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
HARVARD 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 Resistivity Quantum matter without quasiparticles ⇢ + AT ↵ ⇠ 0 T BaFe2(As1-xPx)2 0 Strange TSDW
Metal Tc ! ⇢ T " ⇠ 2 ⇢ h/e 2.0
AF 1.0 +nematicSDW 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, 184519 (2010) 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 Fermi surface coupled to a gauge field
ky
kx
2 ( i~a) 1 2 [ ,a]= † @ ia r µ + ( ~a) L ⌧ ⌧ 2m 2g2 r⇥ ✓ ◆ Fermi surface coupled to a gauge field
[ ,a]= L ± 2 2 +† @⌧ i@x @y + + † @⌧ + i@x @y 1 2 a +† + † + (@ya) 2g2
⇣ M. A. Metlitski and⌘ S. Sachdev, Phys. Rev. B 82, 075127 (2010) Fermi surface coupled to a gauge field
One loop photon self-energy with Nf fermion flavors:
d2k d⌦ 1 ⌃ (~q, !)=Nf (a) 4⇡2 2⇡ [ i(⌦ + !)+k + q +(k + q )2] i⌦ k + k2 Z x x y y x y Nf ! = | | Landau-damping⇥ ⇤ 4⇡ q (a) | y| (b)
Electron self-energy at order 1/Nf :
2 ~ 1 d q d! 1 (b) ⌃(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 No p3N 4⇡ | | f ✓ ◆ quasiparticles Breakdown of quasiparticles requires strong coupling to a low energy col- • lective 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~ =0(whereJ is the electrical current) the d.c. resistivity of • the critical theory6 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 correc- • tions 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 resis- tivity. Breakdown of quasiparticles requires strong coupling to a low energy col- • lective 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~ =0(whereJ is the electrical current) the d.c. resistivity of • the critical theory6 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 correc- • tions 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 resis- tivity. 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 quasiparti- cle 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 quasiparti- cle 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 • A strongly correlated metal built from Sachdev-Ye-Kitaev models
Xue-Yang Song,1, 2 Chao-Ming Jian,2, 3 and Leon Balents2 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- 1, 2 2, 3 2 particles. Here we study a lattice of complex-fermionXue-Yang SYK dots Song, withChao-Ming random Jian, inter-siteand Leonquadratic Balents hopping. 1International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China Combining the imaginary time path integral2Kavli with Institutereal of Theoreticaltime 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,-dimensional 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 + tij,xx c† c (1) modes and nearly temperature-4 independentTitle:behavior Spin-Orbit owingof toa non-FermiDimers and ixNon-Collinear jxliquidkx lx Phases in 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 Cornell University Library We gratefully acknowledge support from the Simons Foundation and member institutions arXiv.org > search 2 t = 0, the U2 term is invariant under ⌧ b⌧ and c b 1/4c, = 1/2, and U2 b 1U2 is irrelevant. Since the SYK 0 0 ! ! 0 ! 0 2 c¯ b 1/4c¯, fixing the scaling dimension = 1/4 of the Hamiltonian (i.e.,U = 0) is quadratic, the disordered free ! 0 (Help | Advanced search) fermion fields. Under this scalingc ¯@⌧c term is irrelevant. fermion model supports quasi-particles and defines a Fermi Yet upon addition of two-fermion coupling, under rescaling, liquid limit. For t U , E defines a crossover scale be- 0 ⌧ 0 c t2 bt2, so two-fermion coupling is a relevant perturba- tween SYK -like non-Fermi liquid and the low temperature 0 ! 0 4 arXiv.org Search Resultstion. By standard reasoning, this implies a cross-over from Fermi liquid. the SYK4-like model to another regime at the energy scale At the level of thermodynamics,2 this crossover can be rig- Back to Search form | Next 25where results the hopping perturbation becomes dominant, which is orously established using imaginary time formalism. Intro- 2= 2 ⌧ ⌧ 1/4 = / 2 1 2 1 Ec = tt0 /U0,0 the.U We0 term expect is invariant the under renormalizationb and c b flowc, is to the1 2, and Uducing0 b aU0 compositeis irrelevant. field Since theGx SYK(⌧, ⌧20) = cix⌧c¯ix⌧ and a La- 0 1/4 ! ! ! N i 0 The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1c¯ b c¯, fixing the scaling dimension = 1/4 of the Hamiltonian (i.e.,U0 = 0) is quadratic, the disordered free ! ⌃ ⌧, ⌧ SYK2 regime.fermion fields. Indeed Under keeping this scaling thec ¯@ SYKc term2 isterm irrelevant. invariantfermion fixes model supportsgrange quasi-particles multiplier andx( defines0) enforcing a Fermi the previous identity, one ⌧ NS P Yet upon addition of two-fermion coupling, under rescaling, liquid limit. Forobtainst0 U0Z,¯E=c defines[dG a][ crossoverd⌃]e scale, with be- the action 2 2 ⌧ Showing results 1 through 25t (of bt181, so total) two-fermion for coupling au:Balents is a relevant perturba- tween SYK4-like non-Fermi liquid and the low temperature 0 ! 0 tion. By standard reasoning, this implies a cross-over from Fermi liquid. R the SYK4-like model to another regime at the energy scale At the level of thermodynamics, this crossover can be rig- 1. arXiv:1707.02308 [pdf, ps, other] where the hopping perturbation becomes dominant, which is orously established using imaginary2 time formalism. Intro- = 2/ U⌧, ⌧ = 1 Title: Fracton topological phasesEc fromt0 U 0strongly. We expect coupled the renormalization spin chains flow is to the ducing a composite field Gx( 0 0) i cix⌧2c¯ix⌧0 and a La-2 S = ln det (@⌧ µ) (⌧1 ⌧2) + ⌃x(⌧1, ⌧2) + d⌧1d⌧2 Gx(⌧N1, ⌧2) Gx(⌧2, ⌧1) + ⌃x(⌧1, ⌧2)Gx(⌧2, ⌧1) SYK 2 regime. Indeed keeping the SYK 2 term invariant fixes grange multiplier ⌃ x(⌧, ⌧0) enforcing4 the previous identity, one Authors: Gábor B. Halász, Timothy xH. Hsieh, Leon Balents 0 x NS P obtainsZ Z¯ = [dG][d⌃]e 2 , with the action 3 Comments: 5+1 pages, 4 figures X ⇥ ⇤ ⇣ X 6 7 +t2 G (⌧ , ⌧ )G (⌧ , ⌧ ) . R 6 7 (3) Subjects: Strongly Correlated Electrons0 x0 (cond-mat.str-el);1 2 x 2 1 Quantum Physics (quant-ph) 4 5 Seexx0 also A. Georges and O. Parcollet PRB 59, 5341 (1999) 2 2. arXiv:1705.00117 [pdf, other] hXi ⌘ U0 2 2 S = ln det (@⌧ µ) (⌧ ⌧ ) + ⌃ (⌧ , ⌧ ) + d⌧ d⌧ G (⌧ , ⌧ ) G (⌧ , ⌧ ) + ⌃ (⌧ , ⌧ )G (⌧ , ⌧ ) 1 2 x 1 2 1 2 4 x 1 2 x 2 1 x 1 2 x 2 1 Title: A strongly correlated metal built fromx Sachdev-Ye-Kitaev modelsZ0 x 2 3 X ⇥ ⇤ ⇣ X 6 7 Authors: Xue-Yang Song, Chao-Ming+t2 JianG (⌧, ,Leon⌧ )G (⌧ Balents, ⌧ ) . 6 7 (3) Self-consistent 0equationsx0 1 2 x 2 1 4 5 xx Comments: 17 pages, 6 figures hX0i ... The large N limit is controlled⌘ by the saddle point conditions Subjects: Strongly Correlated Electrons (cond-mat.str-el) S/ G = S/ ⌃ = 0, satisfied by Gx(⌧, ⌧0) = G(⌧ ⌧0), 3. arXiv:1703.08910Self-consistent [pdf, other] equations2 t ... U⌃x(⌧, ⌧0The) = large⌃4N(⌧limit is⌧0 controlled) + zt G by(⌧ the saddle⌧0)( pointz is conditions the coordination 0 Title: Anomalous Hall effect and S/ topologicalG = S/ ⌃ = defects0, satisfied in by antiferromagneticGx(⌧, ⌧0) = G(⌧ ⌧0), Weyl semimetals: Mn number of the lattice of SYK2 dots), which obey Sn/Ge t U⌃x(⌧, ⌧0) = ⌃4(⌧ ⌧0) + zt G(⌧ ⌧0)(z is the coordination 0 Authors: Jianpeng Liu, Leon Balentsnumber of the1 lattice of SYK dots), which obey 2 G(i!n) = i!n + µ ⌃4(i!n) zt0G(i!n), 1 2 ... Subjects: Mesoscale and Nanoscale... PhysicsG(i! ) = (cond-mat.mes-hall);i! + µ ⌃ (i! ) zt G (i! ) ,Materials Science (cond-mat.mtrl- n n 2 42 n 0 n ...... ⌃4(⌧) = U20G(⌧2 ) G( ⌧), (4) sci) ⌃4(⌧) = U G(⌧) G( ⌧), (4) 0 4. arXiv:1611.00646 [pdf, ps, other] where where!n¯=!Gn¯(2=(in!(2¯+n)=+1)1)˜⇡tG˜⇡/ (isiis! the) the Matsubara Matsubara frequency. frequency. We We Title: Spin-Orbit Dimers and Non-CollinearsolveG them(i!¯ numerically)= PhasestG and(i re-insert in! ) into Cubic (3) to Double obtain the Perovskites ... solve themfree energy, numerically hence the full and thermodynamics[32]. re-insert into (3) Consider to obtain the Authors:... Judit Romhányi, Leon Balents, George Jackeli free energy,the entropy henceS. A key the feature full of thermodynamics[32]. the SYK4 solution is an ex- Consider tensive ( N) entropy[13] in the T 0 limit, an extreme the entropy S/. A key feature of the! SYK4 solution is an ex- non-Fermi liquid feature. This entropy must be removed over tensivestrongthe ( narrow similaritiesN) temperature entropy[13] to window DMFT in set the by equationsT the the coherence0 limit, en- an extremeFIG. 1. The entropy and specific heat(inset) collapse to universal / ! T functions of , given t0, T U0(z = 2). C 0(0)T/Ec as ergy E . Consequently, we expect that S/N = (T/E ) for Ec non-Fermi liquidc feature. This entropy mustS bec removed over ⌧ ! S T, E U , where the universal function ( = 0) = 0 in- T/Ec 0. Solid curves are guides to the eyes. mathematical structure appearedthe narrow in cearly⌧ temperature 0study of doped window t-J model set byS withT the the coherence! en- FIG. 1. The entropy and specific heat(inset) collapse to universal strong similaritiesdicating no zero temperature to DMFT entropy in aequations Fermi liquid, and T double large N and infinite dimension limits: O. Parcollet+A. Georges, 1999 functions of , given t0, T U0(z = 2). C 0(0)T/Ec as ergy Ec.( Consequently,) = 0.4648 we, recovering expect the that zeroS/ temperatureN = (T/Ec) for Ec ⌧ ! S S T !1 ··· S T/Ec 0. Solid curves are guides to the eyes. , entropy of the SYK4 model. The universal scaling collapse= = mathematical structure appearedT inE cearlyU 0study, where of the doped universal t-J function model with( 0) in SYK0 in-4 model, we expect! that K is only weakly perturbed is confirmed by numerical solution, as shown in Fig. 1. This ⌧ S T = dicating no zero temperature entropy in a Fermi liquid,by small andt0. For t0 U0, we indeed have K K t0=0 implies also that the specific heat NC = (T/Ec) 0(T/Ec), and ⌧ ⇡ | double large N and infinite dimension limits: O. Parcollet+A. Georges,S 1999 c/U0 with the constant c 1.04 regardless of T/Ec. For ( hence the) = low-temperature0.4648 Sommerfeld, recovering coe cient the zero temperature ⇡ S T !1 ··· free fermions, the compressibility and Sommerfeld coe cient entropy of the SYK4 model.C The0(0) universal scaling collapseare both proportional to the single-particle density of states lim = S (5) (DOS), and in particularin SYK /4Kmodel,= ⇡2/3 for we free expect fermions. that Here K is only weakly perturbed is confirmed by numerical⌘ T 0 solution, as shown in Fig. 1. This ! T Ec 2 we find /K = by( 0(0) small/c)U0/tE0c. For(U0/tt00) 1.U This0, we can indeed have K K = S ⇠ t0=0 impliesis alsolarge thatdue to the the specific smallness of heatE .NC Specifically,= (T/E comparedc) 0(T/Eonlyc), and be reconciled with Fermi liquid theory by⌧ introducing a ⇡ | c S c/U0 with the constant c 1.04 regardless of T/Ec. For hence thewith low-temperature the Sommerfeld coe cient Sommerfeld in the weak interaction coe cient 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 thefree interaction fermions,fab via the "a compressibility= fab nb, where and Sommerfeld coe cient 0 0 0 b enhancement” of m⇤/m t0/Ec U0/t0. Thus the low tem- a, b label quasiparticleare both states. proportional For a di↵usive disordered to the Fermi single-particle density of states ⇠ C⇠ 0(0) P perature state is a heavy Fermilim liquid=. S liquid,(5) we take f(DOS),ab = F/g(0), and where in particularg(0) is the quasi-particle /K = ⇡2/3 for free fermions. Here To establish that the⌘ lowT temperature0 state is truly a strongly DOS, and F is the dimensionless Fermi liquid interaction pa- ! T Ec 2 rameter. The standardwe find result / ofK Fermi= ( liquid0(0) theory[32],/c)U /E is (U /t ) 1. This can renormalized Fermi liquid with large Fermi liquid parame- S 0 c ⇠ 0 0 ters, we compute the compressibility, NK = @ /@µ T . Be- that is una↵ected by F but K is renormalized, leading to is large due to the smallness of Ec. Specifically,N | compared2 only be reconciled with Fermi liquid theory by introducing a cause the compressibility has a smooth low temperature limit /K = ⇡ (1 + F). We see that F (U /t )2 1, so that the with the Sommerfeld coe cient in the weak interaction limit3 large Landau⇠ interaction0 0 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 = fab nb, 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 Cornell University Library We gratefully acknowledge support from 2 the Simons Foundation 11 and member institutions t = 0, the U2 term is invariant under ⌧ b⌧ and c b 1/4c, = 1/2, and U2 b 1U2 is irrelevant. Since the SYK 0 0 ! ! 