Electric Forces in the Charged SYK Model

Leonard Susskind

Stanford Institute for Theoretical Physics and Department of Physics,

Stanford University, Stanford, CA 94305-4060, USA

and

Sandbox@Alphabet,

Mountain View, CA

The connection between gravitational force and operator growth, reported in
earlier papers, is generalized to include electromagnetic forces. It is shown
how in the $U(1)$-invariant SYK system electric forces emerge through the same
mechanisms—the momentum-size correspondence, and operator growth— that give
rise to gravitational force. The unit of electric charge implied by the
$U(1)$-SYK theory is consistent with a Kaluza-Klein radius inverse to the
characteristic energy scale of the SYK theory.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 The Bulk
    2. 2.2 The Boundary
    3. 2.3 Null Trajectory
  3. 3 Classical Particle Motion
    1. 3.1 Lagrangian, Momentum, and Hamiltonian
    2. 3.2 Force
  4. 4 Relation to SYK
    1. 4.1 Qi-Streicher Formula
    2. 4.2 Specific Heat at T=0: Relating $\mu$ and ${\cal{J}}$
    3. 4.3 Mass of the Boundary
    4. 4.4 Summary of Preliminaries
  5. 5 Operator Growth in Majorana-SYK
    1. 5.1 Exponential Growth in Circuit Time
    2. 5.2 Why Circuit Time Slows Down
    3. 5.3 Momentum and Size
    4. 5.4 Boundary Time and Circuit Time
    5. 5.5 Momentum and Force
    6. 5.6 Ladder Graphs
  6. 6 Charged Particles in U(1)-invariant SYK
    1. 6.1 The Chemical Potential
    2. 6.2 The Rate Equations
    3. 6.3 Relation to OTOCS and Regge Behavior
    4. 6.4 The Evolution of Size for Charged Operators
    5. 6.5 Note on Charges
  7. 7 Motion of Charged Particles
    1. 7.1 A Puzzle
    2. 7.2 Comparing SYK and GR
    3. 7.3 SYK and KK
    4. 7.4 Boundary Particle Formulation
  8. 8 Extrapolation to Strong Coupling
  9. 9 Conclusion

##  1 Introduction

The size of an operator [1] is a measure of the average number of elementary
operators that are present in its expansion. For times shorter than the
scrambling time111We will assume that the Hamiltonian is of the fast-
scrambling type such as all-to-all k-local Hamiltonians or sparse Hamiltonians
of the type discussed in [2] it is also a measure of the number of elementary
operations that are required to prepare the operator, i.e., operator
complexity. During the early phases of evolution, complexity and size are
indistinguishable. This paper is about that early phase; therefore I will use
the symbol ${\cal{C}}$ for both.

While the concepts of size and complexity have not been precisely formulated
for a continuum quantum field theory, they are well-defined for systems
composed of a finite number of qubits, or as in the case of the SYK model, a
finite number of fermionic operators.

The momentum-size correspondence [3][4][5] is a holographic quantum-mechanical
relation between the size of an operator and the radial momentum $P$ of the
system created by acting with that operator. Roughly speaking, it says that
size and momentum are proportional to one another, with a dimensional
proportionality factor $\tilde{\beta}$ which may depend on the radial location
of the particle. It is also possible to think of $\tilde{\beta}$ as a function
of the time by following it along a light-like trajectory. We will return to
the meaning of $\tilde{\beta}$.

In the context of the SYK model an alternative formulation of the momentum-
size correspondence was given in [6][5],

$P=\frac{d{\cal{C}}}{du}$ (1.1)

where $u$ is the ordinary time conjugate to the SYK Hamiltonian.

Operator size is not conserved and in many contexts one sees a tendency for it
to increase with time [1], a phenomenon called operator growth222Because
quantum mechanical evolution is reversible operators can also shrink, but we
will include this possibility under the general heading of operator growth..
Since the force on a system is the time-rate-of-change of its momentum, the
momentum-size correspondence implies a relationship between force and operator
growth, and therefore between dynamics and quantum information. This has been
illustrated for gravitational force, but it should also apply to other forces
such as electromagnetic.

Recent studies of these connections [3][4][5] have concentrated on the
gravitational force of attraction on a particle falling into the long AdS(2)
throat of a near-extremal SYK black hole. Although the SYK black hole is in
many respects very similar to a near-extremal Reissner-Nordstrom (NERN) black
hole, the usual Majorana-SYK system does not carry a conserved $U(1)$ charge,
and therefore does not provide an opportunity to explore the mechanism leading
to electric forces. However the model can be generalized to have a global
$U(1)$ symmetry [7] by replacing the real fermionic degrees of freedom by
complex fermions, $\psi$ and $\psi^{{\dagger}}$ [7][8]. The model will be
$U(1)$ invariant if each term in the Hamiltonian has an equal number of
$\psi$s and $\psi^{{\dagger}}$s.

Standard lore says that if a holographic theory has a global symmetry then the
bulk system that it describes will have a gauge symmetry. In the case of the
$U(1)$-SYK system the bulk dual is the long throat of a near extremal black
hole. As such it is a $(1+1)$-dimensional system with no propagating gauge
degrees of freedom, but there will be one-dimensional Coulomb forces between
charges. In particular a charged black hole should exert forces on charged
particles in the throat—attractive for opposite-sign and repulsive for like-
sign charges. If we combine this observation with the momentum-size
correspondence it predicts a difference in the growth rates for positively and
negatively charged fermionic operators. The purpose of this paper is to see
the “nuts and bolts” of how this occurs in the quantum mechanics of the
$U(1)$-SYK system.

In [4][5] two calculations were compared. The first was a general relativity
calculation of the proper spatial momentum $P(t)$ of a neutral particle
falling into a NERN background. The second was a microscopic calculation [9]
of the size of the Heisenberg operator $\psi(t)$ in standard SYK which made no
use of any assumption of a gravitational dual. Using the momentum-size
correspondence the calculations were shown to agree in detail.

In this paper we generalize this strategy to the $U(1)$-SYK model by
calculating the motion of a charge $\pm 1$ particle falling through the long
throat of a near extremal $U(1)$-charged black hole. The acceleration (rate of
change of momentum) of the particle depends on whether the particle and black
hole have the same or opposite sign charges. We then study the operator growth
of the charged operators $\psi(t)$ and $\psi^{{\dagger}}(t)$. For a given
charge on the black hole we find that the operator growths of $\psi(t)$ and
$\psi^{{\dagger}}(t)$ differ in just the way predicted by the momentum-size
correspondence.

##  2 Preliminaries

###  2.1 The Bulk

In previous papers [4][5] the Majorana-SYK model was compared with the
dynamics of neutral particles in a background of a dimensionally reduced
Reissner-Nordstrom black hole. In this paper we will replace the Reissner-
Nordstrom background with the closely related but slightly simpler solution of
the $(1+1)$-dimensional dilaton-gravity theory discussed in [10]. The model is
a variant of the Jackiw-Teitelboim (JT) theory described by the action,

$I=-\frac{1}{16\pi G}\left[\int
d^{2}x\sqrt{g}\left(\Phi_{0}+\phi\right)R+\frac{1}{\mu^{2}}\sqrt{g}\phi+\rm
BT+MT\it\right]$ (2.1)

where $\Phi_{0}$ is a dimensionless constant, $\phi$ is a dynamical field, BT
stands for a boundary term, and MT stands for matter terms. The term
proportional to $\Phi_{0}$ is topological and determines the zero-temperature
extremal entropy. We will define the dilaton field $\Phi$ by,

$\Phi(x)=\Phi_{0}+\phi(x).$ (2.2)

The parameter $\mu$ has units of length and determines the radius of curvature
of the $AdS(2)$ solution.

