aainstitutetext: Center for Gravitational Physics, Department of Space
Science, Beihang University, Beijing 100191, Chinabbinstitutetext: Peng Huanwu
Collaborative Center for Research and Education, Beihang University, Beijing
100191, China

# Probing Krylov Complexity in Scalar Field Theory with General Temperatures

Peng-Zhang He a,b , Hai-Qing Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Krylov complexity characterizes the operator growth in quantum many-body
systems or in quantum field theories. The existing literatures have studied
the Krylov complexity in the low temperature limit in quantum field theory. In
this paper, we extend and systematically study the Krylov complexity and
Krylov entropy in a scalar field theory with general temperatures. To this
end, we propose a new method to calculate the Wightman power spectrum which
allows us to compute the Lanczos coefficients and then to study the Krylov
complexity (entropy) extensively. Compared to the previous studies in the low
temperature limit, we find that the Lanczos coefficients and Krylov complexity
(entropy) in the high temperature limit will behave somewhat differently.
Finally, we also discuss the Krylov complexity (entropy) in the symmetry
breaking phase with the Higgs scalar field at finite temperatures.

##  1 Introduction

In recent years, complexity Nielsen:2006cea ; Jefferson:2017sdb ;
Parker:2018yvk  has become a crucial notion to understand the chaotic
properties of a quantum system or of a spacetime PhysRevLett.116.191301 ;
Susskind:2014rva . It provides a new insight and an analytical approach to
probe the chaotic behaviors of a physical system. The complexity is originated
as a means to quantify the difficulty of a quantum system which can transit
from one state to another. With the deepening of the research, complexity has
received wide applications in the fields such as quantum chaos, quantum
information, quantum field theory, and holographic theory, etc.

Recent studies on complexity has led to the emergence of Krylov complexity as
an important concept in quantum physics. Initially, it was developed to
explore the growth of operator in the Heisenberg picture and to distinguish
between chaotic and integrable systems Parker:2018yvk . Usually, this operator
is defined in a local Hilbert space called Krylov space, which is the span of
nested commutators (see the following section 2). The Lanczos coefficients can
be constructed according to the basis in the Krylov space. Consequently,
Krylov complexity can be computed from the Schrödinger-type equation of the
Lanczos coefficients. The authors Parker:2018yvk  found that the time
evolution of the Krylov complexity satisfy an exponential growth as time goes
by, which indicates a signature of chaos in quantum many-body systems, such as
the spin chains and Sachdev-Ye-Kitaev (SYK) model Sachdev:1992fk ; kitaev . At
finite temperature, the authors further found that there is an upper bound for
this exponential growth rate of the Krylov complexity.

In nowadays, Krylov complexity has received increasingly widespread
applications in many other quantum systems Liu:2022god ; Hashimoto:2023swv ;
Adhikari:2022whf ; Avdoshkin:2022xuw . Specifically, the study of Krylov
complexity has unveiled some astonishing phenomena. In chaotic systems, its
growth initially exhibits an exponential trend, gradually transits to linear,
and eventually reaches a plateau, indicating a saturation, as the system size
is finite Parker:2018yvk ; Rabinovici:2020ryf . However, even in integrable
quantum field theories, such as free scalar field theory Camargo:2022rnt  or
conformal field theory Dymarsky:2021bjq , the Lanczos coefficients display
linear growth, while the Krylov complexity exhibits an exponential growth.
This implies that there are still some subtle aspects of Krylov complexity
that we have not yet fully understood. So far, the study of Krylov complexity
has permeated multiple fields such as free and interacting field theory,
random matrix theory, and open quantum systems etc Balasubramanian:2023kwd ;
Dymarsky:2019elm ; Dymarsky:2021bjq ; Camargo:2022rnt ; Vasli:2023syq ;
PhysRevD.106.126022 ; Iizuka:2023pov ; Iizuka:2023fba ; Erdmenger:2023wjg ;
Bhattacharjee:2022ave ; Bhattacharjee:2023uwx ; Bhattacharya:2023zqt ;
Bhattacharyya:2023grv ; Malvimat:2024vhr ; Caputa:2024vrn ; Afrasiar:2022efk ;
Tan:2024kqb ; Li:2024iji ; Li:2024kfm ; Vardian:2024fsp . Moreover, the
concept of Krylov complexity has been successfully applied to various specific
problems, including the SYK model Rabinovici:2020ryf ; He:2022ryk ,
generalized coherent states Patramanis:2021lkx ; Caputa:2021sib , Ising and
Heisenberg models Cao:2020zls ; Trigueros:2021rwj ; Heveling:2022hth ,
topological phases of matter Caputa:2022eye . Interested readers can refer to
the recent seminal review Nandy:2024htc .

In this paper, we extend the study of Krylov complexity in five dimensional
free field theory from Camargo:2022rnt  to general temperatures. In
Camargo:2022rnt  the authors only studied the case in the low temperature
limit which satisfies $\beta m\gg 1$, which can simplify the Wightman power
spectrum in this limit. However, in our paper we find a new method to compute
the Wightman power spectrum with general $\beta m$, i.e., with general
temperatures. As a result, the Lanczos coefficients can be subsequently
computed and the Krylov complexity and Krylov entropy can also be obtained
accordingly. We found that the Lanczos coefficients $b_{n}$ exhibit the
‘staggering’ behavior, which separates the odd and even $n$ into two families.
For large $n$ one can linearly fit the Lanczos coefficients as
$b_{n}\sim\alpha n+\gamma$. The linear coefficient $\alpha$ is approximately
identical to $\pi/\beta$. The differences $\Delta\gamma$ between $\gamma$ of
odd $n$ and even $n$ is linearly proportional to the mass of the field as
$\beta m\gg 1$. These behaviors are consistent with previous studies in
Camargo:2022rnt . However, in the high temperature limit $\beta m\ll 1$, we
observe that the linear behavior of the difference $\Delta\gamma$ to the mass
will break down, which may come from the effect of the high temperature. We
also found that in the high temperature limit, the Krylov complexity and
Krylov entropy behaves like those in conformal field theories Dymarsky:2021bjq
. The exponential growth rate of the Krylov complexity in the high temperature
limit is roughly $2\pi/\beta$, however, in the low temperature limit the rate
is smaller than $2\pi/\beta$. Therefore, totally for the general temperatures,
the exponential growth rate of the Krylov complexity has an upper bound
$2\pi/\beta$ which is consistent with previous reports. For the Krylov
entropy, we found that in the late time it is linearly proportional to time.
This is consistent with the claims that Krylov entropy is proportional to
logarithmic of the Krylov complexity. Finally, we also investigate the Krylov
complexity and Krylov entropy in the symmetry breaking phase with a Higgs
field theory. We argue that the procedures are similar to those in the free
scalar field theory, by only replacing the mass parameter in the free scalar
field theory with the new mass which has received the quantum corrections in
the broken phase.

This paper is arranged as follows: In section 2 we will give a brief review of
Krylov space and Krylov complexity; In section 3, we will first introduce a
new method to compute the Wightman power spectrum in general temperatures, and
then compare the Krylov complexity and Krylov entropy for general temperatures
in the free scalar field theory; In section 4, we will discuss the Krylov
complexity and Krylov entropy in the spontaneous symmetry breaking phase. Then
we draw our conclusions in section 5. Appendix A will provide some details
about the spontaneous symmetry breaking in the real scalar field theory.

##  2 Overview on Krylov Space and Krylov Complexity

In this section, we will give a brief review on the basics of Krylov space and
Krylov complexity. Interested readers can refer to Refs. Parker:2018yvk ;
Camargo:2022rnt ; Nandy:2024htc  for more details.

###  2.1 Lanczos algorithm

In quantum mechanics, time evolution of an operator $\mathcal{O}$ is
determined by the Heisenberg equation

$\partial_{t}\mathcal{O}(t)=i[H,\mathcal{O}(t)],$ (1)

where $H$ is the Hamiltonian of the system. The solution of the above equation
is

$\mathcal{O}(t)=e^{iHt}\mathcal{O}(0)e^{-iHt}.$ (2)

where $\mathcal{O}(0)$ is the value of the operator at time $t=0$. From the
well-known Baker-Campbell-Hausdorff formula wachter2011relativistic

$e^{A}Be^{-A}=\sum_{n=0}^{\infty}\frac{1}{n!}[A^{(n)},B],$ (3)

where $[A^{(n)},B]\equiv[\underbrace{A,\cdots,[A,[A}_{n},B]]$, the Eq.(2) can
be rewritten as

$\mathcal{O}(t)=\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}[H^{(n)},\mathcal{O}]\equiv\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}\tilde{\mathcal{O}}_{n},$
(4)

where we have defined $\tilde{\mathcal{O}}_{n}\equiv[H^{(n)},\mathcal{O}]$ and
set $\mathcal{O}\equiv\mathcal{O}(0)$. This equation describes how a “simple”
operator may become increasingly “complex” as time evolves. Then one can
introduce a super-operator called Liouvillian

$\mathcal{L}:=[H,\cdot],$ (5)

which is a linear map in the space of operators such that

$\mathcal{L}\mathcal{O}=[H,\mathcal{O}].$ (6)

Obviously we have

$\mathcal{L}^{n}\mathcal{O}=\tilde{\mathcal{O}}_{n},$ (7)

and the Eq.(2) can be reformulated as

$\mathcal{O}(t)=\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}\mathcal{L}^{n}\mathcal{O}=e^{it\mathcal{L}}\mathcal{O}.$
(8)

Compared to the Schrödinger’s picture of ordinary wave function, the above
equation can be explained as the operator’s “wave function” expanded in some
local basis of “states” $\tilde{\mathcal{O}}_{n}$ belonging to a local Hilbert
space $\mathcal{H}_{\mathcal{O}}$, known as Krylov space. Formally, Eq.(8)
resembles the wave function in the Schrödinger picture, with $\mathcal{L}$
playing a role as the Hamiltonian. In the following, we will use the symbol
$\left|A\right)$ to represent a state in the Krylov space, i.e.
$|A)\in\mathcal{H}_{\mathcal{O}}$.

