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The extended diffusive Sachdev-Ye-Kitaev model as a sort of"strange metal"

The 0+1-d Sachdev-Ye-Kitaev (SYK) fermionic model attracts nowadays a wide spread interest of the Condensed Matter community, as a benchmark toy model for strong electron correlation and non Fermi Liquid behavior. It is exactly solvable in the infrared limit and reproduces the linear dependence of the resistivity on temperature T, in linear response, typical of the strange metal phase of High Temperature Superconducting (HTS) materials. The breaking of its conformal symmetry requires ultraviolet corrections for a faithful description of its pseudo Goldstone Modes. Extension of the model to higher space dimension includes a local U(1) phase generating collective bosonic excitations driven by the additional ultraviolet contribution to the action. These excitations are studied here, in a temperature window of incoherent dynamics in which the expected chaotic regime has not yet taken over. We identify them as neutral diffusive energy excitations with temperature dependent lifetime h / k_B T. They provide thermalization of the system and contribute to the T dependence of the transport coefficients. The linear unbound T increase of particle current is confirmed by our hydrodynamic modelization. A quantum liquid in interaction with this system would become a Marginal Fermi Liquid.


I. INTRODUCTION
After almost three decades the strange high temperature metal phase of the various families of materials which become superconducting at a high critical temperature T c and are denoted as High Tc Superconductors (HTS) is still a puzzle. The great number of families of materials which undergo superconductivity at high temperature, cuprates, iron based material and picnides, possibly nickelates, present rather general and basic surprising features, which require an advancement in the understanding of strongly correlated fermion systems. The most striking of these features is the linear increase with temperature of the electrical conductivity in the normal metal phase, up to ∼ 1000 K, without any sign of saturation 1-3 and the constancy of the thermal conductivity 4 . The second puzzling feature is the misterious mechanism for superconducting electron pairing at "high" temperature.
The phenomenon of superconductivity has always been an icebreaker vessel for the interpretation of the quantum nature of the metal realm. The BCS quantum model came out more than fourty years after the discovery of superconductivity in metals, most likely because the full fledge Quantum Field Theory was required to provide a convincing tool box. Something similar is happening today with the HTS, because we do not have full understanding of strong correlation in many body systems. It was immediately realized by P.W.Anderson 5 that the doped Mott insulator, which appears as a good starting point for interpreting the normal metal phase of HTS, breaks the frame of the Landau Fermi Liquid (FL) theory 6,7 , a theory which has dominated successfully up to the eighties. A brand new approach to strong correlation in metals, the Non Fermi Liquid Theory (NFL) 8,9 , is required, which is still unsettled.
A burst of interest comes now from the appearance of a new exactly solvable model in 0 + 1 − d dimension, the Sachdev-Ye-Kitaev (SYK) 10-12 model, which describes random all-to-all J − interaction between N Majorana fermions. The SYK model has become highly popular as a holographic dual for gravity theories of black holes [13][14][15] . Although it appears as the farthest possible from a realistic description of any real material, its extension to higher space dimensions could make it a useful prototype model for NFL.
Lattice anisotropy which characterises the cuprates and their 2 − d CuO planes would motivate preference for two space dimensions. We are unable to bridge "bad" or "strange" metals and Quantum Spin Liquids on one side [27][28][29] with the SYK model on the other side 14,18,[30][31][32][33][34] , but we concentrate on the circumstance of approximate symmetry breaking in the infrared (IR) limit, when there is scaling to strong interaction J → ∞, N → ∞, but finite βJ /N ratio. Fermionic pseudo Goldstone modes (pGm) arise and additional bosonic excitations, when the hopping on a 2−d space lattice also includes the first UV corrections 35 .
In fact, in the IR limit, hopping in the lattice can be dealt with within lowest perturbative order. The response of the fermionic excitations, in the conformal symmetry limit, to an external driving to be specified, gives rise to the celebrated linear temperature dependence of the resistivity and to the constancy in temperature of the thermal conductivity. However, the kinetic term added to the model requires the addition of a complex U (1) phase to the real fields, which couples with the fermionic pGm beyond the IR limit and gives rise to bosonic collective and diffusive modes 19,35,36 . The original part of this work focuses on these modes, which we call Q-excitations. We set up an hydrodynamical description of these energy modes which, by transporting energy, are responsible for the thermalization of the system in a temperature window T coh T T 0 . The temperature T 0 , with βT 0 ∼ O (βJ /N ) (the notation O(...) means "order of "), is given by Eq.(62) and is assumed to be the temperature threshold above which, on the basis of what found in the 0 + 1 − d SYK model, a "scrambled phase" leading to quantum chaos is expected in the SYK system.
On the other side, T coh denotes the temperature below which hopping in the lattice is coherent. If W is the bandwidth which arises from the hopping in the lattice, T coh ∼ W 2 /N J , as argued in Section II. To guarantee that hopping is a marginal perturbation in the J , N → ∞ limits, is βT coh ∼ O (βJ /N ), as well.