The variational equation for $\phi$ requires the curvature to be constant,

$R=-1/\mu^{2}.$ (2.3)

and the equations generated by variation with respect to the metric are,

$\nabla_{\mu}\nabla_{\nu}\phi-g_{\mu\nu}\nabla^{2}\phi+g_{\mu\nu}\phi=0$ (2.4)

which, together with boundary conditions, determine $\phi.$

The solution of the equations of motion for the metric is $AdS(2)$ which can
be written in the form,

$ds^{2}=-\frac{r^{2}-\mu^{2}}{\mu^{2}}dt^{2}+\frac{\mu^{2}}{r^{2}-\mu^{2}}dr^{2}.$
(2.5)

The point $r=\mu$ represents the horizon but unlike in higher dimensions the
coordinate $r$ does not signify the radius of a sphere. Nor is its value at
the horizon related to entropy. In JT-gravity that role is played by the
dilaton field $\Phi(r)$ whose value at the horizon determines the entropy
according to,

$S=\frac{\Phi(\mu)}{4G}.$ (2.6)

The JT equations of motion of the dilaton field are solved by the sum of the
constant term $\Phi_{0}$ and a varying part $\phi(r)$ which varies according
to,

$\phi=c(\beta)\ r.$ (2.7)

The factor of proportionality $c(\beta)$ in is temperature dependent and will
be fixed in section 4.

The proper distance from the horizon will be called $\sigma,$

$\sigma=\int_{\mu}^{r}dr^{\prime}\frac{\mu}{\sqrt{r^{2}-\mu^{2}}},$ (2.8)

from which we find,

$r=\mu\cosh{(\sigma/\mu)}.$ (2.9)

The metric takes the form,

$ds^{2}=-\sinh^{2}(\sigma/\mu)dt^{2}+d\sigma^{2}.$ (2.10)

Defining $y=\sigma/\mu$ and $t/\mu=i\theta$ the Euclidean continuation has the
familiar form of the hyperbolic disc,

$ds^{2}=\mu^{2}\left[(\sinh^{2}{y})\ \ d\theta^{2}+dy^{2}\right].$ (2.11)

###  2.2 The Boundary

In the model of [10] the geometry ends at a boundary located at the value of
the radial variable $y$ at which the circumference of the circle
$(0<\theta\leq 2\pi)$ is the inverse temperature $\beta,$

$\sinh{y_{b}}=\frac{\beta}{2\pi\mu}.$

The subscript $b$ stands for boundary. (See figure 1)

Figure 1: Euclidean AdS(2) can be represented by the Poincare disc. The
geometry at finite temperature is cut off by a boundary shown in red. The bulk
geometry is the shaded portion of the disc.

In terms of $\sigma$ the boundary is at

$\sinh{(\sigma_{b}/\mu)}=\frac{\beta}{2\pi\mu}.$ (2.12)

For large $\beta$ (low temperature),

$\sigma_{b}\approx\mu\log{\left(\frac{\beta}{\pi\mu}\right)}.$ (2.13)

The time coordinate $t$ is not the proper time of an observer at the boundary.
Following [10] we denote the boundary proper time by $u.$ The relation between
$t$ and $u$ is,

$u=\left(\frac{\beta}{2\pi\mu}t\right)$ (2.14)

When considering the relation between JT gravity and the holographic SYK
system, the usual convention is that the time conjugate to the SYK Hamiltonian
is $u.$

It is convenient when studying the motion of a particle that falls in from the
boundary to introduce a coordinate which measures proper distance from the
boundary. Calling it $\rho,$

$\rho=(\sigma_{b}-\sigma).$ (2.15)

The $AdS(2)$ spatial geometry and the coordinates $\sigma\ \rho$ are shown in
figure 2.

Figure 2: The spatial geometry of a near-extremal black hole consists of a
long throat whose length diverges in the extremal limit. The throat for non-
extremal black holes is bounded from both ends. At the near-horizon end the
AdS geometry gives way to a Rindler region and at the far end the throat is
terminated at a boundary. The coordinates $\sigma$ and $\rho$ represent proper
distance from the horizon and the boundary respectively.

###  2.3 Null Trajectory

A particle dropped from the boundary will very quickly become relativistic and
move on an almost light-like trajectory. A light-like trajectory originating
at the boundary ($\rho=0$) at time $u=0$ will satisfy,

$u=\beta\log{\left(\frac{\beta+2\pi\mu e^{\rho/\mu}}{\beta-2\pi\mu
e^{\rho/\mu}}\right)}-\beta\log{\left(\frac{\beta+2\pi\mu}{\beta-2\pi\mu}\right)}$
(2.16)

It is easy to show that it will reach the Rindler region in a time
$\beta/2\pi.$ Assuming $\beta>>\mu$ then for almost all of that time the
trajectory will be in the throat region.

Equation 2.16 can be used to convert a function of spatial position $f(\rho)$
to a function of time $f(u)$ along an infalling trajectory. In particular the
function $\tilde{\beta}(\rho)$ introduced in [4][5] may be thought of as a
function of time along the trajectory of an infalling relativistic particle.

##  3 Classical Particle Motion

###  3.1 Lagrangian, Momentum, and Hamiltonian

We will begin by deriving the equation of motion of a particle moving in the
metric,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}$ (3.1) $\displaystyle=$
$\displaystyle-f(r)dt^{2}+d\sigma^{2}$ (3.3)

and an abelian gauge field of the form,

$\displaystyle A_{\sigma}$ $\displaystyle=$ $\displaystyle 0$ (3.4)
$\displaystyle A_{t}(\sigma)$ $\displaystyle=$ $\displaystyle-(r-r_{b})E.$
(3.6)

Here $E$ is constant and represents a uniform electric field and $r_{b}$ is
the value of $r$ at the boundary.

The standard Lagrangian is,

${\cal{L}}=-m\sqrt{f(r)-\dot{\rho}^{2}}+eA_{t}(\rho).$ (3.7)

where dot means derivative with respect to $t.$

The momentum conjugate to $\rho$ is given by,

$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{\partial{\cal{L}}}{\partial\dot{\rho}}$ (3.8)
$\displaystyle=$ $\displaystyle\frac{\dot{\rho}}{\sqrt{f(r)-\dot{\rho}^{2}}},$
(3.10)

and the Hamiltonian by,

$H_{t}=P\dot{\rho}-{\cal{L}}=\frac{mf}{\sqrt{f(r)-\dot{\rho}^{2}}}-eA_{t}(\rho).$
(3.11)

Note that the Hamiltonian $H_{t}$ is conjugate to $t$ and not to the boundary
time $u.$ The Hamiltonian conjugate to boundary time will simply be denoted by
$H$ without any subscript. The relation between $H$ and $H_{t}$ is,

$H=H_{t}\ \frac{dt}{du}=H_{t}\left(\frac{2\pi\mu}{\beta}\right).$ (3.12)

###  3.2 Force

The radial force $F_{t}$ on the particle is defined to be $dP/dt$ and is given
by,

$\displaystyle F_{t}$ $\displaystyle=$
$\displaystyle\frac{\partial{\cal{L}}}{\partial\rho}$ (3.13) $\displaystyle=$
$\displaystyle-\frac{m}{2}\frac{\partial_{\rho}f}{\sqrt{f-\dot{\rho}^{2}}}+e\frac{\partial
A_{t}}{\partial\rho}$ (3.15) $\displaystyle=$
$\displaystyle-\frac{m}{2}\frac{\partial_{r}f}{\sqrt{f-\dot{\rho}^{2}}}\frac{dr}{d\rho}+e\frac{\partial
A_{t}}{\partial r}\frac{dr}{d\rho}$ (3.17) $\displaystyle=$
$\displaystyle\frac{m}{2}\frac{\partial_{r}f}{\sqrt{f-\dot{\rho}^{2}}}\sqrt{f}+e\frac{\partial
A_{t}}{\partial r}\sqrt{f}$ (3.19)

From 3.11 we find,

$F_{t}=\frac{\partial_{r}f}{2\sqrt{f}}(H_{t}+eA_{t})-e\frac{\partial
A_{t}}{\partial r}\sqrt{f}$ (3.20)

The force $F_{t}$ represents the rate of change of momentum $P$ with respect
to $t,$

$F_{t}=\frac{dP}{dt}.$ (3.21)

The rate of change of $P$ with respect to boundary time will be denoted
without a subscript,

$F=\frac{dP}{du}=F_{t}\left(\frac{2\pi\mu}{\beta}\right).$ (3.22)

Thus one finds,

$\displaystyle F$ $\displaystyle=$ $\displaystyle F_{G}+F_{E}$ (3.23)
$\displaystyle F_{G}$ $\displaystyle=$
$\displaystyle\frac{\partial_{r}f}{2\sqrt{f}}(H+eA_{u})$ (3.25) $\displaystyle
F_{E}$ $\displaystyle=$ $\displaystyle-e\frac{\partial A_{u}}{\partial
r}\sqrt{f},$ (3.27)

where

$A_{u}=A_{t}\frac{dt}{du}=A_{t}\left(\frac{2\pi\mu}{\beta}\right).$

The subscripts $G$ and $E$ refer to gravitational and electric.

