Quantum Matter Without Quasiparticles

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

HARVARD Ubiquitous “Strange”,

“Bad”,

or “Incoherent”,

metal has a resistivity, ⇢, which obeys

⇢ T , ⇠ and

in some cases ⇢ h/e2 (in two dimensions), where h/e2 is the quantum unit of resistance. Strange metals just got stranger… B-linear magnetoresistance!?

Ba-122 LSCO

I. M. Hayes et. al., Nat. Phys. 2016

P. Giraldo-Gallo et. al., arXiv:1705.05806 Strange metals just got stranger… Scaling between B and T !?

Ba-122 Ba-122

I. M. Hayes et. al., Nat. Phys. 2016 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 ex- ist. Need to add irrelevant (umklapp) e↵ects to obtain a • non-zero resistivity, but this not yield a large linear- in-T resistivity. Theories of metallic states without quasiparticles in the presence of disorder Well-known perturbative theory of disordered met- • als 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 intermedi- • ate disorder and strong interactions without quasiparticle excitations. Although disorder is present, it largely self-averages at long scales. SYK models • Theories of metallic states without quasiparticles in the presence of disorder Well-known perturbative theory of disordered met- • als 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 intermedi- • ate disorder and strong interactions without quasiparticle excitations. Although disorder is present, it largely self-averages at long scales. SYK models • The Sachdev-Ye-Kitaev (SYK) model

Pick a set of random positions The SYK model

Place electrons randomly on some sites The SYK model

Entangle electrons pairwise randomly The SYK model

This describes both a strange metal and a black hole! 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 N H = U c†c†c c µ c†c (2N)3/2 ij;k` i j k ` i i i i,j,k,X`=1 X cicj + cjci =0 ,cicj† + cj†ci = ij 1 = c†c Q N i i i X 2 2 Uij;k` are independent random variables with Uij;k` = 0 and Uij;k` = U N yields critical strange metal. | | !1

S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015) The SYK model

Feynman graph expansion in Jij.., and graph-by-graph average, yields exact equations in the large N limit: 1 G(i!)= , ⌃(⌧)= U 2G2(⌧)G( ⌧) i! + µ ⌃(i!) G(⌧ =0)= . Q Low frequency analysis shows that the solutions must be gapless and obey 1 A ⌃(z)=µ pz + ... , G(z)= A pz

i⇡/4 2 1/4 where A = e (⇡/U ) at half-ﬁlling. The ground state is a non-Fermi liquid, with a continuously variable density . Q

S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) The SYK model T = 0 fermion Green’s function is singular: • 1 G(⌧) at large ⌧. ⇠ p⌧

(Fermi liquids with quasiparticles have G(⌧) 1/⌧) ⇠ S. Sachdev and J. Ye, PRL 70, 3339 (1993) T>0 Green’s function has conformal invariance, • and is ‘dephased’ at the characteristic scale kBT/~, which is independent of U. ⇠

1/2 2⇡ T ⌧ T G e E ⇠ sin(⇡k T ⌧/ ) ✓ B ~ ◆ measures particle-hole asymmetry. E The SYK model T = 0 fermion Green’s function is singular: • 1 G(⌧) at large ⌧. ⇠ p⌧

(Fermi liquids with quasiparticles have G(⌧) 1/⌧) ⇠ T>0 Green’s function has conformal invariance, • and is ‘dephased’ at the characteristic scale kBT/~, which is independent of U. ⇠

1/2 2⇡ T ⌧ T G e E ⇠ sin(⇡k T ⌧/ ) ✓ B ~ ◆ measures particle-hole asymmetry. E A. Georges and O. Parcollet PRB 59, 5341 (1999) S. Sachdev, PRX, 5, 041025 (2015) The SYK model T = 0 fermion Green’s function is singular: • 1 G(⌧) at large ⌧. ⇠ p⌧

(Fermi liquids with quasiparticles have G(⌧) 1/⌧) ⇠ T>0 Green’s function has conformal invariance, • and is ‘dephased’ at the characteristic scale kBT/~, which is independent of U. ⇠

1/2 2⇡ T ⌧ T G e E ⇠ sin(⇡k T ⌧/ ) ✓ B ~ ◆ measures particle-hole asymmetry. E The last property indicates ⌧eq ~/(kBT ), and this • has been found in a recent numerical⇠ study.

A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, arXiv:1706.07803 The SYK model

GR(!)GA(!)

R ReGR(!) ImG (!)

! Green’s functions away from half-filling FIG. 1. Plots of the Green’s functions in Eq. (3) for =1/4, q = 1, T = 1, A = 1, =1/4 with E R R ~ = kB = 1. Note that while neither ImG (!) or ReG (!) have any definite properties under ! !, $ the product GR(!)GA(!) becomes an even function of ! after a shift by ! =2⇡q T = ⇡/2. So the Green’s functions display thermal ‘damping’S E at a where is thescale Bekenstein-Hawking set by T alone, entropy which densiy is of theindependent AdS horizon. of Indeed,U. Eq. (8)isa SBH 2 general consequence of the classical Maxwell and Einstein equations, and the conformal invariance of the AdS2 horizon, as we shall show in Section III B. Moreover, a Legendre transform of the identity in Eq. (8)wasestablishedbySen[15, 16] for a wide class of theories of gravity in the Wald formalism [17–21], in which is generalized to the Wald entropy. SBH The main result of this paper is the identical forms of the relationship Eq. (7)forthestatistical entropy of the SY state, and Eq. (8) for the Bekenstein-Hawking entropy of AdS2 horizons. This result is strong evidence that there is a gravity dual of the SY state with a AdS2 horizon. Con- versely, assuming the existence of a gravity dual, Eqs. (7)and(8)showthatsuchacorrespondence is consistent only if the black hole entropy has the Bekenstein-Hawking value, and endow the black hole entropy with a statistical interpretation [30]. It is important to keep in mind that (as we mentioned earlier) the models considered here have adi↵erent ‘equation of state’ relating to : this is specified for the SY state in Eq. (A5), for the E Q planar black hole in Eq. (58), and for the spherical black hole in Eq. (B8).

