aainstitutetext: Department of Physics,
University of California, Berkeley,
California 94720, USA
bbinstitutetext: Department of Physics and Astronomy,
University of Kentucky,
505 Rose Street,
Lexington, KY, USA 40506
ccinstitutetext: The Racah Institute of Physics, The Hebrew University of
Jerusalem,
Jerusalem 91904, Israel

# Krylov complexity in quantum field theory, and beyond

Alexander Avdoshkin b Anatoly Dymarsky c Michael Smolkin

###### Abstract

We study Krylov complexity in various models of quantum field theory: free
massive bosons and fermions on flat space and on spheres, holographic models,
and lattice models with the UV-cutoff. In certain cases we find asymptotic
behavior of Lanczos coefficients, which goes beyond previously observed
universality. We confirm that in all cases the exponential growth of Krylov
complexity satisfies the conjectural inequality, which generalizes the
Maldacena-Shenker-Stanford bound on chaos. We discuss temperature dependence
of Lanczos coefficients and note that the relation between the growth of
Lanczos coefficients and chaos may only hold for the sufficiently late, truly
asymptotic regime governed by the physics at the UV cutoff. Contrary to
previous suggestions, we show scenarios when Krylov complexity in quantum
field theory behaves qualitatively differently from the holographic
complexity.

##  1 Introduction

Dynamics in Krylov space, and the associated K(rylov) complexity, have
recently emerged as a new probe of the chaotic dynamics of quantum systems.
Starting from an autocorrelation function $C(t)$ of a sufficiently simple,
e.g. local operator $A$, via recursion method one can define Lanczos
coefficients $b_{n}$, which control and characterize the operator growth in
the Krylov subspace. The original work Parker_2019  proposed the universal
operator growth hypothesis, which connects the asymptotic behavior of $b_{n}$
with the type of dynamics exhibited by the underline system. Namely, for a
generic physical systems without apparent or hidden symmetries $b_{n}$ will
exhibit maximal possible growth consistent with locality, $b_{n}\propto n$ for
spatially extended systems in $D>1$. This hypothesis is essentially the
quantum version of an earlier observation, that relates the high-frequency
tail of the power spectrum $f^{2}(\omega)$ – the Fourier of $C(t)$ – with
presence or lack of integrability in classical spin models
elsayed2014signatures . Similarly, the hypothesis can be understood as a
statement that in typical non-integrable lattice systems the norm of nested
commutators $[H,[H,\dots A]]$, or equivalently the norm of $A$ subject to
Euclidean time evolution, grows with the maximal speed allowed by universal
geometric constraints Avdoshkin_2020 . There is non-trivial evidence
supporting the connection between the behavior of $b_{n}$ and
integrability/chaos, yet it does not seem to be universal. In particular a
possible stronger formulation, relating the linear growth of $b_{n}$
specifically to chaotic behavior of the underlying systems is apparently wrong
Dymarsky:2021bjq ; Bhattacharjee_2022 . We observe that for continuous systems
and local operator $A$, Lanczos coefficients always exhibits linear growth.
The situation is changed if a UV-cutoff is introduced, in which case we
propose the asymptotic behavior of $b_{n}$ would probe integrability or lack
thereof in the lattice model underlying the UV regime.

Besides being a probe of chaos, the Krylov complexity has a very intriguing
and non-trivial connection with the exponent controlling the growth of the Out
of Time Ordered Correlator (OTOC). Namely, the original work Parker_2019
conjectured (and proved for the case of infinite temperature) that the
exponent of Krylov complexity bounds the exponent of OTOC, $\lambda_{\rm
OTOC}\leq\lambda_{\rm K}$. We further propose this inequality being a part of
a stronger relation, which generalizes the Maldacena-Shenker-Stanford bound on
chaos MSS ,

$\displaystyle\lambda_{\rm OTOC}\leq\lambda_{\rm K}\leq{2\pi\over\beta}.$ (1)

In the most cases considered so far first inequality in (1) is non-trivial,
while the second one trivially becomes an equality Avdoshkin_2020 . A proof of
(1) in such a scenario was recently given in Gu_2022 . Below we give examples
of free massive bosons and fermions, when the first inequality is trivial,
while the second one becomes non-trivial. We numerically show that in this
case $\lambda_{\rm K}\leq{2\pi\over\beta}$ is satisfied, thus providing
further evidence for (1).

The third virtue of Krylov complexity is its potential connection to other
measures of complexity in quantum systems, in particular, holographic
complexity Susskind:2014rva ; Brown:2015bva ; Brown:2015lvg ; Ben-Ami:2016qex
; Chapman:2016hwi ; Carmi:2017jqz ; Belin:2021bga . Earlier studies of K
complexity, specifically, in the SYK model Jian_2021 , suggested its
qualitative behavior matches that one of holographic complexity. In
particular, complexity grows linearly in earlier times and then saturates at
the exponential values Susskind:2014rva ; Carmi:2017jqz . We analyze
K-complexity for fields compactified on spheres and show its behavior is
different from the one described above. Hence, we concluded that K complexity,
despite its name, in field theories can be qualitatively very different from
its computational or holographic ‘‘cousins.’’111Krylov complexity draws its
name from the fact that its definition matches certain axioms of complexity
Parker_2019 , but there was no strong reason to expect it would match
computational or holographic complexities, which characterize the whole
quantum state of the system, not merely an operator growth in Krylov space. An
attempt to reconcile these discrepancies is recently taken in
https://doi.org/10.48550/arxiv.2202.06957 .

The connection between K complexity and open questions of quantum dynamics
make it a popular subject of study dymarsky2020quantum ;
PhysRevLett.124.206803 ; Yates2020 ; Yates:2021lrt ; Yates:2021asz ;
Dymarsky:2021bjq ; Noh2021 ; Trigueros:2021rwj ;
https://doi.org/10.48550/arxiv.2207.13603 ; Fan_2022 ; Kar_2022 ;
Barbon:2019wsy ; Rabinovici_2021 ; Rabinovici_2022 ; Rabinovici2022 ;
https://doi.org/10.48550/arxiv.2109.03824 ; Caputa_2021 ;
https://doi.org/10.48550/arxiv.2205.05688 ; PhysRevE.106.014152 ;
Bhattacharjee_2022 ; https://doi.org/10.48550/arxiv.2203.14330 ;
https://doi.org/10.48550/arxiv.2204.02250 ;
https://doi.org/10.48550/arxiv.2205.12815 ;
https://doi.org/10.48550/arxiv.2207.05347 ; H_rnedal_2022 ;
https://doi.org/10.48550/arxiv.2208.13362 ;
https://doi.org/10.48550/arxiv.2208.10520 ;
https://doi.org/10.48550/arxiv.2208.08452 ;
https://doi.org/10.48550/arxiv.2210.06815 ; Bhattacharjee:2022lzy ;
Alishahiha:2022anw , though with a few exceptions most of the literature is
focusing on discrete models. In this paper we continue the study of K
complexity in quantum field theory, initiated in Dymarsky:2021bjq . There we
considered only conformal field theories in flat space. Now we consider models
with mass, compact spatial support and/or UV-cutoff and find rich behavior,
supporting main conclusions outlined above. Interestingly, each deformation we
considered – mass, compact space or UV-cutoff have their own imprint on
Lanczos coefficients $b_{n}$, though universality of these imprints is
unclear. Our paper sheds light on temperature dependence of $b_{n}$ as well as
the role of locality. It frames QFT as an IR limit of some discrete system,
connection the behavior of $b_{n}$ between different limiting cases.

This paper is organized as follows. In section 2 we give preliminaries of
Lanczos coefficients and Krylov complexity. We proceed in section 3 to
consider freem massive scalar and fermions to observe the effect of mass,
which cases the exponent of Krylov comlexity to reduce. In section 4 we
compactify CFTs on a sphere to notice this results in a capped Krylov
complexity, at least in certain scenarious. Section 5 continues with the role
of temperature and UV cutoff. We conclude with a discussion in section 6.

