Violation of the viscosity/entropy bound in translationally invariant non-Fermi liquids
Xian-Hui Ge, Shao-Kai Jian, Yi-Li Wang, Zhuo-Yu Xian, Hong Yao
VViolation of the viscosity/entropy bound in translationally invariant non-Fermi liquids
Xian-Hui Ge, Shao-Kai Jian,
2, 3
Yi-Li Wang, Zhuo-Yu Xian, and Hong Yao
2, 6, 7, ∗ Department of Physics and Shanghai Key Laboratory of High TemperatureSuperconductors, Shanghai University, Shanghai, 200444, China Institute for Advanced Study, Tsinghua University, Beijing 100084, China Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, Maryland 20742, USA Department of Physics, Shanghai University, Shanghai, 200444, China Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China Department of Physics, Stanford University, Stanford, CA 94305, USA (Dated: April 7, 2020)Shear viscosity is an important characterization of how a many-body system behaves like a fluid.Here we study the shear viscosity of a strongly-interacting solvable model in two spatial dimensions,consisting of coupled Sachdev-Ye-Kitaev (SYK) islands. As temperature is lowered, the model ex-hibits a crossover from an incoherent metal with local criticality to a marginal fermi liquid. Wefind that while the shear viscosity to entropy density ratio satisfies the Kovtun-Son-Starinets (KSS)bound in the marginal Fermi liquid regime, it can strongly violate the KSS bound within a finite androbust temperature range in the incoherent metal regime, implying nearly perfect fluidity of the inco-herent metal with local criticality. To the best of our knowledge, it provides the first translationallyinvariant example violating the KSS bound with known gauge-gravity correspondence.
I. INTRODUCTION
Fluid mechanics is among the oldest and the most fun-damental subjects in physics. A generic many-body sys-tem with globally conserved quantities, such as mass, en-ergy, and momentum, will exhibit fluidity if the localthermalization time scale is much less than the relax-ation time scale of the conserved quantities. As a result,universal properties of a fluid can provide extremely use-ful insights in understanding correlated many-body sys-tems with complicated interactions between their consti-tutes, like ultra-cold Fermi gases in the unitary regimeand quark-gluon plasma (QGP) produced in relativis-tic heavy-ion collisions, where no control parameter ex-ists [1]. More recently, owing to the advances of experi-mental techniques, quantum fluid behaviors are also wit-nessed in correlated electrons in lattice systems [2–4]. In-terestingly, the theory of fluids also receives a boost fromthe development of holographic principles [5, 6]. A fun-damental characterization of fluids is the shear viscositythat measures the resistance of a fluid to shear stress.Since viscosity generates entropy and causes dissipation,a good fluid should have small shear viscosity. However,the viscosity cannot be arbitrarily small. Namely, like theuncertainty principle, the fundamental laws of nature puta lower bound on the ratio of shear viscosity to entropy.Based on the AdS/CFT correspondence, Kovtun, Sonand Starintes conjectured a lower bound (KSS bound) onthe ratio of shear viscosity to entropy in strongly couplednon-quasiparticle systems [7], i.e., η/ S ≥ / π , where η and S refer to shear viscosity and entropy density, re-spectively.The closer the ratio, η/ S , of a many-body system is to the KSS bound, the better it behaves as a perfect fluid.Thus, it is of great interest and importance to explorescarce examples that saturate, or even violate the bound.Among holographic systems, the KSS bound is obeyed inEinstein gravity with both rotational and translationalsymmetries, while a weaker bound [8–14] is obeyed inhigher-derivative gravity theory. When rotational sym-metry is broken, like the anisotropic black branes [15–17],certain component of shear viscosity tensor may violatethe KSS bound in a parametric manner which was re-cently illustrated in an anisotropic Dirac fluid [18]. More-over, the black brane solution for Gauss-Bonnet massivegravity and Rastall AdS massive gravity show violationof KSS bound [19]. For isotropic black branes with lin-ear axion fields, the KSS bound can also be violated; butshear viscosity does not have a hydrodynamic interpre-tation since momentum is no longer conserved [20–26].For many-body systems, the minimal of the ratio η/ S normally occurs at the fixed point exhibiting emergentconformal symmetry, where the quasiparticle descriptionoften invalidates. When the fixed point locates at zerotemperature, the ratio should be a universal number as-sociated with the universality class of the fixed point.Such examples include the electron fluid in graphene [27],the Luttinger-Abrikosov-Beneslavskii phase in three di-mensional quadratic band touching semimetal [28], andIsing nematic quantum critical point in 2D metals [29].However, if the fixed point locates at finite temperature,the ratio shows a non-universal behavior as a function oftemperature. The well-studied unitary quantum gasesand the QGP fall into this class [30–34]. In unitaryquantum gases, the minimal of the ratio occurs at anintermediate temperature range associated with the su-perfluid transition, providing possible examples violating a r X i v : . [ h e p - t h ] A p r rijkl J rijkl g rijkl J (a) ~ � ��� ~ � �� � π η / � ∝ � - � ���� - � ∝ � � / � MFL IM Classical (b)