The holographic link between the SY state and the AdS2 horizons of charged black branes has been conjectured earlier [22–24], based upon the presence of a non-vanishing zero temperature entropy density and the conformal structure of correlators. The results above sharpen this link by establishing a precise quantitative connection for the Bekenstein-Hawking entropy [31, 32]ofthe black hole with the UV complete computation on the microscopic degrees of freedom of the SY

5 A strongly correlated metal built from Sachdev-Ye-Kitaev models

Xue-Yang Song,1, 2 Chao-Ming Jian,2, 3 and Leon Balents2 arXiv.org Search 8/3/17, 11(25 PM 1International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China 2Kavli Institute of Theoretical Physics, University of California, Santa Barbara, CA 93106, USA 3Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA (Dated: May 8, 2017)

Strongly correlated metals comprise an enduringCornell puzzle University at the heartLibrary of condensed matter physics. Commonly a highly renormalized heavy Fermi liquidWe gratefully occurs below acknowledge a small support coherence from scale, while at higher temperatures a broad incoherent regime pertainsthe Simons in which Foundation quasi-particle description fails. Despite the ubiquity of this phenomenology, strong correlationsand and member quantum institutions fluctuations make it challenging to study. The Sachdev-Ye-Kitaev(SYK) model describes a 0 + 1D quantum cluster with random all-to-all four-fermion interactions among N Fermion modes which becomes exactly solvable as N , exhibiting A strongly correlatedarXiv.org metal built > fromsearch Sachdev-Ye-Kitaev!1 models a zero-dimensional non-Fermi liquid with emergent conformal symmetry and complete absence of quasi- Xue-Yang Song,1, 2 Chao-Ming Jian,2, 3 and Leon Balents2 particles. Here we study a lattice of complex-fermion SYKSearch dotsor Article with ID random inter-site quadratic hopping. 1International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China Combining the imaginary time path integral2Kavli with Institutereal of TheoreticalAlltime papers path Physics, integral University formulation, of California, Santa we Barbara, obtain CA 93106, a heavy USA Fermi liquid to incoherent metal crossover in3 fullStation detail, Q, Microsoft including Research, Santa thermodynamics, Barbara, California 93106-6105, low temperature USA (Dated: May 23, 2017) Landau quasiparticle interactions, and both electrical(Help and | Advanced thermal search conductivity) at all scales. We find ⇢ linear in temperature resistivity in the incoherentProminent systems regime, like the high-T andc cuprates a Lorentz and heavy ratio fermionsL displayvaries intriguing between features going two beyond the quasiparticle description. The Sachdev-Ye-Kitaev(SYK) model⌘ describesT a 0 + 1D quantum cluster with universal values as a function of temperature.random all-to-all Ourfour-fermion work exemplifies interactions among anN Fermion analytically modes which controlled becomes exactly study solvable of as aN arXiv.org Search Results ! , exhibiting a zero-dimensional non-Fermi liquid with emergent conformal symmetry and complete absence strongly correlated metal. 1 of quasi-particles. Here we study a lattice of complex-fermion SYK dots with random inter-site quadratic hopping. Combining theBack imaginary to Search time path form integral | Next with 25real resultstime path integral formulation, we obtain a heavy Fermi liquid to incoherent metal crossover in full detail, including thermodynamics, low temperature Landau quasiparticle interactions,The URL and for both this electrical search and is thermalhttp://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1 conductivity at all scales. We find linear in Prominent systems like the high-Tc cupratestemperature and heavy resistivity in theFermi incoherent liquid, regime, and which a Lorentz ratio is howeverL ⇢ varies between highly two universalrenormalized values by the ⌘ T fermions display intriguing features going beyondas a the function quasi- of temperature.strong Our work interactions. exemplifies an analytically For T controlled> E , the study system of a strongly enters correlated the incoher- metal. Showing results 1 through 25 (of c181 total) for au:Balents particle description[1–9]. The exactly soluble SYK models ent metal regime and the resistivity ⇢ depends linearly on tem- 1. arXiv:1707.02308 [pdf, ps, other] provide a powerful framework to study such