Since we are interested in the effects of finite temperature, we can define
the inner product in Krylov space as the Wightman inner product

$\left(A|B\right):=\expectationvalue{e^{\beta H/2}A^{\dagger}e^{-\beta
H/2}B}_{\beta}\equiv\frac{1}{\mathcal{Z}_{\beta}}\tr(e^{-\beta
H/2}A^{\dagger}e^{-\beta H/2}B),$ (9)

where $\beta$ is the inverse of the temperature $T=\beta^{-1}$ and
$\mathcal{Z}_{\beta}=\tr(e^{-\beta H})$ is the thermal partition function.
With the definition of the Wightman inner product, the Liouvillian
$\mathcal{L}$ is Hermitian, i.e, $(A|\mathcal{L}B)=(\mathcal{L}A|B)$. Note
that Eq.(8) can be considered as the expansion of
$\left|\mathcal{O}(t)\right)$ in the basis
$\\{\mathcal{L}^{n}\left|\mathcal{O}\right)\\}$, which are not necessarily
orthonormal. However, we can use the Gram-Schmidt orthogonalization procedure
gram1883ueber ; schmidt1907theorie  to obtain a set of orthonormal basis
$\\{\left|\mathcal{O}_{n}\right)\\}$ called the Krylov basis Parker:2018yvk .
Suppose the initially given operator $\mathcal{O}$ itself is a normalized
state in the Krylov space. We can define

$\left|\mathcal{O}_{0}\right):=\left|\mathcal{O}\right).$ (10)

Afterwards, we can continuously construct vectors that are orthonormal to the
existing basis vectors to obtain the Krylov basis. 111Besides, we assume
$\mathcal{O}$ is a Hermitian operator $\mathcal{O}^{\dagger}=\mathcal{O}$, so
that it is an observable. Thus, $\left|\mathcal{O}_{1}\right)$ can be
constructed as follows 222From the inner product Eq.(9) and Eq.(6), it is
readily to get that
$\left(\mathcal{O}_{0}\right|\mathcal{L}\left|\mathcal{O}_{0}\right)=0$.

$b_{1}\left|\mathcal{O}_{1}\right)=\mathcal{L}\left|\mathcal{O}_{0}\right)-\left|\mathcal{O}_{0}\right)\left(\mathcal{O}_{0}\right|\mathcal{L}\left|\mathcal{O}_{0}\right)=\mathcal{L}\left|\mathcal{O}_{0}\right),$
(11)

where
$b_{1}:=(\mathcal{L}{\mathcal{O}_{0}}|\mathcal{L}{\mathcal{O}_{0}})^{1/2}$.
For $n>1$, $\left|\mathcal{O}_{n}\right)$ can be constructed as

$b_{n}\left|\mathcal{O}_{n}\right):=\left|A_{n}\right)=\mathcal{L}\left|\mathcal{O}_{n-1}\right)-b_{n-1}\left|\mathcal{O}_{n-2}\right),$
(12)

where $b_{n}=(A_{n}|A_{n})^{1/2}$. The sequence $\\{b_{n}\\}$ is called the
Lanczos coefficients. The above procedure is also known as Lanczos algorithm.

###  2.2 Krylov complexity

The information about the growth of the operator $\mathcal{O}(t)$ is contained
in the sequence Eq.(12). We can expand $\left|\mathcal{O}(t)\right)$ in terms
of Krylov basis $\\{\left|\mathcal{O}_{n}\right)\\}$ as

$\left|\mathcal{O}(t)\right)=\sum_{n=0}^{\infty}\left|\mathcal{O}_{n}\right)\left(\mathcal{O}_{n}\right|\left.\mathcal{O}(t)\right)\equiv\sum_{n=0}^{\infty}i^{n}\varphi_{n}(t)\left|\mathcal{O}_{n}\right),$
(13)

where $\varphi_{n}(t)$ are the probability amplitudes and satisfy

$\sum_{n=0}^{\infty}\absolutevalue{\varphi_{n}(t)}^{2}=1.$ (14)

Combined with Eq.(1), we can obtain a discrete “Schrödinger” equation

$\partial_{t}\varphi_{n}(t)=b_{n}\varphi_{n-1}(t)-b_{n+1}\varphi_{n+1}(t).$
(15)

According to Eq.(13), one can find that $\varphi_{n}(0)=\delta_{n,0}$ and from
the recursion Eq.(15) we can define $\varphi_{-1}(t)\equiv 0$. The Krylov
complexity of an operator $\mathcal{O}$ is defined as

$K(t):=\left(\mathcal{O}(t)|n|\mathcal{O}(t)\right)=\sum_{n=0}^{\infty}n\absolutevalue{\varphi_{n}(t)}^{2}.$
(16)

For convenience, the definition of Krylov complexity we are going to use will
differ slightly from Eq.(16). In order to be consistent with the Krylov
complexity in Dymarsky:2021bjq , we redefine it as

$K(t):=1+\sum_{n=0}^{\infty}n\absolutevalue{\varphi_{n}(t)}^{2}.$ (17)

In the study of Krylov complexity, there exists a very important quantity,
i.e., the autocorrelation function or thermal Wightman 2-point function,

$\displaystyle C(t)$
$\displaystyle:=\varphi_{0}(t)\equiv(\mathcal{O}(t)\mid\mathcal{O}(0))$ (18)
$\displaystyle\equiv\left\langle
e^{i(t-i\beta/2)H}\mathcal{O}^{\dagger}(0)e^{-i(t-i\beta/2)H}\mathcal{O}(0)\right\rangle_{\beta}$
$\displaystyle=\left\langle\mathcal{O}^{\dagger}(t-i\beta/2)\mathcal{O}(0)\right\rangle_{\beta}:=\Pi^{W}(t).$

It is not difficult to find that as long as the derivatives of the
autocorrelation function at all orders are known, the Krylov complexity can
also be determined. To this end, we can expand the autocorrelation function
around $t=0$,

$\displaystyle\Pi^{W}(t)$
$\displaystyle=\sum_{n=0}^{\infty}\left(\mathcal{O}\right|\frac{(-it)^{n}}{n!}\mathcal{L}^{n}\left|\mathcal{O}\right)$
(19) $\displaystyle=\sum_{n=0}^{\infty}\mu_{2n}\frac{(it)^{2n}}{(2n)!},$

in which we have used the facts that
$(\mathcal{O}|\mathcal{L}^{p}|\mathcal{O})=0$ with $p$ an odd number.
Therefore, only terms with even powers of $\mathcal{L}$ are left. In the above
Eq.(19), $\\{\mu_{2n}\\}$ is called the moments with the expression

$\mu_{2n}:=\left(\mathcal{O}\right|\mathcal{L}^{2n}\left|\mathcal{O}\right)=\left.\frac{1}{i^{2n}}\frac{d^{2n}}{dt^{2n}}\Pi^{W}(t)\right|_{t=0}.$
(20)

These moments can also be obtained from the Wightman power spectrum
$f^{W}(\omega)$, which is related to the Wightman 2-point function via a
Fourier transformation

$f^{W}(\omega)=\int_{-\infty}^{\infty}dte^{i\omega t}\Pi^{W}(t).$ (21)

From Eqs.(20) and (21), the moments are related to $f^{W}(\omega)$ via

$\mu_{2n}=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\omega^{2n}f^{W}(\omega).$
(22)

In particular, there is a nonlinear recursive relationship between moments and
Lanczos coefficients, $b_{1}^{2n}\cdots
b_{n}^{2}=\det\left(\mu_{i+j}\right)_{0\leq i,j\leq n}$, where $\mu_{i+j}$ is
a Hankel matrix built from the moments viswanath1994recursion . Alternatively,
this expression can be reformulated in the following recursive relations,

$\displaystyle b_{n}=\sqrt{M_{2n}^{(n)}},$ (23)

where,

$\displaystyle
M^{(j)}_{2l}=\frac{M^{(j-1)}_{2l}}{b_{j-1}^{2}}-\frac{M^{(j-2)}_{2l-2}}{b^{2}_{j-2}},\qquad
l=j,\dots,n,$ (24)

with $M^{(0)}_{2l}=\mu_{2l},b_{-1}\equiv b_{0}:=1,M^{(-1)}_{2l}=0$.