0 ! 0 2 c¯ b 1/4c¯, fixing the scaling dimension = 1/4 of the Hamiltonian (i.e.,U = 0) is quadratic, the disordered free ! 0 where z is the coordination numberarXiv.org of the lattice > fermionsearch under fields. consideration Under this scaling and wec ¯@⌧c haveterm is regularized irrelevant. thefermion free model energy supports by subtracting quasi-particles the and defines a Fermi 1 µ/T part for free fermion, i.e.,G (i! ) = , and addingYet upon back additionT ofln(1 two-fermion+e ). coupling, One switches under rescaling, to Helmholtzliquid limit. free energy For t0 whichU0, Ec dependsdefines a crossover scale2 be- 0 n i!n 2 2 ⌧ 2 t0 bt0, so two-fermion coupling is a relevant perturba- tween SYK4-like non-Fermi liquid11 and the low temperature on “universal” particle number density /N by! a legendre transformation2 = ⌦/N + µ 1//4N, and obtain entropy2 1 2 density by tion. By standardt0 2= 0, reasoning, the U0 term this is invariant implies under a cross-over⌧ b⌧ and1/4c fromb c,Fermi = liquid.1/2,2 and U01 2 b U0 is irrelevant. Since the SYK2 @ N t0 = 0, the U0 term1 is/4 invariant under ⌧ Fb⌧ and!c b c!, N = 1/2, and U0 b U!0 is irrelevant. Since the SYK2 S/N = F . The entropy for SYK4 (i.e. vanishingthe SYKt0)4-like agrees1/4c¯ modelb with toc¯, another fixing the results the regime scaling! in at dimension Ref the energy! 5 and= scale1/ entropy4 of the At (FigHamiltonian the 1)level approaches! (i.e., of thermodynamics,U0 = 0) is identically quadratic, this the crossover disordered can free be rig- @T c¯ b c¯,! fixing the scaling dimension = 1/4 of the Hamiltonian (i.e.,U0 = 0) is quadratic, the disordered free where z is the coordination number of the! latticefermion under fields. consideration Under this scaling and wec ¯@⌧c haveterm is regularized irrelevant. thefermion free model energy supports by subtracting quasi-particles the and defines a Fermi wherefermion the hopping fields. perturbation Under this scaling becomesc ¯@ c dominant,term is irrelevant. which is fermionorously model established supports quasi-particles using imaginary and defines time a formalism. Fermi Intro- regardless of t0/U0 the universal(Help | Advanced ln 2 for search high) temperature1 Yet upon (not addition shown of two-fermion in theµ/⌧ T coupling, figure). under The rescaling, entropyliquid is limit. significantly For t U , E reduceddefines a crossover for scale be- part for free fermion, i.e.,G0(i!n) = = ,2/ and adding back T ln(1+e ). One switches to Helmholtz free energy0 which0 c⌧ depends, ⌧ = 1 Ec i!Yetnt0 uponU0. addition We2 expect of2 two-fermion the renormalization coupling, under flow rescaling, is to the liquidducing limit. a For compositet0 U0, field⌧Ec definesGx( a0 crossover) N scalei cix⌧ be-c¯ix⌧0 and a La- small temperature by the presence of two-fermion2 hopping.2 t0 bt0, so two-fermion coupling is a relevant perturba- tween SYK4-like⌧ non-Fermi liquid and the low temperature on “universal” particle number densitySYKt regime./btN,by so Indeed! two-fermion a legendre keeping coupling transformation the SYK is a termrelevant invariantperturba-= ⌦/ fixesN +tweenµ grange/N SYK, and4 multiplier-like obtain non-Fermi⌃ entropy(⌧, liquid⌧0) enforcing density and the low by the temperature previous identity, one 20 ! 0 tion. By2 standard reasoning, this2 implies a cross-over from Fermi liquid. x @ arXiv.org Search1 @ Resultstion.N By standard@ reasoning, this implies a cross-overF from FermiN liquid. ¯ NS P The compressibilityS/N = is obtainedF . The entropy as K for= SYKN4 (i.e.or K vanishing= the1/ SYK(t0)24-likeF agrees). model The with to plot another the results in regime Fig. in at 1 Ref the shows energy 5 and scale the entropyobtains resultsAt (FigZ the= using 1)level[ approachesdG of][ thermodynamics, thed⌃]e first identically, with derivative this the crossover action can be rig- @T N @µ @ N the SYK4-like where model the hoppingN to another perturbation regime at becomes the energy dominant, scale which isAt theorously level of established thermodynamics, using imaginary this crossover time formalism. can be rig- Intro- regardless of t0/U0 the universal ln 2 for high temperature (not shown in the figure). The entropy is significantly2 reduced for Back to Search form | Next 25where results the hopping= 2/ perturbationq becomes dominant, which is orously establishedR using imaginary⌧, time⌧ = formalism.1 Intro- method (which agrees with that found in Ref 1 as well asEc a large-t0 U0. Wecalculation expect the renormalization (unpublished)). flow is to the ducing a11 composite field Gx( 0) N i cix⌧c¯ix⌧0 and a La- small temperature by the presence of two-fermion2= 2 hopping. ⌧ ⌧ 1/4 = / 2 1 2 1 Ec = tt0 /U0,0 the.U We0 term expect is invariant the under renormalizationb and c b flowc, is to1 the2, and Uducing0 b U a0 compositeis irrelevant. field Since theGx SYK(⌧, ⌧20) = cix⌧c¯ix⌧ and a La- 0 SYK1/4 2 regime. Indeed2 keeping! the! SYK2 term invariant fixes! grange multiplier ⌃x(⌧, ⌧0) enforcingN i the previous0 identity, one The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1c¯ 1 @b c¯, fixing the scaling dimension@ = 1/4 of the Hamiltonian (i.e.,U0 = 0) is quadratic,2 the disordered free SYK2 regime.! Indeed keeping the SYK2 term invariant fixes grange multiplierU¯ ⌃=x(⌧, ⌧0) enforcing⌃ NS the previousP identity, one The compressibilitywhere z is the coordination is obtained number asof theK lattice= fermion underN fields.or considerationK Under= this1 scaling/ and( 2 F wec ¯@⌧).c haveterm The isregularized irrelevant. plot in the Fig.fermion free 1 model energy shows supports by the subtracting quasi-particlesobtains resultsZ0 the and using defines[dG][ the ad Fermi]e first , with derivative the action N @µ @ N NS 2 P2 1S = Yet uponln addition det (@ of⌧ two-fermion µ)µ/ (T⌧1 coupling,N ⌧2) under+ ⌃ rescaling,x(⌧1, ⌧2) liquid+ limit.d⌧ For1dt⌧2 U ,¯E=defines aG crossoverx⌃(⌧1, ⌧ scale2) be-Gx(⌧2, ⌧1) + ⌃x(⌧1, ⌧2)Gx(⌧2, ⌧1) part for free fermion, i.e.,G0(i!n) = , and adding back T ln(1+e ). One switches to Helmholtz free energyobtains0 which0 Z c depends[dG][d ]e , with the action i!n 2 2 ⌧ 4 method (which agrees with that found in Reft 1bt as, so well two-fermion as a coupling large- isq a relevantcalculationperturba- (unpublished)).tween SYK0 4-like non-Fermi liquid and theR low temperature Showing results 1 through 25 0(ofx! 1810 total) for au:Balents 11Z x 2 3 on “universal” particleVI. number HEAVY density FERMI/N tion.byX a By legendre standard LIQUID reasoning, transformation this PHENOMENOLOGY implies a= cross-over⌦/N + fromµ /NFermi, and liquid. obtain entropyX density by @ N ⇥ F N ⇤ ⇣ R6 7 S/ = 2 6 2 7 N @TF . The entropy for SYK4 (i.e.+t vanishingtheG SYKxt0()4⌧-like agrees1, ⌧ model2)G withx to(⌧ another the2, ⌧ results1) regime. in at Ref the energy 5 and scale entropyAt (Fig the 1)level approaches of thermodynamics, identically6 thisU crossover can be rig- 7 (3) 1. arXiv:1707.02308 [pdf, ps, other0 ] where0 the hopping perturbation becomes dominant, which is orously established using imaginary4 time0 formalism. Intro-2 2 5 where z is the coordinationregardless number of of thet0/ latticeU0 the under universal consideration ln 2 for high and we temperature haveS regularized= (not shownln the det free( in@⌧ energy theµ figure).) ( by⌧1 subtracting⌧ The2) + entropy⌃x(⌧ the1, ⌧ 2 is) significantly+ d⌧1d⌧2 reduced2 for Gx(⌧1, ⌧2) Gx(⌧2, ⌧1) + ⌃x(⌧1, ⌧2)Gx(⌧2, ⌧1) xx 2 U 1 1 µ/T 0Ec = t /U . We expect the renormalization flow is to the ducing a composite field Gx(⌧ ,0⌧0) = 4 cix⌧2c¯ix⌧ and a La-2 part for free fermion, i.e.,G0(i!n) = Title:, and addingFracton back topologicalT ln(1+ ephasesh ).i One from switches0 0strongly to Helmholtzx coupled freespin energy chains which depends 11 0 x N i 0 small temperaturei!n by the presence of two-fermionXS = hopping.ln det (@⌧ µ) (⌧⌘1 ⌧2) + ⌃x(⌧1, ⌧2) + d⌧1d⌧Z2 ⌃ ⌧, ⌧ Gx(2⌧1, ⌧2) Gx(⌧2, ⌧1) + ⌃x(⌧1, ⌧2)Gx(⌧2, ⌧1) 3 VI.SYK HEAVY2 regime. Indeed FERMIX2 keeping LIQUID the SYK2 term PHENOMENOLOGY invariant fixes grange multiplier x( 0) enforcing⇣ X the6 previous identity, one 7 on “universal” particle number densityAuthors:/N by A.Gábor a legendre Quasi-particle B. Halász transformation, Timothy1 @ =H. residue⌦ Hsieh/2N +@,µ Leon and /N, Balents and⇥ “Bad” obtain entropy Fermi density liquid by0 ⇤ NS 4 6 P 7 The compressibility is obtained as K = N or Kx =+ 1/( F ).⌧ The, ⌧ plot in⌧ , Fig.⌧ 1. shows theobtains resultsZ¯ = using[dG][ thed⌃]xe first 2 , with derivative the6 action 3 7 @ N N @µ F t0 @2GNx0 ( 1 2)Gx( 2 1) Z 46 57 (3) S where z is the coordination number of the lattice under considerationX and weNN have regularized the free energy by subtracting the ⇣ X /N = @TF . The entropy for SYK4 (i.e.Comments: vanishing t0 5+1) agrees pages, with 4 the figures results2 in Ref 5 and⇥ entropy (Fig 1) approaches identically⇤ 6 7 1 + ⌧ µ/,Txx⌧ 0 ⌧ , ⌧ . R 6 7 methodpart for (which free fermion, agrees i.e., withG0(i!n that) = found, and adding in Ref backt 10 asT wellGln(1x0 ( as+e1 ahX large-).2i) OneGxq( switchescalculation2 1) to Helmholtz (unpublished)). free energy which depends 46 57 (3) regardless of t0/U0 the universalCoherence ln 2 forSubjects: high temperature Stronglyi! nCorrelated (not shownA. inElectrons Quasi-particle the figure).scale (cond-mat.str-el); The entropy residue is significantly Quantum and “Bad”⌘ Physics reduced Fermi (quant-ph) for liquid 2 on “universal” particle number density /N by a legendreSeexx0 also transformation A. Georges and =O. ⌦Parcollet/N + µ PRB/N ,59 and, 5341 obtain (1999) entropy density2 by ... ˜ small temperature by the presence2. arXiv:1705.00117 of@ two-fermion hopping. [pdf, Nother] hXi F N U0 2 2 pz t S/N = F . The entropy for SYK (i.e.The vanishing2 larget )N agreeslimit with is the controlled results in Ref 5 by and⌘ the entropy saddle (Fig 1) pointapproaches conditions identically 0 @T 1 @ 4 @ 0 S = ln det (@⌧ µ) (⌧1 ⌧2) + ⌃x(⌧1, ⌧2) + d⌧1d⌧2 Gx(⌧1, ⌧2) Gx(⌧2, ⌧1) + ⌃x(⌧1˜, ⌧2)Gx(⌧2, ⌧1) ˜ The saddle point condition forK imaginary-time= N K = / F Green’s function is (assuming zero0 chemical 4 potential,t0 t0, Ec ) The compressibility is obtainedregardless as ofTitle:t0/UN0 theA@µ strongly universalor ln correlated1 2 for( @2 high). Thetemperature metal plot built in (not Fig. shownxfrom 1 shows inSachdev-Ye-Kitaev the the figure). results The using entropy the models is first significantly11Z derivative reducedx 2 for p 3 U0 Coherence S/NN VI.G = HEAVY S/ FERMI⌃X= 0, LIQUID satisfied scale PHENOMENOLOGY by Gx(⌧, ⌧0) = G(⌧X ⌧0), 2 ⇣ 2 2 ⇥ ⇤ 6 ⌘ 7 ⌘ small temperature by the presence of... two-fermion hopping. 