physics. The perature. These results are2 strikingly/ similar to those of Par- Introduction - Strongly correlated metalsTitle: comprise Fracton an en- topologicalscale E phasesc t0 Ufrom0[21, strongly 27, 28] betweencoupled a spin heavy chains Fermi liquid ⌘ < most-studied SYK4 model, a 0 + 1D quantumduring puzzle cluster at the heart of of condensedN collet matterAuthors: physics. and Georges[28],Gábor Com- B. Halászand an, incoherent whoTimothy studied H. metal. Hsieh For a, variantLeonT BalentsEc, SYK the SYK model2 induces ob- a monly a highly renormalized heavy FermiComments: liquid occurs 5+1 be- pages,Fermi 4 figures liquid, which is however highly renormalized by the Majorana fermion modes with random all-to-all four-fermion tained in a double limit of infinite dimension> and large N. Our low a small coherence scale, while at higherSubjects: temperatures Strongly a Correlatedstrong interactions. Electrons For (cond-mat.str-el)T Ec, the system; entersQuantum the incoher- Physics (quant-ph) broad incoherent regime pertains in which quasi-particle de- ent metal regime and the resistivity ⇢ depends linearly on tem- interactions[10–18] has been generalized to SYKq models 2.model arXiv:1705.00117 is simpler, [pdf and, other does] notSee also require A. Georges infinite and O. Parcollet dimensions. PRB 59, 5341 (1999) We scription fails[1–9]. Despite the ubiquity of this phenomenol- perature. These results are strikingly similar to those of Par- with q-fermion interactions. Subsequent works[19, 20] ex- alsoTitle: obtain A strongly further correlated results metal on built the from thermal Sachdev-Ye-Kitaev conductivity models, en- ogy, strong correlations and quantum fluctuations make it collet and Georges[29], who studied a variant SYK model ob- tended the SYK model to higher spatialchallenging dimensions to study. by The cou- exactly solubletropyAuthors: SYK density models Xue-Yang pro- and Lorentz tainedSong, in Chao-Ming a double ratio[29, limit Jian of 30], infinite Leon in dimensionBalents this crossover. and large N. ThisOur vide a powerful framework to study such physics.Comments: The most- 17Building pages,model 6 figures is simpler, and a does metal not require infinite dimensions. We pling a lattice of SYK4 quantum clusters by additional four- workSubjects: bridges Strongly traditional Correlated... Fermi Electrons liquid (cond-mat.str-el) theory and the hydrody- studied SYK4 model, a 0 + 1D quantum cluster of N Ma- also obtain further results on the thermal conductivity , en- fermion “pair hopping” interactions. Theyjorana obtained fermion modes electrical with random3.namical all-to-allarXiv:1703.08910 four-fermion descriptiont [pdf, tropyother of density] an incoherentU and Lorentz ratio[30, metallic 31] in system. this crossover. This Title: Anomalous Hall effect and topological defects in antiferromagnetic Weyl semimetals: Mn and thermal conductivities completely governedinteractions[10–18] by di has↵usive been generalizedSYK to SYK modelq models andwork Imaginary-time bridges traditional formulation Fermi liquid theory - andWe the consider hydrody- Sn/Ge modes and nearly temperature-independentwith behaviorq-fermion interactions. owing to Subsequenta d works[19,-dimensional3 20] ex- arraynamical of description quantum of an dots, incoherent each metallic with system.N species tended the SYK model to higher spatialSachdev-Ye-Kitaev dimensionsAuthors:... byJianpeng cou- LiuSYK, Leon model Balents and... Imaginary-time model formulation - We consider the identical scaling of the inter-dot andpling intra-dot a lattice couplings. of SYK4 quantum clustersof fermionsbySubjects: additional Mesoscale four-labeleda andd by-dimensional Nanoscalei, j, k array Physics, of quantum (cond-mat.mes-hall) dots, each with;N Materialsspecies Science (cond-mat.mtrl- fermion “pair hopping” interactions. They obtainedsci) electrical of fermions labeled··· by i, j, k , Here, we take one step closer to realism by considering a A toy exactly soluble model··· and thermal conductivities completely4. arXiv:1611.00646 governed by di↵usive [pdf,... ps, other] lattice of complex-fermion SYK clusters with SYK intra- = Ui jkl,xc† c† c c + t1ij,xx c† c (1) modes and nearly temperature-4 independentTitle:behavior Spin-Orbit owingof toa non-FermiDimers and ixNon-Collinear jxliquidkx lx Phases in d Cubic0 i, xDoublej,x0 Perovskites H = Ui jkl,xc† c† c c + tij,xx c† c (1) the identical scaling of the inter-dot and intra-dot couplings.x i< j,k

