IPM/P-2023/nnn

Krylov Complexity in Lifshitz-type Scalar Field Theories

M. J. Vaslia, K. Babaei Velnia,c, M. R. Mohammadi Mozaffara, A. Mollabashib
and M. Alishahihac

a Department of Physics, University of Guilan, P.O. Box 41335-1914, Rasht,
Iran
b Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,
Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
c School of Physics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box 19395-5531, Tehran, Iran

E-mails: vasli@phd.guilan.ac.ir,{babaeivelni<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
We investigate various aspects of the Lanczos coefficients in a family of free
Lifshitz scalar theories, characterized by their integer dynamical exponent,
at finite temperature. In this non-relativistic setup, we examine the effects
of mass, finite ultraviolet cutoff, and finite lattice spacing on the behavior
of the Lanczos coefficients. We also investigate the effect of the dynamical
exponent on the asymptotic behavior of the Lanczos coefficients, which show a
universal scaling behavior. We carefully examine how these results can affect
different measures in Krylov space, including Krylov complexity and entropy.
Remarkably, we find that our results are similar to those previously observed
in the literature for relativistic theories.

###### Contents

  1. 1 Introduction
  2. 2 A brief review of Lanczos algorithm
  3. 3 Lanczos coefficients and Krylov complexity in Lifshitz scalar theory
    1. 3.1 Massless case ($m=0$)
    2. 3.2 Massive case
  4. 4 Lanczos coefficients and Krylov complexity in the presence of UV cutoff
    1. 4.1 Krylov complexity with hard UV cutoff
    2. 4.2 Krylov complexity for the Lifshitz harmonic model
  5. 5 Conclusions

##  1 Introduction

Quantum chaos is an interesting subject though it is difficult to understand.
This is due to the fact that the time evolution of quantum mechanics is local
and unitary and thus, in general, it is hard to study the emergence of ergodic
behavior in quantum systems. Therefore it is of great interest to understand
thermal behavior at the quantum level in which the eigenstate thermalization
hypothesis plays an important role [1, 2, 3, 4].

At the classical level, the chaotic behavior may be associated with the
sensitivity of trajectories in the phase space to the initial conditions.
Indeed, in chaotic systems two initially nearby trajectories separate
exponentially fast characterized by the Lyapunov exponent. Having dealt with
trajectories, it is then clear why the generalization of chaos to the quantum
level should not be straightforward.

Nonetheless, to probe the nature of quantum chaos certain quantities have been
introduced in the literature. These include, for example, out-of-time-order
correlators (OTOCs) [5, 6]. Semiclassically due to the butterfly effect, OTOCs
exhibit an exponential growth characterized by the Lyapunov exponent which is
conjectured to be bounded [7]. The bound saturates for certain strongly
interacting models which have holographic descriptions such as the Sachdev-Ye-
Kitaev (SYK) model [8, 9, 10]. We note, however, that the exponential growth
of OTOCs is not a generic feature of chaotic systems[11, 12].

It is an interesting problem to explore a possible uniform approach that could
describe the chaotic nature of chaotic systems. Recently, it has been proposed
that the operator growth in many body systems may have enough information to
make a distinction between chaotic and non-chaotic systems [13]. In a general
many-body quantum system, the evolution of an operator is given by the
Heisenberg equation, ${\cal O}(t)=e^{-iHt}{\cal O}e^{iHt}$, by which a simple
operator may become rather a complex operator as time evolves. Here $H$ is the
Hamiltonian of the system. More precisely, at any time the operator may be
expanded in terms of nested operators $[H,[H,\cdots,[H,{\cal O}]]]$ as follows

${\cal O}(t)=e^{-iHt}{\cal
O}e^{iHt}=\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}{\cal L}_{n}\,,$ (1.1)

where ${\cal L}_{n}=\\{{\cal O},[H,{\cal O}],[H,[H,{\cal O}]],\cdots\\}$.
These nested operators, given a proper inner product in the space of
operators, are not orthogonal and normalized, although it is possible to
construct an orthogonal and ordered basis known as the Krylov basis. The
procedure by which the basis is constructed is known as the Gram-Schmidt
process. The construction of the Krylov basis via a recursion method amounts
to defining Lanszoc coefficients $b_{n}$ that contain information on the
dynamics of the operator in the Krylov space. It is then natural to define a
measure that probes the growth of the operator over the Krylov basis which
could be thought of as an indicator of how complex the operator might become
as time evolves. The corresponding measure is known as the Krylov complexity
or K-complexity, which has been the subject of a wide regain interest from
many body systems to holography [14, 15, 16, 17, 18, 19, 20, 22, 21, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40].

The authors of [13] proposed a universal operator growth hypothesis that
relates the asymptotic behavior of the Lanczos coefficients to the nature of
the dynamics of the system under consideration. More precisely, for a chaotic
many-body quantum system (for dimensions greater than one) without symmetry
the Lanczos coefficients, asymptotically, should grow linearly

$\displaystyle b_{n}=\alpha
n+\gamma+O(1),\,\,\,\,\,\,\,\;\;\;\;\;\;\;\;\;\;\;{\rm for}\,\,\;\;\;n\gg 1,$
(1.2)

where $\alpha>0$ is a real constant referred as the growth rate and $\gamma$
is also a constant. In this case, the K-complexity exhibits an exponential
growth with an exponent $\lambda=2\alpha$. This hypothesis is motivated by the
behavior of the power spectrum (that is the Fourier transformation of auto-
correlation function) at high frequency limit [41]. Indeed the linear growth
of the Lanczos coefficients is equivalent to the exponential decay of the
power spectrum which in turn is equivalent to a pole in the auto-correlation
function.

Although, for chaotic systems considered in the literature, the Lanczos
coefficients exhibit asymptotic linear growth, it seems that the above
proposal is not universal in the sense that the linear growth may not be
directly related to the chaotic nature of the system. Indeed, the asymptotic
linear growth may also occur even in non-chaotic models [24, 25]. Actually,
for continuous systems such as quantum field theory, the situation is even
worst in the sense that for a local operator the Lancsoz coefficients always
exhibit linear growth unless we add extra ingredients to the system, such as
adding a hard cutoff or putting the theory on a compact space [42, 43].
Intuitively, this is because, for any field theory, the singularity of the
two-point function when the operators approach each other yields an
exponential decay in the power spectrum, which automatically results in an
asymptotic linear growth for Lanczos coefficients, preventing it from being a
good probe for chaos.

As we mentioned the asymptotic linear growth of the Lanczos coefficients
implies an exponential growth of the K-complexity in the asymptotic limit
$t\rightarrow\infty$, i.e,

$\displaystyle K_{\mathcal{O}}(t)\propto e^{\lambda_{K}t},$ (1.3)

where the exponent $\lambda_{K}$ controls the rate of change of
$K_{\mathcal{O}}(t)$. It was shown in [13] that in local quantum many-body
systems at infinite temperature with finite-dimensional Hilbert spaces,
$\lambda_{K}$ bounds the Lyapunov exponent, i.e.,
$\lambda_{L}\leq\lambda_{K}$, which conjecturally applies even at finite
temperature, which would put even tighter bound on chaos. Moreover, from the
general behavior of the exponent $\lambda_{K}$ for systems at finite
temperature, it is plausible to conjecture the following inequality [15]

$\displaystyle\lambda_{L}\leq\lambda_{K}\leq 2\pi T.$ (1.4)

The aim of this article is to further explore the behavior of Lanczos
coefficients, K-complexity, and the above conjectural bound for certain
systems with Lifshitz scaling symmetry acting as