###  2.3 Krylov complexity in scalar field theory with low temperatures $\beta
m\gg 1$

In this subsection, we will give a brief review on the Krylov complexity in
low-temperature scalar field theory based on Camargo:2022rnt . Consider a real
massive scalar field $\phi$ in five-dimensional spacetime, we can write its
Lagrangian in the following form

$L_{\phi}=\frac{1}{2}(\partial_{\mu}\phi\partial^{\mu}\phi-m^{2}\phi^{2}),$
(25)

where $\mu=0,1,2,3$ and $x^{0}=-i\tau$ with $\tau$ the Euclidean time. Then
the Wightman power spectrum $f^{W}(\omega)$ can be obtained at the finite
temperature $\beta^{-1}=T$ as,

$f^{W}(\omega)=\tilde{N}(m,\beta)(\omega^{2}-m^{2})\Theta(\absolutevalue{\omega}-m)\bigg{/}\sinh\left(\frac{\beta|\omega|}{2}\right),$
(26)

where the symbol $\Theta$ represents the Heaviside step function and
$\tilde{N}(m,\beta)$ is the normalization factor satisfying

$\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega{f^{W}(\omega)}=1.$ (27)

In order to get the Krylov complexity, we need to calculate the Lanczos
coefficients $b_{n}$ by solving the discrete Schrödinger equation (15). In
general, this problem is not easy to do. However, it is relatively simple at
low temperatures Camargo:2022rnt . It is worth noting that in finite
temperature field theories, low temperature means $\beta m\gg 1$, i.e.,
temperature is much less than the mass of the field, rather than simply
$\beta\gg 1$ Kapusta:2006pm . Therefore, even if $\beta$ is not very large, it
may still be considered as low temperature if $m$ is sufficiently large. In
the low temperature limit $\beta m\gg 1$, we can rewrite the Wightman power
spectrum Eq.(26) as

$f^{W}(\omega)\approx
N(m,\beta)e^{-\beta\absolutevalue{\omega}/2}(\omega^{2}-m^{2})\Theta(\absolutevalue{\omega}-m),\qquad\beta
m\gg 1,$ (28)

in which we have absorbed the extra term $2$ into the normalization factor
$N(m,\beta)$, i.e., $N(m,\beta)=2\tilde{N}(m,\beta)$. Using Eq.(27),
$N(m,\beta)$ can be determined analytically,

$N(m,\beta)=\frac{\pi\beta^{3}e^{\beta m/2}}{16+8\beta m},\qquad~{}\beta m\gg
1.$ (29)

Therefore, the moments $\mu_{2n}$ in Eq.(22) can be directly computed as

$\mu_{2n}=\frac{2^{-2}e^{\frac{\beta m}{2}}}{2+\beta
m}\left(\frac{2}{\beta}\right)^{2n}\left[4\tilde{\Gamma}\left(3+2n,\frac{\beta
m}{2}\right)-\beta^{2}m^{2}\tilde{\Gamma}\left(1+2n,\frac{\beta
m}{2}\right)\right],$ (30)

where $\tilde{\Gamma}(n,z)$ is the incomplete Gamma function.

Following the non-linear recursive relation in Eqs.(23)-(24), the Lanczos
coefficients $\\{b_{n}\\}$ can be obtained from the Eq.(30). For example, we
show the behavior of the Lanczos coefficients in Figure 1 with $\beta=1$ and
$m=50$. We see that $b_{n}$ exhibits staggering behavior, meaning that it is
divided into two groups: one is with odd $n$ (blue dots) while the other is
with even $n$ (brown dots). Each group is a monotonically increasing function
with respect to $n$. In Camargo:2022rnt  the authors also observed that the
separation $\Delta b_{n}:=|b_{n}^{\rm odd}-b_{n}^{\rm even}|$ is of the order
of the mass $m$.

Figure 1: Lanczos coefficients $b_{n}$ for $\beta=1,m=50$. The blue dots are
for odd $n$ while the brown dots are for even $n$.

The Krylov complexity can be computed from the auto-correlation function
$\varphi_{0}(t)$ in Eq.(18) and the Lanczos coefficients by means of Eq.(15).
Performing the Fourier transformation of $f^{W}(\omega)$ one can yield the
auto-correlation function as

$\displaystyle\varphi_{0}(t)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega e^{-i\omega
t}f^{W}(\omega)$ (31)
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}d\omega\cos(\omega
t)f^{W}(\omega)$ $\displaystyle=\frac{\beta^{3}\left((\beta^{3}(2+\beta
m)-24\beta t^{2}-16mt^{4})\cos(mt)-4t(\beta^{2}(3+\beta m)+4(\beta
m-1)t^{2})\sin(mt)\right)}{(2+\beta m)(\beta^{2}+4t^{2})^{3}}.$

in which we have used the fact that $\varphi_{0}(t)$ is a real function and
$f^{W}(\omega)$ is an even function of $\omega$. Afterwards, we can use the
fourth Runge-Kutta method to numerically solve Eq.(15) and consequently obtain
the Krylov complexity $K(t)$ and the Krylov entropy $S_{K}(t)$, which is
defined by Barbon:2019wsy

$S_{K}(t):=-\sum_{n=0}^{\infty}\absolutevalue{\varphi_{n}(t)}^{2}\log\absolutevalue{\varphi_{n}(t)}^{2}.$
(32)

Figure 2: Time evolution of Krylov complexity (left panel) and Krylov entropy
(right panel) of the free scalar field theories for $\beta=1,m=50$ in five
dimensional spacetime. The vertical axis is in a logarithmic scale for $K(t)$
in the left panel.

We show the time evolution of $K(t)$ and $S_{K}(t)$ in Figure 2 as an example
with $\beta=1,m=50$. In the left panel of Figure 2, it is found that the
Krylov complexity oscillates when time is short. As was explained in
Camargo:2022rnt , these oscillations are due to the trigonometric functions in
the auto-correlation function (31). The oscillatory behavior of
$\varphi_{0}(t)$ will subsequently pass on to $\varphi_{n}(t)$ according to
the discrete Schrödinger equation (15). In the early time the period of the
oscillations is roughly $\pi/m$, which is consistent with the plots in the
left panel of Figure 2 where the period is roughly $0.06$. In the late time,
the amplitudes of the oscillations becomes smaller, which is due to the
cancellation of $\varphi_{n}(t)$ coming from various $n$. However, in the
early time only small number of $n$ will contribute to $\varphi_{n}(t)$. Thus,
the oscillation in the early time is more significant than in the late time.
We will study the behaviors of $K(t)$ and $S_{K}(t)$ more in the later
subsection 3.4.

The above discussions only applies to the low temperature case of $\beta m\gg
1$ since Eq.(28) merely works for $\beta m\gg 1$. Therefore, we cannot use it
for more general temperatures. In the next section, we will provide detailed
studies of Krylov complexity and Krylov entropy for more general situations of
$\beta m$.

##  3 Krylov complexity and Krylov entropy with General Temperatures

In this section, we will relax the constraint on $\beta m$ to study the Krylov
complexity and Krylov entropy in a more general situation. The key idea is
that we can rewrite the Wightman power spectrum $f^{W}(\omega)$ (26) in a
series of summation with general $\beta m$, which subsequently makes the
computation of Krylov complexity and Krylov entropy feasible.

###  3.1 Truncations of the Wightman power spectrum

For general values of $\beta m$, the Wightman power spectrum (26) can be
rewritten as an infinite series. For convenience, we consider only the part
for $\omega>m$ and the other part for $-\omega>m$ can be obtained from the
fact that Eq.(26) is an even function with respect to $\omega$. Therefore, for
$\omega>m$ we have

$\displaystyle f^{W}(\omega)$
$\displaystyle=\tilde{N}(\beta,m)(\omega^{2}-m^{2})/\sinh(\frac{\beta\omega}{2})$
(33)
$\displaystyle=2\tilde{N}(\beta,m)(\omega^{2}-m^{2})\frac{e^{-\frac{\beta\omega}{2}}}{1-e^{-\beta\omega}}$
$\displaystyle=2\tilde{N}(\beta,m)(\omega^{2}-m^{2})e^{-\frac{\beta\omega}{2}}\sum_{k=0}^{\infty}e^{-\beta\omega
k}$
$\displaystyle=2\sum_{k=0}^{\infty}\tilde{N}(\beta,m)(\omega^{2}-m^{2})e^{-\beta\omega(k+1/2)},$

in which we have used the series expansion
$\frac{1}{1-x}=\sum_{k=0}^{\infty}x^{k}$. Let $\beta_{k}\equiv\beta(2k+1)$ and
absorb the factor $2$ into $N(\beta,m)$ we obtain

$f^{W}(\omega)=N(\beta,m)\sum_{k=0}^{\infty}(\omega^{2}-m^{2})e^{-\beta_{k}\omega/2},$
(34)

where $N(\beta,m)$ can be calculated from the normalization conditions (27)
and its exact form is

$N(\beta,m)=\frac{\pi\beta^{3}e^{\frac{3\beta m}{2}}}{2\beta m\Phi(e^{-\beta
m},2,3/2)+2e^{\beta m}(4\beta m+\Phi(e^{-\beta m},3,1/2))},$ (35)

in which $\Phi(z,s,a)\equiv\sum_{k=0}^{\infty}z^{k}/(k+a)^{s}$ is the Lerch
transcendent function lerch . Therefore, the series of the Wightman power
spectrum in Eq.(34) can in principle work for general $\beta m$’s. It has much
broader applications than that in Eq.(28) from Camargo:2022rnt , which can
only apply in the low temperature limit.

