Transport in non-fermi liquids

Transcription

1 HARVARD Transport in non-fermi liquids Theory Winter School National High Magnetic Field Laboratory, Tallahassee Subir Sachdev January 12, 2018 Talk online: sachdev.physics.harvard.edu

2 Quantum matter without quasiparticles Strange metal Entangled electrons lead to strange SM temperature dependence of resistivity and FL other properties p (hole/cu) Figure: K. Fujita and J. C. Seamus Davis

3 Quantum matter without quasiparticles Resistivity 0 + AT Strange Metal T BaFe 2 (As 1-x P x ) 2 T 0 T SD T c! " 2.0 AF SDW +nematic 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, PRB 81, (2010)

4 Ubiquitous Strange, Bad, or Incoherent, metal has a resistivity,, which obeys T, and in some cases h/e 2 (in two dimensions), where h/e 2 is the quantum unit of resistance.

5 Strange metals just got stranger B-linear magnetoresistance!? Ba-122 LSCO I. M. Hayes et. al., Nat. Phys P. Giraldo-Gallo et. al., arxiv:

6 Strange metals just got stranger Scaling between B and T!? Ba-122 Ba-122 I. M. Hayes et. al., Nat. Phys. 2016

7 Fermi surface coupled to a gauge field k y k x ia (r i~a) 2 2m µ + 1 2g 2 (r ~a)2

8 Fermi surface coupled to a gauge field L[ ±,a]= i@ y i@ y 2 a g 2 (@ ya) 2 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010)

9 Fermi surface coupled to a gauge field One loop photon self-energy with N f fermion flavors: Z d 2 k d (~q,!) = N f = N f 4! q y 1 [ i( +!)+k x + q x +(k y + q y ) 2 ] i k x + ky 2 Landau-damping Electron self-energy at order 1/N f : ( ~ k, ) = 1 N f Z d 2 q d! [ i(! + )+k x + q x +(k y + q y ) 2 ] 1 " # q 2 y g 2 +! q y = i 2 g 2 2/3 p sgn( ) 2/3 3Nf 4 No quasiparticles

10 Breakdown of quasiparticles requires strong coupling to a low energy collective mode In all known cases, we can write down the singular processes in terms of a continuum field theory of the fermions near the Fermi surface coupled to the collective mode. In all known cases, the continuum critical theory has a conserved total (pseudo-) momentum, P ~, which commutes with the Hamiltonian. This momentum may not be equal to the crystal momentum of the underlying lattice model. As long as J, ~ P ~ 6=0(where J ~ is the electrical current) the d.c. resistivity of the critical theory is exactly zero. This is the case even though the electron self energy can be highly singular and there are no fermionic quasiparticles (many well-known papers on non-fermi liquid transport ignore this point.) We need to include additional (dangerously) irrelevant umklapp corrections to obtain a non-zero resistivity. Because these additional corrections are irrelevant, it is di cult to see how they can induce a linear-in-t resistivity.

11 Breakdown of quasiparticles requires strong coupling to a low energy collective mode In all known cases, we can write down the singular processes in terms of a continuum field theory of the fermions near the Fermi surface coupled to the collective mode. In all known cases, the continuum critical theory has a conserved total (pseudo-) momentum, P ~, which commutes with the Hamiltonian. This momentum may not be equal to the crystal momentum of the underlying lattice model. As long as J, ~ P ~ 6=0(where J ~ is the electrical current) the d.c. resistivity of the critical theory is exactly zero. This is the case even though the electron self energy can be highly singular and there are no fermionic quasiparticles (many well-known papers on non-fermi liquid transport ignore this point.) We need to include additional (dangerously) irrelevant umklapp corrections to obtain a non-zero resistivity. Because these additional corrections are irrelevant, it is di cult to see how they can induce a linear-in-t resistivity.

12 Theories of metallic states without quasiparticles in the presence of disorder Well-known perturbative theory of disordered metals has 2 classes of known fixed points, the insulator at strong disorder, and the metal at weak disorder. The latter state has long-lived, extended quasiparticle excitations (which are not plane waves). Needed: a metallic fixed point at intermediate disorder and strong interactions without quasiparticle excitations. Although disorder is present, it largely self-averages at long scales. SYK models

13 Theories of metallic states without quasiparticles in the presence of disorder Well-known perturbative theory of disordered metals has 2 classes of known fixed points, the insulator at strong disorder, and the metal at weak disorder. The latter state has long-lived, extended quasiparticle excitations (which are not plane waves). Needed: a metallic fixed point at intermediate disorder and strong interactions without quasiparticle excitations. Although disorder is present, it largely self-averages at long scales. SYK models

14 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Building Song, Chao-Ming a Jian, metal Leon Balents Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) t... U X X H = X x X... U ijkl,x c ix c jx c kx c lx + X X t ij,xx 0c i,x c j,x 0 i<j,k<l hxx 0 i i,j U ijkl 2 = 2U 2 N 3 Gaussian distrib d t ij,x,x 0 2 = t 2 0 /N. sm, one studies

