††institutetext: Department of Physics, Shinshu University, 3-1-1 Asahi,
Matsumoto 390-8621, Japan

# End of the world brane in double scaled SYK

Kazumi Okuyama<EMAIL_ADDRESS>
###### Abstract

We study the end of the world (EOW) brane in double scaled SYK (DSSYK) model.
We find that the boundary state of EOW brane is a coherent state of the
$q$-deformed oscillators and the associated orthogonal polynomial is the
continuous big $q$-Hermite polynomial. In a certain scaling limit, the big
$q$-Hermite polynomial reduces to the Whittaker function, which reproduces the
wavefunction of JT gravity with an EOW brane. We also compute the half-
wormhole amplitude in DSSYK and show that the amplitude is decomposed into the
trumpet and the factor coming from the EOW brane.

##  1 Introduction

The double scaled SYK (DSSYK) model Cotler:2016fpe ; Berkooz:2018jqr  is a
useful toy model for the study of quantum gravity and holography. As shown in
Berkooz:2018jqr , DSSYK is exactly solvable by using the technique of the
chord diagrams and the transfer matrix. DSSYK allows us to explore the high
energy regime of holography and quantum gravity, beyond the realm of JT
gravity and the Schwarzian modes at low energy. The bulk dual of DSSYK has a
peculiar feature: the length scale is quantized in units of the coupling
$\lambda$ and the geodesic length is replaced by the discrete chord number $n$
Lin:2022rbf . In Jafferis:2022wez , it is also suggested that the length of
geodesic loop $b$ is replaced by a discrete length $\mathfrak{b}$ as well.

To understand the bulk dual of DSSYK better, in this paper we consider the end
of the world (EOW) brane in DSSYK. EOW branes in JT gravity were very useful
for the study of black hole information problem and the Page curve of Hawking
radiation Penington:2019kki . If we quantize JT gravity in the presence of an
EOW brane, the problem boils down to a quantum mechanical problem of a
particle moving in the Morse potential, and the wavefunction is given by the
Whittaker function Gao:2021uro . We will show that the DSSYK analogue of the
Whittaker function is the continuous big $q$-Hermite polynomial
$H_{n}(x,a|q)$, where $a$ is related to the tension $\mu$ of EOW brane by

$\displaystyle a=q^{\mu+\frac{1}{2}}.$ (1)

We also find that the boundary state $|B_{a}\rangle$ of the EOW brane is a
coherent state of the $q$-deformed oscillator $A_{\pm}$

$\displaystyle A_{-}|B_{a}\rangle=a|B_{a}\rangle,$ (2)

where $A_{\pm}$ acts on the chord number state $|n\rangle$ as a creation and
annihilation operator of chords. We find that the JT gravity amplitude of the
half-wormhole ending on the EOW brane Gao:2021uro  has a complete parallel in
DSSYK. We find that the half-wormhole amplitude in DSSYK is decomposed into
the trumpet and the factor $\frac{a^{\mathfrak{b}}}{1-q^{\mathfrak{b}}}$
coming from the EOW brane

$\beta$EOW$=$$\beta$$\mathfrak{b}$$Z_{\text{trumpet}}$$\times~{}\frac{a^{\mathfrak{b}}}{1-q^{\mathfrak{b}}}$,$\displaystyle\sum_{\mathfrak{b}=0}^{\infty}$
(3)

where the trumpet partition function is given by

$\beta$$\mathfrak{b}$$Z_{\text{trumpet}}$$\displaystyle=~{}~{}\int_{0}^{\pi}\frac{d\theta}{2\pi}e^{-\beta
E(\theta)}2\cos(\mathfrak{b}\theta)$. (4)

Note that the integral over the length $b$ of geodesic loop is replaced by the
sum over the discrete length $\mathfrak{b}$ in (3).

This paper is organized as follows. In section 2, we review the transfer
matrix formalism of DSSYK. In section 3, we review the EOW brane in JT gravity
and the computation of the half-wormhole amplitude in Gao:2021uro . In section
4, we consider the EOW brane in DSSYK. We find that the wavefunction in the
presence of EOW brane is given by the big $q$-Hermite polynomial. We also find
that the boundary state of EOW brane is a coherent state of the $q$-deformed
oscillators. In section 5, we compute the half-wormhole amplitude in DSSYK.
Along the way, we find the trumpet partition function in DSSYK. In section 6,
we study the cylinder amplitude in DSSYK, which is obtained by gluing two
trumpets. Finally we conclude in section 7. In appendix A we summarize useful
formulae of the $q$-Pochhammer symbols. In appendix B we summarize useful
properties of the continuous big $q$-Hermite polynomial.

##  2 Review of DSSYK

We first review the result of DSSYK Berkooz:2018jqr . SYK model is defined by
the Hamiltonian for $N$ Majorana fermions $\psi_{i}~{}(i=1,\cdots,N)$ with
all-to-all $p$-body interaction

$\displaystyle H=\mathrm{i}^{p/2}\sum_{1\leq i_{1}<\cdots<i_{p}\leq
N}J_{i_{1}\cdots i_{p}}\psi_{i_{1}}\cdots\psi_{i_{p}},$ (5)

where $J_{i_{1}\cdots i_{p}}$ is a random coupling drawn from the Gaussian
distribution. DSSYK is defined by the scaling limit

$\displaystyle
N,p\to\infty\quad\text{with}\quad\lambda=\frac{2p^{2}}{N}:\text{fixed}.$ (6)

SYK model is exactly solvable in this double scaling limit. As shown in
Berkooz:2018jqr , the ensemble average of the moment $\operatorname{tr}H^{k}$
reduces to a counting problem of the intersection number of chord diagrams

$\displaystyle\langle\operatorname{tr}H^{k}\rangle_{J}=\sum_{\text{chord
diagrams}}q^{\\#(\text{intersections})},$ (7)

where $q$ is given by

$\displaystyle q=e^{-\lambda}.$ (8)

One can introduce a matter field in the bulk which is dual to an operator in
DSSYK. One simple example is the length $s$ strings of Majorana fermions

$\displaystyle\mathcal{O}_{\Delta}=\mathrm{i}^{s/2}\sum_{i_{1}\cdots
i_{s}}K_{i_{1}\cdots i_{s}}\psi_{i_{1}}\cdots\psi_{i_{s}}$ (9)

with Gaussian random coefficients $K_{i_{1}\cdots i_{s}}$ which is drawn
independently from the random coupling $J_{i_{1}\cdots i_{p}}$ in the SYK
Hamiltonian. The effect of this operator can be made finite by taking the
scaling limit

$\displaystyle
p,s\to\infty\quad\text{with}\quad\Delta=\frac{s}{p}:\text{fixed}.$ (10)

In this limit, the random average of the correlator
$\operatorname{tr}(e^{-\beta_{2}H}\mathcal{O}_{\Delta}e^{-\beta_{1}H}\mathcal{O}_{\Delta})$
can be computed using the technique of the chord diagram

$\displaystyle\langle\operatorname{tr}(e^{-\beta_{2}H}\wick{\c{\cO}_{\Delta}e^{-\beta_{1}H}\c{\cO}_{\Delta}})\rangle_{J}=\sum_{\text{chord
diagrams}}q^{\\#(H\text{-}H~{}\text{intersections})}q^{\Delta\\#(H\text{-}\mathcal{O}~{}\text{intersections})}.$
(11)

As shown in Berkooz:2018jqr , the counting problem of the chord diagram can be
solved by introducing the transfer matrix $T$

$\displaystyle T=\frac{A_{-}+A_{+}}{\sqrt{1-q}},$ (12)

where $A_{\pm}$ are the $q$-deformed oscillators

$\displaystyle A_{-}A_{+}-qA_{+}A_{-}=1-q.$ (13)

$A_{\pm}$ acts on the states $\\{|n\rangle\\}_{n=0,1,\cdots}$ as

$\displaystyle A_{-}|n\rangle=\sqrt{1-q^{n}}|n-1\rangle,\quad
A_{+}|n\rangle=\sqrt{1-q^{n+1}}|n+1\rangle,$ (14)

where $n$ labels the number of chords and $A_{\pm}$ creates/annihilates the
chord. Then the disk partition function of DSSYK is written as

$\displaystyle Z(\beta)=\langle\operatorname{tr}e^{-\beta
H}\rangle_{J}=\langle 0|e^{\beta T}|0\rangle.$ (15)

One can diagonalize $T$ by using the $q$-Hermite polynomial

$\displaystyle\langle\theta|n\rangle=\frac{H_{n}(\cos\theta|q)}{\sqrt{(q;q)_{n}}},$
(16)

which is orthogonal with respect to the measure $\mu(\theta)$ defined by
111See appendix A for the definition of the $q$-Pochhammer symbol.

$\displaystyle\mu(\theta)=(q,e^{\pm 2\mathrm{i}\theta};q)_{\infty},$ (17)

and the orthogonality relation reads

$\displaystyle\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)\langle
n|\theta\rangle\langle\theta|m\rangle=\langle n|m\rangle=\delta_{n,m}.$ (18)

Note that $|\theta\rangle$ is normalized as

$\displaystyle\langle\theta|\theta^{\prime}\rangle=\frac{2\pi}{\mu(\theta)}\delta(\theta-\theta^{\prime}),$
(19)

and the resolution of the identity is written as

$\displaystyle 1=\sum_{n=0}^{\infty}|n\rangle\langle
n|=\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)|\theta\rangle\langle\theta|.$
(20)

The transfer matrix $T$ is diagonal in the $|\theta\rangle$-basis

$\displaystyle T|\theta\rangle=-E(\theta)|\theta\rangle,$ (21)

where the eigenvalue $E(\theta)$ is given by

$\displaystyle E(\theta)=-u_{0}\cos\theta,\quad u_{0}=\frac{2}{\sqrt{1-q}}.$
(22)

One can see that the energy spectrum is supported in the range
$E\in[-u_{0},u_{0}]$ and $u_{0}$ specifies the edge of the spectrum. Now the
disk partition function (15) becomes

$\displaystyle Z(\beta)=\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)e^{-\beta
E(\theta)}.$ (23)

One can also compute the correlator of matter operators in DSSYK using the
technique of the chord diagrams and the transfer matrix $T$. After the Wick
contraction of $K_{i_{1},\cdots,i_{s}}$ in (9), the bi-local operator
$\mathcal{B}_{\beta,\Delta}=\wick{\c{\cO}_{\Delta}e^{-\beta
H}\c{\cO}_{\Delta}}$ becomes Berkooz:2018jqr

