Figure 1: sketch of the temperature dependence of the thermal conductivity
$\kappa$ and of the resistivity $\rho$ in the incoherent phase of the strongly
interacting fermionic model presented here, which is a sort of extension to
higher space dimensions of the $0+1-d$ SYK model in its pre-chaotic phase.

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 $\sim\hbar/k_{B}T$. 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.

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

A.Tagliacozzo1,2,3 1 INFN-Sezione di Napoli, Complesso Universitario di Monte
S. Angelo Edificio 6, via Cintia, I-80126 Napoli, Italy  2 Dipartimento di
Fisica "E. Pancini", Università degli Studi di Napoli Federico II, via Cintia,
I-80126 Napoli, Italy  3 CNR-SPIN, Monte S.Angelo via Cintia, I-80126 Napoli,
Italy

##  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 $\sim 1000\>K$, without any sign of saturationGurvitch and Fiory
(1987); Daou et al. (2008); Cha et al. (2020a) and the constancy of the
thermal conductivityCohn et al. (1992). 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.AndersonAnderson (2013) 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) theoryHill et al. (2001); Parcollet and
Georges (1999), 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)Schofield (1999); Varma et al. (2002a), 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)Sachdev and Ye (1993);
Kitaev ; Kitaev (2006) model, which describes random all-to-all ${\cal{J}}-$
interaction between $N$ Majorana fermions. The SYK model has become highly
popular as a holographic dual for gravity theories of black holes Maldacena
(1999); Sachdev (2010); Policastro et al. (2002). 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.

Generalized SYK models have been proposed with extension to higher space
dimensions Gu et al. (2017); Davison et al. (2017); Song et al. (2017);
Berkooz et al. (2017); Chew et al. (2017); Haldar et al. (2018); Patel et al.
(2018a); Chowdhury et al. (2018), most of them having in mind applications to
High Critical Temperature ($HT_{c}$) superconducting materials. There are
complex fermion versions of the SYK modelFu and Sachdev (2016); Cha et al.
(2020a); Davison et al. (2017); Klebanov et al. (2020). 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 sideVarma et al. (2002b);
Carlson et al. (2002); Jain and Anderson (2009) with the SYK model on the
other sideSachdev (2010); Patel et al. (2018b); Haldar and Shenoy (2018);
Chowdhury and Berg (2020a, b); Davison et al. (2017); Cha et al. (2020b), but
we concentrate on the circumstance of approximate symmetry breaking in the
infrared (IR) limit, when there is scaling to strong interaction
${\cal{J}}\to\infty,N\to\infty$, but finite $\beta{\cal{J}}/N$ 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 correctionsSalvati and Tagliacozzo (2021).

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 modesSong et al. (2017); Cai et
al. (2018); Salvati and Tagliacozzo (2021). 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 $T_{coh}\lesssim T\lesssim T_{0}$. The temperature $T_{0}$, with $\beta
T_{0}\sim{\cal{O}}\left(\beta{\cal{J}}/N\right)$ (the notation
${\cal{O}}(...)$ means "order of "), is given by Eq.(65) 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, $T_{coh}\sim W^{2}/N{\cal{J}}$, as argued in Section II. To
guarantee that hopping is a marginal perturbation in the
${\cal{J}},N\to\infty$ limits, is $\beta
T_{coh}\sim{\cal{O}}\left(\beta{\cal{J}}/N\right)$, as well.

Figure 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. Back-interaction between
the SYK sheets induced by the coupling with the MFL is not considered. The
interaction of the Q-excitations 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.

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 $\mu$, which is related to the time
derivative of the phase $\partial_{t}\varphi$, 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 resistance with temperature and the constancy of the
thermal conductivity $\kappa$ are obtained in this way, as reported in Section
II.C.

The phase $\varphi$ also arises in a reparametrization of the fermionic
propagator in the IR limit. This allows to extract the first correction to the
action $\sim{\cal{O}}\left(N/\beta{\cal{J}}\right)$, 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
$\tau_{0}\sim\hbar\beta$ (where $\beta=1/k_{B}T$). A temperature independent
diffusion coefficient $\tilde{D}_{Q}$ can be derived in this frame, which
turns out to be ${\cal{O}}\left(\beta{\cal{J}}/N\right)$, and allows for the
definition of a mean square diffusion length $\tilde{a}_{\ell}\sim T_{0}/T$.
In this approximation, the gapless diffusive Q-excitation modes contribute to
the thermal conductivity with a term $\propto\sqrt{T}$, which is
${\cal{O}}\left([\beta{\cal{J}}/N]^{1/2}\right)$ (Section III.B).

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, are not well defined when
the local interaction with the $0+1-d$ SYK systems is turned on. We show that
the qL becomes a Marginal Fermi Liquid (MFL). Inserting the quasiparticle
lifetime of the MFL in an approximate expression for the thermal conductance,
Eq.(74), provides again constancy of $\kappa$ 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
mantained, 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 $\sim T^{2}$ 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. On the
contrary resistivity turns out to mantain the linear $T$ dependence as it is
experimentally measuredGurvitch and Fiory (1987); Cohn et al. (1992).

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 ${\cal{H}}_{0}$ for the uncoupled $0+1$-d SYK dots is:

$\displaystyle{\cal{H}}_{0}=\sum_{x}H_{x}=\frac{1}{4!}\>\sum_{x}\sum_{klmn}{\cal{J}}_{klmn}\chi_{x,k}\chi_{x,l}\chi_{x,m}\chi_{x,n}.$
(1)

Lattice position is denoted by the subscript $x$ in the continuum limit.
$\chi_{x,l}$ are Majorana fermion operators of $N$ fermionic flavors
(${klmn}\in 1,..,N$) on each site $x$ .

In each uncoupled dot, the all to all interaction ${\cal{J}}_{klmn}$ is
averaged over. After this averaging, the incoherent dynamics produced by the
random interaction and disorder is parametrized by an average interaction
strength ${\cal{J}}$. The large ${\cal{J}}$ 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 $\widetilde{SL}(2,\mathbb{R})$ group symmetryKitaev (2018), with
pseudo-Goldstone modes (pGm) which make the four point function divergent. The
two point function, the single particle Green’s function,
$G^{\beta}_{c}(\tau)={\bf
Tr}\left\\{T_{t}\left[\chi_{b1}(\tau)\chi_{b1}(0)\right]\right\\}$ plays the
role of an order parameter and is non local in imaginary time $\tau$. However,
symmetry breaking is induced anyhow, when the first ultraviolet correction is
included to cure the divergencyMaldacena and Stanford (2016); Kitaev and Suh
(2018). An additional term of ${\cal{O}}\left(N/\beta{\cal{J}}\right)$ is
added to the action, as a result of reparametrization of the"quasi-order-
parameter" 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,
$G_{x}(\tau_{1},\tau_{2}),\>\Sigma_{x}(\tau_{1},\tau_{2})$, have a space
dependent modulus and phase, with small fluctuations around the saddle point
values $G_{c}(\tau_{12}),\>\Sigma_{c}(\tau_{12})$ of the $0+1-d$ SYK dot at
each site $x$ (here $\tau_{12}\equiv\tau_{1}-\tau_{2}$). We write:

$\displaystyle
G_{x}(\vartheta_{1},\vartheta_{2})=\left[G_{c}(\vartheta_{12})+\delta
G(x,\vartheta_{12},\vartheta_{+})\right]\>e^{i\>\varphi_{x}(\vartheta_{+})},$
$\displaystyle\Sigma_{x}(\vartheta_{1},\vartheta_{2})=\left[\Sigma_{c}(\vartheta_{12})+\delta\Sigma(x,\vartheta_{12},\vartheta_{+})\right]\>e^{i\varphi_{x}(\vartheta_{+})}.$
(2)

Here $\vartheta=2\pi\tau/\beta$ is a dimensionless time and
$\vartheta_{12}=\vartheta_{1}-\vartheta_{2}$, while
$\vartheta_{+}=(\vartheta_{1}+\vartheta_{2})/2$ is a local time in the space
evolution (however, we will most of the times denote the dimensionless time as
$\tau$ unless differently specified).

The local correction $\delta
G\left(x,\tau_{1}-\tau_{2},\tau_{+}\right)\>e^{i\>\varphi_{x}(\tau_{+})}$
arises from nearest neighbour hopping in a lattice of spacing ${\tilde{a}}$. 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 $\tau(\equiv\tau_{+})\to\tau^{\prime}=\tau/s$ with $s\to{\cal{J}}/T$.
Majorana fields scale as $\chi^{\prime}\sim\chi\>s^{1/4}$ (scaling dimension
is $\Delta=1/4$). The kinetic energy due to the lattice is naively written as

$\displaystyle t_{0}\int d\tau\>\chi_{x,l}\chi_{x^{\prime},m}\to
t^{\prime}_{0}\int
d\tau^{\prime}\>{\chi^{\prime}}_{x,l}{\chi^{\prime}}_{x^{\prime},m},$ (3)

with $t^{\prime}_{0}\sim s^{1/2}t_{0}$. This is a relevant perturbation.
Scaling up to $s_{T}={\cal{J}}/T$ implies

$\displaystyle t_{0}\to
t_{0}(s_{T})=t_{0}\times\left(\frac{{\cal{J}}}{T}\right)^{\frac{1}{2}},$ (4)

which is a weak hopping, up to $t_{0}(s_{T})\sim{\cal{J}}$. This defines a
threshold ( ’coherence’ ) temperature $T_{coh}$ when:

$\displaystyle{\cal{J}}\sim
t_{0}\times\left(\frac{{\cal{J}}}{T_{coh}}\right)^{\frac{1}{2}}\>\>\to\>T_{coh}\sim\frac{t_{0}^{2}}{{\cal{J}}}.$
(5)

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 $c_{x}^{\dagger},c_{x}$ for
the particle can be represented in terms of two flavours of the neutral
fermions on the same site:

$\displaystyle
c_{x}=\frac{1}{\sqrt{2}}\left(\chi_{x1}+i\>\chi_{x2}\right),\>\>\>c_{x}^{\dagger}=\frac{1}{\sqrt{2}}\left(\chi_{x1}-i\>\chi_{x2}\right).$
(6)

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
$e^{i\>\varphi_{x}(\tau_{+})}$. It can be shownSalvati and Tagliacozzo (2021)
that the change of energy is, for one single realization of disorder and
replica:

$\displaystyle{\cal{H}}_{Kx}(\tau)=\\!i\>\frac{1}{3!}\sum_{lm}~J_{12lm}e^{i\>\varphi_{x}(\tau_{+})}\chi_{x,l}(\tau)\chi_{x,m}(\tau)+h.c.$
(7)

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
${\cal{J}},N\to\infty$. To guarantee that the kinetic term is marginal also in
the incoherent phase limit, it is required that $\beta
T_{coh}\sim{\cal{O}}\left(\frac{\beta{\cal{J}}}{N}\right)$, where $T_{coh}$ is
defined as $\frac{t_{0}^{2}}{N{\cal{J}}}$, as suggested by Eq.(5).

