1 ⇣ ⌘ Ek > 0 ± Metal with Ek+ > 0 hole and Ek < 0 electron pockets ~' =0 h i6

Ek < 0 ± Fermion dispersions: "k1 = v1 k and "k2 = v2 k · 2 · "k1 + "k2 "k1 "k2 2 2 Ek = + ~' ± 2 ± 2 | | s✓ ◆ Tuesday, January 8, 13 Hertz action.

Upon integrating the fermions out, the leading term in the ⇧ e⇤ective action is ⇥(q, ⌅ ) ⇧ (q, ⌅ ) 2,where⇥(q, ⌅ )isthe n | n | n fermion polarizability. This is given by a simple fermion loop diagram

d d k dn 1 ⇥(q, ⌅n)= d . (2⇤) 2⇤ [ i⇥(n + ⌅n)+v1 (k + q)][ i⇥n + v2 k] · (1) · We deﬁne oblique co-ordinates p = v k and p = v k.Itis 1 1 · 2 2 · then clear that the integrand in (1) is independent of the (d 2) d 2 transverse momenta, whose integral yields an overall factor (in d = 2 this factor is precisely 1). Also, by shifting the integral Tuesday, January 8, 13 over k1 we note that the integral is independent of q.Sowehave d 2 dp1dp2d⇥n 1 ⇥(q, ⌃n)= 3 . v1 v2 8⌅ [ i⇤(⇥n + ⌃n)+p1][ i⇤⇥n + p2] | ⇥ | ⌃ (2) Next, we evaluate the frequency integral to obtain d 2 dp dp [sgn(p ) sgn(p )] ⇥(q, ⌃ )= 1 2 2 1 n ⇤ v v 4⌅2 i⇤⌃ + p p | 1 ⇥ 2| ⌃ n 1 2 ⌃ d 2 = | n| . (3) 4⌅ v v | 1 ⇥ 2| In the last step, we have dropped a frequency-independent, cuto⇤-dependent constant which can absorbed into a redeﬁnition of r. Notice also that the factor of ⇤ has cancelled. Inserting this fermion polarizability in the e⇤ective action for ⌦⌥ ,we obtain the Hertz action for the SDW transition: d d k 1 = T k2 + ⌃ + s ⌦⌥ (k, ⌃ ) 2 SH (2⌅)d 2 | n| | n | ⌃ n ⇧ ⇤ ⌅ u 2 + d d xd⇧ ⌦⌥ 2(x, ⇧) . (4) 4 Tuesday, January 8, 13 ⌃ ⇥ Exercise: Perform a tree-level RG rescaling on H .Nowwe zS rescale co-ordinates as x0 = xe and 0 = e .Herez is the dynamic critical exponent. Show that the gradient and non-local terms become invariant for z = 2 (previous theories considered here had z = 1). Then show that the transformation of the quartic (2 d) term is u0 = ue . This led Hertz to conclude that the SDW quantum critical point was described by a Gaussian theory for the SDW order parameter in d 2.

Tuesday, January 8, 13 Fate of the fermions.

Let us, for now, assume the validity of the Hertz Gaussian action, and compute the leading correction to the electronic Green’s function. This is given by the following Feynman graph for the electron self energy, . At zero momentum for the ⌃1 fermion we have

d 2 d q d⇥n 1 1(0, ⌥n)=⌅ d 2 . (2⇧) 2⇧ [q + ⇥n ][ i⇤(⇥n + ⌥n)+v2 q ] | | ·(5) We ﬁrst perform the integral over the q direction parallel to v2, while ignoring the subdominant dependence on this momentum in the boson propagator. The dependence on ⇤ immediately Tuesday, January 8, 13 disappears, and we have

⇤2 d d 1q d⇥ sgn(⇥ + ⇧ ) (0, ⇧ )=i n n n 1 n v (2⌅)d 1 2⌅ q 2 + ⇥ | 2| ⇤ ⇤ | | | n| ⇤2 d d 1q q 2 + ⇧ = i sgn(⇧ ) ln | | | n| . (6) ⌅ v n (2⌅)d 1 q 2 | 2| ⇤ | | ⇥ Evaluation of the q integral shows that

