RIKEN-iTHEMS-Report-21

YITP-21-124

Factorizing Wormholes
in a Partially Disorder-Averaged SYK Model

Kanato Gotoa, Kenta Suzukib and Tomonori Ugajinb

aRIKEN Interdisciplinary Theoretical and Mathematical Sciences
(iTHEMS), Wako, Saitama 351-0198, Japan
bCenter for Gravitational Physics,
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

In this paper, we introduce a “partially disorder-averaged” SYK model. This
model has a real parameter that smoothly interpolates between the ordinary
totally disorder-averaged SYK model and the totally fixed-coupling model. For
the large $N$ effective description, in addition to the usual bi-local
collective fields, we also introduce a new additional set of local collective
fields. These local fields can be understood as “half” of the bi-local
collective fields, and in the totally fixed-coupling limit, they represent the
“half-wormholes” which were found in recent studies. We find that the large
$N$ saddles of these local fields vanish in the total-disorder-averaged limit,
while they develop nontrivial profiles as we gradually fix the coupling
constants. We argue that the bulk picture of these local collective fields
represents a correlation between a spacetime brane and the asymptotic AdS
boundary. This illuminates how the half-wormhole saddles emerge in the SYK
model with fixed couplings.

###### Contents

  1. 1 Introduction
  2. 2 Partial Disorder Average
  3. 3 Partially Disorder-Averaged Partition Function
    1. 3.1 Zero-dimensional SYK
    2. 3.2 Total disorder-averaging
    3. 3.3 Slightly fixed-coupling
    4. 3.4 Bulk interpretation of $G_{\sigma}$ and $\Sigma_{\sigma}$
    5. 3.5 Comments on general $\sigma$ case
  4. 4 Two-point Function $\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}$
    1. 4.1 Zero-dimensional SYK
      1. 4.1.1 Saddle-point solutions
      2. 4.1.2 On-shell action
    2. 4.2 Adding a coupling
    3. 4.3 Brownian SYK
  5. 5 Spectrum
  6. 6 Conclusions and Discussions
  7. A External couplings as another universe fermions
  8. B Integration over the fermions
  9. C Another formalism for slightly fixed-coupling
  10. D Two-point function of $q=2$, 0d SYK
  11. E Direct computation of $\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}$ at $\sigma=0$

##  1 Introduction

Recent studies suggested a holographic dual of a bulk gravitational theory has
to involve an ensemble average of theories with random couplings. The Sachdev-
Ye-Kitaev (SYK) model [1, 2] provides such an example of the duality. It is a
quantum mechanical many-body system with all-to-all interactions on fermionic
$N$ sites ($N\gg 1$). Its coupling constant is random with a Gaussian
probability distribution. The model provides an interesting, highly nontrivial
example of the AdS/CFT duality and a potential framework for quantum black
holes, with related out-of-time-order correlators exhibiting quantum chaos and
the maximal Lyapunov exponent characteristic of black holes [2, 3, 4, 5]. A
central role is played by the emergent Schwarzian mode [4, 6, 7, 8, 9, 10,
11], which dominates the dynamics of the model in the low-temperature limit.
The dual of the Schwarzian mode in the gravity theory is well understood as
the boundary graviton of the Jackiw-Teitelboim (JT) gravity [12, 13] in the
context of nearly AdS2/CFT1 correspondence [14, 15, 16, 17].

The relationship between a boundary ensemble average and a bulk gravity theory
becomes more evident in the recent calculation of the JT gravity partition
function at a non-perturbative level, defined by summing over all higher genus
topologies. For instance, it was shown that the non-perturbative JT gravity
partition coincides with a random Hermitian matrix integral [18]. In this
correspondence, the central role is played by Euclidean wormholes in the
gravitational path integrals. For instance, in the presence of multiple
conformal boundaries, the natural rule to define semi-classical gravitational
path integral is to include all Euclidean manifolds consistent with the
boundary condition. This includes Euclidean wormhole geometries where several
conformal boundaries get connected, resulting in the random matrix integral,
as mentioned above. It was also shown that even in a very simple topological
model of gravity, once we include all possible topologically distinct
manifolds respecting the boundary condition, its boundary naturally realizes
an ensemble average [19]. Its generalization to supersymmetric cases can be
found in [20].

One manifestation of the emergence of ensemble averages in the presence of
wormholes is the non-factorization of partition functions. Let us consider the
simplest example, where we have two disjoint conformal boundaries. Then the
gravitational path integral receives contributions from wormholes connecting
the two boundaries. This is puzzling from the boundary field theory point of
view, because on these two disjoint boundaries, its partition function gets
factorized [21]. The resolution is to regard its dual as an ensemble of
theories living on the boundaries. The relation between ensemble averages and
the physics of wormholes was extensively discussed a few decades ago [22, 23],
and recent discussions mainly focus on its implications for holographic
duality [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and the black hole
information paradox [36, 37, 38, 39, 40, 41, 42].

Given these developments, a natural question arises: How do we get a bulk
gravity description of a fixed boundary field theory without averaging ? We
are particularly interested in the fate of Euclidean wormholes in such a
setup. There have been several attempts toward this goal. For example, it was
shown in [43] that by introducing “eigenbranes” in the bulk of JT gravity, one
can fix several eigenvalues of the dual random matrix. This question was
studied in [33] for the SYK model with one time point (i.e., zero-dimensional
SYK model) with a fixed coupling constant. There, in the calculation of a
product of partition functions $Z_{L}Z_{R}$ without averaging, two types of
large $N$ saddles are found. One is the usual wormhole saddles which connect
two boundaries $L$ and $R$, and give the averaged two-point function $\langle
Z_{L}Z_{R}\rangle$ over the random coupling. In addition to these, there is
another type of saddles called “half-wormholes”. From a bulk gravity point of
view, these saddles can be interpreted as geometries starting from a conformal
boundary and ending somewhere in the bulk. Quite remarkably, the contributions
of these half-wormholes are essential for $Z_{L}Z_{R}$ to get factorized. This
idea is further developed in [44, 45, 46, 47, 48, 49, 50, 51] for more
details.

To further investigate these ideas, wormholes vs. half-wormholes and ensemble
averaging vs. fixed-coupling (or theory), in this paper, we introduce a
partially disorder-averaged SYK model. This model has a real parameter
$\sigma$ that smoothly interpolates between the ordinary totally disorder-
averaged SYK model ($\sigma=\infty$) and the totally fixed-coupling model
($\sigma=0$). This is archived by modifying the probability distribution of
the coupling constant. In the $\sigma=\infty$ limit, the probability
distribution is reduced to the ordinary Gaussian form. In contrast, in the
$\sigma=0$ limit, the probability distribution becomes a product of delta
functions that enforces each component of the coupling constant to a fixed
external value. This type of interpolation between the totally disorder-
averaged and fixed-coupling cases is previously studied in the context of
matrix integrals in [45].

One convenient way to study the conventional SYK model in the large $N$ limit
is introducing bi-local collective fields
$\\{G(\tau_{1},\tau_{2}),\Sigma(\tau_{1},\tau_{2})\\}$ depending on two
boundary times. This formalism has a further advantage for studies of two-
point (or higher-point) functions of the partition functions. When each time
corresponds to each boundary theory, the bi-local fields have a natural bulk
interpretation for the corresponding gravitational configurations, including
Euclidean wormholes in a suitable setup. Below we will argue that in the large
$N$ analysis of our deformed SYK model, we need to introduce additional local
collective fields $\\{G_{\sigma}(\tau),\Sigma_{\sigma}(\tau)\\}$. These local
fields are analogs of “half-wormholes” because they depend only on one
boundary time. We will study in detail the behavior of these local fields
$\\{G_{\sigma}(\tau),\Sigma_{\sigma}(\tau)\\}$ as we change the deformation
parameter $\sigma$, from $\sigma=\infty$ whose low energy part has a complete
gravitational description in terms of JT gravity, to the totally fixed
coupling model $\sigma=0$ whose gravitational description we would like to
know better.

The remainder of the paper is organized as follows. In section 2, we define a
partially disorder-averaged SYK model by modifying the probability
distribution of the coupling constants. We show that this correctly
interpolates between the totally disorder-averaged and totally fixed coupling
case.

In section 3, we study one-point function of the partition function after this
partial disorder averaging. For the large $N$ effective description, in
addition to the usual bi-local collective fields, we also introduce a new
additional set of local collective fields. We explain that these local fields
can be understood as the “half” of the bi-local collective fields. Before
studying the one-dimensional model, in section 3.1, we study the zero-
dimensional model (i.e., SYK model with one time point). We show that the
partially disorder-averaged partition function is proportional to the
“hyperpfaffian” of the external coupling together with a coefficient that
vanishes in the total disorder-averaged limit while giving a finite value in
the total fixed coupling limit. Then, we go back to the one-dimensional model.
After reviewing the totally disorder-averaged case (which is well-studied in
the literature) in section 3.2, we study the slightly fixed coupling case in
section 3.3. We show that for the large $N$ collective field description, the
effect of slightly fixed coupling is given by the same form as the chemical
potential contribution in the complex SYK model. In section 3.4, we explain a
possible bulk picture of the new local collective fields and in section 3.5 we
comments on the general $\sigma$ (i.e. general partial disorder-averaged)
case.

In section 4, we study two-point function of the partition functions after the
partial disorder averaging. In section 4.1, focusing on the zero-dimensional
model, we study the wormhole and half-wormhole saddles in detail. We also
comment on the case when we add explicit interactions between the left and
right SYK’s in section 4.2. Our discussion on the Brownian SYK model is given
in section 4.3.

We study the transition of the spectral density as well as the spectral form
factor from the totally fixed coupling limit to the totally disorder-averaged
limit in section 5. Our conclusions and further discussions are presented in
section 6. Supplemental computations and discussion are gathered in
appendices.

##  2 Partial Disorder Average

In this section, we define our partially disorder-averaged SYK model, which
smoothly interpolates between the ordinary totally disorder-averaged SYK model
and the totally fixed coupling model by introducing a real parameter $\sigma$.
The original SYK model [1, 2] is defined by the Hamiltonian

$\displaystyle
H\,=\,i^{\frac{q}{2}}\sum_{i_{1}<i_{2}<\cdots<i_{q}}^{N}J_{i_{1}\cdots
i_{q}}\,\chi_{i_{1}}\cdots\chi_{i_{q}}\,,$ (2.1)

where the one-dimensional Majorana fermions satisfy

$\displaystyle\\{\chi_{i},\chi_{j}\\}\,=\,\delta_{ij}\,.$ (2.2)

The coupling $J_{i_{1}\cdots i_{q}}$ is random, and drawn from Gaussian
ensemble with the vanishing mean value $\langle J_{i_{1}\cdots
i_{q}}\rangle=0$. Its Lagrangian is given by

$\displaystyle
L\,=\,\frac{1}{2}\sum_{i=1}^{N}\chi_{i}\partial_{\tau}\chi_{i}\,-\,i^{\frac{q}{2}}\sum_{i_{1}<\cdots<i_{q}}^{N}J_{i_{1}\cdots
i_{q}}\,\chi_{i_{1}}\cdots\chi_{i_{q}}\,.$ (2.3)

Now we introduce a partial disorder averaging by deforming the probability
distribution of the random coupling as

$\displaystyle P(J_{i_{1}\cdots
i_{q}})\,=\,\exp\left[-\frac{N^{q-1}}{2(q-1)!}\sum_{i_{1}<\cdots<i_{q}}^{N}\left(\frac{J_{i_{1}\cdots
i_{q}}^{2}}{J^{2}}\,+\,\frac{(J_{i_{1}\cdots i_{q}}-J_{i_{1}\cdots
i_{q}}^{(0)})^{2}}{\sigma^{2}}\right)\right]\,,$ (2.4)

with

$\displaystyle\big{\langle}\mathcal{O}\big{\rangle}_{J}\,\equiv\,\mathcal{N}_{\sigma}^{-1}\int\prod_{i_{1}<\cdots<i_{q}}^{N}dJ_{i_{1}\cdots
i_{q}}\,P(J_{i_{1}\cdots i_{q}})\,\mathcal{O}\,.$ (2.5)

Here $J_{i_{1}\cdots i_{q}}^{(0)}$ is the external fixed coupling, and the
parameter $\sigma$ controls how much we take disorder averaging. This type of
partial disorder averaging was studied in [45] for a matrix model. As in [45],
$\sigma=\infty$ corresponds to the total disorder-averaging coincides with the
ordinary SYK model, while $\sigma\to 0$ corresponds to totally fixing the
coupling constant $J_{i_{1}\cdots i_{q}}=J_{i_{1}\cdots i_{q}}^{(0)}$. In the
following, it is convenient to introduce a new set of effective couplings by

$\displaystyle\frac{1}{\widetilde{J}^{2}}\,\equiv\,\frac{1}{J^{2}}\,+\,\frac{1}{\sigma^{2}}\,,\qquad
J_{\sigma}\,\equiv\,\frac{\widetilde{J}^{2}J_{0}}{\sigma^{2}}\,,$ (2.6)

where $J_{0}$ is defined in (3.4) and we assume $\widetilde{J}\geq 0$. In the
extreme limits of the parameter $\sigma$, these two couplings behave as

$\displaystyle\sigma\,\to\,\infty\,:$
$\displaystyle\qquad\widetilde{J}\,\to\,J\,,\qquad J_{\sigma}\,\to\,0\,,$
(2.7) $\displaystyle\sigma\,\to\,0\,:$
$\displaystyle\qquad\widetilde{J}\,\to\,\sigma\,,\qquad
J_{\sigma}\,\to\,J_{0}\,.$ (2.8)

The normalization factor $\mathcal{N}_{\sigma}$ is defined by

$\displaystyle\mathcal{N}_{\sigma}\,\equiv\,\int\prod_{i_{1}<\cdots<i_{q}}^{N}dJ_{i_{1}\cdots
i_{q}}\,P(J_{i_{1}\cdots i_{q}})\,,$ (2.9)

and this gives us

$\displaystyle\mathcal{N}_{\sigma}^{-1}\,=\,\left(\frac{N^{q-1}}{2(q-1)!\pi\widetilde{J}^{2}}\right)^{\frac{1}{2}\binom{N}{q}}\,\exp\left(\frac{N^{q-1}}{2(q-1)!(J^{2}+\sigma^{2})}\sum_{i_{1}<\cdots<i_{q}}^{N}\Big{(}J_{i_{1}\cdots
i_{q}}^{(0)}\Big{)}^{2}\right)\,.$ (2.10)

For the usual total-disorder-averaged SYK model, this factor does not play any
crucial role, but for our partially disorder-averaged SYK model, we have to
keep at least nontrivial $\sigma$-dependence coming from this normalization
factor. For example, in the totally-fixed-coupling limit ($\sigma\to 0$),
combined with this normalization factor, the probability distribution gives
delta functions:

$\displaystyle\lim_{\sigma\to 0}\ \mathcal{N}_{\sigma}^{-1}\,P(J_{i_{1}\cdots
i_{q}})\,=\,\prod_{i_{1}<\cdots<i_{q}}^{N}\delta\left(J_{i_{1}\cdots
i_{q}}-J_{i_{1}\cdots i_{q}}^{(0)}\right)\,.$ (2.11)

This type of delta function projection of the coupling constant was also
studied in [50].

This definition of partial disorder averaging does not give a vanishing mean
value of the coupling unlike the ordinary SYK model, but we find

$\displaystyle\big{\langle}J_{i_{1}\cdots
i_{q}}\big{\rangle}_{J}\,=\,\frac{\widetilde{J}^{2}}{\sigma^{2}}\,J_{i_{1}\cdots
i_{q}}^{(0)}\,,\qquad\big{\langle}J_{i_{1}\cdots
i_{q}}^{2}\big{\rangle}_{J}\,=\,\frac{(q-1)!\widetilde{J}^{2}}{N^{q-1}}\left(1+\frac{\widetilde{J}^{2}N^{q-1}}{(q-1)!\sigma^{4}}\,\big{(}J_{i_{1}\cdots
i_{q}}^{(0)}\big{)}^{2}\right)\,.$ (2.12)

In the total disorder-averaging limit, we recover the ordinary SYK results:
$\big{\langle}J_{i_{1}\cdots i_{q}}\big{\rangle}_{J}^{\sigma=\infty}=0$ and
$\big{\langle}J_{i_{1}\cdots
i_{q}}^{2}\big{\rangle}_{J}^{\sigma=\infty}=(q-1)!J^{2}/N^{q-1}$, while in the
totally fixed-coupling limit, we find $\big{\langle}J_{i_{1}\cdots
i_{q}}\big{\rangle}_{J}^{\sigma=0}=J^{(0)}_{i_{1}\cdots i_{q}}$ and
$\big{\langle}J_{i_{1}\cdots
i_{q}}^{2}\big{\rangle}_{J}^{\sigma=0}=\big{(}J^{(0)}_{i_{1}\cdots
i_{q}}\big{)}^{2}$.