$t\rightarrow\lambda^{z}t,\;\;\;\;\;\;\;\;x_{i}\rightarrow\lambda x_{i},$
(1.5)

where $t$ is time and $x_{i}$’s are spatial directions of the space-time.
Moreover, $z$ denotes the dynamical critical exponent that determines the
anisotropy between time and space such that for $z=1$ the relativistic scaling
is recovered. A quantum field theory that respects the above symmetry is a
Lifshitz field theory111Further discussions on different aspects of Lifshitz
symmetries can be found in e.g. [44, 45, 46]. (see [47] for a review). In
particular, we consider a $d$-dimensional scalar theory which is a
generalization of relativistic Klein-Gordon theory and respects Lifshitz
scaling symmetry in the massless limit with the following action [47]

$\displaystyle S=\frac{1}{2}\int
dt\,d^{d-1}x\left(\dot{\phi}^{2}-\sum_{i=1}^{d}\left(\partial_{i}^{z}\phi\right)^{2}+m^{2z}\phi^{2}\right),$
(1.6)

where the dot indicates derivative with respect to $t$. The corresponding
dispersion relation takes the form

$\displaystyle\epsilon_{k}^{2}=k^{2z}+m^{2z},$ (1.7)

where $k^{2z}=\sum_{i=1}^{d}k_{i}^{2z}$. By replacing the space continuum with
a discrete mesh of lattice points the above expression can be transformed into
a discrete counterpart as follows [50]

$\displaystyle\epsilon_{k}^{2}=\sum_{i=1}^{d-1}\left(2\sin\frac{\pi
k_{i}}{N}\right)^{2z}+m^{2z},$ (1.8)

where we assume a hypercubic lattice with length $N$ in every spatial
direction. Recently, there have been many attempts to investigate various
properties of information measures, including entanglement entropy, in such
theories. These investigations have led to a remarkably rich and varied range
of new insights, e.g., [50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. Related
investigations attempting to better understand quantum chaos, computational
complexity, and entanglement measures in the context of Lifshitz holography
have also been reported in [60, 61, 62].

The remainder of our paper is organized as follows: In Sec. 2, we give the
general framework in which we are working, establishing our notation and the
general form of the Lanczos coefficients, K-complexity, and other related
quantities in the Krylov space. In Sec. 3, we consider the continuum case and
study the properties of Lanczos coefficients and K-complexity numerically. We
present a combination of numerical and analytic results on the scaling of
these quantities. To get a better understanding of the results, we will also
compare the behavior of complexity to other measures including K-entropy. In
Sec. 4, we extend our studies in the presence of a UV cutoff, either by
introducing a finite UV cutoff in continuous momentum space or considering a
discretized version of our model with finite lattice spacing. We review our
main results and discuss their physical implications in Sec. 5, where we also
indicate some future directions.

##  2 A brief review of Lanczos algorithm

In this section, we employ the Lanczos algorithm to find the Lanczos
coefficients, by which we may compute several interesting quantities, such as
K-complexity. As we already mentioned, in order to study operator growth in
the Krylov space, one needs to define a proper inner product. Since we are
interested in a system at a finite temperature, the appropriate inner product
may be defined by the Wightman inner product

$\displaystyle(\mathcal{O}|\mathcal{O}^{\prime})\equiv\langle e^{\frac{\beta
H}{2}}\mathcal{O}^{\dagger}e^{-\frac{\beta
H}{2}}\mathcal{O}^{\prime}\rangle_{\beta}.$ (2.1)

Using this inner product, one could construct the Krylov space starting from
an initial operator ${\cal O}(0)$. Denoting the Krylov basis by $\\{|{\cal
O}_{n})\\}$, the evolved operator at a given time may be expanded in this
basis as follows

$\displaystyle|\mathcal{O}(t))=\sum_{n=0}^{\infty}i^{n}\phi_{n}(t)|\mathcal{O}_{n}),$
(2.2)

where due to the normalization condition we have
$\sum_{n=0}^{\infty}|\phi_{n}(t)|^{2}=1$. The probability amplitudes
$\phi_{n}(t)$ may be computed recursively from the following Schrödinger
equation

$\displaystyle\frac{d\phi_{n}}{dt}=b_{n}\phi_{n-1}-b_{n+1}\phi_{n+1},$ (2.3)

with the boundary conditions $\phi_{n}(0)=\delta_{n0},\,\phi_{-1}(t)\equiv 0$.

Having found the probability amplitudes $\phi_{n}(t)$, one may define several
physical quantities that could probe the nature of the operator growth, which
in turn could give us information about the nature of the dynamics of the
system under study. The most famous quantity in this context is the
K-complexity, defined by

$\displaystyle K_{\mathcal{O}}(t)=\sum_{n=0}^{\infty}n|\phi_{n}(t)|^{2}.$
(2.4)

Moreover, motivated by [63], in order to gain a better insight into the
properties of $K_{\mathcal{O}}(t)$, one may also define the $k$-th order
K-variance

$\displaystyle\delta_{\mathcal{O}}(t)=\frac{\left(\sum_{n}n^{k}|\phi_{n}(t)|^{2}-K_{\mathcal{O}}(t)^{k}\right)^{\frac{1}{k}}}{K_{\mathcal{O}}(t)},$
(2.5)

which measures the fluctuations around the average. Of course, in the present
paper, we will mainly consider the case of $k=2$.

One can also extract further properties of the distribution of probability
amplitudes $\phi_{n}(t)$ by studying entropic measures such as operator
entropy, or K-entropy, which is defined through the von Neumann entropy of the
probabilities as follows [14]222More generically, one can compute Renyi
K-entropies, which is defined through the moments of the probability
distribution, see e.g., [64].

$\displaystyle
S_{\mathcal{O}}(t)=-\sum_{n}|\phi_{n}(t)|^{2}\;\log|\phi_{n}(t)|^{2}.$ (2.6)

Clearly, if the amplitudes are very peaked at a particular value of $n$, the
K-entropy is small, while for uniform distribution, it becomes large.

From the definition of the above quantities, we see that they can be computed
if we know the explicit form of the probability amplitudes. Of course, to
obtain the probability amplitudes, one needs to know the explicit form of the
Lanczos coefficients. Therefore, one may conclude that all information about
the operator growth is, indeed, encoded in the Lanczos coefficients. Thus the
aim is to see how these coefficients can be computed for a given system. In
order to find the Lanczos coefficients $b_{n}$ it is convenient to define the
moments $\\{\mu_{2n}\\}$

$\displaystyle\mu_{2n}=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\;\omega^{2n}f(\omega),$
(2.7)

where $f$ is the power spectrum which is defined as the Fourier transformation
of auto-correlation function, i.e.,
$\phi_{0}(t)=(\mathcal{O}(t)|\mathcal{O}(0))$, as follows

$\displaystyle f(\omega)=\int_{-\infty}^{\infty}d\omega\;e^{i\omega
t}\phi_{0}(t)\,.$ (2.8)

Indeed, the moments $\mu_{2n}$’s are the Maclaurin expansion coefficients of
the auto-correlation function. As shown in [49], having known the moments, the
Lanczos coefficients can be computed using the following recursion relation

$\displaystyle b_{n}=\sqrt{M_{2n}^{(n)}},\hskip 28.45274ptb_{-1}=b_{0}\equiv
1,\hskip 28.45274ptM_{2\ell}^{(-1)}=0,\hskip
28.45274ptM_{2\ell}^{(0)}=\mu_{2\ell},$ (2.9) $\displaystyle
M_{2\ell}^{(j)}=\frac{M_{2\ell}^{(j-1)}}{b_{j-1}^{2}}-\frac{M_{2\ell-2}^{(j-2)}}{b_{j-2}^{2}},\hskip
56.9055pt\ell=j,\cdots,n.$

Let us emphasize that, in general, it is not possible to find a closed-form
expression for $\phi_{n}(t)$, and hence finding the full-time profile of the
quantities defined above requires some numerical treatment. This amounts to
modifying, for example, the Eq. (2.4) and the normalization condition as
follows

$\displaystyle K_{\mathcal{O}}(t)\approx\sum_{n=0}^{n_{\rm
max}}n|\phi_{n}(t)|^{2},\hskip 28.45274pt{\rm with}\hskip
28.45274pt\sum_{n=0}^{n_{\rm max}}|\phi_{n}(t)|^{2}\approx 1.$ (2.10)

This means that, actually, we are approximately computing K-complexity using
the above equation for some finite $n_{\rm max}$.