Section II collects the properties of the 0 + 1 − d SYK model perturbed by the lattice, in the conformal symmetry limit, including the hopping term in the action. Fermionic quasiparticles can be derived which smear the Fermi Surface (FS), but the most relevant contribution to particle and thermal conductivities comes from the dominant pGm which arise from the approximate symmetry breaking. These incoherent excitations are the sources of space gradients of the chemical potential µ, which is related to the time derivative of the phase ∂ t ϕ, introduced by the hopping term in the extended action. We characterize these phase modes and the related response of the system with the help of a hydrodynamical dissipative approach. The linearity of the 2 − d resistance σ −1 with temperature and the constancy of the thermal conductivity κ are obtained in this way, as reported in Section FIG. 2: A caricature of the toy system modelized in this work. 2 − d sheets of strongly interacting Majorana Fermions are described by a SYK model which includes a hopping term in the continuum limit approximation. The SYK action is given by Eq. (8). In the temperature window considered here the SYK system is incoherent and fully thermalized, but not chaotic. Linear response of the SYK system in the IR limit gives a thermal conductivity which is T independent and a linear in T resistivity. Breaking of the conformal symmetry which is present in the IR limit require UV corrections for regularization. Bosonic collective diffusive Q-excitations arise, which contribute to the transport coefficients. A 3 − d single band Fermi Liquid (FL) of complex fermions ("electrons" represented as stars in the picture) becomes a Marginal Fermi Liquid (MFL), when perturbative interaction with the SYK sheets in the incoherent IR limit is turned on. Backinteraction between the SYK sheets induced by the coupling with the MFL is not considered. The interaction of the Qexcitations with the MFL do not change the linear T dependence of the particle current derived in the IR limit, but adds extra T -dependence to the thermal conductance.
II.C. They are O (βJ /N ) in the large N strong J limit.
From now on we investigate the single particle and collective excitations of the model as well as their interactions to give an estimate of further contributions to the transport coefficient arising from these extra excitations beyond the IR limit. The bottom line of this analysis is that all extra contributions to κ and σ −1 coming from these extra excitations are are subdominant with respect to the ones of Section II.C and can be ignored in the strong coupling limit. From Section III onward, corrections beyond the IR approximation are considered.
The phase ϕ arises in a reparametrization of the fermionic propagator in the IR limit. This allows to extract the first correction to the action ∼ O (N/βJ ), i.e. the one coming from ultraviolet (UV) corrections. This extra term entails a diffusive dynamics for the phase and its fluctuations. In the absence of coherence, hydrodynamics can help to extract the collective properties of the phase beyond the conformal limit, as discussed in Section III. It is assumed that thermalization occurs at a Planck rate, with lifetime of the collective excitations τ 0 ∼ β (where β = 1/k B T ). A temperature independent diffusion coefficientD Q can be derived in this frame, which turns out to be O (βJ /N ), and allows for the definition of a mean square diffusion lengthã ∼ T 0 /T . In this approximation, the gapless diffusive Q-excitation modes contribute to the thermal conductivity with a term In Section IV we consider an hypotetical 3 − d quantum Liquid (qL), in which SYK sheets are embedded, which would be a Fermi Liquid of the same bandwidth W , when considered in isolation(see Figure 2). The complex fermionic quasiparticles of the qL, close to the Fermi energy become ill defined when the local interaction with the 0 + 1 − d SYK systems is turned on the qL becomes a Marginal Fermi Liquid (MFL). Inserting the quasiparticle lifetime of the MFL in an approximate expression for the thermal conductance, Eq.(71), provides again a subdominant contribution to κ, which is constant with temperature. In Section IV.B, the additional scattering rate induced by the presence of the Q-excitations is derived by means of the Fermi Golden rule, and compared to the case of the usual electron-phonon interaction. Competing effects produce a big drop of the exponent in the T −power law, with respect to the electron-phonon case. On the one hand incoherent scattering, in which thermalization is very effective, so that no bookkeeping for particle number and momentum is maintained, should imply higher scattering possibilities and higher scattering rates. On the other hand, the very local nature of the interaction reduces the energy matrix element and the scattering rate. The quasiparticle inverse lifetime acquires a FL-like contribution ∼ T 2 to be summed to the MFL scattering rate in a sort of Matthiessen's rule. This T 2 inverse scattering rate implies a 1/T contribution to thermal conductivity which is however subdominant once more. Similarly for the resistivity contribution, which, though subdominant, turns out to maintain the experimentally measured linear T dependence 1 .
In the Appendix A, the derivation of the hydrodynamic relativistic equations used in the text is reported.

II. THE SYK MODEL AND ITS EXTENSION TO HIGHER SPACE DIMENSIONS
The Hamiltonian H 0 for the uncoupled 0 + 1-d SYK dots is: Lattice position is denoted by the subscript x in the continuum limit. χ x,l are Majorana fermion operators of N fermionic flavors (klmn ∈ 1, .., N ) on each site x . In each uncoupled dot, the all to all interaction J klmn is averaged over. After this averaging, the incoherent dynamics produced by the random interaction and disorder is parametrized by an average interaction strength J . The large J and N limit (infrared (IR) limit) is considered, which is exactly solvable. In the IR limit, the model develops an approximate conformal symmetry, because it is invariant under a full reparametrization symmetry. Spontaneous breaking of this symmetry occurs, down to the SL(2, R) group symmetry 37 , with pseudo-Goldstone modes (pGm) which make the four point function divergent. The two point function, the time ordered single particle Green's function, G β c (τ ) = Tr {T t [χ x1 (τ )χ x1 (0)]} plays the role of an order parameter and is non local in imaginary time τ . However, symmetry breaking is induced anyhow, when the first ultraviolet correction is included to cure the divergency 38,39 . An additional term of O (N/βJ ) is added to the action, as a result of reparametrization of the"quasi-orderparameter" field. While the latter is bilocal in time, the added correction to the action is local in time and adds local dynamics which becomes extended in space when the model is extended to higher space dimensionality.