In order to compute the moments $\mu_{2n}$ (22) we first define

$I(n)\equiv\int_{m}^{\infty}\omega^{n}e^{-\frac{\beta_{k}\omega}{2}}d\omega$
(36)

which can be integrated by parts as,

$\displaystyle
I(n)=\frac{2}{\beta_{k}}e^{-\frac{\beta_{k}m}{2}}m^{n}+\frac{2n}{\beta_{k}}I(n-1).$
(37)

In particular, for $n=0$ we can get it directly from the integration Eq.(36),

$\displaystyle
I(0)=\int_{m}^{\infty}e^{-\frac{\beta_{k}\omega}{2}}d\omega=\frac{2}{\beta_{k}}e^{-\frac{\beta_{k}m}{2}}.$
(38)

These equations imply that

$\displaystyle I(n)$
$\displaystyle=\left(\frac{2}{\beta_{k}}\right)^{n+1}\tilde{\Gamma}\left(n+1,\frac{\beta_{k}m}{2}\right).$
(39)

Therefore, from the definition of moments Eq.(22), we have

$\displaystyle\mu_{2n}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\omega^{2n}}{2\pi}\left[N(\beta
m)\sum_{k=0}^{\infty}(\omega^{2}-m^{2})e^{-\beta_{k}\omega/2}\right]$ (40)
$\displaystyle=\frac{N(\beta,m)}{\pi}\sum_{k=0}^{\infty}\left[I(2(n+1))-m^{2}I(2n)\right]$
$\displaystyle=N(\beta,m)\sum_{k=0}^{\infty}\frac{2^{2n+1}\beta_{k}^{-(2n+3)}}{\pi}\left[4\tilde{\Gamma}\left(2n+3,\frac{\beta_{k}m}{2}\right)-\beta_{k}^{2}m^{2}\tilde{\Gamma}\left(2n+1,\frac{\beta_{k}m}{2}\right)\right].$

Unfortunately, this formula is too complex to be computed analytically.
Therefore, we resort to numerical computations. To this end, we can truncate
the summation of $\mu_{2n}$ in Eq.(40) at a suitable $k$ to obtain approximate
results. In order to ensure sufficient computational speed and accuracy, the
truncation position of $k$ should not be too large or too small. We find that
a suitable choice is to make the normalization factor of the truncated
Wightman power spectrum close to the exact form in Eq.(35). In the remainder
of this paper, we always take

$f^{W}\approx
N(\beta,m)\sum_{k=0}^{k_{\text{max}}}(\omega^{2}-m^{2})e^{-\beta_{k}\omega/2},$
(41)

in which $k_{\rm max}$ is a large cut-off of $k$. We can see that the exact
$f^{W}(\omega)$ in Eq.(34) satisfies $k_{\text{max}}=\infty$, while the
approximated $f^{W}(\omega)$ in Eq.(28) in the low temperature limit satisfies
$k_{\text{max}}=0$. In our case we choose $k_{\text{max}}=200$ in Eq.(41),
because in most cases it is sufficient to obtain accurate numerical results.
In Figure 3 we compare the exact Wightman power spectrum $f^{W}$ to the
approximated functions with $\beta=1$ and $m=0.01$, i.e. $\beta m\ll 1$ in the
limit of high temperature. In the left panel of Figure 3, we see that
$f^{W}(\omega)$ for $k_{\rm max}=0$ (blue dash-dotted line) is only consistent
with the exact form (black line with $k_{\rm max}=\infty$) in the regime of
large frequency; However, they will deviate from each other as $\omega\lesssim
12$, which means the formula in Eq.(28) is not accurate in the low frequency
regime with high temperatures. On the contrary, still in the left panel of
Figure 3, we can see that there is no distinguishable differences for the
Wightman power spectrum $f^{W}(\omega)$ between $k_{\text{max}}=\infty$ (black
line) and $k_{\text{max}}=200$ (red dashed line) in the whole regime of the
frequency, which verifies that our choice of $k_{\text{max}}=200$ is
sufficient for the accuracy.

Figure 3: Relation between the Wightman power spectrum $f^{W}(\omega)$ against
the frequency $\omega$ with $\beta=1,\omega=0.01$. (Left) The black line
($k_{\text{max}}=\infty$) corresponds to Eq.(34), while the blue dash-dotted
line ($k_{\text{max}}=0$) corresponds to Eq.(28). The two lines deviate from
each other as $\omega\lesssim 12$. The red dashed line ($k_{\text{max}}=200$)
corresponds to Eq.(41). It matches with the black line in the whole regime of
the frequency, reflecting that taking $k_{\text{max}}=200$ is accurate enough
for the numerical calculations; (Right) The logarithmic plot of
$f^{W}(\omega)$ against $\omega$ in the limit of large frequency. The three
lines (black, red dashed and blue dash-dotted) for different $k_{\rm max}$
collapse together and the green dashed line is the best fit of them. They
share the same scaling law as $f^{W}(\omega)\sim 1611.14\times
e^{-0.4919\omega}$.

In the right panel of Figure 3 we show the logarithmic relations between
$f^{W}(\omega)$ and $\omega$ in the large frequency limit. We can find that in
the large frequency limit, the Wightman power spectrum with various $k_{\rm
max}$’s will overlap together and decay exponentially with the frequency,

$\displaystyle f^{W}(\omega)\sim
e^{-\omega/\omega_{0}},\qquad~{}~{}\omega\to\infty.$ (42)

The fitted line (green dashed line) in the right panel of Figure 3 shows that
this decay rate is $1/\omega_{0}\approx 0.4919$ as $\omega\to\infty$.
Therefore, we get $\omega_{0}\approx 2.0329$, which is consistent with
discussions in Camargo:2022rnt  that the leading exponential decay of the
power spectrum is $\omega_{0}=2/\beta$ where we have set $\beta=1$.

###  3.2 Lanczos coefficients with truncations

Before studying the Krylov complexity, let’s first look at the behaviors of
the Lanczos coefficients $b_{n}$ with the truncations $k_{\rm max}=200$.

Figure 4: Lanczos coefficients $b_{n}$ for $\beta=1,m=10$ (dots) and
$\beta=1,m=50$ (squares) with $k_{\text{max}}=200$. They are grouped in two
families with odd $n$ (blue) and even $n$ (brown).

In Figure 4, we present the behaviors of $b_{n}$ for $\beta=1,m=10$ (dots) and
$\beta=1,m=50$ (squares). These coefficients also exhibit clear behavior of
“staggering”, which are grouped in two families with odd (in blue) and even
$n$ (in brown). As $n$ is relatively large, $b_{n}$ becomes linearly
proportional to $n$. Intuitively, we can find that the separation $\Delta
b_{n}$ between $b_{n}$ for the odd $n$ and even $n$ with $\beta=1,m=50$ are
greater than those with $\beta=1,m=10$. This phenomenon is consistent with the
discussions in the preceding section that $\Delta b_{n}$ is of the order of
the mass $m$ Camargo:2022rnt . In order to analyze them quantitatively, we fit
$b_{n}$ linearly for odd $n$ and even $n$ separately similar to
Camargo:2022rnt :

$\displaystyle
b_{n}\sim\alpha_{\text{odd}}n+\gamma_{\text{odd}},\qquad\text{odd $n$},$ (43)
$\displaystyle
b_{n}\sim\alpha_{\text{even}}n+\gamma_{\text{even}},\qquad\text{even $n$},$
(44)

where $\alpha_{\text{odd}},\alpha_{\text{even}},\gamma_{\text{odd}}$ and
$\gamma_{\text{even}}$ are constants which are independent of $n$.
Specifically, we perform the linear fitting for $b_{n}$ in the range of
$n\in[100,450]$.

Figure 5: (Left) Mass-dependence of the slope $\alpha_{\text{odd}}$ and
$\alpha_{\text{even}}$ in Eqs.(43)-(44); (Right) Mass-dependence of the
difference $\gamma_{\text{odd}}-\gamma_{\text{even}}$ in Eqs.(43)-(44). The
inset plot exhibits the difference in small frequency regime.