##  2 Preliminaries: dynamics in Krylov space and chaos

All three formulations of the ‘‘signature of chaos’’ mentioned in the
introduction: through the high-frequency behavior of the power spectrum
elsayed2014signatures

$\displaystyle f^{2}(\omega)={1\over 2\pi}\int dte^{i\omega t}C(t),$ (2)

through the growth of $|A(it)|$ Avdoshkin_2020 , and through the asymptotic
behavior of $b_{n}$ Parker_2019 , are essentially mathematically equivalent,
but there are some important subtleties. The exponential asymptotic

$\displaystyle f^{2}(\omega)\sim e^{-\omega/\omega_{0}}$ (3)

is equivalent to a pole of $C(t)$ at $t=i/\omega_{0}$, which is the same as
the divergence of the Frobenius norm of $e^{-\beta H/4}A(t)e^{-\beta H/4}$ at
$t=i/(2\omega_{0})$, as follows from the definition

$\displaystyle C(t)\propto{\rm Tr}(e^{-\beta H/2}A(t)e^{-\beta H/2}A(0)).$ (4)

This is equivalent to asymptotic linear growth

$\displaystyle b_{n}\sim{\pi\over 2\omega_{0}}n$ (5)

assuming asymptotically $b_{n}$ is a smooth function of $n$. Thus (5) implies
(3), but the other way around may not be true provided $b_{n}$ exhibits some
complicated behavior, for example $b_{n}$ split into two branches for even and
odd $n$. The reference Avdoshkin_2020  provided a mathematical example of this
kind, when $b_{n}$ asymptotically split into two branches,

$\displaystyle b_{n}^{2}\sim\left\\{\begin{array}[]{c}n^{2},\quad
n\,\,\,\,{\rm is\,even},\\\ n,\qquad n\,\,{\rm is\,odd},\end{array}\right.$
(8)

while corresponding power spectrum exhibits asymptotic behavior
$f^{2}\propto\omega^{-\omega}$, which is an analog of (3) for 1D systems.222
For 1D systems two-point function $C(t)$ is analytic in the entire complex
plane Araki:1969ta . Accordingly, the slowest possible decay of the power
spectrum for large $\omega$ is, schematically, $f^{2}\propto\omega^{-\omega}$.
This corresponds to maximal possible growth of the norm of nested commutators
$|[H,[H,\dots A]]|$ allowed by 1D geometry Avdoshkin_2020 . Corresponding
maximal growth of Lanczos coefficients is $b_{n}\propto n/\ln(n)$ Parker_2019
.

For any field theory and a local operator $A(t)$ (or a local operator
integrated over some region), singularity of the two-point function when the
operators collide immediately implies (3) with $\omega_{0}=2/\beta$. Hence,
trivially, asymptotic behavior of $f^{2}(\omega)$ can not be used as a probe
of chaos. This prompts the question is if perhaps in field theory $b_{n}$ may
exhibit some complex behavior such that asymptotic behavior of $b_{n}$ may
distinguish integrability from chaos, while (3) wold always be satisfied. A
study of conformal field theories in flat space in Dymarsky:2021bjq  revealed
that nothing of the sort happens and $b_{n}$ demonstrate a ‘‘vanilla’’
asymptotic behavior fixed by the singularity of $C(t)$ at $t=i\beta/2$,

$\displaystyle b_{n}={\pi\over\beta}(n+\Delta+1/2)+o(n),\qquad n\gg 1,$ (9)

where $\Delta$ is the dimension of the operator $A$ controlling the pole
singularity at $t=i\beta/2$,

$\displaystyle C(t)\sim{1\over(t-i\beta/2)^{2\Delta}}.$ (10)

Now we ask if this is perhaps a feature of conformal models in flat space that
$b_{n}$ always grow as $b_{n}\propto n$, while for more complicated QFTs
Lanczos coefficients may exhibit more complex behavior. And indeed, as we see
below, once scale is introduced through a mass or space volume, coefficients
$b_{n}$ may split into even and odd branches, with different asymptotic
behavior, while (3) with $\omega_{0}=2/\beta$ is always satisfied.

##  3 Free massive fields in flat space

In this section we consider free massive scalar and Dirac fermion in $d$
spacetime dimensions, for which the correlation functions are given by, see
Appendix A,

$\displaystyle C(t)={\rm Tr}(e^{-\beta H/2}\phi(t,x)e^{-\beta
H/2}\phi(0,x))\propto\int_{\tilde{m}}^{\infty}(y^{2}-\tilde{m}^{2})^{d-3\over
2}{\cosh(y\,\tau)\over\sinh(y/2)}dy$

and

$\displaystyle C(t)={\rm Tr}(e^{-\beta H/2}\psi^{\dagger}(t,x)e^{-\beta
H/2}\psi(0,x))\propto\int_{\tilde{m}}^{\infty}dy\,(y^{2}-\tilde{m}^{2})^{d-3\over
2}{y\cosh(y\tau)\pm\tilde{m}\sinh(y\tau)\over\cosh\big{(}{y/2}\big{)}},$

where $\tau\equiv it/\beta$, and $\tilde{m}=\beta m$. In the first case
Lanczos coefficients split into even and odd branches, both growing linearly
with $n$ albeit with different intercepts, as shown for $d=4$ in the left
panel of Fig. 5. We call this behavior ‘‘persistent staggering’’: $b_{n}$ grow
linearly, with the universal slope $\pi n/\beta$, but even and odd branches
have different finite terms:

$\displaystyle
b_{n}=\left\\{\begin{array}[]{c}{\pi\over\beta}(n+c_{1})+o(n),\qquad{\rm
even}\,n.\\\ {\pi\over\beta}(n+c_{2})+o(n),\qquad{\rm
odd}\,n,\end{array}\right.\qquad n\gg 1.$ (13)

In the second case of free massless fermions $C(t)$ is not an even function,
hence besides $b_{n}$, Lanczos coefficients also include $a_{n}$, see Appendix
B. In this particular case coefficients $a_{n}=\pm(-1)^{n}\tilde{m}$ can be
expressed analytically, while $b_{n}$ exhibit standard ‘‘vanilla’’ asymptotic
(9) $\beta b_{n}=\pi n$, see the right panel of Fig. 5.

Both the persistent staggering and oscillating $a_{n}$ have similar effects on
Krylov complexity, which we calculate in both cases numerically, see Fig. 2.
Namely, $\ln K(t)$ continue to grow linearly, but with the slope $\lambda_{K}$
smaller than $2\pi/\beta$. This provides a non-trivial test of the second
inequality of (1). The first inequality is satisfied trivially because we
assign $\lambda_{\rm OTOC}=0$ in free field theories.

From the mathematical point of view, it is not entirely clear which property
of $C(t)$ is a cause for persistent staggering. (We are focusing on the case
of even $C(t)$ here, when $a_{n}\equiv 0$.). Based on a handful of examples
reference viswanath2008recursion  claims, apparently erroneously, this is due
to periodicity of $C(t)$. We propose that persistent staggering is a
reflection of the ‘‘mass gap’’ of $f^{2}(\omega)$, that it is zero for
$|\omega|$ smaller than some finite value $\omega_{m}$. Proving this statement
or ruling it wrong, is an interesting mathematical question.

For free fields non-zero particle mass automatically translates into ‘‘mass
gap’’ at the level of $f^{2}$. When particles are massive but interact,
$f^{2}(\omega\rightarrow 0)$ will not be zero, although will be exponentially
small.333We thank Luca Delacretaz for brining this point to our attention. It
is an interesting question to understand if this behavior will have any
obvious imprint on $b_{n}$, and consequently on Krylov complexity.

In the case of persistent staggering exponential growth of $K(t)$ was verified
numerically. We leave it as another open question to develop an analytic
approximation to evaluate $\lambda_{K}$ in terms of $\beta,c_{1},c_{2}$.

Figure 1: Lanczos coefficients for free massive scalar (left) and fermion
(right) in $d=4$ dimensions. Values for the conformal (massless) cases we
discussed in Dymarsky:2021bjq . For the scalar, introduction of
$\tilde{m}=\beta m$ results in ‘‘persistent staggering’’ of $b_{n}$ around the
conformal values (larger $\tilde{m}$ causes larger staggering amplitude) but
does not change the slope $\pi n/\beta$. Lanczos coefficients split into two
branches, see eq. 13. For fermions, asymptotically, $b_{n}$ grow linearly with
the slope $\pi n/\beta$ and the $m$-dependent intercept. The main effect of
mass is non-vanishing $a_{n}=\tilde{m}(-1)^{n}$.

Figure 2: Krylov complexity for the free massive scalar (left) and fermion
(right) in $d=4$ dimensions. For the massless scalar $K=1+2\sinh^{2}(\pi
t/\beta)$ is known analytically. The main effect of mass is the decrease of
Krylov exponent $\lambda_{K}$.

##  4 CFTs on a sphere

In this section we consider a CFT places on a sphere and calculate Lanczos
coefficients and Krylov complexity associated with the Wightman-ordered
thermal two-point function. We consider two cases, 4d free massless scalar on
${\mathbb{S}}^{3}$ and holographic theories.

###  4.1 Free scalar on ${\mathbb{S}}^{3}\times{\mathbb{S}}^{1}$

We first consider a 4d free massless scalar compacted on a three-sphere.
Because we are considering finite-temerature correlation function the 4d
compactification manifold is ${\mathbb{S}}^{3}\times{\mathbb{S}}^{1}$.
Corresponding thermal two-point function is given by, see Appendix C,

$\displaystyle C(\tau,R)$ $\displaystyle\propto$
$\displaystyle\sum_{n,\ell\in{\mathbb{Z}}}{(\tau+1/2+n)^{2}-(R\ell)^{2}\over((\tau+1/2+n)^{2}+(R\ell)^{2})^{2}}=$
(16) $\displaystyle\sum_{\ell\in{\mathbb{Z}}}{\pi^{2}\over
2}\left({1\over\cosh^{2}(\ell\pi R+i\pi\tau)}+{1\over\cosh^{2}(\ell\pi
R-i\pi\tau)}\right)-{2\pi\over R}=\quad$
$\displaystyle\sum_{n\in{\mathbb{Z}}}{\pi^{2}\over
R^{2}}{1\over\sinh^{2}((n+1/2+\tau)\pi/R)}.$

Here $R$ is the radios of ${\mathbb{S}}^{3}$ measured in the units of $\beta$,
which is the radius of ${\mathbb{S}}^{1}$. Euclidean time $\tau$ is defined as
$\tau=it/\beta$. As expected the function is periodic in Euclidean time
$\tau\rightarrow\tau+1$, but it is also periodic in Lorentzian time under
$t\rightarrow 2\beta/R$, which follows e.g. from (16).