While the Green function and the self-energy of the original model are real, those of the extended model, , have a space dependent modulus and phase, with small fluctuations around the saddle point values G c (τ 12 ), Σ c (τ 12 ) of the 0 + 1 − d SYK dot at each site x (here τ 12 ≡ τ 1 − τ 2 ). We write: Here ϑ = 2πτ /β is a dimensionless time and ϑ 12 = ϑ 1 − ϑ 2 , while ϑ + = (ϑ 1 + ϑ 2 )/2 is a local time in the space evolution (however, we will most of the times denote the dimensionless time as τ unless differently specified).
The local correction δG (x, τ 1 − τ 2 , τ + ) e i ϕx(τ+) arises from nearest neighbour hopping in a lattice of spacingã. A simple argument for the determination of the relevance of the kinetic energy term in the scaling to the UV is reported here below.
Let τ (≡ τ + ) → τ = τ /s with s → J /T . Majorana fields scale as χ ∼ χ s 1/4 (scaling dimension is ∆ = 1/4). The kinetic energy due to the lattice is naively written as with t 0 ∼ s 1/2 t 0 . This is a relevant perturbation. Scaling up to s T = J /T implies which is a weak hopping, up to t 0 (s T ) ∼ J . This defines a threshold ( 'coherence' ) temperature T coh when: For T >> T coh the lattice can be dealt with as an irrelevant perturbation on the 0+1−d SYK model, while for T < T coh , the hopping is a marginal correction in the scaling limit.
We introduce now a more careful approach to the kinetic energy term to be added to Eq.(1).
Electron quasiparticles hopping out of site x to one of the neighbouring sites requires complex fermions. In a full paramagnetic system we drop out the spin, and the complex fermionic spinless operator c † x , c x for the particle can be represented in terms of two flavours of the neutral fermions on the same site: The hopping will change the interaction energy of all the Majorana Fermions of site x. As we ignore conservation of the particle number, we focus just on the site which is the origin of the jump. However, the local interference among neighbouring sites involves the local phase factor e i ϕx(τ+) . It can be shown 35 that the change of energy is, for one single realization of disorder and replica: The thermal average including annealed disorder (the average uses the replica trick in the standard SYK model) includes now an additional gaussian average for J 12lm , which mixes terms from different sites, and gives rise to random interdot hopping of energy scale t 0 /N . As usual, we have assumed that the mixing of replicas can be ignored in the saddle point approximation and we have dropped the replica label everywhere. The temperature determines the relevance of the resulting kinetic term in the limit J , N → ∞. To guarantee that the kinetic term is marginal also in the incoherent phase limit, it is re- The next step is the integration over the Majorana fields χ x,l (τ ), with the help of the Hubbard-Stratonovich fields of Eq.(2). The saddle point action in terms of the complex bilocal auxiliary fields G x (τ 1 , τ 2 ) and Σ x (τ 1 , τ 2 ), characterized by the phase ϕ x , is: The last term of the action is the intersite kinetic term, which was also previously derived 19 . The action has become complex due to the additional U (1) minimal coupling in Eq. (7). The real Green function in the saddle point approximation is where iω n are fermionic frequencies.
The conformal limit in the IR corresponds to the dropping of ∂ τ in the determinant , or of −iω n in Eq.(9). In this limit, at low temperature, assuming particle-hole (ph) symmetry, G c (iω n ) is given by: However, ∂ τ plays an important role in the extended model of Section III.
A. quasiparticle energy distribution of the SYK extended model close to the Fermi Surface at T T coh To check the nature of the quasiparticles for energies close to the Fermi Surface (FS), when the SYK model is extended perturbatively to two space dimensions, we calculate the quasiparticle occupation number n k close to the Fermi energy F ∼ µ.
The single particle Green's function of the SYK model in the conformal symmetry limit, The Green function G c (iω n ) of Eq.(10) is local in space. To include the k−dependence in the propagator, for k vectors close to the FS, we add a single band energy k =ṽ F k, in the continuum limit to the right hand side of Eq.(9) and expand to lowest order, obtaining 24 : The occupation number of the complex fermions can be rewritten in terms of the Majorana fermions by means of Eq.(6) as: We evaluate the Matsubara sum of G(k, i ω n ) in the complex frequency plane i ω n → z: G(k, z) has a cut and a pole at the origin. The cut can be taken on the real z > 0 axis and contributes to the integral of Eq.(13) as follows: where ζ 1 2 = −1.46035 is the Riemann function. At zero temperature the contribution of the cut vanishes.
The second term in Eq. (14) contains the pole at the origin. Defining ξ = k −µ J , we get It follows that the final result for Eq. (12), close to the Fermi energy, in the limit βJ → ∞ and low temperature, is: The FS of the SYK model extended to higher space dimensionality is blurred in this perturbative approach and Fermionic quasiparticles are no longer well defined. In Section IV.A we will derive the lifetime of the quasiparticles of a 3 − d quantum Liquid (qL) in perturbative interaction with the 0 + 1 − d SYK system and we find that the qL becomes itself a Marginal Fermi Liquid.