FIG. 1: (a) A cartoon of the model. The red and bluedots represent the conduction electrons ( c fermions), and ( f fermions), respectively. The black dotted lines and orangedashed lines indicate the interactions between between f and c fermions and self-interaction of f fermions, respectively. (b)A schematic plot of the ratio η/ S as a function of tempera-ture. There are three regimes, marginal Fermi liquid (MFL),IM (incoherent) and semi-classical regime, exhibiting differ-ent behaviors. The ratio violates the KSS bound indicated bythe dashed line in the IM regime. the KSS bound [35], while at the zero-temperature limitthe gapless Goldstone modes lead to a divergent ratio.Recently, Patel et al. [36] and Chowdhury et al. [37]constructed a 2D strongly correlated solvable model, con-sisting of coupled Sachdev-Ye-Kitaev (SYK) islands asshown in Fig. 1(a). This model is of great interest dueto the fact that the SYK model is believed to have agravity dual [38–44] with maximal chaos [45], and thatthough the model exhibits marginal Fermi liquid (MFL)with well-defined quasiparticle at low temperature, itexhibits an intermediate-temperature incoherent metal(IM) regime where the quasiparticle description invali-dates. Here, we consider a translationally invariant ver-sion of such model [37], and evaluate the shear viscosityby using the Kubo formula at large- N limit. As indi-cated in Fig. 1(b), in the MFL regime with T < T inc ,we find η/ S ∝ T − ; the ratio obeys a KSS-like boundand diverges at zero-temperature limit. For T > T cl where the system can be treated classically, we have η/ S ∝ T / [46]. Thus, the ratio necessarily exhibits aminimal in the intermediate temperature. Interestingly,the ratio can strongly violate the KSS bound in a robusttemperature range of the IM regime, not only implyinga nearly perfect fluidity of the coupled local critical SYKmodels, but also providing the first translationally androtationally invariant example violating the KSS boundwith known gauge-gravity correspondence. II. THE MODEL
We consider a 2D lattice model with M flavors of con-duction fermions c ri , i =1 , · · · , M , and N flavors of va-lence fermions f rj , j =1 , · · · ,N , on each site r , as shown in Fig. 1(a): H = − (cid:88) rr (cid:48) M (cid:88) i =1 ( t rr (cid:48) c † ri c r (cid:48) i + h.c. ) + (cid:88) r (cid:104) − µ c M (cid:88) i =1 c † ri c ri − µ f N (cid:88) i =1 f † ri f ri + N (cid:88) i,j =1 M (cid:88) k,l =1 g ijkl N M / f † ri f rj c † rk c rl + N (cid:88) i,j,k,l =1 J ijkl N / f † ri f † rj f rk f rl (cid:105) . (1)where t rr (cid:48) is the hopping amplitude of c fermions betweensites r and r (cid:48) , and µ i ( i = c, f ) denote the chemical po-tential of c and f fermions, respectively. The local inter-action strength g ijkl and J ijkl are random numbers whichsatisfies (cid:104)(cid:104) J ijkl J lkij (cid:105)(cid:105) = J and (cid:104)(cid:104) g ijkl g lkij (cid:105)(cid:105) = g and allother (cid:104)(cid:104) ... (cid:105)(cid:105) are vanishing. Here (cid:104)(cid:104) ... (cid:105)(cid:105) means disorder-average. Note that the coupling constants g ijkl and J ijkl on different sites not only have the same distribution,but are identical in each realization. In the following, wechoose the hopping amplitude to be a function depend-ing on | r − r (cid:48) | , for instance, t rr (cid:48) = tδ r (cid:48) ,r +ˆ e i , where ˆ e i isthe primitive lattice vector. As a result, the Hamiltonianis translationally invariant. If g = 0, the model can beviewed as two independent subsystems: the conducting c fermions with a hopping t rr (cid:48) , and the local f fermionswith SYK interaction at each site. Finite g > g , similar to the Kondo lattice model [47–49].We consider large N and M limit, while keep theirratio, M/N , fixed. The Green’s functions are givenby [37], G c ( k , iω ) = [ iω n − (cid:15) k + µ c − Σ cf ( k , iω n )] − and G f ( k , iω n ) = [ iω n + µ f − Σ (cid:48) cf ( k , iω n ) − Σ f ( k , iω n )] − ,where k and ω n denote momentum and Matsubara fre-quency, Σ cf , Σ (cid:48) cf and Σ f refer to self-energies from thecoupling between c and f fermions and self-interaction of f fermions, respectively. Local critical f fermion propa-gator, i.e., G f ( k , iω n ) = G f ( iω n ), is always a consistentsolution to saddle point equations [50]. Especially, in thelimit M/N = 0, the saddle point equations of f fermionsare identical to the zero-dimensional complex SYK modelwith the following conformal-limit solutions [51] G f ( τ ) = − π cosh (2 π E ) J √ e − π E (cid:18) T sin( πT τ ) (cid:19) e − π E T τ , where E is a parameter controlling the particle-hole asym-metry, and τ ∈ [0 , β ] is the imaginary time.Now, moving to the propagator of c fermion, we willfollow Ref. [36] closely. Though the model in Ref. [36]breaks translational symmetry by the locally indepen-dent disorder, we show in appendix that at MN (cid:28)
1, bothmodels have the same saddle point solutions. The trans-lational symmetry and the resulting momentum conser-vation equation are also shown in appendix. In the = + +
FIG. 2: The ladder diagram shows the self-consistent equa-tion for shear viscosity vertex. The black and red solid linesrepresent the Green’s function of c fermions and f fermions,respectively. The dashed line represents disorder average andthe shaded vertex represents full vertex. limit g (cid:29) tJ , there exists a crossover temperature, T inc ∼ t Jg , between the MFL regime in the lower temper-ature and the IM regime in higher temperature. When T (cid:28) T inc , the hopping term between conduction elec-trons dominates, and the self-energy of the c fermionyields [36, 50]Σ MFL cf ( iω n ) = ig T Jt cosh / (2 π E ) π / (cid:18) ω n T ln (cid:18) πT e γ E − J (cid:19) + ω n T ψ (cid:18) − iω n πT (cid:19) + π (cid:19) , (2)where ψ is the digamma function, and γ E = 0 .