Using 2.13, 2.14, 2.15, and 3.6, in the throat region equation 3.27 may be
written as,

$\displaystyle F$ $\displaystyle=$ $\displaystyle F_{G}+F_{E},$ (3.28)
$\displaystyle F_{G}$ $\displaystyle=$
$\displaystyle\frac{H}{\mu}+eE(e^{-\rho/\mu}-1)$ (3.30) $\displaystyle F_{E}$
$\displaystyle=$ $\displaystyle eEe^{-\rho/\mu}$ (3.32)

The factors of $e^{-\rho/\mu}$ in 3.32 go to zero as we move away from the
boundary at $\rho=0.$ In particular this means that $F_{G}$ quickly tends to a
constant value

$F_{G}\to\frac{H}{\mu}-eE$

while $F_{E}$ rapidly decreases as the particle moves away from the boundary.

For an electrically neutral particle the force has the simpler form,

$F=\frac{\partial_{r}f}{2\sqrt{f}}H=\frac{H}{\mu}.$ (3.33)

##  4 Relation to SYK

###  4.1 Qi-Streicher Formula

In the Majorana-SYK theory acting with a fermion operator $\psi$ at time $u=0$
creates a particle at rest at the boundary, which subsequently falls into the
throat. The Heisenberg operator $\psi(u)$ initially has size $1$. The size
grows with time, but the energy is conserved. This is the process described
below by the Qi-Streicher formula 4.2. The energy of such an initial size-$1$
particle is what we have been calling $H$ in equations like 3.33.

As we will discuss in section 5 the momentum-size correspondence can be
expressed in the form [6][5],

$P=\frac{d{\cal{C}}}{du}$ (4.1)

where ${\cal{C}}(t)$ is the size of the operator $\psi(t).$

Qi and Streicher [9] have calculated the size of $\psi(u)$ in Majorana-SYK to
be,

${\cal{C}}(u)=1+2\left(\dfrac{\beta{\cal{J}}}{\pi}\sinh{(\pi
u/\beta)}\right)^{2}.$ (4.2)

From 4.1 and 4.2 we find that in the throat,

$P=4{\cal{J}}^{2}u,$ (4.3)

and the force which is defined as $dP/du$ is given by the constant value,

$F=4{\cal{J}}^{2}.$ (4.4)

From 3.33 and 4.4 we obtain the following expression for $H$,

$H=4\mu{\cal{J}}^{2}.$ (4.5)

To determine $\mu$ in terms of SYK parameters we turn to black hole
thermodynamics and compare the specific heat of a near-extremal black hole
with that of the SYK model.

###  4.2 Specific Heat at T=0: Relating $\mu$ and ${\cal{J}}$

We would like to express $\mu$ in terms of the SYK parameters $N,{\cal{J}},q.$
To do so we will equate the zero-temperature specific heat of the JT gravity
solution with the corresponding SYK result. We begin with the dilaton which
plays the role of area in JT. The dilaton has a constant background part
$\Phi_{0}$ and a varying part $\phi.$

The dilaton profile $\phi(\sigma)$ is determined by solving the equation of
motion and setting the boundary value to a fixed constant of order $\Phi_{0}.$
The precise value is not important but for convenience we will choose,

$\phi_{b}=\frac{\Phi_{0}}{\log{2}}.$ (4.6)

One finds

$\phi(\sigma)=\frac{2\Phi_{0}}{\log{2}}\frac{\pi\mu}{\beta}\cosh{\sigma/\mu}.$
(4.7)

The extremal entropy is given by

$S_{0}=\frac{\Phi_{0}}{4G}.$

For large $q$ it has the value $\frac{1}{2}N\log{2}$. Thus,

$\frac{\Phi_{0}}{2G}=N\log{2}.$ (4.8)

The near-extremal entropy is given by $1/4G$ times the dilaton at the horizon
which from 4.7 is,

$S=\frac{\Phi}{4G}=\frac{\Phi_{0}}{4G}+T\frac{\Phi_{0}\pi\mu}{2\log{2}\ G}$
(4.9)

where $T=1/\beta.$

Using 4.8,

$\frac{dS}{dT}=\pi\mu N.$ (4.10)

To determine $\mu$ we use the SYK value for $dS/dT.$ In the large $q$ limit
the SYK value has been calculated analytically [11],

$(dS/dT)_{SYK}=\frac{\pi^{2}N}{q^{2}{\cal{J}}}.$ (4.11)

Equating 4.10 and 4.11,

$\mu=\frac{\pi}{q^{2}{\cal{J}}}$ (4.12)

for large $q$.

Finally, having obtained $\mu$ in terms of SYK parameters, we may go back to
4.5 and obtain $H$,

$H=4\pi\frac{{\cal{J}}}{q^{2}}.$ (4.13)

###  4.3 Mass of the Boundary

We have treated the boundary as if it were frozen at $\sigma=\sigma_{b}$ with
$\sigma_{b}$ given by 2.12. To put it another way, we have treated the
boundary as if it has infinite mass. In practice that is a good approximation,
but in fact the boundary is a dynamical object which can recoil and move when
a force is applied to it. For example, when a particle is dropped in from the
boundary the boundary will recoil so that its momentum will be equal and
opposite to the momentum of the particle [5]. However, the boundary itself
behaves as a very heavy non-relativistic particle whose mass is,

$M=\frac{N}{\mu}.$ (4.14)

Using 4.12 we may write,

$M=\frac{1}{\pi}Nq^{2}{\cal{J}}.$ (4.15)

The fact that the boundary mass is order $N$ means that the recoil motion is
usually negligible.

###  4.4 Summary of Preliminaries

For $q>>1$ the relation between the JT and SYK variables that we will need is
given by,

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle\frac{\pi}{q^{2}{\cal{J}}}\
\ \ \ \ \ \ \ \ \ \ (a)$ (4.16) $\displaystyle\frac{\Phi_{0}}{2G}$
$\displaystyle=$ $\displaystyle N\log{2}\ \ \ \ \ \ \ \ (b)$ (4.18)
$\displaystyle H$ $\displaystyle=$ $\displaystyle 4\pi\frac{{\cal{J}}}{q^{2}}\
\ \ \ \ \ \ \ \ \ \ (c)$ (4.20) $\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}Nq^{2}{\cal{J}}.\ \ \ \ \ \ \ \ (d)$ (4.22)
$\displaystyle F$ $\displaystyle=$ $\displaystyle 4{\cal{J}}^{2}\ \ \ \ \ \ \
\ \ \ \ \ (e)$ (4.24)

where in 4.24(c,d) the quantites $H$ and $F$ refer to the energy and the force
on a size-1 particle in the throat.