Now we are equipped with all we need to study the behavior of Lanczos
coefficients and thereby other quantities defined in this section for quantum
field theories with the Lifshitz symmetry.

##  3 Lanczos coefficients and Krylov complexity in Lifshitz scalar theory

In this section, following [42, 43], we would like to find the Lanczos
coefficients for the model introduced in equation (1.6) at finite temperature,
which can be used to compute the quantities we introduced in the previous
section.

To start, let us consider the thermal Wightman two-point function, also known
as the auto-correlation function, which is

$\displaystyle\Pi^{W}(t)=\phi_{0}(t)=(\mathcal{O}(t)|\mathcal{O}(0))\equiv\langle\mathcal{O}^{\dagger}\left(t-\frac{i\beta}{2}\right)\mathcal{O}(0)\rangle_{\beta},\hskip
28.45274pt{\rm with}\hskip 28.45274pt\beta=\frac{1}{T}\,.$ (3.1)

Then the Wightman power spectrum, $f^{W}(\omega)$, can be expressed in terms
of the spectral function $\rho(\omega,k)$ as follows

$\displaystyle f^{W}(\omega)=\int_{-\infty}^{\infty}d\omega\;e^{i\omega
t}\Pi^{W}(t)=\frac{1}{\sinh\frac{\beta\omega}{2}}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\rho(\omega,\vec{k}).$
(3.2)

The spectral function $\rho(\omega,\vec{k})$ for the free massive scalar
theory is given by [48]

$\displaystyle\rho(\omega,\vec{k})=\frac{\mathcal{N}}{\epsilon_{k}}\left(\delta(\omega-\epsilon_{k})-\delta(\omega+\epsilon_{k})\right)\,.$
(3.3)

Here $\mathcal{N}$ is a normalization factor and $\epsilon_{k}$ denotes the
energy eigenvalues. In our case where we are dealing with the free Lifshitz
scalar theory the dispersion relation is given by
$\epsilon_{k}=\sqrt{\vec{k}^{2z}+m^{2z}}$ [47]. By making use of Eqs. (3.2)
and (3.3), it is relatively straightforward to evaluate the Wightman power
spectrum

$\displaystyle
f^{W}(\omega)=\mathcal{N}(m,\beta,d,z)\frac{(\omega^{2}-m^{2z})^{\frac{d-1}{2z}-1}}{z\sinh(\frac{\beta|\omega|}{2})}\Theta(|\omega|-m^{z}),$
(3.4)

where the normalization factor $\mathcal{N}(m,\beta,d,z)$ can be determined by
simply evaluating the following normalization condition

$\displaystyle\int\frac{d\omega}{2\pi}f^{W}(\omega)=1.$ (3.5)

###  3.1 Massless case ($m=0$)

To explore the $z$-dependence of the Lanczos coefficients in our model, in
what follows, we will consider the massless case, for which from Eq. (3.4),
one gets

$\displaystyle
f^{W}(\omega)=\frac{\pi\beta}{(2^{\frac{d-1}{z}}-2)\Gamma\left(\frac{d-1}{z}-1\right)\zeta\left(\frac{d-1}{z}-1\right)}\frac{|\beta\omega|^{\frac{d-1}{z}-2}}{\sinh(\frac{\beta|\omega|}{2})}\Theta(|\omega|).$
(3.6)

Here, we have used Eq. (3.5) to fix the normalization factor ${\cal N}$. From
this expression, one observes that in the high-frequency limit, the power
spectrum becomes $f^{W}(\omega\rightarrow\infty)\sim
e^{-\frac{\beta\omega}{2}}\omega^{\frac{d-1}{z}-2}$, which is, indeed, the
scaling behavior we expect to have for an operator with dimension
$2\Delta=\frac{d-1}{z}-1$ in a scale-invariant theory (for the CFT case see
[24]). Plugging this expression into Eq. (2.7), the moments $\\{\mu_{2n}\\}$
are computed as follows

$\displaystyle\mu_{2n}=\frac{\beta^{-2n}\left(2\
2^{1/z}-2^{\frac{d}{z}+2n}\right)\zeta\left(2n+\frac{d-z-1}{z}\right)\Gamma\left(2n+\frac{d-z-1}{z}\right)}{\left(2\
2^{1/z}-2^{d/z}\right)\zeta\left(\frac{d-z-1}{z}\right)\Gamma\left(\frac{d-z-1}{z}\right)},$
(3.7)

where $\zeta$ denotes the zeta function. Although the above expression for
moments $\mu_{2n}$ looks very complicated, one can numerically evaluate the
Lanczos coefficients using Eq. (2.9). The results are depicted in figure 1 for
several values of $z$ and $d$333 The scaling dimension of the scalar field is
given by $\frac{d-z}{2}$. Restricting to positive scaling dimensions, studying
larger values of $z$ is possible in $d>z$ dimensions. Meanwhile $d<z$ are
well-behaving as long as an IR cut-off is introduced (see for instance [50,
51, 53, 54, 55, 56, 57, 58, 59]). .

Figure 1: Lanczos coefficients in the massless regime for different values of
$z$ and $d$. As we see, although the slope is the same for all $z$, the
$y$-intercept depends on $z$. In particular, as one increases $z$, the
$y$-intercept decreases and the difference of $y$-intercepts for odd and even
$n$ becomes less pronounced.

Interestingly enough, looking at the numerical results shown in figure 1, one
observes that the slope is independent of $d$ and $z$. Hence, non-relativistic
scale invariance does not influence the rate of change of the Lanczos
coefficients. Indeed, as far as the slope is concerned, one can see that for
all cases, the best fit is given by $b_{n}=\frac{\pi}{\beta}n+\cdots$.

We note, however, that the $y$-intercept of different cases depends on $z$.
More precisely, as one increases the critical exponent, the $y$-intercept
decreases. Moreover, the staggering effect444The staggering effect is defined
through the fact that the Lanczos coefficients, $b_{n}$, are separated into
two families for odd and even $n$ [24, 42, 43]., which causes to have
different $y$-intercepts for even and odd $n$’s for given cases, becomes less
pronounced as we increase $z$. Actually, for large $n$, the best fit is

$\displaystyle b_{n}=\frac{\pi}{\beta}\left(n+\frac{d-2z-1}{2z}\right).$ (3.8)

Let us now turn to the computation of the K-complexity in this setup using Eq.
(2.4). To proceed, we note that from Eq. (3.6) and using inverse Fourier
transformation, one can find $\phi_{0}(t)$ as follows

$\displaystyle\phi_{0}(t)=\frac{\zeta\left(\frac{d-z-1}{z},\frac{it}{\beta}+\frac{1}{2}\right)+\zeta\left(\frac{d-z-1}{z},\frac{1}{2}-\frac{it}{\beta}\right)}{2\left(2^{\frac{d-z-1}{z}}-1\right)\zeta\left(\frac{d-z-1}{z}\right)}.$
(3.9)

It is also straightforward to obtain a closed form for $n$-th derivative of
$\phi_{0}(t)$

$\displaystyle\frac{d^{n}\phi_{0}(t)}{dt^{n}}=\frac{\Gamma\left(\frac{d-1-z}{z}+n\right)\left(\left(\frac{i}{\beta}\right)^{n}\zeta\left(\frac{d-1-z}{z}+n,\frac{it}{\beta}+\frac{1}{2}\right)+\left(\frac{-i}{\beta}\right)^{n}\zeta\left(\frac{d-1-z}{z}+n,\frac{1}{2}-\frac{it}{\beta}\right)\right)}{(-1)^{n+1}\left(2-2^{\frac{d-1}{z}}\right)\zeta\left(\frac{d-1}{z}-1\right)\Gamma\left(\frac{d-1}{z}-1\right)},$
(3.10)

which could be used to compute probability amplitudes $\phi_{n}(t)$
numerically by making use of Eq. (2.3). It is then possible to compute the
different physical quantities we defined in the previous section. In
particular, figure 2 shows the K-complexity as a function of time in
logarithmic scale for several values of $z$ with $d=11$.