577 is theEuler-Mascheroni constant. The self-energy shows thatthe c fermions exhibit a MFL behavior. Indeed, in thisregime, the model a linear-in- T resistivity as well as a T ln T entropy density [36, 52], i.e., S MFL ∼ g MJt ( T + T ln JT ).On the other hand, when T > T inc , the interactingterm between the conduction and the valence band elec-trons dominates. Since the interacting term is local, the c fermion propagator will also exhibit local critical be-havior [36, 50]. The c fermion self-energy reads [36, 50]Σ IM cf ( iω n ) = iT g Λ ν (0)( − (1 + e π E c ) e π E π J ( i + e π E c ) cosh (2 π E ) × Γ( + i E c + ω n πT )Γ( + i E c + ω n πT ) , (3)where Γ denotes Gamma function, and E c is a pa-rameters related to the conduction band filling. Atsmall µ f /J , µ c /g limit, E (cid:39) − µ f /Jπ / √ and E c (cid:39)− π / cosh / (2 π E ) µ c /g [36]. The form of self-energy in-dicates the quasiparticle does not exist, and the conduc-tion electrons enter the IM regime. As the Green’s func-tions of both c and f fermions are local SYK-type [36],the entropy density scales as S IM ∼ M JTg + N TJ , wherethe first and second term come from c fermions and f fermions, respectively [36]. III. SHEAR VISCOSITY
The shear viscosity is usually evaluated via the Kuboformula η = lim ω → ω Im G xy,xyR ( ω, G xy,xyR isthe retarded Green’s function of xy component of theenergy-momentum tensor, i.e., iG xy,xyR ( ω, p ) = (cid:90) dtd x e i ( ωt − p · x ) θ ( t ) (cid:104) [ T xy ( t, x ) , T xy (0 , (cid:105) . where θ ( t ) denotes the step function such that θ ( t ) = 1for t ≥ ... ] is commutator.In the following, we consider the isotropic dispersion (cid:15) k = k m − Λ2 with − Λ / ≤ (cid:15) ≤ Λ /
2. Generalizationto other dispersions is straightforward, and won’t changeour results qualitatively. Note that the lattice constanthas been taken to be 1, so momentum k becomes dimen-sionless, and we have the relations m ∼ t ∼ ∼ ν (0),where ν (0) denotes the density of states at Fermi level.For the isotropic dispersion, the density of state is a con-stant, ν ( (cid:15) ) ≡ (cid:82) k πδ ( (cid:15) − (cid:15) k ) = ν (0), (cid:82) k ≡ (cid:82) d k (2 π ) , irre-spective of the energy. The tensor T xy of c fermions isgiven by T xy ( p ) = (cid:80) i (cid:82) k c † k i Γ ( p ; k ) c k + p ,i + H . c . , where c k i = (cid:82) d x c x i e i k · x , and Γ ( p ; k ) = ( k x + px )( k y + py ) m forthe isotropic dispersion.As shown in Fig. 2, to the leading nontrivial orderin large- N limit, the self-consistent equation for the fullvertex Γ isΓ( p ; q ) = Γ ( p ; q ) + 1 N (cid:88) i (cid:90) q (cid:48) F ( i ) ( p ; q, q (cid:48) )Γ( p ; q (cid:48) ) , (4)where (cid:82) k ≡ (cid:82) k (cid:82) k , (cid:82) k ≡ T (cid:80) ω n and F ( i ) is representedin the second and third diagram in Fig. 2, i.e., F (1) = − g (cid:90) k G f ( q − q (cid:48) + k ) G f ( q (cid:48) − q − k ) G c ( q (cid:48) ) G c ( p + q (cid:48) ) . Because we are interested in the uniform case, i.e., p = , (cid:90) q (cid:48) F (1) ( , p ; q, q (cid:48) )Γ( , p ; q (cid:48) )= − g (cid:90) k,q (cid:48) G f ( q − q (cid:48) + k ) G f ( q (cid:48) − q − k ) × (cid:90) q (cid:48) G c ( q (cid:48) , q (cid:48) ) G c ( q (cid:48) , p + q (cid:48) )Γ( , p ; q (cid:48) ) , (5)Eq. 5 vanishes since it is odd in q (cid:48) x (or q (cid:48) y ). Owing to thesame reason, we find that F (2) on the right-hand side inFig. 