##  5 Operator Growth in Majorana-SYK

Qi and Streicher [9] have given a rigorous discussion of operator growth in
SYK at low temperature. In this paper we make no pretense of rigor. We rely on
intuitive “epidemic” arguments [12] which reproduce the results of [9] in the
throat region and which are easy to generalize to the charged case. The
argument is best justified in the large $q$ limit in which
$1<<\beta{\cal{J}}<<q^{2}<<N.$ Later we will discuss the extrapolation to
fixed but large $q^{2}$ and $\beta{\cal{J}}>>q^{2}.$

During the passage through the throat which lasts for a boundary-time
$\sim\beta$ the size and complexity of an evolving simple operator are
indistinguishable and we will use the same symbol for both; namely
${\cal{C}}.$

###  5.1 Exponential Growth in Circuit Time

At large $q$ the evolution of a single fermion operator is described by
diagrams of the form shown in figure 3. The diagram represents the “branching
diffusion” pattern shared by SYK and string theory. It is simplified so that
it shows an uniform rate of branching with respect to some yet-to-be-
determined time variable $\tau.$ This is an simplification which is only
correct in an average sense but it’s good enough for our purposes. The time
variable $\tau,$ which we will call “circuit time” is not assumed to be the
boundary time $u$ or even linear333The nonlinear relation between circuit and
ordinary time was also necessary in the quantum-epidemic model of Qi and
Streicher in order to reproduce the results of their calculation of growth. in
$u.$

Figure 3: Branching diffusion. The diagrams controlling the large $q$ behavior
of operator growth in SYK are tree like. They grow exponentially with depth.
The number of leaves of the tree is proportional to the number of vertices.
The tree-like behavior and exponential growth breaks down at the scrambling
time which only happens well after the particle has left the Throat and passed
through the Rindler region.

Identifying the number of endpoints (leaves of the tree) as the size
${\cal{C}}$ of the growing operator, the branching diffusion implies an
exponential growth of size in $\tau$. In the $q$-local version of SYK a
fermion will split into $(q-1)$ offspring. It is natural to write,

$\frac{d{\cal{C}}}{d\tau}=(q-2){\cal{C}}.$

However by redefining $\tau$ we may eliminate the factor $(q-2)$ and write,

$\displaystyle\frac{d{\cal{C}}}{d\tau}$ $\displaystyle=$
$\displaystyle{\cal{C}}$ (5.1) $\displaystyle{\cal{C}}(\tau)$ $\displaystyle=$
$\displaystyle e^{\tau}.$ (5.3)

We have assumed ${\cal{C}}(0)=1$ in line with the fact that the initial
fermion operator in the upcoming SYK calculation has size $1.$

As remarked above, the circuit time $\tau$ is not the usual asymptotic time
$u,$ nor is it linear in $u$. One might think that 5.3 has no content other
than as a definition of $\tau$ but that’s not true. The invariant fact
expressed by figure 3 and equation 5.3 is that the number of leaves of the
tree (the size) is proportional to the number of vertices. By analogy with
quantum circuits the number of vertices represents the growing complexity of
$\psi(t).$ This is the basis for the claim that size and complexity are
proportional to one another.

###  5.2 Why Circuit Time Slows Down

Let us digress for a moment to explain why the circuit time is not
proportional to the time $u$ conjugate to the SYK Hamiltonian. Consider the
energy carried by each line in the diagram of figure 3. An incoming energy
$\omega$ will split at each vertex so that the average energy carried by a
line will decrease exponentially with $\tau.$ This naturally implies that the
time $\Delta u$ to the next vertex will grow exponentially with $\tau.$ In
terms of the epidemic model it means that the rate of contact between
participants slows down in real time. This slowdown was noted in [5] where the
time between vertices $\Delta\tau$ was denoted $\tilde{\beta}.$ It was also
commented on in [9].

The proportionality of size and complexity can not go on forever. It and and
the tree-like nature of the evolution must break down at the scrambling time
when the size becomes comparable to the number of fermionic degrees of
freedom, $N$. At that point the size saturates but the complexity continues to
grow linearly with $u.$ But by this time the particle has long passed through
the throat and is in the Rindler region.

We will assume that boundary time $u$ and circuit time $\tau$ are
monotonically related,

$u=u(\tau),\ \ \ du/d\tau>0.$

Following [5] we define,

$\frac{du}{d\tau}=\tilde{\beta}.$ (5.4)

The first of equations 5.3 takes the form

$\frac{d{\cal{C}}}{du}\frac{du}{d\tau}=\tilde{\beta}\frac{d{\cal{C}}}{du}={\cal{C}}$
(5.5)

###  5.3 Momentum and Size

In [5] a specific form of the momentum-size correspondence was proposed,

$\tilde{\beta}P={\cal{C}}.$ (5.6)

Combining 5.6 with 5.5 implies,

$P=\frac{d{\cal{C}}}{du}.$ (5.7)

Equations 5.6 and 5.7 are two forms of the momentum-size correspondence which
in the scaling region, i.e., the throat, are equivalent. Which of them is more
fundamental is not obvious but the version 5.7 was derived in a rigorous
manner from the $SL(2R)$ symmetry of AdS(2). Moreover it holds not only
through the throat but also through the scrambling process in the Rindler
region.

Note that one may start with 5.7 and reverse the argument to derive 5.6.

###  5.4 Boundary Time and Circuit Time

Let us consider the functional relation $u(\tau).$ One way to obtain it is to
use the approximate scale invariance of SYK at low temperature. It implies
that ${\cal{C}}(u)$ should be a power of $u$. Furthermore it should be time-
symmetric and smooth at $u=0$. The simplest dimensionally consistent
assumption is

${\cal{C}}(u)=c{\cal{J}}^{2}u^{2}$ (5.8)

with $c$ being a constant. Qi and Streicher [9] calculate that at large $q$
the size grows according to 4.2,

$\displaystyle{\cal{C}}(u)$ $\displaystyle=$ $\displaystyle
1+2\left(\frac{\beta{\cal{J}}}{\pi}\sinh{(\pi u/\beta)}\right)^{2}$ (5.9)
$\displaystyle\approx$ $\displaystyle 2{\cal{J}}^{2}u^{2}.\ \ \ \ \ \ \ \ \ \
\ (u<<\beta\ \rm\ in\ throat)$ (5.11)

From 5.11 we read off that the constant in 5.8 is $c=2.$

Next we note that

$\tilde{\beta}=\frac{{\cal{C}}}{d{\cal{C}}/du}$

so that

$\tilde{\beta}=u/2,$ (5.12)

and from $du/{d\tau}=\tilde{\beta}$ we find,

$u={{\cal{J}}}^{-1}e^{\tau/2}.$ (5.13)

where the factor ${{\cal{J}}}^{-1}$ is necessary for dimensional consistency
and to match 5.11.

###  5.5 Momentum and Force

The conjectured momentum-size correspondence, if correct, would imply that the
momentum is given by,

$P=d{\cal{C}}/du=4{\cal{J}}^{2}u.$ (5.14)

This may be interpreted in terms of a constant force $F$ acting on the
particle,

$F=dP/du=4{\cal{J}}^{2}.$ (5.15)

which agrees with 4.4.

###  5.6 Ladder Graphs

At this point we will digress very briefly to comment on the relation between
the rate equation 5.3 and the ladder-sum for four-point out-of-time-order-
correlators. The diagrams of figure 3 represent amplitudes which need to be
squared in order to give probabilities. Accounting for the usual SYK disorder
averaging the leading melon diagrams are like the one in the left panel of
figure 4. In order to calculate the average size at a given time one needs to
count the number of fermionic lines in the intermediate state. This is done by
inserting the number operator444The definition of the number operator that we
use was given by Qi and Streicher [9]. The number operator is defined on two
copies of the $N$ fermion system which are identified with the bra and ket
sides of the diagrams in figure 3.  $[(\psi_{i})_{L},(\psi_{i})_{R}]$
according to the prescription in [9]. Diagrammatically it is denoted by the
open gap on one of the intermediate lines. The open gap can be on any of the
intermediate lines and the sum over such diagrams represents the probability
for a given size.

Figure 4: Diagrams like figure 3 represent amplitudes for a given number of
fermions, in this case $n=27$. To construct probabilities the amplitude must
be squared. In the left diagram the gap in the uppermost line represents the
counting of fermions at a fixed time. The ladder diagram on the right is
exactly the same diagram as on the left, but redrawn to emphasize the ladder
nature of the diagram.