(d 1)/2 (0, ⇧ ) ⇧ (7) 1 n | n| The most important case is d = 2, where we have

⇤2 1(0, ⇧n)=i sgn(⇧n) ⇧n , d =2. (8) ⌅ v2 ⇤ | | | | ⌅

Tuesday, January 8, 13 Strong coupling physics in d =2

The theory so far has the boson propagator 1 ⇤ q2 + ⇤ | | which scales with dynamic exponent zb = 2, and now a fermion propagator 1 . ⇤ i⇥⇤ + c ⇤ (d 1)/2 + v q 1| | · First note that for d < 3, the bare i⇥⇤ term is less important than the contribution from the self energy at low frequencies. This indicates that ⇥ is irrelevant in the critical theory, and we can set ⇥ 0. Fortunately, all the loop diagrams evaluated so far are ⌅ independent of ⇥.

Setting ⇥ = 0, we see that the fermion propagator scales with dynamic exponent z =2/(d 1). For d > 2, z < z ,andsoat f f b small momenta the boson fluctuations have lower energy than the fermion fluctuations. Thus it seems reasonable to assume that the Tuesday, January 8, 13 fermion fluctuations are not as singular, and we can focus on an eective theory of the SDW order parameter ⇤ alone. In other words, the Hertz assumptions appear valid for d > 2.

However, in d = 2, we have zf = zb = 2. Thus fermionic and bosonic ﬂuctuations are equally important, and it is not appropriate to integrate the fermions out at an initial stage. We have to return to the original theory of coupled bosons and fermions. This turns out to be strongly coupled, and exhibits complex critical behavior. For more details, see

M. A. Metlitski and S. Sachdev, arXiv:1005.1288 (Physical Review B 82, 075127 (2010)).

Tuesday, January 8, 13 = † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵ 1 ⇣ s u Order parameter: = ( ~' )2 + (@ ~' )2 + ~' 2 + ~' 4 L' 2 rr 2 ⌧ 2 4 e “Yukawa” coupling: = ~' † ~ + † ~ Lc · 1↵ ↵ 2 2↵ ↵ 1 ⇣ ⌘

Perform RG on both fermions and ⇥ , using a local ﬁeld theory.

Tuesday, January 8, 13 = † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵ 1 ⇣ s u Order parameter: = ( ~' )2 + (@ ~' )2 + ~' 2 + ~' 4 L' 2 rr 2 ⌧ 2 4 e “Yukawa” coupling: = ~' † ~ + † ~ Lc · 1↵ ↵ 2 2↵ ↵ 1 ⇣ ⌘ ` z` Under the rescaling x0 = xe , ⌧ 0 = ⌧e , the spatial gradients are ﬁxed if the ﬁelds transform as

(d+z 2)`/2 (d+z 1)`/2 ~' 0 = e ~' ;“ 0 = e .

Then the Yukawa coupling transforms as

(4 d z)`/2 0 = e

For d = 2, with z = 2 the bare time-derivative terms ⇣, ⇣ are irrelevant, but the Yukawa coupling is invariant. Thus we have to work at ﬁxed = 1, and cannot expand in powers of : criticale theory is strongly coupled.

Tuesday, January 8, 13 Critical point theory is strongly coupled in d =2 Results are independent of coupling

v1 v2

ky 1 Gfermion ⇠ ip v.k k x

A. J. Millis, Phys. Rev. B 45, 13047 (1992) Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004)

Tuesday, January 8, 13 Critical point theory is strongly coupled in d =2 Results are independent of coupling

k ? ky k k

Z(k ) Gfermion = k ,Z(k ) vF (k ) k i vF (k )k k ⇠ k ⇠ k k ? M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) Tuesday, January 8, 13 Outline