15 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models X Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Self-consistent equations... t... U G(i! n ) 1 = i! n + µ 4 (i! n ) zt G(i! n ) 1 = i! n + µ 4 (i! n ) zt G(i! 0 2 G(i! n), n), 4 ( ) = U0 2 G( )2 G( ), 4 ( ) = U 2 0 G( )2 G( ), Ḡ(i!) = tg(i!)... strong similarities to DMFT equations mathematical structure appeared in early study of doped t-j model with double large N and infinite dimension limits: O. Parcollet+A. Georges, 1999

16 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models X Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Coherence scale G(i! n ) 1 = i! n + µ 4 (i! n ) zt G(i! n ) 1 = i! n + µ 4 (i! n ) zt G(i! 0 2 G(i! n), n), 4 ( ) = U0 2 G( )2 G( ), 4 ( ) = U 2 0 G( )2 G( ), Rescaling.! =! Ẽ c, = Ẽ c, atted as... Rescaling Ḡ(i!)! =! = tg(i!), = Ẽ c (i!), Ḡ(i!) = tg(i!) (i!) = (i!)/ t t = (i!)/ t t = z = z 2 2 t 2 Ẽ c matted as Ḡ(i!) = t t U i! (i!) (i!) Ḡ(i!) = tg(i!) ( ) = For t U, a single universal coherence scale appears c Ḡ( ) 2 Ḡ( ) + 2Ḡ( ), 1 Ẽ c = t 2 U 1 2 t

17 arxiv: [pdf, other] See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures Coherence scale We solve these equations in a real time Keldysh formulation numerically and determine asymptotics analytically. Narrow coherence peak appears in spectral function: heavy quasiparticles form for T E c! = E c Quasiparticle weight Z ~ t/u

18 arxiv: [pdf, other] See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures X 2 64 Entropy 3 75 Level repulsion: entropy is released for T<E c! S(T ) T/E c Universal scaling forms S/N = S(T/E c ) on S(T 0) 0 C/N = T/E c S 0 (T/E c ) C lim T!0 T = S0 (0) E c m /m U/t Sommerfeld enhancement

19 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Conductivity From the Kubo formula, we have the conductivity Z Re[ (!)] / t 2 f(! + ) 0 d f( )! where A(!) = Im[G R (!)]. At T >E c this yields A( )A(! + ) e2 h t 2 0 U 1 T

20 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Low coherence scale E c t2 0 U (a)

21 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Low coherence scale E c t2 0 U For E c <T <U, the resistivity,, and entropy density, s, are (a) h e 2 T E c, s = s 0

22 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Low coherence scale E c t2 0 U For E c <T <U, the resistivity,, and entropy density, s, are (a) h e 2 T E c, s = s 0

23 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) Low coherence scale E c t2 0 U For T <E c, the resistivity,, and entropy density, s, are = h e 2 " c 1 + c 2 T E c 2 # (a) s s 0 T E c

24 arxiv: [pdf, other] Title: Magnetotransport in a model of a disordered strange metal Authors: Aavishkar A. Patel, John McGreevy, Daniel P. Arovas, Subir Sachdev Aavishkar Patel

25 H = + Infecting a Fermi liquid and making it SYK Mobile electrons (c, green) interacting with SYK quantum dots (f, blue) with exchange interactions. This yields the first model agreeing with magnetotransport in strange metals! t 1 NM 1/2 MX hrr 0 i; i=1 NX r; i,j=1 (c ri c r 0 i +h.c.) µ c M X MX k,l=1 r; i=1 c ri c ri µ g r ijklf ri f rjc rk c rl + 1 N 3/2 NX r; i=1 NX r; i,j,k,l=1 f ri f ri J r ijklf ri f rj f rkf rl. A. A. Patel, J. McGreevy, D. P. Arovas and S. Sachdev, arxiv:

26 Infecting a Fermi liquid and making it SYK ( 0 )= J 2 G 2 ( 0 )G( 0 ) G(i! n )= M N g2 G( 0 )G c ( 0 )G c ( 0 ), 1 i! n + µ (i! n ), (f electrons) c ( 0 )= g 2 G c ( 0 )G( 0 )G( 0 ), G c (i! n )= X 1 i! n k + µ c c (i! n ). (c electrons) k Exactly solvable in the large N,M limits! Low-T phase: c electrons form a Marginal Fermi-liquid (MFL), f electrons are local SYK models

27 Infecting a Fermi liquid and making it SYK High-T phase: c electrons form an incoherent metal (IM), with local Green s function, and no notion of momentum; f electrons remain local SYK models G c ( ) = G( ) = C c p 1+e 4 E c C p 1+e 4 E 1/2 T e 2 E ct, sin( T ) 1/2 T e 2 ET, 0 apple < sin( T )