2 also vanishes. Therefore, the vertex correctionsvanish, Γ( , p ; q ) = Γ ( ; q ) = q x q y m . Thus, to leadingorder in 1 /N , the shear viscosity is given by the sumover the set of ladder diagrams shown in Fig. 3, and thespectral representation of shear viscosity is [50] η = M π (cid:90) + ∞−∞ dω (cid:18) − ∂n F ( ω ) ∂ω (cid:19) (cid:90) + ∞−∞ d(cid:15) Θ xy ( (cid:15) ) A c ( ω, (cid:15) ) , (6) = + O (cid:0) N (cid:1) FIG. 3: The Feynman diagram for the calculation of (cid:104) T xy T xy (cid:105) at leading order in 1 /N , where the vertex correction vanishes.The black lines represent the Green’s function of c fermions. where n F ( ω ) = 1 / ( e βω + 1) is the Fermi-Dirac distribu-tion, A c ( ω, (cid:15) ) = − G c ( iω n → ω + i + , (cid:15) )] denotes thespectral function, and Θ xy ( (cid:15) ) = (cid:82) d k (2 π ) ( k x k y m ) δ ( (cid:15) − (cid:15) k )is the transport density of states for shear viscosity. IV. SHEAR VISCOSITY IN MFL REGIME
In the MFL regime, the Fermi surface is well definedand the leading temperature-dependence contribution toviscosity comes from the states near Fermi surface, (cid:15) = 0.This allows us to approximate Θ xy ( (cid:15) ) by the value atFermi surface, i.e., Θ xy ( (cid:15) ) ≈ k F πm ν (0), and to extendthe range of the integral of (cid:15) to infinity [50]. Finally, wehave η MFL ( T ) = M ν (0)64 m T (cid:90) ∞−∞ dω π sech ( ω T ) 1 | Im Σ MFL cf ( ω ) |≈ . M t Jg T cosh (2 π E ) . (7)Dividing the viscosity by the entropy density contributedby c fermions, S MFL c ∼ g MJt T ln JT , the shear viscosity toentropy ratio at low temperature scales as η MFL S MFL c ∼ cosh (2 π E ) J t g T ln( JT ) . (8)Since T (cid:28) T inc in MFL regime, the ratio is larger than aconstant, η MFL / S MFL c (cid:29) / ln( J/T inc ) = 1 / g/t ). Atzero temperature limit, η MFL / S MFL c diverges, as shownin Fig. 1(b).For the system with (marginally) well-defined quasi-particle, the shear viscosity is actually proportional tothe lifetime of quasiparticle, as indicated in Eq. (7). Thequasiparticle lifetime in the MFL is τ ∝ T − , which leadsthe scaling form of shear viscosity η ∝ T − (up to log-arithmic corrections). Note that for Fermi liquid, thequasiparticle lifetime, ∝ T − , leads to the well-knownresult η ∝ T − . More concretely, the inverse lifetime of c fermions in the MFL regime is [36] γ = g Tπ tJ cosh (2 π E ) . (9)Then we can estimate the viscosity to be η MFL ≈ εγ − ∼ M t Jg T cosh (2 π E ) , (10) where ε is the energy density which scales as ε ∼ M t ,agreeing with the result in Eq. (7).