$\displaystyle\mathcal{B}_{\beta,\Delta}$
$\displaystyle=\sum_{n,m,i=0}^{\infty}(q^{2\Delta};q)_{i}\sqrt{\Bigl{[}~{}\begin{matrix}n+i\\\
i\end{matrix}~{}\Bigr{]}_{q}\Bigl{[}~{}\begin{matrix}m+i\\\
i\end{matrix}~{}\Bigr{]}_{q}}|m+i\rangle\langle
m|q^{\Delta\widehat{N}}e^{\beta T}q^{\Delta\widehat{N}}|n\rangle\langle n+i|,$
(24)

where $\widehat{N}$ is the number operator

$\displaystyle\widehat{N}|n\rangle=n|n\rangle.$ (25)

As shown in Berkooz:2018jqr , $\mathcal{B}_{\beta,\Delta}$ is invariant under
the time translation and commutes with $T$

$\displaystyle\bigl{[}T,\mathcal{B}_{\beta,\Delta}\bigr{]}=0.$ (26)

This implies that $T$ and $\mathcal{B}_{\beta,\Delta}$ are simultaneously
diagonalized in the $|\theta\rangle$-basis and the eigenvalue of
$\mathcal{B}_{\beta,\Delta}$ is given by

$\displaystyle\mathcal{B}_{\beta,\Delta}|\theta\rangle$
$\displaystyle=|\theta\rangle\langle\theta|q^{\Delta\widehat{N}}e^{\beta
T}|0\rangle$
$\displaystyle=|\theta\rangle\int_{0}^{\pi}\frac{d\theta^{\prime}}{2\pi}\langle\theta|q^{\Delta\widehat{N}}|\theta^{\prime}\rangle
e^{-\beta E(\theta^{\prime})},$ (27)

where the factor $\langle\theta|q^{\Delta\widehat{N}}|\theta^{\prime}\rangle$
in (27) is given by

$\displaystyle\langle\theta|q^{\Delta\widehat{N}}|\theta^{\prime}\rangle=\frac{(q^{2\Delta};q)_{\infty}}{(q^{\Delta}e^{\mathrm{i}(\pm\theta\pm\theta^{\prime})};q)_{\infty}}.$
(28)

This factor also appears in the two-point function of matter operators
$\mathcal{O}_{\Delta}$

$\displaystyle\langle
0|e^{\beta_{1}T}\mathcal{B}_{\beta_{2},\Delta}|0\rangle=\langle
0|e^{\beta_{1}T}q^{\Delta\widehat{N}}e^{\beta_{2}T}|0\rangle=\int\prod_{i=1,2}\frac{d\theta_{i}}{2\pi}e^{-\beta_{i}E(\theta_{i})}\langle\theta_{1}|q^{\Delta\widehat{N}}|\theta_{2}\rangle.$
(29)

See also Goel:2023svz ; Mukhametzhanov:2023tcg ; Okuyama:2023bch  for the
semi-classical limit of matter correlators in DSSYK.

##  3 EOW brane in JT gravity

In this section, we review the wavefunction of JT gravity in the presence of
an EOW brane Gao:2021uro . As shown in Gao:2021uro , quantization of JT
gravity with an EOW brane leads to the Hamiltonian with Morse potential and
the eigenvalue equation for this Hamiltonian reads

$\displaystyle-k^{2}\Psi=\left[\partial_{L}^{2}-\mu
e^{-L}-\frac{1}{4}e^{-2L}\right]\Psi,$ (30)

where $L$ denotes the renormalized geodesic length between the EOW brane and
the $AdS_{2}$ boundary where SYK lives, and $\mu$ is the tension of the brane.
Note that $L$ can be negative since $L$ is a renormalized length. One can show
that (30) is solved by the Whittaker function

$\displaystyle\Psi_{k}(L)=e^{\frac{1}{2}L}W_{-\mu,\mathrm{i}k}(e^{-L}),$ (31)

which is normalized as

$\displaystyle\int_{-\infty}^{\infty}dL\frac{\rho(k)}{2\pi}|\Gamma(\mu+1/2+\mathrm{i}k)|^{2}\Psi_{k}(L)\Psi_{k^{\prime}}(L)$
$\displaystyle=\delta(k-k^{\prime}),$ (32)
$\displaystyle\int_{0}^{\infty}\frac{dk}{2\pi}\rho(k)|\Gamma(\mu+1/2+\mathrm{i}k)|^{2}\Psi_{k}(L)\Psi_{k}(L^{\prime})$
$\displaystyle=\delta(L-L^{\prime}),$

where $\rho(k)$ is the density of states of the Schwarzian theory
Stanford:2017thb

$\displaystyle\rho(k)=\frac{2k\sinh(2\pi k)}{\pi}.$ (33)

As discussed in Gao:2021uro , one can extract the coupling between the trumpet
and the EOW brane by considering a half-wormhole ending on the EOW brane. The
half-wormhole amplitude is defined by

$\displaystyle Z_{\text{half-
wormhole}}(\beta,\mu)=\int_{-\infty}^{\infty}dL\int_{0}^{\infty}\frac{dk}{2\pi}\rho(k)|\Gamma(\mu+1/2+\mathrm{i}k)|^{2}\Psi_{k}(L)e^{-\beta
k^{2}}\Psi_{k}(L),$ (34)

which is schematically depicted as

$=$$\displaystyle\int_{-\infty}^{\infty}dL$$L$$L$EOW$\beta$EOW$\beta$ (35)

The integral over $L$ represents a trace on the Hilbert space spanned by the
states $\\{\Psi_{k}(L)\\}$ and the top and the bottom of the left hand side of
(35) are periodically identified. Thus the corresponding spacetime is a half-
wormhole as shown on the right hand side of (35). This trace is obviously
divergent due to the first relation in (32). However, as demonstrated in
Gao:2021uro , one can rewrite this amplitude as an integral over the length
$b$ of geodesic loop and interpret this divergence as a UV divergence coming
from $b=0$.

Here we review the computation in Gao:2021uro , which serves as a warm-up for
a similar computation of DSSYK in section 5. As we will see in section 5, the
computation below has a complete parallel in DSSYK. First, we rewrite
$\Psi_{k}(L)^{2}$ appearing in (34) by using the formula 6.647-1 in
gradshteyn2014table

$\displaystyle|\Gamma(\mu+1/2+\mathrm{i}k)|^{2}\Psi_{k}(L)^{2}$
$\displaystyle=2\int_{0}^{\infty}dxx^{\mu-\frac{1}{2}}(2z+x)^{-\mu-\frac{1}{2}}e^{-x-z}K_{2\mathrm{i}k}\Bigl{(}\sqrt{x(2z+x)}\Bigr{)},$
(36)

where $z=e^{-L}$ and $K_{\nu}(x)$ denotes the modified Bessel function of the
second kind. If we further make a change of variable

$\displaystyle x=\frac{2z}{e^{b}-1},$ (37)

then (36) becomes

$\displaystyle|\Gamma(\mu+1/2+\mathrm{i}k)|^{2}\Psi_{k}(L)^{2}$
$\displaystyle=\int_{0}^{\infty}db\,e^{-\mu b}\mathcal{G}(b,e^{-L},k),$ (38)

with

$\displaystyle\mathcal{G}(b,z,k)=\frac{1}{2\sinh\frac{b}{2}}2K_{2\mathrm{i}k}\left(\frac{z}{\sinh\frac{b}{2}}\right)e^{-z\coth\frac{b}{2}}.$
(39)

Then the half-wormhole amplitude (34) is written as

$\displaystyle Z_{\text{half-
wormhole}}(\beta,\mu)=\int_{-\infty}^{\infty}dL\int_{0}^{\infty}db\,e^{-\mu
b}\mathcal{K}_{\beta}(b,e^{-L})$ (40)

where $\mathcal{K}_{\beta}(b,z)$ is the so-called boundary particle propagator
Suh:2020lco ; Yang:2018gdb ; Kitaev:2018wpr 222See also (2.42) in
Kolchmeyer:2023gwa . Our $\mathcal{K}_{\beta}(b,e^{-w})$ corresponds to
$\mathcal{K}^{t}_{\beta}(b,w|0,w)$ in Kolchmeyer:2023gwa .

$\displaystyle\mathcal{K}_{\beta}(b,z)=\int_{0}^{\infty}\frac{dk}{2\pi}\rho(k)e^{-\beta
k^{2}}\mathcal{G}(b,z,k).$ (41)

The integral over $L$ can be performed by using the integration formula of the
modified Bessel function

$\displaystyle\int_{0}^{\infty}\frac{dx}{x}2K_{2\mathrm{i}k}(x)e^{-x\cosh\frac{b}{2}}=\frac{2\cos(bk)}{\rho(k)},$
(42)

which was heavily utilized in the study of minimal string theory
Martinec:2003ka . Using this formula, we find

$\displaystyle\int_{-\infty}^{\infty}dL\,\mathcal{G}(b,e^{-L},k)=\frac{1}{2\sinh\frac{b}{2}}\frac{2\cos(bk)}{\rho(k)}.$
(43)

Using (43), the $L$ integral of the boundary particle propagator in (41)
becomes 333The fact that the $L$ integral of the boundary particle propagator
$\mathcal{K}_{\beta}(b,e^{-L})$ is proportional to the trumpet was also
pointed out in Kolchmeyer:2023gwa .

$\displaystyle\int_{-\infty}^{\infty}dL\mathcal{K}_{\beta}(b,e^{-L})=\frac{1}{2\sinh\frac{b}{2}}Z_{\text{trumpet}}(\beta,b),$
(44)

where the trumpet partition function is given by

$\displaystyle Z_{\text{trumpet}}(\beta,b)$
$\displaystyle=\int_{0}^{\infty}\frac{dk}{2\pi}e^{-\beta
k^{2}}2\cos(bk)=\frac{e^{-\frac{b^{2}}{4\beta}}}{\sqrt{4\pi\beta}}.$ (45)

This agrees with the known result of trumpet in JT gravity Saad:2019lba , as
expected. Note that $\rho(k)$ is canceled in this computation and the trumpet
is independent of $\rho(k)$. Finally, we arrive at the desired expression of
the half-wormhole amplitude

$\displaystyle Z_{\text{half-
wormhole}}(\beta,\mu)=\int_{0}^{\infty}db\frac{e^{-\mu
b}}{2\sinh\frac{b}{2}}Z_{\text{trumpet}}(\beta,b).$ (46)

This amplitude has a divergence coming from $b=0$, which is interpreted as a
UV divergence in the bulk gravitational theory.

##  4 EOW brane in DSSYK

In this section, we consider the EOW brane in DSSYK. It turns out that the big
$q$-Hermite polynomial $H_{n}(x,a|q)$ plays the role of the Whittaker function
in (31).