##  3 Partially Disorder-Averaged Partition Function

Having specified the model of our interest, in this section we study the
partially disorder-averaged partition function of this model in the large $N$
limit, by introducing collective fields. We see that to study the partially
disordered model with finite $\sigma$, in addition to ordinary bi-local fields
$\\{G(\tau_{1},\tau_{2}),\Sigma(\tau_{1},\tau_{2})\\}$ in the totally averaged
theory at $\sigma=\infty$, it is convenient to introduce an additional set of
collective fields denoted by $\\{G_{\sigma}(\tau),\Sigma_{\sigma}(\tau)\\}$.
We will explain a possible bulk picture of these local collective fields in
section 3.4 and for the two-point function we study in section 4, they
eventually correspond to the degrees of freedom of “half-wormhole”.

The original partition function with fixed coupling $J_{i_{1}\cdots i_{q}}$ is
given by

$\displaystyle Z\,$ $\displaystyle=\,\int D\chi_{i}\,e^{-S}$
$\displaystyle=\,\int D\chi_{i}\,\exp\left[-\frac{1}{2}\int
d\tau\sum_{i=1}^{N}\chi_{i}\partial_{\tau}\chi_{i}\,+\,i^{\frac{q}{2}}\int
d\tau\sum_{i_{1}<\cdots<i_{q}}^{N}J_{i_{1}\cdots
i_{q}}\,\chi_{i_{1}}\cdots\chi_{i_{q}}\right]\,.$ (3.1)

Taking the partial disorder averaging of this partition function with respect
to the probability distribution (2.4), we obtain

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,=\,\int D\chi_{i}\,e^{-S_{\rm
eff}[\chi]}\,,$ (3.2)

where

$\displaystyle S_{\rm eff}[\chi]\,$ $\displaystyle=\,\frac{1}{2}\int
d\tau\sum_{i=1}^{N}\chi_{i}\partial_{\tau}\chi_{i}\,-\,\frac{\widetilde{J}^{2}}{2qN^{q-1}}\int
d\tau_{1}d\tau_{2}\left(\sum_{i=1}^{N}\chi_{i}(\tau_{1})\chi_{i}(\tau_{2})\right)^{q}$
$\displaystyle\quad-\,\frac{i^{\frac{q}{2}}\widetilde{J}^{2}}{\sigma^{2}}\int
d\tau\sum_{i_{1}<\cdots<i_{q}}^{N}J_{i_{1}\cdots
i_{q}}^{(0)}\,\chi_{i_{1}}\cdots\chi_{i_{q}}\,.$ (3.3)

The normalization factor $\mathcal{N}_{\sigma}$ is canceled out by the
Gaussian integral of the coupling. We emphasize that even though we integrated
$J_{i_{1}\cdots i_{q}}$ to obtain $\langle Z\rangle_{J}$, this still can be
the totally fixed-coupling partition function in the $\sigma\to 0$ limit.
Indeed, we can see that in the totally fixed-coupling limit ($\sigma\to 0$),
the second term vanishes, while the last term survives because
$\widetilde{J}^{2}/\sigma^{2}\to 1$. The result is nothing but the totally
fixed-coupling action (2.3). On the other hand, in the totally disorder-
averaged limit ($\sigma\to\infty$) the last term vanishes and the remaining
first two terms are the usual disorder-averaged SYK effective action.

We also note that besides the last term, the original SYK effective action has
the global $O(N)$ gauge symmetry: $\chi_{i}\to\sum_{j}O_{ij}\chi_{j}$.
However, this global $O(N)$ symmetry is now broken by the last term by fixing
the coupling constant $J_{i_{1}\cdots i_{q}}^{(0)}$. In addition, this term
contains a factor $i^{q/2}$, so in order to keep the action real, it looks
like we need a complex $J_{i_{1}\cdots i_{q}}^{(0)}$, such that the entire
last term is still real. This type of complex coupling is similar to the one
discussed in [31, 46], but the precise relation is unclear.

Fixed-couplings as an external source
Here we will explain how to introduce collective fields in our partially
disorder-averaged SYK model. For the first line in (3.3), we introduce a set
of bi-local fields $(G_{LR},\Sigma_{LR})$ as usual. To introduce a new set of
collective fields for the second line in (3.3), we will give the following
interpretation of the external fixed couplings.

Let us now assume a specific form of the external coupling $J_{i_{1}\cdots
i_{q}}^{(0)}$. We choose it to be proportional to the totally anti-symmetric
tensor $\varepsilon_{{i_{1},\cdots i_{q}}}$ for $i_{k}=1,\cdots,N$, with
$\varepsilon_{1,2,\cdots,q}=1$, $\varepsilon_{2,3,\cdots,q+1}=1$, e.t.c. From
now on, we interpret the second term in (3.3) as the couplings between the
fermions $\chi_{i}$ in the SYK model and the external fermionic sources
$\theta_{i}$ by decomposing the anti-symmetric tensor
$\varepsilon_{{i_{1},\cdots i_{q}}}$ into the products of $\theta_{i}$,

$\displaystyle\qquad J_{i_{1}\cdots
i_{q}}^{(0)}\,=\,\frac{(q-1)!J_{0}}{i^{\frac{q}{2}}N^{q-1}}\,\theta_{i_{1}}\cdots\theta_{i_{q}}\,,\qquad\quad{\rm
with}\qquad\\{\theta_{i},\theta_{j}\\}\,=\,\delta_{ij}\,,$ (3.4)

where the coefficient was chosen for later convenience. 111Here we introduced
a non-vanishing square of the variables $\theta_{i}$ in order to keep the
second term of $\langle J_{i_{1}\cdots i_{q}}^{2}\rangle_{J}$ in (2.12) non-
zero (or to keep non-vanishing $\langle J_{i_{1}\cdots
i_{q}}^{2}\rangle_{J}^{\sigma=0}$). If we regard $\theta_{i}$ as external
Grassmann variables, it is natural to have $\\{\theta_{i},\theta_{j}\\}=0$,
but we can think that the external coupling originally had a slight time-
dependence, and we smeared over that time-dependence, so we have
$\theta_{i}^{2}=1/2$. A similar argument was also given in footnote 9 of [24].

This variable $\theta_{i}$ can be interpreted as non-dynamical fermions living
in another universe. We discuss this interpretation in Appendix A. After using
this decomposition, we have the effective action

$\displaystyle S_{\rm eff}[\chi]\,$ $\displaystyle=\,\frac{1}{2}\int
d\tau\sum_{i=1}^{N}\chi_{i}\partial_{\tau}\chi_{i}\,-\,\frac{\widetilde{J}^{2}}{2qN^{q-1}}\int
d\tau_{1}d\tau_{2}\left(\sum_{i=1}^{N}\chi_{i}(\tau_{1})\chi_{i}(\tau_{2})\right)^{q}$
$\displaystyle\quad-\,\frac{J_{\sigma}}{qN^{q-1}}\int
d\tau\left(\sum_{i=1}^{N}\theta_{i}\chi_{i}(\tau)\right)^{q}\,,$ (3.5)

with $J_{\sigma}$ defined in (2.6). Now the term depending on the fixed
couplings is rewritten as a local term similar to the usual bi-local term in
the first line, which enables us to introduce the collective fields to this
term. Now we are considering the SYK model coupled to the external fields
$\theta_{i}$. We will analyze this model instead of the original partially
disorder-averaged SYK model. After all computations, we replace all the
dependence of $\theta_{i}$ by the original fixed-couplings $J_{i_{1}\cdots
i_{q}}^{(0)}$, then we expect to recover the result in the original partially
disorder-averaged SYK model.

Now it is useful to employ the following Hubbard–Stratonovich trick:

$\displaystyle 1\,$ $\displaystyle=\,\int DG\int
DG_{\sigma}\,\delta\left(G-\frac{1}{N}\sum_{i=1}^{N}\chi_{i}\chi_{i}\right)\delta\left(G_{\sigma}-\frac{1}{N}\sum_{i=1}^{N}\theta_{i}\chi_{i}\right)$
(3.6) $\displaystyle=\,\int DGD\Sigma
DG_{\sigma}D\Sigma_{\sigma}\exp\left[\frac{1}{2}\textrm{Tr}\,\Sigma\cdot\left(NG-\sum_{i=1}^{N}\chi_{i}\chi_{i}\right)-\textrm{Tr}\,\Sigma_{\sigma}\cdot\left(NG_{\sigma}-\sum_{i=1}^{N}\theta_{i}\chi_{i}\right)\right]\,,$

We insert the above identity to the partially disorder-averaged partition
function (3.2) with $S_{\rm eff}[\chi]$ given by (3.5). Rewriting all terms in
$S_{\rm eff}[\chi]$ (3.5) by the bi-local $G$ and the local $G_{\sigma}$
fields, we can perform the fermion integrals (see Appendix B for detail) and
obtain

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,=\,\mathcal{N}_{\sigma}^{-1}\int
DGD\Sigma DG_{\sigma}D\Sigma_{\sigma}\,e^{-S_{\rm
eff}[G,\Sigma,G_{\sigma},\Sigma_{\sigma}]}\,$ (3.7)

with

$\displaystyle S_{\rm
eff}[G,\Sigma,G_{\sigma},\Sigma_{\sigma}]\,=\,-\frac{N}{2}\textrm{Tr}\log(\Sigma)-\frac{N}{2}\int
d\tau\big{[}\partial_{\tau}G(\tau,\tau^{\prime})\big{]}_{\tau^{\prime}=\tau}$
$\displaystyle\qquad\qquad\qquad+\frac{N}{2}\int
d\tau_{1}d\tau_{2}\left(\Sigma(\tau_{1},\tau_{2})G(\tau_{1},\tau_{2})-\frac{\widetilde{J}^{2}}{q}G(\tau_{1},\tau_{2})^{q}\right)$
(3.8) $\displaystyle\qquad+N\int
d\tau\left(\Sigma_{\sigma}(\tau)G_{\sigma}(\tau)-\frac{J_{\sigma}}{q}G_{\sigma}(\tau)^{q}\right)\,-\,\frac{N}{4}\int
d\tau_{1}d\tau_{2}\,\Sigma_{\sigma}(\tau_{1})G(\tau_{1},\tau_{2})\Sigma_{\sigma}(\tau_{2})\,.$

Before studying this effective action in detail, let us first consider the
meaning of the local field $G_{\sigma}$ we introduced here in addition to the
usual SYK bi-local fields. If we consider the combination of
$G_{\sigma}^{q}(\tau_{1})G_{\sigma}^{q}(\tau_{2})$, this is written as

$\displaystyle\Big{(}G_{\sigma}(\tau_{1})G_{\sigma}(\tau_{2})\Big{)}^{q}\,=\,\frac{i^{q}}{\big{(}(q-1)!\big{)}^{2}J_{0}^{2}N^{2}}\,$
$\displaystyle\sum_{i_{1},\cdots,i_{q}}^{N}J_{i_{1}\cdots
i_{q}}^{(0)}\,\chi_{i_{1}}(\tau_{1})\cdots\chi_{i_{q}}(\tau_{1})$
$\displaystyle\times\sum_{j_{1},\cdots,j_{q}}^{N}J_{j_{1}\cdots
j_{q}}^{(0)}\,\chi_{j_{1}}(\tau_{2})\cdots\chi_{j_{q}}(\tau_{2})\,,$ (3.9)

where we used the decomposition of $J_{i_{1}\cdots i_{q}}^{(0)}$ (3.4). If we
take the total disorder averaging of $J_{i_{1}\cdots i_{q}}^{(0)}$ for this
quantity with using the same form of probability distribution as (2.4) for
$J_{i_{1}\cdots i_{q}}^{(0)}$, we find

$\displaystyle\Big{\langle}G_{\sigma}^{q}(\tau_{1})G_{\sigma}^{q}(\tau_{2})\Big{\rangle}_{J_{0}}\,=\,\frac{i^{q}\widetilde{J}^{2}}{(q-1)!J_{0}^{2}N}\,G(\tau_{1},\tau_{2})^{q}\,.$
(3.10)

This relation implies that we can think of the local field $G_{\sigma}(\tau)$
as the “half” of the bi-local field $G(\tau_{1},\tau_{2})$. Furthermore, one
can regard the anti-symmetric variable $\theta_{i}$ as fermionic degrees of
freedom living in the surface at which the bulk spacetime ends. Since the new
variable $G_{\sigma}(\tau)$ is the propagator between the original SYK
fermions $\chi_{i}$ and the new ones $\theta_{i}$, this also suggests that
$G_{\sigma}(\tau)$ corresponds to the correlation between the boundary and the
interior of the bulk. These interpretations will be useful in the following
discussion.

Since all terms in the effective action (3.8) are proportional to $N$, we can
use large $N$ saddle-point evaluation. Variation of each field leads to the
following saddle-point equations

$\displaystyle\delta G\,:$
$\displaystyle\qquad\Sigma(\tau_{1},\tau_{2})\,=\,\widetilde{J}^{2}\,G(\tau_{1},\tau_{2})^{q-1}\,+\,\partial_{\tau_{1}}\delta(\tau_{1}-\tau_{2})\,+\,\frac{1}{2}\,\Sigma_{\sigma}(\tau_{1})\Sigma_{\sigma}(\tau_{2})\,,$
(3.11) $\displaystyle\delta\Sigma\,:$
$\displaystyle\qquad\Sigma^{-1}(\tau_{2},\tau_{1})\,=\,G(\tau_{1},\tau_{2})\,,$
(3.12) $\displaystyle\delta G_{\sigma}\,:$
$\displaystyle\qquad\Sigma_{\sigma}(\tau)\,=\,J_{\sigma}\,G_{\sigma}(\tau)^{q-1}\,,$
(3.13) $\displaystyle\delta\Sigma_{\sigma}\,:$ $\displaystyle\qquad
G_{\sigma}(\tau)\,=\,\frac{1}{2}\int
d\tau^{\prime}\,G(\tau,\tau^{\prime})\Sigma_{\sigma}(\tau^{\prime})\,,$ (3.14)

where the inverse is defined by

$\displaystyle\int
d\tau_{3}\,\Sigma(\tau_{1},\tau_{3})\Sigma^{-1}(\tau_{3},\tau_{2})\,=\,\delta(\tau_{1}-\tau_{2})\,.$
(3.15)

From (3.11), in order to keep the time translation symmetry, the only possible
solution for the local field $\Sigma_{\sigma}$ is a time-independent constant
solution. Such a constant term in (3.11) has the same structure as the
chemical potential contribution in the complex SYK model as we will discuss
more in detail in section 3.3.

###  3.1 Zero-dimensional SYK

Before studying this model in detail, let us first consider the zero-
dimensional case discussed in [33, 44]. This model is not dynamical, but it
might capture some important topological contributions. The treatment of this
case is slightly different from the one-dimensional case we discussed above.
This is because in the effective action (3.3), the second term identically
vanishes in this case, and of course, the kinetic term also does not exist:

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,=\,\int
d\chi_{i}\,\exp\left[\frac{J_{\sigma}}{qN^{q-1}}\bigg{(}\sum_{i=1}^{N}\theta_{i}\chi_{i}\bigg{)}^{q}\right]\,.$
(3.16)

Therefore, in this case, we cannot introduce the bi-local fields, which are
identically zero, so at most, we can introduce the local fields only:

$\displaystyle
1\,=\,\int_{-\infty}^{\infty}dG_{\sigma}\int_{-i\infty}^{i\infty}\frac{d\Sigma_{\sigma}}{2\pi
i/N}\,\exp\bigg{[}\Sigma_{\sigma}\Big{(}NG_{\sigma}-\sum_{i=1}^{N}\theta_{i}\chi_{i}\Big{)}\bigg{]}\,.$
(3.17)

Hence, in this case, after integrating out the fermions (see Appendix B for
detail) we have

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,=\,\frac{N}{2\pi
i}\,\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\int
dG_{\sigma}d\Sigma_{\sigma}\,e^{-S_{\rm eff}[G_{\sigma},\Sigma_{\sigma}]}\,,$
(3.18)

with

$\displaystyle S_{\rm
eff}[G_{\sigma},\Sigma_{\sigma}]\,=\,-\,N\log(-\Sigma_{\sigma})\,-\,N\left(\Sigma_{\sigma}G_{\sigma}+\frac{J_{\sigma}}{q}G_{\sigma}^{q}\right)\,,$
(3.19)

where the $\log(-\Sigma_{\sigma})$ term comes from expanding the linear in the
fermion term in the Hubbard–Stratonovich trick (3.17).