V. SHEAR VISCOSITY IN IM REGIME
In the IM regime, the c fermions exhibit local criti-cal behavior, and there is no notion of Fermi surface.Thus, in contrast to the case of MFL, we should calculateΘ xy ( (cid:15) ) in the full spectrum instead of approximating it atthe fermi surface [50], Θ xy ( (cid:15) ) = m π (cid:0) (cid:15) + Λ2 (cid:1) θ (cid:0) Λ2 − | (cid:15) | (cid:1) .A technical advantage occurs owing to the local criticalform of c fermions’ propagator in the IM regime, namely,the spectral function is independent of (cid:15) , A c IM ( ω, (cid:15) ) = A c IM ( ω ). As a result, the shear viscosity splits into twoindependent integrations, η IM = M πT (cid:90) d(cid:15) Θ xy ( (cid:15) ) (cid:90) dω sech ( ω T ) A c IM ( ω ) , (11)both of which can be evaluated directly [50], and the finalresult is η IM ( T ) = M π
24 Λ Jg T cosh (2 π E )cosh(2 π E c ) . (12)In the IM regime, the entropy density corresponds to c fermions is given by S IM c ∼ M JTg , so the ratio betweenshear viscosity and entropy density is given by η IM S IM c ∼ cosh (2 π E )cosh(2 π E c ) Λ T . (13)If Λ (cid:28) J , there exists a robust temperature window inthe IM regime, i.e., Λ (cid:28) T (cid:28) min( J, g /J ), such thatthe KSS bound is strongly violated!In fact, the scaling form of the shear viscosity obtainedin the IM regime, η ∝ T − , is a universal property forlocal critical systems. In local critical regime, the localinteraction dominates over hoppings, and in turn dictatesthe scaling dimension of fermions. The most generic localinteraction allowed by U (1) symmetry is of quartic order.Thus, the local critical freedoms, i.e., the c fermions inour case, have scaling dimension 1/4, and consequentlythe spectral weight A ∝ T − / . Furthermore, the localcriticality also renders the vertex correction vanishing,and leads to the spectral representation of shear viscos-ity, as shown in Eq. (6). These reasons lead to the scalingform of shear viscosity η ∝ T − . Note that though thescaling form is the same in the MFL regime, the ori-gins behind them are different, i.e., the shear viscosity isdetermined by quasiparticle lifetime in the MFL as dis-cussed before. The essential point for the violation ofthe KSS bound is that the scaling form in the IM regimecan survive in an intermediate-temperature range, whichlead to a robust energy window violating the bound,as indicated in Fig. 1(b). In the discrete translation-ally symmetric system considered here, the only processthat can relax the momentum is electron-electron umk-lapp scattering. However, the c -fermion density can betuned small enough to suppress the umklapp process inlow energy and long-distance, so that the system is essen-tially momentum-preserving and hydrodynamics emergesin both the MFL and the IM regimes [53]. It calls for fur-ther experiments to establish whether or not the electronfluids in strange metals are in the hydrodynamic regime VI. DISCUSSION AND CONCLUSIONS
Though a similar violation of the KSS bound is also re-ported in unitary quantum gases by dynamic mean fieldtheory calculation [35], the SYK model has a better holo-graphic interpretation [39, 42] and analytical controllabil-ity than the model used in Ref. [35]. Thus our calcula-tions provide the first translationally invariant exampleviolating the KSS bound with known gauge-gravity cor-respondence. Moreover, as indicated in Ref. [37, 54], wealso expect that the model in this paper has a descriptionof semi-holography: f fermions form the bulk geometrywhile c fermions live on the boundary. From this pointof view, the η/ S c we calculate here is different from theone calculated in those full-holographic models, wherethe entropy is black hole entropy. To compare our resultwith those full-holographic results, one should replacethe S c in η/ S c by the entropy density of the whole sys-tem consisted of both f fermions and c fermions. Since S f ∝ N (cid:29) S c ∝ M , we have η/ S f ∝ M/N →
0, at the