For any finite $R$, Lanczos coefficients split into even and odd branches,
which grow linearly with $n$ but with different slopes, see Fig. 3. This can
be seen analytically in the limit $R\rightarrow 0$ when $C(\tau)$ is
exponentially small (unless $|\tau|=1/2$) and Lanczos coefficients, at leading
order, are given by

$\displaystyle b_{n}^{2}=\left({2\pi\over
R}\right)^{2}\left\\{\begin{array}[]{rc}(n+1)^{2}/4,&\,\,n=1,3,5,\dots\\\
\frac{4n(n+1)^{2}}{n+2}e^{-\pi/R},&\,\,n=2,4,6,\dots\end{array}\right.$ (19)

Numerically, this approximation is good already for $R\lesssim 1$.

In the opposite limit of large ${\mathbb{S}}^{3}$ the correlation function
approaches that one of flat space, plus a correction $-2\pi/R$, plus the
exponentially suppressed terms

$\displaystyle C\propto{\pi^{2}\over\cos^{2}(\pi\tau)}-{2\pi\over
R}+{\pi^{2}\over\cosh^{2}(\pi R-i\pi\tau)}+{\pi^{2}\over\cosh^{2}(\pi
R+i\pi\tau)}+\dots$ (20)

The exponentially suppressed terms are important for the asymptotic behavior
of $b_{n}$. Neglecting them, i.e. keeping only first two terms in (20) yields
the following expression for Lanczos coefficients, which we denote
$b^{0}_{n}$,

$\displaystyle(b_{n}^{0})^{2}$ $\displaystyle=$
$\displaystyle\pi^{2}n(n+1)-(-1)^{n}{c_{n}^{2}\over c_{n}R+d_{n}},\qquad n\geq
1,$ (21) $\displaystyle c_{n}$ $\displaystyle=$
$\displaystyle\pi\left(2n+1-(-1)^{n}\right),\qquad
d_{n}=4\left\lfloor\frac{n+1}{2}\right\rfloor\left(\psi^{(0)}\left(\frac{3}{2}\right)-\psi^{(0)}\left(\left\lfloor\frac{n+1}{2}\right\rfloor+\frac{1}{2}\right)-2\right).$

Here $\psi^{0}$ is a polygamma function. This expression has incorrect
asymptotic behavior for large $n$. Taking into account two more terms in (20)
leads to

$\displaystyle b_{n}^{2}=(b_{n}^{0})^{2}+8\pi^{2}(-1)^{n}n(n+1)(2n+1)e^{-2\pi
R}+O(e^{-\pi R}/R),\qquad n\geq 1,$ (22)

which still has incorrect the asymptotic. We thus conclude that to reproduce
‘‘two slopes’’ behavior

$\displaystyle
b_{n}=\left\\{\begin{array}[]{c}\alpha_{1}(n+c_{1})+o(n),\qquad{\rm
odd}\,n.\\\ \alpha_{2}(n+c_{2})+o(n),\qquad{\rm
even}\,n,\end{array}\right.\qquad n\gg 1.$ (25)

would require taking into account an infinite serious of ever more
exponentially suppressed terms into account. For all finite $R$,
$\alpha_{1}>\alpha_{2}$; the opposite would contradict
$C(t\rightarrow\infty)\rightarrow 0$.444This follows from existence of a
normalized Liuvillian’s zero mode if asymptotically, for odd $n$,
$b_{n}/b_{n+1}<1$. The ratio $\alpha_{1}/\alpha_{2}$ grows with $R^{-1}$, when
radius is large two slopes are almost equal to each other and $\pi n/\beta$,
but for small $R$ they are significantly different, with $\alpha_{2}$ quickly
approaching zero.

Now we discuss Krylov complexity, which we calculate numerically. We find that
$K(t)$ first increases, but then reaches its maximum and starts oscillating
quasi-periodically, see the right panel of Fig. 3. This behavior is more
pronounced for small $R$, while for large radius the behavior of $K(t)$
initially follows the flat space counterpart $K(t)=1+2\sinh^{2}(\pi t)$,
before peaking at some finite value.

Figure 3: Left: Lanczos coefficients for free massless scalar compactified on
${\mathbb{S}}^{3}$ of radius $R=1$. Right: Krylov complexity $K(t)$ for radii
$R=1$, flat space $K=1+2\sinh^{2}(\pi t)$, and $R=3/2$.

The two slopes behavior of $b_{n}$ is a novel phenomenon, not observed for
physical systems previously. This goes beyond the universal operator growth
hypothesis of Parker_2019 , which assumes a smooth asymptote of $b_{n}$.
Similarly, capped Krylov complexity with the maximal value dependent on $R$
(the ratio of ${\mathbb{S}}^{3}$ and ${\mathbb{S}}^{1}$ radii) but not on the
UV-cutoff is in tension with the proposal that qualitatively $K(t)$ is similar
to holographic and computational complexities Barbon:2019wsy ; Rabinovici_2022
; Rabinovici2022 . In the latter case, for a QFT on compact space, complexity
would grow up to exponentially large values, controlled by the volume of
${\mathbb{S}}^{3}$ measured in the units of the UV cutoff. In case of $K(t)$
this growth was presumably supposed to come from the region of $b_{n}$ which
is controlled by UV physics (and where $b_{n}$ are approximately constant
while $K(t)$ grows linearly, see sections 5.2, 5.3 and Figs. 5 and 6). But the
example above shows that the operator can be confined at the beginning of
‘‘Krylov chain,’’ in which case the values of $b_{n}$ for large $n$ do not
matter.

Physically, it is tempting to relate two slopes behavior of $b_{n}$ and
bounded $K(t)$ to finite volume of the space QFT is placed on, but we will see
in the next section this is not true in full generality.

Mathematically, the two slopes behavior of $b_{n}$ is probably due to
periodicity of $C(t)$ in Lorentzian time, or more broadly due to
$f^{2}(\omega)$ being a sum of delta-functions for a not necessary periodic
grid of values of $\omega$. We leave a proper investigation of this question
for the future, together with the question of quantitatively relating
$\alpha_{1},\alpha_{2}$ to some properties of $C(t)$. Here we only make one
step in this direction and use the integral over Dyck paths developed in
Avdoshkin_2020  to relate a particular combination of $\alpha_{1},\alpha_{2}$
to the location of pole of $C(t)$ in the complex plane, see Appendix E.
Another question, which we also leave for the future, is to analytically
relate maximal or time-averaged value of $K(t)$ to $\alpha_{i}$.

###  4.2 Holographic theories

Next we consider a holographic theory with the two-point function of heavy
operators given by the sum over geodesic lengths in thermal AdS space, or
black hole background, below and above the Hawking-Page transition
correspondingly. In the former case the two-point function is555We thank
Matthew Dodelson for collaboration laying foundation for results of this
section.

$\displaystyle
C(t)\propto\sum_{n}{1\over(\cosh((\tau+1/2+n)\beta)-1)^{\Delta}}$ (26)

where the radius of boundary sphere is taken to be one. This expression is
valid for all$\Delta$, not necessarily large, see e.g. Alday:2020eua . This
expression is valid in all dimensions $d\geq 3$. We notice that, again,
besides periodicity in Euclidean time $\tau\rightarrow\tau+1$, this function
is periodic in Lorentzian time $t\rightarrow t+2\pi$. This is presumably the
reason why $b_{n}$ behave qualitatively the same as in the previous subsection
– they split into two branches for even and odd $n$, with both exhibiting
asymptotic linear growth (25). The behavior of Krylov complexity also follows
the pattern of free scalar on ${\mathbb{S}}^{3}$, it first grows but then
oscillates quasi-periodically. We thus conclude that in holographic theories
Krylov complexity can be UV-independent, thus making it qualitatively
different from the holographic complexity Susskind:2014rva ; Brown:2015bva ;
Brown:2015lvg ; Ben-Ami:2016qex ; Chapman:2016hwi ; Carmi:2017jqz ;
Belin:2021bga .

Figure 4: Left: Lanczos coefficients for thermal AdS background (26) with
$\beta=2\pi$ and $\Delta=1$. Right: Krylov complexity for these parameters
(blue) superimposed with $K(t)$ for flat space 2d CFT result
$K=1+2\sinh^{2}(\pi t)$ (orange).