Now we can simply integrate out both fields as

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,$
$\displaystyle=\,\frac{N}{2\pi
i}\,\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\int
dG_{\sigma}d\Sigma_{\sigma}\,\left(-N^{-1}\partial_{G_{\sigma}}\right)^{N}e^{N\left(\Sigma_{\sigma}G_{\sigma}+\frac{J_{\sigma}}{q}G_{\sigma}^{q}\right)}$
$\displaystyle=\,\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\int
dG_{\sigma}\,e^{\frac{NJ_{\sigma}}{q}G_{\sigma}^{q}}\left(-N^{-1}\partial_{G_{\sigma}}\right)^{N}\,\delta(G_{\sigma})$
$\displaystyle=\,\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\left(\frac{J_{\sigma}N}{q}\right)^{\frac{N}{q}}\frac{N!}{N^{N}(N/q)!}\,,$
(3.20)

where for the last line, we assume $N$ is a multiple of $q$. For the total-
disorder-averaging limit ($\sigma\to\infty$), we have $J_{\sigma}\to 0$ (2.8).
Therefore, in this limit we have $\langle Z\rangle_{J}=0$. On the other hand,
for finite $\sigma>0$, we have a nontrivial partition function, which is
proportional to the “hyperpfaffian” of the external coupling [44]:

$\displaystyle{\rm PF}\Big{(}J_{i_{1}\cdots i_{q}}^{(0)}\Big{)}\,$
$\displaystyle\equiv\,\sideset{}{{}^{\prime}}{\sum}_{A_{1}<\cdots<A_{p}}\textrm{sgn}(A)\,J_{A_{1}}^{(0)}\cdots
J_{A_{p}}^{(0)}$
$\displaystyle=\,\frac{N!}{p!(q!)^{p}}\,J_{1\,\cdots\,q}^{(0)}\cdots
J_{N-q\,\cdots\,N}^{(0)}$
$\displaystyle=\,\frac{N!}{p!N^{N}}\left(\frac{NJ_{0}}{q}\right)^{\frac{N}{q}}\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\,,$
(3.21)

where $p=N/q$ and $A$’s denote ordered $q$-subsets $A={i_{1}<\cdots<i_{q}}$.
For the third line, we used (3.4) and assumed $N\in 8\mathbb{Z}$ to eliminate
the factor coming from the $i^{q/2}$. Therefore, we find

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,=\,\bigg{(}\,\frac{\widetilde{J}}{\sigma}\,\bigg{)}^{\frac{2N}{q}}{\rm
PF}\Big{(}J_{i_{1}\cdots i_{q}}^{(0)}\Big{)}\,.$ (3.22)

Going back to the expression of (3.20), for the numerical coefficient, besides
the $J_{\sigma}$ dependence, we find in the large $N$ approximation that

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,\approx\,\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\big{(}J_{\sigma}\big{)}^{\frac{N}{q}}\sqrt{q}\,e^{-(1-\frac{1}{q})N}\,.$
(3.23)

We can also evaluate this partition function in the large $N$ saddle-point
approximation. From the effective action, we have the large $N$ saddle-point
equations

$\displaystyle\frac{1}{\Sigma_{\sigma}}\,=\,-\,G_{\sigma}\,,\qquad\Sigma_{\sigma}\,=\,-\,J_{\sigma}\,G_{\sigma}^{q-1}\,,$
(3.24)

and there are $q$ solutions:

$\displaystyle G_{\sigma}^{(s)}\,=\,e^{\frac{2m\pi
i}{q}}\,J_{\sigma}^{-\frac{1}{q}}\,,\qquad\Sigma_{\sigma}^{(s)}\,=\,-\,e^{-\frac{2m\pi
i}{q}}\,J_{\sigma}^{\ \frac{1}{q}}\,,$ (3.25)

where $m=1,2,\cdots,q$. Evaluating the on-shell action for these large $N$
saddle-point solutions, we find

$\displaystyle S_{\scriptsize\mbox{on-shell}}\,=\,-\,\frac{N}{q}\log
J_{\sigma}\,+\,N\left(1-\frac{1}{q}\right)\,+\,\frac{2m\pi iN}{q}\,,$ (3.26)

and

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,=\,\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\big{(}J_{\sigma}\big{)}^{\frac{N}{q}}\,\sum_{m=1}^{q}e^{-(1-\frac{1}{q})N-\frac{2m\pi
iN}{q}}\,.$ (3.27)

We obtained the same $J_{\sigma}$\- and $N$-dependence as the above exact
computation. The remaining sum can also be evaluated as

$\displaystyle\sum_{m=1}^{q}e^{-2mp\pi i}\,=\,\frac{1-e^{2pq\pi i}}{e^{2pq\pi
i}(1-e^{2p\pi i})}\,=\,q\,,$ (3.28)

where we put $p=N/q$ and for the first equality we assumed $p\neq\mathbb{Z}$,
but for the second equality we took $p\to\mathbb{Z}$ and evaluated by using
l’Hôpital’s rule. The $1/\sqrt{q}$ difference from the exact result (3.23) is
accounted by the one-loop contribution [33]. To evaluate this contribution, we
expand the fields around the saddle-point solutions:

$\displaystyle G_{\sigma}\,=\,G_{\sigma}^{(s)}\,+\,\frac{\delta
G_{\sigma}}{\sqrt{N}}\,,\qquad\Sigma_{\sigma}\,=\,\Sigma_{\sigma}^{(s)}\,+\,\frac{\delta\Sigma_{\sigma}}{\sqrt{N}}\,.$
(3.29)

Then, the quadratic action is written as

$\displaystyle S_{(2)}\,$
$\displaystyle=\,\frac{1}{2}\Big{[}J_{\sigma}^{-\frac{2}{q}}e^{\frac{4m\pi
i}{q}}\,\delta\Sigma_{\sigma}^{2}\,-\,2\delta\Sigma_{\sigma}\delta
G_{\sigma}\,-\,(q-1)J_{\sigma}^{\ \frac{2}{q}}e^{-\frac{4m\pi i}{q}}\,\delta
G_{\sigma}^{2}\Big{]}$
$\displaystyle=\,\frac{1}{2}\Big{[}\delta\tilde{\Sigma}_{\sigma}^{2}\,+\,2i\delta\tilde{\Sigma}_{\sigma}\delta\tilde{G}_{\sigma}\,+\,(q-1)\delta\tilde{G}_{\sigma}^{2}\Big{]}\,,$
(3.30)

where in the second line we change the variables by
$\tilde{G}_{\sigma}=iJ_{\sigma}^{1/q}e^{-\frac{2m\pi i}{q}}\delta G_{\sigma}$
and $\tilde{\Sigma}_{\sigma}=J_{\sigma}^{-1/q}e^{\frac{2m\pi
i}{q}}\delta\Sigma_{\sigma}$ to make the Gaussian integral convergent.
Therefore, the one-loop contribution is given by
$\big{\langle}Z\big{\rangle}_{J}^{\scriptsize\mbox{one-loop}}=1/\sqrt{q}$.
Finally, including this one-loop contribution, we find

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,=\,\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\big{(}J_{\sigma}\big{)}^{\frac{N}{q}}\sqrt{q}\,e^{-(1-\frac{1}{q})N}\,.$
(3.31)

This precisely agrees with (3.23).

###  3.2 Total disorder-averaging

Now let us come back to the one-dimensional model (3.8). In the totally
disorder-averaging limit ($\sigma\to\infty$), we have $J_{\sigma}\to 0$. This
means that the variation of $G_{\sigma}$ (3.13) sets $\Sigma_{\sigma}=0$.
Therefore, the remaining dynamics is described by the bi-local fields
$\\{G,\Sigma\\}$, which is nothing but the effective system of the ordinary
SYK model.

Combining (3.11) and (3.12), the Schwinger-Dyson equation for
$G(\tau_{1},\tau_{2})$ is now obtained as

$\displaystyle-\delta(\tau_{1}-\tau_{2})\,$ $\displaystyle=\,\int
d\tau_{3}\,G(\tau_{1},\tau_{3})\Sigma(\tau_{3},\tau_{2})$
$\displaystyle=\,\partial_{\tau_{1}}G(\tau_{1},\tau_{2})\,+\,J^{2}\int
d\tau_{3}\,G(\tau_{1},\tau_{3})G(\tau_{3},\tau_{2})^{q-1}\,.$ (3.32)

The free solution, which is obtained by setting $J=0$, is found as

$\displaystyle G_{\rm
free}(\tau_{1},\tau_{2})\,=\,\frac{\textrm{sgn}(\tau_{12})}{2}\,,$ (3.33)

where $\tau_{12}\equiv\tau_{1}-\tau_{2}$. On the other hand, the IR solution,
which is obtained by eliminating the kinetic term, is given by

$\displaystyle
G^{\textrm{IR}}_{\beta=\infty}(\tau_{1},\tau_{2})\,=\,\frac{b^{\Delta}}{J^{2\Delta}}\,\frac{\textrm{sgn}(\tau_{12})}{|\tau_{12}|^{2\Delta}}\,,$
(3.34)

for zero temperature case where $\Delta=1/q$ and

$\displaystyle
b^{\Delta}\,=\,\left(\frac{1-2\Delta}{2}\right)\,\tan(\pi\Delta)\,.$ (3.35)

This solution can be determined by using the scaling ansatz and the Fourier
transform

$\displaystyle\frac{\textrm{sgn}(\tau)}{|\tau|^{\alpha}}\,=\,c(\alpha)\int\frac{d\omega}{2\pi}\,e^{-i\omega\tau}\,|\omega|^{\alpha-1}\textrm{sgn}(\omega)\,,\qquad
c(\alpha)\,=\,i2^{1-\alpha}\sqrt{\pi}\,\frac{\Gamma(1-\frac{\alpha}{2})}{\Gamma(\frac{1}{2}+\frac{\alpha}{2})}\,.$
(3.36)

Now we realize that the IR effective action

$\displaystyle S_{\rm
IR}[G]\,=\,\frac{N}{2}\textrm{Tr}\log(G)-\frac{NJ^{2}}{2q}\int
d\tau_{1}d\tau_{2}\,G(\tau_{1},\tau_{2})^{q}\,,$ (3.37)

is invariant under the conformal transformation $\tau\to f(\tau)$ together
with the bi-local field transformation

$\displaystyle
G(\tau_{1},\tau_{2})\,\to\,G_{f}(\tau_{1},\tau_{2})\,\equiv\,|f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})|^{\Delta}\,G(f(\tau_{1}),f(\tau_{2}))\,.$
(3.38)

For the IR solution (3.34), a general transformation gives

$\displaystyle
G^{\textrm{IR}}_{f}(\tau_{1},\tau_{2})\,=\,\frac{b^{\Delta}}{J^{2\Delta}}\,\frac{|f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})|^{\Delta}}{|f(\tau_{1})-f(\tau_{2})|^{2\Delta}}\,\textrm{sgn}(\tau_{12})\,.$
(3.39)

In particular, the finite temperature solution can be obtained by this
transformation with $f(\tau)=(\pi/\beta)\tan(\pi\tau/\beta)$ as

$\displaystyle
G^{\textrm{IR}}_{\beta}(\tau_{1},\tau_{2})\,=\,\frac{b^{\Delta}}{J^{2\Delta}}\,\left|\frac{\pi}{\beta\sin\frac{\pi\tau_{12}}{\beta}}\right|^{2\Delta}\,\textrm{sgn}(\tau_{12})\,.$
(3.40)

###  3.3 Slightly fixed-coupling

In this subsection, we study the effect of slightly fixing the coupling
constant. For a slightly fixed coupling (which means large $\sigma$), we have

$\displaystyle\widetilde{J}^{2}\,$
$\displaystyle=\,\frac{J^{2}\sigma^{2}}{J^{2}+\sigma^{2}}\,=\,J^{2}\left[1\,-\,\frac{J^{2}}{\sigma^{2}}\,+\,\mathcal{O}(\sigma^{-4})\right]\,,$
(3.41) $\displaystyle J_{\sigma}\,$
$\displaystyle=\,\frac{J^{2}J_{0}}{J^{2}+\sigma^{2}}\,=\,\frac{J^{2}J_{0}}{\sigma^{2}}\,+\,\mathcal{O}(\sigma^{-4})\,.$
(3.42)

Let us first consider the free limit $T\gg J$, where $T$ is temperature. We
would like to keep small $J_{\sigma}$ corrections while totally neglect
$\widetilde{J}$ contribution. This means that we have

$\displaystyle J_{0}\,\sim\,\sigma\,\gg\,T\,\gg\,J\,.$ (3.43)

Therefore, we can only keep $J_{\sigma}$ corrections and ignore the terms
proportional to $\widetilde{J}^{2}$. In this limit, we start from the zero-th
order (in $J_{\sigma}$), where the free bi-local sector is given by
$\Sigma(\tau_{1},\tau_{2})\,=\,\partial_{\tau_{1}}\delta(\tau_{12})$ and
$\Sigma_{\sigma}=0$. To get the first order, inverting the equation (3.14), we
find

$\displaystyle\Sigma_{\sigma}(\tau)\,$ $\displaystyle=\,-2\int
d\tau^{\prime}\,\Sigma(\tau,\tau^{\prime})G_{\sigma}(\tau^{\prime})$
$\displaystyle=\,-2\,\partial_{\tau}G_{\sigma}(\tau)\,.$ (3.44)

Therefore, equating with (3.13), the differential equation for
$G_{\sigma}(\tau)$ is found as

$\displaystyle\partial_{\tau}G_{\sigma}(\tau)\,=\,-\,\frac{J_{\sigma}}{2}\,G_{\sigma}(\tau)^{q-1}\,,$
(3.45)

and the solution of this equation is

$\displaystyle G_{\sigma}(\tau)\,$
$\displaystyle=\,\left(\frac{2}{(q-2)(J_{\sigma}\tau+c)}\right)^{\frac{1}{q-2}}$
$\displaystyle=\,\left(\frac{2}{c(q-2)}\right)^{\frac{1}{q-2}}\left[1\,-\,\frac{J_{\sigma}\tau}{c(q-2)}\,+\,\mathcal{O}(J_{\sigma}^{2})\right]\,,$
(3.46)

where $c$ is an integration constant. From (3.44), we also find

$\displaystyle\Sigma_{\sigma}(\tau)\,$
$\displaystyle=\,-\,J_{\sigma}\left(\frac{2}{(q-2)(J_{\sigma}\tau+c)}\right)^{-\frac{q-1}{q-2}}$
$\displaystyle=\,-\,\left(\frac{2}{c(q-2)}\right)^{-\frac{q-1}{q-2}}\Big{[}J_{\sigma}\,+\,\mathcal{O}(J_{\sigma}^{2})\Big{]}\,.$
(3.47)

Focusing on the leading contribution (in $J_{\sigma}$) for this
$\Sigma_{\sigma}$, the last term in (3.11) gives a constant contribution for
$\Sigma(\tau_{1},\tau_{2})$. Fourier transforming the bi-local time to
frequency space, such a constant contribution gives a contribution
$\Sigma(\omega)=\mu\delta(\omega)\,+\,\cdots$, where the ellipsis denotes
contributions from other terms. Such a contribution localized at $\omega=0$ is
the same structure as a chemical potential of the complex SYK model 222Here we
emphasize that we are just talking about the structural similarity with the
complex SYK model, but we are not claiming the model becomes the actual
complex model. [52, 53, 54]. Hence, the saddle-point equation for
$G(\tau_{1},\tau_{2})$ becomes

$\displaystyle-\,\delta(\tau_{1}-\tau_{2})\,=\,\big{(}\partial_{\tau_{1}}-\mu\big{)}G(\tau_{1},\tau_{2})\,.$
(3.48)

This is nothing but Green’s equation for a free fermion with a chemical
potential $\mu$, and the solution is given by

$\displaystyle G_{\rm
free}(\tau_{1},\tau_{2})\,=\,-\,\frac{\textrm{sgn}(\tau_{12})\,e^{\mu|\tau_{12}|}}{1+e^{\,\textrm{sgn}(\tau_{12})\beta\mu}}\,,$
(3.49)

where $\beta$ is the inverse temperature.