The basic features can2 be determined by a simple power- counting. Considering for2 simplicity µ = 0, starting from U0 t0 S = d⌧ c¯ ⌧(@⌧ µ)c ⌧ d⌧ d⌧ c¯ ⌧ c¯ ⌧ c ⌧ c ⌧ c¯ ⌧ c¯ ⌧ c ⌧ c ⌧ + c¯ ⌧ c ⌧ c¯ ⌧ c ⌧ . (2) c ix ix 1 2 4N3 ix 1 jx 1 kx 1 lx 1 lx 2 kx 2 jx 2 ix 2 N ix 1 jx0 1 jx0 2 ix 2 x 0 0 x xx X Z Z h X hX0i i

The basic features can be determined by a simple power- counting. Considering for simplicity µ = 0, starting from arXiv.org Search 8/3/17, 11(25 PM

Cornell University Library We gratefully acknowledge support from the Simons Foundation and member institutions

A strongly correlatedarXiv.org metal built > fromsearch Sachdev-Ye-Kitaev models 2 Xue-Yang Song,1, 2 Chao-Ming Jian,2, 3 and Leon Balents2 Search or Article ID 1 International Center for Quantum Materials, School2 of Physics, Peking University, Beijing, 100871, China 1/4 2 1 2 2Kavli Institute of TheoreticalAllt papers0 = 0, Physics, the U University0 term of is California, invariant Santa under Barbara,⌧ CA 93106,b⌧ USAand c b c, = 1/2, and U0 b U0 is irrelevant. Since the SYK2 3Station Q, Microsoft Research,1/4 Santa Barbara, California 93106-6105, USA! ! ! c¯ b c¯, fixing the scaling dimension = 1/4 of the Hamiltonian (i.e.,U0 = 0) is quadratic, the disordered free ! (Dated: May 23, 2017) (Helpfermion | Advanced fields. search) Under this scalingc ¯@⌧c term is irrelevant. fermion model supports quasi-particles and defines a Fermi Prominent systems like the high-Tc cuprates and heavy fermions display intriguing features going beyond the quasiparticle description.Yet The Sachdev-Ye-Kitaev(SYK) upon addition of model two-fermion describes a 0 + coupling,1D quantum cluster under with rescaling, liquid limit. For t0 U0, Ec defines a crossover scale be- random all-to-all four-fermion2 interactions2 among N Fermion modes which becomes exactly solvable as N ⌧ arXiv.orgt bt ,Search so two-fermion Results coupling is a relevant!perturba- tween SYK -like non-Fermi liquid and the low temperature , exhibiting a zero-dimensional0 non-Fermi0 liquid with emergent conformal symmetry and complete absence 4 1 ! of quasi-particles. Here wetion. study a lattice By standard of complex-fermion reasoning, SYK dots this with random implies inter-site a cross-overquadratic from Fermi liquid. hopping. Combining theBack imaginary to Search time path form integral | Next with 25real resultstime path integral formulation, we obtain a heavy Fermi liquid to incoherentthe metal SYK crossover4-like in model full detail, to including another thermodynamics, regime low at temperature the energy scale At the level of thermodynamics, this crossover can be rig- Landau quasiparticle interactions,Thewhere URL and for both the this electrical hopping search and is thermalhttp://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1 perturbation conductivity becomesat all scales. We dominant, find linear in which is orously established using imaginary time formalism. Intro- temperature resistivity in the incoherent regime, and a Lorentz ratio L ⇢ varies between two universal values = 2/ ⌘ T ⌧, ⌧ = 1 as a function of temperature.E Ourc workt0 exemplifiesU0. We an analytically expect controlled the renormalization study of a strongly correlated flow is to the ducing a composite field Gx( 0) N i cix⌧c¯ix⌧0 and a La- metal. Showing results 1 through 25 (of 181 total) for au:Balents SYK2 regime. Indeed keeping the SYK2 term invariant fixes grange multiplier ⌃x(⌧, ⌧0) enforcing the previous identity, one NS P 1. arXiv:1707.02308 [pdf, ps, other] obtains Z¯ = [dG][d⌃]e , with the action Introduction - Strongly correlated metals comprise an en- scale E t2/U [21, 27, 28] between a heavy Fermi liquid Title: Fracton topological phasesc ⌘ 0 from0 strongly coupled spin chains during puzzle at the heart of condensed matterAuthors: physics. Gábor Com- B. Halászand an, incoherentTimothy H. metal. Hsieh For, LeonT < BalentsEc, the SYK2 induces a R monly a highly renormalized heavy FermiComments: liquid occurs 5+1 be- pages,Fermi 4 figures liquid, which is however highly renormalized by the low a small coherence scale, while at higher temperatures a strong interactions. For T > E , the system enters the incoher- Subjects: Strongly Correlated Electrons (cond-mat.str-el)c ; Quantum Physics (quant-ph) 2 broad incoherent regime pertains in which quasi-particle de- ent metal regime and the resistivity ⇢ depends linearly on tem- U 2. arXiv:1705.00117 [pdf, other] See also A. Georges and O. Parcollet PRB 59, 5341 (1999) 0 2 2 scription fails[1–9]. Despite the ubiquity of thisS phenomenol-= ln detperature.(@⌧ Theseµ) results(⌧1 are⌧2 strikingly) + ⌃x(⌧ similar1, ⌧2) to those+ ofd Par-⌧1d⌧2 Gx(⌧1, ⌧2) Gx(⌧2, ⌧1) + ⌃x(⌧1, ⌧2)Gx(⌧2, ⌧1) Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models 4 ogy, strong correlations and quantum fluctuations makex it collet and Georges[29], who studied a variant SYK model0 ob- x 2 3 Authors: Xue-YangX Song, Chao-Ming Jian, Leon Balents Z ⇣ X challenging to study. The exactly soluble SYK models2 pro- tained⇥ in a double limit of infinite dimension and⇤ large N. Our 6 7 vide a powerful framework to study such physics.Comments:+t The most- 17GBuilding xpages,(⌧1model, ⌧62 figures)G isx simpler,(⌧2, ⌧ and1 )a. does metal not require infinite dimensions. We 6 7 (3) 0 0 ... 4 5 Subjects: Strongly Correlated Electrons (cond-mat.str-el) studied SYK4 model, a 0 + 1D quantum cluster ofxxN0 Ma- also obtain further results on the thermal conductivity , en- jorana fermion modes with random3. all-to-allarXiv:1703.08910 four-fermionhXi t [pdf, tropyother density] U and Lorentz⌘ ratio[30, 31] in this crossover. This interactions[10–18] has been generalizedTitle: to SYK Anomalousq models Hallwork effect bridges and traditional topological Fermi defects liquid theoryin antiferromagnetic and the hydrody- Weyl semimetals: Mn with q-fermion interactions. Subsequent works[19,3 Sn/Ge 20] ex- namical description of an incoherent metallic system. Self-consistenttended the SYK model to higher spatial dimensionsAuthors:... byJianpeng cou-equations Liu, Leon Balents... SYK model and Imaginary-time formulation - We consider pling a lattice... of SYK4 quantum clustersThe bySubjects: additional large MesoscaleN four-limit isa andd controlled-dimensional Nanoscale array byPhysics the of quantum (cond-mat.mes-hall) saddle dots, point each conditions with;N Materialsspecies Science (cond-mat.mtrl- fermion “pair hopping” interactions. They obtainedsci) electrical of fermions labeled by i, j, k , ( ) S/G = S/⌃ = 0, satisfied by G···x(⌧, ⌧0) = G(⌧ ⌧0), and thermal conductivities completely4. arXiv:1611.00646 governed by di↵usive [pdf,... ps, other] S T 2 modest and nearly temperature-independentU⌃xTitle:behavior(⌧, ⌧ 0Spin-Orbit) owing= ⌃ to4(⌧ Dimers⌧0) and+ ztNon-CollinearG(⌧ ⌧0)( Phasesz is in the d1 coordination Cubic Double Perovskites = 0 Ui jkl,xc† c† c c + tij,xx c† c (1) the identical scaling of the inter-dot and intra-dot couplings. H ix jx kx lx 0 i,x j,x0 numberAuthors: of Judit the Romhányi lattice of, Leon SYKx i