M/N (cid:28) S f , with U (1) symmetry at each site.In conclusion, we investigated the shear viscosity ina translationally invariant, strongly correlated solvablemodel [36, 37]. By using Kubo formula, we obtained theinteresting behaviors of shear viscosity as a function oftemperature. In the MFL regimes, the shear viscosity isrelated to the quasiparticle lifetime; in the IM regimes,the result is more general and can be inferred from lo-cal criticality. As shown in Fig. 1(b), we further find aninteresting robust temperature range in the IM regimewhere the ratio of shear viscosity to entropy density, η/ S ,can strongly violate the KSS bound. To the best of ourknowledge, it is for the first time that the perfect fluid-ity behaviors are discovered in the coupled local criticalSYK models in an intermediate-temperature range. Webelieve that our results could shed new light to under-standing shear viscosity of strongly correlated systems. ACKNOWLEDGEMENT
We would like to thank Wei-Jia Li, Hong L¨ u , Sang-JinSin, Yu Tian and Shao-Feng Wu for helpful discussions.This work is partly supported by NSFC (No.11875184 & No.11805117), NSFC under Grant No. 11825404 (S.-K.J. and H.Y.), the Simons Foundation via the It FromQubit Collaboration (S.-K.J.), and NSFC under GrantNo. 11575195 (Z.-Y.X.). Z.-Y.X. is also supported by theNational Postdoctoral Program for Innovative TalentsBX20180318. X.-H.G. would also like to thank HanyangUniversity for the hospitality during the APCTP focusprogram “Holography and Geometry of Quantum Entan-glement”. APPENDIX A: SADDLE POINT SOLUTIONS
Summing the relevant Feynman diagrams in the large- N limit [37], the saddle-point equations are given by G c ( k , iω ) = 1 iω n − (cid:15) k + µ c − Σ cf ( k , iω n ) , (A1) G f ( k , iω n ) = 1 iω n + µ − Σ (cid:48) cf ( k , iω n ) − Σ f ( k , iω n ) , (A2)Σ cf ( k , iω n ) = − g (cid:90) k (cid:48) G c ( k (cid:48) , iω n (cid:48) )Π f ( k + k (cid:48) , iω n + iω n (cid:48) ) , (A3)Σ (cid:48) cf ( k , iω n ) = − MN g (cid:90) k (cid:48) G f ( k (cid:48) , iω n (cid:48) )Π c ( k + k (cid:48) , iω n + iω n (cid:48) ) , (A4)Σ f ( k , iω n ) = − J (cid:90) k (cid:48) G f ( k (cid:48) , iω n (cid:48) )Π f ( k + k (cid:48) , iω n + iω n (cid:48) ) , (A5)Π f ( q , i Ω n ) = (cid:90) k G f ( k , iω n ) G f ( q + k , i Ω n + iω n ) , (A6)Π c ( q , i Ω n ) = (cid:90) k G c ( k , iω n ) G c ( q + k , i Ω n + iω n ) , (A7)where k and ω n denote momentum and Matsubara fre-quency, G i , i = c, f refers to the Green’s function of c and f fermion, respectively, and (cid:82) k ≡ (cid:82) k (cid:82) k , (cid:82) k ≡ T (cid:80) ω n , (cid:82) k ≡ (cid:82) d k (2 π ) . It is easy to check from the saddle pointequations that local critical f fermion propagator, i.e., G f ( k , iω n ) = G f ( iω n ), is always a consistent solution tothe saddle point equations. Indeed, at the M/N → f fermion propagator is [51] G f ( τ ) = − π cosh (2 π E ) J √ e − π E (cid:18) T sin( πT τ ) (cid:19) e − π E T τ , (A8)where E is a parameter controlling the particle-hole asym-metry, and τ ∈ [0 , β ] is the imaginary time. For finite M/N , a local critical form of f fermion propagator is stillconsistent with the full saddle point equations. More-over, according to Ref. [36, 37], finite M/N correction issubleading. Thus, we assume the local critical solutionholds at a small but finite
M/N , and focus on the case
M/N (cid:28) c fermion propagators, we will followRef. [36] closely. The self-energy of c fermion is givenby Eq. A3. Since G f is local critical, we can see fromEqs. (A3) and (A6) that Σ cf is also independent ofmomentum, i.e., Σ cf ( k , iω n ) = Σ cf ( iω n ), and conse-quently Σ cf ( τ ) = − g G c ( τ ) G f ( τ ) G f ( − τ ), with G c ( τ ) ≡ T (cid:80) ω n G c ( iω n ) and G c ( iω n ) ≡ (cid:82) k G c ( k , iω n ). Thenwith the assumption sgn(Im[Σ cf ( iω n )]) = − sgn( ω n ),and in the limit of infinite bandwidth Λ → ∞ (i.e.,bandwidth is the largest energy scale), G c ( iω n ) ≈ ν (0) (cid:82) + ∞−∞ d(cid:15) π iω n − (cid:15) − Σ cf ( k ,iω n ) = − i ν (0)sgn( ω n ), and G c ( τ ) = − ν (0) T πT τ ) , where ν (0) is the density of stateat fermi level. The self-energy of the c fermion yields [36]Σ MFL cf ( iω n ) = ig T Jt cosh / (2 π E ) π / (cid:18) ω n T ln (cid:18) πT e γ E − J (cid:19) + ω n T ψ (cid:18) − iω n πT (cid:19) + π (cid:19) , (A9)where ψ is the digamma function, and γ E = 0 .