The next step is the integration over the Majorana fields $\chi_{x,l}(\tau)$,
with the help of the Hubbard-Stratonovich fields of Eq.(2). The saddle point
action in terms of the complex bilocal auxiliary fields
$G_{x}(\tau_{1},\tau_{2})$ and $\Sigma_{x}(\tau_{1},\tau_{2})$, characterized
by the phase $\varphi_{x}$, is:

$\displaystyle\frac{I_{ex}}{N}=\sum_{x}\left[-\frac{1}{2}\ln
det\left[\partial_{\tau}\\!-\Sigma_{x}\right]\\!+\\!\int\\!d\tau\>d\tau^{\prime}\\!\left\\{\\!-\frac{J^{2}}{4}\left|G_{x}(\tau,\tau^{\prime})\right|^{4}\\!+\\!\Sigma_{x}(\tau,\tau^{\prime})G_{x}^{*}(\tau,\tau^{\prime})-\frac{t_{0}^{2}}{N}\\!\sum_{x^{\prime}\in
nn}G_{x}(\tau,\tau^{\prime})G_{x^{\prime}}^{*}(\tau,\tau^{\prime})\\!\right\\}\right].$
(8)

The last term of the action is the intersite kinetic term, which was also
previously derivedSong et al. (2017). The action has become complex due to the
additional $U(1)$ minimal coupling in Eq.(7).

The single dot $0+1$-d SYK action can be recovered by dropping the sum over
sites, $\delta\Sigma$ and the last term. The real Green function in the saddle
point approximation is

$\displaystyle\frac{1}{G(i\omega_{n})}=-i\omega_{n}-\Sigma(i\omega_{n}).\>\>\>\>$
(9)

The conformal limit in the IR corresponds to the dropping of $\partial_{\tau}$
in the Pfaffian, or of $-i\omega_{n}$ in Eq.(9), with
$\Sigma(i\omega_{n})\to\Sigma_{c}^{0}(i\>\omega_{n})=\sqrt{{\cal{J}}}\sqrt{-i\>\omega_{n}}$.

On the contrary, $\partial_{\tau}$ plays an important role in the extended
model of Section III.

###  II.1 quasiparticle energy distribution of the SYK extended model close to
the Fermi Surface at $T\lesssim T_{coh}$

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 $\epsilon_{F}\sim\mu$.

The single particle Green’s function of the SYK model in the conformal
symmetry limit, at low temperature, assuming particle-hole (p-h) symmetry, is
given by:

$\displaystyle
G_{c}(i\omega_{n})=i\>\frac{sign(\omega_{n})}{\sqrt{{\cal{J}}}\sqrt{|\omega_{n}|}},$
(10)

where $i\omega_{n}$ are fermionic frequencies. The Green function
$G_{c}(i\omega_{n})$ 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 $\epsilon_{\vec{k}}=\tilde{v}_{F}k$, in the continuum limit
to the right hand side of Eq.(9) and expand to lowest order,
obtainingChowdhury et al. (2018):

$\displaystyle
G(\vec{k},i\omega_{n})\approx\left[\frac{1}{\Sigma_{c}^{0}(i\>\omega_{n})}+\frac{\epsilon_{\vec{k}}}{\left|\Sigma_{c}^{0}(i\>\omega_{n})\right|^{2}}\right].$
(11)

The occupation number of the complex fermions can be rewritten in terms of the
Majorana fermions by means of Eq.(6) as:

$\displaystyle n_{k}=\langle
c^{\dagger}_{k}c_{k}\rangle=\frac{1}{2}\left[1+i\>\langle\chi_{2}\chi_{1}\rangle_{k}\right]\hskip
56.9055pt$ (12)
$\displaystyle=\frac{1}{2}\left(1+\frac{1}{\hbar\beta}\>\sum_{n}e^{i\>\omega_{n}0^{+}}~\>G(k,i\>\omega_{n})\right).$

We evaluate the Matsubara sum of $G(k,i\>\omega_{n})$ in the complex frequency
plane $i\>\omega_{n}\to z$:

$\displaystyle\frac{1}{\hbar\beta}\>\sum_{n}e^{i\>\omega_{n}0^{+}}~G(k,i\>\omega_{n})$
$\displaystyle=\frac{1}{2\pi i}\oint
e^{z0^{+}}~\frac{dz}{e^{\beta{\cal{J}}z}+1}\>G(k,z),$ (13) $\displaystyle
G(k,z)\approx-\frac{1}{\sqrt{{\cal{J}}}}\frac{sign[z]}{\sqrt{-z}}+\frac{\epsilon_{k}-\mu}{{\cal{J}}z}.$
(14)

$G(k,z)$ 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:

$\displaystyle-\frac{1}{2\pi
i}\left\\{\int_{+\infty}^{0}\frac{dz}{e^{\beta{\cal{J}}z}+1}\>\frac{e^{-i\pi/2}}{\sqrt{z}}+\int^{+\infty}_{0}\frac{dz}{e^{\beta{\cal{J}}z}+1}\>\frac{e^{i\pi/2}}{\sqrt{z}}\right\\}$
$\displaystyle=2\frac{1}{\pi}\int^{+\infty}_{0}e^{-s0^{+}}\frac{ds}{e^{\beta{\cal{J}}s^{2}}+1}=-\frac{(\sqrt{2}-1)\>\zeta\left[\frac{1}{2}\right]}{\sqrt{\pi\beta{\cal{J}}}}>0,$

where $\zeta\left[\frac{1}{2}\right]=-1.46035$ 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
$\xi=\frac{\epsilon_{k}-\mu}{{\cal{J}}}$, we get

$\displaystyle-\frac{1}{2\pi i}\xi\>\lim_{\eta\to 0^{+}}\oint
e^{z0^{+}}\>\frac{dz}{e^{\beta{\cal{J}}z}+1}\>\frac{1}{\eta-z}=-\xi.$ (15)

It follows that the final result for Eq.(12), close to the Fermi energy, in
the limit $\beta{\cal{J}}\to\infty$ and low temperature, is:

$\displaystyle
n_{k}=\frac{1}{2}\left(1-\frac{\epsilon_{k}-\mu}{{\cal{J}}}\right).$ (16)

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.

###  II.2 Fluctuations around the IR saddle point solution

Back to Eq.(8), we can expand the action to second order and integrate out the
$\delta\Sigma$ fluctuations. The resulting functional integral, in terms of
the fluctuations of the $G$ field, $\delta g(\tau_{12},\tau_{+})$, isSalvati
and Tagliacozzo (2021)

$\displaystyle{\cal{Z}}\left[\partial_{\tau}\varphi_{p}(\tau_{+})\right]\propto\int
D\\!\left(\Pi\delta g^{*}_{\tau_{12},\tau_{+}}\right)D\\!\left(\Pi\delta
g_{\tau_{12},\tau_{+}}\right)\>e^{\frac{N}{4}\left[\langle\delta
g|K_{c}^{-1}-1|\delta g\rangle\right]}\>e^{-\frac{N}{2}\Re
e\left\\{\left\langle-i\>R_{c}^{-1}\partial_{\tau}\varphi_{p}|\delta
g\right\rangle\right\\}}$ $\displaystyle\times
e^{\frac{N}{4}\frac{{t_{0}}^{2}}{N}p^{2}\left[\langle-i\>\partial_{\tau}\varphi_{p}|R_{c}^{-1}\Lambda_{c}R_{c}^{-1}|-i\>\partial_{\tau}\varphi_{p}\rangle\right]}$
(17) $\displaystyle
g(\tau_{1},\tau_{2})=R_{c}(\tau_{1},\tau_{2})\>G(\tau_{1},\tau_{2}),\>\>\>\>\>R_{c}(\tau_{1},\tau_{2})=\beta{\cal{J}}\sqrt{3}\left|\tilde{G}_{c}(\tau_{1},\tau_{2})\right|$
$\displaystyle
K_{c}(\vartheta_{1},\vartheta_{2},\vartheta_{3},\vartheta_{4})=R_{c}(\vartheta_{1},\vartheta_{2}){G}_{c}(\vartheta_{1},\vartheta_{3}){G}_{c}(\vartheta_{4},\vartheta_{2})R_{c}(\vartheta_{3},\vartheta_{4})\hskip
85.35826pt$
$\displaystyle=(\beta{\cal{J}})^{2}\>(q-1)\left|{G}_{c}(\vartheta_{1},\vartheta_{2})\right|^{\frac{q-2}{2}}{G}_{c}(\vartheta_{1},\vartheta_{3}){G}_{c}(\vartheta_{4},\vartheta_{2})\left|{G}_{c}(\vartheta_{3},\vartheta_{4})\right|^{\frac{q-2}{2}}.$
$\displaystyle\langle\partial_{\tau}\varphi_{p}|R_{c}^{-1}\Lambda_{c}R_{c}^{-1}|\partial_{\tau}\varphi_{p}\rangle=\int
d\tau_{12}\int d\tau_{+}\left[\delta G_{c,x}(\tau_{12},\tau_{+})\>\delta
G_{c,x^{\prime}}^{*}(\tau_{12},\tau_{+})\right],$ (18)

where $q=4$, as in the usual SYK notation and

$\displaystyle\delta G_{c,x}(\tau_{12},\tau_{+})\delta
G_{c,x^{\prime}}^{*}(\tau_{34},\tau_{+}^{\prime})\hskip 28.45274pt$
$\displaystyle=\left[G_{x}(\tau_{1},\tau_{2})\>G_{x^{\prime}}^{*}(\tau_{3},\tau_{4})-G_{c}(\tau_{1}-\tau_{2})\>G_{c}(\tau_{3}-\tau_{4})\right]$
$\displaystyle\approx\frac{1}{2}\>G_{c}(\tau_{12})\>\left(e^{-i{\tilde{a}}\cdot\nabla_{x}\left[\varphi_{x}(\tau_{+})-\varphi_{x}(\tau_{+}^{\prime})\right]}-1\right)\>G_{c}(\tau_{34})+c.c.$

Here we keep only quadratic terms in the phase difference
$\varphi_{x}(\tau_{+})-\varphi_{x}(\tau_{+}^{\prime})$, because the first
order drops out by adding the $c.c.$ and we obtain:

$\displaystyle\to-\frac{1}{2}\>G_{c}(\tau_{12})\left({\tilde{a}}\cdot\nabla_{x}\left[\varphi_{x}(\tau_{+})-\varphi_{x}(\tau_{+}^{\prime})\right]\right)\\!\\!^{2}\>G_{c}(\tau_{34}).$
(19)

We now approximate
$\left[\varphi_{x}(\tau_{+})-\varphi_{x}(\tau_{+}^{\prime})\right]\approx$
$\left(\tau_{+}-\tau_{+}^{\prime}\right)\partial_{\tau}\varphi_{x}(\tau_{+})$
in the kernel $\Lambda_{c}$ of Eq.(18). Owing to the self-averaging
established for the SYK model at large $N$, translational invariance allows
space Fourier transform ($p$ is a wavevector).