We suppose the above chemical potential contribution also influences on the IR
region as in the complex SYK model. For the IR region, we stay at the strong
coupling limit $J\gg T$. On top of that, we would like to keep the leading
term of $J_{\sigma}$ as a small parameter and neglect the sub-leading
contribution of $\widetilde{J}$. All this means that in this subsection we
study the regime of

$\displaystyle\sigma\,\gg\,J_{0}\,\sim\,J\,\gg\,T\,.$ (3.50)

Including the potential contribution, the saddle-point equation for
$G(\tau_{1},\tau_{2})$ can be written as

$\displaystyle-\delta(\tau_{1}-\tau_{2})\,=\,\big{(}\partial_{\tau_{1}}-\mu\big{)}G(\tau_{1},\tau_{2})\,+\,J^{2}\int
d\tau_{3}\,G(\tau_{1},\tau_{3})G(\tau_{3},\tau_{2})^{q-1}\,.$ (3.51)

This equation is the saddle-point equation of the complex SYK model with a
chemical potential $\mu$. We will show that the same structure also appears in
another formalism of the slightly fixed coupling SYK model in Appendix C. It
is very interesting that even though we started with Majorana fermions, after
slightly fixing the random couplings, we obtain the induced chemical
potential. This might be related to the branes discussed in [18, 43, 48] in
order to fix the boundary theory.

In the IR limit, the kinetic term and the chemical potential do not contribute
(since they are delta functional local in $\Sigma(\tau_{1},\tau_{2})$ [52,
54]) and the saddle-point solution is given by

$\displaystyle
G^{\textrm{IR}}_{\beta=\infty}(\tau_{1},\tau_{2})\,=\,-\,\textrm{sgn}(\tau_{12})\,\frac{b^{\Delta}\,e^{\textrm{sgn}(\tau_{12})\pi\mathcal{E}}}{|\tau_{12}|^{2\Delta}}\,,$
(3.52)

at zero temperature. In this case, the coefficient is defined by

$\displaystyle
b\,=\,\left(\frac{1-2\Delta}{4\pi}\right)\,\frac{\sin(2\pi\Delta)}{\cos\pi(\Delta+i\mathcal{E})\cos\pi(\Delta-i\mathcal{E})}\,,$
(3.53)

and the “spectral asymmetry” $\mathcal{E}$ is related to the chemical
potential and it cannot be determined in the IR limit, but it requires
interpolation to the UV solution [52, 53, 54].

###  3.4 Bulk interpretation of $G_{\sigma}$ and $\Sigma_{\sigma}$

In this subsection, we consider a bulk interpretation of the local fields
$\\{G_{\sigma},\Sigma_{\sigma}\\}$. Before considering these fields, let us
first remind ourselves about the scalar field coupled to JT gravity in the
(near) AdS2 [14, 15, 16, 17]. The bulk scalar field is described by the action

$\displaystyle S[\chi]\,=\,\frac{1}{2}\int
d^{2}x\sqrt{g}\,\Big{(}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi\,+\,m^{2}\chi^{2}\Big{)}\,,$
(3.54)

on the Poincare coordinates

$\displaystyle ds^{2}\,=\,\frac{d\tau^{2}+dz^{2}}{z^{2}}\,.$ (3.55)

The boundary action is given by

$\displaystyle S[\chi]\,=\,-\,\frac{1}{2}\int d\tau\,\chi\partial_{z}\chi\,.$
(3.56)

The bulk field is now expressed in terms of the bulk-to-boundary propagator

$\displaystyle
K_{\Delta}(\tau,z;\tau^{\prime})\,=\,C_{\Delta}\left(\frac{z}{z^{2}+(\tau-\tau^{\prime})^{2}}\right)^{\Delta}\,,$
(3.57) 333where
$C_{\Delta}=\frac{\Gamma(\Delta)}{\pi^{\frac{1}{2}}\Gamma(\Delta-\frac{1}{2})}$,

with $\Delta=1/2+\sqrt{m^{2}+1/4}$, as

$\displaystyle\chi(\tau,z)\,=\,\int
d\tau^{\prime}\,K_{\Delta}(\tau,z;\tau^{\prime})j(\tau^{\prime})\,,$ (3.58)

where the boundary source is defined by

$\displaystyle j(\tau)\,=\,\lim_{z\to 0}z^{\Delta-1}\chi(\tau,z)\,.$ (3.59)

Using this expression and evaluating the boundary action (3.56), one finds

$\displaystyle S[j]\,=\,-\,(\Delta-\tfrac{1}{2})C_{\Delta}\int
d\tau_{1}d\tau_{2}\,\frac{j(\tau_{1})j(\tau_{2})}{|\tau_{1}-\tau_{2}|^{2\Delta}}\,.$
(3.60)

If we consider the conformal transformation of the boundary $\tau\to f(\tau)$,
we also have [14, 15, 16, 17]

$\displaystyle S[j]\,=\,-\,(\Delta-\tfrac{1}{2})C_{\Delta}\int
d\tau_{1}d\tau_{2}\,\frac{|f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})|^{\Delta}}{|f(\tau_{1})-f(\tau_{2})|^{2\Delta}}\,j(\tau_{1})j(\tau_{2})\,.$
(3.61)

Now we will see that the same effective action is obtained for the fluctuation
of the $\Sigma_{\sigma}$ field in our partially disorder-averaged SYK model.
444The discussion below is rather formal in that the mass of a bulk scalar
field with $\Delta=1/q$ seems to violate the BF bound. To see this, we
consider $1/N$ fluctuations of the fields around the totally disorder-averaged
IR saddle-point solutions:

$\displaystyle G(\tau_{1},\tau_{2})\,$
$\displaystyle=\,G_{\beta=\infty}^{\textrm{IR}}(\tau_{1},\tau_{2})\,+\,\frac{1}{\sqrt{N}}\,\delta
G(\tau_{1},\tau_{2})\,,$ $\displaystyle\Sigma(\tau_{1},\tau_{2})\,$
$\displaystyle=\,\Sigma_{\beta=\infty}^{\textrm{IR}}(\tau_{1},\tau_{2})\,+\,\frac{1}{\sqrt{N}}\,\delta\Sigma(\tau_{1},\tau_{2})\,,$
$\displaystyle G_{\sigma}(\tau)\,$
$\displaystyle=\,G_{\sigma}^{\textrm{IR}}(\tau)\,+\,\frac{1}{\sqrt{N}}\,\delta
G_{\sigma}(\tau)\,,$ $\displaystyle\Sigma_{\sigma}(\tau)\,$
$\displaystyle=\,\Sigma_{\sigma}^{\textrm{IR}}(\tau)\,+\,\frac{1}{\sqrt{N}}\,\delta\Sigma_{\sigma}(\tau)\,,$
(3.62)

where $G_{\sigma}^{\textrm{IR}}=\Sigma_{\sigma}^{\textrm{IR}}=0$. Substituting
these expansions into the action (3.8), this generates several interactions
among the fluctuations and the background. The term we are interested in right
now is the interaction between the background of the bi-local fields and the
fluctuations of the local fields. This term is given by

$\displaystyle S_{\rm
int}\big{[}G_{\sigma}^{\textrm{IR}},\delta\Sigma_{\sigma}\big{]}\,=\,-\,\frac{1}{4}\int
d\tau_{1}d\tau_{2}\,\delta\Sigma_{\sigma}(\tau_{1})G_{\beta=\infty}^{\textrm{IR}}(\tau_{1},\tau_{2})\delta\Sigma_{\sigma}(\tau_{2})\,.$
(3.63)

Using the explicit form of the background solution (3.34), we find

$\displaystyle S_{\rm
int}\big{[}G_{\sigma}^{\textrm{IR}},\delta\Sigma_{\sigma}\big{]}\,=\,-\,\frac{b^{\Delta}}{4J^{2\Delta}}\int
d\tau_{1}d\tau_{2}\,\frac{\delta\Sigma_{\sigma}(\tau_{1})\delta\Sigma_{\sigma}(\tau_{2})}{|\tau_{1}-\tau_{2}|^{2\Delta}}\,.$
(3.64)

The leading interaction term between the Schwarzian mode and the local field
$\delta\Sigma_{\sigma}$ is given by simply transforming $G^{\textrm{IR}}\to
G^{\textrm{IR}}_{f}$ (3.39) as

$\displaystyle S_{\rm
int}[f,\delta\Sigma_{\sigma}]\,=\,-\,\frac{b^{\Delta}}{4J^{2\Delta}}\int
d\tau_{1}d\tau_{2}\,\frac{|f^{\prime}(\tau_{1})f^{\prime}(\tau_{2})|^{\Delta}}{|f(\tau_{1})-f(\tau_{2})|^{2\Delta}}\,\delta\Sigma_{\sigma}(\tau_{1})\delta\Sigma_{\sigma}(\tau_{2})\,.$
(3.65)

This form of the boundary action precisely agrees with (3.61). This implies
that the bulk interpretation of the local field $\delta\Sigma_{\sigma}$ is a
boundary source of extra bulk fields. Then, it is natural to interpret the
dual field $\delta G_{\sigma}$ as the expectation value of the corresponding
boundary operator.

###  3.5 Comments on general $\sigma$ case

In this subsection, we make some comments on the general $\sigma$ case. As we
mentioned below (3.15), the time translation symmetry of
$\Sigma(\tau_{1},\tau_{2})$ requires from (3.11) that the only possible
solution for the local field $\Sigma_{\sigma}$ is a time-independent constant
solution. Then, as we explained in section 3.3, such a constant term in (3.11)
has the same structure as the chemical potential contribution in the complex
SYK model, and the solution of $G(\tau_{1},\tau_{2})$ is given by (3.49) for
free case or by (3.52) for the IR case.

Given these facts, we can evaluate the integral in (3.14) for example for
$G_{\rm free}$ as

$\displaystyle\int_{0}^{\beta}d\tau^{\prime}\,G_{\rm
free}(\tau,\tau^{\prime})\,=\,\frac{1-e^{\mu\tau}}{\mu(1+e^{\beta\mu})}\,+\,\frac{e^{\mu(\beta-\tau)}-1}{\mu(1+e^{-\beta\mu})}\,.$
(3.66)

This seems to indicate that we can still have a nontrivial time-dependent
solution for $G_{\sigma}(\tau)$ even though we should have a time-independent
constant solution for $\Sigma_{\sigma}$. At the same time, we also have to
make the solutions consistent with the other equation (3.13). This seems quite
nontrivial for the general $\sigma$ case. It would be very interesting to
study this general $\sigma$ case more in detail, but we leave this to future
work.

##  4 Two-point Function $\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}$

In this section, we study the partially disorder-averaged two-point partition
functions in the large $N$ limit. Before the partial disorder averaging, we
have the two-point function of the partition functions

$\displaystyle Z_{L}Z_{R}\,=\,\int D\chi_{i}^{L}D\chi_{i}^{R}\,\exp\bigg{[}$
$\displaystyle-\frac{1}{2}\sum_{a=L,R}\int
d\tau\,\sum_{i=1}^{N}\chi_{i}^{a}\partial_{\tau}\chi_{i}^{a}$
$\displaystyle\qquad+\,i^{\frac{q}{2}}\sum_{a=L,R}\int
d\tau\,\sum_{i_{1}<\cdots<i_{q}}^{N}J_{i_{1}\cdots
i_{q}}\,\chi_{i_{1}}^{a}\cdots\chi_{i_{q}}^{a}\bigg{]}\,.$ (4.1)

Using the duplicated Hubbard–Stratonovich transformation

$\displaystyle 1\,$
$\displaystyle=\,\int\prod_{a,b}^{L,R}DG_{ab}\,\delta\left(G_{ab}-\frac{1}{N}\sum_{i=1}^{N}\chi_{i}^{a}\chi_{i}^{b}\right)\int\prod_{a}^{L,R}DG_{\sigma}^{a}\,\delta\left(G_{\sigma}^{a}-\frac{1}{N}\sum_{i=1}^{N}\theta_{i}\chi_{i}^{a}\right)$
$\displaystyle=\,\int\prod_{a,b}^{L,R}DG_{ab}D\Sigma_{ab}\exp\left[-\frac{1}{2}\textrm{Tr}\,\Sigma_{ab}\cdot\left(NG_{ab}-\sum_{i=1}^{N}\chi_{i}^{a}\chi_{i}^{b}\right)\right]$
$\displaystyle\qquad\quad\times\int\prod_{a}^{L,R}DG_{\sigma}^{a}D\Sigma_{\sigma}^{a}\exp\left[-\textrm{Tr}\,\Sigma_{\sigma}^{a}\cdot\left(NG_{\sigma}^{a}-\sum_{i=1}^{N}\theta_{i}\chi_{i}^{a}\right)\right]\,,$
(4.2)

we now find

$\displaystyle\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}\,=\,\int\prod_{a,b}^{L,R}DG_{ab}D\Sigma_{ab}\int\prod_{a}^{L,R}DG_{\sigma}^{a}D\Sigma_{\sigma}^{a}\
e^{-S_{\rm eff}[G_{ab},\Sigma_{ab},G_{\sigma}^{a},\Sigma_{\sigma}^{a}]}\,,$
(4.3)

with

$\displaystyle S_{\rm
eff}[G_{ab},\Sigma_{ab},G_{\sigma}^{a},\Sigma_{\sigma}^{a}]\,$
$\displaystyle=\,-\frac{N}{2}\sum_{a,b}\textrm{Tr}\log(\Sigma_{ab})-\frac{N}{2}\sum_{a}\int
d\tau\big{[}\partial_{\tau}G_{aa}(\tau,\tau^{\prime})\big{]}_{\tau^{\prime}=\tau}$
$\displaystyle\quad+\frac{N}{2}\sum_{a,b}\int
d\tau_{1}d\tau_{2}\left(\Sigma_{ab}(\tau_{1},\tau_{2})G_{ab}(\tau_{1},\tau_{2})-\frac{\widetilde{J}^{2}}{q}G_{ab}(\tau_{1},\tau_{2})^{q}\right)$
$\displaystyle\quad+N\sum_{a}\int
d\tau\left(\Sigma_{\sigma}^{a}(\tau)G_{\sigma}^{a}(\tau)-\frac{J_{\sigma}}{q}G_{\sigma}^{a}(\tau)^{q}\right)$
$\displaystyle\quad-\,\frac{N}{4}\sum_{a,b}\int
d\tau_{1}d\tau_{2}\,\Sigma_{\sigma}^{a}(\tau_{1})G_{ab}(\tau_{1},\tau_{2})\Sigma_{\sigma}^{b}(\tau_{2})\,.$
(4.4)

Similar to (3.11) - (3.14), the large $N$ saddle-point equations are

$\displaystyle\delta G_{ab}\,:$
$\displaystyle\qquad\Sigma_{ab}(\tau_{1},\tau_{2})\,=\,\widetilde{J}^{2}\,G_{ab}(\tau_{1},\tau_{2})^{q-1}+\delta_{ab}\,\partial_{\tau_{1}}\delta(\tau_{1}-\tau_{2})\,+\,\frac{1}{2}\Sigma_{\sigma}^{a}(\tau_{1})\Sigma_{\sigma}^{b}(\tau_{2})\,,$
$\displaystyle\delta\Sigma_{ab}\,:$
$\displaystyle\qquad\Sigma_{ab}^{-1}(\tau_{1},\tau_{2})\,=\,G_{ab}(\tau_{1},\tau_{2})\,,$
$\displaystyle\delta G_{\sigma}^{a}\,:$
$\displaystyle\qquad\Sigma_{\sigma}^{a}(\tau)\,=\,J_{\sigma}\,G_{\sigma}^{a}(\tau)^{q-1}\,,$
$\displaystyle\delta\Sigma_{\sigma}^{a}\,:$ $\displaystyle\qquad
G_{\sigma}^{a}(\tau)\,=\,\frac{1}{4}\sum_{b}\int
d\tau^{\prime}\,G_{ab}(\tau,\tau^{\prime})\Sigma_{\sigma}^{b}(\tau^{\prime})\,.$
(4.5)