B. Fluctuations around the IR saddle point solution
Back to Eq.(8), we can expand the action to second order and integrate out the δΣ fluctuations. The resulting functional integral, in terms of the fluctuations of the G field, δg(τ 12 , τ + ), is 35 where q = 4, as in the usual SYK notation and Here we keep only quadratic terms in the phase difference ϕ x (τ + ) − ϕ x (τ + ), because the first order drops out by adding the c.c. and we obtain: Owing to the self-averaging established for the SYK model at large N , translational invariance allows space Fourier transform (p is a wavevector).
To highlight the physical meaning of the source term in the functional of Eq.(17), we define the energy density current J E in a hydrodynamic relativistic approach 40 (see Appendix A): Here w = + p is the enthalpy with pressurep and T α i is the energy-momentum tensor (italic indices i = 0, 1, 2, 3, while greek indices α = 1, 2, 3). J is the average particle current density and we will adopt the linear response approach, so that the fluctuating current ν is a small contribution, ν << J. Here u α are the components of the velocity and u 0 = 1.
The starting point of Appendix A is the conservation of the energy-momentum tensor T k i , which includes dissipative processes. The viscosity and the thermal dissipation are accounted for by the stress tensor of Navier-Stokes origin, τ k i . Here u i τ k i = 0, because the dissipative components of the stress-energy tensor and current are orthogonal to u. In the absence of external forces and in the case of J = 0, it can be shown in Eq.(A8) that the change of entropy (per unit volume) is given by: Here D Dt = u i ∂ ∂x i , is the covariant derivative, which includes the convective derivative. The last two terms on the right hand side are dissipative terms. When we integrate over space and time and we substitute ν/n → J E /k B T from Eq.(20), we get a source term to be added to the action of Eq. (33).
which reminds of the source term ∝ J · E of electrodynamics. T plays here the role of the background average constant temperature in linear response and v b is a small volume. As µ(x, τ + ) = ∂ τ+ ϕ x , it follows that fluid correlations contributing to energy density currents, , are generated by the functional of Eq. (33). This allows us to discuss thermal transport in the extended SYK model within the conformal limit, what is done in the next Subsection.
Meanwhile, it is found that the approximate conformal symmetry of the SYK model is spontaneously broken down to the SL(2, R) group symmetry 37 . The dominant contribution to the functional integration in Eq. (17) comes from the pseudo Goldstone modes (pGm) δg τ12 . At this level of approximation, the δg's do not depend on τ + and are eigenfunctions of the Kernel K c with eigenvalue 1. The signature of the symmetry breaking is the disappearance of the action of the δg's ∝ δg|K −1 c −1|δg in Eq. (17), what implies that the functional integral is actually divergent. Regularization requires ultraviolet (UV) corrections. In the next Subsection we are going first to discuss transport in the conformal symmetric limit by considering the saddle point for the action only, which suffices for a linear response approximation. The divergence of the functional integration will be cured in Section III.
C. Transport in the IR conformal symmetry limit In the conformal symmetry limit, we keep only the lowest order in the kinetic term by approximating derived from Eq. (19). The function ∂ τ ϕ p is canonically conjugated to the particle fluxṄ p (τ + ) = 1  17), δ 2 δφ 2 ln Z. Moving back to space and real time, the correlator in the hopping between sites, at saddle point, is where the labels L, R refer to left and right side in the hopping process and we have lumped in t 0 all prefactors. Here where A ± are the fermionic spectral functions with A β + (ω) = e βω A β − (ω). The A ± 's are real and positive. Let Ṅ (t) ← → be the currents to the Left/Right, which differ for a chemical potential µ imbalance. From Eq.(24), we have: The spectral functions are actually independent of the side: A β ±L (ω) = A β ±R (ω). According to Eq. (22), the difference between Left and Right fluxes over ω, in the limit ω → 0, is connected to the response function to an external perturbation of the chemical equilibrium: Indeed, by comparing with Eq. (26), provides the real part of the conductivity, with the density of the incoming states − m G Rβ (ν) /π.
The thermal conduction arises from the response to the energy current in place of the particle current. It is enough to substitute A(ν) → ν A(ν) in the final expression to obtain the thermal conduction response Re{κ}. On the other hand, at equilibrium, the imaginary part of the response function D Rβ (ω) and the retarded function G Rβ (ω) are simply related by We obtain The SYK Green function in the conformal limit which was approximated in Eq.(10), is: and, integrating by parts, we get: where we have introduced the definition of T coh .
By the same token, from Eq.s (27,28), the particle current can be derived following the same steps as before.
Ignoring the fact the current is neutral in this Majorana fermion system, an hypothetical electrical real conductivity σ would be 19 : Eq.(32) is the celebrated linear T dependence of the electrical resistivity offered by the SYK model. In the next Section we discuss the contribution of the collective Qmodes to the T −dependence of the transport coefficients. These modes arise when the UV corrections to the action are included.