tioned heavy fermion... systems. We assume t0 U0, which self-averaging established for the SYK model at large N, it is ⌧ S implies strong interactions, and focus onfree the correlated energy, regime hencesu thecient full to study thermodynamics[32].Z¯ = [dc¯][dc]e c , with (repeated Consider species T U0. We show the system has a coherence temperatureS indices are summed over) ⌧ the entropy . A key feature of theR SYK4 solution is an ex- tensive ( N) entropy[13] in the T 0 limit, an extreme / ! 2 2 non-FermiU liquid0 feature. This entropy mustt0 be removed over S = d⌧ c¯ ⌧(@⌧ µ)c ⌧ d⌧ d⌧ c¯ ⌧ c¯ ⌧ c ⌧ c ⌧ c¯ ⌧ c¯ ⌧ c ⌧ c ⌧ + c¯ ⌧ c ⌧ c¯ ⌧ c ⌧ . (2) c ix ix 1 2 4N3 ix 1 jx 1 kx 1 lx 1 lx 2 kx 2 jx 2 ix 2 N ix 1 jx0 1 jx0 2 ix 2 FIG. 1. The entropy and specific heat(inset) collapse to universal x 0 strong0the narrow similaritiesx temperature to window DMFT set by equations thexx the coherence en- X Z Z X hX0i T h i functions of , given t0, T U0(z = 2). C 0(0)T/Ec as ergy Ec. Consequently, we expect that S/N = (T/Ec) for Ec ⌧ ! S S T/Ec 0. Solid curves are guides to the eyes. mathematical structure appearedT ,inE cearlyU 0study, where of the doped universal t-J function model with( = 0) = 0 in- ! The basic features can be determined by a simple⌧ power- counting. Considering for simplicityS Tµ = 0, starting from double large N and infinite dimensiondicating limits: no zero O. Parcollet+A. temperature entropy Georges, in a Fermi 1999 liquid, and ( ) = 0.4648 , recovering the zero temperature S T !1 ··· entropy of the SYK model. The universal scaling collapse 4 in SYK model, we expect that K is only weakly perturbed is confirmed by numerical solution, as shown in Fig. 1. This 4 by small t0. For t0 U0, we indeed have K K t0=0 = implies also that the specific heat NC = (T/Ec) 0(T/Ec), and ⌧ ⇡ | S c/U0 with the constant c 1.04 regardless of T/Ec. For hence the low-temperature Sommerfeld coecient ⇡ free fermions, the compressibility and Sommerfeld coecient

C 0(0) are both proportional to the single-particle density of states lim = S (5) (DOS), and in particular /K = ⇡2/3 for free fermions. Here ⌘ T 0 T Ec ! we ﬁnd /K = ( (0)/c)U /E (U /t )2 1. This can S0 0 c ⇠ 0 0 is large due to the smallness of Ec. Speciﬁcally, compared only be reconciled with Fermi liquid theory by introducing a with the Sommerfeld coecient in the weak interaction limit large Landau interaction parameter. In Fermi liquid theory, 1 t U , which is of order t , there is an “e↵ective mass one introduces the interaction fab via "a = fabnb, where 0 0 0 b enhancement” of m⇤/m t0/Ec U0/t0. Thus the low tem- a, b label quasiparticle states. For a di↵usive disordered Fermi ⇠ ⇠ P perature state is a heavy Fermi liquid. liquid, we take fab = F/g(0), where g(0) is the quasi-particle To establish that the low temperature state is truly a strongly DOS, and F is the dimensionless Fermi liquid interaction pa- renormalized Fermi liquid with large Fermi liquid parame- rameter. The standard result of Fermi liquid theory[32], is ters, we compute the compressibility, NK = @ /@µ T . Be- that is una↵ected by F but K is renormalized, leading to N | 2 cause the compressibility has a smooth low temperature limit /K = ⇡ (1 + F). We see that F (U /t )2 1, so that the 3 ⇠ 0 0 arXiv.org Search 8/3/17, 11(25 PM