###  4.1 Zero-dimensional SYK

####  4.1.1 Saddle-point solutions

The large $N$ saddle-point equations are a little complicated to solve, so let
us first consider the zero-dimensional case again. In this case, we only have
$\\{G_{LR},\Sigma_{LR}\\}$ for the bi-local fields. For the following
discussion, it is more convenient to combine the two couplings into one
effective coupling that appears in the interaction term between the local and
bi-local sectors. This can be implemented by rescaling the
Hubbard–Stratonovich fields as $G_{LR}\to\widetilde{J}^{-2/q}G_{LR}$,
$\Sigma_{LR}\to\widetilde{J}^{2/q}\Sigma_{LR}$, $G_{\sigma}\to
J_{\sigma}^{-1/q}G_{\sigma}$ and $\Sigma_{\sigma}\to
J_{\sigma}^{1/q}\Sigma_{\sigma}$. After all of these rescalings, the effective
action of this case reads

$\displaystyle S_{\rm
eff}[G_{LR},\Sigma_{LR},G_{\sigma}^{a},\Sigma_{\sigma}^{a}]\,$
$\displaystyle=\,-N\log(\Sigma_{LR})\,+\,N\left(\Sigma_{LR}G_{LR}-\frac{1}{q}\,G_{LR}^{q}\right)$
$\displaystyle\quad+N\sum_{a}\left(\Sigma_{\sigma}^{a}G_{\sigma}^{a}-\frac{1}{q}\big{(}G_{\sigma}^{a}\big{)}^{q}\right)\,-\,N\lambda_{\sigma}G_{LR}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\,,$
(4.6)

where we introduced the coupling between the local and bi-local sectors by

$\displaystyle\lambda_{\sigma}\,\equiv\,\frac{1}{2}\left(\frac{J_{\sigma}}{\widetilde{J}}\right)^{\frac{2}{q}}\,.$
(4.7)

This coupling behaves $\lambda_{\sigma}\sim\frac{1}{2}(J_{0}/\sigma)^{2/q}$ as
$\sigma\rightarrow 0$ i.e., in the totally fixed-coupling limit and
$\lambda_{\sigma}\sim\frac{1}{2}(JJ_{0}/\sigma^{2})^{2/q}$ as
$\sigma\rightarrow\infty$ i.e., in the total disorder-averaged limit. Here
from the action, we excluded the constant contribution
$-\frac{2N}{q}\log\widetilde{J}$ coming from the rescaling of $\Sigma_{LR}$,
so that the two-point function of the partition functions is now written as

$\displaystyle\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}\,=\,\widetilde{J}^{\,\frac{2N}{q}}\int
dG_{LR}d\Sigma_{LR}\int\prod_{a}^{L,R}dG_{\sigma}^{a}d\Sigma_{\sigma}^{a}\
e^{-S_{\rm eff}[G_{LR},\Sigma_{LR},G_{\sigma}^{a},\Sigma_{\sigma}^{a}]}\,.$
(4.8)

Now the saddle-point equations are written as

$\displaystyle\Sigma_{LR}\,=\,\big{(}G_{LR}\big{)}^{q-1}+\,\lambda_{\sigma}\,\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\,,\qquad\frac{1}{\Sigma_{LR}}\,=\,G_{LR}\,,$
(4.9)
$\displaystyle\Sigma_{\sigma}^{a}\,=\,\,\,\big{(}G_{\sigma}^{a}\big{)}^{q-1}\,,\qquad
G_{\sigma}^{L}\,=\,\lambda_{\sigma}\,G_{LR}\Sigma_{\sigma}^{R}\,,\qquad
G_{\sigma}^{R}\,=\,\lambda_{\sigma}\,G_{LR}\Sigma_{\sigma}^{L}\,.$ (4.10)

Let us first solve for $\\{G_{\sigma},\Sigma_{\sigma}\\}$ in terms of
$G_{LR}$. From (4.10), we can find one trivial solution and $q(q-2)$
nontrivial solutions 555These $q(q-2)$ solutions degenerate into $q$
independent solutions after plugging the solutions of $G_{LR}$.

$\displaystyle\big{\\{}G_{\sigma}^{L}\,,G_{\sigma}^{R}\big{\\}}\,=\,\\{0,0\\}\,,\quad\left\\{e^{\frac{2n\pi
i}{q(q-2)}}\big{(}\lambda_{\sigma}G_{LR}\big{)}^{-\frac{1}{q-2}},\,e^{\frac{(q-1)2n\pi
i}{q(q-2)}}\big{(}\lambda_{\sigma}G_{LR}\big{)}^{-\frac{1}{q-2}}\right\\}\,,$
(4.11)

with $n=1,2,\cdots,q(q-2)$. The corresponding solutions of
$\Sigma_{\sigma}^{a}$ are given by the first equation of (4.10). The
nontrivial solutions are valid for $q>2$. In the rest of this subsection, we
focus on $q>2$ case, and we will present a discussion for $q=2$ in Appendix D.
We will call the first trivial solution
$\\{G_{\sigma}^{L}\,,G_{\sigma}^{R}\\}=\\{0,0\\}$ “wormhole” saddle-point
solution, while the other $q(q-2)$ nontrivial solutions are analogous to the
“half-wormhole” saddle-point solutions. 666Even though in the main text, we
often call these nontrivial solutions “half-wormhole” solutions, we emphasize
that these solutions are analogous to the half-wormhole saddle-point solutions
found in [33] and discussed further in [44]. There are some differences
between our nontrivial solutions and their half-wormhole saddle-point
solutions, as we will discuss below and in section 6.  The reasoning for these
names will be explained below.

Figure 1: The wormhole saddles for $q=12$.

For the wormhole saddles $\\{G_{\sigma}^{L}\,,G_{\sigma}^{R}\\}=\\{0,0\\}$,
the remaining equations (4.9) are solved as

$\displaystyle\qquad\big{\\{}G_{LR}\,,\Sigma_{LR}\big{\\}}\,=\,\left\\{\,e^{\frac{2m\pi
i}{q}}\,,\,e^{-\frac{2m\pi i}{q}}\,\right\\}\,,\qquad{\rm with}\qquad
m\,=\,1,2,\cdots,q\,.$ (4.12)

There are $q$ solutions and they are located at $|G^{LR}|=1$ and
$|\Sigma^{LR}|=1$, which agree with the wormhole saddles found in [33]. This
is why we call the trivial solution
$\\{G_{\sigma}^{L}\,,G_{\sigma}^{R}\\}=\\{0,0\\}$ the wormhole saddle-point
solution. These wormhole solutions are depicted in Figure 1.

For the half-wormhole saddles, we look for solutions that behave differently
from the above wormhole solutions in the totally fixed-coupling limit
($\sigma\to 0$). Hence, in the $\sigma\to 0$ limit for the first equation, we
look for solutions satisfy
$\Sigma_{LR}\approx\lambda_{\sigma}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}$.
This is solved as

$\displaystyle\left\\{G_{LR}\,,\Sigma_{LR}\right\\}\,\approx\,\left\\{\lambda_{\sigma}^{-1}\,e^{\frac{2n\pi
i}{q}}\,,\,\lambda_{\sigma}\,e^{\frac{-2n\pi i}{q}}\right\\}\,,$ (4.13)

and

$\displaystyle\left\\{\big{\\{}G_{\sigma}^{L}\,,G_{\sigma}^{R}\big{\\}}\,,\
\big{\\{}\Sigma_{\sigma}^{L}\,,\Sigma_{\sigma}^{R}\big{\\}}\right\\}\ \approx$
$\displaystyle\ \left\\{\left\\{1,\,e^{\frac{2n\pi i}{q}}\right\\}\,,\
\left\\{1,\,e^{\frac{-2n\pi i}{q}}\right\\}\right\\}$ $\displaystyle\ {\rm
or}$ $\displaystyle\ \left\\{\left\\{e^{\frac{2n\pi i}{q}},\,1\right\\}\,,\
\left\\{e^{\frac{-2n\pi i}{q}}\,,1\right\\}\right\\}\,.$ (4.14)

Notice that we have $2q$ independent solutions similarly to the wormhole
solutions. Since in the totally fixed-coupling limit ($\sigma\to 0$), we have
$\lambda_{\sigma}^{-1}\propto\sigma^{-2/q}\to 0$, we have $G_{LR}\to 0$ in
this limit ($\sigma\to 0$). In contrast to the wormhole solutions, this means
that we have no correlation between the two SYK systems (L and R) and they are
analogous to the half-wormhole saddle-point solutions found in [33] and
discussed further in [44]. These half-wormhole solutions are depicted in
Figure 2.

Figure 2: The half-wormhole saddles in the totally fixed-coupling limit for
$q=12$.

On the other hand, in the disorder-averaged limit ($\sigma\to\infty$), the
coupling $\lambda_{\sigma}$ becomes small, and $G_{LR}$ becomes large.
Therefore, the approximation
$\Sigma_{LR}\approx\lambda_{\sigma}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}$ we
used is no longer valid in this limit, and we should treat as the extra term
$\lambda_{\sigma}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}$ as a perturbation
around the wormhole solutions.

####  4.1.2 On-shell action

In this subsection, we study the $\sigma$-dependence of the on-shell action of
the saddle-point solutions found above.

Let us first consider the on-shell action by taking a derivative with respect
to explicit $\lambda_{\sigma}$ dependence as in [4]

$\displaystyle\frac{\partial}{\partial\lambda_{\sigma}}\,S_{\rm
eff}\,=\,-\,NG_{LR}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\,.$ (4.15)

For the wormhole solutions, it is obvious that

$\displaystyle\frac{\partial}{\partial\lambda_{\sigma}}\,S^{\scriptsize\mbox{on-
shell}}_{\scriptsize\mbox{WH}}\,=\,0\,,$ (4.16)

thus the on-shell action of the wormhole solutions does not have
$\lambda_{\sigma}$ dependence. For the half-wormhole solutions (4.13) for
small $\sigma$, we can write the derivative of the on-shell action as

$\displaystyle\frac{\partial}{\partial\lambda_{\sigma}}\,S^{\scriptsize\mbox{on-
shell}}_{\scriptsize\mbox{HW}}\,\approx\,-N\lambda^{-1}_{\sigma}\,.$ (4.17)

Let us also study the value of the on-shell action directly (without taking
derivative with respect to $\lambda_{\sigma}$). For the wormhole solutions,
this is given by

$\displaystyle S^{\scriptsize\mbox{on-shell}}_{\scriptsize\mbox{WH}}\,$
$\displaystyle=\,-N\log\Sigma_{LR}\big{|}_{\scriptsize\mbox{on-
shell}}\,+\,N\left(1-\frac{1}{q}\right)\,G_{LR}^{q}\big{|}_{\scriptsize\mbox{on-
shell}}$ $\displaystyle=\,N\left(1-\frac{1}{q}\right)\,+\,\frac{2m\pi
iN}{q}\,.$ (4.18)

For the half-wormhole solutions at small $\sigma$, we have $G_{LR}\approx 0$.
Therefore, it is given by

$\displaystyle S^{\scriptsize\mbox{on-shell}}_{\scriptsize\mbox{HW}}\,$
$\displaystyle\approx\,\left[-N\log\Sigma_{LR}\,+\,N\left(1-\frac{1}{q}\right)\sum_{a}\big{(}G_{\sigma}^{a}\big{)}^{q}\right]_{\scriptsize\mbox{on-
shell}}$
$\displaystyle\approx\,-\,N\log(\lambda_{\sigma})\,+\,2N\left(1-\frac{1}{q}\right)\,+\,\frac{2n\pi
iN}{q}\,.$ (4.19)

We can see that the $\lambda_{\sigma}$ dependence agrees with that obtained by
integrating (4.17).

For $\sigma\to\infty$, we have only wormhole contributions. Therefore,
combining with the prefactor in (4.8), the two-point function is given by

$\displaystyle\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}\,$
$\displaystyle\approx\,J^{\,\frac{2N}{q}}\sum_{m=1}^{q}e^{-N(1-\frac{1}{q})-\frac{2m\pi
iN}{q}}$ $\displaystyle=\,q\,J^{\,\frac{2N}{q}}\,e^{-N(1-\frac{1}{q})}\,,$
(4.20)

where we used $\widetilde{J}\to J$ in the $\sigma\to\infty$ limit, and the
summation (3.28). On the other hand, for $\sigma\to 0$, we have
$\widetilde{J}\to 0$. Therefore, the wormhole contributions, which do not have
$\sigma$ dependence, do not contribute. For the half-wormhole contributions,
the $\sigma$ dependence coming from (4.19) precisely cancels with the $\sigma$
dependence coming from the prefactor in (4.8) and we find

$\displaystyle\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}\,$
$\displaystyle\approx\,2^{-N}J_{0}^{\frac{2N}{q}}\sum_{n=1}^{q}e^{-2N\left(1-\frac{1}{q}\right)-\frac{2n\pi
iN}{q}}$
$\displaystyle=\,q\,2^{-N}J_{0}^{\frac{2N}{q}}\,e^{-2N\left(1-\frac{2}{q}\right)}\,.$
(4.21)

We can see that the $J$ and $J_{0}$ dependence of the two-point function
precisely agree with the results of [44] in both limits. Also the above
$\sigma\to 0$ result precisely agrees with the square of (3.23) with
$\theta_{i}^{2}=1/2$ in the $\sigma\to 0$ limit. Furthermore, at $\sigma=0$,
we can compute the two-point function directly (which is presented in Appendix
E) and the result agrees with the above computation.

The large $N$ saddles contributing to this totally fixed-coupling two-point
function look different from the conclusion of [33, 44], but this is simply
due to the difference of the “linked” half-wormhole and the “unlinked” half-
wormhole, as we will explain further in section 6.

###  4.2 Adding a coupling

Let us make some comments what happens if we introduce a coupling between $L$
and $R$ systems in zero-dimensional SYK model

$\displaystyle S_{\rm int}=\mu\sum_{i=1}^{N}\chi_{i}^{L}\chi_{i}^{R}\,,$
(4.22)

according to [26, 33]. The saddle-point equations become

$\displaystyle\Sigma_{LR}\,=\,\big{(}G_{LR}\big{)}^{q-1}+\,\lambda_{\sigma}\,\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}-\mu\,,\qquad\frac{1}{\Sigma_{LR}}\,=\,G_{LR}\,,$
(4.23)
$\displaystyle\Sigma_{\sigma}^{a}\,=\,\,\,\big{(}G_{\sigma}^{a}\big{)}^{q-1}\,,\qquad
G_{\sigma}^{L}\,=\,\lambda_{\sigma}\,G_{LR}\Sigma_{\sigma}^{R}\,,\qquad
G_{\sigma}^{R}\,=\,\lambda_{\sigma}\,G_{LR}\Sigma_{\sigma}^{L}\,.$ (4.24)

If we consider the strong coupling limit near the totally fixed coupling
region such that $1\ll\lambda_{\sigma}\ll\mu$, the interaction term can be
neglected. In this case, the half-wormhole saddle and the wormhole saddle
degenerates, and we have a solution

$\displaystyle G_{LR}\approx\mu^{\frac{1}{q-1}},\
\Sigma_{LR}\approx\mu^{-\frac{1}{q-1}}\,,$ (4.25)

with $G^{L,R}_{\sigma}\approx 0,\Sigma^{L,R}_{\sigma}\approx 0$. Therefore,
even near the fixed coupling region, the wormhole contribution becomes
dominant. This is consistent with the conclusions [33, 44].