6 pz7 t˜ method (which agrees with that foundAuthors: in Ref 1 as Xue-Yang well as a large- Songq ,calculation Chao-Ming+Thet (unpublished)). Jian largeGx0 (⌧1,, LeonN⌧2)Glimitx(⌧ 22Balents, ⌧ is1) controlled. by the saddle point 46 conditions 57 (3) 0 Self-consistentwhere z is the coordination number of⌃ the lattice⌧1,@⌧ under consideration= ⌃ 0equations⌧ and@2 we have⌧ regularized+ the free⌧ energy⌧ by subtracting the ˜ , ˜ The saddle point condition for imaginary-timex=( 0) = 4/( Green’s0) zt functionG( 0)( isz (assumingis the coordination zero chemical potential,t0 t0 Ec ) The compressibility is obtained as1K N @Nµ or K 1 (µ/Txx2 F0 ). The plot in0 Fig. 1 shows the results using the first derivative p U part forComments: free fermion, i.e.,1G 170(i! npages,) = i! , and adding6 figures back T ln(1+ eS@hX/).NN i OneG switches= S to/ Helmholtz ⌃ = free20, energy satisfied which2 depends by G (⌧, ⌧0) =2G(⌧ ⌧0), 2 0 ... n ⌘ x ⌘ ⌘ method (whichon “universal”G agrees(i particle! with) number that= found densityi! in/ RefN byThe 1A.⌃ a as legendre( welli large Quasi-particle! transformation as), aN large-limit⌃q (calculation is=⌧ residue⌦) controlled/N=+ µ and (unpublished))./N,U and “Bad” byobtainG the entropy( Fermi⌧2 saddle) densityG liquid( by point⌧) conditions+ 2t˜ G(⌧ ). (6.1) @ numberN of the latticeF of SYKN dots),0 which obey 0 S/N Subjects:= F . The entropy Strongly for SYK (i.e. Correlated vanishing t ) agrees Electrons with the⌃x results(⌧, ⌧ in(cond-mat.str-el)0) Ref= 5 and⌃4 entropy(⌧ (Fig⌧0 1)) approaches+ zt G identically(⌧ ⌧0)(z is the coordination @T 4 S0/ G1 = S/ ⌃ = 0, satisfied by 0Gx(⌧2, ⌧0) =2 G(⌧ ⌧0), 2 regardlessVI. of HEAVYt0/U0 the universalCoherence FERMI ln 2 LIQUIDforG high(i temperature!) PHENOMENOLOGY = (noti shown! in the ⌃ figure).scale(i!) The, entropy⌃( is⌧ significantly) = U reducedG for(⌧) G( ⌧) + 2t˜ G(⌧). (6.1) 3. arXiv:1703.08910 [pdf, other] number of the lattice2 of SYK dots),0 which obey 0 2 ... ˜ small temperature by the presence of two-fermion⌃ hopping.⌧, ⌧ = ⌃ 1 ⌧ ⌧ + ⌧ ⌧ 2 pz t0 x( 0The) 2 large 4N( limit is0 controlled) zt G by( the saddle0)( pointz is conditions the coordination The saddlet point condition for imaginary-time1 U@ G(i@ ! Green’s) = functioni! + µ is0 (assuming⌃ (i! ) zerozt chemicalG(i! ), potential, t˜ t , E˜ ) Rescaling functions as The compressibility is obtained as KVI.= HEAVYN or K = FERMI1/( 2 F ).n LIQUID The plot in PHENOMENOLOGY Fig. 1n shows the results4 using then first derivative0 n 0 0 c Title: Anomalous HallN @effectµ and S@ /N topologicalG = S/ ⌃ = defects in antiferromagnetic G ⌧, ⌧ = G ⌧ ⌧ Weyl semimetals:p2 Mn U0 N 0, satisfied1 by x( 0) ( 0), 2 ⌘ ⌘ ... methodA. (which Quasi-particle agrees with that found residue in Ref... 1 asnumber and well “Bad” as a large- ofq Fermi thecalculation lattice liquid (unpublished)).G of(i! SYK) 2 = dots),i! + whichµ ⌃ obey(i! ) zt G(i! ), Rescaling functionsSn/Ge as ⌃x(⌧, ⌧0) = ⌃4(⌧ ⌧0)2+nzt G(⌧2 ⌧0n)(z is the coordination4 n 0 n ... 1 ⌃4(...⌧) = U0G(0⌧) 2G ( 2⌧), 2 (4) GA.(i! Quasi-particle) = i!number residue⌃( ofi! the) and, lattice “Bad”⌃(⌧ of) SYK Fermi= U dots), liquid0G which(⌧)2 G obey( 2⌧) + 2t˜0G(⌧). (6.1) Authors: Jianpeng! Liu, Leon Balents 1 ⌃4( ⌧) = U 0G(⌧ ) G(2 ⌧), ⌃(i!) (4) VI. HEAVY FERMI LIQUIDG(i PHENOMENOLOGY!n) = i!n + µ ⌃4(i!n) zt0G(i!n),t˜2 ! 1 2 pz ¯ 0 ⌃(i!) The saddle point condition for imaginary-time! = Green’s, function⌧ = ⌧ is (assuming˜ , G zero¯(i! )! chemical = i!=+ µ˜ potential, ⌃ (i! !) t˜zt,G (i! t)⌃, , E˜ ! =) t˜2 . ... Rescaling functionsSubjects:¯ as Mesoscale¯ and Nanoscale...Ec GPhysics(in ¯˜) (cond-mat.mes-hall);n 2 t¯0G42 (in 0)0 p n 0Materials(ic ¯p)zU¯ Science0 (cond-mat.mtrl- (6.2) The saddle... point condition for imaginary-timewhere!¯ =! Green’sn ,=... function(2⌧¯n=+⌧ is1)E (assumingc⇡,/ isG zero ( thei! chemical¯ ) Matsubara= t˜⌘ potential,0G2(i!t˜0) frequency., ⌘ t⌃0, (E˜ic!¯ ) =) We . (6.2) ˜ A. Quasi-particle¯ residue andwhere “Bad”⌃4(⌧ Fermi)!=n liquid˜=U2(20Gn(⌧2+) 1)G⇡(/ ⌧)is, the Matsubarap2 frequency.U(4)0 ˜ We sci) E G˜ (i!¯)=⌃4(⌧)¯=tGU0(G(i⌧)!G()⌧˜), ⌘(4) ⌘ t0 ˜ 1 c ! E2 c 2 G2 ( i!¯)= tG (i!) ⌃(i!) t0 solve1 them numerically2˜ and2 re-insert2 intot˜2 (3) to obtain the G(i!) = i! ⌃(i!), !⌃(⌧)==! U⌃G!(⌧,) G⌃solve(⌧ ⌧=) + them2t G⌧ numerically(⌧). ⌧ + ˜ and⌧pz .¯ re-insert0 into(6.1) (3) to obtain the The saddle point condition forG imaginary-time(i ) !¯ =i Green’s,0(i function⌧¯) = ⌧ is(E (assuming˜) , UG zero¯0(G0i!( chemical¯ ) G=(t˜ potential,G) (i2!tt˜00)G, ( )t⌃0, (E˜ic!¯ ) =) . (6.1) (6.2) 4. arXiv:1611.00646... [pdf, ps, other] c 0 p U ... where! =! n¯= n (2+n + 1)⇡⇡/ is the Matsubara⌘ 2 frequency.⌘ 0 We whereE˜c n¯freeG(2( energy,i!¯)=1)˜tG˜ hence(iis!) the the Matsubarafull thermodynamics[32]. frequency.t˜0 We Consider free1 energy,solveG hence them(i2 !¯ numerically2)= the fulltG˜2 and( thermodynamics[32].i re-insert!) into (3) to obtain the Consider The saddleRescaling point functions equationThe as saddle isRescaling point formatted functions equationTitle: asSpin-Orbit as is formattedG (Dimersi!) = i! and⌃(i as!), Non-Collinear⌃(⌧) = U0G(⌧) G( ⌧) + Phases2t0G(⌧). in Cubic Double(6.1) Perovskites Rescaling ... solve them numerically and re-insert into (3) to obtain1 the freeS energy,the entropy hence theS. full A thermodynamics[32]. key feature of the Consider SYK1 4 solution is an ex- The saddleRescaling point functions equation... as is formattedthe entropy as . A key feature of the SYK4 solutionz is an ex- Authors: Judit Romhányi! , Leon Balents, George ¯Jackeli⌃(i!) 1 ˜ 2 ! !¯ = , free⌧¯ = ⌧ energy,E˜ the, entropyG¯(Gi¯!¯( hencei)!¯=)=S.t˜ AGtG˜ (⌃ keyi( the!i(!i)!), feature) full⌃⌃¯((ii!¯!¯ of) thermodynamics[32].)== the⌃ SYK(i!).4/t˜solution isz ant2 ex-= Considert (6.2) ˜ ¯ ¯! ˜ c tensive⌃¯¯(i!¯)=0 ( ⌃(N⌃i(!i)!˜)) entropy[13]/t˜ int˜z =2 the T t0 limit,2 an extreme Rescaling!¯ = , ⌧¯ = ⌧Ec, !¯G=(Gi!E¯˜(,i)!¯=⌧¯)==t˜⌧0EG˜tG, (i(!Gi¯!(Gi)¯!¯),(i)!¯=)=t˜⌃GtG˜((i(!i!!¯)),) =⌃⌃¯¯((ii!¯!¯))== ⌃E(i!.c)./t˜ t˜t˜0 =t˜0 t (6.2)2 (6.2) tensive˜ c c ( tensiveN)˜0 entropy[13] ( N)˜ entropy[13]1/ ˜˜˜ in in the theT T 0 limit,2 0 an limit, extreme! an extreme Rescaling RescalingE˜c Ec the entropyE¯ c1S/. AEc keyt˜0 featuret0t0t0 of the! SYK¯ 4 solution¯ is an ex- 1/G¯non-Fermi(i!¯G)non-Fermi (i!=¯ liquid) ( feature.i=!¯ liquid This(⌃¯ feature.i( entropy!¯i!¯ )) must This⌃¯!(i⌃ be!¯( entropy) removedi,!¯ )) over must⌃(i be!¯ ) removed, over The saddleThe point saddle equation point equation is formatted is formattedG¯non-Fermi( as asi!¯ ) = liquid( feature.i!¯ This˜˜ ⌃¯( entropyi!¯ ))˜ must⌃¯(i be!¯ ) removed, over The saddle point equation is formatted as tensivethe ( narrowtheN) narrow temperature entropy[13]t˜0 temperature t windowE0 c in set the by window⇡TE thec the coherence0 set limit, by ⇡ en- the an the extremeFIG. coherence 1. The entropy en- and specificFIG. heat(inset) 1. The collapse entropy to universal and specific heat(inset) collapse to universal ˜t˜ E˜ /˜ t˜ ˜ ! T T 1 c E t˜0 t˜ ⇡ functions of , given t0, T U0(z = 2). C 0(0)T/Ec as ¯ t ¯G¯(i!¯ ) = 1 (ergyi!¯ c E0⌃c¯.(i!¯ Consequently,)) 0⌃¯(i!¯ ), E wec2 expect that S/N = (T/Ec) for Ec FIG.functions 1. The of entropy, given andt0, T specificU0(z heat(inset)= 2). C collapse0(0)T/E toc as universal ¯ G(i!¯)= i!¯the⌃ narrow(¯Ginon-Fermi¯!¯(i)!¯ ) ˜ = temperature˜(ergyi liquid!¯ ¯ E⌃c feature.¯.(i!¯ Consequently,)) window¯ ⌃¯( Thisi!¯ )¯, entropy set we by2¯ expect must the the be that removed coherenceS/N = over(T en-/Ec) for ⌧ Ec! S G(i!¯)= i!¯ ⌃(i!¯)t0 Ec ⌃(¯⇡⌧) = ¯G(¯⌧) G( ⌧¯¯) + 2G(¯⌧¯)S, ¯T/Ec 0. Solid curves are guides to the(6.3) eyes. T ⌧ ! S ˜ EU˜ t˜ T, tE˜ c U0,˜ where⌃ the(¯⇡⌧ universal) = functionG(¯⌧) (G(= 0)⌧¯=)0+ in-2G(¯⌧), (6.3) t 1 c 0 ¯ ¯0 2 ¯ Ec ¯ S! FIG.functions 1. TheT/Ec entropy of0. Solid and, given curves specifict are0, heat(inset)T guides toU the0( collapsez eyes.= 2). to universalC 0(0)T/Ec as ¯ ¯G¯(i!¯ ) = U (ergyi!¯strong theE⌃c¯.⌃( narrowi(¯⌧!¯ ) Consequently,similarities=)) G(¯⌧⌃)T¯⌧ temperatureG(,i(!E¯⌧¯)c)+, 2G(¯⌧U), 0 ,to we where window 2DMFT expect the set universal that by equationsS T theS/(6.3) thefunctionN coherence= (T( /E= en-c)0) for= 0 in- ! Ec G(i!¯)= i!E¯˜c ⌃(i!¯) dicating¯¯ no zero2 temperature¯ entropy¯ in a Fermi liquid,¯ and T ⌧ ! S ˜ ˜ ˜ ⌃⌃(¯⌧)⌧= G=¯(¯⌧)⌧G¯( ⌧¯) +⌧2G¯(¯⌧), ⌧ + 2 ⌧ ,S T (6.3) Ec t ⇡ G G ˜ G that, given˜ that,˜ given 1, is1, is an an equation equation0 set with set only withEdimensionlessc only(¯parameters.dimensionless) As we argued(¯parameters. in) the text, the( low As¯ energy) we behaviorSt argued2/ is=(¯ in)S the/ text, the lowfunctions energyT/E behaviorc of E 0., given Solid is t0 curves, T areU0( guidesz = 2). toC the(6.3) eyes.0(0)T/Ec as Ec U t0 t˜0 ergy Ec.( Consequently,dicating ) = 0 no. 4648 zero we, recovering temperature expect the that˜ zero entropy temperatureN in( aT FermiEc) for liquid, and c ˜ For⌧ t⌧≪U, a singleT universal, Ec UcoherenceS0,T where!1 scale the universal appears··· functionEc = ( = 0) =2 0 in- ! ⌧ ! S that, given in theE1,c realm is of an Fermi equation liquid theory.¯ Then set the¯ spectral with2 ¯ weightentropy onlyA¯(¯!) should¯ ofdimensionless the contain SYK a quasiparticlemodel.¯ The contribution, universal parameters. which scalingU because collapse AsS wet˜ arguedT in/Ec the0. text, Solid the curves low are guides energy to the behavior eyes. is inthat, thet˜ realm given t˜ of Fermi1, is an liquid equation⌃(¯ theory.⌧ set) = withG Then only(¯⌧T)⌧,GdimensionlessE the( ⌧¯) spectral+U2G((¯parameters.