III. COLLECTIVE BOSONIC EXCITATIONS FROM UV CORRECTIONS
In the previous Section, based on the spontaneous conformal symmetry breaking, we acted as including just the Goldstone modes in the functional integration and regarded Eq.(17) as a generating functional for the Goldstone mode correlations, having ∂ τ+ ϕ p as a source field: We have swept under the carpet the fact that these massless Goldstone modes are responsible for the divergence of the functional integral of Eq. (17). Moreover, ϕ x (τ + ) was given as an external source with no dynamics. However, symmetry breaking is only approximately spontaneous, because the term ∼ ∂ τ in Eq.(8) produces itself a symmetry breaking, which becomes relevant in the UV. The modes of the gaussian action are, so to say, pseudo Goldstone modes (pGm) and display a small mass. In the IR, a time reparametrization of the conformal Green's function under the diffeomorphism e iϑ → e i ϕx(ϑ) produces a local term to the action, which gives dynamics to the ϕ x (τ + ) fluctuations and generates diffusive collective bosonic excitations which we call Q-excitations. As the 0+1−SYK dots at the lattice sites are sinks of charge and momentum, the surviving dynamical variable is energy density and we interpret the Q-excitations as energy density currents. The IR Green's function under reparametrization reads (ϕ ≡ ∂ ϑ ϕ, ∆ = 1 4 ): which, when expanded to lowest order in ϑ 12 , gives ≈G β→∞ (ϑ 1 , ϑ 2 ) 1 + ∆ 6 (ϑ 12 ) 2 Sch e i ϕ(ϑ+) ,ϑ + , where Sch e i ϕ(ϑ+) , ϑ + is the Schwarzian 11 . The fluctuations δg τ12 ∼ R c G IR −G β→∞ can be substituted in the source term of Eq.(33), giving rise to a local action of order 1/βJ : which adds dynamics to the field ϕ x (τ + ). The constant term in Eq.(35) gives a contribution ∝ β, which is just a correction to the energy and will appear in Eq.(60). The added local action of Eq. (35) is O (βJ −1 ) and a correction of the same order should be included to the action 35,38 , coming from the δg|K −1 c − 1|δg term appearing in Eq. (17). We remind that ∂ τ ϕ x (τ + ) corresponds to changes in the chemical potential µ, according to Eq. (22).
The UV correction of the 0 + 1 − d SYK model is local in time. To extend the model to higher space dimensions we use the macroscopic continuity equation where J E is the energy current density of Eq. (20). This will help us in characterizing the collective bosonic Qexcitations. In the next Subsection we derive the response function to J E − J E fluctuations, starting from the equation of motion, which is assumed to be diffusive: where g is an external stochastic source. Diffusivity, besides being physically justified in the incoherent regime, can also be justified starting from the full action I[∂ τ ϕ p ] in the large J limit 35 .

A. Hydrodynamic response to energy current density fluctuations
According to Eq. (22), the gradients of the chemical potential can be obtained as a response to energy current density fluctuations in the 2 + 1 − d SYK system. This is of importance for us, as the perturbation of the local chemical potential in the 3 − d qL, induced by the energy current density fluctuations of the 2 + 1 − d SYK system, is our final goal. While in Section II.C we have presented the response of the energy current density J E to chemical potential gradients ∇µ, in this Subsection we study the response of the chemical potential to the J E fluctuations, starting from temperature gradients and from the continuity equation, Eq.(36).
Let us define the canonical conjugate variables x a , X a , appearing in the continuity equation, Eq.(36): in terms of which the usual linearized response equation takes the form:ẋ (here g b is the external stochastic source and a, b denote elementary space cells of volume v a , v b ).
Space integration of Eq.(36), together with Eq.(37), gives:Ṡ It follows from Eq.s (38,39,40) thaṫ The hydrodynamical approach developed in Appendix A, provides Eq.(A8), which reads: with ν defined in Eq. (20). When the particle current vanishes, the right hand side contributes with dissipative terms which can be cumulatively expressed as − ∂w ∂t . The term µ T ∇ν provides the diffusive contribution to the change in time of the entropy. Eq.(A10) shows that − w n ≈ −κ∇ 2 T . Introducing the diffusive constant D = κ/nC, where C is the specific heat, and posing C∇ 2 T ≈ ∇ 2 w, we obtain: Fourier transforming in space and inverting this equation, we get the response equation that we are looking after (µ → µ(ω ≈ 0)): The source X b,−ω in this equation can be interpreted as an elementary energy current density source: has the dimension of a thermal conductance 41 [C/time]. We obtain: In the following we consider the thermal average of dω J E −q,ω−ω J E q,ω , as inducing a response on the space and time dependent chemical potential energy density µ n. We will use this result when discussing the interaction between the low energy qL and the bosonic Q-excitations. In the Subsection B, the thermal average dω J E −q,ω−ω J E q,ω will be denoted as Classically, the symmetrizied correlation derived from from Eq.(39) is 42 1 2 {g a (t), g b (t )} v b = (γ ab + γ ba ) δ(t− t ) and, with the correspondence of Eq. (41), it becomes in the quantum case: where κ ∼ lim t→∞ Re{κ} t = lim ω→0 |ω| e {κ(ω)} , so that, substituting the J E −J E fluctuations with the imaginary part of the response via the fluctuation dissipation theorem, we have: where The response function of Eq.(47) will be evaluated in the next Subsection for damped Q-excitations within a restricted 2−d volume in which correlation is non vanishing. The space volume integration is explicit in Eq. (45) for the J E −J E fluctuations and implicit in the k−integral of Eq.(47). We will denote this restricted 2 − d volume asã 2 in Subsection B.

B. Lagrangian approach to gapless diffusive excitation modes
Eq.(37) is a classical diffusion equation of a non conserving system. We now construct a Lagrangian of excitation modes which is conserving but, assuming a relaxation time τ 0 for these modes, it reproduces diffusive motion. We will quantize this Hamiltonian and derive the fluctuations of these modes from the response function by means of the fluctuation-dissipation theorem. The canonical conjugate variables and the corresponding La-grangian (in 2-d) are:θ Note that the dimension of the two terms in square brack- From the equation of motion: the classical motion equation of Eq. (37) can be recovered if we put τ 0JE ≈ J E .