Figure 2: Evolution of K-complexity in the massless regime for various values
of the dynamical exponent. The complexity decreases as one increases $z$,
though for the late time, the slope is the same for all cases, which is given
by $\frac{2\pi}{\beta}$.

To produce these plots, we have used Eq. (2.10) with $n_{\rm max}=250$ to
approximately compute K-complexity.

From these numerical results, one observes that $K_{\mathcal{O}}(t)$ decreases
with the dynamical exponent, which is perfectly consistent with the results
illustrated in figure 1. Moreover, since the vertical axis is in the
logarithmic scale, the linear growth corresponds to an exponential growth for
the complexity. From Eq. (1.4), representing our best fit for the Lanczos
coefficients, one would expect that the slope of these curves is the same for
all cases and is equal to $\frac{2\pi}{\beta}$. Indeed, our numerical results
confirm our expectation for different values of $d$ and $z$.

To close this subsection, we present our numerical results for K-variance and
K-entropy, defined in equations (2.5) and (2.6), respectively, in figure 3.
Note that all of the curves for $\delta_{\mathcal{O}}(t)$ stabilize to a
constant value at late times, and further, the fluctuations become less
pronounced for larger values of the dynamical exponent. As for K-entropy, one
gets late time linear growth with the same slope for different $z$. Indeed,
our numerical results show that the slope is again given by
$\frac{2\pi}{\beta}$.

Figure 3: K-variance (left) and K-entropy (right) as a function of time in the
massless regime for various values of the dynamical exponent.

###  3.2 Massive case

In this section, we consider a massive scalar field to explore how the non-
zero mass could affect the results presented in the previous section. To
highlight these effects, in what follows we will consider large mass limit in
low temperature regime. To be more concrete, we will consider the case in
which $\beta m^{z}\gg 1$ and hence, from Eq. (3.4), the normalized Wightman
power spectrum reads

$\displaystyle
f^{W}(\omega)=\frac{z\pi^{3/2}\beta^{\frac{d-z-1}{2z}}}{2^{\frac{d-z-1}{z}}m^{\frac{d-z-1}{2}}K_{\frac{d-z-1}{2z}}\left(\frac{\beta
m^{z}}{2}\right)\Gamma\left(\frac{d-1}{2z}\right)}\frac{(\omega^{2}-m^{2z})^{\frac{d-1}{2z}-1}}{z}e^{-\frac{\beta|\omega|}{2}}\Theta(|\omega|-m^{z}).$
(3.11)

It is found useful to express our results in terms of the dimensionless
parameter $s=\beta m^{z}$. Plugging the above expression into Eq. (2.7), one
arrives at

$\displaystyle\mu_{2n}=$
$\displaystyle\frac{\sqrt{\pi}\left(\frac{s}{4}\right)^{\frac{d+z-1}{2z}}}{K_{\frac{d-z-1}{2z}}\left(\frac{s}{2}\right)}\left(\frac{2}{\beta}\right)^{2n}\Bigg{(}\frac{\Gamma\left(\frac{1+z-d}{2z}-n\right)}{\Gamma\left(\frac{1}{2}-n\right)}\left(\frac{s}{2}\right)^{2n-1}{}_{p}F_{q}\left(n+\frac{1}{2};\frac{1}{2},\frac{d-1}{2z}+n+\frac{1}{2};\left(\frac{s}{4}\right)^{2}\right)$
(3.12)
$\displaystyle+2\frac{\Gamma\left(\frac{d-1}{z}+2n-1\right)}{\Gamma\left(\frac{d-1}{2z}\right)}\left(\frac{s}{2}\right)^{\frac{1-d}{z}}{}_{p}F_{q}\left(1+\frac{1-d}{2z};1-n+\frac{1-d}{2z},\frac{1-d}{2z}+\frac{3}{2}-n;\left(\frac{s}{4}\right)^{2}\right)$
$\displaystyle-\frac{\Gamma\left(\frac{1-d}{2z}-n\right)}{\Gamma\left(-n\right)}\left(\frac{s}{2}\right)^{2n}{}_{p}F_{q}\left(n+1;\frac{3}{2},\frac{d-1}{2z}+n+1;\left(\frac{s}{4}\right)^{2}\right)\Bigg{)},$

where ${}_{p}F_{q}$ is the regularized generalized hypergeometric function.
This expression can be used to obtain the Lanczos coefficients numerically
from Eq. (2.9). The numerical results are shown in figure 4.

Figure 4: Lanczos coefficients in the large mass regime. In this case, both
slope and $y$-intercept are affected by non-zero mass.

As it is evident from this figure, the Lanczos coefficients for the massive
case exhibit qualitatively similar behavior as that of the massless one with
two interesting features. First, we note that, similar to the massless case,
the slope of the curves for odd and even $n$ are the same. Actually, in the
present case, the best fit is given by

$\displaystyle b_{n}=\;\bigg{\\{}\begin{array}[]{rcl}&\alpha_{{\rm
o}}\;n+\gamma_{{\rm o}}&\,\,\,{\rm odd}\;n,\cr&\alpha_{{\rm
e}}\;n+\gamma_{{\rm e}}&\,\,\,{\rm even}\;n,\end{array}$ (3.15)

where based on our numerics we have $\alpha_{{\rm e}}=\alpha_{{\rm
o}}\equiv\alpha$. To explore different features of $\alpha$, we have evaluated
it in terms of $s$ for different values of the dynamical exponent in $d=5$.
The results are depicted in figure 5.

Figure 5: Mass-dependence of $\alpha$ for different values of $z$ for $d=5$.
These coefficients seem to be independent of $z$ and always obey the bound
$\alpha\leq\pi/\beta$.

Using the numerical results, one observes that $\alpha$ is monotonically
decreasing function of mass similar to that of relativistic case [43].
Moreover, $\alpha$ is $z$ independent and only depends on mass. This function
is bounded from above and the bound is given by its value at zero mass limit
$\alpha\leq\pi/\beta$.

The second new feature we find for the massive case is that, unlike the
massless case, the difference of $y$-intercepts for even and odd curves
increases as one increases the critical exponent, indicating that for non-zero
mass the difference $|\gamma_{{\rm o}}-\gamma_{{\rm e}}|$ is of the order of
$m^{z}$. This observation may be confirmed by evaluating $b_{n}$s in a series
expansion for $s\gg n$. To be concrete for $z=2$ and $d=5$ in this limit, one
finds

$\displaystyle
b_{n}=m^{2}\;\Bigg{\\{}\begin{array}[]{rcl}&1+\frac{2n}{s}+\frac{2n^{2}}{s^{2}}-\frac{2n(n^{2}+1)}{s^{3}}+\cdots&\,\,\,n\in
2k+1,\\\
&\frac{2n}{s}+\frac{2n^{2}}{s^{2}}-\frac{2n^{3}}{s^{3}}+\cdots&\,\,\,n\in
2k,\end{array}$ (3.18)

resulting in $|\gamma_{{\rm o}}-\gamma_{{\rm e}}|\approx m^{2}$.