1. Weak coupling theory of SDW ordering, and d-wave superconductivity

2. Universal critical theory of SDW ordering

3. Emergent pseudospin symmetry, and quadrupolar density wave

4. Quantum Monte Carlo without the sign problem

Tuesday, January 8, 13 Outline

1. Weak coupling theory of SDW ordering, and d-wave superconductivity

2. Universal critical theory of SDW ordering

3. Emergent pseudospin symmetry, and quadrupolar density wave

4. Quantum Monte Carlo without the sign problem

Tuesday, January 8, 13 Emergent [SU(2)]4 pseudospin symmetry

= † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵ 1 ⇣ s u Order parameter: = ( ~' )2 + (@ ~' )2 + ~' 2 + ~' 4 L' 2 rr 2 ⌧ 2 4 e “Yukawa” coupling: = ~' † ~ + † ~ Lc · 1↵ ↵ 2 2↵ ↵ 1 ⇣ ⌘

Tuesday, January 8, 13 Emergent [SU(2)]4 pseudospin symmetry

= † (⇣@ iv ) + † (⇣@ iv ) Lf 1↵ ⌧ 1 · rr 1↵ 2↵ ⌧ 2 · rr 2↵ 1 ⇣ s u Order parameter: = ( ~' )2 + (@ ~' )2 + ~' 2 + ~' 4 L' 2 rr 2 ⌧ 2 4 e “Yukawa” coupling: = ~' † ~ + † ~ Lc · 1↵ ↵ 2 2↵ ↵ 1 Introduce the spinors ⇣ ⌘

1↵ 2↵ 1↵ = , 2↵ = ✏ † ✏ † ✓ ↵ 1 ◆ ✓ ↵ 2 ◆ Then the Lagrangian is invariant under the SU(2) transformation U with

U , U 1 ! 1 2 ! 2 Note that U can be chosen independently at the 4 pairs of hotspots.

This symmetry relies on the linearization of the fermion dispersion about the hot spots. Tuesday, January 8, 13 ck† c† k⇥ = ⇥(cos kx cos ky) D E

Unconventional pairing at and near hot spots

Tuesday, January 8, 13 ck† Q/2,ck+Q/2, = (cos kx cos ky) D E After Q is ‘2kF ’ pseudospin wavevector rotation

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

K. B. Efetov, H. Meier, and C. Pepin, arXiv:1210.3276

Unconventional particle-hole pairing at and near hot spots

Tuesday, January 8, 13 ck† Q/2,ck+Q/2, = (cos kx cos ky) D E After Q is ‘2kF ’ pseudospin wavevector rotation

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

K. B. Efetov, H. Meier, and C. Pepin, arXiv:1210.3276 corresponds to a 2kF bond-nematic or a quadrupole density wave

Unconventional particle-hole pairing at and near hot spots

Tuesday, January 8, 13 Quadrupole density wave

ck† Q/2,ck+Q/2, = (cos kx cos ky) Tuesday, January 8, 13D E Quadrupole density wave

"1 “Bond density” measures amplitude for electrons to be in spin-singlet valence bond.

No modulations on sites, cr†cs is modulated only for r = s. 6 ⌦ ↵

ck† Q/2,ck+Q/2, = (cos kx cos ky) Tuesday, January 8, 13D E Quadrupole density wave

Local Ising nematic order with an envelope which oscillates

No modulations on sites, cr†cs is modulated only for r = s. 6 ⌦ ↵

ck† Q/2,ck+Q/2, = (cos kx cos ky) Tuesday, January 8, 13D E Quadrupole density wave

Local Ising nematic order with an envelope which oscillates

No modulations on sites, cr†cs is modulated only for r = s. 6 ⌦ ↵

ck† Q/2,ck+Q/2, = (cos kx cos ky) Tuesday, January 8, 13D E Quadrupole density wave

"1 “Bond density” measures amplitude for electrons to be in spin-singlet valence bond.