The approximate conformal symmetry of the SYK model is spontaneously broken
down to the $\widetilde{SL}(2,\mathbb{R})$ group symmetryKitaev (2018). The
dominant contribution to the functional integration in Eq.(17) comes from the
pseudo Goldstone modes (pGm) $\delta g_{\tau_{12},\tau_{+}}$. At the present
level of approximation, the $\delta g$’s do not depend on $\tau_{+}$ 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 $\delta g$’s in
Eq.(17), what implies that the functional integral is divergent.
Regularization requires ultraviolet (UV) corrections. In the next Subsection
we are going first to discuss transport in the conformal symmetric limit. We
put here aside the matter of the divergence of the functional integration,
which will be cured in Section III and regard Eq.(17) as a generating
functional for the pGm correlations in the conformal limit, having
$\partial_{\tau_{+}}\varphi_{p}$ as a source field:

$\displaystyle{\cal{Z}}\left[\partial_{\tau}\varphi_{p}\right]=\int D\delta
g_{\tau_{12}}\>\>e^{-\frac{N}{2}\left\langle-i\>R_{c}^{-1}\partial_{\tau}\varphi_{p}|\delta
g\right\rangle}$ $\displaystyle\times
e^{\frac{N}{4}\frac{{t_{0}}^{2}}{N}p^{2}\left[\langle-i\>\partial_{\tau}\varphi_{p}|R_{c}^{-1}\Lambda_{c}R_{c}^{-1}|-i\>\partial_{\tau_{+}}\varphi_{p}\rangle\right]}.$
(20)

To highlight the physical meaning of the source term in the functional of
Eq.(20), we define the energy density current $\vec{J}^{\cal{E}}$ in a
hydrodynamic relativistic approachLandau and Lifshitz (1987):

$\displaystyle\vec{J}^{\cal{E}}=\frac{(\epsilon+p)}{n}\>\left(\vec{J}-\vec{\nu}\right)=T_{0}^{\alpha}\equiv
w\>u^{\alpha}.$ (21)

Here $w=\epsilon+p$ is the enthalpy with pressure $p$ and $T_{i}^{\alpha}$ is
the energy-momentum tensor (italic indices $i=0,1,2,3$, while greek indices
$\alpha=1,2,3$). $J$ is the average particle current density and we will adopt
the linear response approach, so that the fluctuating current $\vec{\nu}$ is a
small contribution, $\vec{\nu}<<J$. Here $u^{\alpha}$ are the components of
the velocity and $u^{0}=1$.

Our starting point is the conservation of the energy-momentum tensor
$T_{i}^{k}$, which includes dissipative processes. The viscosity and the
thermal dissipation are accounted for by the stress tensor of Navier-Stokes
origin, $\tau_{i}^{k}$. Here $u^{i}\tau_{i}^{k}=0$, because the dissipative
components of the stress-energy tensor and current are orthogonal to
$\vec{u}$. In the absence of external forces and in the case of $J=0$, it can
be shown (see Eq.(88) of Appendix A) that the change of entropy is given by
(small letters refer to quantities per unit volume):

$\displaystyle T\>\frac{Ds}{Dt}=\mu\nabla\nu-\tau_{i}^{k}\frac{\partial
u^{i}}{\partial x^{k}}$
$\displaystyle=-\nu\cdot\nabla\mu+\nabla\cdot\left(\mu\vec{\nu}\right)-\tau_{i}^{k}\frac{\partial
u^{i}}{\partial x^{k}}.$ (22)

Here $\frac{D}{Dt}=u^{i}\frac{\partial}{\partial x^{i}}$, 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 $\nu/n\to\vec{J}\>^{\cal{E}}/k_{B}T$ from Eq.(21), we
get a source term to be added to the action of Eq.(20).

$\displaystyle[\delta S]_{source}=\int dt\int
dv_{b}\>\>\frac{1}{k_{B}T}\vec{J}\>^{\cal{E}}\cdot\vec{\nabla}\mu,$ (23)

which reminds of the source term $\propto\int J\cdot E$ of electrodynamics.
$T$ plays here the role of the background average constant temperature in
linear response and $v_{b}$ is a small volume. As
$\mu(x,\tau_{+})=\hbar\>\partial_{\tau_{+}}\varphi_{x}$, it follows that fluid
correlations contributing to energy density currents,
$\frac{1}{(k_{B}T)^{2}}\langle\vec{J}^{\cal{E}}_{p}\>\vec{J}^{\cal{E}}_{p}\rangle\>\equiv\langle\dot{\cal{N}}_{p}(\tau_{+})\dot{\cal{N}}_{p}(0)\rangle$,
are generated by the functional of Eq.(20). This allows us to discuss thermal
transport in the extended SYK model within the conformal limit, what is done
in the next Subsection.

###  II.3 Transport in the IR conformal symmetry limit

In the conformal symmetry limit, we keep only the lowest order in the kinetic
term by approximating
$G_{x}(\vartheta_{1},\vartheta_{2})\approx{G}_{c}(\vartheta_{1},\vartheta_{2})\>e^{i\varphi_{x}(\tau_{+})}$
in Eq.(2). The kinetic term operator appearing in Eq.(18),
$\frac{1}{2}\frac{p^{2}{\tilde{a}}^{2}}{\hbar^{2}}R_{c}^{-1}\Lambda_{c}R_{c}^{-1}$,
is the space Fourier Transform of

$\displaystyle\frac{1}{2}({\tilde{a}}\cdot\overset{\leftarrow}{\nabla}_{x})G_{c}(\tau_{12})(\tau_{+}-\tau_{+}^{\prime})^{2}G_{c}(\tau_{34})({\tilde{a}}\cdot\overset{\rightarrow}{\nabla}_{x^{\prime}})$
(24)

derived from Eq.(19). The function $\hbar\partial_{\tau}\varphi_{p}$ is
canonically conjugated to the particle flux density
$\dot{{\cal{N}}}_{p}(\tau_{+})=\frac{\delta
I_{ex}}{\delta\hbar\partial_{\tau}\varphi_{p}}$, where
$I_{ex}[\partial_{\tau}\varphi_{p}]$ originates from the action of Eq.(8).
From second variation of the generating functional of Eq.(20),
$\frac{\delta^{2}}{\delta{\dot{\varphi}}^{2}}\ln{\cal{Z}}$, moving back to
space and real time, the current-current correlator in the hopping between
sites is

$\displaystyle\frac{t_{0}^{2}}{2\hbar}\langle\dot{\cal{N}}_{p}(t)\dot{\cal{N}}_{p}(0)\rangle\to\frac{t_{0}^{2}}{2\hbar}\left(i\>G_{cL}^{\beta}(t)\right)\left(-i\>G_{cR}^{\beta}(-t)\right),$
(25)

where the labels $L,R$ refer to left and right side in the hopping process and
we have lumped in $t_{0}$ all prefactors. The response function to an external
perturbation of the chemical equilibrium reads:

$\displaystyle D^{R\beta}(\omega)=-i\>\int
dt\>\theta(t)\left\langle\left[\dot{\cal{N}}(t),\dot{\cal{N}}(0)\right]\right\rangle\>e^{i\>\omega
t},$

with

$\displaystyle\left\langle\left[\dot{\cal{N}}(t),\dot{\cal{N}}(0)\right]\right\rangle\approx\frac{t_{0}^{2}}{4\hbar}\left[\left(i\>G_{L}^{\beta}(t)\right)\left(-i\>G_{R}^{\beta}(-t)\right)\right.$
$\displaystyle\left.-\left(-i\>G_{L}^{\beta}(-t)\right)^{*}\left(i\>G_{R}^{\beta}(t)\right)^{*}\right].$
(26)

Here

$\displaystyle i\>G^{\beta}(t)=\left\\{\begin{array}[]{cc}\int
d\omega\>A_{+}(\omega)\>e^{-i\>\omega t}&{\rm for}\>\>t>0\\\ -\int
d\omega\>A_{-}(\omega)\>e^{-i\>\omega t}&{\rm for}\>\>t<0\end{array}\right.$
(29)

where $A_{\pm}$ are the fermionic spectral functions with
$A_{+}^{\beta}(\omega)=e^{\beta\omega}\>\>A_{-}^{\beta}(\omega)$. We can drop
the stars in Eq.(26), because the $A_{\pm}$ are real and positive. Let
$\langle\dot{\cal{N}}(t)\rangle_{\begin{subarray}{c}\\\ \leftarrow\\\
\rightarrow\end{subarray}}$ be the currents to the Left/Right, which differ
for a chemical potential $\mu$ imbalance. From Eq.(25), we have:

$\displaystyle\frac{\delta}{\delta\dot{\varphi}}\langle\dot{\cal{N}}(t)\rangle_{\begin{subarray}{c}\\\
\leftarrow\\\ \rightarrow\end{subarray}}=-\frac{1}{4\hbar}t_{c}^{2}N\>\int
d\omega\>\int d\nu\>\left[A_{+L}(\nu)\>A_{-R}(\nu\mp\omega)\right.$
$\displaystyle\left.-A_{-L}(\nu)\>A_{+R}(\nu\mp\omega)\right]\>e^{\pm
i\>\omega t},\>\>\>{\rm for}\>\>t>0.\hskip 28.45274pt$ (31)

The spectral functions are independent of the side: $A_{\pm
L}^{\beta}(\omega)=A_{\pm R}^{\beta}(\omega)$ and $-\Im
m\left\\{G^{R\beta}(\nu)\right\\}/\pi$ is the density of the incoming states.
Therefore, the difference between Left and Right fluxes over $\omega$, in the
limit $\omega\to 0$, provides the real part of the conductivity:

$\displaystyle\lim_{\omega\to
0}\left[\frac{\delta}{\delta\dot{\varphi}}\>\left\langle\left(\dot{\cal{N}}_{\leftarrow}-\dot{\cal{N}}_{\rightarrow}\right)\right\rangle\right]_{\omega}$
$\displaystyle=-\frac{t_{0}^{2}\tau_{0}}{2\pi^{2}\hbar}N\>\int d\nu\>\Im
m\left\\{D^{R\beta}(\nu)\right\\}\>\frac{\partial}{\partial\nu}\Im
m\left\\{G^{R\beta}(\nu)\right\\},\>\>\>$ (32)

where we have defined
$\tau_{0}=\sqrt{\langle\left(\tau_{+}-\tau_{+}^{\prime}\right)^{2}\rangle}$ as
an average timescale of the correlations in time, according to Eq.(24). The
thermal conduction arises from the response to the energy current in place of
the particle current. It is enough to substitute $A(\nu)\to\hbar\nu\>A(\nu)$
in the final expression to obtain the thermal conduction response
$Re\\{\kappa\\}$. On the other hand, the imaginary part of the response
function $D^{R\beta}(\omega)$ and the retarded function $G^{R\beta}(\omega)$
are simply related by

$\displaystyle\Im
m\left\\{D^{R\beta}(\omega)\right\\}=\tanh\frac{\beta\omega}{2}\>\Im
m\left\\{G^{R\beta}(\omega)\right\\}.$ (33)

We obtain

$\displaystyle\frac{\kappa}{k_{B}}=\lim_{t\to\infty}\left[\frac{Re\\{\kappa(t)\\}}{k_{B}}\right]=\lim_{\omega\to
0}|\omega|\>\Re e\left\\{\kappa(\omega)\right\\}\hskip 42.67912pt$
$\displaystyle=N\>\frac{t_{0}^{2}}{2\hbar}\int d\nu\>{\rm
tanh}\frac{\beta\nu}{2}\>\frac{\partial}{\partial\nu}\left[-\frac{\nu}{\pi}\>\Im
m\left\\{G^{R\beta}(\nu)\right\\}\right]^{2}.$

The SYK Green function in the conformal limit is:

$\displaystyle
G^{R\beta}(\omega)=-i\>b\>\beta\>\frac{1}{\sqrt{\pi}}\>\frac{\Gamma\left(\frac{1}{4}-i\>\frac{\beta\omega}{2\pi}\right)}{\Gamma\left(\frac{3}{4}-i\>\frac{\beta\omega}{2\pi}\right)},\>\>b=\frac{\pi^{1/4}}{\sqrt{2\beta{\cal{J}}}}$
(34)

and, integrating by parts, we get:

$\displaystyle\lim_{t\to\infty}\left[\frac{Re\\{\kappa\\}}{k_{B}}\right]_{t}=\frac{1}{\pi^{3/2}\hbar}\>t_{0}^{2}\frac{N}{2{\cal{J}}}\hskip
71.13188pt$ (35) $\displaystyle\times\int dx\>{\rm sech}^{2}(\pi x)\>\left[\Re
e\left\\{\frac{\Gamma\left(\frac{1}{4}-i\>x\right)}{\Gamma\left(\frac{3}{4}-i\>x\right)}\right\\}\right]^{2}\propto
N\>\frac{k_{B}T_{coh}}{\hbar}.$

The imaginary part of the susceptibility is
$\chi^{\prime\prime}(\omega)=-\omega\>\Re
e\left\\{\kappa(\omega)\right\\}\propto\>-sign(\omega)$ as for a NFLShekhter
and Varma (2009).

By the same token, from Eq.s(26,32), the particle current is

$\displaystyle\lim_{t\to\infty}\left\langle\left[\dot{\cal{N}}(t),\dot{\cal{N}}(0)\right]\right\rangle\hskip
128.0374pt$ $\displaystyle=-\frac{t_{0}^{2}}{2\pi^{2}\hbar}N\>\int d\nu\>\Im
m\left\\{D^{R\beta}(\nu)\right\\}\>\frac{\partial}{\partial\nu}\Im
m\left\\{G^{R\beta}(\nu)\right\\}.$

Following the same steps as before, an hypothetical electrical conductivity
would beSong et al. (2017):

$\displaystyle\lim_{\omega\to
0}\left[\frac{Re\\{\sigma\\}}{e^{2}}\right]_{\omega}=N\frac{\beta}{\pi^{3/2}}\>\frac{t_{0}^{2}}{{\cal{J}}}\hskip
71.13188pt$ $\displaystyle\times\int dx\>{\rm sech}^{2}(\pi x)\>\left[\Re
e\left\\{\frac{\Gamma\left(\frac{1}{4}-i\>x\right)}{\Gamma\left(\frac{3}{4}-i\>x\right)}\right\\}\right]^{2}\propto
N\frac{T_{coh}}{T}.$ (36)

which implies 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 Q-modes 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 we have ignored that the conformal symmetry
spontaneous breaking gives rise to massless Goldstone modes which are
responsible for the divergence of the functional integral of Eq.(17).
Moreover, $\varphi_{x}(\tau_{+})$ was given as an external source with no
dynamics. However, the term $\sim\partial_{\tau}$ in Eq.(8) produces itself a
symmetry breaking which becomes relevant in the UV. The Goldstone modes are
only pGm. In the IR, a time reparametrization under the diffeomorphism
$e^{i\vartheta}\to e^{i\>\varphi_{x}(\vartheta)}$ of the conformal Green’s
function produces a local term to the action, which gives dynamics to the
$\varphi_{x}(\tau_{+})$ 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
($\varphi^{\prime}\equiv\partial_{\vartheta}\varphi$, $\Delta=\frac{1}{4}$):

$\displaystyle\tilde{G}_{IR}~(\vartheta_{1},\vartheta_{2})=\tilde{G}_{c}\left(\varphi(\vartheta_{1}),\varphi(\vartheta_{2})\right)\>\varphi^{\prime}(\vartheta_{1})^{\Delta},\varphi^{\prime}(\vartheta_{2})^{\Delta},$

which, when expanded to lowest order in $\vartheta_{12}$, gives

$\displaystyle\approx\tilde{G}_{\beta\to\infty}~\\!\\!(\vartheta_{1},\vartheta_{2})\\!\left[\\!1+\\!\frac{\Delta}{6}(\vartheta_{12})^{2}Sch\\!\left(e^{i\>\varphi(\vartheta_{+})},\\!\vartheta_{+}\right)\right],$
(37)

where $Sch\left(e^{i\>\varphi(\vartheta_{+})},\vartheta_{+}\right)$ is the
SchwarzianKitaev . We get a local action of order $1/\beta{\cal{J}}$:

$\displaystyle\frac{I_{local}}{N}~[\partial_{\tau}\varphi]=-2\pi\>\frac{\alpha_{S}}{\beta{\cal{J}}}\\!\int\frac{d\vartheta}{2\pi}\left[\frac{1}{2}-\frac{(\varphi^{\prime\prime})^{2}-\\!(\varphi^{\prime})^{2}}{2}\right].$
(38)

which adds dynamics to the field $\varphi_{x}(\tau_{+})$ (the constant term in
Eq.(38) gives a contribution $\propto\beta$, which is just a correction to the
energy and will appear in Eq.(63)). This correction is
${\cal{O}}\left((\beta{\cal{J}}^{-1})\right)$ and a correction of the same
order should be included to the actionMaldacena and Stanford (2016); Salvati
and Tagliacozzo (2021), coming from the $\langle\delta g|K_{c}^{-1}-1|\delta
g\rangle$ term appearing in Eq.(17). We remind that
$\partial_{\tau}\varphi_{x}(\tau_{+})$ corresponds to changes in the chemical
potential $\mu$, according to Eq.(23).

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 continuity equation

$\displaystyle\rho\>T\>\frac{D}{Dt}s=-\nabla\cdot J^{\cal{E}},$ (39)

where $J^{\cal{E}}$ is the energy current density of Eq.(21). This will help
us in characterizing the collective bosonic Q-excitations. In the next
Subsection we derive the response function to $J^{\cal{E}}-J^{\cal{E}}$
fluctuations, starting from the equation of motion, which is assumed to be
diffusive:

$\displaystyle J^{\cal{E}}=-\kappa\>\nabla T+g,$ (40)

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 $I[\partial_{\tau}\varphi_{p}]$ in the large ${\cal{J}}$
limitSalvati and Tagliacozzo (2021).

###  III.1 Hydrodynamic response to energy current density fluctuations

According to Eq.(23), 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 $3-d$ qL
induced by the energy current density fluctuations of the $2+1-d$ SYK system,
via chemical potential gradients, is our final goal. In this Subsection we
study the response of the chemical potential to the energy current density,
$J^{\cal{E}}$, fluctuations, starting from temperature gradients and from the
continuity equation, Eq.(39).

Let us define the canonical conjugate variables $x_{a},X_{a}$, appearing in
the continuity equation, Eq.(39):

$\displaystyle\dot{S}=-\dot{x}_{a}\frac{\partial S}{\partial x_{a}},\>\>\>{\rm
with}\>\>X_{a}=-\frac{\partial S}{\partial x_{a}},$ (41)

in terms of which the usual linearized response equation takes the form:

$\displaystyle\dot{x}_{a}=\sum_{b}\left(\gamma_{ab}X_{b}+y_{b}\right)$ (42)

(here $y_{b}$ is the external stochastic source and $a,b$ denote elementary
space cells of volume ${v}_{a},{v}_{b}$).

Space integration of Eq.(39), together with Eq.(40), gives:

$\displaystyle\frac{\dot{S}}{k_{B}}=\int
dv_{b}\>\frac{1}{k_{B}T}\>\nabla\cdot\left(\kappa\nabla T\right)\hskip
56.9055pt$ $\displaystyle=-\int
d{v}_{b}\>{\left\\{\frac{J^{\cal{E}}}{k_{B}}\right\\}_{b}\>\left\\{\frac{\nabla
T}{T^{2}}\right\\}}_{b}.$ (43)

It follows from Eq.s(41,42,43) that

$\displaystyle\dot{x}=\frac{J^{\cal{E}}}{k_{B}},\>\>\>X=\frac{\nabla
T}{T^{2}},\>\>\>\gamma_{ab}=-\frac{\kappa\>T^{2}}{k_{B}}.$ (44)

The hydrodynamical approach developed in Appendix A, provides Eq.(88), which
reads:

$\displaystyle\frac{Ds}{Dt}-\frac{\mu}{T}\nabla\nu=T\>\nu\\!\cdot\\!\nabla\frac{\mu}{T}-\tau_{i}^{k}\frac{\partial
u^{i}}{\partial x^{k}}.$ (45)

Here $D/Dt$ is the covariant total derivative. When the particle current
vanishes, the right hand side contributes with dissipative terms which can be
cumulatively expressed as $-\frac{\partial w}{\partial t}$. The diffusive
contribution to the change in time of the entropy, given by Eq.(90), is:

$\displaystyle
k_{B}\>\dot{x}_{b}\>X_{b}=-\frac{Ds_{b}}{Dt}=\frac{n}{w}\frac{\mu}{T}\left(\tilde{\cal{D}}\>\nabla^{2}w-\frac{\partial
w}{\partial t}\right)_{b}.$ (46)

Here $\tilde{\cal{D}}$ is the diffusion coefficient defined in Eq.(93).
Fourier transforming in space and inverting this equation, we get the response
equation that we are looking after ($\mu\to\mu(\omega\approx 0)$):

$\displaystyle\mu\>n_{q}={k_{B}T}\>\sum_{\omega}\dot{x}_{b,\omega}\>\frac{-i\omega}{\tilde{\cal{D}}\>q^{2}-i\omega}\>X_{b,-\omega}.\hskip
170.71652pt$

The source $X_{b,-\omega}$ in this equation can be interpreted as an
elementary energy current density source:
${j^{\cal{E}}_{-q,-\omega}}=\frac{Tk_{B}^{2}}{h}[\nabla_{x^{\prime}}T]_{q}$.
In fact $\kappa_{0}=\frac{Tk_{B}^{2}}{h}$ has the dimension of a thermal
conductancePendry (1983) $[C/time]$. We obtain:

$\displaystyle\mu\>n_{q}=\frac{h}{(k_{B}T)^{2}}\>\sum_{\omega}j^{\cal{E}}_{-q,-\omega}J^{\cal{E}}_{q,\omega}\>\frac{-i\>\omega}{\tilde{\cal{D}}\>q^{2}-i\omega}.$
(47)

In the following we consider the thermal average of
$J^{\cal{E}}_{-q,-\omega}J^{\cal{E}}_{q,\omega}\>$, as inducing a response on
the space and time dependent chemical potential energy density $\mu\>n_{q}$.
We will use this result when discussing the interaction between the low energy
qL and the bosonic Q-excitations.