577 isthe Euler-Mascheroni constant. The self-energy indicatethat in the large bandwidth limit, the c fermions exhibita MFL behavior.On the other hand, in the limit where | iω n + µ c − Σ c ( iω n ) | (cid:29) Λ, one can find local critical solutions ofSYK type for both c and f fermions [36] at conformallimit. Namely, the f fermion propagator is still given byEq. (A8), while the c fermion will enter the IM regime,whose propagator reads [36] G c ( iω n ) ≈ π ( µ c − Σ cf ( iω n )) , (A10)where the self-energy is given byΣ cf ( iω n ) = iT g Λ ν (0)( − (1 + e π E c ) e π E π J ( i + e π E c ) cosh (2 π E )Γ( + i E c + ω n πT )Γ( + i E c + ω n πT ) , (A11)where E (cid:39) − µ/Jπ / √ and E c (cid:39) − π / cosh / (2 π E ) µ c g at small µ f /J , µ c /g limit. Note Eq. (A10) is only valid provided T (cid:29) T inc and g (cid:29) Λ J . APPENDIX B: SYMMETRY AND NOETHERCURRENTS
In the following, we use Lagrangian formalism to de-fine the energy-momentum tensor in the long wavelength limit. The Lagrangian density of our model is given by L = (cid:88) l c † l ( x )( ∂ τ − ∇ m − µ c ) c l ( x )+ (cid:88) n f † n ( x )( ∂ τ − µ f ) f n ( x )+ (cid:88) i,j,k,l g ijkl N M / f † i ( x ) f j ( x ) c † k ( x ) c l ( x )+ (cid:88) i,j,k,l J ijkl N / f i ( x ) † f j ( x ) † f k ( x ) f l ( x ) , (B1)The Lagrangian L is invariant under the translationalsymmetry (cid:126)x → (cid:126)x + (cid:126)a , τ → τ + a . Following the standardNoether procedure, one obtains the energy-momentumtensor T µν = ∂ L ∂∂ µ ψ ∂ ν ψ − δ µν L , (B2)from which we can get the momentum operator P i ≡ T i P i ( x ) = − i (cid:88) l c † l ( x ) ∂ i c l ( x ) , (B3)which is conserved due to the translational symmetry.Note that the immobile f -fermions do not contribute tothe total momentum. More importantly, the stress tensorused to evaluate the shear viscosity is given by T xy ( x ) = − m (cid:88) l c † l ( x ) ∂ x ∂ y c l ( x ) . (B4)Indeed, the interacting part of the Lagrangian densitydoes not show up in the stress tensor. Only the diagonalpart is modified, T xx ( x ) = − m (cid:88) l c † l ( x ) ∂ x c l ( x ) − L ( x ) (B5)= − (cid:88) l c † l ( x ) (cid:16) ∂ τ − − ∂ x + ∂ y m − µ c (cid:17) c l ( x ) − (cid:88) n f † n ( x )( ∂ τ − µ f ) f n ( x ) − L I , (B6)where the last term is the interacting part. APPENDIX C: THE DERIVATION OF SHEARVISCOSITY IN TERMS OF SPECTRALFUNDTION
We prove that the shear viscosity defined via the Kuboformula η = lim ω → ω Im G Rxy,xy ( ω, ,G Rxy,xy ( ω,
0) = − i (cid:90) dtd(cid:126)xe iωt θ ( t ) (cid:104) [ T xy ( t, (cid:126)x ) , T xy (0 , (cid:105) , (C1)is equivalent to (6) in terms of spectral functions.The xy -component of the uniform energy-momentumtensor for c -fermions is given by T xy = (cid:90) d k (2 π ) c † ki k x k y m c ki . (C2)To obtain the retarded Green function, we first use theimaginary time formula. In the tree level, we have G xy,xy ( i Ω ,
0) = − M T (cid:88) ω n (cid:90) d k (2 π ) (cid:18) k x k y m (cid:19) G c ( iω n , k ) G c ( iω n + i Ω n , k ) . (C3)Using the spectral representation, G ( z ) = (cid:82) dω π A c ( ω ) z − ω , oneis able to sum over Matsubara frequencies and continueto real frequencyIm T (cid:88) ω n G ( iω n ) G ( iω n + Ω + iδ )= − (cid:90) dω (cid:48) π A c ( ω (cid:48) ) A c ( ω (cid:48) + Ω)[ n F ( ω (cid:48) ) − n F ( ω (cid:48) + Ω)] . (C4)We obtain the imaginary part of the retarded Green’sfunctionIm G Rxy,xy (Ω , M (cid:90) d k (2 π ) (cid:18) k x k y m (cid:19) (cid:90) dω π A c ( ω, k ) A c ( ω + Ω , k )[ n F ( ω ) − n F ( ω + i Ω)] . (C5)The shear viscosity is then given by η = M (cid:90) ∞−∞ dω π (cid:18) − ∂n F ∂ω (cid:19) (cid:90) ∞−∞ d(cid:15) Θ xy ( (cid:15) ) A c ( ω, (cid:15) ) , (C6)where Θ xy ( (cid:15) ) ≡ (cid:82) d k (2 π ) (cid:0) k x k y m (cid:1) δ ( (cid:15) − (cid:15) k ). APPENDIX D: SHEAR VISCOSITY INMARGINAL FERMI LIQUID
In MFL regime, the well-defined fermi surface allowsus to approximate the density of states ν ( (cid:15) ) at energy (cid:15) by density of states at fermi surface ν (0). Then we haveΘ xy ( (cid:15) ) = m v F (cid:90) d k (2 π ) cos θ sin θδ ( (cid:15) − (cid:15) k ) ≈ m v F π ν (0) ≈ ν (0)16 πm , (D1) where in the last step, we use the relation v F ∼ m in theisotropic dispersion. The shear viscosity is given by η MFL = M πT (cid:90) dω sech ( ω T ) (cid:90) d(cid:15) Θ xy ( (cid:15) ) A c MFL ( ω, (cid:15) ) = M πT m v F π ν (0) (cid:90) dω sech ( ω T ) (cid:90) d(cid:15)A c MFL ( ω, (cid:15) ) = M m v F ν (0)128 πT (cid:90) dω sech ( ω T ) | Im Σ MFL cf ( ω ) |≈ . M t Jg T cosh (2 π E ) , (D2)where in the last line, we have used the relation v F ∼ m ∼ ν (0) ∼ t in the isotropic dispersion. APPENDIX E: SHEAR VISCOSITY ININCOHERENT METAL