###  4.1 Triple scaling limit

The continuous big $q$-Hermite polynomial $H_{n}(x,a|q)$ is a one parameter
generalization of the $q$-Hermite polynomial $H_{n}(x|q)$. $H_{n}(x,a|q)$
reduces to $H_{n}(x|q)$ when $a=0$. $H_{n}(x,a|q)$ is defined recursively by
the three-term recursion relation

$\displaystyle
2xH_{n}(x,a|q)=H_{n+1}(x,a|q)+aq^{n}H_{n}(x,a|q)+(1-q^{n})H_{n-1}(x,a|q),$
(47)

with the initial condition $H_{-1}(x,a|q)=0,H_{0}(x,a|q)=1$. An important
property of $H_{n}(x,a|q)$ is the orthogonality relation

$\displaystyle\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)\frac{1}{(ae^{\pm\mathrm{i}\theta};q)_{\infty}}H_{n}(\cos\theta,a|q)H_{m}(\cos\theta,a|q)=(q;q)_{n}\delta_{n,m}.$
(48)

It is convenient to introduce the normalized wavefunction

$\displaystyle\psi_{n}(\theta)=\frac{H_{n}(\cos\theta,a|q)}{\sqrt{(q;q)_{n}}},$
(49)

which satisfies the orthogonality relation

$\displaystyle\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)\frac{1}{(ae^{\pm\mathrm{i}\theta};q)_{\infty}}\psi_{n}(\theta)\psi_{m}(\theta)=\delta_{n,m}.$
(50)

We will show that $\psi_{n}(\theta)$ is the DSSYK analogue of the wavefunction
$\Psi_{k}(L)$ (31) of JT gravity with an EOW brane. In terms of the
wavefunction $\psi_{n}(\theta)$ in (49), the three-term recursion relation
(47) is written as

$\displaystyle
2\cos\theta\psi_{n}(\theta)=\sqrt{1-q^{n+1}}\psi_{n+1}(\theta)+aq^{n}\psi_{n}(\theta)+\sqrt{1-q^{n}}\psi_{n-1}(\theta).$
(51)

To see the semi-classical picture of (51), we consider the triple scaling
limit

$\displaystyle\lambda\to 0,\quad a=q^{\mu+\frac{1}{2}},\quad q^{n}=\lambda
e^{-L},\quad\theta=\lambda k,$ (52)

with $\mu,L,$ and $k$ fixed. If we denote $\psi_{n}(\theta)=\Psi_{k}(L)$
444Readers should not be confused by the fact that the argument and the
subscript are exchanged between $\psi_{n}(\theta)$ and $\Psi_{k}(L)$. in this
limit, the right hand side of (51) becomes

$\displaystyle\sqrt{1-\lambda e^{-L-\lambda}}\Psi_{k}(L+\lambda)+\lambda
e^{-\lambda(\mu+1/2)-L}\Psi_{k}(L)+\sqrt{1-\lambda e^{-L}}\Psi_{k}(L-\lambda)$
(53) $\displaystyle=$
$\displaystyle\,2\Psi_{k}(L)+\lambda^{2}\left[\partial_{L}^{2}-\mu
e^{-L}-\frac{1}{4}e^{-2L}\right]\Psi_{k}(L)+\mathcal{O}(\lambda^{3}).$

Similarly, the left hand side of (51) is expanded as

$\displaystyle 2\cos(\lambda
k)\Psi_{k}(L)=2\Psi_{k}(L)-\lambda^{2}k^{2}\Psi_{k}(L)+\mathcal{O}(\lambda^{3}).$
(54)

Finally, equating the $\mathcal{O}(\lambda^{2})$ term of the both sides of
(51) we find

$\displaystyle-k^{2}\Psi_{k}(L)=\left[\partial_{L}^{2}-\mu
e^{-L}-\frac{1}{4}e^{-2L}\right]\Psi_{k}(L),$ (55)

which is exactly the equation (30) for JT gravity with an EOW brane! This
strongly suggests that $\psi_{n}(\theta)$ in (49) is the DSSYK analogue of the
EOW wavefunction $\Psi_{k}(L)$. For the small $a$ expansion to be well-
defined, from (52) we see that $\mu$ should satisfy the inequality
$\mu>-\frac{1}{2}$. This inequality also appears in JT gravity with an EOW
brane Gao:2021uro .

As another consistency check, let us see that the measure factor correctly
reduces to that of the Whittaker function. As shown in Berkooz:2018jqr , in
the scaling limit (52), $\mu(\theta)$ reduces to the Schwarzian density of
states $\rho(k)$ in (33)

$\displaystyle\mu(\theta)~{}\to~{}\rho(k).$ (56)

There is an extra factor $1/(ae^{\pm\mathrm{i}\theta};q)_{\infty}$ in (48),
which becomes

$\displaystyle\frac{1}{(ae^{\pm\mathrm{i}\theta};q)_{\infty}}=\frac{1}{(q^{\mu+1/2\pm\mathrm{i}k};q)_{\infty}}.$
(57)

We observe that (57) is proportional to the $q$-Gamma function

$\displaystyle\Gamma_{q}(z)=(1-q)^{1-z}\frac{(q;q)_{\infty}}{(q^{z};q)_{\infty}}.$
(58)

Since $\Gamma_{q}(z)$ reduces to the ordinary Gamma function $\Gamma(z)$ in
the limit $q\to 1$, the scaling limit of (57) becomes

$\displaystyle\frac{1}{(ae^{\pm\mathrm{i}\theta};q)_{\infty}}~{}\to~{}|\Gamma(\mu+1/2+\mathrm{i}k)|^{2},$
(59)

up to an overall factor. This correctly reproduces the measure factor of
Whittaker function in (32).

###  4.2 Boundary state of EOW brane

Let us consider a physical interpretation of the extra measure factor (57). We
propose that the extra factor (57) is the $\theta$-representation of the
boundary state $|B_{a}\rangle$ of the EOW brane

$\displaystyle\frac{1}{(ae^{\pm\mathrm{i}\theta};q)_{\infty}}=\langle\theta|B_{a}\rangle.$
(60)

Using the formula (122), the left hand side of (60) is expanded as

$\displaystyle\frac{1}{(ae^{\pm\mathrm{i}\theta};q)_{\infty}}=\sum_{n=0}^{\infty}\frac{a^{n}}{(q;q)_{n}}H_{n}(\cos\theta|q)=\sum_{n=0}^{\infty}\frac{a^{n}}{\sqrt{(q;q)_{n}}}\langle\theta|n\rangle,$
(61)

where we used (16) in the last equality. From (60) and (61), $|B_{a}\rangle$
is given by

$\displaystyle|B_{a}\rangle$
$\displaystyle=\sum_{n=0}^{\infty}\frac{a^{n}}{\sqrt{(q;q)_{n}}}|n\rangle.$
(62)

Then the partition function in the presence of the extra factor
$\langle\theta|B_{a}\rangle$ is written as

$\displaystyle\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)\langle\theta|B_{a}\rangle
e^{-\beta E(\theta)}=\langle 0|e^{\beta T}|B_{a}\rangle.$ (63)

This expression suggests that $|B_{a}\rangle$ is the boundary state of EOW
brane and (63) is the partition function of half-disk ending on the EOW brane

$\langle 0|e^{\beta T}|B_{a}\rangle=$EOW$\beta$ (64)

where the blue thick line represents the worldline of the EOW brane.

The boundary state $|B_{a}\rangle$ in (62) has an interesting
characterization: it is a coherent state of the $q$-deformed oscillator

$\displaystyle A_{-}|B_{a}\rangle=a|B_{a}\rangle,$ (65)

which can be shown by using the first relation of (14). $|B_{a}\rangle$ is
also written as

$\displaystyle|B_{a}\rangle$
$\displaystyle=\sum_{n=0}^{\infty}\frac{a^{n}A_{+}^{n}}{(q;q)_{n}}|0\rangle=\frac{1}{(aA_{+};q)_{\infty}}|0\rangle,$
(66)

where we used the relation

$\displaystyle A_{+}^{n}|0\rangle=\sqrt{(q;q)_{n}}|n\rangle,$ (67)

and the formula (112).

In the $a\to 0$ limit, $|B_{a}\rangle$ reduces to the Hartle-Hawking vacuum
$|0\rangle$ Okuyama:2022szh

$\displaystyle\lim_{a\to 0}|B_{a}\rangle=|0\rangle,$ (68)

and the half-disk amplitude (63) reduces to the disk amplitude (15)

$\displaystyle\lim_{a\to 0}\langle 0|e^{\beta T}|B_{a}\rangle=\langle
0|e^{\beta T}|0\rangle.$ (69)

Thus, $a=0$ corresponds to the absence of EOW brane. In fact, the big
$q$-Hermite polynomial $H_{n}(x,a|q)$ reduces to the $q$-Hermite polynomial
$H_{n}(x|q)$ when $a=0$. From the relation $a=q^{\mu+\frac{1}{2}}$, the
absence of EOW brane corresponds to $\mu=\infty$, not $\mu=0$. This is similar
to the situation considered in Berkooz:2020uly : a heavy object in the bulk
corresponds to a projection to the vacuum $|0\rangle$. For instance, the large
dimension limit of the two-point function (29) of matter operator
$\mathcal{O}_{\Delta}$ becomes

$\displaystyle\lim_{\Delta\to\infty}\langle
0|e^{\beta_{1}T}q^{\Delta\widehat{N}}e^{\beta_{2}T}|0\rangle=\langle
0|e^{\beta_{1}T}|0\rangle\langle 0|e^{\beta_{2}T}|0\rangle.$ (70)

Note that the right hand side is the product of two disk amplitudes. (70) is
schematically depicted as

$\begin{aligned} \leavevmode\hbox to244.82pt{\vbox
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One might naively think that the heavy operator corresponds to a black hole in
the bulk, but this is not the case in DSSYK: if we try to create a black hole
by putting two heavy operators on the boundary, then the bulk spacetime is
pinched and splits into two pieces and the black hole never forms!