Classically, the symmetrizied correlation derived from from Eq.(42) isWeiss
(2021)
$\frac{1}{2}\left\langle\left\\{y_{a}(t),\>y_{b}(t^{\prime})\right\\}\right\rangle\>v_{b}=\left(\gamma_{ab}+\gamma_{ba}\right)\>\delta(t-t^{\prime})$
and, with the correspondence of Eq.(44), it becomes in the quantum case:

$\displaystyle\frac{1}{2}\int d^{2}r\int
dt\>\left\langle\left\\{J^{{\cal{E}}}(r,t),J^{{\cal{E}}}(r,t^{\prime})\right\\}\right\rangle\>e^{i\>\omega(t-t^{\prime})}$
$\displaystyle=-\hbar\omega\>T\kappa\>\coth\left(\frac{\hbar\omega}{2k_{B}T}\right).\hskip
56.9055pt$ (48)

where $\kappa\sim\lim_{t\to\infty}Re\\{\kappa\\}_{t}=\lim_{\omega\to
0}|\omega|\>\Re e\left\\{\kappa(\omega)\right\\}\>$, so that, substituting the
$J^{{\cal{E}}}-J^{{\cal{E}}}$ fluctuations with the imaginary part of the
response via the fluctuation dissipation theorem, we have:

$\displaystyle\omega\>\Re e\\{\kappa(\omega)\\}=\frac{1}{2}\>\Im
m\left\\{D^{\beta}_{J^{{\cal{E}}},J^{{\cal{E}}}}\\!(\omega)\right\\}\\!\frac{1}{\hbar\omega
T}\\!\tanh\left(\frac{\hbar\omega}{2k_{B}T}\right),$ (49)

where

$\displaystyle D^{\beta}_{J^{{\cal{E}}},J^{{\cal{E}}}}(\omega)=-i\int dt\>\int
dt^{\prime}\>\Theta(t-t^{\prime})\>$ $\displaystyle\times\int
d^{2}k\left\langle\left[J^{{\cal{E}}}_{-k}(t),\>J^{{\cal{E}}}_{k}(t^{\prime})\right]\right\rangle\>e^{i\>\omega(t-t^{\prime})}.$
(50)

Both the $J^{{\cal{E}}}-J^{{\cal{E}}}$ fluctuations of Eq.(48) and the
response function of Eq.(50) are integrated over the space volume. The
response function of Eq.(50) will be evaluated in the next Subsection for
damped Q-excitations within a restricted $2-d$ volume in which correlation is
non vanishing.

###  III.2 Lagrangian approach to gapless diffusive excitation modes

Eq.(40) 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 $\tau_{0}$ 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 Lagrangian (in 2-d) are:

$\displaystyle\dot{\theta}=\left(\frac{\kappa C}{\hbar
T}\right)^{1/2}\>\frac{J_{\cal{E}}\tau_{0}}{k_{B}},\>\>$ (51)
$\displaystyle\nabla\theta=\left(\frac{\hbar}{\kappa
C\>T}\right)^{1/2}\>\frac{\kappa}{T}\nabla T$
$\displaystyle{\cal{L}}=\frac{1}{2}\int
d^{2}x\>\left[\frac{k_{B}}{T}\>\left(\frac{J_{\cal{E}}\tau_{0}}{k_{B}}\right)^{2}+\frac{\hbar}{\kappa
C}\>\left(\frac{\kappa}{T}\nabla T\right)^{2}\right]$
$\displaystyle\equiv\frac{1}{2}\int
d^{2}x\>\left[A\>\dot{\theta}^{2}+B\>(\nabla\theta)^{2}\right]$ (52)
$\displaystyle A=\frac{\hbar k_{B}}{\kappa C},\>\>\>B=T.$

Note that the dimension of the two terms in square brackets is
$[{\cal{E}}/\ell^{2}]$.

From the equation of motion:

$\displaystyle\frac{d}{dt}\left(\frac{\partial
L}{\partial\dot{\theta}}\right)-\frac{\partial
L}{\partial\nabla\theta}=A\ddot{\theta}+B\nabla\theta=0,$ (53)

the classical motion equation of Eq.(40) can be recovered if we put
$\tau_{0}\dot{J}_{\cal{E}}\approx{J}_{\cal{E}}$.

Although $\hbar$ is already in the Lagrangian, we proceed to quantize the
theory. Fourier transforming the Lagrangian:

$\displaystyle{\cal{L}}=\frac{1}{2}\sum_{k}\>\left[A\>{\dot{\theta}}_{-k}{\dot{\theta}}_{k}-B\>k^{2}\theta_{-k}\theta_{k}\right],\hskip
28.45274pt$ (54)

the canonical momentum for $\theta_{k}$ is
$\pi_{k}=\frac{\partial{\cal{L}}}{\partial\dot{\theta}_{k}}=A\dot{\theta}_{-k}$,
and $\left[\theta_{k},\pi_{k^{\prime}}\right]=i\>\delta_{kk^{\prime}}$. It
follows that

$\displaystyle\pi_{k}=-i\>T^{1/2}\frac{1}{(2\eta_{k})^{1/2}}|k|\>\left(\hat{a}_{-k}-{\hat{a}}^{\dagger}_{k}\right),$
$\displaystyle\theta_{k}=T^{-1/2}\frac{(2\eta_{k})^{1/2}}{|k|}\>\left(\hat{a}_{k}+{\hat{a}}^{\dagger}_{-k}\right),$
$\displaystyle{\cal{H}}=\sum_{k}\eta_{k}\>{\hat{a}}^{\dagger}_{k}\hat{a}_{k}+cnst,\>\>\>\eta_{k}=\tilde{a}\left[\frac{\kappa}{k_{B}}\frac{CT}{\hbar}\right]^{1/2}\\!\\!\\!\\!|k|\equiv
v\>|k|,$ (55)

what defines the velocity $v$ in terms of a lattice parameter $\tilde{a}$. The
approach is similar to the one for phonons, but $\pi(x)$ plays the role of the
lattice displacement $d(x,t)$, while $\nabla\theta$ plays the role of the
corresponding momentum $\pi_{ph}(x,t)$. The atomic mass $M$ of the phonon
problem corresponds to $1/T$ here. With the definitions given in Eq.s(51,55)

$\displaystyle\frac{1}{2}\left\langle\left\\{J^{{\cal{E}}}_{-k},J^{{\cal{E}}}_{k}\right\\}\right\rangle_{\omega}=\frac{1}{\tau_{0}^{2}}\frac{\kappa
CT}{2\hbar}\left\langle\left\\{\pi_{-k},\pi_{k}\right\\}\right\rangle_{\omega}.$
(56)

The space integral of the response function, $D^{\beta}_{\pi\pi}(\omega)$,
corresponding to Eq.(50), but referring to the symmetrized correlation
$\left\langle\left\\{\pi_{-k},\pi_{k}\right\\}\right\rangle_{\omega}$, 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 $\eta_{k}$ defined in
Eq.(55). To account for thermalization, we introduce the inverse lifetime of
the collective mode appearing in Eq.(51), as a damping of the mode,
$\Gamma=\hbar/\tau_{0}$ in the response function:

$\displaystyle D^{\beta}_{\pi\pi}(\omega)=\frac{\pi T}{v^{4}\hbar}\hskip
85.35826pt$
$\displaystyle\times\>\tilde{a}_{\ell}^{2}\int_{0}^{\infty}\eta^{2}d\eta\>\left(\frac{1}{\omega+\eta+i\>\Gamma}-\frac{1}{\omega-\eta+i\>\Gamma}\right)\>e^{-\eta
0^{+}}$ (57)

so that,

$\displaystyle\Im m\left\\{D^{\beta}_{\pi\pi}(\omega)\right\\}\to-\frac{2\pi
T}{v^{4}}\>\tilde{a}_{\ell}^{2}\>\omega\frac{\Gamma}{\hbar}{\rm\>\>\>\>\>\>for\>\>\Gamma/\omega<<1}.$
(58)

Here we have assumed that the correlations range a space area
$\tilde{a}_{\ell}^{2}>>\tilde{a}$. Using Eq.(56), we get

$\displaystyle\frac{1}{2}\>\tilde{a}_{\ell}^{2}\int
kdk\left\langle\left\\{J^{{\cal{E}}}_{-k},J^{{\cal{E}}}_{k}\right\\}\right\rangle_{\omega}=-\hbar\>\omega\>T\>\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)$
$\displaystyle\>\times\left[\frac{\tilde{a}_{\ell}^{2}}{\tau_{0}^{2}}\frac{1}{2}k_{B}\>\frac{2\pi}{v^{2}}\frac{\Gamma}{\hbar}\right].$
(59)

Comparing Eq.(48) with Eq.(59) we see that the square bracket provides an
expression for $\kappa$:

$\displaystyle\kappa=\frac{k_{B}\Gamma}{\hbar}\left(\frac{\tilde{a}^{2}_{\ell}}{\tilde{a}^{2}}\>\frac{\pi\Gamma}{CT}~\right)^{1/2},$
(60)

where the velocity given by Eq.(55) has been inserted.