In fact the symmetry of the diagram is such that the sum just gives a
multiplicative factor equal to the number of fermion lines.

The relation with four-point ladder diagrams is clear if we redraw the left
panel in the equivalent form in the right panel. The two diagrams are
equivalent but the right panel has been drawn to emphasize the ladder-like
nature of the graphs.

One question that figure 4 raises is why the lower rungs of the ladder have
more internal structure than the upper rungs. The answer is that in the model
with uniform spacing of vertices in $\tau$ the circuit time between the
beginning and end of a rung is greatest at the bottom of the ladder and grows
smaller as we proceed up the ladder. The actual ladder kernal is a sum of
diagrams, but which diagram dominates depends on the time-interval between
where it connects with the rails.

From the ladder diagram point of view the exponential growth of the size is
characteristic of the graviton Regge trajectory.

##  6 Charged Particles in U(1)-invariant SYK

Now we turn to the main subject of the paper—operator growth and its relation
to electric forces in the $U(1)$-invariant version of the SYK model. The model
has a global ungauged conserved charge. Positive and negative charged fermions
are coupled in a $q$-local manner. The $U(1)$ symmetry requires that each term
in the Hamiltonian has $q/2$ fermions of each sign. Schematically,

$H=\sum
j_{....}\psi_{1}^{{\dagger}}...\psi_{\frac{q}{2}}^{{\dagger}}\psi_{1}...\psi_{\frac{q}{2}}$
(6.1)

with555The definition of ${\cal{J}}$ is the same as in Majorana-SYK except
that the combinitorial factor $q!$ in M-SYK is replaced by $(q/2)!^{2}$ in
$U(1)SYK.$

$\langle
j^{2}\rangle=\frac{2^{q-1}{\cal{J}}^{2}}{q^{2}N^{q-1}}\left(q/2\right)!^{2}$
(6.2)

and ${\cal{J}}$ is to be held fixed as $N$ and $q$ are varied.

The model we will use for studying operator growth is a simple variant of the
perturbative epidemic model [12] which is justified in the limit
$(1<<\beta{\cal{J}}<<q^{2}<<N.)$ Later we will discuss its extrapolation to
$(1<<q^{2}<<\beta{\cal{J}}<<N.)$

###  6.1 The Chemical Potential

The vertices governing the growth of size are shown in figure 5.

Figure 5: $q=8$ “pitchfork diagrams” representing the basic unit of operator
growth in the U(1)-SYK model.The expressions under each vertex indicate its
weighting.

The diagrams in figure 5 are to be read from left to right, the propagators
being time-ordered.

To vary the charge of the black hole a chemical potential $\nu$ is
introduced666In our convention the chemical potential is dimensionless in
contrast to the usual convention in which it has units of energy.  into the
thermal density matrix as an extra factor $e^{\nu(n_{+}-n_{-})},$ where
$(n_{+}-n_{-})$ is the total fermion charge. It’s effect is to modify the bare
propagators connecting the vertices. A line labeled $\pm$ carries a factor
$e^{\mp\nu}$ as in figure 5

Let us see how these factors comes about. The bare propagators are defined in
the non-interacting theory (${\cal{J}}=0$) in which the different fermionic
coordinates do not interact. Therefore it is sufficient to concentrate on a
single complex fermionic degree of freedom. We may think of the fermionic
system in terms of Pauli matrices,

$\displaystyle\psi^{{\dagger}}$ $\displaystyle=$
$\displaystyle\sigma_{+}=(\sigma_{x}+i\sigma_{y})/2$ (6.3) $\displaystyle\psi$
$\displaystyle=$ $\displaystyle\sigma_{-}=(\sigma_{x}-i\sigma_{y})/2$ (6.5)
$\displaystyle n$ $\displaystyle=$ $\displaystyle\sigma_{z}/2$ (6.7)

The bare propagator is defined in terms of the non-interacting two-point
function,

$\langle\psi^{{\dagger}}\psi\rangle=\frac{1}{2}\ \rm Tr\it\
\rho\psi^{{\dagger}}\psi$ (6.8)

where the density matrix is

$\rho=e^{\nu n}=e^{\nu\sigma_{z}/2}$ (6.9)

One finds,

$\displaystyle\langle\psi^{{\dagger}}\psi\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{\nu/2}$ (6.10)
$\displaystyle\langle\psi\psi^{{\dagger}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\nu/2}$ (6.12)

$\displaystyle\langle\psi\psi^{{\dagger}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\nu/2}$ (6.13)
$\displaystyle\langle\psi^{{\dagger}}\psi\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{\nu/2}$ (6.15)

Thus a line representing a positive fermion carries a factor
$\frac{e^{\nu/2}}{2}$ and a line representing a negative fermion carries a
factor $\frac{e^{-\nu/2}}{2}$.

Figure 6: Splitting and redistributing $e^{\nu}$. A propagator representing a
positive fermion is shown weighted by a factor $e^{-\nu}$ in the left panel.
In the right panel that same factor is redistributed to the vertices connected
by the propagator.

It is convenient to redistribute these factors by splitting each
$e^{\pm\nu/2}$ into two factors $e^{\pm\nu/4}e^{\pm\nu/4}$ and bringing them
to the vertices as in figure 6. when this is done each vertex with an incoming
$\pm$ line will be weighted with a factor $e^{\pm\nu/2}.$ In addition the
entire graph will have an overall factor $e^{\nu/2}$ which does not depend on
the graph. The coupling ${\cal{J}}$ governing the strength of each vertex is
thereby “renormalized ” from ${\cal{J}}$ to $e^{\pm\nu/2}{\cal{J}}.$ This is
indicated in figure 5.

The factors $e^{\pm\nu/2}$ occur in amplitudes—roughly speaking, the amplitude
that the initial fermionic perturbation evolves to a given number of positive
and negative constituents—but the rate equations in the next section are about
probabilities. This means that in the weighting of each graph the factors
$e^{\pm\nu/2}$ must be squared to give a weight $e^{\pm\nu}$ to each vertex.

###  6.2 The Rate Equations

Equation 5.3 is the simplest possible rate equation that one can write down.
In the case of $U(1)$-SYK the equations are similar but more complicated since
one has to keep track of two species of fermions—those with positive charge
and those with negative charge.

Define the partial sizes $n_{\pm}$ to be the average number of $\pm$ charged
fermions in an operator. The total size $(n_{+}+n_{-})$ will again be denoted
${\cal{C}}.$ If the process is initiated by a $\pm$ charged fermion operator,
then the size after circuit time $\tau$ will be denoted by
${\cal{C}}(\tau)_{\pm}$. Specifically the size of $\psi^{{\dagger}}(\tau)$ is
${\cal{C}}(\tau)_{+}$ and similarly the size of $\psi(\tau)$ is
${\cal{C}}(\tau)_{-}$.

The rate equations for the tree-like graphs that dominate the large $q$ model
should have several properties:

  1. 1.

They should be linear. Once a splitting has occurred the different branches do
not interact.

  2. 2.

For reasons explained above, the splitting of a $\pm$ fermion should be
weighted by a probability $e^{\mp\nu.}$ This is indicated in figure 5.

  3. 3.

The equations should conserve the value of $(n_{+}-n_{-}).$ This is to insure
charge conservation.