The discussion above applies to temperatures small enough, below the Hawking-
Page transition. As the temperature increases the dual geometry is given by
the BTZ background, assuming we focus specifically on the case of 2d theories.
The two-point function in this background is given by Keski-Vakkuri:1998gmz ;
Maldacena:2001kr

$\displaystyle
C(t)\propto\sum_{n}{1\over\left(\cos(2\pi\tau)+\cosh(4\pi^{2}n/\beta)\right)^{\Delta}}.$
(27)

This function is, of course, periodic under $\tau\rightarrow\tau+1$, but there
is no periodicity in Lorentzian time. As a result Lanczos coefficients have
the ‘‘vanilla’’ behavior, growing linearly with the asymptote (9). In fact
leading contribution comes from the $n=0$ term in (27), which is the same as
the flat-space expression, with other terms for $\beta<2\pi$ giving very small
corrections. Thus, numerically, $b_{n}^{2}$ are very close to
$\pi^{2}(n+1)(n+2\Delta)/\beta^{2}$ Dymarsky:2021bjq . Accordingly, Krylov
complexity grows exponentially with $\lambda_{K}=2\pi/\beta$.

The calculation in the BTZ background shows that two slopes behavior and
capped $K(t)$ are not universal features of QFT on compact spaces, and the
unbounded exponential growth of $K(t)$ is possible in the holographic
settings. This prompts the question of how the inequality (1) fairs in
different holographic scenarios. We see that above the Hawking-Page transition
Krylov exponent $\lambda_{K}$ is well defined and equal to $2\pi/\beta$. This
means second inequality in (1) is saturated and reduces to Maldacena-Shenker-
Stanford bound, which is also saturated by holographic theories. Below the
transition, Krylov complexity is bounded, hence $\lambda_{K}$ is not well-
defined, or we can formally take it equal to zero. Exactly in the same
scenario the OTOC correlator exhibits periodic recurrences, which also makes
$\lambda_{\rm OTOC}$ ill-defined (or vanish) Anous:2019yku . In other words,
(1) remains valid in both cases.

##  5 Temperature and UV-cutoff dependence of Lanczos
coefficients

We have seen in the previous sections that giving particles mass or placing a
CFT on a compact background both have an explicit imprint on Lanczos
coefficients and Krylov complexity. One feature, which nevertheless seems to
be lost in the QFT case is the sensitivity of $b_{n}$ to chaos. Indeed, in all
cases above $b_{n}$ exhibit linear growth (albeit sometimes by splitting into
two branches), even though many considered theories are not chaotic. This
mirrors the behavior we previously observed for CFTs in flat space
Dymarsky:2021bjq  and complements recent observation that non-chaotic systems
with saddle-dominated scrambling also exhibit linear growth of $b_{n}$
Bhattacharjee_2022 .

This could be our final conclusion – that asymptotic of Lanczos coefficients
is not a proper probe of chaos, but apparent success of this approach in case
of discrete systems with finite local Hilbert space, in particular spin
chains, hints this conclusion could be premature. As we mentioned above, for
1D lattice systems slowest possible decay of $f^{2}(\omega)$ is not
exponential but super-exponential $f^{2}(\omega)\sim\omega^{-\omega}$
Avdoshkin_2020 . Yet for integrable spin chains $f^{2}(\omega)$ decays faster,
as a Gaussian (which correspond to asymptotic growth $b_{n}\propto n^{1/2}$)
or even becomes zero for $\omega$ exceeding certain size-independent
threshold. This is is the case in all known examples, including preliminary
numerical evidence for the XXZ model PhysRevE.100.062134 . On the contrary for
the non-integrable 1d Ising model in traverse field it was recently proved in
Cao_2021  that $f^{2}$ will decay with the slowest possible asymptotic,
$f^{2}(\omega)\propto\omega^{-\omega}$. These results constitute non-trivial
evidence for the ‘‘asymptotic of $b_{n}$ as a probe of chaos’’ stronger
version of the universal operator growth hypothesis.666The original hypothesis
claimed that a generic system would feature maximal possible growth of
$b_{n}$. We should mention, there is an additional evidence for the this
hypothesis for D>1 lattice models: an analytic proof of $f^{2}\propto
e^{-\omega/\omega_{0}}$ asymptotic for a non-integrable 2D lattice model
bouch2015expected , and for the limit of SYK model Parker_2019 .

Apparent success with the spin chains and other discrete models and failure in
field theoretic models suggest the issue could be related to the continuous
nature of the latter. We elaborate on this in the next section.

###  5.1 Asymptotic of $f^{2}(\omega)$ and locality

While for the spin chains high frequency asymptotic of $f^{2}$ apparently
contains some dynamical information, for continuous systems it is completely
universal. Let’s consider the Fourier of the two-point correlation function in
field theory. Then $f^{2}(\omega)$ has a simple interpretation as the sum of
transition amplitudes between the one-particle states with energies $E_{i}$
and energies $E_{f}=E_{i}+\omega$. Its high-frequency asymptotic can be
deduced from the following consideration. Let us consider a free quantum-
mechanical particle propagating in 1D. Assuming the particle is fully
localized, Heisenberg uncertainly principle implies the transition amplitude
to a state with arbitrarily high energy is not suppressed (the Fourier of
delta-function is a constant). After restoring $\beta$-dependent factors this
means $f^{2}(\omega)$ will be proportional to $e^{-\beta\omega/2}$. Here
locality of the initial state was crucial for the transition amplitude to
states with large final energy to be unsuppressed. Going back to the two-point
function $C(t)$, this translates into locality of the operator $A$. To see
this in more detail, let us consider a same example, free quantum-mechanical
particle with the Hamiltonian $H={p^{2}\over 2m}$, and an operator in the
Heisenberg picture

$\displaystyle A=e^{-x^{2}/a}.$ (28)

We would like to evaluate (4) and find

$\displaystyle f^{2}(\omega)=K_{0}(\beta(\beta+4am)\omega/2).$ (29)

It’s clear that when $a$ is large, which corresponds to the de-localized wave-
packet, the decay of $f^{2}$ is faster then the kenematicaly fixed value
$f^{2}\propto e^{-\beta\omega/2}$ associated with the point-like operator
$a\rightarrow 0$.

Provided the logic above is correct we may expect that in the case of a
translationally-invariant lattice models similar behavior will emerge in the
continuous (long wave) approximation, emerging in the low energy limit. We
test this below in case of bosonic and fermionic lattice models. For the fixed
value of coupling constants, effective wavelength is specified by inverse
temperature $\beta$. Thus we expect that for large $\beta$ an approximate
continuous description will emerge, leading to universal high frequency tail
of $f$ and as well as universality of $b_{n}$ behavior. On the contrary, for
small $\beta$ we expect to recover the diversity of $b_{n}$ behaviors
previously observed in the literature, and in particular connection between
the asymptotic of $b_{n}$ and chaos.

###  5.2 Integrable $XY$ model at finite temperature

The integrable periodic $XY$ spin chain

$\displaystyle
H=\sum_{i=1}^{N}(1+\gamma)S_{i}^{x}S_{i+1}^{x}+(1-\gamma)S_{i}^{y}S_{i+1}^{y}-h\,S_{i}^{z}$
(30)

is solvable through the Jordan-Wigner transform and the two-point function at
finite temperature

$\displaystyle C(t)=\langle S^{z}(t)S^{z}(0)\rangle^{W}_{\beta}={\rm
Tr}(e^{-\beta H/2}S^{z}(t)e^{-\beta H/2}S^{z})$ (31)

is known explicitly, see Appendix D. We are interested in the thermodynamic
limit of infinite $N$ and start with the case of isotropic XY model with
$\gamma=h=0$, in which case

$\displaystyle C(\tau)={1\over
4\pi^{2}}\left(\int_{-1}^{1}{dc\over\sqrt{1-c^{2}}}{\cosh(c\,\tau)\over\cosh(c\,\beta/2)}\right)^{2}.$
(32)

Calculating $b_{n}$ numerically, for different values of $\beta$ reveals the
following behavior. Initially Lanczos coefficients grow linearly as
$b_{n}\approx\pi n/\beta$, and then saturate at some universal value
$b_{n}\approx 1$, see Fig. 5. Clearly, initial region of linear growth
increases for larger $\beta$. This has the following interpretation, at small
temperatures the isotropic XY model becomes the free fermion CFT, hence
Lanczos coefficients exhibit universal vanilla CFT behavior (9). For large
$n$, when $b_{n}$ (which have dimension of energy) become of the order of UV-
cutoff (which is of order one in our case), the universal QFT behavior is
substituted by the true asymptotic behavior reflecting the dynamics of the
lattice model. Since the isotropic XY model is free, Lanczos coefficients
saturate at a fixed value, in full agreement with the previous observations
that the asymptotic of $b_{n}$ is a probe of chaos (or lack thereof).

The main takeaway here is that to probe chaos, one should consider true
asymptotic behavior of $b_{n}$ which is controlled by UV-cutoff physics. In
the QFT models with an infinite cutoff, such as conformal field theories, the
true asymptote is not accessible, giving way to the essentially universal
vanilla behavior (of one or two linearly growing branches of $b_{n}$).

Figure 5: Both panels: Lanczos coefficients for the isotropic XY model (32).