The diffusivity $\tilde{D}_{Q}$ defined in Eq.(92) is connected to the thermal
conductivity $\kappa$ by the Einstein relation:

$\displaystyle\tilde{D}_{Q}=\frac{\kappa}{\rho_{0}C}.$ (61)

The "particle density" $\rho_{0}$ 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
$\rho_{0}=\Gamma^{2}/v^{2}\hbar^{2}$. Inserting Eq.(60) in Eq.(61) and using
$\rho_{0}$ with the velocity given in Eq.(55), gives:

$\displaystyle\tilde{D}_{Q}=\tilde{a}^{2}\>\pi^{2}\Gamma\>\frac{k_{B}T}{\hbar}\>\frac{\tilde{a}^{2}_{\ell}}{\tilde{a}^{2}}\>\frac{1}{CT}.$
(62)

As the energy diffusion is mainly due to heat transport in a highly
thermalizable environment, an estimate of
$CT\>\tilde{a}^{2}/\tilde{a}^{2}_{\ell}$, appearing in Eq.s(60,62), can be
derived from the saddle point contribution to the energy of the SYK
modelMaldacena and Stanford (2016):

$\displaystyle\ln{\cal{Z}}=-\beta
E_{0}+S_{0}+\frac{c}{2\beta}+...,\>\>\>\frac{c}{2}=\frac{2\pi^{2}\alpha_{S}N}{{\cal{J}}}.$
(63)

By posing $\frac{\tilde{a}^{2}}{\tilde{a}_{\ell}^{2}}CT\sim\partial_{\beta}\ln
Z={c}/{2\beta^{2}}$, we get

$\displaystyle\tilde{D}_{Q}=\frac{\Gamma}{\hbar}\left(\frac{\beta{\cal{J}}}{\pi\alpha_{S}N}~\right)\>\tilde{a}^{2}.$
(64)

We conclude that $\tilde{a}_{\ell}^{2}$ is the mean squared diffusive length,
such that $\tilde{D}_{Q}\tau_{0}\sim\tilde{a}_{\ell}^{2}$. From Eq.(64), a
temperature $T_{0}$ can be defined, such that
$\tilde{a}_{\ell}^{2}\sim\tilde{a}^{2}\frac{T_{0}}{T}$, with

$\displaystyle
k_{B}T_{0}\sim\Gamma\left(\frac{\beta{\cal{J}}}{\pi\alpha_{S}N}~\right).$ (65)

$T_{0}$ is the threshold temperature for thermalization to be efficient and is
independent of temperature if $\tau_{0}\sim\hbar\Gamma^{-1}\sim\hbar\beta$.
Indeed, a "Planck" relaxation time scale is expected when the full incoherent
SYK regime sets inHartnoll (2015).

Eq(65) confirms that $T_{0}$ is
${\cal{O}}\left(\frac{\beta{\cal{J}}}{N}\right)$, if just the zero order for
$\Gamma$ is retained. Assuming that $\Gamma\propto T$ both $\tilde{D}_{Q}$ of
Eq.(64) and $T_{0}$ of Eq.(65) are temperature independent. If we make the
same substitution for $CT\>\tilde{a}^{2}/\tilde{a}^{2}_{\ell}$ in Eq.(60) that
we did in Eq.(64), we get:

$\displaystyle\kappa=k_{B}\>\frac{\Gamma}{\hbar}\left(\beta\Gamma\right)^{1/2}\>\left(\frac{\beta{\cal{J}}}{\pi\alpha_{S}N}~\right)^{1/2}.$
(66)

This contribution to the thermal conductance is
$\sim{\cal{O}}\left(\sqrt{T}\right)$ and is reported in the sketch of Figure
1.

The definition of $\tilde{a}_{\ell}^{2}\sim\tilde{a}^{2}\>T_{0}/T$, emerged in
a previous study Salvati and Tagliacozzo (2021), by including scaling to
finite $\beta{\cal{J}}/N$. Both $T_{coh}$ and $T_{0}$ are
${\cal{O}}\left(\frac{\beta{\cal{J}}}{N}\right)$ because they are parameter
scales of marginal perturbations in the large $N$, large ${\cal{J}}$ limit.

##  IV  $3-d$ quantum Liquid in interaction with the SYK system in the
temperature range $(T_{coh},T_{0})$

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.(65), 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 $T\lesssim T_{coh}$). 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).

###  IV.1 marginal Fermi liquid (MFL) for $T\gtrsim T_{coh}$

In a perturbative approachChowdhury et al. (2018), let us assume a local
interaction strength ${\cal{J}}$ in the qL. The quasiparticles of the low
energy FL have a residue $Z$, such that $Z\nu_{0}\sim{\cal{J}}^{-1}$. The
single particle energy is $\epsilon_{\vec{k}}=\tilde{v}_{F}k$ in the continuum
limit, in the vicinity of the FS, with a renormalized physical velocity
$v_{F}^{*}=Z\tilde{v}_{F}$. The isotropic self-energy arising from the
interaction, for $\vec{k}$ on the Fermi surface, is:

$\displaystyle\Sigma(k_{F},i\omega)\hskip 170.71652pt$
$\displaystyle={\cal{J}}\sum_{\vec{q}}\int\frac{d\Omega}{2\pi}\>G(\epsilon_{\vec{k}_{F}+\vec{q}}-\epsilon_{\vec{k}_{F}},i\omega+i\>\Omega)\>\Pi(q,i\>\Omega)$
$\displaystyle\approx{\cal{J}}\sum_{\vec{q}}\int\frac{d\Omega}{2\pi}\>\frac{1}{iZ^{-1}(\omega+\Omega)-\tilde{v}_{F}q\cos\theta}\>\Pi(q,i\>\Omega),$
(67)

where $\Pi(q,i\>\Omega)$ is the polarization function

$\displaystyle\Pi(q,i\>\Omega)=\sum_{p}\sum_{\omega_{n}}G(\epsilon_{p},i\omega_{n})\>G(\epsilon_{p+q},i\>\omega_{n}+i\>\Omega_{m}).$

In Eq.(67), $\theta$ is the angle between $\vec{q}$ and $\vec{k}=\vec{k}_{F}$
and, for $|q|<<k_{F}$, we have approximated
$\epsilon_{\vec{k}_{F}-\vec{q}}-\epsilon_{\vec{k}_{F}}\approx\tilde{v}_{F}q\cos\theta$.

In the range of frequencies $\Omega\sim T_{coh}=W^{2}/{\cal{J}}$, 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:

$\displaystyle\Im
m\\{\Sigma^{(1)}(k_{F},\omega)\\}\\}\approx\alpha\>Z^{-2}\>sign[\omega]\>\omega^{2}$
(68)

($\alpha$ is a parameter of order one). The contribution coming from
interaction with the extended SYK system, $\Pi^{2}(i\>\Omega)$, 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),

$\displaystyle\Pi^{2}(i\>\Omega)=-\frac{1}{{\cal{J}}}\\!\\!\left\\{\\!\int_{-{\cal{J}}}^{-T_{coh}}\\!\\!\\!\\!+\\!\int_{T_{coh}}^{{\cal{J}}}\right\\}d\omega\>\frac{[sign(\omega+\Omega)\>sign(\omega)]}{\sqrt{|\omega|}\sqrt{|\omega+\Omega|}}$
$\displaystyle\approx-\frac{4}{{\cal{J}}}\>\ln\left(\frac{{\cal{J}}}{max[\Omega,T_{coh},T]}\right)\to-\frac{8}{{\cal{J}}}\>\ln\left(\frac{{{W}}}{T_{coh}}\right).\hskip
28.45274pt$ (69)

The final result is:

$\displaystyle\Im
m\\{\Sigma^{(2)}(\omega)\\}\approx-\>\frac{\epsilon_{F}}{2T_{coh}}\>|\omega|\>\ln\left(\frac{{\cal{J}}}{T_{coh}}\right)\>sign(\omega).$
(70)

For $T>>T_{coh}$ we should put $\ln{{\cal{J}}}/{T_{coh}}\to\ln({\cal{J}}/T)$
in Eq.(70). $\Sigma^{(1,2)}(k_{F},\omega)$ changes sign at $\omega=0$ when the
quasi-particle becomes a quasi-hole. Putting both terms together, the
quasiparticle relaxation rate is:

$\displaystyle\frac{1}{\tau}\sim-Z\>\left|\Im m\Sigma(k_{F},\omega)\right|$
$\displaystyle=|\omega|\frac{Z\epsilon_{F}}{k_{B}T_{coh}}\ln\left(\frac{{\cal{J}}}{T_{coh}}\right)\\!+\\!\frac{\alpha}{Z}\>\nu_{0}\>|\omega|^{2},$
(71)

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 Salvati and Tagliacozzo (2021). 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 $\kappa$ 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.(48) can be approximated as

$\displaystyle\frac{1}{2}\int d^{2}r\int
dt\>\left\langle\left\\{J^{{\cal{E}}}(r,t),J^{{\cal{E}}}(r,t^{\prime})\right\\}\right\rangle\>e^{i\>\omega(t-t^{\prime})}$
$\displaystyle\approx\overline{n}_{MFL}\>\frac{L}{2\pi}\tilde{v}_{F}^{2}\>\langle\left(\delta{\cal{E}}\right)^{2}\rangle\>\tau_{MFL}$
(72)

The saddle point contribution to energy of the SYK model of Eq.(63) provides
us with the energy fluctuation, given by:

$\displaystyle\langle\left(\delta{\cal{E}}\right)^{2}\rangle=\partial_{\beta}^{2}\ln
Z=\frac{c}{\beta^{3}}.$ (73)

By plugging Eq.(72), with Eq.(73), in Eq.(48) we get, in the limit
$\hbar\omega/k_{B}T<<1$,

$\displaystyle\kappa\approx
k_{B}\>\frac{L}{2\pi}\overline{n}_{MFL}\>\tilde{v}_{F}^{2}\>\frac{1}{(k_{B}T)^{2}}\frac{4\pi^{2}\alpha_{S}N}{\beta^{3}{\cal{J}}}\>\tau_{MFL}.$
(74)

The lifetime of the MFL, from Eq.(71), is $\tau_{MFL}\sim T^{-1}$, so that
$T-$ power-counting gives $\kappa$ independent of $T$ at this perturbative
level. It is interesting that this is a contribution to the thermal
conductivity $\kappa\sim{\cal{O}}\left([\beta{\cal{J}}/N]^{0}\right)$, while
the contribution coming from the IR limit of Eq.(36) is
${\cal{O}}\left(\beta{\cal{J}}/N\right)$ .