Although is already in the Lagrangian, we proceed to quantize the theory. Fourier transforming the Lagrangian: the canonical momentum for θ k is π k = ∂L ∂θ k = Aθ −k , and [θ k , π k ] = i δ kk . It follows that what defines the velocity v in terms of a lattice parameter a. The approach is similar to the one for phonons, but π(x) plays the role of the lattice displacement d(x, t), while ∇θ plays the role of the corresponding momentum π ph (x, t). The atomic mass M of the phonon problem corresponds to 1/T here. With the definitions given in Eq.s(48,52) The space integral of the response function, D β ππ (ω), corresponding to Eq. (47), but referring to the symmetrized correlation {π −k , π k } ω , can be expressed in terms of the convolution in k space. In turn, the latter can be transformed into an integral over the mode energy η k defined in Eq.(52). To account for thermalization, we introduce the inverse lifetime of the collective mode appearing in Eq. (48), as a damping of the mode, Γ = /τ 0 in the response function: so that, Here we have assumed that the correlations range a space areaã 2 >>ã. Using Eq.(53), we get Comparing Eq. (45) with Eq.(56) we see that the square bracket provides an expression for κ: where the velocity given by Eq.(52) has been inserted.
The diffusivityD Q defined in Eq.(A12) is connected to the thermal conductivity κ by the Einstein relation: The "particle density" ρ 0 appearing here is associated to the Q-excitations and can be defined consistently, on the basis of the parameters used in the hydrodynamical model of this Section, as ρ 0 = 1/v 2 τ 2 0 . Inserting Eq.(57) in Eq.(58) and using ρ 0 with the velocity given in Eq.(52), gives:D As the energy diffusion is mainly due to heat transport in a highly thermalizable environment, an estimate of CTã 2 /ã 2 , appearing in Eq.s(57,59), can be derived from the saddle point contribution to the energy of the SYK model 38 : By posingã 2 a 2 CT ∼ ∂ β ln Z = c/2β 2 , we get Eq.(61) allows to defineã 2 as the mean squared diffusive length, such thatD Q τ 0 ∼ã 2 . In turn, a temperature T 0 can be defined, T 0 is the threshold temperature for thermalization to be efficient and is independent of temperature if τ 0 ∼ Γ −1 ∼ β. Indeed, a "Planck" relaxation time scale is expected when the full incoherent SYK regime sets in 35,43 . In this case, bothD Q of Eq.(61) and T 0 of Eq.(62) are temperature independent.
Eq(62) confirms that T 0 is O βJ N , if just the zero order for Γ is retained. If we make the same substitution for CTã 2 /ã 2 in Eq.(57) that we did in Eq.(61), we get: In this Section we consider a 3 − d quantum Liquid (qL), which would be a single band FL if in isolation, and we consider its interaction with our extended 2 + 1 − d SYK model. In Section II and III we have recognized that there is a temperature threshold T coh given by Eq.(5), below which a 3 − d qL in interaction with the SYK system acquires coherence and a threshold temperature T 0 given by Eq.(62), at which the Q-excitations due to the UV corrections, become dominant. In this Section we discuss how the fermionic SYK excitations and the collective bosonic Q-excitations influence the coherence of the qL. Our aim is to derive changes in the lifetime of the quasiparticles of the qL due to the interaction. We start with the correction to the selfenergy of the FL induced by the energy distribution of the fermionic SYK excitations in the conformal limit (for T T coh ). We will find that the qL becomes a Marginal Fermi Liquid (MFL) (Subsection A). Next we derive the lifetime reduction induced by the interaction with the bosonic Q-excitations in the temperature range (T coh , T 0 ) (Subsection B).

A. marginal Fermi liquid (MFL) for T T coh
In a perturbative approach 24 , let us assume a local interaction strength J in the qL. The quasiparticles of the low energy FL have a residue Z, such that Zν 0 ∼ J −1 . The single particle energy is k =ṽ F k in the continuum limit, in the vicinity of the FS, with a renormalized physical velocity v * F = Zṽ F . The isotropic self-energy arising from the interaction, for k on the Fermi surface, is: where Π(q, i Ω) is the polarization function In Eq.(64), θ is the angle between q and k = k F and, for |q| << k F , we have approximated In the range of frequencies Ω ∼ T coh = W 2 /J , where W is the bandwidth (k B = 1 here), the contribution to the polarization, coming from the virtual FL quasiparticles gives the well known result: (α is a parameter of order one). The contribution coming from interaction with the extended SYK system, Π 2 (iΩ), is evaluated from the single particle Green's function of the SYK model, in the conformally symmetric limit, which is q− independent and reported in Eq. (10), The final result is: For T >> T coh we should put ln J /T coh → ln(J /T ) in Eq.(67). Σ (1,2) (k F , ω) changes sign at ω = 0 when the quasi-particle becomes a quasi-hole. Putting both terms together, the quasiparticle relaxation rate is: which shows that, to the lowest approximation, the perturbed FL is a Marginal Fermi Liquid. The lowest lying collective excitations of the MFL can be also derived in the present perturbative frame 35 . An hydrodynamic collective excitation, the would-be acoustic plasmon, is rather well defined. The acoustic plasmon is on the verge to emerge as a bound state at low energies, splitted off the p-h continuum. Its dispersion tends to the boundary of the p-h continuum, where the imaginary part vanishes. In Section II.C we have derived a thermal conductance κ in the conformal limit, which is independent of T . This T −independence appears to be consistent with the present result. Semiclassically, the left hand side of Eq.(45) can be approximated as (ω ∼ 0): The saddle point contribution to energy of the SYK model of Eq.(60) provides us with the energy fluctuation, given by: By plugging Eq.(69) in Eq. (45) with Eq.(70), we get, in the limit ω/k B T << 1, The The Q-excitations introduced in Section III.B are an additional scattering mechanism which contributes to the reduction of the lifetime of the quasiparticles of the qL.