The most general rate equation with these properties is,

$\frac{d}{d\tau}\begin{bmatrix}n_{+}\\\ n_{-}\\\
\end{bmatrix}\quad=\begin{bmatrix}e^{-\nu}\ &e^{\nu}\\\ e^{-\nu}\ &e^{\nu}\\\
\end{bmatrix}\quad\begin{bmatrix}n_{+}\\\ n_{-}\\\ \end{bmatrix}\quad$ (6.16)

The eigenvalues and eigenvectors777The ket-like notation here does not
indicate a quantum state, but only the real column vector whose entries are
$n_{+},n_{-}.$ of the matrix in 6.16 are,

$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle 0$ (6.17)
$\displaystyle|\lambda_{1})$ $\displaystyle=$
$\displaystyle\begin{bmatrix}e^{\nu}\\\ -e^{-\nu}\\\ \end{bmatrix}\quad$
(6.19) $\displaystyle\lambda_{2}$ $\displaystyle=$ $\displaystyle 2\cosh{\nu}$
(6.21) $\displaystyle|\lambda_{2})$ $\displaystyle=$
$\displaystyle\begin{bmatrix}1\\\ 1\\\ \end{bmatrix}\quad$ (6.23)

We will be interested in solutions of the rate equations in which the initial
state is either one positive or one negative fermion. We denote these initial
states by $|\psi(0))_{\pm}.$ By definition,

$\displaystyle|\psi(0))_{+}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}1\\\ 0\\\ \end{bmatrix}\quad$ (6.24)
$\displaystyle|\psi(0))_{-}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}0\\\ 1\\\ \end{bmatrix}\quad$ (6.26)

The solution of the rate equations with these initial conditions is,

$|\psi(\tau))_{\pm}=\frac{1}{2\cosh{\nu}}\begin{bmatrix}e^{\tau{\mp}\nu}+e^{\pm\nu}\\\
e^{\tau{\mp}\nu}-e^{\mp\nu}\\\ \end{bmatrix}=\begin{bmatrix}n_{+}\\\ n_{-}\\\
\end{bmatrix}_{\pm}$ (6.27)

and finally,

${\cal{C}}_{\pm}=n_{+}+n_{-}=\frac{1}{2\cosh{\nu}}(2e^{\tau\mp\nu}+e^{\pm\nu}-e^{\mp\nu}).$
(6.28)

In general when we compare with the GR calculation the presence of a chemical
potential will back-react and influence the metric, but by working to lowest
order in in $\nu$ back-reaction can be ignored. Thus from now on we will work
to linear order in $\nu,$

${\cal{C}}_{\pm}=e^{\tau}(1{\mp}\nu){\pm}\nu$ (6.29)

This is the total size for an initial state with one ${\pm}$ fermion.

Using 5.13 we find the sizes as a function of $u$,

${\cal{C}}_{\pm}(u)\approx(1\mp\nu){\cal{J}}^{2}u^{2}\pm\nu.$ (6.30)

For ${\cal{J}}u>1$ the second term may be ignored compared to the first,

${\cal{C}}_{\pm}(u)\approx(1\mp\nu){\cal{J}}^{2}u^{2}\ \ \ \ \ \ \ \ \ \ \
({\cal{J}}u>1)$ (6.31)

As an example consider the size of a positively charged operator
${\cal{C}}_{+}$ assuming the chemical potential is positive. From 6.31 we see
that the growth is slowed relative to the case with $\nu=0.$ Since with our
convention a positive chemical potential will induce a positive charge, the
result implies that like-sign charges retard the operator growth. Similarly
opposite charge increases the growth rate. This of course is the holographic
equivalent of like-signs repel, unlike signs attract.

###  6.3 Relation to OTOCS and Regge Behavior

The time dependence of size is rigorously quantified by the evolution of the
4-point out-of-order-correlator. In SYK these OTOCS are dominated by ladder
diagrams which are evaluated by iterating a ladder kernel. The ladders are
described by Regge behavior with Regge intercepts given in terms of the
eigenvalues of the kernel.

The matrix in the rate equations 6.16 is a simplified version of this kernel.
In the limit $\nu\to 0$ the two eigenvalues in 6.23 are $\lambda_{1}=0$ and
$\lambda_{2}=1.$ The second is already present in Majorana-SYK and corresponds
to the graviton trajectory. The first is a new feature of $U(1)$-SYK and
corresponds to the photon trajectory.

###  6.4 The Evolution of Size for Charged Operators

In the large $q^{2},$ fixed $\beta{\cal{J}}$ limit the charge of the system
and the chemical potential are related by,

$Q=\frac{1}{4}N\nu.$ (6.32)

Equation 6.32 together with

$M=q^{2}{\cal{J}}N$ (6.33)

imply that equation 6.31 can be re-written in terms of the black hole mass and
the integer-valued $U(1)$ generator $Q.$

${\cal{C}}_{\pm}\approx{\cal{J}}^{2}u^{2}\left(\frac{M}{{\cal{J}}Nq^{2}}\pm\frac{4Q}{N}\right)$
(6.34)

Concerning equation 6.32, it is true in the weak coupling limit and therefore
in the limit $q^{2}>>\beta{\cal{J}}.$ It may be derived in the free fermion
limit. To see that consider the partition function for a single free fermion
mode,

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\rm Tr\it e^{\nu n}$ (6.35)
$\displaystyle=$ $\displaystyle\rm Tr\it e^{\nu\sigma_{z}/2}$ (6.37)
$\displaystyle=$ $\displaystyle 2\cosh{\nu/2}$ (6.39)

The charge $Q$ to first order in $\nu$ is given by,

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{\partial\log{Z}}{\partial\nu}$ (6.40) $\displaystyle=$
$\displaystyle\frac{1}{4}\nu$ (6.42)

Extending this to $N$ fermionic modes gives 6.32.

Equation 6.32 is correct only correct for $q^{2}>>\beta J$ and must be
significantly modified in the more interesting limit of fixed $q$ and very
large $\beta{\cal{J}}.$ We will discuss this in section 8.

By using the momentum-size correspondence we will compare 6.34 with a general
relativity calculation of the motion of charges in a NERN background. For the
purpose of that comparison we note that the momentum-size correspondence
applied to 6.34 would give,

$P=\frac{d{\cal{C}}}{du}\approx{\cal{J}}^{2}u\left(\frac{M}{{\cal{J}}Nq^{2}}\pm\frac{4Q}{N}\right)$
(6.43)

###  6.5 Note on Charges

In equation 6.32 $Q$ is not exactly the electric charge. $Q$ is the difference
$(n_{+}-n_{-})$ which we do expect to be proportional to the electric charge,
but with a factor of proportionality yet to be determined. Let us call the
electric charge $\tilde{Q}.$ It differs from the integer-valued $U(1)$
generator by a factor $|e|$ representing the fundamental unit of charge,

$\tilde{Q}=|e|Q.$ (6.44)

##  7 Motion of Charged Particles

Returning to equations 3.32, the force on a charged particle in the throat is
given by the sum of two terms: a gravitational force $F_{G}$ and an electric
force $F_{E}.$

$\displaystyle F_{G}$ $\displaystyle=$
$\displaystyle\frac{H}{\mu}+eE(e^{-\rho/\mu}-1)$ (7.1) $\displaystyle F_{E}$
$\displaystyle=$ $\displaystyle eEe^{-\rho/\mu}$ (7.3)

As the particle moves away from the boundary (at $\rho=0$) the electric term
quickly becomes negligible, so throughout most of the throat the force is
constant and equal to,

$F=\frac{H}{\mu}-eE$ (7.4)

The Energy $H$ is conserved and may be evaluated at the start of the infall
when the particle is at rest at the black hole boundary. From 4.24(c),

$H=4\pi\frac{{\cal{J}}}{q^{2}}.$ (7.5)

The electric field $E$ is related to the electric charge of the black hole
which we denote $\tilde{Q},$

$E=\tilde{Q}$ (7.6)

The final form for the force on the charged particle in the throat is,

$F=4{\cal{J}}^{2}\pm\tilde{Q}e$ (7.7)

It follows that the momentum of an infalling charged fermion is given by,

$P(u)=u\left(4{\cal{J}}^{2}\pm\tilde{Q}e\right)$ (7.8)

###  7.1 A Puzzle

There is something puzzling about equations 3.19 and 3.20 (it puzzled me);
namely the presence of the term

$\frac{\partial_{r}f}{2\sqrt{f}}eA_{u}$

in the expression for the gravitational force. Being proportional to the
charge it would seem to be part of the electric force, but that would be very
odd since the usual expression for electric force involves only the spatial
derivative of $A_{u}.$ In fact the term,

$e\frac{\partial A_{u}}{\partial\rho}=-e\frac{\partial A_{u}}{\partial
r}\sqrt{f}$

was identified as the electric force in 3.27. It is also the term we dropped
from 3.32 because it quickly becomes negligible as the particle moves away
from the boundary. However, it would be a mistake to ignore it altogether.