###  IV.2 Influence of the Q-excitations on the lifetime of the MFL (
$T\gtrsim T_{coh})$

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.(23), $J^{\cal{E}}-J^{\cal{E}}$
fluctuations couple to the change of the chemical potential, as can be read
off Eq.(47). 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.(47) can be
newly defined as:

$\displaystyle{G}_{+}^{LQ}(k,q,i\>\omega)=\frac{h}{\rho_{0}}\frac{\left\langle
J^{\cal{E}}_{-k,-\omega}J^{\cal{E}}_{k,\omega}\right\rangle}{(k_{B}T)^{2}}\>\frac{-i\>\omega}{\tilde{\cal{D}}\>q^{2}-i\omega-\frac{\omega^{2}}{c^{2}}}.$
(75)

In real time ${G}_{+}^{LQ}$ is a response function $G_{+}(t)=\Theta(t)G^{(}t)$
and has the dimensions of $[{\cal{E}}]$.

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
$\langle J^{\cal{E}}_{-k,-\omega}J^{\cal{E}}_{k,\omega}\rangle$ term. We
overcame the same difficulty in Eq.s(48,50), by considering the convolution
(with $k-$ integration) as in Eq.(59) of Subsection III.B, which corresponds
to averaging in space, but by assuming that the fluctuation correlations are
localized within a volume $\sim\tilde{a}_{\ell}^{2}$. We eventually get

$\displaystyle
G_{+}^{LQ}(q,i\>\omega)=2\pi\frac{h}{\rho_{0}v^{2}\tau_{0}^{2}}\>\frac{\tilde{a}_{\ell}^{2}}{\tilde{a}^{2}}\>\frac{\Gamma}{\hbar}\>\frac{-i\omega}{-i\>\omega+\hbar\tilde{\cal{D}}q^{2}}.$
(76)

If we assume that the $T-$power counting of $\rho_{0}v^{2}\tau_{0}^{2}\sim
T^{0}$, with $\hbar/\tau_{0}\sim\Gamma\sim T$, the prefactor of the energy
$G_{+}^{LQ}$, provides a squared coupling $|g|^{2}\sim T^{0}$ while, in the
case of the e-ph interaction in a metal, is $|g|^{2}\sim T$.

Eq.(76) 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 rateSchofield (1999):

$\displaystyle\frac{1}{\tau_{\kappa}}=\frac{2\pi}{\hbar}\\!\\!\left(\frac{L}{2\pi}\right)^{d}\\!\\!\int_{0}^{\epsilon}\\!\\!\nu_{0}\>\omega\>d\omega\int_{\omega/\hbar
v_{F}}^{2k_{F}}\\!\\!\frac{q^{d-1}\>dq\left[{G}_{+}^{LQ}(q,i\>\omega)\right]^{2}}{(\hbar
v_{F}q)^{2}}$ (77)

($\omega$ has dimension $[{\cal{E}}]$ from now on, here). $\nu_{0}$ is the
density of states at $\epsilon_{F}$ for a $3-d$ volume. The integration over
the direction of the momentum has already been performed and gives the factor
of $(\hbar v_{F}q)^{2}$ 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 $\omega$, $\omega$ appears also in the lower limit
of the momentum integral. The integral over $\omega$ is simply the number of
possible hole excitations that can be created in the qL close to the FS. If
$G_{+}^{LQ}(q,\omega)$ is independent of $q$ and $\omega$, the $q$ integral is
not sensitive to the value of the lower limit and is independent of $\omega$.
The subsequent integration over $\omega$ recovers the usual $3-d$ result,
${\tau_{\epsilon}}^{-1}\sim\epsilon^{2}$, which is expected for a FL.

Inserting $G_{+}^{LQ}$ of Eq.(76) in Eq.(77), we have

$\displaystyle\frac{1}{\tau_{\kappa}}=\frac{2\pi}{\hbar}\>\left(\frac{L}{2\pi}\right)^{2}\int_{0}^{\epsilon}\nu_{F}\omega\>d\omega\left(\frac{\tilde{a}_{\ell}}{\tilde{a}}\right)^{4}\left(\frac{h}{\rho_{0}v^{2}\tau_{0}^{2}}\right)^{2}\left(\omega\frac{\Gamma}{\hbar}\right)^{2}\>\frac{L}{2\pi}\int_{\omega/\hbar
v_{F}}^{2k_{F}}\frac{dq}{(\hbar
v_{F})^{2}}\left|\frac{1}{\hbar\tilde{\cal{D}}\>q^{2}-i\omega}\right|^{2},$

which gives

$\displaystyle\frac{1}{\tau_{\kappa}}=\frac{2\pi}{\hbar}\\!\\!\left(\frac{L}{2\pi\hbar
v_{F}}\right)^{3}\\!\\!\\!\int_{0}^{\epsilon}\nu_{F}\omega^{2}d\omega\>\left(\frac{\tilde{a}_{\ell}}{\tilde{a}}\right)^{4}\\!\\!\\!\\!\left(\frac{h}{\rho_{0}v^{2}\tau_{0}^{2}}\right)^{2}\\!\\!\\!\\!\left(\frac{\Gamma}{\hbar}\right)^{2}\\!\\!\\!\\!.$
(78)

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 ${\cal{O}}\left(\epsilon^{4}\right)$. To proceed to $T-$power
counting, we assume that the volume is factorized into an in-plane
contribution $L^{2}\sim{\tilde{a}_{\ell}^{2}}$ where the diffusion dynamics
takes place, times an out of plane one, represented by the term $\nu_{0}\hbar
v_{F}L/\tilde{a}_{\ell}^{2}$. Assuming again $\rho_{0}v^{2}\tau_{0}^{2}\sim
T^{0}$, we get:

$\displaystyle\frac{1}{\tau_{\kappa}}\propto
T^{2}\>\left(\frac{~T_{0}}{T}\right)^{3}\epsilon^{3}\sim T^{2}.$ (79)

This $T-$dependence cannot be distinguished from the FL one, although it
arises from a very different origin, as we have seen here, that is from the
correction of the MFL inverse quasiparticle lifetime due to the scattering by
the Q-excitations. The electron -phonon scattering rate, given here for
comparison isAshcroft and Mermin (1976): ${\tau_{e-ph}}^{-1}\sim T^{3}$ for
$T<<\theta_{D}$.

If we use Eq.(74) as an approximate expression for the thermal conductance, in
which we insert the result of Eq.(78), we get $\kappa\to 1/T$ as in the FL
case. In fact, in the FL case, $\kappa=C\>v_{F}\ell=C\>v_{F}^{2}\tau_{0}$. As
$\tau_{0}\sim T^{-2}$ and $C\sim T$, then $\kappa\sim T^{-1}$. This
contribution, which is $\sim{\cal{O}}\left(\beta{\cal{J}}/N]^{0}\right)$, as
the one given by quasiparticle scattering in the MFL of Eq.(74), is however
subdominant, with respect to the one $\sim\sqrt{T}$ given by Eq.(66).

On the other hand, resistivity is ruled by a relaxation time in which the
forward scattering is subtracted from the response appearing in Eq.(76). For
very small $\omega$, both integrals are convergent and we have:

$\displaystyle\frac{L}{2\pi}\int_{\omega/\hbar v_{F}}^{2k_{F}}\frac{dq}{(\hbar
v_{F})^{2}}\left|\frac{-i\omega}{\hbar\tilde{\cal{D}}\>q^{2}-i\omega}-1\right|^{2}=\frac{1}{2\pi}\>\frac{2k_{F}L}{(\hbar
v_{F})^{2}},\>$

where we have excluded the bottom limit of integration. Hence:

$\displaystyle\frac{1}{\tau_{\sigma}}=\frac{1}{\hbar}\>\frac{\tilde{a}^{2}_{\ell}L}{(\hbar
v_{F})^{3}}\\!\\!\int_{0}^{\epsilon}\nu_{F}\omega
d\omega\left(\frac{\tilde{a}_{\ell}}{\tilde{a}}\right)^{4}\\!\\!\left(\frac{1}{\rho_{0}v^{2}\tau_{0}^{2}}\right)^{2}\\!\\!2\>\Gamma^{2}\hbar
v_{F}k_{F}.$ (80)

Here the $T-$power counting provides $\frac{1}{\tau_{\sigma}}\propto
T^{2}\>\left(\frac{~T_{0}}{T}\right)^{3}\epsilon^{2}$ $\sim T$ which points to
the unbound dependence $\propto T$ 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.

##  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 modelLantagne-Hurtubise et al. (2021), when describing
the electronic carriers. Indeed Majorana Fermions are neutral and interaction
${\cal{J}}$ 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" $G_{c}(\tau)$ 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 $\tau_{0}\propto\hbar\beta$ and we infer by scalingSalvati and
Tagliacozzo (2021) that they have mean square space extension
$\tilde{a}_{\ell}^{2}\sim T_{0}/T$. 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 $\rho$ and of the
thermal conductivity $\kappa$ derived in our approach are reported in Figure 1
for the temperature window $T_{coh}\lesssim T\lesssim T_{0}$, in which there
is a crossover to the incoherent regime. For $T>T_{0}$ the SYK model saturates
a bound on chaosMaldacena et al. (2016) 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. The generating functional of the soft mode correlations, derived
in the gaussian approximationSalvati and Tagliacozzo (2021), 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 $\kappa$ and the resistivity $\rho$
of the model in the conformal symmetry limitSong et al. (2017).

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 $2-d$ and to produce energy current density
fluctuations which become dominant for $T\sim T_{0}$. 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 and their
quantization is presented in Section III.B. The $T$ dependence of their
contribution to transport for $T\sim T_{0}$ follows from the choice of their
lifetime $\tau_{0}\sim\hbar\beta$, which implies full thermalization of the
SYK system.

Section IV is dedicated to the $3-d$ qL and to its interaction both with the
single particle fermionic excitationsChowdhury et al. (2018) 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 of the resistivity and thermal conductivity
for $T\sim T_{0}$. The estimates of the temperature dependence of $\kappa$ and
$\rho$ are visualized in the sketch of Figure 1. The model confirms the
temperature dependence of the resistivity that is found in the measurements
also beyond the IR limit. The pivotal quantity of our hydrodynamic approach to
the collective Q-excitations is their mean square distance for diffusion
$\tilde{a}_{\ell}^{2}$. In the scaling to strong interaction limit, the
diffusion coefficient $\tilde{D}_{Q}$ is temperature independent while
$\tilde{a}_{\ell}^{2}\sim T^{-1}$. This fact locates our result
straightforwardly out of the Mott-Ioffe-Regel resistivity limitHussey II et
al. (2004). On the other hand, the thermal conductivity $\kappa$ acquire sa
$\sqrt{T}$ dependence as a contribution coming directly from Q-excitations
beyond the IR limit, for $T\sim T_{0}$, of
${\cal{O}}\left([\beta{\cal{J}}/N]^{1/2}\right)$. We have also singled out
three other contributions to $\kappa$ which we list here. Two perturbative
contributions coming from quasiparticles of the MFL scattered off by energy
fluctuations: $\kappa_{1}\sim T^{0}$ of Eq.(74) and $\kappa_{2}\sim T^{-1}$ of
Eq.(79). They are both ${\cal{O}}\left([\beta{\cal{J}}/N]^{0}\right)$. In
addition there is one non perturbative contribution $\kappa_{3}\sim T^{0}$ in
the IR limit (Eq.(36)), of ${\cal{O}}\left(\beta{\cal{J}}/N\right)$.