No modulations on sites, cr†cs is modulated only for r = s. 6 ⌦ ↵

ck† Q/2,ck+Q/2, = (cos kx cos ky) Tuesday, January 8, 13D E Strength of instability at quantum criticality

BCS theory

!D 1+e-ph log } ! ⇣ ⌘ Cooper logarithm

Tuesday, January 8, 13 Strength of instability at quantum criticality

BCS theory ! 1+ log D e-ph ! ⇣ ⌘ Debye Electron-phonon frequency coupling

Implies T ⇥ exp ( 1/) c ⇠ D

Tuesday, January 8, 13 Strength of instability at quantum criticality

Spin density wave quantum critical point

E 1+ log2 F ⇥(1 + 2) ⇤ ✓ ◆

M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) Y. Wang and A. Chubukov, arXiv:1210.2408 Tuesday, January 8, 13 Strength of instability at quantum criticality

Spin density wave quantum critical point

E 1+ log2 F ⇥(1 + 2) ⇤ ✓ ◆ Fermi energy = tan ⇥,where2⇥ is the angle between Fermi lines. Independent of interaction strength U in 2 dimensions.

M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) Y. Wang and A. Chubukov, arXiv:1210.2408 Tuesday, January 8, 13 2✓

Tuesday, January 8, 13 M. A. Metlitski and S. Sachdev, k ? Phys. Rev. B 85, ky 075127 (2010) k k kx

Z(k ) Gfermion = k ,Z(k ) vF (k ) k i vF (k )k k ⇠ k ⇠ k k ? 1 Z2(k ) k2 k k dk 2 log k k vF (k ) Z k ✓ k ◆

Tuesday, January 8, 13 M. A. Metlitski and S. Sachdev, k ? Phys. Rev. B 85, ky 075127 (2010) k k kx

Z(k ) Gfermion = k ,Z(k ) vF (k ) k i vF (k )k k ⇠ k ⇠ k k ? 1 Z2(k ) k2 k k dk 2 log k k vF (} k ) Z k ✓ k ◆ Cooper logarithm

Tuesday, January 8, 13 M. A. Metlitski and S. Sachdev, k ? Phys. Rev. B 85, ky 075127 (2010) k k kx

Z(k ) Gfermion = k ,Z(k ) vF (k ) k i vF (k )k k ⇠ k ⇠ k k ? 1 Z2(k ) k2 k k dk 2 log k k vF (} k ) Z k ✓ k ◆ Spin ﬂuctuation Cooper propagator logarithm

Tuesday, January 8, 13 Enhancement of pairing susceptibility by interactions Spin density wave quantum critical point

E 1+ log2 F ⇥(1 + 2) ⇤ ✓ ◆

log2 singularity arises from Fermi lines; • singularity at hot spots is weaker. Interference between BCS and quantum-critical logs. • Momentum dependence of self-energy is crucial. • Not suppressed by 1/N factor in 1/N expansion. •

Tuesday, January 8, 13 Enhancement of susceptibility by interactions

Spin density wave quantum critical point

E 1+ log2 F 3⇥(1 + 2) ⇤ ✓ ◆ Emergent pseudospin symmetry of low • energy theory also induces log2 in a single “d-wave” particle-hole channel. Fermi-surface curvature reduces prefactor by 1/3.

corresponds to a 2kF bond-nematic or a • quadrupole density wave

M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) K. B. Efetov, H. Meier, and C. Pepin, arXiv:1210.3276

Tuesday, January 8, 13 Outline

1. Weak coupling theory of SDW ordering, and d-wave superconductivity

2. Universal critical theory of SDW ordering

3. Emergent pseudospin symmetry, and quadrupolar density wave

4. Quantum Monte Carlo without the sign problem

Tuesday, January 8, 13 Outline

1. Weak coupling theory of SDW ordering, and d-wave superconductivity

2. Universal critical theory of SDW ordering

3. Emergent pseudospin symmetry, and quadrupolar density wave

4. Quantum Monte Carlo without the sign problem

Tuesday, January 8, 13 Low energy theory for critical point near hot spots

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

Hot spots in a single band model

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

E. Berg, M. Metlitski, and S. Sachdev, K Science 338, 1606 (2012).

Hot spots in a two band model

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

Faithful realization of the E. Berg, M. Metlitski, and generic S. Sachdev, K Science 338, 1606 universal (2012). low energy theory for the onset of antiferromagnetism. Hot spots in a two band model