###  5.3 Free bosons on the lattice

Next we consider free oscillators on the 1D lattice with periodic boundary
conditions,

$\displaystyle H={1\over
4}\sum_{i=1}^{N}\pi_{i}^{2}+(\phi_{i+1}-\phi_{i})^{2}+4\mu^{2}\phi_{i}^{2},$
(33)

which becomes 1D free massive scalar theory in the continuous limit. We can
think of this model as the 1+1 free scalar QFT with a cutoff. In the
thermodynamic limit $N\rightarrow\infty$, when the mass $\mu$ and temperature
$\beta^{-1}$ are chosen to be much smaller than the cutoff, $\beta\gg
1\gg\mu$, we expect the CFT in flat space behavior with $f^{2}\propto
e^{-\beta\omega/2}$ and linearly growing Lanczos coefficients $b_{n}\sim\pi
n/\beta$. At high energies we are dealing with the discrete integrable model
of non-interacting particles with the energies belonging to a finite width
zone. Accordingly $f^{2}(\omega)$ vanishes for $\omega$ exceeding certain
value $\omega^{*}$. Thus, similarly to the isotropic XY model considered
above, Lanczos coefficients are expected to approach a constant, signaling
lack of chaos.

If $N$ is finite, in the long-wave limit the system is is described by free
massive QFT placed on ${\mathbb{S}}^{1}$. In other words, the model (33)
includes all three deformations we discuss in the paper: the mass, compact
spatial manifold and the UV-cutoff. Accordingly, depending on the interplay
between $\beta,\mu$ and $N$, at the level of $b_{n}$ we expect to see the
combinations of some or all three features: persistent staggering, two-slopes
behavior (since the theory is free, and the spectrum of excitations is
equally-spaced), and field theory behavior below and saturation of $b_{n}$
near the UV-cutoff. An explicit calculation of (4) yields

$\displaystyle C(\tau)\propto{1\over N}\sum_{{\rm
k}=1}^{N}{1\over\epsilon_{k}}{\cosh(\epsilon_{k}\tau)\over\sinh(\epsilon_{k}\beta/2)},\qquad\epsilon_{k}=\sqrt{\sin^{2}(k)+\mu^{2}},\quad
k={\pi{\rm k}/N}.$ (34)

and the behavior of Lanczos coefficients for different $\beta$, $N$ and
$\mu\ll 1$ is shown in Figure 6.

As expected, for essentially infinite $N$ and large $\beta$ initially $b_{n}$
grow linearly, as $b_{n}\approx\pi n/\beta$, but once the value of $b_{n}$
becomes the order of the cutoff (which is of order one in our case), they
saturate to a constant. The punchline here is clear: for small temperatures
the prolonged universal linear growth of $b_{n}$, dictated by the locality and
Heisenberg uncertainty principle, will eventually give way to true asymptotic
behavior governed by the discrete model emerging at the UV-cutoff scale. When
$N$ is finite we observe the two slopes behavior, on top of the persistent
staggering due to mass, and the transition toward true asymptote due to UV-
cutoff.777In this case $C(t)$ is not periodic in Lorentzian time, but for any
finite $N$, $f^{2}(\omega)$ has finite support, which is perhaps behind the
two slopes behavior, clearly visible in Fig. 6.

Figure 6: Lanczos coefficients for free bosons on the lattice (33) for
different values of $\beta,\mu,N$ parameters. For large $\beta$ initial
behavior of $b_{n}$ is that one in filed theory – coefficients grow linearly,
either demonstrating persistent staggering and/or two-slopes behavior, see
left panel. When $b_{n}$ approach UV-cutoff value the QFT behavior changes
into one governed by the UV physics, see right panel.

##  6 Discussion

In the paper we calculated Lanczos coefficients $b_{n}$ and Krylov complexity
$K(t)$ of local operators in several models of quantum field theory, free
massive scalars and fermions, massless scalars compactified on a sphere, and a
few holographic examples. Our calculations revealed that all three
deformations of CFTs in flat space: giving particle mass, placing theory on a
compact space, and introducing finite UV-cutoff has a clear imprint on $b_{n}$
and $K(t)$. Namely, mass leads to persistent staggering of $b_{n}$ (13) and
decreases the Krylov exponent $\lambda_{K}$, while $K(t)$ still grows
exponentially. Compact space, at least in certain cases, leads to two sloped
behavior (25) and a capped Krylov complexity. Finite cutoff introduces a new
asymptotic regime, where $b_{n}$ behavior is controlled by the lattice model
at the UV scale.

These examples help us formulate several takeaway messages, clarifying the
universal operator growth hypothesis of Parker_2019  and the role of Krylov
complexity. First, asymptotic behavior of $b_{n}$ in physical systems goes
beyond universality previously discussed in the literature, namely $b_{n}$ can
split into two branches for even and odd $n$, each with its own asymptotic
behavior. In particular this means that asymptotic behavior of $b_{n}$ is not
mathematically equivalent or fully controlled by the high frequency asymptote
of $f^{2}(\omega)$. Second, we clarified the role of temperature for $b_{n}$
in lattice models. Third, in field theory with an infinite UV-cutoff Lanczos
coefficients do not probe chaotic behavior. But when a finite cutoff
$\Lambda_{\rm UV}$ is introduced, by embedding the QFT into a lattice model, a
true asymptotic regime of $b_{n}\gtrsim\Lambda_{\rm UV}$ emerges, which is
conjecturally probing chaos in the underlying lattice model, modulo issues
raised in Bhattacharjee_2022 . Here it is worth noting that a chaotic QFT can
not emerge as a long wave limit from an integrable lattice model. Thus, with
help of a finite UV-cutoff, universal operator growth hypothesis can be
extended to quantum field theories.

Another takeaway message is the generalization of the Maldacena-Shenker-
Stanford bound (1), which we tested in several non-trivial settings.

Finally, the examples of theories demonstrating two slopes behavior of $b_{n}$
and a capped $K(t)$ clearly show, Krylov complexity can have qualitatively
different behavior than the holographic complexity Susskind:2014rva ;
Brown:2015bva ; Brown:2015lvg ; Ben-Ami:2016qex ; Chapman:2016hwi ;
Carmi:2017jqz ; Belin:2021bga .

Our work raise numerous questions, including a number of mathematical
questions about the relation between $b_{n}$ and $C(t)$. Among them the
features of $C(t)$ which lead to asymptotic persistent staggering (13) or two
slopes (25) behavior and how to deduce $\alpha_{1},\alpha_{2},c_{1},c_{2}$
from $C(t)$. Here we presume a simple generalization of (9) is possible,
though in section 4.1 we saw the example when the asymptotic of $b_{n}$ was
dependent on the exponentially suppressed contributions to $C(t)$. There are
many other questions, how to quantitatively relate $\lambda_{K}$ to
$c_{1},c_{2}$ in case of persistent staggering and maximal or time-averaged
$K(t)$ to $\alpha_{1},\alpha_{2}$ in case of two slopes behavior. There are
also questions about physics of Krylov complexity. Can we prove generalized
Maldacena-Shenker-Stanford bound (1) in full generality? Is there any
holographic counterpart of $K(t)$? What is the behavior of $b_{n}$ and $K(t)$
in interacting QFTs? We leave these and other questions for the future.

## Acknowledgments

We thank Dmitri Trunin for collaboration at the early stages of this project
and Luca Delacretaz, Matthew Dodelson, Oleg Lychkovskiy, Julian Sonner, Subir
Sachdev, Adrian Sanchez-Garrido, and Erez Urbach for discussions. AA
acknowledges support from a Kavli ENSI fellowship and the NSF under grant
number DMR-1918065. AD is supported by the National Science Foundation under
Grant No. PHY 2013812.

##  Appendix A Massive free theories

###  A.1 Massive scalar

We start with a free massive scalar in ${\mathbb{R}}^{d}$ and calculate
thermal two-point function

$\displaystyle\langle\phi(\tau)\phi(-1/2)\rangle_{\beta}=\sum_{n=-\infty}^{\infty}{(-1)^{n}\over\beta}\int{d^{d-1}k\over(2\pi)^{d-1}}{e^{i2\pi
n\tau}\over\big{(}{2\pi n\over\beta}\big{)}^{2}+k^{2}+m^{2}}.$ (35)

Here $\tau=it/\beta$ is the Euclidean time. One operator is placed at
$\tau=-1/2$, thus this is a Wightman-ordered correlator .