Let us now compute K-complexity for the massive case. To do so, one may
compute the auto-correlation function from Eq. (3.11) as follows

$\displaystyle\phi_{0}(t)=\frac{\left(1-\frac{2it}{\beta}\right)^{\frac{1-d+z}{2z}}K_{\frac{1-d+z}{2z}}\left(m^{z}\frac{\beta-2it}{2}\right)+\left(1+\frac{2it}{\beta}\right)^{\frac{1-d+z}{2z}}K_{\frac{1-d+z}{2z}}\left(m^{z}\frac{\beta+2it}{2}\right)}{2K_{\frac{1-d+z}{2z}}\left(\frac{\beta
m^{z}}{2}\right)},$ (3.19)

where $K_{n}(x)$ is the Bessel function of the second kind. It is then
straightforward to compute K-complexity numerically in the large mass limit by
making use of Eqs. (2.3) and (2.4). To perform our computations, we have used
Eq. (2.10) with $n_{\rm max}=100$, which is a good approximation for the time
interval we have considered. The corresponding numerical results for different
$m$ and $z$ are shown in figure 6. Similarly, one could also compute K-entropy
numerically, and the corresponding result is depicted in figure 7.

Figure 6: K-complexity in the large mass limit for different values of $m$ and
$z$ with $d=5$. Here we set $\beta=1$. At early times the results are
independent of the critical exponent, though at late times it has significant
effects.

Figure 7: Evolution of K-entropy for several values of the mass and dynamical
exponent.

From our numerical results shown in these figures, one observes that the
general behavior for K-complexity and K-entropy are the same as those in the
massless case. Namely, while complexity has exponential growth, the K-entropy
has linear growth at late times. We note, however, at early times, there is an
oscillatory behavior that is associated with non-zero mass. Actually, these
oscillations are originating from the oscillatory behavior of probability
amplitudes $\phi_{n}(t)$ [43]. To see this point better, it is useful to write
the explicit form of the auto-correlation function Eq. (3.19) for $z=2$

$\displaystyle\phi_{0}(t)=\frac{\beta^{2}\cos(m^{2}t)-2\beta
t\sin(m^{2}t)}{\beta^{2}+4t^{2}}.$ (3.20)

It is also worth noting that while at early times the results are independent
of $z$, at late times the critical exponent has significant effects such that
as one increases $z$, both complexity and entropy decrease. This is due to the
fact that the slope of the curves is affected by the non-zero mass so that it
is always smaller than that of the massless case, moreover, from dimensional
analysis one finds that the mass dependence is in the form of $m^{z}$. Note
also that the period of the oscillations in the oscillatory region is given by
$m^{-z}$ so that the amplitude of oscillation becomes less pronounced at later
times. Let us examine in more detail the mass-dependence of K-complexity as
shown in figure 8. We see that by increasing the mass parameter, the
K-complexity decreases. For the relativistic case with $z=1$, this is
intuitive in the sense that the correlation length of a massive field is
decreased with the mass parameter and so is its complexity. Similarly, for the
larger values of the dynamical exponent, we expect that the effective length
of quantum fluctuations is decreased by mass, and hence the K-complexity
decreases.

Figure 8: Evolution of K-complexity for several values of the mass with $z=2$.

Although we have not presented the results for K-variance, we have numerically
computed it and its general behavior exhibits similar features to those
discussed for complexity and entropy.

##  4 Lanczos coefficients and Krylov complexity in the presence of UV cutoff

In this section, we will study the effect of having a dimensionful scale in
the model on the behavior of Lanczos coefficients and, consequently,
K-complexity. There are several ways one may have a non-trivial dimensionful
scale. In the previous section, we have already studied the case where the
dimensionful parameter is given by non-zero mass, which acts as an IR cutoff,
and its effects on the slopes of the linear growth of Lanczos coefficients at
large values of $n$ were observed.

We note that there are several other ways to have a new scale in the model.
For example, we can achieve this by putting the model in a compact space or
discretizing the model by putting it on a lattice. In these cases, the scale
is given by the curvature radius of the compact space or lattice spacing,
respectively. Another way to have a non-trivial scale is to add a hard or soft
cutoff to the theory.

In all cases, we expect that the behavior of Lanczos coefficients will be
influenced by the dimensionful scale, the nature of which may depend on the
way the scale is added to the model. In what follows, we will study the effect
of having a non-zero hard cutoff and the case where the model is put on a
lattice.

###  4.1 Krylov complexity with hard UV cutoff

In this section, we will examine the effects of a finite UV cutoff in
continuous momentum space on the behaviors of Lanczos coefficients and
K-complexity. In particular, we will consider a UV cutoff on the integral
upper bound in Eq. (3.2). Hence, the Wightman power spectrum will undergo the
following change

$\displaystyle
f^{W}(\omega)=\frac{1}{\sinh\frac{\beta\omega}{2}}\int_{0}^{\Lambda^{z}}\frac{d^{d-1}k}{(2\pi)^{d-1}}\rho(\omega,\vec{k})=\mathcal{N}\frac{(\omega^{2}-m^{2z})^{\frac{d-2z-1}{2z}}}{z|\sinh(\frac{\beta\omega}{2})|}\Theta(|\omega|-m^{z})\Theta(\Lambda^{z}-|\omega|).$
(4.1)

In the limit of $1\ll\beta m^{z}\ll\beta\Lambda^{z}$, the above equation reads

$\displaystyle
f^{W}(\omega)=\mathcal{N}(m,\beta,d)(\omega^{2}-m^{2z})^{\frac{d-1}{2z}-1}e^{-\frac{\beta|\omega|}{2}}\Theta(|\omega|-m^{z})\Theta(\Lambda^{z}-|\omega|)\,.$
(4.2)

By making use of Eq. (2.7) it is straightforward to compute the moments
$\\{\mu_{2n}\\}$. In particular, for $z=2$ and $d=5$, one finds

$\displaystyle\mu_{2n}=2^{2n}\beta^{-2n}\frac{\Gamma\left(2n+1,\frac{m^{2}\beta}{2}\right)-\Gamma\left(2n+1,\frac{\beta\Lambda^{2}}{2}\right)}{e^{-\frac{1}{2}\left(\beta
m^{2}\right)}-e^{-\frac{1}{2}\left(\beta\Lambda^{2}\right)}}.$ (4.3)

Although it is impossible to find an analytic expression for Lanczos
coefficients for a general choice of the parameters, one can use the recursion
relation Eq. (2.9) to find them numerically. The corresponding numerical
results for different values of $z$ with the finite $\Lambda$ are shown in
figure 9.

Figure 9: Lanczos coefficients in $d=5$ for different values of $z$, $m$ and
$\Lambda$.

It is clear from this figure that the presence of a UV cutoff significantly
modifies the behavior of Lanczos coefficients. The situation appears to be
quite similar to that of a relativistic case [42, 43]. Specifically, the
system undergoes a phase of linear growth as described by Eq. (3.15), but
eventually saturates to a constant value as $n$ becomes large. This saturation
value is proportional to the UV cutoff. More precisely, the saturation value
can be expressed as $b_{\rm s}\approx\frac{\Lambda^{z}\pm m^{z}}{2}$, where
the sign should be chosen as $+(-)$ for odd (even) values of $n$. In addition,
the transition occurs at a sharp saturation point $n=n_{\rm s}$, which can be
estimated as 555Here we use the fact that $\alpha_{{\rm o}}\approx\alpha_{{\rm
e}}\equiv\alpha$ and $\gamma_{\rm e,o}\propto m^{z}\ll\Lambda^{z}$.

$\displaystyle n_{s}=\frac{\Lambda^{z}-\gamma_{{\rm o}}-\gamma_{{\rm
e}}}{2\alpha}\approx\frac{\Lambda^{z}}{2\alpha},$ (4.4)

which perfectly matches the numerical results.

In the right panel of figure 9, we plot Lanczos coefficients using a different
choice of the mass and UV cutoff, which depends on the dynamical exponent,
i.e., $m^{z}$ and $\Lambda^{z}$. It can be observed that the Lanczos
coefficients exhibit linear growth with increasing $n$, followed by saturation
to a constant value. Furthermore, at a given value of $n$, both the growth
rate and the saturation value appear to be approximately independent of the
critical exponent. In fact, their dependence on $z$ is implicitly determined
by the choice of mass and UV scale.