For the dispersion relation (cid:15) k = k m − Λ2 with band-width (cid:15) k ∈ [ − Λ2 , Λ2 ], we haveΘ xy ( (cid:15) ) = (cid:90) d k (2 π ) (cid:18) k x k y m (cid:19) δ ( (cid:15) − (cid:15) k )= 1(2 πm ) (cid:90) dθ cos θ sin θ (cid:90) dkk δ ( (cid:15) − (cid:15) k )= m π (cid:18) (cid:15) + Λ2 (cid:19) θ (cid:18) Λ2 − | (cid:15) | (cid:19) , (E1)where θ ( x ) is the unit step function. One can also findΘ xy using Fourier transform [35, 55], which exactly givesthe same result. The spectral function of c fermion in IMregion is given by [36], A c ( ω, (cid:15) ) = A c ( ω )= − Re (cid:104) e i π π / J / cosh / (2 π E )( i + e π E c ) gT / √ e π E c Γ( − i βω − π E c π )Γ( − i βω − π E c π ) (cid:105) , (E2)which is independent of (cid:15) as a result of local criticality.Then the shear viscosity is given by η = M πT (cid:90) d(cid:15) Θ xy ( (cid:15) ) (cid:90) dω sech ( βω A c ( ω ) = M πT Λ π π / J cosh / (2 π E ) g cosh(2 π E c )= M π /
24 Λ Jg T cosh / (2 π E )cosh(2 π E c ) , (E3)where we have used (cid:82) d(cid:15) Θ xy ( (cid:15) ) = Λ π , and (cid:90) dω sech ( βω A c ( ω ) = 16 π / J cosh / (2 π E ) g T
12 cosh(2 π E c ) (cid:90) dω (cid:16) sech( βω − π E c )Γ( + i βω − π E c π )Γ( − i βω − π E c π ) (cid:17) = 8 π / J cosh / (2 π E ) g cosh(2 π E c ) (cid:90) dx (cid:16) sech( x )Γ( + i x π )Γ( − i x π ) (cid:17) = 8 π / J cosh / (2 π E ) g cosh(2 π E c ) . (E4) APPENDIX F: THERMAL DIFFUSIONCONSTANT
We calculate the thermal diffusion coefficient in bothregimes by using the results given in Ref. [36]. The ther-mal diffusivity can be given by Einstein’s relation D = κ c V , (F1)where κ is the ‘closed-circuit’ thermal conductivity and c V is the specific heat.In MFL regime, from Ref. [36], we have κ MF L ∼ M Jt /g and c MF LV ∼ M ( g /t )( T /J ) ln(
J/T ), wherewe have set E = 0 in the following calculations. Thethermal diffusion constant scales as D MF L ∼ J t g T ln( JT ) . (F2)Note that as T →
0, the thermal diffusion constantbecomes divergent same as the shear viscosity. Since T (cid:28) T inc , we conclude that D MF Lκ (cid:29) t g J g/t ) . Similarly, in the IM regime, one has κ IM ∼ M J Λ /g and c IMV ∼ M JT /g Ref. [36]. The thermal diffusionconstant scales as D IM ∼ π / Λ T . (F3)Due to the IM existing only at temperature above T inc ,we always have D IM (cid:28) π / g J . In the MFL regime, thethermal diffusion has a 1 /T dependence due to local criti-cality. It was argued that the fast ‘Planckian’ dissipationtogether with the causality of diffusion results in an up-per bound of diffusivity [56]. The results found in thiswork strongly implies that the shear viscosity and theupper bound of diffusivity maybe deeply connected. APPENDIX G: RELATION TO THE DCCONDUCTIVITY
In MFL regime, similar to case of shear viscosity, theinverse lifetime Eq. (9) also gives rise to the T − depen- dence of DC conductivity. From [36], one has σ MF LDC ∼ Mmγ ∼ M Jt T g . (G1)From uncertainty principle, the metallic conductivity in2D is bounded below by the Mott-Ioffe-Regel (MIR)limit, and σ = nτ /m ∼ ( k F l )1 / (cid:126) ≥ / (cid:126) , where l is theelectronic mean free path and the charge unit is omit-ted. The conductivity obtained here can be lower thanthe MIR limit 1 / (cid:126) numerically by tuning parameters, al-though the MFL is not rigorously a bad metal.In the IM regime, the DC conductivity reads σ IMDC ∼ M Λ Jg T cosh / (2 π E )cosh(2 π E c ) , (G2)which shares the same scaling form with the shear vis-cosity in Eq. (10). It is not surprising. Firstly, becauseof local criticality, the spectral density is independent ofmomentum. Secondly, the vertex of shear viscosity andconductivity has the same scaling, which is 1 /m ∼ t . Thecombination of above two features completely determinethe scaling form.Both of shear viscosity and DC conductivity vanishwhen T (cid:29) T inc due to the same scaling forms in Eqs. (10)and (G2). To reach T (cid:29) T inc , one can consider the de-couple limit t → S c contributed by c-fermion keep fixed under the decouplelimit, which is equal to the entropy of the SYK modelwith J IM = g /J . From this point of view, the violationof the KSS bound of η/ S c here shares the same reasonwith the deviation from the MIR limit of σ DC in theincoherent metal regime.These two bounds can be understood from the inverselifetime for the c fermions Eq. (9). 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