Quantum matter without quasiparticles

Transcription

1 HARVARD Quantum matter without quasiparticles Frontiers in Many Body Physics: Memorial for Lev Petrovich Gor kov National High Magnetic Field Laboratory, Tallahassee Subir Sachdev January 13, 2018 Talk online: sachdev.physics.harvard.edu

2 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.

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

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

5 Theories of metallic states without quasiparticles without disorder Breakdown of quasiparticles arises from long-wavelength coupling of electrons to some bosonic collective mode. In all cases this can be written in terms of a continuum theory with a conserved momentum. The critical theory has zero resistance, even though the electron quasiparticles do not exist. Need to add irrelevant (umklapp) e ects to obtain a non-zero resistivity, but this not yield a large linearin-t resistivity.

6 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.

7 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

8 The Sachdev-Ye-Kitaev (SYK) model Pick a set of random positions

9 The SYK model Place electrons randomly on some sites

10 The SYK model Entangle electrons pairwise randomly

11 The SYK model Entangle electrons pairwise randomly

12 The SYK model Entangle electrons pairwise randomly

13 The SYK model Entangle electrons pairwise randomly

14 The SYK model Entangle electrons pairwise randomly

15 The SYK model Entangle electrons pairwise randomly

16 The SYK model This describes both a strange metal and a black hole!

17 H = The SYK model (See also: the 2-Body Random Ensemble in nuclear physics; did not obtain the large N limit; T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, and S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981)) 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` c i c j c k c` µ X i c i c j + c j c i =0, c i c j + c j c i = ij Q = 1 X c i N c i i c i c i U ij;k` are independent random variables with U ij;k` = 0 and U ij;k` 2 = U 2 N!1yields critical strange metal. S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Kitaev, unpublished; S. Sachdev, PRX 5, (2015)

18 The SYK model Feynman graph expansion in J ij.., and graph-by-graph average, yields exact equations in the large N limit: G(i!) = 1 i! + µ (i!) G( =0 )=Q., ( ) = U 2 G 2 ( )G( ) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

19 The SYK model T = 0 fermion Green s function is singular: G( ) 1 p at large. (Fermi liquids with quasiparticles have G( ) 1/ ) S. Sachdev and J. Ye, PRL 70, 3339 (1993)

20 The SYK model T = 0 fermion Green s function is singular: G( ) 1 p at large. (Fermi liquids with quasiparticles have G( ) 1/ ) T > 0 Green s function has conformal invariance, and is dephased at the characteristic scale k B T/~, which is independent of U. G e 2 ET 1/2 T sin( k B T /~) E measures particle-hole asymmetry. A. Georges and O. Parcollet PRB 59, 5341 (1999) S. Sachdev, PRX, 5, (2015)

21 The SYK model T = 0 fermion Green s function is singular: G( ) 1 p at large. (Fermi liquids with quasiparticles have G( ) 1/ ) T > 0 Green s function has conformal invariance, and is dephased at the characteristic scale k B T/~, which is independent of U. G e 2 ET 1/2 T sin( k B T /~) E measures particle-hole asymmetry. The last property indicates eq ~/(k B T ), and this has been found in a recent numerical study. A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arxiv:

22 The SYK model G R (!)G A (!) ReG R (!) ImG R (!)! Green s functions away from half-filling So the Green s functions display thermal damping at a scale set by T alone, which is independent of U.

23 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

24 arxiv: [pdf, other] Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models Authors: Xue-Yang Song, Chao-Ming Jian, Leon Balents Building a metal Comments: 17 pages, 6 figures See also A. Georges and O. Parcollet PRB 59, 5341 (1999) t... U Large N equations G(i! n ) 1 = i! n + µ 4 (i! n ) zt 2 0 G(i! n), 4 ( ) = U 2 0 G( )2 G( ),

25 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) There is a low coherence scale E c t 2 0/U, with SYK criticality at T>E c and heavy Fermi liquid behavior for T<E c. Z

26 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) There is a low coherence scale E c t 2 0/U, with SYK criticality at T>E c and heavy Fermi liquid behavior for T<E c. From the Kubo formula, we have the conductivity Re[ (!)] / t 2 0 Z f(! + ) d f( )! where A(!) =Im[G R (!)]. A( )A(! + ) At T>E c,usinga(!)! 1/2 F (!/T ), this yields the bad metal behavior e2 h t 2 0 U 1 T ; h e 2 T E c

27 34th Jerusalem Winter School in Theoretical Physics: New Horizons in Quantum January 5, 2017, this https URL ;(v2) added refs, figures, remarks Subjects: Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics Quantum Physics (quant-ph) 2. arxiv: [pdf, other] Title: Magnetotransport in a model of a disordered strange metal Authors: Aavishkar A. Patel, John McGreevy, Daniel P. Arovas, Subir Sachdev Comments: 18+epsilon pages + Appendices + References, 4 figures Subjects: Strongly Correlated Electrons (cond-mat.str-el); Disordered Systems and (cond-mat.dis-nn); High Energy Physics - Theory (hep-th) 3. arxiv: [pdf, other] Title: Topological order in the pseudogap metal Authors: Mathias S. Scheurer, Shubhayu Chatterjee, Wei Wu, Michel Ferrero, An Sachdev Comments: 8+15 pages, 10 figures ps://arxiv.org/find Aavishkar Patel

28 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: Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear

29 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 Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear

30 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 ) T sin( T ) 1/2 e 2 ET, 0 apple < Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear

31 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) Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear

32 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 Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear

33 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 E1 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 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!

34 Magnetotransport with mesoscopic homogeneity No macroscopic momentum, so equations describing charge transport are just r I(x) =0, I(x) = (x) E(x), E(x) = r (x). A. M. Dykhne, Anomalous plasma resistance in a strong magnetic field, Journal of Experimental and Theoretical Physics 32, 348(1971). D. Stroud, Generalized e ective-medium approach to the conductivity of an inhomogeneous material, Phys. Rev. B 12, 3368(1975). M. M. Parish and P. B. Littlewood, Non-saturating magnetoresistance in heavily disordered semiconductors, Nature 426, 162(2003), cond-mat/ M. M. Parish and P. B. Littlewood, Classical magnetotransport of inhomogeneous conductors, Phys. Rev. B 72, (2005), cond-mat/ V. Guttal and D. Stroud, Model for a macroscopically disordered conductor with an exactly linear high-field magnetoresistance, Phys. Rev. B 71, (2005), cond-mat/ J. C. W. Song, G. Refael, and P. A. Lee, Linear magnetoresistance in metals: Guiding center di usion in a smooth random potential, Phys. Rev. B 92, (2015), arxiv: [cond-mat.mes-hall]. N. Ramakrishnan, Y. T. Lai, S. Lara, M. M. Parish, and S. Adam, Equivalence of E ective Medium and Random Resistor Network models for disorder-induced unsaturating linear magnetoresistance, ArXiv e-prints (2017), arxiv: [cond-mat.mes-hall].

35 Magnetotransport with mesoscopic homogeneity No macroscopic momentum, so equations describing charge transport are just r I(x) =0, I(x) = (x) E(x), E(x) = r (x). 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. M.M. Parish and P.B. Littlewood, Nature 426, 162 (2003)

36 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 MFL 0a b MFL 0b = 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!

37 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.

38 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. Mesoscopic disorder then leads to linear-in-b magnetoresistance, and a combined dependence which scales as p B 2 + T 2

39 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. Mesoscopic 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.

40 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?

41 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)