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

E. Berg, M. Metlitski, and S. Sachdev, K Science 338, 1606 (2012).

Hot spots in a two band model

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

Sign E. Berg, problem is M. Metlitski, and S. Sachdev, absent as K Science 338, 1606 long as K (2012). connects hotspots in distinct bands

Hot spots in a two band model

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

Sign E. Berg, problem is M. Metlitski, and S. Sachdev, absent as K Science 338, 1606 long as K (2012). connects hotspots in Particle-hole or point-group distinct symmetries or bands commensurate densities not required !

Hot spots in a two band model

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

Electrons with dispersion "k interacting with ﬂuctuations of the antiferromagnetic order parameter ~' .

= c ~' exp ( ) Z D ↵D S Z @ = d⌧ c† " c S k↵ @⌧ k k↵ Z Xk ✓ ◆ 1 r + d⌧d2x ( ~' )2 + ~' 2 + ... 2 rx 2 Z  x d⌧ ~' ( 1) i c† ~ c i · i↵ ↵ i i Z X

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism (x) (y) Electrons with dispersions "k and "k interacting with fluctuations of the antiferromagnetic order parameter ~' . E. Berg, (x) (y) = c↵ c↵ ~' exp ( ) M. Metlitski, and Z D D D S S. Sachdev, Z Science 338, 1606 (x) @ (x) (x) (2012). = d⌧ c † " c S k↵ @⌧ k k↵ Z Xk ✓ ◆ (y) @ (y) (y) + d⌧ c † " c k↵ @⌧ k k↵ Z Xk ✓ ◆ 1 r + d⌧d2x ( ~' )2 + ~' 2 + ... 2 rx 2 Z  x (x) (y) d⌧ ~' ( 1) i c †~ c + H.c. i · i↵ ↵ i Z i Tuesday, January 8, 13 X QMC for the onset of antiferromagnetism (x) (y) Electrons with dispersions "k and "k interacting with fluctuations of the antiferromagnetic order parameter ~' . E. Berg, (x) (y) = c↵ c↵ ~' exp ( ) M. Metlitski, and Z D D D S S. Sachdev, Z Science 338, 1606 (x) @ (x) (x) (2012). = d⌧ c † " c S k↵ @⌧ k k↵ Z Xk ✓ ◆ (y) @ (y) (y) + d⌧ c † " c k↵ @⌧ k k↵ Z Xk ✓ ◆ 1 r No sign problem ! + d⌧d2x ( ~' )2 + ~' 2 + ... 2 rx 2 Z  x (x) (y) d⌧ ~' ( 1) i c †~ c + H.c. i · i↵ ↵ i Z i Tuesday, January 8, 13 X QMC for the onset of antiferromagnetism (x) (y) Electrons with dispersions "k and "k interacting with fluctuations of the antiferromagnetic order parameter ~' . E. Berg, (x) (y) = c↵ c↵ ~' exp ( ) M. Metlitski, and Z D D D S S. Sachdev, Z Science 338, 1606 (x) @ (x) (x) (2012). = d⌧ c † " c S k↵ @⌧ k k↵ k Applies without Z X ✓ ◆ (y) @ (y) (y) changes to the + d⌧ c † " c k↵ @⌧ k k↵ microscopic band Z Xk ✓ ◆ structure in the 1 r iron-based + d⌧d2x ( ~' )2 + ~' 2 + ... 2 rx 2 superconductors Z  x (x) (y) d⌧ ~' ( 1) i c †~ c + H.c. i · i↵ ↵ i Z i Tuesday, January 8, 13 X QMC for the onset of antiferromagnetism (x) (y) Electrons with dispersions "k and "k interacting with fluctuations of the antiferromagnetic order parameter ~' . E. Berg, (x) (y) = c↵ c↵ ~' exp ( ) M. Metlitski, and Z D D D S S. Sachdev, Z Science 338, 1606 (x) @ (x) (x) (2012). = d⌧ c † " c S k↵ @⌧ k k↵ Z Xk ✓ ◆ Can integrate out ~' to (y) @ (y) (y) obtain an extended + d⌧ c † " c k↵ @⌧ k k↵ Hubbard model. The interactions in this model Z Xk ✓ ◆ 1 r only couple electrons in + d⌧d2x ( ~' )2 + ~' 2 + ... separate bands. 2 rx 2 Z  x (x) (y) d⌧ ~' ( 1) i c †~ c + H.c. i · i↵ ↵ i Z i Tuesday, January 8, 13 X QMC for the onset of antiferromagnetism