###  4.3 Brownian SYK

Next, we study the Brownian SYK case discussed in [24]. From the effective
action (4.4), the Brownian SYK corresponds to reducing the bi-local fields to
local fields. Furthermore, in this reduction, the diagonal fields in the $a$
index reduce to constants due to the anti-commutation relation of the fermion
(2.2), so we only need to keep $\\{G_{LR},\Sigma_{LR}\\}$. Therefore, the
effective action is now given by

$\displaystyle S_{\rm
eff}[G_{LR},\Sigma_{LR},G_{\sigma}^{a},\Sigma_{\sigma}^{a}]\,=\,-\frac{N}{2}\textrm{Tr}\log(\Sigma_{LR})+\frac{N}{2}\int
d\tau\left(\Sigma_{LR}(\tau)G_{LR}(\tau)-\frac{\widetilde{J}^{2}}{q}G_{LR}(\tau)^{q}\right)$
$\displaystyle\qquad+N\sum_{a}\int
d\tau\left(\Sigma_{\sigma}^{a}(\tau)G_{\sigma}^{a}(\tau)-\frac{J_{\sigma}}{q}G_{\sigma}^{a}(\tau)^{q}\right)\,-\,\frac{N}{4}\int
d\tau\,G_{LR}(\tau)\Sigma_{\sigma}^{L}(\tau)\Sigma_{\sigma}^{R}(\tau)\,.$
(4.26)

In general, it is still hard to search for time-dependent saddle-point
solutions. Hence, let us now consider static (time-independent) saddle-point
solutions. As explained in [24], in this case, we have

$\displaystyle\frac{1}{2}\textrm{Tr}\log(\Sigma_{LR})\,\to\,\log\left(2\cosh\left(\frac{\beta\Sigma_{LR}}{4}\right)\right)\,\approx\,\pm\frac{\beta\Sigma_{LR}}{4}\,,$
(4.27)

where for the rightmost equation, we take the low-temperature limit (i.e.,
large $\beta$) and the plus/minus sign depending on the real part of
$\Sigma_{LR}$. Then, the large $N$ saddle-point equations are

$\displaystyle\Sigma_{LR}\,=\,\widetilde{J}^{2}\,G_{LR}^{q-1}\,+\,\frac{1}{2}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\,,\qquad\pm\frac{1}{2}\,=\,G_{LR}\,,$
(4.28)
$\displaystyle\Sigma_{\sigma}^{a}\,=\,J_{\sigma}\,\big{(}G_{\sigma}^{a}\big{)}^{q-1}\,,\qquad
G_{\sigma}^{L}\,=\,\frac{1}{4}\,G_{LR}\Sigma_{\sigma}^{R}\,,\qquad
G_{\sigma}^{R}\,=\,\frac{1}{4}\,G_{LR}\Sigma_{\sigma}^{L}\,,$ (4.29)

We can easily see that the equations for
$\\{G_{\sigma}^{a},\Sigma_{\sigma}^{a}\\}$ (4.29) are structurally identical
to the 0d SYK case (4.10). Therefore, the wormhole and half-wormhole solutions
are given by

$\displaystyle\big{\\{}G_{\sigma}^{L}\,,G_{\sigma}^{R}\big{\\}}\,=\,\\{0,0\\}\,,\quad\left\\{e^{\frac{2n\pi
i}{q(q-2)}}\left(\frac{J_{\sigma}}{8}\right)^{-\frac{1}{q-2}},\,e^{\frac{(q-1)2n\pi
i}{q(q-2)}}\left(\frac{J_{\sigma}}{8}\right)^{-\frac{1}{q-2}}\right\\}\,,$
(4.30)

where we also used $G_{LR}=\pm 1/2$.

For the wormhole solution $\\{G_{\sigma}^{L}\,,G_{\sigma}^{R}\\}=\\{0,0\\}$,
the remaining equations (4.28) are solved as

$\displaystyle\qquad\big{\\{}G_{LR}\,,\Sigma_{LR}\big{\\}}\,=\,\bigg{\\{}\,\pm\frac{1}{2}\
,\ \pm\frac{\widetilde{J}^{2}}{2^{q-1}}\,\bigg{\\}}\,.$ (4.31)

These wormhole solutions agree with the ones found in [24]. 777The imaginary
factor $i$ difference is due to the fact that we study the model in Euclidean
signature while [24] used Lorentzian time. We emphasize that the wormhole
solutions are obtainable without continuing to Lorentzian time, but simply
taking the low-temperature limit in Euclidean time. For the half-wormhole
solutions, the $\Sigma_{LR}$ solution is obtained as

$\displaystyle\Sigma_{LR}\,=\,\pm\frac{\widetilde{J}^{2}}{2^{q-1}}\,+\,\frac{1}{2}\left(\frac{J_{\sigma}}{8}\right)^{-\frac{2}{q-2}}e^{\frac{2n\pi
i}{q-2}}\,.$ (4.32)

The on-shell actions for these solutions are computed as

$\displaystyle S^{\scriptsize\mbox{on-shell}}_{\scriptsize\mbox{WH}}\,$
$\displaystyle=\,\mp\,\frac{N\beta}{4}\Sigma_{LR}\big{|}_{\scriptsize\mbox{on-
shell}}\,+\,\frac{N\beta\widetilde{J}^{2}}{2}\left(1-\frac{1}{q}\right)\,G_{LR}^{q}\big{|}_{\scriptsize\mbox{on-
shell}}$ $\displaystyle=\,-\,\frac{N\beta\widetilde{J}^{2}}{q\,2^{q+1}}\,,$
(4.33)

for the wormhole solutions and

$\displaystyle S^{\scriptsize\mbox{on-shell}}_{\scriptsize\mbox{HW}}\,$
$\displaystyle=\,\left[\mp\,\frac{N\beta}{4}\Sigma_{LR}+\frac{N\beta\widetilde{J}^{2}}{2}\left(1-\frac{1}{q}\right)G_{LR}^{q}+N\beta
J_{\sigma}\left(1-\frac{1}{q}\right)\sum_{a}\big{(}G_{\sigma}^{a}\big{)}^{q}\right]_{\scriptsize\mbox{on-
shell}}$
$\displaystyle=\,-\,\frac{N\beta\widetilde{J}^{2}}{q\,2^{q+1}}\,\mp\,\frac{N\beta}{8}\left(\frac{J_{\sigma}}{8}\right)^{-\frac{2}{q-2}}\,+\,8N\beta\left(1-\frac{1}{q}\right)\left(\frac{J_{\sigma}}{8}\right)^{-\frac{2}{q-2}}\Big{(}e^{\frac{2n\pi
i}{q-2}}+e^{\frac{(q-1)2n\pi i}{q-2}}\Big{)}\,.$ (4.34)

for the half-wormhole solutions.

For the total disorder-average limit ($\sigma\to\infty$), in order to keep the
large $\beta$ approximation (4.27) valid, we need $\beta J^{2}$ large.
Therefore,both the wormhole and the half-wormhole solutions contribute equally
to the disorder-averaged two-point function in this limit. This behavior is
different from the one found in the zero-dimensional case in section 4.1.
Again, we expect that this difference is associated with the difference of the
“linked” and “unlinked” half-wormhole, as we will explain further in section
6.

##  5 Spectrum

In this section, we study the spectrum of the partially disorder-averaged SYK
model. To see the spectrum, it is more convenient to incorporate the
$\sigma$-dependence into the Hamiltonian itself, such that the probability
distribution of the newly defined random coupling has the original
distribution form. The $\sigma$-dependence in the probability distribution can
be eliminated as

$\displaystyle\mathcal{N}_{\sigma}^{-1}P(J_{i_{1}\cdots i_{q}})\,$
$\displaystyle=\,\mathcal{N}_{\sigma}^{-1}\exp\left[-\frac{N^{q-1}}{2(q-1)!}\sum_{i_{1}<\cdots<i_{q}}^{N}\left(\frac{J_{i_{1}\cdots
i_{q}}^{2}}{J^{2}}\,+\,\frac{(J_{i_{1}\cdots i_{q}}-J_{i_{1}\cdots
i_{q}}^{(0)})^{2}}{\sigma^{2}}\right)\right]$
$\displaystyle=\,\left(\frac{N^{q-1}}{2(q-1)!\pi\widetilde{J}^{2}}\right)^{\frac{1}{2}\binom{N}{q}}\exp\left[-\frac{N^{q-1}}{2(q-1)!}\sum_{i_{1}<\cdots<i_{q}}^{N}\frac{\hat{J}^{2}_{i_{1}\cdots
i_{q}}}{J^{2}}\right]\,,$ (5.1)

by introducing a new coupling constant

$\displaystyle\hat{J}_{i_{1}\cdots
i_{q}}\equiv\sqrt{1+\frac{J^{2}}{\sigma^{2}}}\left[J_{i_{1}\cdots
i_{q}}-\frac{1}{\left(1+\frac{\sigma^{2}}{J^{2}}\right)}J^{(0)}_{i_{1}\cdots
i_{q}}\right]\,.$ (5.2)

The Hamiltonian of the SYK model can be written in terms of
$\hat{J}_{i_{1}\cdots i_{q}}$ as 888A similar (but different) deformation of
the SYK model is studied in the context of $T\bar{T}$ deformation in [55].

$\displaystyle
H\,=\,\frac{i^{\frac{q}{2}}}{\sqrt{1+\frac{J^{2}}{\sigma^{2}}}}\sum_{i_{1}<i_{2}<\cdots<i_{q}}^{N}\hat{J}_{i_{1}\cdots
i_{q}}\,\chi_{i_{1}}\cdots\chi_{i_{q}}\,+\,\frac{i^{\frac{q}{2}}}{\left(1+\frac{\sigma^{2}}{J^{2}}\right)}\sum_{i_{1}<i_{2}<\cdots<i_{q}}^{N}J^{(0)}_{i_{1}\cdots
i_{q}}\,\chi_{i_{1}}\cdots\chi_{i_{q}}\,.$ (5.3)

This Hamiltonian manifests the transition between the usual disorder-averaged
SYK model and the fixed-coupled SYK model, i.e.,

$\displaystyle H\,$
$\displaystyle\underset{\sigma\rightarrow\infty}{\rightarrow}\,H_{\hat{J}}\equiv
i^{\frac{q}{2}}\sum_{i_{1}<i_{2}<\cdots<i_{q}}^{N}\hat{J}_{i_{1}\cdots
i_{q}}\,\chi_{i_{1}}\cdots\chi_{i_{q}}\,,$ $\displaystyle H\,$
$\displaystyle\underset{\sigma\rightarrow 0}{\rightarrow}\,H_{J^{(0)}}\equiv
i^{\frac{q}{2}}\sum_{i_{1}<i_{2}<\cdots<i_{q}}^{N}J^{(0)}_{i_{1}\cdots
i_{q}}\,\chi_{i_{1}}\cdots\chi_{i_{q}}\,.$ (5.4)

In the $\sigma\rightarrow 0$ limit, the Hamiltonian becomes independent of
$\hat{J}_{i_{1}\cdots i_{q}}$, thus the average over the random coupling is
performed trivially as

$\displaystyle\mathcal{N}_{\sigma}^{-1}\int\prod_{i_{1}<\cdots<i_{q}}^{N}dJ_{i_{1}\cdots
i_{q}}\,P(J_{i_{1}\cdots i_{q}})=1\,.$ (5.5)

Figure 3: Density of states $\rho(E)$ with $\sigma=0.03$(blue), $0.3$(orange),
$3$(green). We take $N=8$, $q=4$ and $J=1$, and take $50000$ samples for
averaging. In this plot, the external coupling $J^{(0)}_{i_{1}\cdots i_{q}}$
is chosen randomly from the Gaussian probability distribution
$P(J^{(0)}_{i_{1}\cdots
i_{q}})\,=\,\exp\left(-\frac{N^{q-1}}{2(q-1)!J^{2}}\sum_{i_{1}<\cdots<i_{q}}^{N}(J^{(0)}_{i_{1}\cdots
i_{q}})^{2}\right)$ with $N=8,q=4$ and $J=1$.

The partition function $\langle Z(it)\rangle_{J}$ of the partially disorder-
averaged SYK model in the $\sigma\rightarrow 0$ limit simply becomes that in
the SYK model with a fixed coupling $J^{(0)}_{i_{1}\cdots i_{q}}$, i.e.,

$\displaystyle\langle Z(it)\rangle_{J}=\sum_{E_{J^{(0)}}}{\rm
exp}\left[-itE_{J^{(0)}}\right]\,,$ (5.6)

where $E_{J^{(0)}}$ are the eigenvalues of the Hamiltonian $H_{J^{(0)}}$. Then
the density of states, which is given by the Fourier transform of $\langle
Z(it)\rangle_{J}$, becomes a sum of the delta functions

$\displaystyle\langle\rho(E)\rangle_{J}$
$\displaystyle=\int^{\infty}_{-\infty}\frac{dt}{2\pi}e^{-itE}\langle
Z(it)\rangle_{J}$
$\displaystyle=\sum_{E_{J^{(0)}}}\int^{\infty}_{-\infty}\frac{dt}{2\pi}e^{-it(E-E_{J^{(0)}})}=\sum_{E_{J^{(0)}}}\delta(E-E_{J^{(0)}})\,.$
(5.7)

In Figure 3, we numerically computed the density of states for various
$\sigma$. In the $\sigma\rightarrow 0$ limit, the spectrum has sharp
$L=2^{N/2}$ delta-function-like peaks as expected.

Figure 4: A log-log plot of the spectral form factor $g(t)$ with inverse
temperature $\beta=0.0001$, and $\sigma=0.03$(blue), $0.3$(orange), $3$(green)
in the partially disorder-averaged SYK model. We take $N=8$, $q=4$ and $J=1$,
and take $50000$ samples for averaging. We have chosen the same value of the
external coupling as Figure 3.

We also computed the $\sigma$-dependence of the spectrum form factor

$\displaystyle g(t)=\frac{\big{\langle}Z(\beta+it)Z(\beta-
it)\big{\rangle}_{J}}{\big{\langle}Z(\beta)^{2}\big{\rangle}_{J}}$ (5.8)

where

$\displaystyle Z(\beta+it)\equiv{\rm Tr}\left(e^{-\beta H-iHt}\right)\,,$
(5.9)

which probes the discreteness of the spectrum as depicted in Figure 4. We can
observe erratic oscillations at late times for small values of $\sigma$, which
is characteristic of the SYK model with a fixed coupling [5].

##  6 Conclusions and Discussions

In this paper, we studied a partially disorder-averaged SYK model. We
introduced a partially disorder-averaged SYK model by modifying the
probability distribution of the coupling constants with a real parameter
$\sigma$. The probability distribution becomes the usual Gaussian form in the
$\sigma\to\infty$ limit while in the $\sigma\to 0$ limit, it becomes a product
of delta functions which enforces each component of the coupling constant to a
fixed external value. Given this partial disorder averaging, we studied the
one- and two-point functions of the partition functions as well as spectral
density and the spectral form factor. For the large $N$ effective description,
in addition to the usual bi-local collective fields, we introduced a new
additional set of local collective fields. We explained that these local
fields can be understood as the “half” of the bi-local collective fields.

Bulk dual to the half-wormhole saddles
For the study of the two-point function of the zero-dimensional model
partition function, we found that it contains the wormhole and half-wormholes
as large $N$ saddles, and their configurations change as we gradually vary
$\sigma$. A natural question is what is the gravity dual to the half-wormhole
for each value of $\sigma$. As we saw in section 4.1, in the totally disorder-
averaged limit $\sigma=\infty$, the half-wormhole and the wormhole saddles
degenerate, so in this sense, the dual geometry is just a smooth wormhole
geometry drawn as the left picture in Figure 5. Such geometry can be found in
the one-dimensional model [24] as well as in the dual JT gravity set-up for
example in [18].

Let us consider the half-wormhole saddles in the slightly fixed coupling
region. As we explained section 4.1, in this region, the half-wormhole saddle
can be obtained by considering a perturbation around the wormhole saddles by
adding the small effect of the half-bi-local fields. The interpretation of
this perturbation in the gravity side is considered in section 3.4. We found
that adding the small effect of the half-bi-local fields $G_{\sigma}$
corresponds to having a nontrivial classical configuration of the bulk field
dual to $G_{\sigma}$ on the original wormhole background. As we also pointed
out in Appendix A, the $G_{\sigma}$ can be interpreted as the correlation
between the fermions $\chi_{i}$ on the boundary and other fermioninc degrees
of freedom corresponding to $\theta_{i}$, which represents the fixed
couplings, living on “another universe”. We interpret the gravity dual of
“another universe” as a bulk brane, whose structure strongly depends on the
microscopic details of the bulk gravity theory. Such a “UV-sensitive brane”
might appear in the low-energy effective gravity from the condensation of some
stringy modes in string theory as pointed out in [51]. We leave it as an
interesting research direction for the future.

Similar proposals are also mentioned in [45, 46]. The intuitive picture can be
drawn as the middle picture of Figure 5. In this picture, the spacetime ends
at the bulk brane placed in the middle of the background wormhole geometry.
$G_{\sigma}$ represents the correlation between the spacetime boundary and
this brane. Since we have a nontrivial profile for the bi-local field $G_{LR}$
in this parameter region, the left and right boundaries are still connected by
the wormhole. We imagine that as we gradually lower the value of $\sigma$, the
“gash” made by the brane is getting bigger, and in the totally fixed-coupling
limit $\sigma=0$, the wormhole will be eventually torn into two pieces (right
picture in Figure 5). This is consistent with the fact that the half-wormhole
saddle in this limit represents no correlation between the left and right
boundaries, i.e., $G_{LR}=0$ while there is a nontrivial profile for the half-
bi-local fields $G_{\sigma}$ and $\Sigma_{\sigma}$. This is also consistent
with the bulk picture of the half-wormhole in [33].