The resolution of the puzzle is the following: The particle initially
experiences an electric force $-e\partial_{\rho}A_{u}$ as an impulse as it
starts out at $\rho=0.$ The impulse, although short-lived, affects the
momentum and kinetic energy of the particle. By the time the particle has
moved a distance $\sim\mu,$ depending on the relative sign of the particle and
black hole, the electric impulse will have resulted in the particle having
greater or lesser energy than a corresponding neutral particle. After the
initial kick kinetic energy will be $H+eA_{u}.$

On the other hand the gravitational force on the particle is proportional to
its kinetic energy, so depending on the sign of the charge, once the particle
gets away from the boundary the gravitational force will either be less than
or greater than that of the corresponding neutral particle. We see that the
origin of the term proportional to $(H+eA_{u})$ is gravitational.

That the electric force $-e\partial_{\rho}A_{u},$ decreases quickly compared
to the gravitational force is expected from Regge theory888From the Regge-pole
point of view, gravity corresponds to a trajectory with intercept $2$ while
electromagnetism corresponds to a trajectory with intercept $1.$ In the
current context this implies a dominance of gravitational force over electric
force as the particle gains energy in the throat. .

###  7.2 Comparing SYK and GR

Now we come to the main point: the comparison of the forces and acceleration
of charged particles in the throat, and the growth of operator size in
$U(1)$-SYK. From the general relativity calculation 7.7 we conclude that the
momentum $P$ of an infalling charge is given by 7.8,

$P=u\left(4{\cal{J}}^{2}\pm\tilde{Q}e\right).$

On the other hand let us return to the rate equations in section 6. The result
was equation 6.34 for the growth of size, and (assuming the momentum-size
correspondence) 6.43 for the growth of momentum. It is evident that 7.8 and
6.43 have similar form. Equating them we find,

$e^{2}=\frac{4{\cal{J}}^{2}}{N}.$ (7.9)

Equation 7.9 is new information that fixes the fundamental unit electric
charge, in other words the charge of a fundamental $U(1)$-SYK fermion. The
fact that it is order $1/N$ means that electric forces are $1/N$ effects which
may seem surprising. In the next section we will see that this has a simple
explanation.

###  7.3 SYK and KK

While 7.9 may seem surprising it is in fact entirely natural. What it suggests
is that the $U(1)$-SYK system is a realization of Kaluza-Klein theory with a
compactification scale which we might have guessed.

Let us add an additional compact KK direction so that the geometry becomes
$AdS(2)\times U(1),$ with the compact circle having radius $R_{k}.$ The unit
of electric charge in such a theory would be given by,

$e^{2}=\frac{G}{\Phi_{0}R_{k}^{2}}$ (7.10)

From 4.8 we see that

$e^{2}\sim\frac{1}{NR_{k}^{2}}.$ (7.11)

Comparing this with 7.9 we find,

$R_{k}\sim 1/{\cal{J}}.$ (7.12)

The scaling of $R_{k}$ with $1/{\cal{J}}$ is hardly surprising since
${\cal{J}}^{-1}$ is the only length scale that appears in SYK. In principle
$R_{k}$ might have had some uncancelled $N$ dependence, but that did not
happen. The lack of $q$ dependence in $R_{k}$ is also interesting. We may
compare 7.12 with 4.12,

$\frac{R_{k}}{\mu}\sim q^{2}.$ (7.13)

We see that $R_{k}$ grows relative to the $AdS(2)$ length scale as $q$
increases. The significance of this is not entirely clear and one might wonder
if it is an artifact of weak coupling, not to be trusted for fixed $q$ and
large $\beta{\cal{J}}.$ In the next section999The argument of section 7.4 is
due to Henry Lin and Geoff Penington.  evidence will be given that 7.13 is
robust in the strong coupling limit $1<<q^{2}<<\beta{\cal{J}}.$

###  7.4 Boundary Particle Formulation

The arguments of the previous sections were derived from the weak coupling
limit in which $\beta{\cal{J}}$ is held fixed and $q^{2}\to\infty.$
Nevertheless I believe the results—in particular the scaling of the KK radius
7.12—may be extrapolated to the strong coupling limit in which $q^{2}$ is held
fixed and $\beta{\cal{J}}\to\infty.$

A method which is valid at strong coupling is the boundary particle
description [10]. In this section we will use the boundary particle
description to compute $R_{k}$ and see that it is indeed given by 7.12.

In the Majorana case the boundary particle has a single degree of freedom,
i.e., its proper distance from the horizon. It moves nonrelativistically with
a large mass, $M=N{\cal{J}}q^{2}$ (see 4.24-d). In $U(1)$-SYK the particle has
a second $U(1)$ angular degree of freedom. The radial direction is frozen by a
potential but the angular direction is a zero-mode. The quantized momentum
conjugate to the angular coordinate is the integer-valued charge $Q.$

Because of the large mass of the boundary particle the Hamiltonian for the
angular motion is non-relativistic,

$H=p^{2}/2M=n^{2}/2R_{k}^{2}M.$ (7.14)

We consider the thermal ensemble,

$\rho=e^{-\beta H-\nu n}=e^{-\frac{\beta n^{2}}{2MR_{k}^{2}}-\nu n}$ (7.15)

Completing the square and integrating over $n$ gives a partition function,

$Z=e^{\frac{\nu^{2}R_{k}^{2}M}{2\beta}}.$ (7.16)

The charge $Q$ is given by,

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{\partial{\log{Z}}}{\partial\nu}=\mu\frac{R_{k}^{2}M}{2\beta}$
(7.17) $\displaystyle=$ $\displaystyle\nu Nq^{2}R_{k}^{2}{\cal{J}}/\beta$
(7.19)

In the weak coupling limit the charge and chemical potential are related by
6.32 but this relation breaks down badly in the strong coupling limit. For
$\beta{\cal{J}}>>q^{2}$ the correct relation [7] is,

$Q=\frac{\nu Nq^{2}}{16\beta{\cal{J}}}$ (7.20)

Equating these expressions for $Q$ gives,

$R_{k}^{2}=\frac{Nq^{2}}{8{\cal{J}}M}$

and finally using 4.24(d) we find the $q$ and $N$ dependences cancel,

$R_{k}=\sqrt{\frac{\pi}{8}}\ \frac{1}{{\cal{J}}}\sim\frac{1}{{\cal{J}}}$
(7.21)

in agreement with 7.12.

On the other hand the relation $\mu=\pi/(q^{2}{\cal{J}})$ (equation 4.12 in
section 4.2) made no use of weak coupling. Therefore the mismatch between the
scaling of $R_{k}$ and $\mu$ is not a weak coupling artifact.

To reiterate, the scaling of the KK radius with either $1/{\cal{J}}$ was
derived in two ways: the weak coupling method featuring rate equations,
applicable for $q^{2}>>\beta{\cal{J}}$; and the strong coupling boundary
particle method. This suggests that at least for some quantities it is
possible to calculate by first fixing $\beta{\cal{J}}$ and going to large
$q^{2},$ followed by an extrapolation to large but fixed $q$ and much larger
$\beta{\cal{J}}.$

##  8 Extrapolation to Strong Coupling

In this section we will speculate on why the weak coupling analysis used in
most of this paper gives results which are applicable for the strongly coupled
limit $1<<q^{2}<<\beta{\cal{J}}.$

Let us consider the consequences of a hypothetical “renormalization” of the
chemical potential $\nu$ in which it is replaced by a renormalized value
$\bar{\nu},$

${\bar{\nu}}=f(q,\beta{\cal{J}})\nu$ (8.22)

with $f$ being an arbitrary function of the dimensionless SYK parameters $q$
and $\beta{\cal{J}}.$ In extrapolating from weak to strong coupling there is
no reason why $f(q,\beta{\cal{J}})$ should not change a great deal.