###  4.3 Representation of the $q$-oscillator algebra

Let us consider the action of $A_{\pm}$ on the continuous big $q$-Hermite
polynomials. To this end, it is convenient to introduce the notation

$\displaystyle H_{n}(\cos\theta|q)=\langle\theta|H_{n}\rangle,\quad
H_{n}(\cos\theta,a|q)=\langle\theta|H_{n},a\rangle.$ (72)

The action of $A_{\pm}$ on the basis $|H_{n}\rangle$ is given by

$\displaystyle A_{+}|H_{n}\rangle=|H_{n+1}\rangle,\quad
A_{-}|H_{n}\rangle=(1-q^{n})|H_{n-1}\rangle.$ (73)

From (118), the big $q$-Hermite state $|H_{n},a\rangle$ is written as a linear
combination of the $q$-Hermite state $|H_{k}\rangle$

$\displaystyle|H_{n},a\rangle=\sum_{k=0}^{n}\Bigl{[}~{}\begin{matrix}n\\\
k\end{matrix}~{}\Bigr{]}_{q}(-a)^{k}q^{\frac{1}{2}k(k-1)}|H_{n-k}\rangle.$
(74)

From (73), one can deduce the action of $A_{\pm}$ on the big $q$-Hermite basis

$\displaystyle
A_{+}|H_{n},a\rangle=|H_{n+1},a\rangle+aq^{n}|H_{n},a\rangle,\quad
A_{-}|H_{n},a\rangle=(1-q^{n})|H_{n-1},a\rangle,$ (75)

from which one can easily check that the $q$-oscillator algebra (13) is
realized on the big $q$-Hermite basis

$\displaystyle(A_{-}A_{+}-qA_{+}A_{-})|H_{n},a\rangle=(1-q)|H_{n},a\rangle.$
(76)

We should stress that the parameter $a$ does not deform the algebra itself.
Instead, $a$ labels a representation of the $q$-oscillator algebra (13).

From (75), one can see that $A_{+}+A_{-}$ acts on the big $q$-Hermite basis as

$\displaystyle(A_{+}+A_{-})|H_{n},a\rangle=|H_{n+1},a\rangle+aq^{n}|H_{n},a\rangle+(1-q^{n})|H_{n-1},a\rangle,$
(77)

which is exactly the same combination appearing in the three-term recursion of
the big $q$-Hermite polynomial (47). Thus the transfer matrix
$T=\frac{A_{+}+A_{-}}{\sqrt{1-q}}$ is diagonal on the basis of
$H_{n}(\cos\theta,a|q)$ as well, with the same eigenvalue
$\frac{2\cos\theta}{\sqrt{1-q}}$ as the original one without EOW brane.
Namely, we can use the common transfer matrix
$T=\frac{A_{+}+A_{-}}{\sqrt{1-q}}$ for both with and without the EOW brane.
The only difference is the presence of the extra factor
$\langle\theta|B_{a}\rangle$ for the case with the EOW brane

$\displaystyle\text{pure~{}DSSYK}:$
$\displaystyle\quad\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)e^{-\beta
E(\theta)}(\cdots),$ (78) $\displaystyle\text{DSSYK with EOW brane}:$
$\displaystyle\quad\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)\langle\theta|B_{a}\rangle
e^{-\beta E(\theta)}(\cdots).$

##  5 Half-wormhole in DSSYK

Let us consider the DSSYK analogue of the computation in section 3. Using the
dictionary

$\displaystyle\int_{-\infty}^{\infty}dL{}{}$
$\displaystyle\to~{}~{}\sum_{n=0}^{\infty},$ (79)
$\displaystyle\int_{0}^{\infty}\frac{dk}{2\pi}{}{}$
$\displaystyle\to~{}~{}\int_{0}^{\pi}\frac{d\theta}{2\pi},$
$\displaystyle\rho(k){}{}$ $\displaystyle\to~{}~{}\mu(\theta),$
$\displaystyle|\Gamma(\mu+1/2+\mathrm{i}k)|^{2}{}{}$
$\displaystyle\to~{}~{}\langle\theta|B_{a}\rangle,$
$\displaystyle\Psi_{k}(L){}{}$ $\displaystyle\to~{}~{}\psi_{n}(\theta),$
$\displaystyle e^{-\beta k^{2}}{}{}$ $\displaystyle\to~{}~{}e^{-\beta
E(\theta)},$

the half-wormhole amplitude (34) is translated into DSSYK as

$\displaystyle Z_{\text{half-
wormhole}}(\beta,a)=\sum_{n=0}^{\infty}\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)\langle\theta|B_{a}\rangle\psi_{n}(\theta)e^{-\beta
E(\theta)}\psi_{n}(\theta).$ (80)

This is schematically depicted as

$=$$\displaystyle\sum_{n=0}^{\infty}$$n$$n$$\beta$EOW$\beta$EOW (81)

The sum over $n$ in (80) is divergent due to the relation
$\sum_{n}\psi_{n}(\theta)^{2}\propto\delta(\theta-\theta)$. However, we can
rewrite the amplitude (80) as a sum over the discrete length $\mathfrak{b}$ of
geodesic loop, in a similar manner as the computation in the JT gravity case
in section 3. In analogy with (38), we define
$\mathcal{G}(\mathfrak{b},n,\theta)$ by

$\displaystyle\langle\theta|B_{a}\rangle\psi_{n}(\theta)^{2}$
$\displaystyle=\sum_{\mathfrak{b}=0}^{\infty}a^{\mathfrak{b}}\mathcal{G}(\mathfrak{b},n,\theta).$
(82)

Using the small $a$ expansion of $\langle\theta|B_{a}\rangle$ in (61) and the
expansion of big $q$-Hermite polynomial in (118), together with the
linearization formula of the $q$-Hermite polynomials (124), after some algebra
we find

$\displaystyle\mathcal{G}(\mathfrak{b},n,\theta)=\sum_{k=0}^{n}\frac{(q;q)_{n}}{(q;q)_{n-k}^{2}(q;q)_{k}}\frac{q^{k\mathfrak{b}}}{(q;q)_{\mathfrak{b}}}H_{2n-2k+\mathfrak{b}}(\cos\theta|q).$
(83)

Using the DSSYK analogue of the relation (42) 555 (86) can be proved as
follows. Expanding the left hand side of (86) as
$\displaystyle\frac{2\cos(\mathfrak{b}\theta)}{\mu(\theta)}=\sum_{n=0}^{\infty}c_{n}\frac{H_{n}(\cos\theta|q)}{(q;q)_{n}},$
(84) $c_{n}$ is obtained by using the orthogonality relation of
$H_{n}(\cos\theta|q)$ $\displaystyle
c_{n}=\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)\left(\sum_{j=0}^{\infty}c_{j}\frac{H_{j}(\cos\theta|q)}{(q;q)_{j}}\right)H_{n}(\cos\theta|q)=\int_{0}^{\pi}\frac{d\theta}{2\pi}2\cos(\mathfrak{b}\theta)H_{n}(\cos\theta|q).$
(85) Finally, using the expansion of $H_{n}(\cos\theta|q)$ in (119) we find
(86).

$\displaystyle\frac{2\cos(\mathfrak{b}\theta)}{\mu(\theta)}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{H_{2k+\mathfrak{b}}(\cos\theta|q)}{(q;q)_{k}(q;q)_{k+\mathfrak{b}}},$
(86)

we find

$\displaystyle\sum_{n=0}^{\infty}\mathcal{G}(\mathfrak{b},n,\theta)=\frac{1}{1-q^{\mathfrak{b}}}\frac{2\cos(\mathfrak{b}\theta)}{\mu(\theta)}.$
(87)

This is the analogue of (43). It is also natural to define the DSSYK analogue
of the boundary particle propagator

$\displaystyle\mathcal{K}_{\beta}(\mathfrak{b},n)=\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)e^{-\beta
E(\theta)}\mathcal{G}(\mathfrak{b},n,\theta).$ (88)

In analogy with (44), the sum over $n$ of
$\mathcal{K}_{\beta}(\mathfrak{b},n)$ is written as

$\displaystyle\sum_{n=0}^{\infty}\mathcal{K}_{\beta}(\mathfrak{b},n)$
$\displaystyle=\frac{1}{1-q^{\mathfrak{b}}}Z_{\text{trumpet}}(\beta,\mathfrak{b})$
(89)

where we introduced the DSSYK analogue of the trumpet

$\displaystyle
Z_{\text{trumpet}}(\beta,\mathfrak{b})=\int_{0}^{\pi}\frac{d\theta}{2\pi}e^{-\beta
E(\theta)}2\cos(\mathfrak{b}\theta).$ (90)

Note that in this computation $\mu(\theta)$ is canceled and
$Z_{\text{trumpet}}(\beta,\mathfrak{b})$ is independent of $\mu(\theta)$.
Finally, the half-wormhole amplitude (80) is written as

$\displaystyle Z_{\text{half-wormhole}}(\beta,a)=$
$\displaystyle\sum_{\mathfrak{b}=0}^{\infty}\frac{a^{\mathfrak{b}}}{1-q^{\mathfrak{b}}}Z_{\text{trumpet}}(\beta,\mathfrak{b}).$
(91)

See (3) for a pictorial representation of (91). If we set
$a=q^{\mu+\frac{1}{2}}$, (91) becomes

$\displaystyle Z_{\text{half-wormhole}}(\beta,q^{\mu+\frac{1}{2}})=$
$\displaystyle\sum_{\mathfrak{b}=0}^{\infty}\frac{q^{\mu\mathfrak{b}}}{q^{-\frac{1}{2}\mathfrak{b}}-q^{\frac{1}{2}\mathfrak{b}}}Z_{\text{trumpet}}(\beta,\mathfrak{b}),$
(92)

which is the DSSYK analogue of (46) where the integral over $b$ is replaced by
the sum over $\mathfrak{b}$. Note that the $\mathfrak{b}=0$ term in (92) is
divergent: this is an analogy of the fact that the integral in (46) has a UV
divergence coming from $b=0$.

##  6 Trumpet and cylinder in DSSYK

Plugging $E(\theta)=-u_{0}\cos\theta$ (22) into (90), the trumpet of DSSYK
becomes 666From the usual large $N$ counting, the genus-$g$ with $n$-boundary
amplitude comes with the factor of $e^{(2-2g-n)S_{0}}$, where
$e^{S_{0}}=2^{N/2}$ is the dimension of the Hilbert space of $N$ Majorana
fermions. For instance, the trumpet with $(g,n)=(0,2)$ scales as $e^{0S_{0}}$
which is suppressed relative to the disk with $(g,n)=(0,1)$ of the order
$e^{S_{0}}$. In this paper, we have suppressed this factor $e^{(2-2g-n)S_{0}}$
for simplicity.

$\displaystyle
Z_{\text{trumpet}}(\beta,\mathfrak{b})=\int_{0}^{\pi}\frac{d\theta}{2\pi}e^{\beta
u_{0}\cos\theta}2\cos(\mathfrak{b}\theta)=I_{\mathfrak{b}}(\beta u_{0}),$ (93)

where $I_{\nu}(x)$ denotes the modified Bessel function of the first kind. As
discussed in Jafferis:2022wez , the cylinder amplitude is obtained as a
“double trumpet” by gluing two trumpets 777See also Berkooz:2020fvm  for the
study of non-planar corrections in DSSYK.