⌧) weight, 4 )A As=(¯! we0). should4648 argued in contain the, recovering text, atheS quasiparticle lowT= energy the˜ (6.3) zero= behavior contribution, temperature is which because ˜ mathematical0 it contains0 ⌧structure no parameters, mustappeared have a residue of inO(1). cearly From the 0 scalingstudy, where in (6.2), of the it doped follows universal that the t-J width function model of the “coherence with( E0)c in= SYK0 in-4 model, we expect! that K is only weakly perturbed Ec For⌧ t≪U, a singledicating universal ¯ no ⌧is confirmedzero coherenceS T˜¯ temperature by!1 numerical scale solution, entropy asappears··· shown in in aFig.S Fermi 1.T This liquid, and ˜ it containsin the realmregion” no of parameters, attributed Fermi liquid to quasiparticle theory. must formation Then have the in aA residuespectral(¯!) is multiplied weight of byO(1).EAc(¯in!)A From( should!) (i.e. in contain the physical scaling a units) quasiparticle and in the (6.2), quasiparticle contribution, it follows2 which thatby smallU because thet width. For t of theU , “coherence we indeed have K K = Ec dicating noentropy zero temperature of the˜ SYK entropy4 model.¯ in The a Fermi universal˜ liquid, scaling and 0 collapse0 0 t0=0 that, given 1,in is the an realm equation of Fermi set liquid with theory. only dimensionless Thenimplies the spectral also that theparameters. specificE weightc t˜0 heat NCA=(¯(!T/)Ec As should) 0(T/ weEc), and containt argued a in quasiparticle the⌧ text, the contribution, low⇡ energy| which behavior because is that,˜ given 1, is an equationit contains setresidue no with parameters, of our only modeldimensionless (i.e., must the integral have a of residueA(parameters.!) within of theO(1). “coherence As From we region”) the argued scaling is Z in˜ in= the (6.2), text,1 it which follows the is characteristic low that energy the of width a behavior of the “coherence is in SYK model, we expect that K is only weakly perturbed t0 t˜0 double large N and infinite dimension( limits:) ¯=! O.0.4648 Parcollet+A.⇠ t0 ,U0 recovering⌧ ˜ Georges,! theS ˜ zero1999 temperaturec/U0 with the constant c 1.04 regardless4 of T/Ec. For ⌧ ⌧≪ region” attributed“bad” Fermi liquid. to quasiparticle formation( ¯ hence in A(¯is the) confirmed)= low-temperature is0 multiplied.4648˜ by Sommerfeld numerical, by recoveringEc in coeA solution, ( cient the)E (i.e.c zero as= in shown temperature physical in Fig. units) 1. and This the quasiparticle⇡ For tit containsU, a singleregion” no parameters, attributed universal to quasiparticle must formationcoherenceS haveT¯ ! in!1 aA residue(¯!) is multiplied scale of byO(1).E cappears···in A From(!) (i.e. in the physical scaling units)˜ and in thet˜ (6.2), quasiparticlefree fermions, it follows the compressibility that and the Sommerfeldt width coet of cient theU “coherence K K = in the realm of Fermi liquid theory. Then the spectral weight A(¯S)T should!1 contain a quasiparticle··· contribution,˜ whichEc U because0 by small 0. For 0 0, we indeed have t0=0 residue of our model (i.e., the integralentropy of A of(!) the within SYK the4 “coherencemodel.¯ ! The region”)Ec universalt˜0 is Z NC scaling==T/E collapse1 whichT/E is characteristic of a in the realm of Fermi liquidresidue theory. of our model Then (i.e., the integral the of spectralA(!entropy) within the of weight “coherenceimplies the SYK also region”)4Amodel.(¯ that is) theZ should The specific= universal heat contain1 which scalingt˜0 is( characteristicU0 a collapsearec) quasiparticle both0( of proportional ac), and to the single-particle contribution, density of states⌧ which because ⇡ | it contains no parameters, must have a residue of O(1). From the scaling in (6.2),¯ it follows thatC thet˜00(0) widthU0 of⇠ the˜ “coherence⌧ in SYKc/U 4withmodel, the constant we expectc that1.04K regardlessis only of weaklyT/E . Forperturbed region” attributed to quasiparticleB. Grand canonical potential formation in Fermi liquid theory, in compressibilityA(¯!) and is Sommerfeldlim multiplied=⇠ coeS cient ⌧ by Ec in(5) A((DOS),!S) (i.e. and in particular inin physical SYK /4K model,= ⇡02/3 units) for we free expect fermions. and the that Here quasiparticleK is only weakly perturbedc “bad”“bad” Fermi Fermi liquid. is confirmed byhence numerical the low-temperature⌘ T solution,0 T E as Sommerfeld shown in coe Fig. cient 1. This ⇡ region” attributed to quasiparticle formation in A¯(¯!) is multipliedis byconfirmedE˜c in A( by!) (i.e. numerical in physical! solution, units)c as and shown the quasiparticle in Fig. 1. This 2 it contains no parameters, must have a residue of O(1). From the scaling in (6.2), itwe followsfind /K ˜= ( 0(0) thatby/˜c)U smallfree/E thec fermions,(Ut width/.t ) For the1.t compressibility ofThis can theU , “coherence we and indeed Sommerfeld have coeK= cientK = In Landau’s Fermi liquid theory, the energy is a functional of a series of “quasi-particle”˜ states labeled by a, b, we have Ec by smallt0 0 t0. For0 t00 U00, we indeed0 have K K t0=0 t0=0 impliesimplies also also that! that the the specificE specificc t˜0 heat heatNCNC = ((TT//EE) ) (T(/TE/E), and), and S= ⇠ residue of ourresidue model (i.e., of the our integral model of A(! (i.e.,) within the the integral “coherence of region”)Ais (large) isdue withinZ to the smallness the= “coherence of 1Ec which. Specifically, is characteristic region”) comparedc c 0 0 is ofonlyc aZc be reconciled˜ with Fermiare liquid both1 which theory proportional by⌧ introducing is characteristic⌧ to a the single-particle of a⇡ density| of⇡ states| t˜0 U0 t0 / U0 . / 1 0 ¯ 0 0 1 ˜ 2C 0(0)S c U0c/withU with the constant the constantc 1 04c regardless1.04 regardless of T Ec. For of T/E . For E µ = " n + f (n nA)(n !n ) µ n = const + ⇠ (" f¯ n ⌧µ)n + Ef¯( n ) A(6.4) !S large Landau⇠ interaction parameter.⌧0 In Fermi liquid theory, 2 c region” attributed to quasiparticle formationB. Granda a canonicalab in potentiala a b(¯ in Fermi)with is liquida the multiplied Sommerfeld theory, compressibilitya coe cient bya inand the Sommerfeld weakc ain interaction coe ( cient limit) (i.e. in physical units) and the quasiparticle “bad” Fermi liquid. N 2 hence hence theb low-temperature the low-temperature Sommerfeldb Sommerfeld 2 lim coe coe= S cientcient (5) (DOS), and in particular⇡ /K⇡= ⇡ /3 for free fermions. Here “bad” Fermi liquid. B. Granda canonicala,b potentiala in Fermi liquida theory,b compressibility1 a and Sommerfeld coe cient X X t0X U0, whichX is ofX order t , there⌘ XT is an0 T “e↵ectiveE massc one introduces thefree interaction fermions,fab via the "a compressibility= b fab nb, where and Sommerfeld coe cient 0 ! E˜c t˜0 free fermions, the compressibility and Sommerfeld2 coe cient , 0 enhancement” of m /m t /E U /t . Thus the low tem- a, b label quasiparticle states.we For find a di↵usive /K disordered= ( 0(0) Fermi/c)U0/Ec (U0/t0) 1. This can residue of our model (i.e., theIn Landau’s integralwhere n Fermia na denotes liquid of the theory, occupationA(! the number) energy within of the is quasiparticle a functional the state of “coherenceand a superscriptseries⇤ of 0 “quasi-particle” denotes0 thec occupation region”)0 0 states number labeled of the is by aZ, b, we have = are both1 proportional which isto the characteristic single-particle density of a of states " , µ⇠= C⇠ = 00(0) ˜ are both proportionalP S to the single-particle⇠ density of states “reference” state one starts with to define a fab, and we takeperature it here to state be the is state a heavy with Fermi0,i.e., liquidna µ.=0 na. In the second liquid,t0 we take Ufab0= F/g(0), where g(0) is the quasi-particle 2 ¯ is large due tolim theC smallnessh i= S0(0) of Ec. Specifically,(5) compared, only be reconciled with Fermi liquid theory by introducing a B. GrandIn canonical Landau’sidentity potential we Fermi use f to in liquid replace Fermi1fab theory,for liquid simplicity. theory,the energy compressibility is a functional and Sommerfeld of a series of coe “quasi-particle” cient 1 states⇠ labeled by(DOS),a b,⌧ we and have in particular /K = ⇡ /3 for free2 fermions. Here ¯ 0 0 0 To establish that the⌘ lowlimT temperature0 T =¯ S stateE0 c is truly a strongly¯ 2DOS, and (5)F is the dimensionless Fermi liquid interaction pa- / ⇡ / “bad” Fermi liquid. E Defineµ =Ea = "a" nf +b nb, we havef for(n the partitionn )(n functionn ) in grandµwith canonicaln the= const ensemble Sommerfeld+ as (introduce(" coef a hubbard-stratonovich n cientµ)n in+ thef ( weakn ) interaction(6.4) limit(DOS),large and Landau in particular interaction parameter.K = 2 In3 Fermi for free liquid fermions. theory, Here a a ab a a b b a ⌘ T !0 Ta Eb a a rameter. The standard result / of Fermi= liquid theory[32],/ / is / variable N ) 2 renormalized Fermi liquid with large Fermic liquid1 parame-2 we find K ( 0 (0) c)U0 Ec (U0 t0) 1. This2 can B. Granda P canonicala,b1 potential0 a in0 Fermi liquid!a theory,b compressibilitya 0 ↵ and1 Sommerfeld2 one introduces coeS cient the interaction⇠ f via " = f n , where X X ters, f¯ wet20X computeU0 the, which compressibility,X is ofX orderNK =t @ ,/@ thereµ ¯. Be-X is anthat “e ectiveis una↵ected¯ mass byweF but findK is renormalized,/K = ( leading0(0)/ toc)U0/abEc (aU0/t0b) ab b 1. This can E µ = " n + (E µ ) f (n (Ea µ)nna )(2 (na na) n ) µ n = const + (0" fT n µ)n + f ( n ) (6.4) In Landau’s Fermi liquid theory, the energy is= aa functionalae N = ofab aise series a large ea of due “quasi-particle”b to b the smallness statesa labeled of Ec. by Specifically,a,Nab, we| have comparedb a ⇡2 onlya be reconciled with Fermi liquid theory by introducing a 0 N Z 2 cause the compressibility has a smooth low temperature limit /K = (12+ F). We see that, F (U /t )2 1,S so that the ⇠↵ where na, n denotes the occupation numberna=0,1 a of the quasiparticleP enhancement” state and superscript of m⇤/m 0 denotest0/E thec occupationU0/t0. Thus number the of3 lowthe tem- a b label0 quasiparticle0 states. For a di usive disordered Fermi a a X ais,bXlargeY due to the smallnessa of E .a Specifically,b compared a only be⇠ reconciled with Fermi liquid theory by introducing a X X " ,with the SommerfeldX coe cient inµ⇠c the=X weak⇠ interactionX= 0 limit largeX Landau interaction parameter. In Fermi liquidP theory, “reference” state one starts with to define a fab ,2 and we take it here to be 2 the state with 0,i.e., na µ=0 na. In the second f = F/g g 1 i a na (E µ)nperature state is a heavy (E µ+i Fermi1 liquid. liquid, we take ab (0), where (0) is the quasi-particle 0 0 2 f¯ a a 2 f¯ 0 a1 h i2 E µ = "ana + fab0(na f¯ n )(n=b fn ) µdwith e nat the= const SommerfeldeU + = ("a d coe f¯e n cient[1 + eµ t)n a in+)] the. f¯( weakn(6.5)a) interaction↵ (6.4) limit one introduceslarge Landau the interaction interactiona, b fab via parameter. "a = Infab Ferminb, where liquid theory, In Landau’sidentity we Fermi use to liquid replacea ab theory,bfors2 simplicity.f¯ theP 0 energy0, which is as2 functional isf¯ of orderb 0 of, there a series is an of “e “quasi-particle”ective mass states labeled by , we have b where na, n denotes thena=0 occupation,1 Z numbera of the quasiparticleZ a state and superscript 0 denotes the occupation numberDOS, of and theF is the dimensionless Fermi liquid interaction pa- N 2 Ea = " f¯ n0X Y To establish Y that the1 low2 temperature state is truly a strongly B. Granda Define canonicala,b a a potentialb b, we have fora in the partitionenhancement” Fermi function liquida in of grandm theory,⇤ canonical/mb t ensemble/E compressibilityU as (introduce/t a. Thus a hubbard-stratonovich the low and tem- Sommerfelda, b 0label quasiparticle coe cient states. For a di↵usive disordered " = Fermi X “reference”X The state saddle one point starts condition with for readst to0X defineU"0a,, whichfab, andX we is of takeX order it here0t toc, be there the0 stateX0 is an with “eµ↵=ective0,i.e., massna µ=0 = n one. Inrameter. the introduces second The the standard interaction result offab Fermivia liquida theory[32],b fab nb is, where variable ) renormalized⇠ Fermi0 liquid⇠ with large Fermi liquidh i parame-a P 0 ¯ 1 peraturei s state1 is a heavy Fermi liquid. liquid, we take fab =1 F/g(0), where g(0) is the quasi-particle identity we use f toP replace f for simplicity. 