Figure 5: The intuitive bulk pictures dual to the half-wormholes for various
values of $\sigma$ in the partially fixed-coupling SYK model. The left picture
corresponds to the totally disorder-averaged limit $\sigma=\infty$, where the
half-wormhole and the wormhole saddles degenerate. The middle corresponds to
the slightly fixed-coupling parameter region, where the half-wormhole saddles
can be considered as a small perturbation of the wormhole saddle. The right
picture represents the half-wormhole in the totally fixed-coupling limit,
where the correlation between the left and the right boundaries totally
vanishes.

Linked and unlinked half-wormholes
We analyzed the saddle point solutions for
$\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}$ for large and small values of
$\sigma$. We found that there exist both wormhole and half-wormhole saddles
except for the totally disorder-averaged limit $\sigma=0$, where these two
types of the saddles degenerate. In the totally disorder-averaged limit, only
the half-wormhole saddles contribute to
$\big{\langle}Z_{L}Z_{R}\big{\rangle}^{\sigma=0}_{J}$, and we can write it
schematically as

$\displaystyle\big{\langle}Z_{L}Z_{R}\big{\rangle}^{\sigma=0}_{J}\approx
e^{-S_{\rm on\mathchar 45\relax shell}^{(L)}({\rm Half\mathchar 45\relax
wormhole})}e^{-S_{\rm on\mathchar 45\relax shell}^{(R)}({\rm Half\mathchar
45\relax wormhole})}\,.$ (6.1)

On the other hand, in [33] they analyzed the saddle point solutions in the
(totally) fixed-coupling SYK model and found that

$\displaystyle Z_{L}Z_{R}\underset{\rm SSSY}{\approx}e^{-S_{\rm on\mathchar
45\relax shell}^{(LR)}({\rm Wormhole})}+e^{-S_{\rm on\mathchar 45\relax
shell}^{(LR)}({\rm(Linked)\,Half\mathchar 45\relax wormhole})}\,,$ (6.2)

i.e., the wormhole saddles and the half-wormhole saddles contribute in the
same order to $Z_{L}Z_{R}$. Here “linked half-wormhole” means the half-
wormhole saddle in the collective field description where both wormhole and
half-wormhole saddle exist. On the other hand, in [33] they introduced another
collective field description without wormholes. In this description, the half-
wormholes are called “(unlinked) half-wormholes” to distinguish them from the
linked half-wormholes. The half-wormhole saddles expressed as the nontrivial
profile of the half-bi-local fields, i.e., $G_{\sigma},\Sigma_{\sigma}\neq 0$
discussed in this paper, would correspond to their unlinked half-wormhole
saddles. In this paper, when writing the effective action for the collective
fields (4.6) in $0$d SYK, we rescaled them to have nontrivial profiles of the
wormhole solutions even in the strictly totally fixed coupling limit. It fits
more into the situation of [33] given the collective fields without rescaling
(4.4). As you can see, since $J_{\sigma}\rightarrow J_{0}$ and
$\tilde{J}\rightarrow 0$ in the totally fixed-coupling limit, we can simply
integrate over the bi-local field $G_{LR}$, forcing us to
$\Sigma_{LR}(\tau_{1},\tau_{2})=\frac{1}{2}\Sigma_{\sigma}^{L}(\tau_{1})\Sigma_{\sigma}^{R}(\tau_{2})$.
As a result, the effective action can be written solely with half-bi-local
fields (as well as bi-local fields of the type
$G_{LL},G_{RR},\Sigma_{LL},\Sigma_{RR}$). In this collective field
description, it is obvious that there are only half-wormholes described by the
half-bi-local fields and no wormholes represented by the solutions $G_{LR}\neq
0,\Sigma_{LR}\neq 0$ with $G_{\sigma}^{L,R}=\Sigma_{\sigma}^{L,R}=0$. This
situation corresponds to the collective field description with no wormhole
saddles discussed in [33].

Future directions
In this paper, we studied a partially disorder-averaged Majorana SYK model.
Many generalizations of the SYK model were studied by various authors. Some
examples include the complex SYK model [52, 53, 54], supersymmetric SYK model
[56, 57, 58], SYK model with global symmetries [59, 60], and higher
dimensional generalization of the SYK model [61, 62, 63, 64]. It would be
interesting to study a partially disorder-averaged version of these
generalizations of the SYK model.

The initial motivation to introduce disorder in the original Sachdev-Ye model
[1, 65, 66] was to simulate zero-temperature quantum phase transition between
the quantum disordered spin-liquid phase and the magnetically ordered spin-
glass phase. It would be interesting to study whether our partially disorder-
averaged model brings any further insights into this aspect.

The original motivation to consider the half-wormholes was to resolve the
tension between the factorization of the partition functions and the existence
of the wormholes in the bulk in the AdS/CFT context. We found that the
wormhole solutions persist even in the fixed coupling limit, but they have
zero measure in the path integral. Instead, the half-wormhole solutions emerge
as the “dissolved”-wormholes, and they dominate the partition function. It
would be interesting to study whether a similar thing could happen in the
gravitational path integral.

## Acknowledgements

We are grateful to Tomoki Nosaka for sharing the Mathematica code for
computing the spectral form factor in the SYK model. We also thank Takanori
Anegawa, Norihiro Iizuka, and Tadashi Takayanagi for useful discussion. KG is
supported by JSPS Grant-in-Aid for Early-Career Scientists 21K13930. The work
of KS is supported by the Simons Foundation through the “It from Qubit”
collaboration. TU was supported by JSPS Grant-in-Aid for Young Scientists
19K14716 and MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A
“Extreme Universe” No.21H05184. This work is also supported by MEXT KAKENHI
Grant Number 21H05182, 21H05184, and 21H05187.

##  Appendix A External couplings as another universe fermions

In this appendix, we discuss the interpretation of the $\theta_{i}$ variables
introduced in (3.4) as non-dynamical fermions living in another universe. To
consider another universe, we start from the two-point function (4.1). Taking
the total disorder-averaging ($\sigma=\infty$) of this function, we find

$\displaystyle\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}\,=\,\int
D\chi^{L}_{i}D\chi^{R}_{i}\,$
$\displaystyle\exp\Bigg{[}-\frac{1}{2}\sum_{a=L,R}\int
d\tau\sum_{i=1}^{N}\chi^{a}_{i}\partial_{\tau}\chi^{a}_{i}$ $\displaystyle\
\qquad+\,\frac{(q-1)!\widetilde{J}^{2}}{2N^{q-1}}\sum_{i_{1}<\cdots<i_{q}}^{N}\left(\int
d\tau\sum_{a=L,R}\chi^{a}_{i_{1}}\cdots\chi^{a}_{i_{q}}\right)^{2}\Bigg{]}\,.$
(A.1)

Now we freeze the dynamics of $\chi^{R}_{i}$ and denote the non-dynamical
fermions as $\theta_{i}\propto\chi^{R}_{i}$. Therefore, now we can interpret
the above two-point function as a one-point function of $Z_{L}$ with external
fermionic sources $\theta_{i}$:

$\displaystyle\big{\langle}Z_{L}\big{\rangle}_{J}\,=\,\int D\chi_{i}\,$
$\displaystyle\exp\Bigg{[}-\frac{1}{2}\int
d\tau\sum_{i=1}^{N}\chi_{i}\partial_{\tau}\chi_{i}\,+\,\frac{J_{\sigma}}{qN^{q-1}}\int
d\tau\left(\sum_{i=1}^{N}\theta_{i}\chi_{i}(\tau)\right)^{q}$
$\displaystyle\qquad\qquad+\,\frac{\widetilde{J}^{2}}{2qN^{q-1}}\int
d\tau_{1}d\tau_{2}\left(\sum_{i=1}^{N}\chi_{i}(\tau_{1})\chi_{i}(\tau_{2})\right)^{q}\Bigg{]}\,,$
(A.2)

where we denoted $\chi_{i}\equiv\chi^{L}_{i}$ and also adjusted the
proportionality constant of $\theta_{i}\propto\chi^{R}_{i}$ to give the
precise second term. This effective action precisely agrees with (3.5), and in
this sense, we can interpret the external variable $\theta_{i}$ as non-
dynamical fermions living in another universe.

##  Appendix B Integration over the fermions

In this appendix, we present some details for the integration over fermions.
This is different from the usual Gaussian integrals of Grassmann variables due
to the external variables $\theta_{i}$ (3.4) and its anti-commutation relation

$\displaystyle\\{\theta_{i},\theta_{j}\\}\,=\,\delta_{ij}\,.$ (B.1)

To implement this anti-commutation relation in the path-integral, we can
regard this $\theta_{i}$ as $N$-dimensional Dirac matrices constructed from
the Pauli matrix (for example see [67]).

With this representation of $\theta_{i}$, we first consider the fermion
integration in the one-point function (of the one-dimensional model) studied
in section 3:

$\displaystyle\int\prod_{i=1}^{N}D\chi_{i}\,\exp\Bigg{[}\frac{1}{2}\int
d\tau_{1}d\tau_{2}\sum_{i=1}^{N}\chi_{i}(\tau_{1})\Sigma(\tau_{1},\tau_{2})\chi_{i}(\tau_{2})\,+\int
d\tau\sum_{i=1}^{N}\chi_{i}(\tau)\Sigma_{\sigma}(\tau)\theta_{i}\Bigg{]}\,.$
(B.2)

For the kinetic term, using the delta function
$\delta(G-N^{-1}\sum_{i}\chi_{i}\chi_{i})$, we rewrite it in terms of the bi-
local field $G(\tau_{1},\tau_{2})$, so that it does not appear in the fermion
integral. As usual Majorana fermion integrals, we introduce complex $N/2$
fermions $c_{i}$ and also complex $N/2$ external variables $\varphi_{i}$ by

$\displaystyle\chi_{2i}\,=\,\frac{c_{i}+\bar{c}_{i}}{\sqrt{2}}\,,\qquad\quad\chi_{2i-1}\,=\,\frac{i(c_{i}-\bar{c}_{i})}{\sqrt{2}}\,,$
(B.3)
$\displaystyle\theta_{2i}\,=\,\frac{\varphi_{i}+\bar{\varphi}_{i}}{\sqrt{2}}\,,\qquad\quad\theta_{2i-1}\,=\,\frac{i(\varphi_{i}-\bar{\varphi}_{i})}{\sqrt{2}}\,.$
(B.4)

Then, the above integral is given by

$\displaystyle\int\prod_{i=1}^{N/2}Dc_{i}D\bar{c}_{i}\,\exp\Bigg{[}\int
d\tau_{1}d\tau_{2}\sum_{i=1}^{N/2}c_{i}(\tau_{1})\Sigma(\tau_{1},\tau_{2})\bar{c}_{i}(\tau_{2})\,+\int
d\tau\sum_{i=1}^{N/2}\Sigma_{\sigma}(\varphi_{i}\bar{c}_{i}+\bar{\varphi}_{i}c_{i})\Bigg{]}$
$\displaystyle=\,\Big{[}\textrm{Pf}\left(\Sigma+\tfrac{1}{2}\Sigma_{\sigma}\Sigma_{\sigma}\right)\Big{]}^{N}\,.$
(B.5)

Exponentiating this into the effective action and sifting
$\Sigma(\tau_{1},\tau_{2})\to\Sigma(\tau_{1},\tau_{2})-\frac{1}{2}\Sigma_{\sigma}(\tau_{1})\Sigma_{\sigma}(\tau_{2})$,
we find

$\displaystyle S_{\rm
eff}[G,\Sigma,G_{\sigma},\Sigma_{\sigma}]\,=\,-\frac{N}{2}\textrm{Tr}\log(\Sigma)-\frac{N}{2}\int
d\tau\big{[}\partial_{\tau}G(\tau,\tau^{\prime})\big{]}_{\tau^{\prime}=\tau}$
(B.6) $\displaystyle\qquad\qquad\qquad+\frac{N}{2}\int
d\tau_{1}d\tau_{2}\left(\Sigma(\tau_{1},\tau_{2})G(\tau_{1},\tau_{2})-\frac{\widetilde{J}^{2}}{q}G(\tau_{1},\tau_{2})^{q}\right)$
$\displaystyle\qquad+N\int
d\tau\left(\Sigma_{\sigma}(\tau)G_{\sigma}(\tau)-\frac{J_{\sigma}}{q}G_{\sigma}(\tau)^{q}\right)\,-\,\frac{N}{4}\int
d\tau_{1}d\tau_{2}\,\Sigma_{\sigma}(\tau_{1})G(\tau_{1},\tau_{2})\Sigma_{\sigma}(\tau_{2})\,.$

For the zero-dimensional case in section 3.1, we do not have the bi-local
fields. Therefore, the fermion integral is simply

$\displaystyle\int\prod_{i=1}^{N}d\chi_{i}\,\exp\bigg{[}-\Sigma_{\sigma}\sum_{i=1}^{N}\theta_{i}\chi_{i}\bigg{]}\,=\,\big{(}-\Sigma_{\sigma}\big{)}^{N}\bigg{(}\prod_{i=1}^{N}\theta_{i}\bigg{)}\,.$
(B.7)

Finally, let us consider the fermion integration for the two-point function in
the zero-dimensional SYK model studied in section 4.1. Since we have

$\displaystyle\quad\int
d\chi^{L}d\chi^{R}\,\exp\Big{[}\Sigma_{LR}\chi^{L}\chi^{R}\Big{]}\exp\Big{[}\Sigma_{\sigma}^{L}\theta\chi^{L}\Big{]}\exp\Big{[}\Sigma_{\sigma}^{R}\theta\chi^{R}\Big{]}$
$\displaystyle=\,\int
d\chi^{L}d\chi^{R}\Big{(}1+\Sigma_{LR}\chi^{L}\chi^{R}+\theta^{2}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\chi^{L}\chi^{R}+\cdots\Big{)}$
$\displaystyle=\,\Sigma_{LR}+\frac{1}{2}\,\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\,,$
(B.8)

where we used $\theta^{2}=1/2$, the integrating out $\chi^{L}_{i}\chi^{R}_{i}$
gives $-N\log(\Sigma_{LR}+\frac{1}{2}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R})$
in the effective action. Sifting
$\Sigma_{LR}\to\Sigma_{LR}-\frac{1}{2}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}$,
we find

$\displaystyle S_{\rm
eff}[G_{LR},\Sigma_{LR},G_{\sigma}^{a},\Sigma_{\sigma}^{a}]\,$
$\displaystyle=\,-N\log(\Sigma_{LR})\,+\,N\left(\Sigma_{LR}G_{LR}-\frac{\widetilde{J}^{2}}{q}\,G_{LR}^{q}\right)$
$\displaystyle\quad+N\sum_{a}\left(\Sigma_{\sigma}^{a}G_{\sigma}^{a}-\frac{J_{\sigma}}{q}\big{(}G_{\sigma}^{a}\big{)}^{q}\right)\,-\,\frac{N}{2}G_{LR}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\,.$
(B.9)

To be precise, we should regard $G_{\sigma}^{a}$ as a matrix as

$\displaystyle\big{[}G_{\sigma}^{L},G_{\sigma}^{R}\big{]}\,$
$\displaystyle=\,\frac{1}{N^{2}}\sum_{i,j}\big{[}\theta_{i}\chi_{i}^{L},\theta_{j}\chi_{j}^{R}\big{]}$
$\displaystyle=\,\frac{1}{N^{2}}\sum_{i,j}(\theta_{i}\theta_{j}+\theta_{j}\theta_{i})\chi_{i}^{L}\chi_{j}^{R}$
$\displaystyle=\,\frac{1}{N^{2}}\sum_{i}\chi_{i}^{L}\chi_{i}^{R}$
$\displaystyle=\,\frac{1}{N}\,G_{LR}\,.$ (B.10)

However, as long as we keep the large $N$ limit, we can treat $G_{\sigma}^{a}$
as a usual $c$-number.

##  Appendix C Another formalism for slightly fixed-coupling

In this appendix, we will show that the same structure we saw in section 3.3
and 3.4 also appears in another formulation of the slightly fixed-coupling SYK
model. For simplicity, in this appendix, we focus on the $q=4$ case.