Let us now consider the K-complexity. The procedure is the same as what was
done in the previous section. The numerical results are depicted in Figure 10.

Figure 10: K-complexity in the presence of a hard UV cutoff for different
values of $z$ with $\beta=1$. In this case, the complexity exhibits
exponential growth followed by linear growth at late times.

As expected, the behavior of the complexity exhibits exponential growth at
early times, followed by linear growth at later times. These phases are
associated with linear and saturation phases of Lanczos coefficients,
respectively.

It is also interesting to compute the K-entropy in the presence of a hard UV
cutoff, as shown in figure 11 for different values of $z$.

Figure 11: K-entropy in the presence of a hard UV cutoff for different values
of $z$ with $\beta=1$.

It exhibits an oscillatory behavior, which is a consequence of the non-zero
mass, and approximately follows a logarithmic scaling due to the presence of a
hard UV cutoff. This particular behavior is related to the linear growth of
complexity at later times and will be discussed further in the Conclusion
section.

###  4.2 Krylov complexity for the Lifshitz harmonic model

In this section, we will study a discretized version of our model with a
finite lattice spacing $a$, which results in a UV cutoff of the form
$\Lambda\sim\frac{1}{a}$. To be specific, we will consider a one-dimensional
lattice with periodic boundary conditions. As mentioned earlier, in this case,
the corresponding dispersion relation is given by Eq. (1.8), and thus the
Wightman power spectrum takes the form

$\displaystyle
f^{W}(\omega)=\mathcal{N}\sum_{k=1}^{N}\frac{1}{\sinh(\frac{\beta\omega}{2})}\frac{1}{\epsilon_{k}}[\delta(\omega-\epsilon_{k})-\delta(\omega+\epsilon_{k})],\qquad{\rm
with}\qquad\epsilon_{k}=\sqrt{\sin(\frac{\pi k}{N})^{2z}+m^{2z}}.$ (4.5)

Using Eq. (2.7), one may compute the moments $\\{\mu_{2n}\\}$ on lattice as
follows

$\mu_{2n}=\frac{\mathcal{N}}{2\pi}\sum_{k=1}^{N}\int_{-\infty}^{\infty}d\omega\frac{\omega^{2n}}{\epsilon_{k}\sinh(\frac{\beta\omega}{2})}[\delta(\omega-\epsilon_{k})-\delta(\omega+\epsilon_{k})]=\frac{\mathcal{N}}{\pi}\sum_{k=1}^{N}\frac{\epsilon_{k}^{2n-1}}{\sinh(\frac{\beta\epsilon_{k}}{2})}$
(4.6)

By substituting the above equation into Equation (2.9), one can numerically
compute the Lanczos coefficients for various values of the model parameters.
The corresponding results for different values of $\beta,m,z$ and $N$ are
depicted in figure 12.

Figure 12: Lanczos coefficients in Lifshitz harmonic model for different
values of the parameters.

We can make several observations regarding these numerical results. First,
similar to the case with a Hard UV cutoff, it is observed that the Lanczos
coefficients grow linearly with $n$ and then saturate to a constant value for
large $n$. Moreover, during the linear growth phase, the slope (approximately)
remains constant, independent of the parameters $\beta$, $m$, and $z$.
Interestingly, a staggering effect is observed, which decreases with
increasing $z$. In the linear phase, a linear fitting for $b_{n}$ can be
proposed, as in Eq. (3.15). Figure 13 shows the mass-dependence of the
corresponding slopes in this phase, which are monotonically decreasing
functions of mass.

Figure 13:  Mass-dependence of $\alpha_{o}$ and $\alpha_{e}$. Here $N=1000$
and $\beta=1000$.

Let us now turn our attention to the computation of the K-complexity in this
setup, using Eq. (2.4). To proceed, we note that from Eq. (4.5), one can find
$\phi_{0}(t)$ as follows

$\displaystyle\phi_{0}(t)=\frac{\mathcal{N}}{\pi}\sum_{k=1}^{N}\frac{1}{\epsilon_{k}}\csch\left(\frac{\beta\epsilon_{k}}{2}\right)\cos(\epsilon_{k}\;t).$
(4.7)

It is then straightforward to compute the different measures in the Krylov
basis using the above expression. By examining the behavior of the Lanczos
coefficients, one can conclude that the K-complexity and K-entropy exhibit
similar behaviors to those studied in the previous subsection. Specifically,
the K-complexity exhibits exponential growth followed by linear growth at
later times, while the K-entropy exhibits logarithmic growth. The oscillatory
behavior is also associated with having a non-zero mass. The corresponding
numerical results are shown in figure 14.

Figure 14:  K-complexity (Left) and K-entropy (right) for the Lifshitz
harmonic model.

##  5 Conclusions

In this paper, we have studied the general behavior of Lanczos coefficients
and K-complexity in a Lifshitz scalar field theory with nontrivial values of
the dynamical critical exponent. Furthermore, we have examined the effects of
mass, temperature, finite UV cutoffs in continuous momentum space, and finite
lattice spacing. In the following, we summarize our main results and discuss
some further problems.

  * •

In a continuum massless Lifshitz scalar theory, the Lanczos coefficients grow
linearly with $n$, which is consistent with the universal operator growth
hypothesis. Interestingly, although the slope is completely independent of the
dynamical exponent, the value of $b_{n}$ decreases with $z$. Hence, non-
relativistic scale invariance does not influence the rate of change of the
Lanczos coefficients. A curious feature that we have observed is that the
staggering effect becomes less pronounced as we increase the dynamical
exponent. The K-complexity exhibits exponential growth with time and decreases
as $z$ is increased, although the slope at late times is the same for all
values of the dynamical exponent and is given by $\frac{2\pi}{\beta}$. In this
case, the K-variance stabilizes to a constant value at late times, and the
fluctuations become less pronounced for larger values of the dynamical
exponent. Furthermore, for K-entropy, we observe late-time linear growth with
the same slope for different values of $z$.

  * •

In a continuum massive theory, the Lanczos coefficients exhibit qualitatively
similar behavior to that of the massless case, with two interesting features.
First, similar to the massless case, the slope of $b_{n}$ for odd and even $n$
are the same. It is a monotonically decreasing function of mass and is
independent of $z$. Moreover, the slope is bounded from above by its value at
the zero mass limit, i.e., $\frac{\pi}{\beta}$. Additionally, in the large
mass regime, the separation of $b_{n}$ between odd and even $n$ increases as
one increases the dynamical exponent and is proportional to $m^{z}$.

  * •

The general behavior of $K_{\mathcal{O}}(t)$ and $S_{\mathcal{O}}(t)$ is the
same as that in the massless case. Specifically, while the complexity exhibits
exponential growth, the entropy exhibits linear growth at late times. While
the behavior is independent of $z$ at early times, the critical exponent has a
significant effect at late times, such that both the complexity and entropy
decrease as $z$ is increased. Furthermore, $K_{\mathcal{O}}(t)$ and
$S_{\mathcal{O}}(t)$ are decreasing functions of the mass parameter. Indeed,
similar to the relativistic case, we expect that the effective length of
quantum fluctuations is decreased by mass, and hence both measures decrease.

  * •

In the presence of a hard UV cutoff, the behavior of the Lanczos coefficients
is significantly modified. Initially, they exhibit a phase of linear growth,
which is followed by saturation to a constant value for large $n$. As a
consequence, the K-complexity exhibits exponential growth at relatively early
times, followed by linear growth at late times. These phases are associated
with the linear and saturation phase of the Lanczos coefficients,
respectively. Furthermore, the K-entropy exhibits approximate logarithmic
growth, which is related to the linear growth of complexity at late times.
Similarly, when considering a discretized version of the model with a finite
lattice spacing, we found similar results.