$\displaystyle\langle\phi(\tau)\phi(-1/2)\rangle_{\beta}\propto{\rm
Tr}(e^{-\beta H/2}\phi(t,x)e^{-\beta H/2}\phi(0,x)).$

To evaluate (35) we substitute the sum over $n$ by a contour integral going
over the singularities of $\sinh\big{(}{\tilde{k}_{0}\over 2}\big{)}$,

$\displaystyle\langle\phi(\tau)\phi(-1/2)\rangle_{\beta}={\beta^{2-d}\over
4\pi
i}\int_{\mathcal{C}}d\tilde{k}_{0}\int{d^{d-1}\tilde{k}\over(2\pi)^{d-1}}{e^{\tilde{k}_{0}\tau}\over\sinh\big{(}{\tilde{k}_{0}\over
2}\big{)}\big{(}\tilde{m}^{2}+\tilde{k}^{2}-\tilde{k}_{0}^{2}\big{)}}.$ (36)

The contour can be deformed and closed through the infinite semicircle at
${\rm Re}\tilde{k}_{0}\rightarrow\infty$, yielding

$\displaystyle C(t)={\rm Tr}(e^{-\beta H/2}\phi(t,x)e^{-\beta
H/2}\phi(0,x))\propto{\beta^{2-d}\over(4\pi)^{d-1\over 2}\Gamma({d-1\over
2})}\int_{\tilde{m}}^{\infty}(y^{2}-\tilde{m}^{2})^{d-3\over
2}{\cosh(y\,\tau)\over\sinh(y/2)}dy.$

###  A.2 Massive fermion

Similarly for the massive fermion

$\displaystyle\langle\psi_{\alpha}^{\dagger}(\tau)\psi_{\sigma}(-1/2)\rangle_{\beta}={1\over\beta}\int{d^{d-1}k\over(2\pi)^{d-1}}\sum_{n=-\infty}^{\infty}{(-1)^{n}\over\beta}{\big{(}\pi(2n+1)\delta_{\alpha\sigma}+i\,\tilde{m}\,\gamma^{0}_{\alpha\sigma}\big{)}e^{i\pi(2n+1)\tau}\over\big{(}{(2n+1)\pi\over\beta}\big{)}^{2}+k^{2}+m^{2}}.$
(37)

We work in Dirac representation such that
$\gamma^{0}_{\alpha\sigma}=\pm\delta_{\alpha\sigma}$ and use the same trick
with the contour integral

$\displaystyle\langle\psi_{\alpha}^{\dagger}(\tau)\psi_{\sigma}(-1/2)\rangle_{\beta}={\beta^{1-d}\over
4\pi
i}\int_{\mathcal{C}}d\tilde{k}_{0}\int{d^{d-1}\tilde{k}\over(2\pi)^{d-1}}{(\tilde{k}_{0}\pm\tilde{m})\,e^{\tilde{k}_{0}\tau}\over\cosh\big{(}{\tilde{k}_{0}\over
2}\big{)}\big{(}\tilde{k}^{2}+\tilde{m}^{2}-\tilde{k}_{0}^{2}\big{)}}.$ (38)

After deforming the integration contour we arrive at

$\displaystyle{\rm Tr}(e^{-\beta H/2}\psi^{\dagger}(t,x)e^{-\beta
H/2}\psi(0,x))\propto{\beta^{1-d}\over(4\pi)^{d-1\over
2}\Gamma\big{(}{d-1\over
2}\big{)}}\int_{|\tilde{m}|}^{\infty}dy\,(y^{2}-\tilde{m}^{2})^{d-3\over
2}{y\cosh(y\tau)\pm\tilde{m}\sinh(y\tau)\over\cosh\big{(}{y\over 2}\big{)}}.$

##  Appendix B Lanczos coefficients for general $C(t)$

In the most general case, when $C(t)$ is not a real-valued even function,
besides coefficients $b_{n}$, Lanczos coefficients also include $a_{n}$. Their
definition in terms of Krylov basis can be found in e.g. dymarsky2020quantum ,
and the explicit expression in terms of $C(t)$ is as follows. For the
analytically continued $C(\tau)$, $\tau=it$, we first define an $n\times n$
Hankel matrix of derivatives ${\mathcal{M}}^{(n)}_{ij}=C^{(i+j)}(\tau)$ and
variables $\tau_{n}={\rm det}\,{\mathcal{M}}^{(n)}$, $\tau_{0}\equiv 1$. Then

$\displaystyle\left.b_{n}={\tau_{n-1}\tau_{n+1}\over\tau_{n}^{2}}\right|_{\tau=0},$
(39) $\displaystyle\left.a_{n}={d\over
d\tau}(\tau_{n}-\tau_{n-1})\right|_{\tau=0}.$ (40)

##  Appendix C Free scalar on ${\mathbb{S}}^{d-1}\times{\mathbb{S}}^{1}$

Let us consider a compact space $\mathbb{S}^{d-1}$ of radius $R$. The thermal
correlation function of the scalar field living on
$\mathbb{R}\times\mathbb{S}^{d-1}$ can be expressed in terms of the heat
kernel $K_{\mathcal{M}}(t,x,y)=\langle x|\exp(-t\hat{D})|y\rangle$, see e.g.,
Vassilevich:2003xt

$\displaystyle\langle\phi(x)\phi(y)\rangle_{\beta}=\int_{0}^{\infty}dtK(t,x,y)e^{-tm^{2}_{\text{eff}}}~{},$
(41)

where $\mathcal{M}=\mathbb{S}^{1}\times\mathbb{S}^{d-1}$ ($\mathbb{S}^{1}$
represents a thermal circle parametrized by $\tau$), and $\hat{D}$ is the
second order differential operator of the Laplace-Beltrami type,

$\displaystyle\hat{D}=-{d\over d\tau^{2}}-\Delta_{\mathbb{S}^{d-1}}~{},$ (42)

where $\Delta_{\mathbb{S}^{d-1}}$ is the scalar Laplacian on a sphere. Finally
$m$ denotes the ’effective’ mass of the field under study

$\displaystyle m^{2}_{\text{eff}}=m^{2}+\xi\mathcal{R}$ (43)

where $\xi={d-2\over 4(d-1)}$ is the conformal coupling and
$\mathcal{R}=\mathcal{R}(\mathbb{S}^{d-1})={(d-1)(d-2)/R^{2}}$ is the Ricci
scalar of the background geometry.

The wave operators are separable on the product manifold
$\mathbb{S}^{1}\times\mathbb{S}^{d-1}$ and hence the heat kernel can be
expressed as the product of the two individual heat kernels on
$\mathbb{S}^{1}$ and $\mathbb{S}^{d-1}$, i.e.,

$\displaystyle K_{\mathcal{M}}=K_{\mathbb{S}^{1}}\,K_{\mathbb{S}^{d-1}}$ (44)

$K_{\mathbb{S}^{1}}$ can be readily evaluated using the method of images. It
is given by an infinite sum of the scalar heat kernels on $\mathbb{R}$, which
are shifted by integer multiples of $\beta$ with respect to each other to
maintain periodic boundary conditions for the scalar field on a circle, namely

$\displaystyle K_{\mathbb{S}^{1}}(t,\tau)={1\over\sqrt{4\pi
t}}\sum_{n=-\infty}^{\infty}e^{-{(\tau+n\beta)^{2}\over 4t}}$ (45)

In fact, the heat kernel $K_{\mathbb{S}^{d-1}}$ is also known in full
generality Camporesi:1994ga , e.g.,

$\displaystyle K_{\mathbb{S}^{3}}(t,x,x)={e^{t/R^{2}}\over(4\pi
t)^{3/2}}\sum_{n=-\infty}^{\infty}e^{-{\pi^{2}R^{2}n^{2}\over
t}}\Big{(}1-2{\pi^{2}R^{2}n^{2}\over t}\Big{)}~{}.$ (46)

In $d=4$, we thus get

$\displaystyle
C_{\phi}(\tau)=\sum_{n,\ell=-\infty}^{\infty}\int_{0}^{\infty}{dt\,e^{-t\,m^{2}}\over(4\pi
t)^{2}}e^{-{(\tau+n\beta)^{2}+(2\pi R\ell)^{2}\over
4t}}\Big{(}1-2{\pi^{2}R^{2}\ell^{2}\over t}\Big{)}~{},$ (47)

Integrating over $t$, yields

$\displaystyle C_{\phi}(\tau)$ $\displaystyle=$ $\displaystyle
m\sum_{n,\ell=-\infty}^{\infty}{A\,K_{1}(Am)-4B^{2}m\,K_{2}(Am)\over
4\pi^{2}A^{2}}~{}.$ $\displaystyle A^{2}$ $\displaystyle=$
$\displaystyle(\tau+n\beta)^{2}+(2\pi R\ell)^{2}\,,~{}B=\pi R\ell~{}.$ (48)

For $\beta m,Rm\gg 1$ the sum is dominated by a few terms in the vicinity of
$n=\ell=0$.

$C_{\phi}(\tau)$ simplifies in the case of conformally coupled scalar ($m=0$)

$\displaystyle C_{\phi}(\tau)={1\over
4\pi^{2}}\sum_{n,\ell=-\infty}^{\infty}{(\tau+n\beta)^{2}-(2\pi
R\ell)^{2}\over\big{(}(\tau+n\beta)^{2}+(2\pi R\ell)^{2}\big{)}^{2}}~{}.$ (49)

Upon redefinition $\tau\rightarrow\beta\tau$ and $R\rightarrow\beta R$ we
arrive at (16).