According to the source term of Eq. (22), J E − J E fluctuations couple to the change of the chemical potential, as can be read off Eq. (44). Changes in the chemical potential perturb the qL. Our aim in this Subsection is to derive the effect on the lifetime of the quasiparticles of the qL due to this kind of interaction. The thermal average of Eq. (44) can be newly defined as: In real time G LQ + is a response function G + (t) = Θ(t)G ( t) and has the dimensions of [E].
Here we face a problem in the construction of the interaction matrix element between the SYK sheet system and the 3 − d qL. The transferred momentum q is a 3 − d vector in the liquid, but it is only defined in two dimensions in the J E − J E term. We overcame the same difficulty in Eq.s (45,47), by considering the convolution (with k− integration) as in Eq.(56) of Subsection III.B, which corresponds to averaging in space, but by assuming that the fluctuation correlations are localized within a volume ∼ã 2 . We eventually get Assuming in the T −power counting ρ 0 v 2 τ 2 0 ∼ O [T ] 0 , with /τ 0 ∼ Γ ∼ T , the prefactor of the energy G LQ + , provides a squared coupling prefactor |g| 2 ∼ T 0 while, in the case of the e-ph interaction in a metal, |g| 2 ∼ T .
Eq.(73) can play the role of an interaction energy matrix element for the scattering between Q-excitations and a MFL quasiparticle. We plug it in the Fermi Golden Rule and derive the thermalization rate 8 (ω has dimension [E] from now on): ν 0 is the density of states at F for a 3 − d volume. The integration over the direction of the momentum has already been performed and gives the factor of ( v F q) 2 in the denominator, reflecting the increased time available for small deflections. As a minimum momentum must be transferred to give a change in energy of ω, ω appears also in the lower limit of the momentum integral. The integral over ω is simply the number of possible hole excitations that can be created in the qL close to the FS. If G LQ + (q, ω) is independent of q and ω, the q integral is not sensitive to the value of the lower limit and is independent of ω. The subsequent integration over ω recovers the usual 3 − d result, τ −1 ∼ 2 , which is expected for a FL.
Inserting G LQ + of Eq.(73) in Eq.(74), we have In evaluating the integral, we have excluded the upper limit, because it implies very large transferred q values, which we neglect as it gives a contribution O 4 to the result. To proceed to T −power counting, we assume that the volume is factorized into an in-plane contribution L 2 ∼ã 2 where the diffusion dynamics takes place, times an out of plane one, represented by the term ν 0 v F L/ã 2 . Assuming again ρ 0 v 2 τ 2 0 ∼ T 0 , we get: This T −dependence cannot be distinguished from the FL one, although it arises from a very different origin, that is from the scattering of MFL single particle excitations by the Q-excitations. Just for comparison the electron -phonon scattering rate, is 44 : τ e−ph −1 ∼ T 3 for T << θ D .
If we use Eq.(71) as an approximate expression for the thermal conductance, in which we insert the result of Eq.(75), we get κ → 1/T as in the FL case. In fact, in the FL case, κ = C v F = C v 2 F τ 0 . As τ 0 ∼ T −2 and C ∼ T , then κ ∼ T −1 . However, asã 4 appears in Eq.(75), this κ contribution ∼ O [N/βJ ] 3 and it is irrelevant for large βJ .
On the other hand, resistivity is ruled by a relaxation time in which the forward scattering is subtracted from the response appearing in Eq.(73). For very small ω, both integrals are convergent and we have: where we have excluded the bottom limit of integration. Hence: Here the T −power counting provides 1 τσ ∝ T 2 T0 T 3 2 ∼ T which points to the unbound dependence ∝ T of the resistivity, once more. This result should be compared with the fifth T −power arising from the electron-phonon scattering and shows that, although the constraints on momentum conservation has been relaxed, the localized nature of the Q-excitations does not increase resistance with increasing temperature any further. This can also be seen from the presence ofã 6 in Eq.(77) which implies that a contribution to the resistivity would drop at increasing temperature as O [N/βJ ] 3 , again, and soon become irrelevant.

V. SUMMARY AND CONCLUSIONS
The SYK model, borrowed from gravity theories, is solvable at large flavour number N and is intrinsecally NFL. Non vanishing zero temperature entropy is the signature of a crucial role of quantum fluctuations in the build up of the ground state of the system. The model is originally in 0 + 1 − d dimensions and its fermionic fields are Majorana neutral fermions. This feature, if taken seriously, already implies that the charge and spin degrees of freedom have been dropped out of the model 16 , when describing the electronic carriers. Indeed Majorana Fermions are neutral and interaction J grows to infinity but is fully local and momentum is not conserved. As transport on the c-axis could be attributed to tunnelling, we propose in this work an extension of the SYK model to two space dimensions and include its interaction with a three dimensional quantum Liquid (qL).