Another interesting observation that can be made is that, at least over the
range of our numerical computations, it appears that the behavior of the
K-complexity is similar to that of the exponential of the K-entropy. Our
numerical computations show that in the region where the complexity exhibits
exponential growth, the K-entropy grows linearly, while when the complexity
exhibits linear growth, the K-entropy has logarithmic behavior.

Actually, an alternative definition of complexity has been proposed in the
context of spread complexity [21] (see also [22]), where the complexity is
given as ’the exponential of the entropy of the probability distribution of
weights in an orthonormal basis’ [21]: ${\cal C}=e^{\cal S}$.

Of course, in our case, we would not expect to obtain such an exact relation
between complexity and entropy, as it is evident from their definitions in
Eqs. (2.4) and (2.6), respectively. Nonetheless, from our numerical results,
we have found that at least for the massless case, we have
$1+K_{\mathcal{O}}(t)\sim e^{aS_{\mathcal{O}}(t)}$ for some numerical constant
$a<1$, which is not universal. Indeed, its value depends on the dimension and
critical exponent.

It would be interesting to see the precise information that could be obtained
from complexity and entropy in the context of the dynamics of Krylov space.
Indeed, since in this context, all information is encoded in the Lanczos
coefficients, one would suspect that complexity may have additional
information compared to entropy.

## Acknowledgments

We would also like to thank Souvik Banerjee and Pawel Caputa for discussions
on different aspects of Krylov complexity.