In the left panel of Figure 5, we show the relation between $\beta\alpha$
($\alpha$ means $\alpha_{\rm odd}$ (blue) and $\alpha_{\rm even}$ (brown))
against $\beta m$. It is not difficult to observe that the two coefficients
$\alpha_{\text{odd}}$ and $\alpha_{\text{even}}$ will overlap exactly, meaning
that the slope in the fitting relations Eq.(43) and Eq.(44) are exactly
identical. Moreover, we see that as $\beta m\to 0$ the value of
$\alpha_{\text{odd}}$ and $\alpha_{\text{even}}$ are approaching $\pi/\beta$.
However, as $\beta m$ increases, $\alpha_{\text{odd}}$ and
$\alpha_{\text{even}}$ slightly decrease from $\pi/\beta$. This may come from
the selected range of $n$ when fitting $b_{n}$ as argued in Camargo:2022rnt .
It is expected that larger range of $n$ will improve the value of $\alpha$,
and it will finally approach $\pi/\beta$ more closely Camargo:2022rnt .
Therefore, in our case we can speculate that
$\alpha_{\text{odd}}=\alpha_{\text{even}}\approx\pi/\beta$. This analysis is
also consistent with the universal operator growth hypothesis in
Parker:2018yvk  that the Lanczos coefficients $b_{n}$ in a generic chaotic
system will grow as fast as $b_{n}\sim\alpha n+\gamma$ with
$\alpha=\pi\omega_{0}/2$ as $n\to\infty$. In the fitting of the Wightman power
spectrum Eq.(42) we already get $\omega_{0}\approx 2/\beta$, therefore, we can
get $\alpha\approx\pi/\beta$ which is consistent with the left panel of Figure
5.

In the right panel of Figure 5, we show the relation between the difference
$\beta(\gamma_{\text{odd}}-\gamma_{\text{even}})$ with respect to $\beta m$.
For larger $m$, we can see that
$\beta(\gamma_{\text{odd}}-\gamma_{\text{even}})$ is linearly proportional to
$\beta m$ with the ratio
$\beta(\gamma_{\text{odd}}-\gamma_{\text{even}})/(\beta m)\approx 1$, which is
consistent with the separations of $b_{n}$ in Figure 4 that $\Delta b_{n}$ is
of the order of the mass $m$. However, we find that this proportionality will
break down for small $\beta m\in[0,0.1]$, as shown in the inset plot in the
right panel of Figure 5. In the inset plot, the ratio in the range of small
$\beta m$ is roughly $\beta(\gamma_{\text{odd}}-\gamma_{\text{even}})/(\beta
m)\approx 0.2$ which is much smaller than the ratio $1$ in the large $\beta m$
regime. In the region of small $\beta m\ll 1$, the system is in the high
temperature limit, therefore, we speculate that the linear proportionality
between $\gamma_{\text{odd}}-\gamma_{\text{even}}$ and $m$ may not apply due
to the high temperatures. As far as we know, this phenomenon is not explored
previously.

Therefore, we see that high temperature limit will bring out different
phenomena from the studies in Camargo:2022rnt  with low temperatures. Next, we
will first study the Krylov complexity and Krylov entropy in high-temperature
limit, i.e., $\beta m\ll 1$ and then discuss the general cases with general
$\beta m$. After that, we apply the results to the finite-temperature Higgs
scalar field theory.

###  3.3 High-temperature limit with $\beta m\ll 1$

For convenience, in the numerics we always keep $\beta\equiv 1$ and study the
variations of the Krylov complexity and Krylov entropy with respect to $\beta
m$ by adjusting the values of $m$. The high temperature limit implies $\beta
m\ll 1$ Kapusta:2006pm , therefore, the approximation of Wightman power
spectrum in Eq.(28) from Camargo:2022rnt  is in invalid in this case. We need
to use Eq.(41) to numerically study the Wightman power spectrum and then to
study other quantities such as Krylov complexity and Krylov entropy.

Figure 6: (Left) Time evolution of Krylov complexity for $\beta=1$ and various
cases for $m\ll 1$ in the logarithmic scale. They overlap with each other and
scale as $e^{6.368t}$ (as the black reference line indicates) at late time;
(Right) The corresponding time evolution of Krylov entropy and the black line
is the reference line with scalings $\Delta S_{K}(t)/\Delta t=6.310$.

As examples, we will take the cases of $m=0,0.001$, $0.01$ and $0.1$. We have
plotted the time evolution of Krylov complexity $K(t)$ (in logarithmic scale)
and Krylov entropy $S_{K}(t)$ in the Figure 6. In both panels, the red lines
for $m=0$ are taken from the conformal field theory in Dymarsky:2021bjq . From
the figure, we can see that these curves (i.e., $m=0.001$, $0.01$, $0.1$ and
$0$) will overlap together and cannot be distinguished from each other. This
is consistent in numerics itself since in this case $m$ are very small and
close to zero. Figure 6 indicates that at high temperatures $\beta m\ll 1$,
the behaviors of the Krylov complexity and Krylov entropy for a free scalar
field theory are quite similar to those of the conformal field theory.

From the discussions in Parker:2018yvk , Krylov complexity will grow
exponentially at late time as

$\displaystyle K(t)\propto e^{\lambda_{K}t},$ (45)

in which $\lambda_{K}=\pi\omega_{0}=2\pi/\beta$. In the left panel of Figure
6, the black line is the reference line for the Krylov complexity at late
time. Therefore, we see that in the high temperature limit, the exponential
growth rate for Krylov complexity is roughly $\lambda_{K}\approx 6.368$ which
is a little bit greater than $2\pi/\beta$. This is because we are fitting the
line in a finite time region. For larger time, the slope of the exponential
growth will tend to $2\pi/\beta$ which is consistent with the discussions in
Camargo:2022rnt ; Dymarsky:2021bjq .

In the right panel of Figure 6 the black line is the reference line and scale
as $\Delta S_{K}(t)/\Delta t=6.310$, indicating the linear growth of the
Krylov entropy in late time. The linear slope is also close to $\lambda_{K}$,
which is consistent with the results in Fan:2022xaa ; Camargo:2022rnt  that
$S_{K}(t)\sim\log K(t)$ at late time.

###  3.4 Comparison with general temperatures

For the case of $\beta m\sim 1$, the Eq.(28) for the low temperature limit in
Camargo:2022rnt  is also not applicable. Only the approximation in Eq.(41) is
feasible in this case.

(a) Krylov complexity for $\beta=1,m=1$. (b) Krylov entropy for $\beta=1,m=1$.
(c) Krylov complexity for $\beta=1,m=0.1$. (d) Krylov entropy for
$\beta=1,m=0.1$. (e) Krylov complexity for $\beta=1,m=50$. (f) Krylov entropy
for $\beta=1,m=50$.

Figure 7: Time evolutions of Krylov complexity $K(t)$ and Krylov entropy
$S_{K}(t)$ for $\beta=1$ and various values of $m$. The panels (a), (c) and
(e) for Krylov complexity are plotted in logarithmic scales. For all of the
plots, the black lines are for $k_{\text{max}}=200$ corresponding to Eq.(41),
while the red (dashed) lines are for $k_{\text{max}}=0$ corresponding to
Eq.(28). The green lines are the reference lines for the scalings at late
time.

In Figure 7, we show the time evolution of the Krylov complexity $K(t)$ (in
left column) and Krylov entropy $S_{K}(t)$ (in right column) for various $m$’s
by fixing $\beta\equiv 1$. In these plots, the black curves are for $k_{\rm
max}=200$ corresponding to Eq.(41) while the red (dashed) curves are for
$k_{\rm max}=0$ corresponding to Eq.(28). From the panels in the first row,
i.e. $m=1$ ($\beta m\sim 1$), we can see that the red lines will deviate from
the black lines at late time. This reflects that the approximations in Eq.(28)
is not accurate as the approximations in Eq.(41) at late time, which confirms
our assertion that the approximation in Eq.(28) is only applicable in low
temperature limit, i.e., $\beta m\gg 1$. The green lines in the first row are
the reference lines for $K(t)$ and $S_{K}(t)$, respectively. In the panel (a)
of Figure 7 we can see that the exponential growth rate of the Krylov
complexity is roughly $\lambda_{K}\approx 6.216$ which is close to
$2\pi/\beta$. This exponential scaling is consistent with our discussions in
the preceding subsection. In the panel (b) of Figure 7, the linear scaling of
the Krylov entropy is roughly $\Delta S_{K}(t)/\Delta t\approx 6.189$, which
is consistent with the claims that $S_{K}(t)\sim\log K(t)$ at late time as we
discussed in the preceding subsection. The small differences between the two
scalings, i.e., $6.216$ and $6.189$ are due to the fittings of the two
quantities at a finite time. It is expected that much longer time fitting will
reduce this difference.

In the second row of Figure 7 we compare the Krylov complexity and Krylov
entropy with distinct $k_{\rm max}$ (i.e., $k_{\rm max}=200$ and $k_{\rm
max}=0$) in high temperature limit ($\beta m=0.1\ll 1$). Again, the deviations
between the red lines ($k_{\rm max}=0$) and black lines ($k_{\rm max}=200$)
tell us that in the high temperature limit, the approximation in Eq.(28) is
not valid. The late time scalings in $K(t)$ and $S_{K}(t)$ (blue reference
lines) are also close to $2\pi/\beta$ and consistent with the preceding
discussions.