28 Infecting a Fermi liquid and making it SYK Low-T phase: c electrons form a Marginal Fermi-liquid (MFL), f electrons are local SYK models c (i! n )= ig 2 (0)T 2J cosh 1/2 (2 E) 3/2!n T ln 2 Te E 1 J +! n T!n 2 T +, c (i! n )! ig 2 (0) 2J cosh 1/2 (2 E) 3/2! n ln!n e E 1 J,! n T ( (0) 1/t)

29 Linear-in-T resistivity Both the MFL and the IM are not translationally-invariant and have linear-in-t resistivities! MFL 0 = MT 1 J IM 0 =( 1/2 /8) MT 1 J v 2 F g 2 (0)g 2 cosh 1/2 (2 E) cosh(2 E c ). [Can be obtained straightforwardly from Kubo formula in the large-n,m limits] cosh 1/2 (2 E). (v F t) The IM is also a Bad metal with IM 0 1

30 Magnetotransport: Marginal-Fermi liquid Thanks to large N,M, we can also exactly derive the linearresponse Boltzmann equation for non-quantizing magnetic fields Re[ c R(!)])@ t n(t, k,!)+v F ˆk E(t) n 0 f (!)+v F (ˆk Bẑ) r k n(t, k,!) =2 n(t, k,!)im[ c R(!)], MFL L = M v2 F (0) 16T MFL H = M v2 F (0) 16T (B = eba 2 /~) (i.e. flux per unit cell) Z 1 1 Z 1 de 1 2 sech2 1 de 1 2 sech2 Im[ c R (E 1)] 2T Im[ c R (E 1)] 2 +(v F /(2k F )) 2 B 2, (v F /(2k F ))B 2T Im[ c R (E 1)] 2 +(v F /(2k F )) 2 B 2. E1 E1 MFL L T 1 s L ((v F /k F )(B/T )), MFL H BT 2 s H ((v F /k F )(B/T )). s L,H (x!1) / 1/x 2, s L,H (x! 0) / x 0. Scaling between magnetic field and temperature in orbital magnetotransport!

31 Macroscopic magnetotransport in the MFL Let us consider the MFL with additional macroscopic disorder (charge puddles etc.) Figure: N. Ramakrishnan et. al., arxiv: No macroscopic momentum, so equations describing charge transport are just r I(x) =0, I(x) = (x) E(x), E(x) = r (x). Very weak thermoelectricity for large FS, so charge effectively decoupled from heat transport.

32 Physical picture Current path length increases linearly with B at large B due to local Hall effect, which causes the global resistance to increase linearly with B at large B. Exact numerical solution of charge-transport equations in a random-resistor network. (M. M. Parish and P. Littlewood, Nature 426, 162 (2003))

33 Solvable toy model: two-component disorder Two types of domains a,b with different carrier densities and lifetimes randomly distributed in approximately equal fractions over sample. Effective medium equations can be solved exactly I + a 2 e L e 1 ( a e )+ I + b 2 e L e 1 ( b e )=0. e L e H e L e2 L + H e2 e H /B = e2 L + H e2 q (B/m) 2 a 0a MFL b 0b MFL = a b( MFL 0a a + b m a b MFL 0a + 0b MFL a 2 b MFL 0a + MFL 0b MFL 0b ) 1/2 MFL 0a + 0b MFL. (m = k F /v F 1/t) 2, a,b T (i.e. effective transport scattering rates) e L p c 1 T 2 + c 2 B 2 Scaling between B and T at microscopic orbital level has been transferred to global MR!

34 Scaling between B and T n b /n a =0.8 b/ a =0.8 a =0.1k B T t/100 (B =0.0025) (T = t/100) ~ 50 T (a = 3.82 A)

35 Magnetotransport in strange metals Engineered a model of a Fermi surface coupled to SYK quantum dots which leads to a marginal Fermi liquid with a linear-in-t resistance, with a magnetoresistance which scales as B T. Macroscopic disorder then leads to linear-in-b magnetoresistance, and a combined dependence which scales as p B 2 + T 2 Higher temperatures lead to an incoherent metal with a local Green s function and a linear-in-t resistance, but negligible magnetoresistance.

36 This simple two-component model describes a new state of matter which is realized by electrons in the presence of strong interactions and disorder. Can such a model be realized as a fixed-point of a generic theory of strongly-interacting electrons in the presence of disorder? Can we start from a single-band Hubbard model with disorder, and end up with such two-band fixed point, with emergent local conservation laws?

37 10' 0. Electrons in doped silicon appear to separate into two components: localized spin moments and itinerant electrons P 2 [ " T(K) F1G. 1. Temperature dependence of normalized susceptibility g/gp, ~; of three Si:P,B samples with dia'erent normalized electron densities, n/n, =0. 58, I.I, and 1.8. Solid lines through data are a guide to the eye. M. J. Hirsch, D.F. Holcomb, R.N. Bhatt, and M.A. Paalanen, PRL 68, 1418 (1992) M. Milovanovic, S. Sachdev and R.N. Bhatt, PRL 63, 82 (1989) A.C. Potter, M. Barkeshli, J. McGreevy, T. Senthil, PRL 109, (2012)