E. Berg, M. Metlitski, and S. Sachdev, K Science 338, 1606 (2012).

Hot spots in a two band model

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism 1 a)

E. Berg, 0.5 M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

/ π K y 0 k

−0.5

−1 Center Brillouin zone at (π,π,)

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism 1 1 a) b)

E. Berg, 0.5 M. Metlitski,0.5 and S. Sachdev, Science 338, 1606 K (2012). / π / π y 0 y 0 k k

−0.5 −0.5

−1 −1 Move−1 one of the Fermi0 surface by (π1,π,) −1 0 1 Tuesday, January 8, 13 k /π k /π x x QMC for the onset of antiferromagnetism 1 a)

E. Berg, 0.5 M. Metlitski, and S. Sachdev, Science 338, 1606 (2012). / π y 0 k

−0.5

−1 Now hot spots are at Fermi surface intersections

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism 1 1 a) b)

E. Berg, 0.5 0.5 M. Metlitski, and S. Sachdev, Science 338, 1606 K (2012). / π / π y 0 y 0 k k

−0.5 −0.5

−1 −1 −1 0 1 Expected−1 Fermi surfaces 0in the AFM ordered1 phase k /π Tuesday, January 8, 13 k /π x x QMC for the onset of antiferromagnetism

r = −0.5 r = 0 r = 0.5 1 1 1

0.5 0.5 0.5 1.5 π /

y 0 0 0 1 k

−0.5 −0.5 −0.5 0.5

−1 −1 −1 −1 0 1 −1 0 1 −1 0 1 k / k / k / x π x π x π

Electron occupation number nk as a function of the tuning parameter r

E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

a) 0.6 b) L=8 0.4 L=10 0.3 ) L=12 β 0.4 2 L=14 0.2 /(L φ χ 0.2 0.1 Binder cumulant 0 0 −0.5 0 0.5 1 −0.5 0 0.5 1 r r

AF susceptibility, ', and Binder cumulant as a function of the tuning parameter r

E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

Tuesday, January 8, 13 QMC for the onset of antiferromagnetism

−4 x 10 10 _ 8 P _+ L = 10 P_

) 6

max L = 12

↑ x

( 4 | ± P L = 14 2 r c 0

−2 −2 −1 0 1 2 3 r

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

E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

Tuesday, January 8, 13 Conclusions Solved sign-problem for generic universal theory for the onset of antiferromagnetism in two-dimensional metals.

Obtained (ﬁrst ?) convincing evidence for the presence of unconventional superconductivity at strong coupling and near SDW quantum criticality.

Good prospects for studying competing charge orders, and non-Fermi liquid physics at non-zero temperature.

Tuesday, January 8, 13 Conclusions Solved sign-problem for generic universal theory for the onset of antiferromagnetism in two-dimensional metals.

Obtained (ﬁrst ?) convincing evidence for the presence of unconventional superconductivity at strong coupling and near SDW quantum criticality.

Good prospects for studying competing charge orders, and non-Fermi liquid physics at non-zero temperature.

Tuesday, January 8, 13 Conclusions Solved sign-problem for generic universal theory for the onset of antiferromagnetism in two-dimensional metals.

Obtained (ﬁrst ?) convincing evidence for the presence of unconventional superconductivity at strong coupling and near SDW quantum criticality.

Good prospects for studying competing charge orders, and non-Fermi liquid physics at non-zero temperature.

Tuesday, January 8, 13