0 ¯ 0 0, 2 where na, n denotes the occupation number of the quasiparticleab enhancement” state= and superscript of f m⇤2/m 0 denotest0/E thec occupationU0/t0.(6.6) Thus numberNK the of= lowthe@ / tem-@µ ¯ a bthatlabel is quasiparticle una↵ected¯ by states.F but K Foris a renormalized, di↵usive disordered leading to Fermi a ters, (Ea µ+i s we) compute the compressibility, . Be- E µ = " n + (E µ ) f (nf¯ (Ea µ1)nn+a e )(2 (na na) n ) µ n = const + (" fT n µ)n + f ( n ) (6.4) =a ae 0 N = ab e a a ea b b a 0 a b a 2 a Define Ea = ""a , ff¯ b n , we have for theTo partition establishX function that theµ⇠ inlow= grand temperature canonical⇠n state= ensemblen is truly as a (introduce stronglyN | a hubbard-stratonovichDOS, and F is the⇡ dimensionless Fermi liquid interaction2 P pa- “reference” state one starts withN to define Za ab, andb we2 takeperature it here to state be the iscause state aPheavy the with compressibility Fermi 0,i.e., liquida hasµ.=0 a smootha. In the low second temperature limitliquid, /K we= take(12+ Ffab). We= F see/g that(0),F where(U0/gt(0)0) is1, the so quasi-particle that the ¯ a X a,bnXa=0,1 Ya ah i a rameter.b The standard3 resulta of Fermi⇠ liquid theory[32], is In Landau’sidentity we Fermi use f to liquid replacevariablefab theory,for simplicity.) X the energyX is arenormalized functional Fermi of liquid a series withX large of “quasi-particle” Fermi liquidX parame- statesX labeled by Xa, b, we have 0 P Toi establishn 2 that the low temperature 2 state is truly a strongly DOS, and F is the dimensionless Fermi liquid interaction pa- = " ¯ a a 2 f¯ (Ea µ)na ¯ 2 f¯ (Ea µ+i that is una↵ected by F but K is renormalized, leading to Define Ea a f b nb, we have for the partition= function dµ e in grandters, we canonical computee ensembleµ the= f compressibility, as (introduced2 e a[1 hubbard-stratonovich+NKe = @ )]./@µ T . Be-(6.5) 0 (E ¯ ) (Ea )na 2 ( a na¯) 2 = , e s2 f renormalizedN = P e Fermi e liquid s2 f with large Fermi liquidN | parame- rameter.⇡ The standard result of Fermi2 liquid theory[32], is variable ) where na, na denotes thena=0 occupation1 Z cause number thea compressibility of the quasiparticleZ has a smootha state low and temperature superscript limit 0 denotes/K = (1 the+ F occupation). We see that numberF (U0/t0 of) the1, so that the 1 Z X Y P Y 3 1 P 0 ¯ na=0,1 a0 0 0 ⇠2 Theµ saddle point conditionX forµ readsters, f weX2 computeY" , the compressibility, NK = @ /@µ ¯. Be- µ =that is una↵ected¯ = by F but K is renormalized, leading to E µ = “reference”"ana +(E state) onefab starts(na(E witha )nna to)(2 (n defineabna) n a) fabµ, and wena take= const it here+ to be( the"a statefT withn µ0,i.e.,)na +na µf=0( nan.a In) the second(6.4) = e N = e ea b N | b ⇡2 2 Z 2 2 / h i N 2 ¯ = , a causeP the i i compressibilitysa na ¯ 1 (Ea hasµ)na a smooth low temperature¯ (E limita µ+i K = (12+ F). We see that F (U0/t0) 1, so that the identitya we use af,bntoa 0 replace1 = fab for simplicity.d e = 2 f ae = d e a2 f [1 + e (6.6)b )]. 3(6.5) a X X Y ¯ (Ea µ+i s) ⇠ 0 ¯ f 1 + e ¯ X X ¯ = , s2 f P a aX s2 f X a X X Define Ea = "a f nna 0,21 weZ have for the partitionX function2 Z in grand canonical ensemble as (introduce a hubbard-stratonovich i abna Xb¯ (Ea µ)na Y ¯ (Ea µ+i Y = d e 2 f e = d e 2 f [1 + e )]. (6.5) 0 ¯ ¯ , variablen =0,1The) s saddle2 f pointP conditiona for reads s2 f a where na na denotes theXa occupationZ numberY of the quasiparticleZ Y state and superscript 0 denotes the occupation number of the P ¯ The saddle point condition for reads i s 1 f 2 0 “reference” state one starts with to define " (aE, µfab), and we take= (E ita µ here)na 2 to( a bena) the state with µ = 0,i.e., na µ=0 = n . In the second = e N = e e ( E µ+i ) (6.6) a f¯ 1 + e a s h i ¯ Z i s 1 a identity we use f to replace fab for simplicity.= na=0,1 a X P (6.6) ¯ + (Ea µ+i s) 0 X f a 1 e X Y Define Ea = "a f¯ b n , we have for the partitionX function in grand canonical ensemble as (introduce a hubbard-stratonovich b 2 2 i a na ¯ (Ea µ)na ¯ (Ea µ+i = d e 2 f e = d e 2 f [1 + e )]. (6.5) variable ) ¯ ¯ n =0,1 s2 f P a s2 f a P a Z ¯ Z (E µ ) X (E µ)n f ( Yn )2 Y = e N = e a a e 2 a a Z The saddle point condition for reads na=0,1 a P X X Y i s 1 = (E µ+i ) (6.6) 2 f¯ 1 + e a s 2 i a na (E µ)n a (E µ+i = d e 2 f¯ e a Xa = d e 2 f¯ [1 + e a )]. (6.5) s2 f¯ P s2 f¯ nXa=0,1 Z Ya Z Ya The saddle point condition for reads i 1 s = (E µ+i ) (6.6) f¯ 1 + e a s Xa Cornell University Library We gratefully acknowledge support from 3 the Simons Foundation ( µ ) Fermi liquid is extremely strongly interacting. Comparing to with ⇢ and= e memberH N and institutionsU the identity evolution operator the e↵ective mass, one has F (m /m)2. U = e i( µ )(t0 t f )e i( µ )(t f t0) describing evolving for- ⇠ ⇤ H N H N Real time formulation- While imaginary time formula- ward from t t (with Keldysh label +) and backward 3 0 ! f tion is adequate for thermodynamics, it encounters di - (Keldysh label ) identically. Paralleling the imaginary-time ( µ ) Fermi liquid is extremely strongly interacting. Comparing to witharXiv.org⇢ = e H N and U >the identitysearchevolution operator culties in addressing transport due to di culty of analytic development, we introduce collective variables Gx,ss0 (t, t0) = 2 i s s0 i( µ )(t0 t f ) i( µ )(t f t0) continuationthe e↵ective to mass, zero onereal hasfrequencyF (m⇤/ inm) the. presence of the Ui c=c¯ e andH N⌃x,ss withe Hs, sN0 = labelingdescribing Keldysh evolving con- for- ⇠ N ixt ixt0 0 ± emergentReal time low energy formulation- scale EWhile. Instead imaginary we reformulate time formula- the tour,ward and from integratet0 outt f the(with fermionic Keldysh fields label to+ obtain) and backwardZ¯ = c ! problemtion is in adequate real time for using thermodynamics, Keldysh path it integral. encounters The di - [dGP(Keldysh][d⌃]eiNS labelK [32],) with identically. the Keldysh Paralleling action the imaginary-time Tr[⇢U] Keldyshculties formalism in addressing calculates transport the partition due to function di cultyZ of= analytic development, we introduce collective variables G , (t, t0) = Tr[⇢] R x ss0 i s s0 ⌃ , = continuation to zero real frequency in the presence of the N i cixtc¯ixt and x,ss0 with s s0 labeling Keldysh con- 0 ± ¯ emergent low energy scale Ec. Instead we reformulate the tour,(Help and integrate | Advanced out the search fermionic) fields to obtain Z = P iNS K problem in real time using Keldysh path integral.t f The [2dG][d⌃]e [32], with the Keldysh action z Tr[⇢U] U0 2 2 KeldyshiS = formalismln det[ calculates(i@ + µ) (t thet partition0) ⌃ (t, functiont0)] Z = dtdt0 ss0G , (t, t0) G , (t0, t) ⌃ , (t, t0)G , (t0, t) K t x Tr[⇢] R x ss0 x s0 s x ss0 x s0 s x ss t0 x 4 x X X0 Z h X arXiv.org SearchX Results 2 + t0 ss0Gx,ss0 (t, t0)Gx0,s0 s(t0, t) (6) x0 x t f 2 hXi z i UBack0 to Search2 form | Next2 25 results iS = ln det[ (i@ + µ) (t t0) ⌃ (t, t0)] dtdt0 ss0G (t, t0) G (t0, t) ⌃ (t, t0)G (t0, t) K t x x,ss0 x,s0 s x,ss0 x,s0 s x ss t0 x 4 x X X0 Z h X X 2 The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1 + t0 ss0Gx,ss0 (t, t0)Gx0,s0 s(t0, t) (6) where ⌃x in the determinant is to be understood as the ma- tions, defined as 'c/q = ('+ ' )/2 and in Fourier space: x0 x z ± trix [⌃x,ss ]hX andi acts in Keldysh space.i We obtain the nu- t f 0 Showing results 1 through 25 (of 181 total) for au:Balents merical solution to the Green’s functions[32] by solving for iS ' = dtdt0 ⇤1(t t0)@t'c,p(t)@t'q, p(t0) t0 the saddle point of S K. We plot in Fig. 2 the spectral weight Xp Z 1 ⇥ A(!) Im G (!)(G is retarded Green function) at fixed 1. arXiv:1707.02308 ( )⇤ (t t0)' ([t)pdf' , ps(t0), other+ .] (8) where ⌃⇡ in theR determinantR is to be understood as the ma- tions, definedp as '2 / = (' c,p ' q),/2p and in Fourier space: ⌘ x 4 c q + ··· U0/T = 10 for Ezc/T = 0, 0.09, 1, 9, which illustrates the Title: Fracton ±topological phases from strongly coupled spin chains trix [⌃ ] and acts in Keldysh space. We obtain the nu- t f x,ss0 Here the first term arises from the ln det[ ] and⇤ the second from crossover between the SYK4 and Fermi liquid regimes. For 2 · merical solution to the Green’s functions[32] by solving for the hoppingiS ('t =Authors:) term indtdt (6). Gábor0 The⇤1( tB. function tHalász0)@t'c ,p((p,t ))Timothy@ encodest'q, p(t0) theH. Hsieh, Leon Balents 0 ! Ec, we observe the quantum critical form of the SYK4 t0 the saddle point of S K. We plot in Fig. 2 the spectral weight band structureComments: forXp theZ two-fermion 5+1 pages, hopping 4 term,figures dependent model, which1 displays !/T scaling, evident in the figure from ⇥ 2 A(!) ⇡ Im GR(!)(GR is retarded Green function) at fixed on lattice details,Subjects: and (p the)⇤ ellipses2 (Stronglyt t0)' representc,p( tCorrelated)'q, Op((t'0)q)+ terms Electrons. which (8)(cond-mat.str-el); Quantum Physics (quant-ph) the collapse⌘ onto4 a single curve at large !/T. At low fre- ···See also A. Georges and O. Parcollet PRB 59, 5341 (1999) U0/T = 10 for Ec/T = 0, 0.09, 1, 9, which illustrates the do not contribute2. arXiv:1705.00117 to the density correlations [pdf, other (and] are omitted quency, the SYK4 model has A(! T) 1/ pU0T, whose Here the first term arises from the ln det[ ] and⇤ the second from crossover between the SYK4 and⌧ Fermi⇠ liquid regimes. For hereafter –seeTitle: [32]2 for A reasons). strongly The correlated coe · cients metal⇤1( tbuilt) and from Sachdev-Ye-Kitaev models divergence as T 0 is cut-o↵ when T . Ec. This is seen the hopping (t0) term in (6). The function (p) encodes the ! Ec, we observe! the quantum critical form of the SYK4 ⇤2(t) are expressed in terms of saddle point Green’s functions in the reduction of the peak height in Fig. 2, pU0TA(! = 0), band structureAuthors: for the Xue-Yang two-fermion Song hopping, Chao-Ming term, dependent Jian, Leon Balents model, which displays !/T scaling, evident in the figure from in [32]. We remark here that any further approximations,2 e.g., with increasing E /T. On a larger frequency scale (inset), the on lattice details,Comments: and the ellipses17 pages, represent 6 figuresO(' ) terms which the collapse ontoc a single curve at large !/T. At low fre- conformal invariance, are not assumed to arrive at actionq (8), narrow “coherence peak”, associated with the small spectral do not contributeSubjects: to the Strongly density correlations Correlated (and Electrons are omitted (cond-mat.str-el) quency, the SYK4 model has A(! T) 1/ pU0T, whose and hence this derivation applies in all regimes. weight of heavy quasiparticles, is clearly⌧ visible.