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 x Fermi surface with full square lattice symmetry Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering y x Spontaneous elongation along x direction: Ising order parameter φ > 0. Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering y x Spontaneous elongation along y direction: Ising order parameter φ < 0. Tuesday, January 8, 13 Ising-nematic order parameter 2 φ d k (cos k cos k ) c† c ⇠ x − y kσ kσ Z Measures spontaneous breaking of square lattice point-group symmetry of underlying Hamiltonian Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering or φ =0 φ =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 φ =0 φ =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 φ =0 φ =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 φ =0 φ =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 φ =0 φ =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 φ =0 φ =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 φ =0 φ =0 ⇥ ⇤ ⇥ r rc Phase diagram as a function of T and r Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering Effective action for Ising order parameter = d2rd⇥ (⌅ ⇤)2 + c2( ⇤)2 +(λ λ )⇤2 + u⇤4 S⇥ τ ⇤ − c ⇤ ⇥ Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering Effective action for Ising order parameter = d2rd⇥ (⌅ ⇤)2 + c2( ⇤)2 +(λ λ )⇤2 + u⇤4 S⇥ τ ⇤ − c ⇤ ⇥ Effective action for electrons: Nf = dτ c† ⇤ c t c† c Sc iα ⇥ iα − ij iα iα⇥ α=1 i i<j ⌃ ⇧ ⇧ ⇧ Nf ⇤ ⌅ dτc† (⇤ + ⇥ ) c ⇥ kα ⇥ k kα α=1 ⇧ ⇧k ⌃ Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering Coupling between Ising order and electrons Nf φc = g d⌧ φq (cos kx cos ky)c† ck q/2,↵ S − − k+q/2,↵ − ↵=1 Z X Xk,q for spatially dependent φ φ > 0 φ < 0 ⇥ ⇥ Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering = d2rd⇥ (⌅ ⇤)2 + c2( ⇤)2 +(λ λ )⇤2 + u⇤4 S⇥ τ ⇤ − c ⇤ ⇥ Nf = dτc† (⇤ + ⇥ ) c Sc kα ⇥ k kα α=1 k ⇥ Nf φc = g d⌧ φq (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 φ fluctuation at wavevector ~q couples most efficiently to fermions • near ~k . ± 0 Expand fermion kinetic energy at wavevectors about ~k0 and • boson (φ) kinetic energy about ~q = 0. ± Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering φ fluctuation at wavevector ~q couples most efficiently to fermions • near ~k . ± 0 Expand fermion kinetic energy at wavevectors about ~k0 and • boson (φ) kinetic energy about ~q = 0. ± Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering [ ,φ]= L ± 2 2 +† @⌧ i@x @y + + † @⌧ + i@x @y − − − − − 1 2 φ +† + + † + (@yφ) − − − 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 φ +† + + † + (@yφ) − − − 2g2 ⇣ ⌘ One loop φ self-energy with Nf fermion flavors: (a) d2k d⌦ 1 ⌃φ(~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 φ +† + + † + (@yφ) − (a)− − 2g2 ⇣ ⌘ (b) Electron self-energy at order 1/Nf : 2 ~ 1 d q d! 1 ⌃(k, ⌦) = 2 −Nf 4⇡ 2⇡ q2 ! Z [ i(! +⌦)+k + q +(k + q )2] y + | | − x x y y g2 q " | y|# 2 g2 2/3 = i sgn(⌦) ⌦ 2/3 − p3N 4⇡ | | f ✓ ◆ Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering 2 2 = +† @⌧ i@x @y + + † @⌧ + i@x @y L − − − − − 1 2 φ +† + + † + (@yφ) − (a)− − 2g2 ⇣ ⌘ (b) Electron self-energy at order 1/Nf : 2 ~ 1 d q d! 1 ⌃(k, ⌦) = 2 −Nf 4⇡ 2⇡ q2 ! Z [ i(! +⌦)+k + q +(k + q )2] y + | | − x x y y g2 q " | y|# 2 g2 2/3 = i sgn(⌦) ⌦ 2/3 ⌦ d/3 in dimension d. − p3N 4⇡ | | f ✓ ◆ ⇠ | | Tuesday, January 8, 13 Quantum criticality of Ising-nematic ordering 2 2 = +† @⌧ i@x @y + + † @⌧ + i@x @y L − − − − − 1 2 φ +† + + † + (@yφ) − − − 2g2 ⇣ ⌘ Schematic form of φ 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 <zb for d>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 φ +† + + † + (@yφ) − − − 2g2 ⇣ ⌘ Schematic form of φ 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 φ +† + + † + (@yφ) − − − 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 gφ +† + † +( @yφ) − − − − ⇣ ⌘ 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 gφ +† + † +( @yφ) − − − − ⇣ ⌘ Simple scaling argument for z =3/2. Under the rescaling x x/s, y y/s1/2, and ⌧ ⌧/sz,we find invariance provided! ! ! φ φs(2z+1)/4 ! (2z+1)/4 s ! (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 − 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 2 Lower Tc superconductivity in the heavy fermion compounds Generally, the ground state of a Ce heavy-fermion sys- tem is determined by the competition of the indirect Ru- derman Kittel Kasuya Yosida (RKKY) interaction which provokes magnetic order of localized moments mediated by the light conduction electrons and the Kondo interac- tion. 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 interac- tions 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 criti- cal 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 acoexistencephase approaches exist to model the magnetic quantum phase AF+SC exist. 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.