##  Appendix D Thermal 2pt function in XY model

Consider the Hamiltonian of integrable $XY$ model with periodic boundary
conditions, $j=0$ is the same as $j=N$,

$\displaystyle
H=\sum_{j=1}^{N}\left[(1+\gamma)S^{x}_{j}S^{x}_{j+1}+(1-\gamma)S^{y}_{j}S^{y}_{j+1}\right]-h\sum_{j=1}^{N}S_{j}^{z}.$
(50)

This chain is diagonalizable by the Jordan-Wigner transform NIEMEIJER1967377 ,
with the quasiparticle energies given by $\epsilon_{k}=\sqrt{(\cos
k-h)^{2}+\gamma^{2}\sin^{2}k}$. Here $k$ varies between $0$ to $2\pi$. The
autocorreation function at inverse temperature $\beta$ is defined as

$\displaystyle C^{z}_{\beta}(t)\equiv\langle
S^{z}_{0}S^{z}_{0}(t)\rangle_{\beta}={\rm Tr}(e^{-\beta
H}S^{z}_{0}S^{z}_{0}(t))/{\rm Tr}(e^{-\beta H}).$ (51)

Before giving the expression for $C^{z}_{\beta}(t)$ we first thermal
expectation value

$\displaystyle m_{z}(h)=\langle
S^{z}_{0}\rangle_{\beta}=-\frac{1}{2\pi}\int_{0}^{\pi}dk\cos\left\\{\tan^{-1}\left(\frac{\gamma\sin
k}{\cos k-h}\right)\right\\}\tanh\frac{\beta\epsilon_{k}}{2},$ (52)

and one can show that $\lim\limits_{h\to 0}m^{z}(h)=0$. Finally, the
autocorrelation function is given by

$\displaystyle
C^{z}_{\beta}(t)=m^{2}_{z}(h)+\left[\frac{1}{2\pi}\int\limits_{0}^{\pi}dk\left\\{\cos(\epsilon_{k}t)+i\sin(\epsilon_{k}t)\tanh\frac{\beta\epsilon_{k}}{2}\right\\}\right]^{2}$
(53)
$\displaystyle-\left[\frac{1}{2\pi}\int\limits_{0}^{\pi}dk\cos(2\lambda_{k})\left\\{i\sin(\epsilon_{k}t)+\cos(\epsilon_{k}t)\tanh\frac{\beta\epsilon_{k}}{2}\right\\}\right]^{2},$

where $\lambda_{k}=\frac{1}{2}\tan^{-1}\frac{\gamma\sin k}{\cos k-h}$.

This expression becomes particularly simple, when we set $\gamma=0$, $h=0$
which corresponds to isotropic $XY$ model
$H=\sum_{j=1}^{N}\left[S^{x}_{j}S^{x}_{j+1}+S^{y}_{j}S^{y}_{j+1}\right]$,

$\displaystyle
C_{\beta}(t)=\frac{1}{4\pi^{2}}\left(\int\limits_{-1}^{1}\frac{dc}{\sqrt{1-c^{2}}}\left[\cos(tc)-i\sin(tc)\tanh\frac{\beta
c}{2}\right]\right)^{2},$ (54)

which becomes even simpler upon the shift $t\to t-i\beta/2$ that gives the
Whightman-ordered correlator, cf. with (32),

$\displaystyle
C_{\beta}^{W}(t)=\frac{1}{4\pi^{2}}\left[\int\limits_{-1}^{1}\frac{dc}{\sqrt{1-c^{2}}}\frac{\cos(tc)}{\cosh\beta
c/2}\right]^{2}.$ (55)

##  Appendix E Dyck paths integral for two slopes scenario

###  E.1 Integral over Dyck paths formalism for two branches

Here we will generalize the derivation of the asymptotic behavior of moments

$\displaystyle\mu_{2n}=\int d\omega f^{2}(\omega)\omega^{2n}$ (56)

from Avdoshkin_2020  to the case when $b_{n}$ form two continious bracnhes for
large $n$, $b_{2n}=b_{even}(n)$ and $b_{2n+1}=b_{odd}(n)$. We start with the
Dyck path sum representation of moments Parker_2019

$\displaystyle\mu_{2n}=\sum_{\\{h_{k}\\}\in
D_{n}}\prod_{k=0}^{2n-1}b_{(h_{k}+h_{k+1})/2},$ (57)

where $D_{n}$ is the set of all Dyck paths.

We approximate each Dyck path with $h_{i}=1/2+2nf(i/2n)$, where $f(t)$ is a
continious function defined on $0<t<1$ and satisfing $|f^{\prime}(t)|<1$. Let
us consider N consequtive steps in a Dyck path, where $h_{k+1}-h_{k}=1$ for
$n$ distinct values of $k$ so that $f^{\prime}=\frac{2n-N}{2}$. We
additionally require that the index of $b$ is even $d$ times more than it is
odd. We split the interval in pairs and we will denote the steps were value of
$h$ increased (decreased) as "+" or "-" correspondingly.

For pairs "++" or "- -" one factor of $b$ has an odd index and another has an
even one. For pairs "+-" and "-+" both factors will have the same parity that
depends on the current value of $h_{i}$. If we denote the number of "++" pairs
as $m$ and the number of opposite sign pairs that lead to two even factors as
$l$ we obtain $d=4l-2n+4m$. Assuming our interval is small, its weight in (57)
is given by

$\displaystyle
b_{odd}^{N/2-2l+n-2m}b_{even}^{N/2+2l-n+2m}=\sqrt{b_{odd}b_{even}}^{N}(b_{even}/b_{odd})^{2l-n+2m}.$
(58)

The parameters satisfy $0<n<N$, $n-N/2<m<n/2$ and $0<l<n-2m$. The number of
paths for fixed $n$, $m$ and $l$ is given by
$C_{N/2}^{m}C_{N/2-m}^{n-2m}C_{n-2m}^{l}$. This product can be approximated
with the help of the Stirling’s formula

$\displaystyle\log C_{N/2}^{m}C_{N/2-m}^{n-2m}C_{n-2m}^{l}\approx$
$\displaystyle\frac{N}{2}\left[S(\alpha)+(1-\alpha)S\left(\frac{f^{\prime}+1-2\alpha}{1-\alpha}\right)+(f^{\prime}+1-2\alpha)S\left(\frac{\beta}{f^{\prime}+1-2\alpha}\right)\right],$
(59)

where $n=N\frac{f^{\prime}+1}{2}$, $\alpha=\frac{m}{N/2}$,
$\beta=\frac{l}{N/2}$ and

$\displaystyle S(\alpha)=-\alpha\log\alpha-(1-\alpha)\log(1-\alpha).$ (60)

We can evaluate the sum over $l$ and $m$

$\displaystyle\sqrt{b_{odd}b_{even}}^{N}\sum_{m,l}C_{N/2}^{m}C_{N/2-m}^{n-2m}C_{n-2m}^{l}\epsilon^{2l-n+2m},$
(61)

where $\epsilon=b_{even}/b_{odd}$ via the saddle point approximation. The
function to be minimized is

$\displaystyle W(\alpha,\beta,f^{\prime},\epsilon)=$ $\displaystyle
S(\alpha)+(1-\alpha)S\left(\frac{f^{\prime}+1-2\alpha}{1-\alpha}\right)+(f^{\prime}+1-2\alpha)S\left(\frac{\beta}{f^{\prime}+1-2\alpha}\right)+(2\alpha+2\beta-f^{\prime}-1)\log\epsilon.$

The saddle point equations are

$\displaystyle\epsilon^{2}(1-2\alpha-\beta+f^{\prime})^{2}=\alpha(\alpha-f^{\prime}),$
(62) $\displaystyle\epsilon^{2}(1-2\alpha-\beta+f^{\prime})=\beta.$ (63)

It is solved by

$\displaystyle\alpha^{*}=\frac{f^{\prime}(1-\epsilon^{2})^{2}-4\epsilon^{2}+(1+\epsilon^{2})\sqrt{4\epsilon^{2}+(\epsilon^{2}-1)^{2}f^{\prime
2}}}{2(\epsilon^{2}-1)^{2}},$ (64)
$\displaystyle\beta^{*}=\frac{\epsilon^{2}}{(\epsilon^{2}-1)^{2}}(1+\epsilon^{2}-\sqrt{4\epsilon^{2}+(\epsilon^{2}-1)^{2}f^{\prime
2}}).$ (65)

Plugging this back in the expression for $W$, we obtain

$\displaystyle
W_{eff}(f^{\prime},\epsilon)=-\frac{f^{\prime}}{2}\log\frac{\epsilon^{2}f^{\prime
2}+f^{\prime}(f^{\prime}+\sqrt{4\epsilon^{2}+(\epsilon^{2}-1)^{2}f^{\prime
2}})+\epsilon^{2}(2+f^{\prime}\sqrt{4\epsilon^{2}+(\epsilon^{2}-1)f^{\prime
2}})}{2\epsilon^{2}(1-f^{\prime})^{2}}+$
$\displaystyle+\log\frac{(\epsilon^{2}-1)^{2}}{\epsilon+\epsilon^{3}-\epsilon\sqrt{4\epsilon^{2}+(\epsilon^{2}-1)^{2}(f^{\prime})^{2}}}.$

Finally, this gives us a path integral representation of moments (56)

$\displaystyle\mu_{2n}=\int\mathcal{D}f(t)e^{n\int\limits_{0}^{1}dt\left(W(f^{\prime},\epsilon)+2\log
b(f)\right)}.$ (66)

We can verify that this path integral reduces to the path integral for a
single branch developed in Avdoshkin_2020  when we set $\epsilon=1$
($b_{even}=b_{odd}$). While the second term is singular as $\epsilon\to 0$ it
is finite if we carefully take the limit

$\displaystyle\lim\limits_{\epsilon\to
1}W_{\text{eff}}(f^{\prime},\epsilon)=2S\left(\frac{f^{\prime}+1}{2}\right).$
(67)

Alternatively, the saddle point configuration simplifies to

$\displaystyle\alpha^{*}=\frac{(1+f^{\prime})^{2}}{4},~{}~{}\beta^{*}=\frac{(1-f^{\prime})^{2}}{4}.$
(68)

and again we obtain
$W(\alpha^{*},\beta^{*},f^{\prime},1)=2s\left(\frac{f^{\prime}+1}{2}\right)$.
This agrees with the expected result, since $W$ is multiplied by $n$ (it used
to be $2n$ in the single branch case).