For this purpose, instead of (3), we introduce another Hubbard–Stratonovich
trick

$\displaystyle 1\,$ $\displaystyle=\,\int DG\int
DG^{\sigma}_{ij}\,\delta\left(G(\tau_{1},\tau_{2})-\frac{1}{N}\sum_{i=1}^{N}\chi_{i}(\tau_{1})\chi_{i}(\tau_{2})\right)\delta\Big{(}G^{\sigma}_{ij}(\tau)-\chi_{i}(\tau)\chi_{j}(\tau)\Big{)}$
$\displaystyle=\,\int DGD\Sigma\int
DG^{\sigma}_{ij}D\Sigma^{\sigma}_{ij}\exp\Bigg{[}-\frac{1}{2}\int
d\tau_{1}d\tau_{2}\,\Sigma(\tau_{1},\tau_{2})\left(NG(\tau_{1},\tau_{2})-\sum_{i=1}^{N}\chi_{i}(\tau_{1})\chi_{i}(\tau_{2})\right)$
$\displaystyle\hskip 160.0pt-\frac{1}{2}\sum_{i,j=1}^{N}\int
d\tau\,\Sigma^{\sigma}_{ij}(\tau)\Big{(}G^{\sigma}_{ij}(\tau)-\chi_{i}(\tau)\chi_{j}(\tau)\Big{)}\Bigg{]}\,.$
(C.1)

Using this trick for the partially disorder-averaged partition function and
performing the Gaussian integral for $\chi_{i}$, we obtain

$\displaystyle\big{\langle}Z\big{\rangle}_{J}\,=\,\int DGD\Sigma
DG^{\sigma}D\Sigma^{\sigma}\,e^{-S_{\rm
eff}[G,\Sigma,G^{\sigma},\Sigma^{\sigma}]}\,,$ (C.2)

with

$\displaystyle S_{\rm eff}[G,\Sigma,G^{\sigma},\Sigma^{\sigma}]\,$
$\displaystyle=\,-\frac{1}{2}\textrm{Tr}\log(\Delta_{ij})+\frac{N}{2}\int
d\tau_{1}d\tau_{2}\left(\Sigma(\tau_{1},\tau_{2})G(\tau_{1},\tau_{2})-\frac{\widetilde{J}^{2}}{4}G(\tau_{1},\tau_{2})^{4}\right)$
$\displaystyle\qquad+\frac{1}{2}\sum_{i,j=1}^{N}\int
d\tau\left(\Sigma^{\sigma}_{ij}(\tau)G^{\sigma}_{ij}(\tau)+\frac{\widetilde{J}^{2}}{12\sigma^{2}}\sum_{k,l=1}^{N}J_{ijkl}^{(0)}G^{\sigma}_{ij}(\tau)G^{\sigma}_{kl}(\tau)\right)\,,$
(C.3)

where the trace is now taken both for the indices $i,j$ and the bi-local time
$(\tau_{1},\tau_{2})$ with

$\displaystyle\Delta_{ij}(\tau_{1},\tau_{2})\,\equiv\,\delta_{ij}\Big{(}\delta(\tau_{1}-\tau_{2})\partial_{\tau_{2}}-\Sigma(\tau_{1},\tau_{2})\Big{)}-\Sigma^{\sigma}_{ij}(\tau_{1})\delta(\tau_{1}-\tau_{2})\,.$
(C.4)

If this $\Delta_{ij}$ is completely diagonal for the indices $i,j$, we recover
a factor of $N$ in front of the trace term, but for a general $\Delta_{ij}$,
we miss this factor of $N$. The extra local fields
$\\{G^{\sigma},\Sigma^{\sigma}\\}$ represents a non-singlet sector of the
global $O(N)$ symmetry. In general, it is hard to understand the non-singlet
sector of a large $N$ theory [68, 69]. In the context of AdS2/CFT1, these
fields are probably related to the open string degrees of freedom attached to
the boundary (see section 6.2 of [4]).

Even though we miss the factor of $N$ in front of the trace term and the
second line of the above action, let us here pretend that we can use the large
$N$ saddle-point evaluation. Variation of each field leads to saddle-point
equations

$\displaystyle\delta G\,:$
$\displaystyle\qquad\Sigma(\tau_{1},\tau_{2})\,=\,\widetilde{J}^{2}\,G(\tau_{1},\tau_{2})^{3}\,,$
(C.5) $\displaystyle\delta\Sigma\,:$
$\displaystyle\qquad\frac{1}{N}\sum_{i=1}^{N}\Big{[}\Delta^{-1}(\tau_{1},\tau_{2})\Big{]}_{ii}\,=\,G(\tau_{1},\tau_{2})\,,$
(C.6) $\displaystyle\delta G^{\sigma}\,:$
$\displaystyle\qquad\Sigma^{\sigma}_{ij}(\tau)\,=\,-\,\frac{\widetilde{J}^{2}}{6\sigma^{2}}\,\sum_{k,l=1}^{N}\,J_{ijkl}^{(0)}G^{\sigma}_{kl}(\tau)\,,$
(C.7) $\displaystyle\delta\Sigma^{\sigma}\,:$
$\displaystyle\qquad\Big{[}\Delta_{ij}^{-1}\Big{]}(\tau,\tau)\,=\,G^{\sigma}_{ij}(\tau)\,,$
(C.8)

where we defined two kinds of inverse. The first one is inverse for the bi-
local time:

$\displaystyle\int
d\tau_{3}\,\Delta_{ij}(\tau_{1},\tau_{3})\Big{[}\Delta^{-1}(\tau_{3},\tau_{2})\Big{]}_{ij}\,=\,\delta(\tau_{1}-\tau_{2})\,,\qquad\quad({\rm
for\ fixed}\ i,\,j)\,,$ (C.9)

and the second one is for the $O(N)$ index:

$\displaystyle\sum_{k=1}^{N}\,\Delta_{ik}(\tau_{1},\tau_{2})\Big{[}\Delta^{-1}_{kj}\Big{]}(\tau_{1},\tau_{2})\,=\,\delta_{ij}\,,\qquad\quad({\rm
for\ fixed}\ \tau_{1},\,\tau_{2})\,.$ (C.10)

We first note that the saddle-point solution of the diagonal part of
$G^{\sigma}$ must be given by

$\displaystyle
G^{\sigma}_{ii}(\tau)\,=\,\big{\langle}\chi_{i}(\tau)\chi_{i}(\tau)\big{\rangle}\,=\,\frac{1}{2}\,,$
(C.11)

where for the second equality, we used the equal-time anti-commutation
relation of the fermion (2.2). Let us now consider the following external
coupling $J_{ijkl}^{(0)}$

$\displaystyle\qquad
J_{ijkl}^{(0)}\,=\,\frac{J_{0}}{6}\Big{(}\delta_{ij}\delta_{ik}\delta_{il}\,+\,A_{ijkl}^{(0)}\Big{)}\,,$
(C.12)

where $A_{ijkl}^{(0)}$ is the totally anti-symmetric tensor without any
diagonal piece. Namely, if any of the indices coincide, $A_{ijkl}^{(0)}=0$.
Then, from (C.7) the index structure of $\Sigma^{\sigma}_{ij}$ is decomposed
into symmetric and antisymmetric parts as

$\displaystyle\Sigma^{\sigma}_{ij}(\tau)\,=\,\delta_{ij}\Sigma^{\sigma}_{\rm
dig}(\tau)\,+\,\Sigma^{\rm ant}_{ij}(\tau)\,,$ (C.13)

and the equation (C.7) is reduce to

$\displaystyle\Sigma^{\sigma}_{\rm dig}(\tau)\,$
$\displaystyle=\,-\,\frac{J_{\sigma}}{N}\,\sum_{k=1}^{N}\,G^{\sigma}_{kk}(\tau)\,=\,-\frac{J_{\sigma}}{2}\,,$
(C.14) $\displaystyle\Sigma^{\rm ant}_{ij}(\tau)\,$
$\displaystyle=\,-\,\frac{J_{\sigma}}{N}\,\sum_{k,l=1}^{N}\,A_{ijkl}^{(0)}G^{\sigma}_{kl}(\tau)\,.$
(C.15)

Using this expression of $\Sigma^{\sigma}_{ij}$ into (C.4), we can see that
the diagonal components of $\Delta_{ij}$ are independent of the index $i$

$\displaystyle\Delta_{ii}(\tau_{1},\tau_{2})\,$
$\displaystyle=\,\delta(\tau_{1}-\tau_{2})\left(\partial_{\tau_{2}}+\frac{J_{\sigma}}{2}\right)\,-\,\Sigma(\tau_{1},\tau_{2})$
$\displaystyle\equiv\,\Delta_{\textrm{Dig}}(\tau_{1},\tau_{2})\,.$ (C.16)

Therefore, the summation in (C.6) is trivially taken and gives a factor of
$N$, which cancels with the $1/N$ coefficient. Hence, we can simply invert
$\Delta^{-1}(\tau_{1},\tau_{2})$ and this equation is now written as

$\displaystyle\delta(\tau_{1}-\tau_{2})\,=\,\int
d\tau_{3}\,\Delta_{\textrm{Dig}}(\tau_{1},\tau_{3})G(\tau_{3},\tau_{2})\,.$
(C.17)

The modification from the ordinary SYK is the $J_{\sigma}$ term in (C.16),
which acts as a mass or chemical potential term.

##  Appendix D Two-point function of $q=2$, 0d SYK

In this appendix, we study the $q=2$ case of the $0d$ SYK model discussed in
section 4.1. For $q=2$, we have the saddle-point equations

$\displaystyle\Sigma_{LR}\,=\,G_{LR}\,+\,\lambda_{\sigma}\,\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\,,\qquad\frac{1}{\Sigma_{LR}}\,=\,G_{LR}\,,$
(D.1) $\displaystyle\Sigma_{\sigma}^{a}\,=\,G_{\sigma}^{a}\,,\qquad
G_{\sigma}^{L}\,=\,\lambda_{\sigma}G_{LR}\Sigma_{\sigma}^{R}\,,\qquad
G_{\sigma}^{R}\,=\,\lambda_{\sigma}G_{LR}\Sigma_{\sigma}^{L}\,.$ (D.2)

From (D.2), we find equations for $G_{\sigma}^{L}$ as

$\displaystyle\Big{[}\big{(}\lambda_{\sigma}G_{LR}\big{)}^{2}+1\Big{]}G_{\sigma}^{L}\,=\,0\,.$
(D.3)

Therefore, the trivial solution is
$\\{G_{\sigma}^{L}\,,G_{\sigma}^{R}\\}=\\{0,0\\}$, which can be understood as
the “wormhole” saddle-point solution again. The nontrivial solution can be
found only if

$\displaystyle G_{LR}\,=\,\pm\,\lambda_{\sigma}^{-1}\,.$ (D.4)

In this case, $\\{G_{\sigma}^{L}\,,G_{\sigma}^{R}\\}$ are not yet fixed by the
equations (D.2). Since in the totally fixed-coupling limit ($\sigma\to 0$), we
have $\lambda_{\sigma}^{-1}\propto\sigma^{2/q}$, this solution behaves as
$G_{LR}\to 0$ in this limit. Therefore, this nontrivial solution is understood
as the “half-wormhole” saddle-point solution.

For the wormhole case, the solution of the bi-local sector is again given by
(4.12) as $\\{G_{LR},\Sigma_{LR}\\}=\\{\pm 1,\pm 1\\}$. For the half-wormhole
case, solving (D.1), we find

$\displaystyle\\{G_{\sigma}^{L}\,,G_{\sigma}^{R}\\}\,=\,\left\\{\pm\sqrt{1-\lambda_{\sigma}^{-2}}\,,\,\sqrt{1-\lambda_{\sigma}^{-2}}\right\\}\,.$
(D.5)

The on-shell actions are obtained

$\displaystyle S^{\scriptsize\mbox{on-
shell}}_{\scriptsize\mbox{WH}}\,=\,\frac{N}{2}\,+\,m\pi iN\,,$ (D.6)

for the wormhole saddles with $m=1,2$, and

$\displaystyle S^{\scriptsize\mbox{on-
shell}}_{\scriptsize\mbox{HW}}\,\approx\,-\,N\log(\lambda_{\sigma})\,+\,N\sqrt{1-\lambda_{\sigma}^{-2}}\,+\,n\pi
iN\,.$ (D.7)

for the half-wormhole solutions with $n=1,2$.

##  Appendix E Direct computation of
$\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}$ at $\sigma=0$

In this appendix, we directly solve the two-point function (4.8) at
$\sigma=0$. Because the prefactor $\widetilde{J}^{2N/q}\to\sigma^{2N/q}$ in
the $\sigma\to 0$ limit, while the coupling in the effective action (4.6) is
$\lambda_{\sigma}\to\sigma^{-2/q}$ in this limit, so that the action gives a
divergence. To get a finite non-zero result, we have to expand the interaction
term of (4.6) perturbatively and consider the $N$-th order term (which gives
$\lambda_{\sigma}^{N}$). Therefore, the two-point function at $\sigma=0$ is
given by

$\displaystyle\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}^{\sigma=0}\,$
$\displaystyle=\,\frac{N^{N}J_{0}^{\frac{2N}{q}}}{2^{N}N!}\int
dG_{LR}\,\frac{d\Sigma_{LR}}{2\pi
i/N}\int\prod_{a}^{L,R}dG_{\sigma}^{a}\,\frac{d\Sigma_{\sigma}^{a}}{2\pi
i/N}\,\Big{(}G_{LR}\Sigma_{LR}\Sigma_{\sigma}^{L}\Sigma_{\sigma}^{R}\Big{)}^{N}$
$\displaystyle\quad\times\exp\left[-N\left(\Sigma_{LR}G_{LR}-\frac{1}{q}G_{LR}^{q}\right)-N\sum_{a}\left(\Sigma_{\sigma}^{a}G_{\sigma}^{a}-\frac{1}{q}(G_{\sigma}^{a})^{q}\right)\right]\,.$
(E.1)

Now we can see that each set of $\\{G,\Sigma\\}$ decoupled from each other.
Therefore, we can perform the integrals separately. For the bi-local sector,
we have

$\displaystyle\int dG_{LR}\,\frac{d\Sigma_{LR}}{2\pi
i/N}\,\Big{(}G_{LR}\Sigma_{LR}\Big{)}^{N}\exp\left[-N\left(\Sigma_{LR}G_{LR}-\frac{1}{q}G_{LR}^{q}\right)\right]$
$\displaystyle=\,$ $\displaystyle\int
dG_{LR}\,e^{\frac{N}{q}G_{LR}^{q}}\big{(}G_{LR}\big{)}^{N}\big{(}-N^{-1}\partial_{G_{LR}}\big{)}^{N}\delta(G_{LR})$
$\displaystyle=\,$ $\displaystyle\,\frac{N!}{N^{N}}\,,$ (E.2)

while for the local sector, we have

$\displaystyle\int dG_{\sigma}^{L}\,\frac{d\Sigma_{\sigma}^{L}}{2\pi
i/N}\,\big{(}\Sigma_{\sigma}^{L}\big{)}^{N}\exp\left[-N\left(\Sigma_{\sigma}^{L}G_{\sigma}^{L}-\frac{1}{q}(G_{\sigma}^{L})^{q}\right)\right]$
$\displaystyle=\,$ $\displaystyle\int
dG_{\sigma}^{L}\,e^{\frac{N}{q}(G_{\sigma}^{L})^{q}}\big{(}-N^{-1}\partial_{G_{\sigma}^{L}}\big{)}^{N}\delta(G_{\sigma}^{L})$
$\displaystyle=\,$
$\displaystyle\,\frac{N!}{N^{N}}\frac{(N/q)^{N/q}}{(N/q)!}\,.$ (E.3)

Therefore, combining these integral results, we find that the two-point
function at $\sigma=0$ is given by

$\displaystyle\big{\langle}Z_{L}Z_{R}\big{\rangle}_{J}^{\sigma=0}\,$
$\displaystyle=\,\frac{J_{0}^{\frac{2N}{q}}}{2^{N}}\left(\frac{N!}{N^{N}}\frac{(N/q)^{N/q}}{(N/q)!}\right)^{2}$
$\displaystyle\approx\,2^{-N}J_{0}^{\frac{2N}{q}}\,q\,e^{-2N\left(1-\frac{1}{q}\right)}\,,$
(E.4)

where for the last line, we used the large $N$ approximation besides the
factor $2^{-N}J_{0}^{2N/q}$. This result agrees with the saddle-point
evaluation (4.21).

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