$\displaystyle Z_{\text{cylinder}}(\beta_{1},\beta_{2})$
$\displaystyle=\sum_{\mathfrak{b}=0}^{\infty}\mathfrak{b}Z_{\text{trumpet}}(\beta_{1},\mathfrak{b})Z_{\text{trumpet}}(\beta_{2},\mathfrak{b}).$
(94)

This is schematically depicted as

$\beta_{1}$$\beta_{2}$$\mathfrak{b}$$\mathfrak{b}$$\displaystyle\sum_{\mathfrak{b}=0}^{\infty}\mathfrak{b}\times$
(95)

Using the relation

$\displaystyle\frac{2\nu}{x}I_{\nu}(x)=I_{\nu-1}(x)-I_{\nu+1}(x),$ (96)

we can perform the summation over $\mathfrak{b}$ in (94) in a closed form

$\displaystyle Z_{\text{cylinder}}(\beta_{1},\beta_{2})$
$\displaystyle=\frac{\beta_{1}\beta_{2}u_{0}}{2(\beta_{1}+\beta_{2})}\Bigl{[}I_{0}(\beta_{1}u_{0})I_{1}(\beta_{2}u_{0})+I_{1}(\beta_{1}u_{0})I_{0}(\beta_{2}u_{0})\Bigr{]}.$
(97)

Note that (97) is equal to the cylinder amplitude in the Gaussian matrix model
Akemann:2001st ; Giombi:2009ms ; Okuyama:2018aij . This is expected since the
cylinder amplitude is independent of the details of the matrix model potential
and it depends only on the endpoints $\pm u_{0}$ of the spectrum
Ambjorn:1990ji ; Brezin:1993qg . 888As argued in Okuyama:2021eju , trumpet in
topological gravity is also independent of the background $\\{t_{k}\\}$.

By using the asymptotic form of the modified Bessel function

$\displaystyle I_{\nu}(x)\sim\frac{e^{x}}{\sqrt{2\pi x}},\qquad(x\gg 1)$ (98)

the low temperature limit of $Z_{\text{cylinder}}(\beta_{1},\beta_{2})$
becomes

$\displaystyle
Z_{\text{cylinder}}(\beta_{1},\beta_{2})\sim\frac{\sqrt{\beta_{1}\beta_{2}}}{2\pi(\beta_{1}+\beta_{2})}e^{(\beta_{1}+\beta_{2})u_{0}}.$
(99)

This agrees with the known result of cylinder amplitude in JT gravity
Saad:2019lba , up to the factor $e^{(\beta_{1}+\beta_{2})u_{0}}$ coming from
the threshold energy $E=-u_{0}$.

In the semi-classical limit $\lambda\to 0$ with

$\displaystyle\theta=\lambda k,\quad
b=\lambda\mathfrak{b},\quad\beta\to\beta\lambda^{-\frac{3}{2}}$ (100)

the trumpet in (93) reduces to

$\displaystyle
Z_{\text{trumpet}}(\beta,b)=\int_{0}^{\infty}\frac{dk}{2\pi}e^{-\beta
k^{2}}2\cos(bk),$ (101)

up to an overall factor. This reproduces the known result of trumpet in JT
gravity, as expected. In the semi-classical limit (100), the sum over
$\mathfrak{b}$ in (94) is replaced by the integral over $b$

$\displaystyle Z_{\text{cylinder}}(\beta_{1},\beta_{2})$
$\displaystyle=\int_{0}^{\infty}dbbZ_{\text{trumpet}}(\beta_{1},b)Z_{\text{trumpet}}(\beta_{2},b),$
(102)

which is also consistent with the JT gravity result.

One advantage of DSSYK over the continuum approach of JT gravity is that one
can easily incorporate the effect of matter field within the framework of
chord diagrams. According to Jafferis:2022wez , if we include the loop
correction of the matter field with dimension $\Delta$, the cylinder amplitude
is modified as

$\displaystyle Z_{\text{cylinder}}(\beta_{1},\beta_{2},\Delta)$
$\displaystyle=\sum_{\mathfrak{b}=0}^{\infty}\mathfrak{b}I_{\mathfrak{b}}(\beta_{1}u_{0})I_{\mathfrak{b}}(\beta_{2}u_{0})\sum_{n=0}^{\infty}\left(\frac{q^{\Delta\mathfrak{b}}}{1-q^{\mathfrak{b}}}\right)^{n}$
(103)
$\displaystyle=\sum_{\mathfrak{b}=0}^{\infty}\mathfrak{b}I_{\mathfrak{b}}(\beta_{1}u_{0})I_{\mathfrak{b}}(\beta_{2}u_{0})\frac{1-q^{\mathfrak{b}}}{1-q^{\mathfrak{b}}-q^{\Delta\mathfrak{b}}}.$

In the semi-classical limit (100), the denominator is replaced by
$1-e^{-b}-e^{-\Delta b}$ which vanishes at a certain value of $b$. This leads
to a divergence which was interpreted as a Hagedorn divergence in
Jafferis:2022wez . On the other hand, the denominator in (103) does not vanish
for a generic value of $q,\Delta$ with $\mathfrak{b}\in\mathbb{Z}_{\geq 0}$.
Thus we expect that the Hagedorn divergence from the matter loop is
regularized in DSSYK. We do not know how to perform the sum over
$\mathfrak{b}$ in (103) in a closed form, but we can easily evaluate (103)
numerically by truncating the summation up to some cut-off
$\mathfrak{b}\leq\mathfrak{b}_{\text{cut}}$. In figure 1, we show the plot of
spectral form factor (SFF)
$Z_{\text{cylinder}}(\beta+\mathrm{i}t,\beta-\mathrm{i}t,\Delta)$ as a
function of $t$. When $\Delta$ is large, the effect of matter loop is
negligible and the SFF exhibits a linear growth called “ramp” (see figure
1(a)). When $\Delta$ becomes small, the effect of matter loop drastically
changes the behavior of SFF; as we can see from figure 1(b), SFF exhibits an
oscillatory behavior as a function of $t$. It would be interesting to
understand the bulk gravitational interpretation of this oscillation.

(a) $\Delta=2$ (b) $\Delta=\frac{1}{4}$

Figure 1:  Plot of the spectral form factor
$Z_{\text{cylinder}}(\beta+\mathrm{i}t,\beta-\mathrm{i}t,\Delta)$ as a
function of $t$. We set $u_{0}=1,\beta=1,q=1/2$ in (103) and evaluate it
numerically by truncating the summation up to $\mathfrak{b}_{\text{cut}}=100$.

##  7 Conclusions and outlook

In this paper, we have studied the EOW brane in DSSYK. We found that the
boundary state of EOW brane is a coherent state of the $q$-deformed oscillator
and the associated orthogonal polynomial is the continuous big $q$-Hermite
polynomial $H_{n}(x,a|q)$. We showed that $\psi_{n}(\theta)$ in (49) reduces
to the Whittaker function in the triple scaling limit (52). We also computed
the half-wormhole amplitude in DSSYK and found the decomposition (91) into the
trumpet and the factor $\frac{a^{\mathfrak{b}}}{1-q^{\mathfrak{b}}}$, which is
a complete parallel of the decomposition of half-wormhole amplitude (46) in JT
gravity.

There are many interesting open questions. Recall that $(L,b)$ and
$(n,\mathfrak{b})$ are related by (see (52) and (100))

$\displaystyle L=\lambda n+\log\lambda,\quad b=\lambda\mathfrak{b}.$ (104)

Thus the bulk geodesic lengths $L$ and $b$ are quantized in units of $\lambda$
and there is a minimal length $\lambda$ in the bulk spacetime. It would be
interesting to understand the physical meaning of this minimal length. 999In
Berkooz:2022mfk  it is discussed that the bulk spacetime of DSSYK becomes a
non-commutative $AdS_{2}$ with a minimal length scale $\lambda$.

We can generalize our setup by introducing two EOW branes and consider the
amplitude

$\displaystyle\frac{\langle B_{a}|e^{\beta T}|B_{b}\rangle}{\langle
B_{a}|B_{b}\rangle}=\int_{0}^{\pi}\frac{d\theta}{2\pi}\mu(\theta)\frac{\langle
B_{a}|\theta\rangle\langle\theta|B_{b}\rangle}{\langle
B_{a}|B_{b}\rangle}e^{-\beta E(\theta)}.$ (105)

Then the measure factor becomes

$\displaystyle\mu(\theta)\frac{\langle
B_{a}|\theta\rangle\langle\theta|B_{b}\rangle}{\langle
B_{a}|B_{b}\rangle}=\mu(\theta)\frac{(ab;q)_{\infty}}{(ae^{\pm\mathrm{i}\theta},be^{\pm\mathrm{i}\theta};q)_{\infty}},$
(106)

and the associated orthogonal polynomial is the Al-Salam-Chihara polynomial
$Q_{n}(x,a,b|q)$. This type of boundary condition has been considered in
sasamoto1999one  in the context of ASEP (asymmetric simple exclusion process)
under the identification

$\displaystyle\langle W|=\langle B_{a}|,\quad|V\rangle=|B_{b}\rangle.$ (107)

It would be interesting to pursue this analogy with ASEP further. One missing
ingredient in this analogy is the relation to the spin chain. In the context
of ASEP, the problem can be mapped to the matrix product state of open XXZ
spin chain and the quantum group naturally arises in the spin chain picture
(see e.g. Crampe:2014aoa  for a review). It would be interesting to understand
the role of spin chain in DSSYK, if any.

As discussed in Jafferis:2022wez , one can construct various amplitudes of
DSSYK by gluing basic building blocks such as the trumpet, in a similar manner
as the JT gravity matrix model Saad:2019lba . However, we still do not have a
complete set of the “Lego block” of DSSYK. For instance, we do not know the
DSSYK analogue of the Weil-Petersson volume, which played an important role
for the construction of higher genus amplitudes of JT gravity. Also, we have
only briefly studied the effect of matter loop in section 6. Clearly, more
work needs to be done to better understand the structure of DSSYK.

###### Acknowledgements.

The author would like to thank Satoru Odake and Kenta Suzuki for discussion.
This work was supported in part by JSPS Grant-in-Aid for Transformative
Research Areas (A) “Extreme Universe” 21H05187 and JSPS KAKENHI Grant
22K03594.