⇠ hereafter3. arXiv:1703.08910 –see [32] for reasons). [pdf, The other coe] cients ⇤1(t) and divergence as T 0 is cut-o↵ when T . Ec. This is seen ⇤2(t) are expressed in termsCoherence of saddle point Green’s functions scale We now turn to! transport, and for simplicityp focus! on= Title: Anomalous Hall effect and topological defects in antiferromagnetic Weyl semimetals: Mn in the reduction of the peak height in Fig. 2, U0TA( 0), in [32]. We remark here that any further approximations, e.g., particle-hole symmetric/ case hereafter. The strategy is to ob- Sn/Ge with increasing Ec T. On a larger frequency scale (inset), the conformal invariance,We solve are these not assumed equations to arrive in ata actionreal time (8), Keldysh formulation tain electrical and heat conductivities from the fluctuations of Authors: Jianpeng Liu, Leon Balents narrow “coherence peak”, associated with the small spectral and hence this derivation applies in all regimes. chargeweight and of energy,heavy quasiparticles, respectively, using is clearly the Einstein visible. relations. Subjects:numerically Mesoscale and and determine Nanoscale asymptotics Physics (cond-mat.mes-hall); analytically. Materials Science (cond-mat.mtrl- We first consider charge, and study the low-energy U(1) phase We now turn to transport, and for simplicity focus on sci) fluctuation '(x, t), which is the conjugate variable to particle 4. arXiv:1611.00646 [pdf, ps, other] numberparticle-hole density symmetric(x, t), around case hereafter. the saddle The point strategy of the action is to ob- N Title: Spin-Orbit Dimers and NarrowNon-Collinear “coherence Phases peak” in appears Cubic in Double Perovskites StainK. Allowing electrical for and phase heat fluctuations conductivities around from the the saddle fluctuations point of solutioncharge amounts and energy, to taking respectively, using the Einstein relations. Authors: Judit Romhányi, Leonspectral Balents function:, George heavy Jackeli quasiparticles We first consider charge, and study the low-energy U(1) phase form for T≪Ec fluctuation '(x, t), which is the conjugate variable to particle number density (x, t), around thei saddle(' (x,t) ' point(x,t )) of the action s s0 0 Gx,ss0 (t, tN0) Gx,ss0 (t t0)e S K. Allowing for phase! fluctuations i(' around(x,t) ' (x the,t )) saddle point Quasiparticle weight Z ~ t/U s s0 0 ! = Ec ⌃x,ss0 (t, t0) ⌃x,ss0 (t t0)e , (7) solution amounts to! taking 4 FIG. 2. The spectral weight A(!) at fixed U0/T = 10 ,µ = 0, z = 2 for Ec/T = 0, 0.09, 1, 9, corresponding a crossover from SYK4 limit to the “heavy Fermi liquid” regime. Inset shows the comparison of where Gx,ss0 (t t0) and ⌃x,ss0 (t t0) are thei('s( saddlex,t) 's (x point,t0)) solu- green’s function for T/E = 9 with free fermion limit result. Gx ,ss (t, t0) Gx,ss (t t0)e 0 sp c tions. Expanding0 (6) to quadratic0 order in ' , S = S + S , ! i(' (xs,t) 'K (x,t ))K ' ⌃ , 0 ⌃ 0 s s0 0 , yields the lowestx,ss0 (t ordert ) e↵ectivex,ss0 (t actiont )e for the U(1) fluctu-(7) ! FIG. 2. The spectral weight A(!) at fixed U /T = 104,µ = 0, z = 2 ations. This is most conveniently expressed in terms of the In the low frequency limit, the Fourier0 transforms of for E /T = 0, 0.09, 1, 9, corresponding a crossover from SYK limit “classical” and “quantum” components of the phase fluctua- ⇤ (t),⇤ c(t) behave as ⇤ (!) 2iK and ⇤ (!) 2KD !4, 1 to the2 “heavy Fermi liquid”1 regime.⇡ Inset shows2 the⇡ comparison' of where Gx,ss0 (t t0) and ⌃x,ss0 (t t0) are the saddle point solu- green’s function for T/E = 9 with free fermion limit result. sp c tions. Expanding (6) to quadratic order in 's, S K = S K + S ', yields the lowest order e↵ective action for the U(1) fluctu- ations. This is most conveniently expressed in terms of the In the low frequency limit, the Fourier transforms of “classical” and “quantum” components of the phase fluctua- ⇤ (t),⇤ (t) behave as ⇤ (!) 2iK and ⇤ (!) 2KD !, 1 2 1 ⇡ 2 ⇡ ' Cornell University Library We gratefully acknowledge support from the Simons Foundation and member institutions arXiv.org > search 2 2 2 (Help | Advanced search) 1/4 2 1 2 t0 = 0, the U0 term is invariant under ⌧ b⌧ and c b c, = 1/2, and U0 b U0 is irrelevant. Since the SYK2 2 1/4 ! 1!/4 2 1!2 t0 = 0,c¯ the Ub0 termc¯, is fixing invariant the under scaling⌧ dimensionb⌧ and c =b 1/4c, of the = 1Hamiltonian/2, and U0 (i.e.,bU 0U0= is0) irrelevant. is quadratic, Since the the disordered SYK2 free 1!/4 ! ! ! c¯ fermionb c¯, fixing fields. the Under scaling this dimension scalingc ¯ @⌧c=term1/4 is of irrelevant. the Hamiltonianfermion (i.e., modelU = supports0) is quadratic, quasi-particles the disordered and defines free a Fermi ! arXiv.org Search Results 0 fermionYet fields. upon addition Under this of two-fermion scalingc ¯@⌧c coupling,term is irrelevant. under rescaling,fermionliquid model limit. supports For t0 quasi-particlesU0, Ec defines and defines a crossover a Fermi scale be- 2 2 ⌧ Yet upont0 additionbt0, so of two-fermion two-fermion coupling coupling, is under a relevant rescaling,perturba-liquidtween limit. SYK For t40-likeU non-Fermi0, Ec defines liquid a crossover and the low scale temperature be- 2 2 2! Back to Search form | Next 25 results ⌧ t0 tion.bt0, so By two-fermion standard reasoning, coupling this is a impliesrelevant a cross-overperturba- fromtweenFermi SYK4-like liquid. non-Fermi liquid and the low temperature ! t = 0, the U2 term is invariant under ⌧ b⌧ and c b 1/4c, = 1/2, and U2 b 1U2 is irrelevant. Since the SYK tion. Bythe standard SYK4-like reasoning, model to this another implies regime a cross-over at the energy from scaleFermi0 liquid.At0 the level of thermodynamics,! ! this crossover can0 ! be rig-0 2 c¯ b 1/4c¯, fixing the scaling dimension = 1/4 of the Hamiltonian (i.e.,U = 0) is quadratic, the disordered free The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1! 0 the SYKwhere4-like the model hopping to another perturbation regime becomes at the dominant, energy scale which is Atfermion theorously fields. level Under established of thermodynamics, this scaling usingc ¯@⌧c term imaginary is this irrelevant. crossover timefermion formalism. can model be supports rig- Intro- quasi-particles and defines a Fermi 2 Yet upon addition of two-fermion coupling, under rescaling, 1 liquid limit. For t U , E defines a crossover scale be- whereE thec = hoppingt /U0. perturbation We expect becomes the renormalization dominant, which flow is is to theorouslyducing established a composite using imaginaryfield Gx(⌧, ⌧ time0) = formalism. cix⌧c¯ix Intro-⌧ and0 ⌧ a0 La-c 0 t2 bt2, so two-fermion coupling is a relevant perturba- N tweeni SYK -like0 non-Fermi liquid and the low temperature 2 0 ! 0 1 4 = SYK/ regime. Indeed keeping the SYK term invariant fixes tion.grange By standard multiplier reasoning, this⌃ implies(⌧,⌧⌧, ⌧) a enforcing cross-over= from the previousFermi liquid. identity, one Ec t0 U20. We expectShowing the renormalization results2 flow 1 through is to the 25ducing (of a181 composite total) field forGx xau:Balents( 0 0) N i cix⌧c¯ix⌧0 and a La- the SYK -like model to another regime atNS the energy scale PAt the level of thermodynamics, this crossover can be rig- SYK regime. Indeed keeping the SYK term invariant fixes grangeobtains multiplier4 Z¯ =⌃ ([⌧dG, ⌧0][)d enforcing⌃]e , with the previous the action identity, one 2 2 where the hopping perturbationx becomes dominant, which is orously established using imaginary time formalism. Intro- 2 NS P 1 obtainsEc = tZ¯/U=0. We[dG expect][d the⌃]e renormalization , with the flow isaction to the ducing a composite field Gx(⌧, ⌧0) = cix⌧c¯ix⌧ and a La- 0 R N i 0 1. arXiv:1707.02308 [pdf, ps, other] SYK2 regime. Indeed keeping the SYK2 term invariant fixes grange multiplier ⌃x(⌧, ⌧0) enforcing the previous identity, one ¯ NS P Title: Fracton topological phases from stronglyR coupled spin chains obtains Z = [dG][d⌃]e , with the action 2 U R Authors: Gábor B. Halász, Timothy H. Hsieh0 , Leon Balents2 2 S = ln det (@⌧ µ) (⌧1 ⌧2) + ⌃x(⌧1, ⌧2) + d⌧1d⌧2 2 Gx(⌧1, ⌧2) Gx(⌧2, ⌧1) + ⌃x(⌧1, ⌧2)Gx(⌧2, ⌧1) 2 U 4 U2 x Comments: 5+1 pages, 4 figures0 0x 2 2 0 2 2 S = ln det (@⌧ µ) (⌧1 ⌧2) + ⌃x(⌧1, ⌧2) + d⌧1Zd⌧2 S = Gln2x det(⌧1(,@⌧⌧ 2)µ)G (⌧x1 (⌧⌧22,) ⌧+1⌃)x(⌧1+, ⌧2⌃) x+(⌧1,d⌧⌧12d)⌧G2 x(⌧2, ⌧1G)x(⌧1, ⌧2) G3x(⌧2, ⌧1) + ⌃x(⌧1, ⌧2)Gx(⌧2, ⌧1) t = 0, the U2 term isX invariant under ⌧ b⌧ and c b 1/4c, = 1/2, and U2 b 1U2 is irrelevant. Since the SYK X 4 0 0 0 0 2 ⇣ x 6 0 x 7 1/4 2 ⇥! ! ! 0 ⇤ 4 6 Z 2 7 3 c¯ b xc¯, fixing the scaling dimension = Subjects:1/4 of the Hamiltonian Strongly (i.e.,U0 = 0) Correlated is quadratic, the disordered Electrons freex X 6(cond-mat.str-el); Quantum ⇣PhysicsX (quant-ph)7 ! +t Gx (⌧1, ⌧2)Gx(⌧2, ⌧1) . Z 22 6 ⇥ ⇤ 6 3 7 (3) 7 fermionX fields.0 Under this0 scalingc ¯@⌧c term is irrelevant. fermion model supports quasi-particles and defines a FermiX+t G (4⌧ , ⌧ )G (⌧ , ⌧ ) . 6 5 7 (3) ⇣ See 6also0 A.x 0 Georges1 2 x and2 1 O. Parcollet PRB 59, 5341 (1999)4 7 5 2Yet upon addition of two-fermion⇥ coupling, under rescaling, liquid limit. For t ⇤ U , E defines a crossover scale be- 6 7 xx 2. arXiv:1705.00117 0[pdf0 , cother] 6 xx0 7 +t 2 2Gx (⌧0 1, ⌧2)Gx(⌧2, ⌧1) . ⌧ 6 h i 7 (3) 0t bt , soh two-fermion0 i coupling is a relevant perturba- tween SYK4-like non-Fermi liquid and the low temperature 4 X ⌘ 5 0 ! 0 X ⌘ tion. By standard reasoning, this implies a cross-over from Fermi liquid. xx0 Title: A strongly correlated metal built from Sachdev-Ye-Kitaev models thehX SYKi4-like model to another regime at the energy⌘ scale At the level of thermodynamics, this crossover can be rig- where the hopping perturbation becomes dominant,Authors: which is orously Xue-Yang established using imaginary Song time, formalism. Chao-Ming Intro- Jian, Leon Balents = 2/ ⌧, ⌧ = 1 Ec t0 U0. We expect the renormalization flow is to the ducing a composite field Gx( 0) N i cix⌧c¯ix⌧0 andThe a La- large N limit is controlled by the saddle point conditions SYK2 regime. Indeed keeping the SYK2 term invariant fixes grange multiplier ⌃x(⌧, ⌧0) enforcing the previous identity, one/ = / ⌃ = ⌧, ⌧ = ⌧ ⌧ Comments: 17 pages,NS 6 figuresP S G S 0, satisfied by Gx( 0) G( 0), obtains Z¯ = [dG][d⌃]e , with the action 2 The large N limit is controlled by the saddle point conditions ⌃ (⌧, ⌧0) = ⌃ (⌧ ⌧0) + zt G(⌧ ⌧0)(z is the coordination x 4 0 Subjects: StronglyR Correlated Electronsnumber of the lattice (cond-mat.str-el) of SYK dots), which obey The large S/ NGlimit= isS/ controlled ⌃ = 0, by satisfied the saddle by G pointx(⌧, ⌧ conditions0) = G(⌧ ⌧0), U2 0 2 2 1 2 S = ln det (@⌧ µ) (⌧1 3.⌧2) +arXiv:1703.08910⌃x(⌧1, ⌧2) 2+ d⌧1d⌧2 Gx(⌧1 , ⌧[2)pdfGx(⌧2,,⌧ 1other) + ⌃x(⌧1, ⌧]2)GEntropyx(⌧2, ⌧1) G(i! ) =(i! + µ ⌃ (i)! ) zt G(i! ), n n 4 n n S/ G⌃x=(⌧, ⌧ S0)/ =⌃ ⌃= (⌧ ⌧0) + zt G0 (⌧G ⌧⌧,0⌧)(4 z =isG the⌧ coordination⌧ 0 x 4 0, satisfied0 byZ x( x 20) ( 0), 3 S T X ⇥ ⇤ ⇣ X 6 7 2 2 +t2 G (⌧ , ⌧ )G (⌧ , ⌧ ) . 