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1. arXiv:1707.02308 [pdf, ps, other] Title: Fracton topological phases from strongly coupled spin chains Authors: Gábor B. Halász, Timothy H. Hsieh, Leon Balents Comments: 5+1 pages, 4 ﬁgures Subjects: Strongly Correlated Electrons (cond-mat.str-el); Quantum Physics (quant-ph) 2. arXiv:1705.00117 [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 There is a low coherence scale E t2/U, with SYK Subjects:• Strongly Correlated Electrons (cond-mat.str-el)c ⇠ 0 3. arXiv:1703.08910criticality at[pdfT>E, other]c and heavy Fermi liquid behavior Title:for AnomalousTEc,usingA(!) ! F (!/T ), this yields https://arxiv.org/findthe bad metal behavior ⇠ Page 1 of 5 e2 t2 1 h T 0 ; ⇢ ⇠ h U T ⇠ e2 E ✓ ◆ c arXiv.org Search 8/3/17, 11(25 PM

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1. arXiv:1801.01125 [pdf, other] Title: Topological order and Fermi surface reconstruction Authors: Subir Sachdev Comments: 48 pages, 22 figures. Review article, comments welcome. Partly based on lectures at the 34th Jerusalem Winter School in Theoretical Physics: New Horizons in Quantum Matter, December 27, 2016 - January 5, 2017, this https URL ;(v2) added refs, figures, remarks Subjects: Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph) 2. arXiv:1712.05026 [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 Neural Networks (cond-mat.dis-nn); High Energy Physics - Theory (hep-th) 3. arXiv:1711.09925 [pdf, other] Title: Topological order in the pseudogap metal Authors: Mathias S. Scheurer, Shubhayu Chatterjee, Wei Wu, Michel Ferrero, Antoine Georges, Subir Sachdev Comments: 8+15 pages, 10 figures

https://arxiv.org/find Page 1 of 5 Aavishkar Patel 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 ! M M N H = t (c† c +h.c.) µ c† c µ f † f ri r0i c ri ri ri ri rr ; i=1 r; i=1 r; i=1 h X0i X X N M N 1 r 1 r + g f † f c† c + J f † f † f f . NM1/2 ijkl ri rj rk rl N 3/2 ijkl ri rj rk rl r; i,j=1 X k,lX=1 r; i,j,k,lX =1 A. A. Patel, J. McGreevy, D. P. Arovas and S. Sachdev, arXiv:1712.05026

Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear Infecting a Fermi liquid and making it SYK

2 2 M 2 c c ⌃(⌧ ⌧ 0)= J G (⌧ ⌧ 0)G(⌧ 0 ⌧) g G(⌧ ⌧ 0)G (⌧ ⌧ 0)G (⌧ 0 ⌧), N 1 G(i! )= , (f electrons) n i! + µ ⌃(i! ) n n c 2 c ⌃ (⌧ ⌧ 0)= g G (⌧ ⌧ 0)G(⌧ ⌧ 0)G(⌧ 0 ⌧), c 1 G (i!n)= c . (c electrons) i!n ✏k + µc ⌃ (i!n) Xk 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 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

1/2 1/2 c Cc T 2⇡ T ⌧ C T 2⇡ T ⌧ G (⌧)= e Ec ,G(⌧)= e E , 0 ⌧ < . p 4⇡ c sin(⇡T ⌧) p 4⇡ sin(⇡T ⌧)  1+e E ✓ ◆ 1+e E ✓ ◆ 1/2 1/2 c Cc T 2⇡ T ⌧ C T 2⇡ T ⌧ G (⌧)= e Ec ,G(⌧)= e E , 0 ⌧ < . p 4⇡ c sin(⇡T ⌧) p 4⇡ sin(⇡T ⌧)  1+e E ✓ ◆ 1+e E ✓ ◆ Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear 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

2 E 1 c ig ⌫(0)T !n 2⇡Te !n !n ⌃ (i!n)= ln + + ⇡ , 2J cosh1/2(2⇡ )⇡3/2 T J T 2⇡T E ✓ ✓ ◆ ⇣ ⌘ ◆

2 E 1 c ig ⌫(0) !n e ⌃ (i!n) !n ln | | , !n T (⌫(0) 1/t) ! 2J cosh1/2(2⇡ )⇡3/2 J | | ⇠ E ✓ ◆ Similar results in non-random models by Y. Werman, D. Chowdhury, T. Senthil, and E. Berg, to appear Linear-in-T resistivity Both the MFL and the IM are not translationally-invariant and have linear-in-T resistivities!

2 MFL 1 vF 1/2 =0.120251 MT J cosh (2⇡ ). (v t) 0 ⇥ ⇥ g2 E F ✓ ◆ ⇠ 1/2 IM 1/2 1 ⇤ cosh (2⇡ ) =(⇡ /8) MT J E . 0 ⇥ ⇥ ⌫(0)g2 cosh(2⇡ ) ✓ ◆ Ec [Can be obtained straightforwardly from Kubo formula in the large-N,M limits]

The IM is also a “Bad metal” with IM 1 0 ⌧

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

• Thanks to large N,M, we can also exactly derive the linear- response Boltzmann equation for non-quantizing magnetic ﬁelds…

c c (1 @ Re[⌃ (!)])@ n(t, k, !)+v kˆ E(t) n0 (!)+v (kˆ zˆ) n(t, k, !)=2n(t, k, !)Im[⌃ (!)], ! R t F · f F ⇥ B · rk R 2 ( = eBa /~) (i.e. ﬂux per unit cell) B