##  Appendix A Useful formulae

The $q$-Pochhammar symbol is defined by

$\displaystyle(a;q)_{n}=\prod_{i=0}^{n-1}(1-aq^{i}).$ (108)

We also use the notation

$\displaystyle(a_{1},\cdots,a_{r};q)_{n}=\prod_{j=1}^{r}(a_{j};q)_{n}$ (109)

and we often use the shorthand notation such as

$\displaystyle(ae^{\mathrm{i}(\pm\theta_{1}\pm\theta_{2})};q)_{\infty}=\prod_{s_{1},s_{2}=\pm}(ae^{\mathrm{i}s_{1}\theta_{1}+\mathrm{i}s_{2}\theta_{2}};q)_{\infty}.$
(110)

The $q$-binomial coefficient is defined by

$\displaystyle\Bigl{[}~{}\begin{matrix}n\\\
k\end{matrix}~{}\Bigr{]}_{q}=\frac{(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}.$ (111)

Below we summarize useful summation formulas

$\displaystyle\frac{1}{(t;q)_{\infty}}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{t^{n}}{(q;q)_{n}}.$ (112)
$\displaystyle(t;q)_{\infty}$
$\displaystyle=\sum_{n=0}^{\infty}(-1)^{n}q^{\frac{1}{2}n(n-1)}\frac{t^{n}}{(q;q)_{n}}.$
(113)

$\displaystyle(a;q)_{n}$
$\displaystyle=\sum_{k=0}^{n}(-a)^{k}q^{\frac{1}{2}k(k-1)}\Bigl{[}~{}\begin{matrix}n\\\
k\end{matrix}~{}\Bigr{]}_{q}.$ (114)
$\displaystyle\frac{(ta;q)_{\infty}}{(t;q)_{\infty}}$
$\displaystyle=\sum_{n=0}^{\infty}(a;q)_{n}\frac{t^{n}}{(q;q)_{n}}.$ (115)

##  Appendix B Continuous big $q$-Hermite polynomials

We summarize some useful properties of the continuous big $q$-Hermite
polynomial $H_{n}(x,a|q)$. $H_{n}(x,a|q)$ is defined recursively by the three-
term recurrence relation

$\displaystyle
2xH_{n}(x,a|q)=H_{n+1}(x,a|q)+aq^{n}H_{n}(x,a|q)+(1-q^{n})H_{n-1}(x,a|q),$
(116)

with the initial condition $H_{-1}(x,a|q)=0,H_{0}(x,a|q)=1$. When $a=0$, the
continuous big $q$-Hermite polynomial reduces to the $q$-Hermite polynomial

$\displaystyle H_{n}(x,0|q)=H_{n}(x|q).$ (117)

Setting $x=\cos\theta$, we can write down the explicit form of $H_{n}(x,a|q)$

$\displaystyle H_{n}(\cos\theta,a|q)$
$\displaystyle=\sum_{k=0}^{n}\Bigl{[}~{}\begin{matrix}n\\\
k\end{matrix}~{}\Bigr{]}_{q}(ae^{\mathrm{i}\theta};q)_{k}e^{\mathrm{i}(n-2k)\theta}$
(118) $\displaystyle=\sum_{k=0}^{n}\Bigl{[}~{}\begin{matrix}n\\\
k\end{matrix}~{}\Bigr{]}_{q}(-a)^{k}q^{\frac{1}{2}k(k-1)}H_{n-k}(\cos\theta|q).$

When $a=0$, the first line of (118) reduces to

$\displaystyle H_{n}(\cos\theta|q)=\sum_{k=0}^{n}\Bigl{[}~{}\begin{matrix}n\\\
k\end{matrix}~{}\Bigr{]}_{q}e^{\mathrm{i}(n-2k)\theta}.$ (119)

An important property of $H_{n}(x,a|q)$ is the orthogonality relation

$\displaystyle\int_{0}^{\pi}\frac{d\theta}{2\pi}\frac{\mu(\theta)}{(ae^{\pm\mathrm{i}\theta};q)_{\infty}}H_{n}(x,a|q)H_{m}(x,a|q)=(q;q)_{n}\delta_{n,m},$
(120)

where $\mu(\theta)=(q,e^{\pm 2\mathrm{i}\theta};q)_{\infty}$. The generating
function of $H_{n}(x,a|q)$ is given by

$\displaystyle\sum_{n=0}^{\infty}\frac{t^{n}}{(q;q)_{n}}H_{n}(\cos\theta,a|q)=\frac{(ta;q)_{\infty}}{(te^{\pm\mathrm{i}\theta};q)_{\infty}}.$
(121)

As a special case of this relation, we find the generation function of the
$q$-Hermite polynomial

$\displaystyle\sum_{n=0}^{\infty}\frac{t^{n}}{(q;q)_{n}}H_{n}(\cos\theta|q)=\frac{1}{(te^{\pm\mathrm{i}\theta};q)_{\infty}}.$
(122)

Also note that $H_{n}(x,a|q)$ satisfies a $q$-difference equation for the
parameter $a$ floreanini1995algebraic

$\displaystyle H_{n}(x,aq|q)-H_{n}(x,a|q)=a(1-q^{n})H_{n-1}(x,aq|q).$ (123)

The product of $q$-Hermite polynomials is expanded as (see e.g.
ismail1988askey )

$\displaystyle H_{n}(x|q)H_{m}(x|q)$
$\displaystyle=\sum_{k=0}^{\min(n,m)}\frac{(q;q)_{n}(q;q)_{m}}{(q;q)_{n-k}(q;q)_{m-k}(q;q)_{k}}H_{n+m-2k}(x|q),$
(124)

which is called the linearization formula.

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# Localization properties in Lieb lattices and their extensions

Jie Liu<EMAIL_ADDRESS>Xiaoyu Mao<EMAIL_ADDRESS>Jianxin Zhong<EMAIL_ADDRESS>Rudolf A. Römer<EMAIL_ADDRESS>School of Physics and Optoelectronics, Xiangtan University, Xiangtan 411105,
China  Department of Physics, University of Warwick, Coventry, CV4 7AL, United
Kingdom

###### Abstract

We study the localization properties of generalized, two- and three-
dimensional Lieb lattices, $\mathcal{L}_{2}(n)$ and $\mathcal{L}_{3}(n)$,
$n=1,2,3$ and $4$, at energies corresponding to flat and dispersive bands
using the transfer matrix method (TMM) and finite size scaling (FSS). We find
that the scaling properties of the flat bands are different from scaling in
dispersive bands for all $\mathcal{L}_{d}(n)$. For the $d=3$ dimensional case,
states are extended for disorders $W$ down to $W=0.01t$ at the flat bands,
indicating that the disorder can lift the degeneracy of the flat bands
quickly. The phase diagram with periodic boundary condition for
$\mathcal{L}_{3}(1)$ looks similar to the one for hard boundaries [1]. We
present the critical disorder $W_{c}$ at energy $E=0$ and find a decreasing
$W_{c}$ for increasing $n$ for $\mathcal{L}_{3}(n)$, up to $n=3$. Last, we
show a table of FSS parameters including so-called irrelevant variables; but
the results indicate that the accuracy is too low to determine these reliably.

###### keywords:

Localization , Flat band , Phase diagram , Finite size scaling

††journal: Annals of Physics Special Issue: Localisation 2020

##  1 Introduction

Flat band systems, in which the absence of a dispersion in the band structure
leads to a highly dramatic macroscopic degeneracy, have at the flat band
energy an effectively reduced kinetic energy. Hence other terms in the
Hamiltonian can become prominent, such as many-body interactions. This
mechanism leads to a convenient construction of various platform for studying
many-body physics, such as the fractional quantum Hall effect [2, 3, 4], spin
liquids [5, 6], ferromagnetism [7, 8], and superconductivity [9, 10, 11]. In
recent years, artificial lattices [12], for instance, in photonic [13, 14, 15,
16, 17, 18] and cold atom systems [19, 20, 21], allow to realize
experimentally also the probing of the _novel_ many-body problems.

As is well known, states in a flat band are localized [14] because of the high
degeneracy. Hence disorder, which one should expect to destroy this
degeneracy, might also, at least initially, destroy the localization. So what
will happen after disorder is being included in the Hamiltonians describing
these localized flat bands is an interesting question.

Flat band system can be constructed in many models [2, 3, 4, 8, 22]. The Lieb
lattice [10, 23, 24] is one of the simplest and most famous two-dimensional
flat band system. Actually, the $CuO_{2}$ plane of cuprate superconductors is
also a Lieb lattice, namely $\mathcal{L}_{2}(1)$ in our notation. It contains
three atoms per unit cell as shown in Fig. 1(a). In the figure, we also
introduce its extensions $\mathcal{L}_{2}(n)$, $n=2,3$ and $4$, shown in
panels (b)-(d), respectively. The three-dimensional Lieb lattice and its
extensions are also shown in Fig. 1(a)-(d). The number of flat bands is
related to the number of central red atoms between two nearest blue atoms. In
short, for $\mathcal{L}_{d}(n)$, the number of flat band is $n$ with $d-1$
degeneracy. The central energy at $E=0$ is part of a flat band for
$\mathcal{L}_{2}(1)$, $\mathcal{L}_{2}(3)$, $\mathcal{L}_{3}(1)$ and
$\mathcal{L}_{3}(3)$ while it remains in a dispersive band for
$\mathcal{L}_{2}(2)$, $\mathcal{L}_{2}(4)$, $\mathcal{L}_{3}(2)$ and
$\mathcal{L}_{3}(4)$[25, 1].

This paper is organized as follows. In section 2 and section 3 we will discuss
the two-dimensional and three-dimensional Lieb lattices and their extensions,
respectively. The conclusion is given in section 4.

(a) (b) (c) (d)

Figure 1: Schematic representation of Lieb and extended Lieb lattices
$\mathcal{L}_{d}(n)$. The lightly coloured spheres highlight the situation in
2D while together with the fully coloured spheres they represent the 3D
lattices. Blue spheres denote undecimated lattices site while the red sphere
will be decimated in the TMM formulation. The dark lines are guides to the eye
only. The labels (a), (b), (c) and (d) denote
$\mathcal{L}_{2}(1)$/$\mathcal{L}_{3}(1)$,
$\mathcal{L}_{2}(2)$/$\mathcal{L}_{3}(2)$,
$\mathcal{L}_{2}(3)$/$\mathcal{L}_{3}(3)$ and
$\mathcal{L}_{2}(4)$/$\mathcal{L}_{3}(4)$, respectively.

##  2 The two-dimensional Lieb lattice and its extensions

###  2.1 Previous results for disordered $\mathcal{L}_{2}(n)$ lattices

In our previous paper [25], we study the localization properties of
$\mathcal{L}_{2}(n)$ in detail. Using direct diagonalization for small system
sizes, we compute the density of states (DOS). We can see that in the presence
of disorder, the interplay between flat bands and dispersive bands is
prominent for all $\mathcal{L}_{2}(n)$, $n=1,2,3$ and $4$. The disorder can
quickly lift the degeneracy of flat bands and make the states merge with the
neighboring dispersive bands. When $W\gtrsim 2$, the flat band DOS loses its
peaks distinguishing and becomes part of a large bulk DOS.