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.

## Acknowledgements

The author acknowledges useful discussions with F.Salvati, Elisa Ercolessi and
Francesco Tafuri and financial support of Università "Federico II" di Napoli,
project "PlasmJac" E62F17000290005 .

##  Appendix A  Relativistic approach to transport

In a particle system at equilibrium, the free energy per particle is
$\frac{F}{N}=\frac{U}{N}-\frac{TS}{N}$, where $U/N=\epsilon$ is the internal
energy per particle and $s=S/N$ is the entropy per particle. Adding and
subtracting $p/n$ where $p$ is the pressure and $n$ is the particle density,
equilibrium implies:

$\displaystyle\>\>\>\>dF=0\>\>\to\>\>d\left(\frac{w}{n}\right)=T\>d\left(\frac{s}{n}\right)+\frac{dp}{n}.$
(81)

Here $w=\epsilon+p$ is the enthalpy and small letters are thermodynamic
quantities per unit volume.

Besides, in terms of the Gibbs potential, $G=\mu N=F+pV$ we write

$\displaystyle\frac{w}{n}-\frac{T\>s}{n}=\mu,$ (82)

where $\mu$ is the chemical potential.

Here we consider heat flow, even in the absence of matter convection, starting
from $T_{i}^{k}$, the energy-momentum tensor:

$\displaystyle T_{i}^{k}=w\>u_{i}u^{k}+p\>g_{i}^{k}+\tau_{i}^{k},$ (83)

where $g_{i}^{k}$ is the metric and $\tau_{i}^{k}$ is the stress tensor. In
the proper frame, the velocity $u^{\alpha}=0$ and $u^{0}=1$ (italic indices
$i=0,1,2,3$, while greek indices $\alpha=1,2,3$). Separating the center of
mass flux from a small fluctuating contribution, we have, as in Eq.(21):

$\displaystyle\vec{J}=n\vec{u}+\vec{\nu}~\to T_{\alpha}^{0},$
$\displaystyle\vec{J}^{{\cal{E}}}=\frac{(\epsilon+p)}{n}\>\left(\vec{J}-\vec{\nu}\right)=T_{0}^{\alpha}\equiv
w\>u^{\alpha},$

where $\vec{J}$ is the particle current density while $\vec{J}^{{\cal{E}}}$ is
the energy current density. In the proper frame, the vector particle flux
density $nu^{k}+\nu^{k}$ implies $\nu^{k}u_{k}=0$, because its 0 component
must be equal to the particle number density $-n$ :
$n\>u^{k}u_{k}+\nu^{k}u_{k}=-n$, by definition. The continuity equation reads

$\displaystyle\frac{\partial}{\partial x^{k}}\left[n\>u^{k}+\nu^{k}\right]=0.$
(84)

$T_{i}^{k}$, includes dissipative processes (viscosity and thermal diffusion).
However, in the absence of external forces, conservation requires
$\frac{\partial}{\partial x^{k}}\>T_{i}^{k}=0$, or Landau and Lifshitz (1987)

$\displaystyle 0=u^{i}\frac{\partial}{\partial
x^{k}}\>T_{i}^{k}=u^{i}\frac{\partial p}{\partial
x^{k}}\>g_{i}^{k}+u^{i}u_{i}\>n\>u^{k}\frac{\partial w/n}{\partial x^{k}}$
$\displaystyle+u^{i}u_{i}\>\frac{w}{n}\>\frac{\partial nu^{k}}{\partial
x^{k}}+u^{i}w\frac{\partial u_{i}}{\partial
x^{k}}u_{k}+u^{i}\frac{\partial}{\partial x^{k}}\tau_{i}^{k}.$ (85)

Here $u^{i}u_{i}=-1$, so that $u^{i}\frac{\partial u_{i}}{\partial x^{k}}=0$
and the fourth term vanishes. In the third term we substitute $\frac{\partial
n\>u^{k}}{\partial x^{k}}\to-\frac{\partial\nu^{k}}{\partial x^{k}}$,
according to Eq.(84). Next, from Eq.(81), we have:

$\displaystyle u^{i}\left[\frac{\partial p}{\partial x^{i}}-n\>\frac{\partial
w/n}{\partial x^{i}}\right]=-T\>n\>u^{i}\frac{\partial s/n}{\partial x^{i}}$
(86)

and Eq.(85) becomes:

$\displaystyle 0=-Tn\>u^{i}\frac{\partial}{\partial
x^{i}}\left(\frac{s}{n}\right)+\frac{w}{n}\>\frac{\partial\nu^{k}}{\partial
x^{k}}+u^{i}\frac{\partial}{\partial x^{k}}\tau_{i}^{k},$

or, using Eq.(82),

$\displaystyle 0=-T\frac{\partial}{\partial
x^{i}}\left(s\>u^{i}\right)+\mu\>\frac{\partial\nu^{k}}{\partial
x^{k}}+u^{i}\frac{\partial}{\partial x^{k}}\tau_{i}^{k}.$ (87)

The dissipative components of the stress-energy tensor are orthogonal to
$\vec{u}$: $u^{i}\tau_{i}^{k}=0$ and, from Eq.(87), we get:

$\displaystyle T\left\\{\frac{\partial}{\partial
x^{i}}\\!\left(s\>u^{i}\\!-\\!\frac{\mu}{T}\nu^{i}\right)\\!\right\\}=-T\nu^{i}\\!\frac{\partial}{\partial
x^{i}}\left(\frac{\mu}{T}\right)\\!-\\!\tau_{i}^{k}\frac{\partial
u^{i}}{\partial x^{k}}.$ (88)

Eq.(88) is the crucial equation in the absence of external forces, originating
from $u^{i}\frac{\partial}{\partial x^{k}}\>T_{i}^{k}=0$. On the l.h.s. there
is the variation of the entropy flux density which has to be positive. This
qualifies, to lowest order, the expressions for the dissipative components of
the stress-energy tensor $\tau_{i}^{k}$ and the current density $\nu^{k}$. The
component of the current are:

$\displaystyle\nu_{i}=-\kappa\left(\frac{nT}{w}\right)^{2}\>\left[\frac{\partial}{\partial
x^{i}}\>\left(\frac{\mu}{T}\right)+u_{i}u^{k}\frac{\partial}{\partial
x^{k}}\>\left(\frac{\mu}{T}\right)\right].$ (89)

$\kappa$ is the heat current response to $-\nabla T$ in the absence of a
driving particle current. The choice of the prefactor is in agreement with the
Wiedemann-Franz law $\frac{\kappa}{\sigma
T}=\frac{3}{2}\left(\frac{k_{B}}{e}\right)^{2}$. Thermal conduction with no
particle flux implies vanishing of the second term. We have:

$\displaystyle
T_{\alpha}^{0}=wu^{0}u_{\alpha}=-\frac{w}{n}\>\nu_{\alpha}=\frac{\kappa
nT^{2}}{w}\frac{\partial}{\partial x^{\alpha}}\left(\frac{\mu}{T}\right)$
$\displaystyle=-\kappa\left[\frac{\partial T}{\partial
x^{\alpha}}-\frac{T}{w}\frac{\partial}{\partial x^{\alpha}}p\right].$ (90)

The last equality arises from the following manipulations:

$\displaystyle
d\left(\frac{\mu}{T}\right)=-\frac{1}{T^{2}}\frac{w}{n}\>dT-d\left(\frac{s}{n}\right)+\frac{1}{T}\>d\left[\frac{T\>s}{n}+\mu\right]$
$\displaystyle=-\frac{1}{T^{2}}\frac{w}{n}\>dT+\frac{1}{n\>T}\>dp,$

where Eq.(81) has been used.

Eq.(90 is the energy flux, which also includes a $\nabla P$ term that is
absent in the non relativistic resultLandau and Lifshitz (1987).

From linear response, the energy density current $J^{\cal{E}}$ can be written
as:

$\displaystyle J^{\cal{E}}=w\>J+\kappa\>\nabla
T+\frac{w}{n}\frac{\sigma_{\ell\ell}}{e^{2}}\>\left[-\nabla P\right]$
$\displaystyle+\frac{w}{n}\>\hat{\ell}\frac{\sigma_{\ell
i}}{e^{2}}\epsilon_{ij3}J_{j}\frac{1}{c^{2}}\oint d\vec{\ell}\cdot\vec{J},$
(91)

where $\sigma_{ij}$ is the conductivity tensor with
$\hat{i},\hat{j},\hat{\ell}$ unitary vector in real space. We have added the
last term, which is the Magnus force. The Magnus force is present in case
there is a current circulation $\oint d\vec{\ell}\cdot\vec{J}\neq 0$ in the
flow. In the case of an electromagnetic current it would take the form
$w\>eJ\times\frac{B}{n\>e\>c}\frac{\sigma_{H}}{e^{2}}$, when induced by a
magnetic field $B$, with $\sigma_{H}$ being the Hall conductanceHartnoll et
al. (2007). The dimension of $[J^{\cal{E}}]$ is
$\left[\frac{{\cal{E}}}{\ell^{d-1}t}~\right]$.

We can define the compressibility $\chi$ and the pair of diffusion constants
for the Q-excitations $\tilde{D}_{Q}$ and for the particles $\tilde{D}_{e}$,
respectively, according to:

$\displaystyle\chi=-\frac{1}{\cal{V}}\frac{\partial{\cal{V}}}{\partial
P},\>\>\>\>\tilde{D}_{Q}=\frac{\kappa}{\rho_{0}C},\>\>\>\>\tilde{D}_{e}=\frac{\sigma}{\chi},$
(92)

(space isotropy is assumed here for simplicity). The diffusivity
$\tilde{D}_{Q}$ is related to the thermal capacitance $C$ and $\rho_{0}$ is a
particle density associated to the Q-excitations which does not concide with
the particle density $n_{e}$ of the carriers and is defined after Eq.(61). A
generalized diffusion constant $\tilde{D}$ can be defined, according to:

$\displaystyle\frac{1}{n}\>J^{\cal{E}}=\tilde{D}_{Q}\>C\nabla
T+\frac{w}{n^{2}}\frac{\sigma}{\chi
e^{2}}\>\left[\frac{1}{\cal{V}}\frac{\partial{\cal{V}}}{\partial
T}\right]\nabla T\equiv\tilde{\cal{D}}\>C\>\nabla T.$ (93)

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