2 c MFL vF ⌫(0) 1 dE1 2 E1 Im[⌃R(E1)] L = M sech c 2 2 2 , 16T 2⇡ 2T Im[⌃R(E1)] +(vF /(2kF )) Z1 ✓ ◆ B 2 MFL vF ⌫(0) 1 dE1 2 E1 (vF /(2kF )) H = M sech c 2 B 2 2 . 16T 2⇡ 2T Im[⌃R(E1)] +(vF /(2kF )) Z1 ✓ ◆ B

MFL 1 MFL 2 T s ((v /k )( /T )), T s ((v /k )( /T )). L ⇠ L F F B H ⇠B H F F B s (x ) 1/x2,s (x 0) x0. L,H !1 / L,H ! / Scaling between magnetic ﬁeld and temperature in orbital magnetotransport! 22

[8] A. Y. Kitaev, “Talks at KITP, University of California, Santa Barbara,” Entanglement in Strongly-Correlated Quantum Matter (2015). [9] O. Parcollet and A. Georges, “Non-Fermi-liquid regime of a doped Mott insulator,” Phys. Rev. B 59,5341(1999), cond-mat/9806119 .

[10] T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, “Emergent quantum criticality, Fermi surfaces, and AdS2,” Phys. Rev. D 83,125002(2011), arXiv:0907.2694 [hep-th] . [11] S. Sachdev, “Bekenstein-Hawking Entropy and Strange Metals,” Phys. Rev. X 5,041025(2015), arXiv:1506.05111 [hep- th] . [12] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, “Phenomenology of the normal state of Cu-O high-temperature superconductors,” Phys. Rev. Lett. 63,1996(1989). [13] M. Mitrano, A. A. Husain, S. Vig, A. Kogar, M. S. Rak, S. I. Rubeck, J. Schmalian, B. Uchoa, J. Schneeloch, R. Zhong, G. D. Gu, and P. Abbamonte, “Singular density fluctuations in the strange metal phase of a copper-oxide supercon- ductor,” ArXiv e-prints (2017), arXiv:1708.01929 [cond-mat.str-el] . [14] S. Sachdev, “Holographic Metals and the Fractionalized Fermi Liquid,” Phys. Rev. Lett. 105,151602(2010), arXiv:1006.3794 [hep-th] . [15] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh, “Charge transport by holographic Fermi surfaces,” Phys. Rev. D 88,045016(2013), arXiv:1306.6396 [hep-th] . [16] X.-Y. Song, C.-M. Jian, and L. Balents, “Strongly Correlated Metal Built from Sachdev-Ye-Kitaev Models,” Phys. Rev. Lett. 119,216601(2017), arXiv:1705.00117 [cond-mat.str-el] . [17] Y. Gu, X.-L. Qi, and D. Stanford, “Local criticality, di↵usion and chaos in generalized Sachdev-Ye-Kitaev models,” JHEP 11,014(2017), arXiv:1609.07832 [hep-th] . [18] R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen, and S. Sachdev, “Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography,” Phys. Rev. B 95,155131(2017), arXiv:1612.00849 [cond-mat.str-el] . [19] M. Milovanović,S. Sachdev, and R. N. Bhatt, “E↵ective-field theory of local-moment formation in disordered metals,” Phys. Rev. Lett. 63,82(1989). [20] R. N. BhattMagnetotransport and D. S. Fisher, “Absence of spin with di↵usion inmesoscopic most random lattices,” Phys.homogeneity Rev. Lett. 68,3072(1992). [21] A. C. Potter, M. Barkeshli, J. McGreevy, and T. Senthil, “Quantum Spin Liquids and the Metal-Insulator Transition in Doped Semiconductors,”• No macroscopicPhys. Rev. momentum, Lett. 109,077205(2012) so equations, arXiv:1204.1342 describing charge [cond-mat.str-el] transport. are [22] S. Burdin, D. R.just Grempel, and A. Georges, “Heavy-fermion and spin-liquid behavior in a Kondo lattice with magnetic frustration,” Phys. Rev. B 66I,045111(2002)(x)=0, I,(xcond-mat/0107288)=(x) E(x. ), E(x)= (x). [23] D. Ben-Zion and J. McGreevy,r · “Strange metal from local quantum· chaos,” ArXivr e-prints (2017), arXiv:1711.02686 [cond-mat.str-el] . [24] A. M. Dykhne, “Anomalous plasma resistance in a strong magnetic field,” Journal of Experimental and Theoretical Physics 32,348(1971). [25] D. Stroud, “Generalized e↵ective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12,3368(1975). [26] M. M. Parish and P. B. Littlewood, “Non-saturating magnetoresistance in heavily disordered semiconductors,” Nature 426,162(2003), cond-mat/0312020 . [27] M. M. Parish and P. B. Littlewood, “Classical magnetotransport of inhomogeneous conductors,” Phys. Rev. B 72, 094417 (2005), cond-mat/0508229 . [28] V. Guttal and D. Stroud, “Model for a macroscopically disordered conductor with an exactly linear high-field magnetoresistance,” Phys. Rev. B 71,201304(2005), cond-mat/0502162 . [29] 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 92,180204(2015), arXiv:1507.04730 [cond-mat.mes-hall] . [30] N. Ramakrishnan, Y. T. Lai, S. Lara, M. M. Parish, and S. Adam, “Equivalence of E↵ective Medium and Ran- dom Resistor Network models for disorder-induced unsaturating linear magnetoresistance,” ArXiv e-prints (2017), arXiv:1703.05478 [cond-mat.mes-hall] . Magnetotransport with mesoscopic homogeneity

• No macroscopic momentum, so equations describing charge transport are just I(x)=0, I(x)=(x) E(x), E(x)= (x). r · · r • 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) 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