Our focus is on the symmetry breaking of an approximate conformal symmetry of the SYK model in the IR, strong coupling limit. For temperature in the neighborhood of T coh , the symmetry breaking generates a "quasi-order parameter" G c (τ ) and "single" particle neutral fermionic excitations which are responsible for transforming the qL into a marginal Fermi Liquid. The linear dependence of the resistivity on T and the independence of T of the thermal conductivity, derived with the linear response, follow. However, the spontaneous breaking of the conformal symmetry is in reality explicitely produced by UV corrections. The UV corrections are local in time and they are found to be extended, but rather localized in space, when the SYK model is extended to higher space dimensions. They give rise to bosonic collective modes which we nickname as Q-excitations, in interaction with the qL. The Q-excitations are diffusive modes in the lattice. We assume that they have lifetime τ 0 ∝ β and we infer by scaling 35 that they have mean square space extensionã 2 ∼ T 0 /T . In this work we have derived the contribution of these excitations to the lifetime of the qL quasiparticles. The trends in the temperature dependence of the resistivity ρ and of the thermal conductivity κ derived in our approach are reported in Figure 1 for the temperature window T coh T T 0 , in which there is a crossover to the incoherent regime. For T > T 0 the SYK model saturates a bound on chaos 45 and no perturbative approach is any more feasible.
To sum up, we have studied the effect of interactions between the SYK model extended to 2 − d and the quasiparticles of a 3−d quantum Liquid in which the SYK system is embedded. The 2 − d SYK model can be thought of as a collection of disordered dots in a continuum limit, in which strong disorder restores space invariance due to self-averaging. In Section II, we have introduced the Hamiltonian of the SYK model extended to the lattice. The temperature threshold, T coh , below which the hopping in the lattice establishes coherence in the system has been defined in Section II.A and the complex fermion occupation number n k close to the Fermi energy has been derived in the conformal symmetry limit, by including the lattice perturbation. We show that these complex fermion single particle excitations which remind of FL quasiparticles are ill defined at the Fermi Surface, confirming the NFL nature of the extended version of the SYK model. This fact justifies our choice of concentrating on the neutral excitations of the model, which are the fermionic excitations originating from the IR saddle point strong coupling limit and the collective diffusive Qexcitations arising from UV corrections to the IR theory. The generating functional of the soft mode correlations, derived in the gaussian approximation 35 , is reported in Section II.B. An hydrodynamics of energy, instead of particles is introduced in this Section. It is crucial to keep in mind that the underlying physics of the incoherent SYK system implies that the 2 − d system acts as a sink for particles and momentum, so that the usual particle hydrodynamics is not justified. On the contrary, the energy current density has a diffusive dynamics which does not require momentum and particle conservation. The source term to be added to the action is derived in this way and is used in Section II.C, to obtain the thermal conductivity κ and the resistivity ρ of the model in the conformal symmetry limit 19 . These transport coefficients are proportional the temperature scale T coh which is ∼ O (βJ /N ) in the large N , large J limit.
The role of the UV corrections in giving dynamics to the collective bosonic excitations, the Q-excitations, is presented in Section III. The Q-excitations can be shown to be diffusive in 2 − d 35 and to produce energy current density fluctuations which become dominant for T ∼ T 0 . The response of the chemical potential to these fluctuations is derived in Section III.A. A Lagrangian approach to the dynamics of the Q-excitations, which includes diffusivity in an approximate way, and their quantization is presented in Section III.B. The T dependence of their contribution to transport for T ∼ T 0 follows from the choice of their lifetime τ 0 ∼ β, which implies full thermalization of the SYK system. It turns out that this contribution to thermal conductivity is subdominant (∼ O [βJ /N ] 1/2 ) with respect to the one of Section II.C.
Section IV is dedicated to the 3 − d qL and to its interaction both with the single particle fermionic excitations 24 and with the Q-excitations introduced in Section III. The lifetime of the quasiparticles of the qL is derived in Section IV.A. The interaction with the conformally symmetric SYK system is dealt with perturbatively, and makes the liquid a MFL. The contribution of the MFL to the thermal conductivity is derived in a semiclassical approximation and is found to be consistent with the one derived in Section II.C. In Section IV.B the influence of the Q-excitations on the lifetime of the MFL is derived by using the Fermi Golden Rule. These scattering rates allow to estimate the resistivity and thermal conductivity for T ∼ T 0 . The estimates of the temperature dependence of κ and ρ are visualized in the sketch of Figure 1.
The model confirms, also beyond the IR limit, the temperature dependence of the resistivity that is expected in the measurements, if the model does indeed entail features of the real "strange" NFL metals. All additional contributions to the thermal conductivity estimated here are subdominant with respect to the IR limit contribution (Eq.(32)), which is of O (βJ /N ) in the large N , strong interaction J limit. The pivotal quantity of our hydrodynamic approach to the collective Q-excitations is their mean square diffusion areaã 2 ∼ T 0 /T . In the scaling to the strong interaction limit, both T 0 and the diffusion coefficientD Q are temperature independent. The incipient localization with increasing temperature locates our result straightforwardly out of the Mott-Ioffe-Regel resistivity limit 46 . However, for T > T 0 the 0 + 1 − d SYK model enters into a scrambled phase which is fully chaotic. The analysis of this phase in the extended model is out of scope of this work.