However, for $\beta m\gg 1$, both the approximations in Eqs.(28) and (41) are
valid, see for instance the last row in Figure 7, where we have set $m=50$ for
the computation of Krylov complexity and Krylov entropy. In the time region we
have considered, both of $k_{\rm max}=200$ and $k_{\rm max}=0$ collapse
together and cannot be distinguished from each other. This is different from
the cases with $\beta m=1$ and $\beta m=0.1$ in the first two rows of Figure
7. Interestingly, the late time scalings for both Krylov quantities are much
smaller than $2\pi/\beta$, i.e., $\lambda_{K}<2\pi/\beta$ in the low
temperature limit $\beta m\gg 1$. Compared to high temperature limit, we may
conjecture that the exponential growth rate of the Krylov complexity has a
bound $\lambda_{K}\leq 2\pi/\beta$, which is consistent with the discussions
in Camargo:2022rnt .

Another interesting thing is that in the low temperature limit $\beta m\gg 1$,
there are oscillations in $K(t)$ in the short time region. The period is
roughly $\pi/m$. However, in the opposite limit, i.e., $\beta m\lesssim 1$,
there are no such oscillations in $K(t)$. These phenomena are consistent with
the observations in Camargo:2022rnt ; Dymarsky:2021bjq .

##  4 Krylov complexity and Krylov entropy in symmetry breaking phase

In this section, we will study the Krylov complexity and Krylov entropy in the
symmetry breaking phase by adopting a model of a five-dimensional real Higgs
scalar field theory. The steps will be similar to those in the preceding
section except that the mass $m$ will be changed to $M(\varphi)$ which will be
shown later.

Assume $\phi$ is a Higgs scalar field which can be described by the following
Lagrangian,

$L_{\rm
Higgs}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\lambda\phi^{4},$
(46)

in which $m^{2}<0$ and $\lambda>0$ in order to render the potential
$U(\phi)=\frac{1}{2}m^{2}\phi^{2}+\lambda\phi^{4}$ like a double well
potential. According to the Appendix A, we can write $\phi(X)$ as (59)333The
details of the symmetry breaking phase can be referred to Appendix A.

$\phi(X)=\varphi+\sigma(X),$ (47)

where $\varphi$ is the classical value corresponding to the minimal point of
the potential while $\sigma(X)$ is a new dynamical field. $X$ is the Euclidean
coordinates of the spacetime. In the thermal field theory, the propagator of
the $\sigma$ field is (71)

$\mathcal{D}_{\sigma}(K)=\frac{1}{\omega_{n}^{2}+\mathbf{k}^{2}+M^{2}(\varphi)},$
(48)

where

$\displaystyle M^{2}(\varphi)=m^{2}+12\lambda\varphi^{2}.$ (49)

After quantum corrections in $d=5$ Higgs field theory, $\varphi$ here will
take the value as (95)

$\varphi_{\rm
Q.C.}=\bar{\varphi}=\pm\frac{1}{2\pi}\sqrt{\frac{-m^{2}\pi^{2}-3\lambda
T^{3}\zeta(3)}{\lambda}}.$ (50)

In computing the Krylov complexity the key point is to get the Wightman power
spectrum $f^{W}(\omega)$ in Eq.(26). However, $f^{W}(\omega)$ is calculated
from the propagators in the field theory Camargo:2022rnt . Compared to the
free field theory in Camargo:2022rnt , we find that the difference of the
propagators between Higgs field theory and free field theory is the mass.
Therefore, when calculating the Krylov complexity and Krylov entropy in the
symmetry breaking phase, the only thing we need to do is to substitute $m$ in
free field theory in preceding section with the $M(\varphi)$ in Eq.(49). From
this respect, we will not show the figures of Krylov complexity and Krylov
entropy further in the symmetry breaking phase. They are expected to behave
similarly to those in the preceding section by just regarding $m$ as
$M(\varphi)$.

##  5 Conclusions

In this paper, we have performed a detailed study of the Krylov complexity and
Krylov entropy for a free massive scalar field in five-dimensional spacetime
under various temperature conditions. Later, we also discussed the Krylov
complexity and Krylov entropy in the symmetry breaking phase. Previously,
studies on the Krylov complexity of free scalar fields only considers the low-
temperature limit, i.e., $\beta m\gg 1$, since in that case the Wightman power
spectrum can be easily approximated as in Eq.(28). In the general
temperatures, i.e., general values of $\beta m$, we found a new expression for
$f^{W}(\omega)$ in the form of an infinite series in Eq.(34). We further
obtained a good approximation of $f^{W}(\omega)$ by truncating the infinite
summations to a finite summation, which allowed us to study the Krylov
complexity and Krylov entropy in the regimes where $\beta m$ does not
necessarily subject to $\beta m\gg 1$.

We first examined the performance of the Lanczos coefficients $b_{n}$, which
exhibited the ‘staggering’ behavior by grouping $b_{n}$ into odd and even $n$
families. As usual, $b_{n}$ can be linearly fitted separately for odd and even
$n$ for large $n$. The linear coefficients $\alpha$ satisfies the relation
$\alpha\approx\pi/\beta$ which is consistent with the previous findings in the
universal operator growth hypothesis. We further observed that the difference
$\gamma_{\text{odd}}-\gamma_{\text{even}}$ is linearly proportional to the
mass $m$ for larger $m$, however, this proportionality breaks down for small
$m$. This phenomenon was not explored in the existing literatures and we
argued that the breakdown of the linear proportionality may come from the high
temperature limit.

Next, we investigated the cases for general temperatures with the general
values of $\beta m$. In the late time, we found that the Krylov complexity
exhibited an exponential growth $K(t)\sim e^{\lambda_{K}t}$ where the exponent
satisfies the bound $\lambda_{K}\leq 2\pi/\beta$. The Krylov entropy is
linearly proportional to $t$ in the late time and satisfying the relation
$S(t)\sim\log K(t)$. In the high temperature limit $\beta m\ll 1$, the Krylov
complexity exhibits behaviors remarkably similar to those of the conformal
field theory where $\beta m=0$. We also checked that the finite truncations
$k_{\rm max}=200$ in our paper is accurate enough to get the correct results
by comparing it to the previous studies with $k_{\rm max}=0$. Therefore, the
schemes in our paper to study the Krylov complexity and Krylov entropy with
general temperatures is reliable.

In the final section, we discussed that the Krylov complexity and Krylov
entropy in the symmetry breaking phase in a Higgs scalar field theory can also
be studied similar to the steps in the free scalar field theory. The only
difference is that one needs to substitute the previous mass $m$ in the free
field theory with the new mass $M$ in the broken phase.

In conclusion, our study provides a comprehensive analysis of the Krylov
complexity and Krylov entropy in a five-dimensional scalar field theory under
varying temperature conditions. These findings enhance our understanding of
operator growth and complexity in quantum field theories, particularly in
relation to thermal effects. Future work may extend these results to other
types of field theories and different spacetime dimensions, by exploring the
rich structures of Krylov complexity in diverse physical settings.

###### Acknowledgements.

This work was partially supported by the National Natural Science Foundation
of China (Grants No.12175008).

##  Appendix A Recapitulation of Higgs Scalar Field Theory

Consider a real Higgs scalar field theory, which is described by the
Lagrangian

$L_{\rm
Higgs}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\lambda\phi^{4},$
(51)

in which we adopt the metric signature $(+,-,-,\cdots)$. One can check that
the Lagrangian is invariant under $Z_{2}$ symmetry. From (51), in order to
have the symmetry breaking, usually we set $m^{2}<0$ and $\lambda>0$, thus the
potential

$U(\phi)=\frac{1}{2}m^{2}\phi^{2}+\lambda\phi^{4}$ (52)

behaves like a double well potential. This potential has two global minima
located at

$\phi=\pm v=\pm\sqrt{\frac{-m^{2}}{4\lambda}}.$ (53)

If we chose the vacuum expectation value $\expectationvalue{\phi}=v$, we can
write $\phi(X)$444$\phi(X):=\phi(\tau,\bf{x})$ and we have used the imaginary-
time formalism. In this formalism $t$ is replaced by $-i\tau$ and $\bf{x}$
represents the spatial coordinates. as

$\phi(X)=v+\sigma(X).$ (54)

The Lagrangian (51) can be rewritten as

$L_{\rm
Higgs}=\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{2}(m^{2}+12\lambda
v^{2})\sigma^{2}-4\lambda v\sigma^{3}-\lambda\sigma^{4}-U(v),$ (55)

where the tree-level potential reads

$U(v)=\frac{1}{2}m^{2}v^{2}+\lambda v^{4}.$ (56)

Therefore, Eq.(55) describes a scalar field $\sigma$ with mass
$\sqrt{m^{2}+12\lambda v^{2}}$ and it is not symmetric under the
transformation $\sigma\leftrightarrow-\sigma$. The previous $Z_{2}$ symmetry
is spontaneously broken.