###  E.2 Evaluation of the path integral for two slopes

Here we assume that $\epsilon=\text{const}$ and
$\sqrt{b_{odd}(n)b_{even}(n)}=\alpha n$. The action in this case reads

$\displaystyle I=\int\limits_{0}^{1}dt\left(W(f^{\prime},\epsilon)+2\log
f\right),$ (69)

where have neglected the constant term $2\log(2\alpha n)$ in the integral
since it does not affect the shape of the saddle point. Varying $I$ with
respect to $f(t)$, we obtain the EOMs

$\displaystyle-\frac{2}{f(t)}=\frac{f^{\prime\prime}}{1-(f^{\prime})^{2}}\frac{(1+\epsilon^{2}+\sqrt{4\epsilon^{2}+(-1+\epsilon^{2})^{2}(f^{\prime})^{2}})}{\sqrt{4\epsilon^{2}+(-1+\epsilon^{2})^{2}(f^{\prime})^{2}}}.$
(70)

We can lower the order of equation by defining $g(f)=f^{\prime}$:

$\displaystyle\frac{(1+\epsilon^{2}+\sqrt{4\epsilon^{2}+(1-\epsilon^{2})^{2}g^{2}})g^{\prime}g}{(1-g^{2})\sqrt{4\epsilon^{2}+(1-\epsilon^{2})^{2}g^{2}}}=-\frac{2}{f}.$
(71)

And the solution is

$\displaystyle
g^{2}=\frac{\epsilon^{4}+(1-(c_{1}f)^{2})^{2}-2\epsilon^{2}(1+(c_{1}f)^{2})}{(1-\epsilon^{2})^{2}},$
(72)

where $c_{1}$ is some constant. By requiring $g(f(1/2)=f_{0})=0$, we arrive at

$\displaystyle c_{1}f_{0}=1-\epsilon,~{}c_{1}=\frac{1-\epsilon}{f_{0}}.$ (73)

Having found $g$ as a function of $f$, we can now solve for $f$ as a function
of $t$. Setting $g=f^{\prime}$ leads to

$\displaystyle\frac{1}{c_{1}}H(c_{1}f,\epsilon)=\frac{t}{1-\epsilon^{2}}+c_{2},$
(74)

where

$\displaystyle
H(x,\epsilon)=\frac{(1+\epsilon)\sqrt{1-\left(\frac{x}{1-\epsilon}\right)^{2}}\sqrt{1-\left(\frac{x}{1+\epsilon}\right)^{2}}F\left(\arcsin(\frac{x}{1+\epsilon}),\left(\frac{1+\epsilon}{1-\epsilon}\right)^{2}\right)}{\sqrt{\epsilon^{4}+(1-x^{2})^{2}-2\epsilon^{2}(1+x^{2})}},$
(75)

and $F(x,a)$ is the incomplete elliptic integral (we use the same conventions
as Wolfram Mathematica). Taking the limit $x\to 1-\epsilon$ we have

$\displaystyle
H(1-\epsilon,\epsilon)=\frac{K\left[\left(\frac{1-\epsilon}{1+\epsilon}\right)^{2}\right]}{1+\epsilon},$
(76)

where $K$ is the complete elliptic integral.

We can use this expression to find $c_{1}$ and the value of the function at
the maximum ($t=1/2$)

$\displaystyle
2(1-\epsilon)K\left[\left(\frac{1-\epsilon}{1+\epsilon}\right)^{2}\right]=c_{1},~{}f_{0}=\frac{1}{2K\left[\left(\frac{1-\epsilon}{1+\epsilon}\right)^{2}\right]}.$
(77)

As a consistency check we verify that for $\epsilon=1$ we reproduce the old
result $\lim\limits_{\epsilon\to 1}f_{0}=1/\pi$ (the solution for a sinlge
branch was $\frac{\sin(\pi t)}{\pi}$).

The ‘‘on-shell’’ action can be rewritten as

$\displaystyle
I(\epsilon)=2\int\limits_{0}^{f_{0}}\frac{df}{g(f)}\left(W(g(f),\epsilon)+2\log
f\right),$ (78)

where $f_{0}$ and $g(f)$ are given by

$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2K\left[\left(\frac{1-\epsilon}{1+\epsilon}\right)^{2}\right]},$
(79) $\displaystyle g^{2}(f)$ $\displaystyle=$
$\displaystyle\frac{(f^{2}-f^{2}_{0})}{(1+\epsilon)^{2}f_{0}^{4}}\left[(1-\epsilon)^{2}f^{2}-(1+\epsilon)^{2}f_{0}^{2}\right].$
(80)

Written explicitly, this integral is rather cumbersome and difficult to deal
with. The numeric plot of $I$ as a function of $\epsilon$ is shown in Fig. 7.

Figure 7: Numeric values of $-I(\epsilon)$ are shown in blue, the red line is
$-\log\frac{2}{\pi e}$, the value of the action known analytically for a
single branch, that corresponds to $-I(1)$.

The asymptotic behavior of the moments is

$\displaystyle\boxed{\log\mu_{2n}\approx nI(\epsilon)+2n\log(2\alpha n).}$
(81)

If $\epsilon=1$, $I(1)=2\log\frac{2}{\pi e}$ and $\log\mu_{2n}\approx
2n\log\frac{4\alpha n}{\pi e}$.

###  E.3 Small $\epsilon$ expansion

In this subsection, we will determine the small $\epsilon$ behavior of
$I(\epsilon)$. First of all, since we know the exact value of $f(1/2)$ we can
find the limit

$\displaystyle f(1/2)=f_{0}\approx\frac{1}{|\log\epsilon|}.$ (82)

Next, at small $\epsilon$ (72) simplifies to

$\displaystyle f^{\prime}(t)=1-(f(t)/f_{0})^{2},$ (83)

which is solved by $f(t)=f_{0}\tanh(t/f_{0})$. While this solution seems to
not be symmetric under $t\to 1-t$, it should be thought of as only applicable
at $t<1/2$ and defined by symmetry for $t>1/2$. This function is essentially
constant for $t\gg\frac{1}{|\log\epsilon|}$ and, thus, stitching it with a
different function at $t=1/2$ is appropriate.

Now, we expand $W(f^{\prime},\epsilon)$.

$\displaystyle W(f^{\prime},\epsilon)\approx
S(f^{\prime})+|\log\epsilon|(1-f^{\prime}),$ (84) $\displaystyle
I(\epsilon)=2\int\limits_{0}^{1/2}dt(S(f^{\prime})-|\log\epsilon|f^{\prime})+|\log\epsilon|+4\int\limits_{0}^{1/2}dt\log(f),$
(85)
$\displaystyle\int\limits_{0}^{\infty}(S(f^{\prime})-|\log\epsilon|f^{\prime})\approx\frac{\pi^{2}/4-\log(4)}{|\log\epsilon|}-1,$
(86)
$\displaystyle\int\limits_{0}^{1/2}dt\log(f)\approx-\frac{\pi^{2}}{8}\frac{1}{|\log\epsilon|}-\frac{1}{2}\log|\log\epsilon|.$
(87)

Collecting all the terms together,

$\displaystyle I(\epsilon)=|\log\epsilon|-2\log|\log\epsilon|-2-\frac{2\log
4}{|\log\epsilon|}+\cdots.$ (88)

Thus, the leading behavior is

$\displaystyle\log\mu_{2n}\approx
n|\log\epsilon|-2n\log|\log\epsilon|+2n\log(2\alpha n).$ (89)

###  E.4 $1-\epsilon$ expansion

Let us define $\epsilon=1-\delta$. We can expand

$\displaystyle f_{0}\approx\frac{1}{\pi}-\frac{\delta^{2}}{16\pi}.$ (90)

The correction to the action is

$\displaystyle\delta
S=\frac{\delta^{2}}{4}\int\limits_{0}^{1}dt(1-(f^{\prime})^{2}).$ (91)

Since the variation of the action vanishes on the saddle point solution the
contribution coming from the change in the saddle point will be of order
$\delta^{4}$. Thus, to leading order we only need to evaluate the value of
(91),

$\displaystyle\delta S=\frac{\delta^{2}}{8}.$ (92)

Finally,

$\displaystyle\log\mu_{2n}=2n\left(\log\frac{4\alpha n}{\pi
e}+\frac{\delta^{2}}{16}\right)+O(\delta^{4}).$ (93)

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