There are two places, where the chemical potential enters into our analysis.
The first was in the rate equations 6.16 and the second was in equation
relating chemical potential and charge 6.32. The thing to note is that if
$\nu$ were to be replaced by ${\bar{\nu}}$ in both places:

$\frac{d}{d\tau}\begin{bmatrix}n_{+}\\\ n_{-}\\\
\end{bmatrix}\quad=\begin{bmatrix}e^{{\bar{\nu}}}\ &e^{-{\bar{\nu}}}\\\
e^{{\bar{\nu}}}\ &e^{-{\bar{\nu}}}\\\
\end{bmatrix}\quad\begin{bmatrix}n_{+}\\\ n_{-}\\\ \end{bmatrix}\quad$ (8.23)
$Q=\frac{1}{4}N{\bar{\nu}}$ (8.24)

the final outcome would be unchanged. In particular the relations 7.9 and the
relation between the Kaluza-Klein radius and the SYK energy scale ${\cal{J}}$
7.12 would be unchanged.

From 7.20 we see that the function $f$ should be given by

$f(q,\beta{\cal{J}})=\frac{q^{2}}{4\beta{\cal{J}}}.$ (8.25)

in the strong coupling limit. The question is whether the same renormalization
of $\nu$ would correctly describe the strongly coupled rate equations in 8.23.

Let us first consider 8.24. The total charge is given by the equal-time two-
point function,

$Q=\rm Tr\it\ i\rho\
[\psi^{{\dagger}}_{i},\psi_{i}]=i\langle\psi^{{\dagger}}_{i}\psi_{i}-hc\rangle$
(8.26)

with

$\rho=e^{-\beta H+\nu n}.$

To first order in the chemical potential the total charge is proportional to

$\frac{d\langle\psi^{{\dagger}}_{i}\psi_{i}-hc\rangle}{d\nu}$

in other words,

${\bar{\nu}}\sim\frac{d\langle\psi^{{\dagger}}_{i}\psi_{i}-hc\rangle}{d\nu}|_{u=0}$
(8.27)

Now consider the $\nu$ dependence of the rate equation, which originated from
the dependence of the propagators in figure 6 on $\nu.$ In the weakly coupled
limit those propagators had the form given in 6.15 but they will be corrected
by interactions. Assuming that the diagrams of importance in the coupled
theory are the same as those in the weak limit, but with corrected
propagators, then to first order in $\nu$ the rate equations in the
interacting theory depend on

${\bar{\nu}}\sim\frac{d\langle\psi^{{\dagger}}_{i}\psi_{i}-hc\rangle}{d\nu}.$

The difference is that in the case of the rate equations the effective value
of ${\bar{\nu}}$ depends on the two-point functions over a range of time, not
just at $u=0.$ It does however seem reasonable that the renormalized values of
${\bar{\nu}}$ for the two purposes are approximately the same.

##  9 Conclusion

This paper studied the emergence of bulk electric forces in a holographic
theory with a global symmetry —namely the $U(1)$-SYK model. There main
ingredients were:

  1. 1.

The momentum-size correspondence that relates the momentum of an infalling
particle to the size of the operator that created the particle.

  2. 2.

On the SYK side, a simple generalization of the epidemic model for operator
growth that described the evolution of size for charged operators in a
background of non-zero global $U(1)$ charge.

  3. 3.

A general relativity calculation of the time dependence of momentum of an
infalling charged particle in two-dimensional dilaton-Maxwell gravity.

The agreement of the SYK and GR calculations requires a specific normalization
for unit of electric charge carried by the basic fermion operators $\psi.$
That normalization agrees with a Kaluza-Klein mechanism in which the KK
compactification radius is equal to the inverse energy scale of the SYK theory
as in equation 7.12.

The epidemic model was based on weak coupling arguments and does not obviously
apply in the limit $\beta{\cal{J}}>>q^{2}$ where the coupling is strong. Two
arguments were given that the weak coupling results could be extrapolated to
strong coupling. The first was that by using the strong coupling boundary
particle formulation one finds the same result for the KK scale, namely 7.12.

The second argument was that the main effect of strong coupling is to
“renormalize” the chemical potential. If the renormalization is the same in
the two places where the chemical potential occurs it will cancel out and
reproduce 7.12 when $\beta{\cal{J}}>>q^{2}.$

The charged epidemic model of section 6 is crude and a more technically
precise version of it would be helpful, especially in clarifying the
extrapolation to strong coupling.

One last point seems worthy of mention. The gravitational effect on an
infalling particle is truly a bulk phenomenon in that the force is uniform
throughout the throat. By contrast, although the electric field is uniform,
equation 3.32 shows that the electric force on a charged particle is localized
within a distance $\mu$ from the boundary. The difference is due to the
different energy dependence of the coupling to gravity and electromagnetism

Nevertheless, despite its localization near the boundary, the electric force
leaves an imprint on the kinetic energy and therefore affects the subsequent
gravitational force on the particle throughout the bulk.

## Acknowledgements

I am grateful to Adam Brown, Henry Lin, and Geoff Penington for discussions
during the early phases of this work. The boundary-particle formulation of
section 7.4 is due to Henry and Geoff.

## References

  * [1] D. A. Roberts, D. Stanford and A. Streicher, “Operator growth in the SYK model,” JHEP 06, 122 (2018) doi:10.1007/JHEP06(2018)122 [arXiv:1802.02633 [hep-th]].
  * [2] S. Xu, L. Susskind, Y. Su and B. Swingle, “A Sparse Model of Quantum Holography,” [arXiv:2008.02303 [cond-mat.str-el]].
  * [3] L. Susskind, “Why do Things Fall?,” [arXiv:1802.01198 [hep-th]].
  * [4] A. R. Brown, H. Gharibyan, A. Streicher, L. Susskind, L. Thorlacius and Y. Zhao, “Falling Toward Charged Black Holes,” Phys. Rev. D 98, no.12, 126016 (2018) doi:10.1103/PhysRevD.98.126016 [arXiv:1804.04156 [hep-th]].
  * [5] L. Susskind, “Complexity and Newton’s Laws,” Front. in Phys. 8, 262 (2020) doi:10.3389/fphy.2020.00262 [arXiv:1904.12819 [hep-th]].
  * [6] H. W. Lin, J. Maldacena and Y. Zhao, “Symmetries Near the Horizon,” JHEP 08, 049 (2019) doi:10.1007/JHEP08(2019)049 [arXiv:1904.12820 [hep-th]].
  * [7] R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, “Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography,” Phys. Rev. B 95, no.15, 155131 (2017) doi:10.1103/PhysRevB.95.155131 [arXiv:1612.00849 [cond-mat.str-el]].
  * [8] Y. Gu, A. Kitaev, S. Sachdev and G. Tarnopolsky, “Notes on the complex Sachdev-Ye-Kitaev model,” JHEP 02, 157 (2020) doi:10.1007/JHEP02(2020)157 [arXiv:1910.14099 [hep-th]].
  * [9] X. L. Qi and A. Streicher, “Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK,” JHEP 08, 012 (2019) doi:10.1007/JHEP08(2019)012 [arXiv:1810.11958 [hep-th]].
  * [10] J. Maldacena, D. Stanford and Z. Yang, “Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space,” PTEP 2016, no.12, 12C104 (2016) doi:10.1093/ptep/ptw124 [arXiv:1606.01857 [hep-th]].
  * [11] J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Phys. Rev. D 94, no.10, 106002 (2016) doi:10.1103/PhysRevD.94.106002 [arXiv:1604.07818 [hep-th]].
  * [12] L. Susskind and Y. Zhao, “Switchbacks and the Bridge to Nowhere,” [arXiv:1408.2823 [hep-th]].