1 1 a e b e a e b e I + ( )+ I + ( )=0. 2e · 2e · ✓ L ◆ ✓ L ◆

2 2 e 2 MFL MFL 2 2 MFL MFL ( /m) a0a b0b + ab 0a + 0b ⇢e L = B , L e2 e2 q MFL MFL 1/2 MFL MFL ⌘ L + H ab(0a 0b ) 0a +0b e / + ⇢e H B = a b . H ⌘e2 + e2 m MFL + MFL (m = kF /vF 1/t) L H a b 0a 0b ⇠ T (i.e. effective transport scattering rates) a,b ⇠ ⇢e c T 2 + c B2 L ⇠ 1 2 Scaling between B and T at microscopicp orbital level has been transferred to global MR! 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 pB2 + T 2 ⇠ Higher temperatures lead to an incoherent metal • with a local Green’s function and a linear-in-T resistance, but negligible magnetoresistance. 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? Electrons in doped silicon appear to sepa- • rate into two components: localized spin mo- VOLUME 68, NUMBER 9 mentsPHYSICAL and itinerantREVIEW electronsLETTERS 2 MARCH 1992

10' 10 10 0. ~ Si: P, B; n=2.6x10, n/n&=0. 58 Si: P; n=2.9x10,n/pc=0. 78

~ ~~0 '~ 102 10 g++C~ ~ 9o

0 00 10 2— 0~ 10-1 10— Og O OO/w 1.8 0

[ ~ 10 10-' 10 10-" 10 T(K) FIG. 2. Comparison of measured normalized susceptibilities g/gc„, ;; (circles) of insulating Si:P and Si:P,B samples with theoretical calculation (dashed lines) described in the text. 0.1 [ 1P 2 10-1 10" T(K) terms. The ground state is tested for stability against F1G. 1. Temperature dependence of normalized susceptibili- one- and two-electron hops. The susceptibility is then in the manner of BL using the antiferromag- ty g/gp, „~; of three Si:P,B samples with diA'erent normalized calculatedM. Milovanovic, S. Sachdev and R.N. Bhatt, electron densities, n/n, =0.58, I.I, and 1.8. Solid lines through netic spin- & Heisenberg exchange Hamiltonian: data are a guide to the eye. PRL 63, 82 (1989) H QJ(rj)S; S/, A.C. Potter, M. Barkeshli, J. McGreevy, M. J. Hirsch, D.F. Holcomb, R.N. Bhatt, In Fig. 1 we show the enhancement of the susceptiblity where the sumT. Senthil,over i and PRLj includes 109the, 077205electron occupied (2012) andg (relative M.A. toPaalanen,gp.„„~;=3npa/2kaTF) PRL 68, as1418a function (1992)of tem- donor sites. For the exchange constant we use the asymp- perature for all three compensated samples. These data totic hydrogenic result [24] J(r) =Jo(r/a) / exp( —2r/ are qualitatively similar to the uncompensated Si:P data a), where a 16 A (n, / a=0.25 for Si:P) and Jo=I40 [4], i.e., the susceptibility increases towards lower tem- K. The high-temperature curvature of the theoretical peratures approximately as a power law @~T '. As lines in Fig. 2 is due to the asymptotic formula chosen for shown in Fig. 2, this temperature dependence is observed J(r). This formula underestimates J at small r and these over our entire temperature range for insulating samples. are the values relevant at high temperatures. A theoreti- In this figure we have compared the normalized suscepti- cal estimate of the exponent a, obtained from the low- bilities g/gc„„, ec T' ' (gc«,„=npa/3kaT) of compen- temperature behavior of the dashed lines in Fig. 2, is sated and uncompensated Si:P and find, using least- found to be slightly larger in the compensated case. This squares fits, that the exponent a=0.75+ 0.05 for Si:P,B is due to the rearrangement of the electron occupied is somewhat larger than the value of 0.62 ~0.03 for Si:P. donor sites, which results for the compensated case in a The dashed lines in Fig. 2 represent a quantitative distribution differing from the Poisson distribution at theoretical calculation of the susceptibility using no ad- short distances. In summary, the theory with no adjust- justable parameters as explained below. able parameters is in remarkable agreement with the ex- The susceptibility of uncompensated Si:P for n &n, perirnental results for the insulating phase. was explained by Bhatt and Lee (BL) using a quantum The difference between Si:P and Si:P,B is more spin- —, random Heisenberg antiferromagnetic Hamiltoni- dramatic on the metallic side of the MI transition —the an [23]. We have performed a similar computer calcula- susceptibility enhancement is unexpectedly large in tion of the susceptibility of a model appropriate for a Si:P,B at the lowest temperatures even for the very rnetal- compensated doped semiconductor deep in the insulating lic sample n/n, =1.8. As shown in Fig. 3, comparing Si:P phase. The model consists of distributing donor and ac- and Si:P,B samples with similar values of n/n, =1.1, th.e ceptor sites at random in a 3D continuum. The negative- compensated system shows a factor of 3 to 5 larger local ly charged acceptors provide a fixed random Coulomb po- moment fraction than the uncompensated one for T & 0. 1 tential while the electrons are allowed to occupy the K. This is in contrast to the theoretical results of Milo- donor sites with the lowest-self-consistent energies, vanovic, Sachdev, and Bhatt [16) who find for the disor- neglecting quantum-mechanical (hopping, exchange) dered Hubbard model that the fraction of local moments

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