On the other hand, we can perform Fourier transformation for $\phi(X)$

$\phi(X)=\sqrt{\frac{\beta}{V}}\sum_{\color[rgb]{0,0,1}K}e^{-iK\cdot
X}\phi_{K},\qquad\sum_{K}\equiv\sum_{n=-\infty}^{\infty}\sum_{\mathbf{k}},\qquad-K\cdot
X\equiv\omega_{n}\tau+\mathbf{k}\cdot\mathbf{x},$ (57)

where $\omega_{n}\equiv 2\pi n/\beta~{}(n\in\mathbb{Z})$ are called Matsubara
frequency and $K^{\mu}=(-i\omega_{n},\mathbf{k}),X^{\mu}=(-i\tau,\mathbf{x})$.
We can write down the partition function following the conventions in
Kapusta:2006pm  that

$\mathcal{Z}=\int_{-\infty}^{\infty}d\phi_{0}\prod_{K\neq
0}\int_{-\infty}^{\infty}d\real\phi_{K}\int_{-\infty}^{\infty}d\imaginary\phi_{K}\exp(S[\phi]),$
(58)

in which $S[\phi]=\int_{0}^{\beta}d\tau d^{3}\mathbf{x}L_{\rm Higgs}$. By
extracting out the zero mode of $\phi(X)$, we get

$\phi(X)=\sqrt{\frac{\beta}{V}}\sum_{K}\phi_{K}e^{-iK\cdot
X}=\sqrt{\frac{\beta}{V}}\phi_{0}+\sqrt{\frac{\beta}{V}}\sum_{K\neq
0}\phi_{K}e^{-iK\cdot X}=\varphi+\sigma(X),$ (59)

where $\varphi=v=\sqrt{\frac{\beta}{V}}\phi_{0}$ is independent of $X$ and
$\sigma(X)\equiv\sum_{K\neq 0}e^{-iK\cdot X}\phi_{K}=\sum_{K\neq 0}e^{-iK\cdot
X}\sigma_{K}$.

Then we can define the effective potential as

$\mathcal{V}_{\text{eff}}(\varphi)=-\frac{T}{V}\ln(\prod_{K\neq
0}\int_{-\infty}^{\infty}d\real\sigma_{K}\int_{-\infty}^{\infty}d\imaginary\sigma_{K}\exp(S[\varphi+\sigma]))=-\frac{T}{V}\ln(\int\mathcal{D}\sigma
e^{S[\varphi+\sigma]}),$ (60)

where $\int\mathcal{D}\sigma\equiv\prod_{K\neq
0}\int_{-\infty}^{\infty}d\real\sigma_{K}\int_{-\infty}^{\infty}d\imaginary\sigma_{K}$
and $S[\varphi+\sigma]=S[\phi]$. Then we have

$\mathcal{Z}=\sqrt{VT}\int_{-\infty}^{\infty}d\varphi\exp(-\frac{V}{T}\mathcal{V}_{\text{eff}}(\varphi)).$
(61)

Without loss of generality, we assume that the minimum of effective potential
is located at $\varphi=\bar{\varphi}>0$. In the thermodynamic limit
$V\rightarrow\infty$, the partition function can be written nearby
$\bar{\varphi}$ as

$\mathcal{Z}=\exp(-\frac{V}{T}\mathcal{V}_{\text{eff}}(\bar{\varphi}))\sqrt{\frac{2\pi
T^{2}}{\mathcal{V}^{\prime\prime}_{\text{eff}}(\bar{\varphi})}}.$ (62)

Next, we need to compute $\bar{\varphi}$. To this end, we need to calculate
the effective potential (60) first. Note that $S[\varphi+\sigma]$ is actually
the spacetime integral of the Lagrangian (55) with $v=\varphi$. For
convenience, we rewrite the Lagrangian as,

$\mathcal{L}[\sigma,\varphi]=\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{2}M^{2}(\varphi)\sigma^{2}-4\lambda\varphi\sigma^{3}-\lambda\sigma^{4}-U(\varphi),$
(63)

where

$\displaystyle M^{2}(\varphi)=m^{2}+12\lambda\varphi^{2},$ (64) $\displaystyle
U(\varphi)=\frac{1}{2}m^{2}\varphi^{2}+\lambda\varphi^{4}.$ (65)

Treating $\sigma$ as a dynamical field, the Lagrangian can be divided into a
free part $\mathcal{L}_{0}$ and an interacting part $\mathcal{L}_{I}$ as,

$\mathcal{L}[\sigma,\varphi]=\mathcal{L}_{0}+\mathcal{L}_{I}-U(\varphi),$ (66)

where

$\mathcal{L}_{0}=\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{2}M^{2}(\varphi)\sigma^{2},\qquad\mathcal{L}_{I}=-4\lambda\varphi\sigma^{3}-\lambda\sigma^{4}.$
(67)

Since $\varphi$ is independent of $X$ and $\sigma$, the effective potential
(60) can be decomposed into (notice that $T=\beta^{-1}$)

$\mathcal{V}_{\text{eff}}(\varphi)=U(\varphi)-\frac{T}{V}\ln\left(\int\mathcal{D}\sigma
e^{S_{0}[\sigma]+S_{I}[\sigma]}\right),$ (68)

in which $S_{0}[\sigma]=\int_{0}^{\beta}d\tau d^{3}\mathbf{x}\mathcal{L}_{0}$
and $S_{I}[\sigma]=\int_{0}^{\beta}d\tau d^{3}\mathbf{x}\mathcal{L}_{I}$.

In perturbation theory, the effective potential can be expanded in terms of
$\hbar$ as (we have set $\hbar\equiv 1$)

$\mathcal{V}_{\text{eff}}(\varphi)=\mathcal{V}^{(0)}_{\text{eff}}(\varphi)+\mathcal{V}^{(1)}_{\text{eff}}(\varphi)+\mathcal{V}_{\text{eff}}^{(2)}+\cdots,$
(69)

where the superscript index $i$ in $\mathcal{V}^{(i)}_{\text{eff}}$ indicates
the perturbations in the $i$-th power in $\hbar$. Obviously,
$\mathcal{V}_{\text{eff}}^{(0)}=U(\varphi)$ and

$\displaystyle\mathcal{V}_{\text{eff}}^{(1)}(\varphi)=-\frac{T}{V}\ln\left(\int\mathcal{D}\sigma
e^{S_{0}[\sigma]}\right)=\frac{1}{2}\frac{T}{V}\sum_{K}\ln(\frac{\mathcal{D}_{\sigma}^{-1}(K)}{T^{2}}),$
(70)
$\displaystyle\mathcal{D}_{\sigma}(K)=\frac{1}{\omega_{n}^{2}+\mathbf{k}^{2}+M^{2}(\varphi)}.$
(71)

The value of $\varphi$ needs to be determined by the minimum point of the
effective potential. We will only consider zeroth order and first order terms
of the effective potential, i.e., $\mathcal{V}_{\text{eff}}^{(0)}(\varphi)$
and $\mathcal{V}_{\text{eff}}^{(1)}(\varphi)$. Assuming the spacetime is
$d$-dimensional, then following the steps in thermal field theory
Laine:2016hma , we get

$\displaystyle\mathcal{V}^{(0)}_{\text{eff}}(\varphi)=\frac{1}{2}m^{2}\varphi^{2}+\lambda\varphi^{4},$
(72)
$\displaystyle\mathcal{V}^{(1)}_{\text{eff}}=\int\frac{d^{d-1}\mathbf{k}}{(2\pi)^{d-1}}\left[\frac{E_{\mathbf{k}}}{2}+T\ln(1-e^{-E_{\mathbf{k}}/T})\right],$
(73)

where $E_{\mathbf{k}}=\sqrt{\mathbf{k}^{2}+M^{2}(\varphi)}$. We can recombine
these two equations into a $T$-dependent part and a $T$-independent part

$\displaystyle\mathcal{V}_{0}(\varphi)=\frac{1}{2}m^{2}\varphi^{2}+\lambda\varphi^{4}+\int\frac{d^{d-1}\mathbf{k}}{(2\pi)^{d-1}}\frac{E_{\mathbf{k}}}{2},$
(74)
$\displaystyle\mathcal{V}_{T}(\varphi)=T\int\frac{d^{d-1}\mathbf{k}}{(2\pi)^{d-1}}\ln(1-e^{-E_{\mathbf{k}}/T}).$
(75)

Using the identity Dolan:1973qd

$E_{\mathbf{k}}=\int_{-\infty}^{\infty}\frac{dk_{0}}{2\pi
i}\ln(-k_{0}^{2}+E_{\mathbf{k}}^{2}-i\epsilon),$ (76)

the integration in (74) can be rewritten as

$J(M)\equiv\int\frac{d^{d-1}\mathbf{k}}{(2\pi)^{d-1}}\frac{E_{\mathbf{k}}}{2}=-\frac{i}{2}\int\frac{d^{d}K}{(2\pi)^{d}}\ln(-K^{2}+M^{2}-i\epsilon),$
(77)

where we have set $K^{\mu}=(k^{0},\mathbf{k})$ and $M=M(\varphi)$. We can
rotate the $k_{0}$-integral onto the imaginary axis,let $k_{0}=ik_{4}$,
without crossing the branching cuts of the logarithm. With Euclidean
$d$-momentum variable $K_{E}\equiv(k_{d},\mathbf{k})$ we have