2 Title: Anomalous6 Hall effect and7 (3) topological⌃4(⌧) = U 0defectsG(⌧) G( ⌧), in antiferromagnetic(4) Weyl semimetals: Mn ⌃ (⌧, ⌧number0) =0 ⌃ x of(0 ⌧1 the2 x⌧ lattice02 )1 + zt ofG SYK(⌧ ⌧ dots),0)(z is which4 the coordination obey 5 x xx 4 0 hX0i ⌘ number of the lattice of SYK dots),Sn/Ge whichLevel obey repulsion: entropywhere !isn =released(2n + 1)⇡/ is the for Matsubara T (Help | Advanced search) arXiv.org Search Results Back to Search form | Next 25 results The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1 Showing results 1 through 25 (of 181 total) for au:Balents 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 Subjects: Strongly CorrelatedConductivity Electrons (cond-mat.str-el) 3. arXiv:1703.08910 [pdf, other] Title:From Anomalous the Kubo Hall formula, effect and we topological have the defects conductivity in antiferromagnetic Weyl semimetals: Mn Sn/Ge Authors: Jianpeng Liu, Leon Balents 2 f(! + ⌦) f(⌦) Subjects:Re[ Mesoscale(!)] t andd Nanoscale⌦ Physics (cond-mat.mes-hall);A(⌦)A(! + ⌦ )Materials Science (cond-mat.mtrl- / 0 ! sci) Z 4. arXiv:1611.00646 [pdf, ps, other] where A(!) = Im[GR(!)]. Title: Spin-Orbit Dimers and Non-Collinear Phases in Cubic Double Perovskites Authors:At T >E Juditc this Romhányi yields, Leon Balents, George Jackeli e2 t2 1 0 ⇠ h U T Cornell University Library We gratefully acknowledge support from the Simons Foundation and member institutions arXiv.org > search (Help | Advanced search) arXiv.org Search Results Back to Search form | Next 25 results The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1 Showing results 1 through 25 (of 181 total) for au:Balents 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 4 Subjects: Strongly Correlated ElectronsLow (cond-mat.str-el) ‘coherence’ scale 3. arXiv:1703.08910 [pdf, other] The density-density correlator is expressed as Title: Anomalous Hall effect and topological defects in antiferromagnetict2 Weyl semimetals: Mn DRn(x,t; x0,t0) i✓(t t0) [ (x,t), (x0,t0)] Sn/Ge 0 ⌘ h N N i E i Authors: Jianpeng Liu, Leon Balents c = (x,t) (x0,t0) , (10) ⇠ U 2hNc Nq i Subjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Materials Science (cond-mat.mtrl- N S ' sci) where s , c/q = + (keeping momentum- N ⌘ '˙ s N N ±N independent components- See Sec.B). Adding a contact term 4. arXiv:1611.00646 [pdf, ps, other] to ensure that limp 0 DRn(p, ! , 0) = 0[31], the action (9) Title: Spin-Orbit Dimers and Non-Collinear Phases in Cubic Double Perovskites yields the di↵usive! form [32] Authors: Judit Romhányi, Leon Balents, George Jackeli 2 iNK! NKD' p D (p, !) = + NK = . (11) Rn i! D p2 i! D p2 ' ' From this we identify NK and D' as the compressibility and (a) charge di↵usion constant, respectively. The electric conduc- tivity is given by Einstein relation 1/⇢ = NKD', or, e2 2 d ⌘ restoring all units, = NKD' ~ a (a is lattice spacing). Note the proportionality to N: in the standard non-linear sigma model formulation, the dimensionless conductance is large, suppressing localization e↵ects. This occurs because both U and t interactions scatter between all orbitals, destroy- ing interference from closed loops. The analysis of energy transport proceeds similarly. Since energy is the generator of time translations, one considers the time-reparametrization (TRP) modes induced by ts ts+✏s(t) 1 ! and defines ✏c/q = 2 (✏+ ✏ ). The e↵ective action for TRP modes to the lowest-order± in p, ! reads (Sec. B) + 1 2 2 2 iS ✏ = d!✏c,!(2i ! T p ⇤3(!))✏q, ! + , (12) (b) ··· Xp Z 1 T FIG. 3. (a): For t0, T U0, ⇢'/" “collapse” to R'/"( )/N. (b): The where the ellipses has the same meaning as in (9). At low ⌧ Ec ⇢ ⇡2 ⇡2 frequency, the correlation function integral, given in Sec. B, Lorentz ratio T reaches two constants 3 , 8 , in the two regimes. behaves as ⇤ (!) 2 D T 2!, which defines the energy dif- The solid curves are guides to the eyes. 3 ⇡ ✏ fusion constant D✏ . This identification is seen from the corre- iN S ✏ lator for energy density modes "c/q , ⌘ ✏˙c/q 2 2 R00(0) i NT D✏ p ⇣ where the temperature coe cient of resistivity A⇣ = 2 is D "(p, !) = " " ,! = , (13) 2NEc R 2h c qip i! D p2 ✏ large due to small coherence scale in denominator, charac- where we add a contact term to ensure conservation of energy teristic of a strongly correlated Fermi liquid. Famously, the Kadowaki-Woods ratio, A /(N )2, is approximately system- at p = 0. The thermal conductivity reads = NT D✏ (kB = 1) ' –like , is O(N). independent for a wide range of correlated materials[33, 34]. A' R'00(0) Scaling collapse, Kadowaki-Woods and Lorentz ra- We find here (N )2 = 2[ (0)]2 N3 is independent of t0 and U0! S0 tios – Electric/thermal conductivities are obtained from Turning now to the incoherent metal regime, in limit of lim! 0 ⇤2/3(!)/!, expressed as integrals of real-time corre- large arguments, 1, the generalized resistivities vary lin- ! T lation functions, and can be evaluated numerically for any early with temperature: R⇣ ( ) c⇣ . We analytically obtain T, t0, U0. Introducing generalized resistivities, ⇢' = ⇢, ⇢" = 2 16 T ⇠ T c' = and c" = / (Supplementary Information), implying T/, we find remarkably that for t , T U , they collapse to p⇡ ⇡5 2 0 ⌧ 0 that the Lorenz number, characterizing the Wiedemann-Franz universal functions of one variable, 2 law, takes the unusual value L = ⇡ for E T U . T ! 8 c ⌧ ⌧ 0 1 T More generally, the scaling form (B2) implies that L is a uni- ⇢⇣ (t0, T U0) = R⇣ ( E ) ⇣ ', " , (14) ⌧ N c 2 { } versal function of T/Ec, verified numerically as shown in Fig. 3b. The Lorenz number increases with lower tempera- where R ( ), R ( ) are dimensionless universal functions. ' T " T ture, saturating at T E to the Fermi liquid value ⇡2/3. This scaling collapse is verified by direct numerical calcula- ⌧ c tions shown in Fig. 3a. From the scaling form (B2), we see the Conclusion – We have shown that the SYK model pro- low temperature resistivity obeys the usual Fermi liquid form vides a soluble source of strong local interactions which, when coupled into a higher-dimensional lattice by ordinary ⇢ (T E ) ⇢ (0) + A T 2, (15) but random electron hopping, reproduces a remarkable num- ⇣ ⌧ c ⇡ ⇣ ⇣ Cornell University Library We gratefully acknowledge support from the Simons Foundation and member institutions arXiv.org > search (Help | Advanced search) arXiv.org Search Results Back to Search form | Next 25 results The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1 Showing results 1 through 25 (of 181 total) for au:Balents 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 4 Subjects: Strongly Correlated ElectronsLow (cond-mat.str-el) ‘coherence’ scale 3. arXiv:1703.08910 [pdf, other] The density-density correlator is expressed as Title: Anomalous Hall effect and topological defects in antiferromagnetict2 Weyl semimetals: Mn DRn(x,t; x0,t0) i✓(t t0) [ (x,t), (x0,t0)] Sn/Ge 0 ⌘ h N N i E i Authors: Jianpeng Liu, Leon Balents c = (x,t) (x0,t0) , (10) ⇠ U 2hNc Nq i Subjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Materials Science (cond-mat.mtrl- N S ' sci) where s , c/q = + (keeping momentum- N ⌘ '˙ s N N ±N independent components- See Sec.B). Adding a contact term 4. arXiv:1611.00646 [pdf, ps, other] For Ec 2 iNK! NKD' p entropy density, s, are , ! = + = . DRn(p ) 2 NK 2 (11) i! D' p i! D' p h T From this we identify NK and D' as the compressibility and (a) ⇢ ,s= s ↵ 2 0 charge di usion constant, respectively. The electric conduc- ⇠ e Ec tivity is given by Einstein relation 1/⇢ = NKD', or, ✓ ◆ e2 2 d ⌘ restoring all units, = NKD' ~ a (a is lattice spacing). Note the proportionality to N: in the standard non-linear sigma model formulation, the dimensionless conductance is large, suppressing localization e↵ects. This occurs because both U and t interactions scatter between all orbitals, destroy- ing interference from closed loops. The analysis of energy transport proceeds similarly. Since energy is the generator of time translations, one considers the time-reparametrization (TRP) modes induced by ts ts+✏s(t) 1 ! and defines ✏c/q = 2 (✏+ ✏ ). The e↵ective action for TRP modes to the lowest-order± in p, ! reads (Sec. B) + 1 2 2 2 iS ✏ = d!✏c,!(2i ! T p ⇤3(!))✏q, ! + , (12) (b) ··· Xp Z 1 T FIG. 3. (a): For t0, T U0, ⇢'/" “collapse” to R'/"( )/N. (b): The where the ellipses has the same meaning as in (9). At low ⌧ Ec ⇢ ⇡2 ⇡2 frequency, the correlation function integral, given in Sec. B, Lorentz ratio T reaches two constants 3 , 8 , in the two regimes. behaves as ⇤ (!) 2 D T 2!, which defines the energy dif- The solid curves are guides to the eyes. 3 ⇡ ✏ fusion constant D✏ . This identification is seen from the corre- iN S ✏ lator for energy density modes "c/q , ⌘ ✏˙c/q 2 2 R00(0) i NT D✏ p ⇣ where the temperature coe cient of resistivity A⇣ = 2 is D "(p, !) = " " ,! = , (13) 2NEc R 2h c qip i! D p2 ✏ large due to small coherence scale in denominator, charac- where we add a contact term to ensure conservation of energy teristic of a strongly correlated Fermi liquid. Famously, the Kadowaki-Woods ratio, A /(N )2, is approximately system- at p = 0. The thermal conductivity reads = NT D✏ (kB = 1) ' –like , is O(N). independent for a wide range of correlated materials[33, 34]. A' R'00(0) Scaling collapse, Kadowaki-Woods and Lorentz ra- We find here (N )2 = 2[ (0)]2 N3 is independent of t0 and U0! S0 tios – Electric/thermal conductivities are obtained from Turning now to the incoherent metal regime, in limit of lim! 0 ⇤2/3(!)/!, expressed as integrals of real-time corre- large arguments, 1, the generalized resistivities vary lin- ! T lation functions, and can be evaluated numerically for any early with temperature: R⇣ ( ) c⇣ . We analytically obtain T, t0, U0. Introducing generalized resistivities, ⇢' = ⇢, ⇢" = 2 16 T ⇠ T c' = and c" = / (Supplementary Information), implying T/, we find remarkably that for t , T U , they collapse to p⇡ ⇡5 2 0 ⌧ 0 that the Lorenz number, characterizing the Wiedemann-Franz universal functions of one variable, 2 law, takes the unusual value L = ⇡ for E T U . T ! 