Next, we use a renormalized transfer-matrix method (TMM) and compute the
reduced localization lengths $\Lambda_{M}(E,W)=\lambda(E,W)/M$, where $M$
corresponds to the width of the quasi-1D transfer-matrix strip. For all
$\mathcal{L}_{d}(n)$, we find that for $W\gtrsim t$ all states are localized.
The localization lengths for states at flat band energies are about one order
of magnitude smaller than for states from dispersive bands. We employ
traditional one-parameter finite-size scaling methods to estimate a scaling
parameter $\xi(E,W)$ [26]. The $\Lambda(E,W)/M$ can be described after scaling
by a single scaling branch, corresponding to a fully localized behaviour.
After fitting the $\xi$ with disorder $W$, we use three fitting forms, power
low form as $\xi(W)\propto W^{-2}$[27], a non-universal form
$\xi(W)=aW^{-\alpha}\exp(\beta W^{-\gamma})$ and a constraint form
$\xi(W)=aW^{-2}\exp(\beta W^{-1})$. For $\mathcal{L}_{2}(1)$ and
$\mathcal{L}_{2}(3)$ at the flat band energy $E=0$, we find that the usual
power low form $\xi(W)\propto W^{-2}$ for 1D localization fits well for
disorder around $1<W<2$. However, for $\mathcal{L}_{2}(2)$ and
$\mathcal{L}_{2}(4)$ at $E=0$, which is an energy in a dispersive band for
these lattices, none of the fits gives a convincing result.

###  2.2 Scaling function $\Lambda_{M}$ vs reduced correlation length
$\xi_{M}$ for $\mathcal{L}_{2}(n)$

The scaled localization lengths $\Lambda_{M}(0,W)$ as a function of scaled
correlation length $\xi/M$ for $\mathcal{L}_{2}(n)$, $n=1,2,3$ and $4$ are
shown in Fig. 2(a) at energy $E=0$. This corresponds to flat bands for
$\mathcal{L}_{2}(1)$ and $\mathcal{L}_{2}(3)$ and dispersive band for
$\mathcal{L}_{2}(2)$ and $\mathcal{L}_{2}(4)$. The $\Lambda_{M}(0,W)$ data all
show the behaviour for localized states, scaling as
$\Lambda_{M}(0,W)\propto\xi(0,W)/M$ for large system sizes and large
disorders. In this regime, the behaviour of states in the flat bands and in
the dispersive bands is similar as shown also in the inset graph of said
figure.

(a) (b)

Figure 2: (a) Scaled localization length $\Lambda_{M}(0,W)$ versus $\xi/M$ at
energy $E=0$ for $\mathcal{L}_{2}(1)$ (red $\bigcirc$), $\mathcal{L}_{2}(2)$
(blue $\square$), $\mathcal{L}_{2}(3)$ (green $\triangledown$) and
$\mathcal{L}_{2}(4)$ (purple $\diamondsuit$). For clarity, lines show all data
points while symbols denote only about $15\%$ of all data. Inset: the detail
of small reduced correlation length. (b) Scaling parameters $\xi(0,W)$ for the
same Lieb lattices as in (a). The dashed lines represent the power law fit
functions $ax^{b}$. Error bars are within symbol size in both panels.

###  2.3 Scaling parameter $\xi$ vs disorder $W$ for $\mathcal{L}_{2}(n)$

The disorder dependence of the scaling parameter $\xi$ for small disorders
$t\leq W\leq 2t$ is shown in Fig. 2(b), computed from the $\Lambda_{M}(E=0,W)$
data of Fig. 2(a). We see that the behaviour of $\xi$ for $\mathcal{L}_{2}(1)$
is comparable to $\mathcal{L}_{2}(3)$; both are well-described by a power law
$ax^{b}$ with exponent approximately $b\sim-2$, similar to localization
properties of a standard 1D Anderson model [27]. This might suggest that the
localization behaviour of these flat band states at least for weak disorder is
similar to the 1D behaviour. On the other hand, for the dispersive states of
$\mathcal{L}_{2}(2)$ and $\mathcal{L}_{2}(4)$ at $E=0$, we find that the
values of $\xi$ are orders of magnitude larger than for $\mathcal{L}_{2}(1)$
and $\mathcal{L}_{2}(3)$. The simple power-law also does not fit anymore and
we rather see the more standard behaviour of a 2D Anderson model [27] with a
quickly diverging $\xi$ when $W\rightarrow 0$. Nevertheless, for both flat
band and dispersive band energies, the fits are not very robust and have
rather small $p$ values of $<10^{-10}$. This shows that the true form of the
behaviour of $\xi(W\rightarrow 0)$ is yet to be determined.

###  2.4 Density of states without Gaussian broadening for
$\mathcal{L}_{2}(n)$

$\mathcal{L}_{2}(n)$ $\mathcal{L}_{3}(n)$
(a)
(b)
(c)
(d)

Figure 3:  Normalized bar chart histograms of the $(E,W)$-dependence of the
density of states (DOS) for (a) $\mathcal{L}_{2}(1)$ and $\mathcal{L}_{3}(1)$,
(b) $\mathcal{L}_{2}(2)$ and $\mathcal{L}_{3}(2)$, (c) $\mathcal{L}_{2}(3)$
and $\mathcal{L}_{3}(3)$, and (d) $\mathcal{L}_{2}(4)$ and
$\mathcal{L}_{3}(4)$. The colors denote different DOS values ranging from zero
(deep purple) to maximal (white). Bin widths in $(E,W)$ directions have been
chosen for representational clarity.

The results of the disorder-averaged density of states (DOS), calculated with
direct diagonalization, are shown in Fig. 3. The system sizes are
$M^{2}=13^{2}$, $10^{2}$, $9^{2}$, $8^{2}$ for $\mathcal{L}_{2}(n)$, $n=1,2,3$
and $4$, respectively. The disorder ranges from $W=0$ to $W=5.2$ in step of
$0.05$ using $300$ independent random samples. We can see the prominent peaks
of flat bands close to $W=0$ have been largely vanished when the disorder
reaches up to $W=2$ for all $\mathcal{L}_{2}(n)$, $n=1,2,3$ and $4$.

##  3 Three-dimensional Lieb lattice and its extensions

###  3.1 Previous results for disordered $\mathcal{L}_{3}(n)$ lattices

In Ref. [1], we investigated the DOS, the localization properties and the
phase diagrams for the 3D Lieb lattices $\mathcal{L}_{3}(n)$, $n=1,2,3$ and
$4$ as shown in Fig. 1. Obviously, the main difference to the 2D case is the
existence of an Anderson metal insulator transition (MIT) in 3D Lieb lattices
[1]. Details of the TMM construction for $\mathcal{L}_{3}(n)$ can also be
found in Ref. [1] along with finite-size scaling results for the universal
critical exponent $\nu$ of the localization lengths. We found $\nu$ to be in
good agreement with the currently accepted value of $\nu=1.590(1.579,1.602)$
[28, 29] for the Anderson transition. In the following, we shall elaborate on
the stability of the phase diagrams to a change in boundary conditions,
highlight the various positions of the transitions in the phase diagrams,
comment on the possibility of FSS with irrelevant scaling parameters and also
provide the DOS without Gaussian-broadening.

###  3.2 Phase diagram with periodic boundary condition for
$\mathcal{L}_{3}(1)$

The phase diagrams given in Ref. [1] have been computed for the
$\mathcal{L}_{3}(n)$ lattices with hard wall boundaries. In Fig. 4(a) we now
show a phase diagram for $\mathcal{L}_{3}(1)$ with periodic boundary
condition. The phases have been determined by the scaling behaviour of the
$\Lambda_{M}(E,W)$ for system sizes $M=4$, $M=6$ and $M=8$ with error $\leq
0.1\%$[30]. Comparing this to the results obtained with hard boundary
conditions, we find that it looks very similar as expected although the
extended region is a little wider in the $E$ axis. As for the hard wall case,
we can identify a re-entrant region around disorder $W=4$ and a shoulder at
$E\sim 3$ and $W\sim 6$. Hence as expected, the change in boundary conditions
does not change the phase diagrams appreciably already with the small system
sizes and modest disorder averages as used here and in Ref. [1].

(a) (b)

Figure 4:  (a) Phase diagram for $\mathcal{L}_{3}(1)$ in case of periodic
boundaries. The three solid and colored lines represent the approximate
location of the phase boundary estimated from small $M$, i.e. the blue
line/$\bigcirc$ comes from widths $M=4$ and $M=6$, the red line/$\times$ from
$M=4$ and $M=8$, and the green line/$+$ from $M=6$ and $M=8$. The shaded area
in the center contains extended states while states outside the phase boundary
are localized. The dashed lines on both sides are guides-to-the-eye for the
expected continuation of the phase boundary for $W<2$. The red short vertical
line at $E=0$ represents the position of the doubly-degenerate flat band. The
diamonds ($\blacklozenge$) denote the band edges for $W=0$, i.e.
$E_{\mathrm{min}}=-2\sqrt{3}$ and $E_{\mathrm{max}}=2\sqrt{3}$. The dotted
lines are the theoretical band edges $\pm\left(|E_{\mathrm{min}}|+W/2\right)$
and the forbidden areas below those band edges have been shaded. (b) Scaling
parameters $\xi$ versus disorder $W$ for $\mathcal{L}_{3}(1)$(red $\bigcirc$),
$\mathcal{L}_{3}(2)$ (blue $\square$) and $\mathcal{L}_{3}(3)$ (green
$\triangledown$) at energy $E=0$. The expansion parameters $n_{r}$, $n_{i}$,
$m_{r}$ and $m_{i}$ are the same as the highlighted line in Table 1.

###  3.3 Localization and extended transition with 0.01 $\leq$ $W$ $\leq$ 2.0

(a) (b)
(c) (d)

Figure 5: (a) Small $W$ behaviour of $\Lambda_{M}$ for $\mathcal{L}_{3}(1)$
with (a) energy $E=0.05$, (b) $E=0.1$, (c) $E=0.15$, and (d) for
$\mathcal{L}_{3}(2)$ at $E=1.05$ with disorder down to $0.01$ in steps of
$0.01$ and with error less than $1.0\%$. System sizes $M$ are $4$ (black
$\oplus$), $6$ (red $\diamond$), $8$ (green $\square$), $10$ (dark-blue
$\times$), $12$ (brown $\triangleleft$), $14$ (purple $\triangleright$). Error
bars are denoted with a solid line. Insets: increased disorder range up to
$W=2$ for the corresponding cases in the $4$ main panels.