$J(M)\equiv\frac{1}{2}\int\frac{d^{d}K_{E}}{(2\pi)^{d}}\ln(K^{2}_{E}+M^{2}).$
(78)

Using the following identities Peskin:1995ev

$\int\frac{d^{d}K_{E}}{(2\pi)^{d}}=\int\frac{d\Omega_{d}}{(2\pi)^{d}}\int_{0}^{\infty}dK_{E}K^{d-1}_{E},\qquad\int
d\Omega_{d}=\frac{2\pi^{d/2}}{\Gamma(d/2)},$ (79)

we have

$J(M)=\frac{1}{2}\frac{2\pi^{d/2}}{\Gamma(d/2)(2\pi)^{d}}\int_{0}^{\infty}dxx^{d-1}\ln(x^{2}+M^{2}).$
(80)

However, this integral is divergent. Therefore, we need to introduce a
truncation $\Lambda$ such that

$\displaystyle J(M)$
$\displaystyle=\frac{1}{2}\frac{2\pi^{d/2}}{\Gamma(d/2)(2\pi)^{d}}\int_{0}^{\Lambda}dxx^{d-1}\ln(x^{2}+M^{2})$
(81)
$\displaystyle=\frac{1}{2d}\frac{2\pi^{d/2}}{\Gamma(d/2)(2\pi)^{d}}\left[\Lambda^{d}\ln(\Lambda^{2}+M^{2})-\int_{0}^{\Lambda}\frac{2x\cdot
x^{d}}{x^{2}+M^{2}}dx\right]$
$\displaystyle=\frac{1}{2d}\frac{2\pi^{d/2}}{\Gamma(d/2)(2\pi)^{d}}\left\\{\Lambda^{d}\ln(\Lambda^{2}+M^{2})-\frac{2\Lambda^{2+d}}{(2+d)M^{2}}{}_{2}F_{1}\left[1,\frac{2+d}{2},\frac{4+d}{2},-\frac{\Lambda^{2}}{M^{2}}\right]\right\\},$

where ${}_{2}F_{1}$ is the hypergeometric function. In the following, we will
compute $J(M)$ with the dimensions with $d=4$ and $d=5$.

###  A.1 $d=4$

If $d=4$, we can expand $J(M)$ in terms of $1/\Lambda$

$J(M)=\left\\{\frac{1}{32\pi^{2}}\left[M^{2}\Lambda^{2}-\frac{M^{4}}{2}\left(\ln\frac{\Lambda^{2}}{M^{2}}+\frac{1}{2}\right)\right]\right\\}+\frac{\Lambda^{4}}{64\pi^{2}}\left[\ln\Lambda^{2}-\frac{1}{2}\right]+O(1/\Lambda^{2}),$
(82)

in which the second term is divergence and independent of $\varphi$,
therefore, we can directly drop it. Let

$\ln\frac{\Lambda^{2}}{M^{2}}=\ln\frac{\Lambda^{2}}{\mu^{2}}+\ln\frac{\mu^{2}}{M^{2}},\qquad{\rm
with}~{}~{}M^{2}=m^{2}+12\lambda\varphi^{2},$ (83)

where $\mu$ is an arbitrary energy scale. Then we have

$\displaystyle J(M)=$
$\displaystyle\left[\frac{\Lambda^{2}m^{2}}{32\pi^{2}}-\frac{m^{4}}{64\pi^{2}}\ln\frac{\Lambda^{2}}{\mu^{2}}+\varphi^{2}\left(\frac{3\Lambda^{2}}{8\pi^{2}}\lambda-\frac{3m^{2}}{8\pi^{2}}\lambda\ln\frac{\Lambda^{2}}{\mu^{2}}\right)-\varphi^{4}\frac{9\lambda^{2}}{4\pi^{2}}\ln\frac{\Lambda^{2}}{\mu^{2}}\right]$
(84)
$\displaystyle-\frac{m^{4}}{128\pi^{2}}-\frac{m^{4}}{64\pi^{2}}\ln\frac{\mu^{2}}{m^{2}+12\varphi^{2}\lambda}-\varphi^{2}\left(\frac{3m^{2}}{16\pi^{2}}\lambda+\frac{3m^{2}\lambda}{8\pi^{2}}\ln\frac{\mu^{2}}{m^{2}+12\varphi^{2}\lambda}\right)$
$\displaystyle-\varphi^{4}\left(\frac{9\lambda^{2}}{8\pi^{2}}+\frac{9\lambda^{2}}{4\pi^{2}}\ln\frac{\mu^{2}}{m^{2}+12\varphi^{2}\lambda}\right).$

The part in the square brackets $[\dots]$ is divergent and can be directly
discarded. In addition, $-\frac{m^{4}}{128\pi^{2}}$ is independent of
$\varphi$, thus can also be discarded without affecting our results.
Therefore, the part of the effective potential (74) that does not depend on
the temperature is

$\mathcal{V}_{0}(\varphi)=\left(1-\frac{3\lambda}{8\pi^{2}}\right)\frac{m^{2}}{2}\varphi^{2}+\left(1-\frac{9\lambda}{8\pi^{2}}\right)\lambda\varphi^{4}+\frac{1}{64\pi^{2}}\left(m^{2}+12\lambda\varphi^{2}\right)^{2}\ln\left(\frac{m^{2}+12\lambda\varphi^{2}}{\mu^{2}}\right).$
(85)

Assume that the coupling constant $\lambda\ll 1$, then Laine:2016hma

$\mathcal{V}_{0}(\varphi)=\frac{m^{2}}{2}\varphi^{2}+\lambda\varphi^{4},$ (86)

where we have discarded all terms that do not depend on $\varphi$, as such
terms will not affect our results. Now consider the part of the effective
potential that depends on temperature (75)

$\displaystyle\mathcal{V}_{T}(\varphi)$
$\displaystyle=T\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\ln(1-e^{-E_{\mathbf{k}}/T})$
(87) $\displaystyle=4\pi
T^{4}\int_{0}^{\infty}\frac{dx}{(2\pi)^{3}}x^{2}\ln(1-e^{-\sqrt{x^{2}+\alpha^{2}}}),$

where $\alpha=M/T$. In the high temperature limit $\alpha\ll 1$,

$\ln(1-e^{-\sqrt{x^{2}+\alpha^{2}}})\approx\ln(1-e^{-x})+\frac{1}{e^{x}-1}\frac{\alpha^{2}}{2x}.$
(88)

Then we get

$\mathcal{V}_{T}(\varphi)=-\frac{\pi^{2}}{90}T^{4}+\frac{T^{2}}{24}(m^{2}+12\varphi^{2}\lambda),$
(89)

in which the terms independent of $\varphi$ can be discarded again. Therefore,
combining Eqs.(86) and (89) the effective potential under weak coupling
$\lambda\ll 1$ and high temperature $\alpha\ll 1$ is Laine:2016hma

$\mathcal{V}_{\text{eff}}(\varphi)=\frac{1}{2}(m^{2}+\lambda
T^{2})\varphi^{2}+\lambda\varphi^{4}.$ (90)

The minimum point of the effective potential is located at

$\bar{\varphi}=\pm\frac{1}{2}\sqrt{\frac{-m^{2}-\lambda T^{2}}{\lambda}}.$
(91)

###  A.2 $d=5$

Following the previous operations, we can obtain the effective potential with
$d=5$. The specific process will not be repeated here. The part of the
effective potential that does not depend on temperature is

$\displaystyle\mathcal{V}_{0}^{(5)}(\varphi)=$
$\displaystyle\left(1+\frac{2\lambda\sqrt{m^{2}+12\lambda\varphi^{2}}}{5\pi^{2}}\right)\frac{m^{2}}{2}\varphi^{2}+\left(1+\frac{6\lambda\sqrt{m^{2}+12\lambda\varphi^{2}}}{5\pi^{2}}\right)\lambda\varphi^{4}$
(92)
$\displaystyle+\frac{m^{4}\sqrt{m^{2}+12\lambda\varphi^{2}}}{120\pi^{2}}.$

The part of the effective potential that depends on temperature is

$\mathcal{V}_{T}^{(5)}(\varphi)=\frac{3\lambda
T^{3}\varphi^{2}\zeta(3)}{2\pi^{2}},$ (93)

where $\zeta$ is the Riemann-zeta function. In the limit of high temperature
and weak coupling, the effective potential is

$\mathcal{V}^{(5)}_{\text{eff}}(\varphi)=\frac{m^{2}}{2}\varphi^{2}+\lambda\varphi^{4}+\frac{3\lambda
T^{3}\varphi^{2}\zeta(3)}{2\pi^{2}}=\frac{1}{2}\left(m^{2}+\frac{3\lambda
T^{3}\zeta(3)}{\pi^{2}}\right)\varphi^{2}+\lambda\varphi^{4}.$ (94)

The minimum point of the effective potential is located at

$\bar{\varphi}=\pm\frac{1}{2\pi}\sqrt{\frac{-m^{2}\pi^{2}-3\lambda
T^{3}\zeta(3)}{\lambda}}.$ (95)

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