8 c ⌧ ⌧ 0 1 T More generally, the scaling form (B2) implies that L is a uni- ⇢⇣ (t0, T U0) = R⇣ ( E ) ⇣ ', " , (14) ⌧ N c 2 { } versal function of T/Ec, verified numerically as shown in Fig. 3b. The Lorenz number increases with lower tempera- where R ( ), R ( ) are dimensionless universal functions. ' T " T ture, saturating at T E to the Fermi liquid value ⇡2/3. This scaling collapse is verified by direct numerical calcula- ⌧ c tions shown in Fig. 3a. From the scaling form (B2), we see the Conclusion – We have shown that the SYK model pro- low temperature resistivity obeys the usual Fermi liquid form vides a soluble source of strong local interactions which, when coupled into a higher-dimensional lattice by ordinary ⇢ (T E ) ⇢ (0) + A T 2, (15) but random electron hopping, reproduces a remarkable num- ⇣ ⌧ c ⇡ ⇣ ⇣ Cornell University Library We gratefully acknowledge support from the Simons Foundation and member institutions arXiv.org > search (Help | Advanced search) arXiv.org Search Results Back to Search form | Next 25 results The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1 Showing results 1 through 25 (of 181 total) for au:Balents 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 4 Subjects: Strongly Correlated ElectronsLow (cond-mat.str-el) ‘coherence’ scale 3. arXiv:1703.08910 [pdf, other] The density-density correlator is expressed as Title: Anomalous Hall effect and topological defects in antiferromagnetict2 Weyl semimetals: Mn DRn(x,t; x0,t0) i✓(t t0) [ (x,t), (x0,t0)] Sn/Ge 0 ⌘ h N N i E i Authors: Jianpeng Liu, Leon Balents c = (x,t) (x0,t0) , (10) ⇠ U 2hNc Nq i Subjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Materials Science (cond-mat.mtrl- N S ' sci) where s , c/q = + (keeping momentum- N ⌘ '˙ s N N ±N independent components- See Sec.B). Adding a contact term 4. arXiv:1611.00646 [pdf, ps, other] For Ec 2 iNK! NKD' p entropy density, s, are , ! = + = . DRn(p ) 2 NK 2 (11) i! D' p i! D' p h T From this we identify NK and D' as the compressibility and (a) ⇢ ,s= s ↵ 2 0 charge di usion constant, respectively. The electric conduc- ⇠ e Ec tivity is given by Einstein relation 1/⇢ = NKD', or, ✓ ◆ e2 2 d ⌘ restoring all units, = NKD' ~ a (a is lattice spacing). Note the proportionality to N: in the standard non-linear sigma model formulation, the dimensionless conductance is large, suppressing localization e↵ects. This occurs because both U and t interactions scatter between all orbitals, destroy- ing interference from closed loops. The analysis of energy transport proceeds similarly. Since energy is the generator of time translations, one considers the time-reparametrization (TRP) modes induced by ts ts+✏s(t) 1 ! and defines ✏c/q = 2 (✏+ ✏ ). The e↵ective action for TRP modes to the lowest-order± in p, ! reads (Sec. B) + 1 2 2 2 iS ✏ = d!✏c,!(2i ! T p ⇤3(!))✏q, ! + , (12) (b) ··· Xp Z 1 T FIG. 3. (a): For t0, T U0, ⇢'/" “collapse” to R'/"( )/N. (b): The where the ellipses has the same meaning as in (9). At low ⌧ Ec ⇢ ⇡2 ⇡2 frequency, the correlation function integral, given in Sec. B, Lorentz ratio T reaches two constants 3 , 8 , in the two regimes. behaves as ⇤ (!) 2 D T 2!, which defines the energy dif- The solid curves are guides to the eyes. 3 ⇡ ✏ fusion constant D✏ . This identification is seen from the corre- iN S ✏ lator for energy density modes "c/q , ⌘ ✏˙c/q 2 2 R00(0) i NT D✏ p ⇣ where the temperature coe cient of resistivity A⇣ = 2 is D "(p, !) = " " ,! = , (13) 2NEc R 2h c qip i! D p2 ✏ large due to small coherence scale in denominator, charac- where we add a contact term to ensure conservation of energy teristic of a strongly correlated Fermi liquid. Famously, the Kadowaki-Woods ratio, A /(N )2, is approximately system- at p = 0. The thermal conductivity reads = NT D✏ (kB = 1) ' –like , is O(N). independent for a wide range of correlated materials[33, 34]. A' R'00(0) Scaling collapse, Kadowaki-Woods and Lorentz ra- We find here (N )2 = 2[ (0)]2 N3 is independent of t0 and U0! S0 tios – Electric/thermal conductivities are obtained from Turning now to the incoherent metal regime, in limit of lim! 0 ⇤2/3(!)/!, expressed as integrals of real-time corre- large arguments, 1, the generalized resistivities vary lin- ! T lation functions, and can be evaluated numerically for any early with temperature: R⇣ ( ) c⇣ . We analytically obtain T, t0, U0. Introducing generalized resistivities, ⇢' = ⇢, ⇢" = 2 16 T ⇠ T c' = and c" = / (Supplementary Information), implying T/, we find remarkably that for t , T U , they collapse to p⇡ ⇡5 2 0 ⌧ 0 that the Lorenz number, characterizing the Wiedemann-Franz universal functions of one variable, 2 law, takes the unusual value L = ⇡ for E T U . T ! 8 c ⌧ ⌧ 0 1 T More generally, the scaling form (B2) implies that L is a uni- ⇢⇣ (t0, T U0) = R⇣ ( E ) ⇣ ', " , (14) ⌧ N c 2 { } versal function of T/Ec, verified numerically as shown in Fig. 3b. The Lorenz number increases with lower tempera- where R ( ), R ( ) are dimensionless universal functions. ' T " T ture, saturating at T E to the Fermi liquid value ⇡2/3. This scaling collapse is verified by direct numerical calcula- ⌧ c tions shown in Fig. 3a. From the scaling form (B2), we see the Conclusion – We have shown that the SYK model pro- low temperature resistivity obeys the usual Fermi liquid form vides a soluble source of strong local interactions which, when coupled into a higher-dimensional lattice by ordinary ⇢ (T E ) ⇢ (0) + A T 2, (15) but random electron hopping, reproduces a remarkable num- ⇣ ⌧ c ⇡ ⇣ ⇣ Cornell University Library We gratefully acknowledge support from the Simons Foundation and member institutions arXiv.org > search (Help | Advanced search) arXiv.org Search Results Back to Search form | Next 25 results The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Balents/0/1/0/all/0/1 Showing results 1 through 25 (of 181 total) for au:Balents 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 4 Subjects: Strongly Correlated ElectronsLow (cond-mat.str-el) ‘coherence’ scale 3. arXiv:1703.08910 [pdf, other] The density-density correlator is expressed as Title: Anomalous Hall effect and topological defects in antiferromagnetict2 Weyl semimetals: Mn DRn(x,t; x0,t0) i✓(t t0) [ (x,t), (x0,t0)] Sn/Ge 0 ⌘ h N N i E i Authors: Jianpeng Liu, Leon Balents c = (x,t) (x0,t0) , (10) ⇠ U 2hNc Nq i Subjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Materials Science (cond-mat.mtrl- N S ' sci) For T 2 2 iNK! NKD' p h T D (p, !) = + NK = . (11) Rn 2 2 ⇢ = c1 + c2 i! D' p i! D' p e2 E " ✓ c ◆ # From this we identify NK and D' as the compressibility and (a) charge di↵usion constant, respectively. The electric conduc- T s s0 tivity is given by Einstein relation 1/⇢ = NKD', or, ⇠ Ec e2 2 d ⌘ ✓ ◆ restoring all units, = NKD' ~ a (a is lattice spacing). Note the proportionality to N: in the standard non-linear sigma model formulation, the dimensionless conductance is large, suppressing localization e↵ects. This occurs because both U and t interactions scatter between all orbitals, destroy- ing interference from closed loops. The analysis of energy transport proceeds similarly. Since energy is the generator of time translations, one considers the time-reparametrization (TRP) modes induced by ts ts+✏s(t) 1 ! and defines ✏c/q = 2 (✏+ ✏ ). The e↵ective action for TRP modes to the lowest-order± in p, ! reads (Sec. B) + 1 2 2 2 iS ✏ = d!✏c,!(2i ! T p ⇤3(!))✏q, ! + , (12) (b) ··· Xp Z 1 T FIG. 3. (a): For t0, T U0, ⇢'/" “collapse” to R'/"( )/N. (b): The where the ellipses has the same meaning as in (9). At low ⌧ Ec ⇢ ⇡2 ⇡2 frequency, the correlation function integral, given in Sec. B, Lorentz ratio T reaches two constants 3 , 8 , in the two regimes. behaves as ⇤ (!) 2 D T 2!, which defines the energy dif- The solid curves are guides to the eyes. 3 ⇡ ✏ fusion constant D✏ . This identification is seen from the corre- iN S ✏ lator for energy density modes "c/q , ⌘ ✏˙c/q 2 2 R00(0) i NT D✏ p ⇣ where the temperature coe cient of resistivity A⇣ = 2 is D "(p, !) = " " ,! = , (13) 2NEc R 2h c qip i! D p2 ✏ large due to small coherence scale in denominator, charac- where we add a contact term to ensure conservation of energy teristic of a strongly correlated Fermi liquid. Famously, the Kadowaki-Woods ratio, A /(N )2, is approximately system- at p = 0. The thermal conductivity reads = NT D✏ (kB = 1) ' –like , is O(N). independent for a wide range of correlated materials[33, 34]. A' R'00(0) Scaling collapse, Kadowaki-Woods and Lorentz ra- We find here (N )2 = 2[ (0)]2 N3 is independent of t0 and U0! S0 tios – Electric/thermal conductivities are obtained from Turning now to the incoherent metal regime, in limit of lim! 0 ⇤2/3(!)/!, expressed as integrals of real-time corre- large arguments, 1, the generalized resistivities vary lin- ! T lation functions, and can be evaluated numerically for any early with temperature: R⇣ ( ) c⇣ . We analytically obtain T, t0, U0. Introducing generalized resistivities, ⇢' = ⇢, ⇢" = 2 16 T ⇠ T c' = and c" = / (Supplementary Information), implying T/, we find remarkably that for t , T U , they collapse to p⇡ ⇡5 2 0 ⌧ 0 that the Lorenz number, characterizing the Wiedemann-Franz universal functions of one variable, 2 law, takes the unusual value L = ⇡ for E T U . T ! 8 c ⌧ ⌧ 0 1 T More generally, the scaling form (B2) implies that L is a uni- ⇢⇣ (t0, T U0) = R⇣ ( E ) ⇣ ', " , (14) ⌧ N c 2 { } versal function of T/Ec, verified numerically as shown in Fig. 3b. The Lorenz number increases with lower tempera- where R ( ), R ( ) are dimensionless universal functions. ' T " T ture, saturating at T E to the Fermi liquid value ⇡2/3. This scaling collapse is verified by direct numerical calcula- ⌧ c tions shown in Fig. 3a. From the scaling form (B2), we see the Conclusion – We have shown that the SYK model pro- low temperature resistivity obeys the usual Fermi liquid form vides a soluble source of strong local interactions which, when coupled into a higher-dimensional lattice by ordinary ⇢ (T E ) ⇢ (0) + A T 2, (15) but random electron hopping, reproduces a remarkable num- ⇣ ⌧ c ⇡ ⇣ ⇣ Cornell University Library We gratefully acknowledge support from the Simons Foundation and member institutions arXiv.org > search (Help | Advanced search) arXiv.org Search Results Back to Search form | Next 25 results The URL for this search is http://arxiv.org:443/find/cond-mat/1/au:+Sachdev/0/1/0/all/0/1 Showing results 1 through 25 (of 333 total) for au:Sachdev 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 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 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 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 ✓ ◆ 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 ✓ ◆ 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 ⌧ 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, !)=2 n(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 )) Z 1 ✓ ◆ 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 )) Z 1 ✓ ◆ 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! Macroscopic magnetotransport in the MFL • Let us consider the MFL with additional macroscopic disorder (charge puddles etc.) Figure: N. Ramakrishnan et. al., arXiv: 1703.05478 • No macroscopic momentum, so equations describing charge transport are just I(x)=0, I(x)= (x) E(x), E(x)= (x). r · · r • Very weak thermoelectricity for large FS, so charge effectively decoupled from heat transport. 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)) 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. 2 e · 2 e · ✓ L ◆ ✓ L ◆ 2 2 e 2 MFL MFL 2 2 MFL MFL ( /m) a 0a b 0b + a b 0a + 0b ⇢e L = B , L e2 e2 q MFL MFL 1/2 MFL MFL ⌘ L + H a b( 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! Scaling between B and T ( =0.0025) B t/100 nb/na =0.8 b/ a =0.8 a =0.1kBT (T = t/100) ~ 50 T (a = 3.82 A) 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 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 interac- tions 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 1419