For $W<1$, it is well known that the convergence of the TMM is very slow.
Hence results for appropriately small errors are hard to compute. Usually,
this is not a problem since, e.g., in the 3D Anderson model, the limit as
$W\rightarrow 0$ belongs trivially to the extended phase. However, for the
$\mathcal{L}_{3}(n)$ lattice, we know that at the flat band energies even at
$W=0$, we except compactly localized states [31, 32]. Hence it is interesting
to see if the localization properties at $W<1$ for flat band energies indicate
any possible “inverse” Anderson transition from extended states at $W\sim 1$
to localized states at small, but finite $W>0$.

In Ref. [1], we had shown that at the flat band energy $E=0$ for
$\mathcal{L}_{3}(1)$ and at $E=1$ for $\mathcal{L}_{3}(2)$, the $\Lambda_{M}$
increases with increasing $M$, indicating extended behaviour, down to
disorders as small as $W=0.01$. In Fig. 5(a-c), we now augment that result by
studying energies close by. For $\mathcal{L}_{3}(1)$ and $E=0.05$ we initially
find localized behaviour, e.g., $\Lambda_{M}$ decreasing with increasing $M$
up to $M=10$, but then reversing to extended behaviour for larger $M$. For
$E=0.1$ the reversal to the extended behaviour already starts at $M=8$ while
for $E=0.15$, only the extended behaviour remains. In Fig. 5(d), we see that
for $\mathcal{L}_{3}(1)$ at $E=1.05$, the $\Lambda_{M}$ decreases with
increasing $M$. This localized behaviour should vanish for larger $M$ values,
but it is at present beyond our computational capabilities.

We conclude that the presence of the compactly localized states at the flat
band energies certainly has an effect at small disorder, but for larger
disorder values, the broadening of the flat bands and the mixing with the
dispersive bands becomes dominant such that the character if the states is
extended down to disorders $W=0.01$. This is usually already true at $W\sim
0.1$.

###  3.4 Divergence of the scaling parameter $\xi(W)$

The behaviour of $\xi(W)$ for $\mathcal{L}_{3}(1)$, $\mathcal{L}_{3}(2)$ and
$\mathcal{L}_{3}(3)$ is given in Fig. 4. We can clearly see how the critical
disorder $W_{c}$ decreases from $8.59$ for $\mathcal{L}_{3}(1)$ to $5.96$ for
$\mathcal{L}_{3}(2)$ and finally to $4.79$ for $\mathcal{L}_{3}(3)$. This
suggests that a larger $n$ in $\mathcal{L}_{3}(n)$, i.e. a larger number of
additional (red, cp. Fig. 1) atoms, leads to stronger localization and hence
an MIT already for smaller values of $W_{c}$. It could be an interesting study
to estimate $W_{c}(n)$, particularly the limiting behaviour when
$n\rightarrow\infty$.

###  3.5 Scaling with irrelevant variables $n_{i}$, $m_{i}$

For high-precision estimates of critical properties, including $\nu$, it is by
now state of the art to include irrelevant scaling contributions, i.e. scaling
as $M^{-y}$ with $y>0$, in the FSS analysis. However, such FSS methods also
require large $M$ values to reliably model the irrelevant scaling. Due to the
complexity of the $\mathcal{L}_{3}(n)$ systems, only values of $M\leq 20$ have
been computed in Ref. [1]. For such sizes, adding irrelevant scaling variables
is usually not a net benefit. In Table 1, we show the results for FSS with and
without scaling. We note that although acceptable $p$ values can be obtained
for the fits with irrelevant scaling exponent $y$ included, in nearly all
cases, this results either in increased error estimates for the relevant
exponent $\nu$. Alternatively, one finds estimates for $y$ with very large
errors or very large values for $y$. Except for one case, the final estimate
for the physical quantity $\nu$ has hardly changed. Hence we conclude that for
the available $\Lambda_{M}$ data, the inclusion of irrelevant scaling
parameter $y$ does not necessarily add towards the accuracy of the estimates
for $\nu$. This confirms having made this choice in Ref. [1].

$\mathcal{L}_{3}(1)$
---
$\Delta M$ | $E$ | $\delta W$ | $n_{r}$ | $n_{i}$ | $m_{r}$ | $m_{i}$ | $W_{c}$ | CI($W_{c}$) | $\nu$ | CI($\nu$) | $y$ | CI($y$) | $p$
16-20 | 0 | 8.25-8.9 | 3 | 0 | 1 | 0 | 8.59 | $(58,61)$ | 1.6 | $(4,7)$ | 0 | 0 | $0.15$
16-20 | 0 | 8.25-8.9 | 2 | 1 | 1 | 1 | ${{8.71}}$ | $(57,84)$ | ${{1.3}}$ | $(0.8,1.8)$ | 4 | $(-2,10)$ | $0.86$
14-20 | 1 | 8.0-8.8 | 3 | 0 | 1 | 0 | 8.44 | $(42,45)$ | 1.6 | $(5,7)$ | 0 | 0 | $0.18$
14-20 | 1 | 8.0-8.8 | 3 | 2 | 1 | 1 | $8.48$ | $(45,50)$ | $1.8$ | $(6,9)$ | $6.9$ | $(6.6,7.1)$ | $0.77$
$\Delta M$ | $W$ | $\delta E$ | $n_{r}$ | $n_{i}$ | $m_{r}$ | $m_{i}$ | $E_{c}$ | CI($E_{c}$) | $\nu$ | CI($\nu$) | $y$ | CI($y$) | $p$
16-20 | 3 | 3.725-3.785 | 2 | 0 | 1 | 0 | 3.75 | $(74,75)$ | 1.7 | $(6,9)$ | 0 | 0 | $0.88$
16-20 | 3 | 3.725-3.785 | 3 | 2 | 1 | 2 | $3.75$ | $(74,75)$ | $1.5$ | $(0.6,2.5)$ | $2$ | $(-3,8)$ | $0.7$
16-20 | 6 | 3.04-3.11 | 1 | 0 | 1 | 0 | 3.08 | $(07,09)$ | 1.5 | $(1.0,2.1)$ | 0 | 0 | $0.14$
16-20 | 6 | 3.04-3.11 | 1 | 1 | 2 | 1 | $3.08$ | $(06,09)$ | $1.5$ | $(0.7,2.4)$ | $47$ | $(44,50)$ | $0.13$
$\mathcal{L}_{3}(2)$
$\Delta M$ | $E$ | $\delta W$ | $n_{r}$ | $n_{i}$ | $m_{r}$ | $m_{i}$ | $W_{c}$ | CI($W_{c}$) | $\nu$ | CI($\nu$) | $y$ | CI($y$) | $p$
12,14,18 | 0 | 5.85-6.05 | 2 | 0 | 2 | 0 | 5.96 | $(95,97)$ | 1.8 | $(1.5,2.0)$ | 0 | 0 | $0.08$
12,14,18 | 0 | 5.85-6.05 | 2 | 1 | 1 | 4 | $5.97$ | $(96,98)$ | $1.7$ | $(1.3,2.1)$ | $9$ | $(2,16)$ | $0.89$
$\Delta M$ | $W$ | $\delta E$ | $n_{r}$ | $n_{i}$ | $m_{r}$ | $m_{i}$ | $E_{c}$ | CI($E_{c}$) | $\nu$ | CI($\nu$) | $y$ | CI($y$) | $p$
10-14 | 4 | 1.6-1.8 | 2 | 0 | 1 | 0 | 1.70 | $(70,71)$ | 1.6 | $(4,7)$ | 0 | 0 | $0.18$
10-14 | 4 | 1.6-1.8 | 1 | 1 | 2 | 1 | $1.72$ | $(67,78)$ | $1.6$ | $(1.1,2.1)$ | $6$ | $(-18,31)$ | $0.38$
$\mathcal{L}_{3}(3)$
$\Delta M$ | $E$ | $\delta W$ | $n_{r}$ | $n_{i}$ | $m_{r}$ | $m_{i}$ | $W_{c}$ | CI($W_{c}$) | $\nu$ | CI($\nu$) | $y$ | CI($y$) | $p$
12-18 | 0 | 4.7–4.875 | 2 | 0 | 1 | 0 | 4.79 | $(78,80)$ | 1.6 | $(4,8)$ | 0 | 0 | $0.43$
12-18 | 0 | 4.7–4.875 | 2 | 1 | 1 | 2 | $4.79$ | $(78,80)$ | $1.6$ | $(4,8)$ | $8284$ | $(0,1)$ | $0.11$
Table 1: Critical parameters at the MIT for $\mathcal{L}_{3}(n)$, $n=1,2$ and
$3$. The columns are denoting the system width $M$, fixed $E$ (or $W$), the
range of $W$ (or $E$). The expansion orders $n_{r}$, $n_{i}$, $m_{r}$, $m_{i}$
are listed as well as the obtained critical disorders $W_{c}$ (or energies
$E_{c}$), their 95$\%$ confidence intervals (CI), the critical exponent $\nu$,
its CI, the irrelevant parameter $y$, its CI, and the goodness of fit
probability $p$. The confidence interval are given with one significant
decimal. For instance, $1.6(4,8)$ marks that the CI is $(1.4,1.8)$

###  3.6 Density of states without Gaussian broadening for
$\mathcal{L}_{3}(n)$

The results of for the DOS, calculated with exact diagonalization and without
applied Gaussian smoothing, are in Fig. 3. The system sizes are $M^{3}=5^{3}$,
$5^{3}$, $4^{3}$, $4^{3}$ for $\mathcal{L}_{3}(n)$, $n=1,2,3$ and $4$,
respectively. The disorder ranges are all from $W=0$ to $W=5.2$ in step of
$0.05$ and with $300$ samples for $n=1,2,3$ but only $100$ samples for
$\mathcal{L}_{3}(4)$ because of computing time limits. Again, the results are
very similar to the Gaussian-broadened DOS shown in Ref. [1].

##  4 Conclusions

We have studied the localization properties of the 2D and 3D extended Lieb
lattices. Clearly, the Lieb lattices exhibit stronger localization than their
more standard square and cubic Anderson lattices. This can be understood by
noting that the transport along the ”extra” sites as shown (by red spheres) in
Fig. 1 is essentially one-dimensional and hence subject to stronger
localization. Consequently, in 2D rather small $W$ values can still be studied
(most earlier TMM studies for the 2D Anderson model stop already around $W\sim
2$, cp. Fig. (3) of Ref. [33]). Details of the resulting FSS curves for small
$W$ are given in Fig. 2. In 3D, we similarly see that $W_{c}(n)$ decreases as
a function of $n$. Results for particularly small $W$ are shown in Fig. 5. Due
to the numerical complexity of the $\mathcal{L}_{d}(n)$ systems, scaling is
more challenging than in the Anderson models and only relatively small $M$
values can be reached. Table 1 shows that for the available data, there is no
need to include irrelevant scaling variables — within the accuracy of the
calculation, all estimates of the critical exponent agree with the value found
for the Anderson universality class [28, 29].

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