## References

  * [1] J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys Rev A.43 2046 (1991), doi :10.1103/PhysRevA.43.2046.
  * [2] M. Srednicki, “Chaos and Quantum Thermalization,” doi:10.1103/PhysRevE.50.888 [arXiv:cond-mat/9403051 [cond-mat]].
  * [3] L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, “From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,” Adv. Phys. 65 (2016) no.3, 239-362 doi:10.1080/00018732.2016.1198134 [arXiv:1509.06411 [cond-mat.stat-mech]].
  * [4] M. Srednicki, “The approach to thermal equilibrium in quantized chaotic systems,” J. Phys. A32 (1999) 1163, doi:10.1088/0305-4470/32/7/007.
  * [5] S. H. Shenker and D. Stanford, “Multiple Shocks,” JHEP 12 (2014), 046 doi:10.1007/JHEP12(2014)046 [arXiv:1312.3296 [hep-th]].
  * [6] S. H. Shenker and D. Stanford, “Black holes and the butterfly effect,” JHEP 03 (2014), 067 doi:10.1007/JHEP03(2014)067 [arXiv:1306.0622 [hep-th]].
  * [7] J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos,” JHEP 08 (2016), 106 doi:10.1007/JHEP08(2016)106 [arXiv:1503.01409 [hep-th]].
  * [8] A. Kitaev,“A Simple Model of Quantum Holography,” http:// online.kitp.ucsb.edu/online/entangled15/kitaev/.
  * [9] S. Sachdev and J. Ye, “Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet,” Phys. Rev. Lett. 70, 3339 (1993).
  * [10] J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Phys. Rev. D 94 (2016) no.10, 106002 doi:10.1103/PhysRevD.94.106002 [arXiv:1604.07818 [hep-th]].
  * [11] B. V. Fine, T. A. Elsayed, C. M. Kropf, and A S. de Wijn, “Absence of Exponential Sensitivity to Small Perturbations in Nonintegrable Systems of Spins 1/2” Phys. Rev. E 89, 012923 (2014).
  * [12] S. Xu and B. Swingle, “Accessing scrambling using matrix product operators,” Nature Phys. 16 (2019) no.2, 199-204 doi:10.1038/s41567-019-0712-4 [arXiv:1802.00801 [quant-ph]].
  * [13] D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, “A Universal Operator Growth Hypothesis,,, Phys. Rev. X 9, 041017 (2019).
  * [14] J. L. F. Barbón, E. Rabinovici, R. Shir and R. Sinha, “On The Evolution Of Operator Complexity Beyond Scrambling,” JHEP 10 (2019), 264 doi:10.1007/JHEP10(2019)264 [arXiv:1907.05393 [hep-th]].
  * [15] A. Avdoshkin and A. Dymarsky, “Euclidean operator growth and quantum chaos,” Phys. Rev. Res. 2 (2020) no.4, 043234 doi:10.1103/PhysRevResearch.2.043234 [arXiv:1911.09672 [cond-mat.stat-mech]].
  * [16] A. Dymarsky and A. Gorsky, “Quantum chaos as delocalization in Krylov space,” Phys. Rev. B 102 (2020) no.8, 085137 doi:10.1103/PhysRevB.102.085137 [arXiv:1912.12227 [cond-mat.stat-mech]].
  * [17] E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, “Operator complexity: a journey to the edge of Krylov space,” JHEP 06 (2021), 062 doi:10.1007/JHEP06(2021)062 [arXiv:2009.01862 [hep-th]].
  * [18] X. Cao, “A statistical mechanism for operator growth,” J. Phys. A 54 (2021) no.14, 144001 doi:10.1088/1751-8121/abe77c [arXiv:2012.06544 [cond-mat.stat-mech]].
  * [19] E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, ‘Krylov localization and suppression of complexity,” JHEP 03 (2022), 211 doi:10.1007/JHEP03(2022)211 [arXiv:2112.12128 [hep-th]].
  * [20] F. B. Trigueros and C. J. Lin, “Krylov complexity of many-body localization: Operator localization in Krylov basis,” SciPost Phys. 13 (2022) no.2, 037 doi:10.21468/SciPostPhys.13.2.037 [arXiv:2112.04722 [cond-mat.dis-nn]].
  * [21] V. Balasubramanian, P. Caputa, J. M. Magan and Q. Wu, “Quantum chaos and the complexity of spread of states,” Phys. Rev. D 106 (2022) no.4, 046007 doi:10.1103/PhysRevD.106.046007 [arXiv:2202.06957 [hep-th]].
  * [22] Z. Y. Fan, “Universal relation for operator complexity,” Phys. Rev. A 105 (2022) no.6, 062210 doi:10.1103/PhysRevA.105.062210 [arXiv:2202.07220 [quant-ph]].
  * [23] R. Heveling, J. Wang and J. Gemmer, “Numerically probing the universal operator growth hypothesis,” Phys. Rev. E 106 (2022) no.1, 014152 doi:10.1103/PhysRevE.106.014152 [arXiv:2203.00533 [cond-mat.stat-mech]].
  * [24] A. Dymarsky and M. Smolkin, “Krylov complexity in conformal field theory,” Phys. Rev. D 104, no.8, L081702 (2021) doi:10.1103/PhysRevD.104.L081702 [arXiv:2104.09514 [hep-th]].
  * [25] B. Bhattacharjee, X. Cao, P. Nandy and T. Pathak, “Krylov complexity in saddle-dominated scrambling,” JHEP 05 (2022), 174 doi:10.1007/JHEP05(2022)174 [arXiv:2203.03534 [quant-ph]].
  * [26] P. Caputa and S. Liu, “Quantum complexity and topological phases of matter,” Phys. Rev. B 106 (2022) no.19, 195125 doi:10.1103/PhysRevB.106.195125 [arXiv:2205.05688 [hep-th]].
  * [27] W. Mück and Y. Yang, “Krylov complexity and orthogonal polynomials,” Nucl. Phys. B 984 (2022), 115948 doi:10.1016/j.nuclphysb.2022.115948 [arXiv:2205.12815 [hep-th]].
  * [28] S. He, P. H. C. Lau, Z. Y. Xian and L. Zhao, “Quantum chaos, scrambling and operator growth in $T\overline{T}$ deformed SYK models,” JHEP 12 (2022), 070 doi:10.1007/JHEP12(2022)070 [arXiv:2209.14936 [hep-th]].
  * [29] N. Hörnedal, N. Carabba, A. S. Matsoukas-Roubeas and A. del Campo, “Ultimate Speed Limits to the Growth of Operator Complexity,” Commun. Phys. 5 (2022), 207 doi:10.1038/s42005-022-00985-1 [arXiv:2202.05006 [quant-ph]].
  * [30] M. Alishahiha, “On quantum complexity,” Phys. Lett. B 842 (2023), 137979 doi:10.1016/j.physletb.2023.137979 [arXiv:2209.14689 [hep-th]].
  * [31] M. Alishahiha and S. Banerjee, “A universal approach to Krylov State and Operator complexities,” [arXiv:2212.10583 [hep-th]].
  * [32] A. Bhattacharya, P. Nandy, P. P. Nath and H. Sahu, “On Krylov complexity in open systems: an approach via bi-Lanczos algorithm,” [arXiv:2303.04175 [quant-ph]].
  * [33] J. Erdmenger, S. K. Jian and Z. Y. Xian, “Universal chaotic dynamics from Krylov space,” [arXiv:2303.12151 [hep-th]].
  * [34] A. A. Nizami and A. W. Shrestha, “Krylov construction and complexity for driven quantum systems,” [arXiv:2305.00256 [quant-ph]].
  * [35] E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, “A bulk manifestation of Krylov complexity,” [arXiv:2305.04355 [hep-th]].
  * [36] S. Nandy, B. Mukherjee, A. Bhattacharyya and A. Banerjee, “Quantum state complexity meets many-body scars,” [arXiv:2305.13322 [quant-ph]].
  * [37] A. Bhattacharyya, D. Ghosh and P. Nandi, “Operator growth and Krylov Complexity in Bose-Hubbard Model,” [arXiv:2306.05542 [hep-th]].
  * [38] H. A. Camargo, V. Jahnke, H. S. Jeong, K. Y. Kim and M. Nishida, “Spectral and Krylov Complexity in Billiard Systems,” [arXiv:2306.11632 [hep-th]].
  * [39] D. Patramanis and W. Sybesma, “Krylov complexity in a natural basis for the Schrödinger algebra,” [arXiv:2306.03133 [quant-ph]].
  * [40] P. Caputa, J. M. Magan, D. Patramanis and E. Tonni, “Krylov complexity of modular Hamiltonian evolution,” [arXiv:2306.14732 [hep-th]].
  * [41] T. A. Elsayed, B. Hess and B. V. Fine, Signatures of chaos in time series generated by many-spin systems at high temperatures, Phys. Rev. E 90 (2014) 022910.
  * [42] A. Avdoshkin, A. Dymarsky and M. Smolkin, “Krylov complexity in quantum field theory, and beyond,” [arXiv:2212.14429 [hep-th]].
  * [43] H. A. Camargo, V. Jahnke, K. Y. Kim and M. Nishida, “Krylov Complexity in Free and Interacting Scalar Field Theories with Bounded Power Spectrum,” [arXiv:2212.14702 [hep-th]].
  * [44] J. Hartong, E. Kiritsis and N. A. Obers, “Field Theory on Newton-Cartan Backgrounds and Symmetries of the Lifshitz Vacuum,” JHEP 08, 006 (2015) doi:10.1007/JHEP08(2015)006 [arXiv:1502.00228 [hep-th]].
  * [45] S. A. Hartnoll, A. Lucas and S. Sachdev, “Holographic quantum matter,” [arXiv:1612.07324 [hep-th]].
  * [46] J. Figueroa-O’Farrill, R. Grassie and S. Prohazka, “Lifshitz symmetry: Lie algebras, spacetimes and particles,” SciPost Phys. 14, 035 (2023) doi:10.21468/SciPostPhys.14.3.035 [arXiv:2206.11806 [hep-th]].
  * [47] J. Alexandre, “Lifshitz-type Quantum Field Theories in Particle Physics,” Int. J. Mod. Phys. A 26, 4523-4541 (2011) doi:10.1142/S0217751X11054656 [arXiv:1109.5629 [hep-ph]].
  * [48] M. Laine and A. Vuorinen, “Basics of Thermal Field Theory,” Lect. Notes Phys. 925, pp.1-281 (2016) Springer, 2016, doi:10.1007/978-3-319-31933-9 [arXiv:1701.01554 [hep-ph]].
  * [49] V. S. Viswanath and G. Müller, The Recursion Method: Application to Many-Body Dynamics. Springer Berlin, Heidelberg, Germany, 1994.
  * [50] M. R. Mohammadi Mozaffar and A. Mollabashi, “Entanglement in Lifshitz-type Quantum Field Theories,” JHEP 07, 120 (2017) doi:10.1007/JHEP07(2017)120 [arXiv:1705.00483 [hep-th]].
  * [51] T. He, J. M. Magan and S. Vandoren, “Entanglement Entropy in Lifshitz Theories,” SciPost Phys. 3, no.5, 034 (2017) doi:10.21468/SciPostPhys.3.5.034 [arXiv:1705.01147 [hep-th]].
  * [52] S. A. Gentle and S. Vandoren, “Lifshitz entanglement entropy from holographic cMERA,” JHEP 07, 013 (2018) doi:10.1007/JHEP07(2018)013 [arXiv:1711.11509 [hep-th]].
  * [53] M. R. Mohammadi Mozaffar and A. Mollabashi, “Logarithmic Negativity in Lifshitz Harmonic Models,” J. Stat. Mech. 1805, no.5, 053113 (2018) doi:10.1088/1742-5468/aac135 [arXiv:1712.03731 [hep-th]].
  * [54] M. R. Mohammadi Mozaffar and A. Mollabashi, “Entanglement Evolution in Lifshitz-type Scalar Theories,” JHEP 01, 137 (2019) doi:10.1007/JHEP01(2019)137 [arXiv:1811.11470 [hep-th]].
  * [55] M. R. Mohammadi Mozaffar and A. Mollabashi, “Universal Scaling in Fast Quenches Near Lifshitz-Like Fixed Points,” Phys. Lett. B 797, 134906 (2019) doi:10.1016/j.physletb.2019.134906 [arXiv:1906.07017 [hep-th]].
  * [56] D. Hartmann, K. Kavanagh and S. Vandoren, “Entanglement entropy with Lifshitz fermions,” SciPost Phys. 11, no.2, 031 (2021) doi:10.21468/SciPostPhys.11.2.031 [arXiv:2104.10913 [quant-ph]].
  * [57] M. R. M. Mozaffar and A. Mollabashi, “Time scaling of entanglement in integrable scale-invariant theories,” Phys. Rev. Res. 4, no.2, L022010 (2022) doi:10.1103/PhysRevResearch.4.L022010 [arXiv:2106.14700 [hep-th]].
  * [58] M. Mintchev, D. Pontello, A. Sartori and E. Tonni, “Entanglement entropies of an interval in the free Schrödinger field theory at finite density,” JHEP 07, 120 (2022) doi:10.1007/JHEP07(2022)120 [arXiv:2201.04522 [hep-th]].
  * [59] M. Mintchev, D. Pontello and E. Tonni, “Entanglement entropies of an interval in the free Schrödinger field theory on the half line,” JHEP 09, 090 (2022) doi:10.1007/JHEP09(2022)090 [arXiv:2206.06187 [hep-th]].
  * [60] M. Alishahiha, A. Faraji Astaneh and M. R. Mohammadi Mozaffar, “Thermalization in backgrounds with hyperscaling violating factor,” Phys. Rev. D 90, no.4, 046004 (2014) doi:10.1103/PhysRevD.90.046004 [arXiv:1401.2807 [hep-th]].
  * [61] D. A. Roberts and B. Swingle, “Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories,” Phys. Rev. Lett. 117, no.9, 091602 (2016) doi:10.1103/PhysRevLett.117.091602 [arXiv:1603.09298 [hep-th]].
  * [62] M. Alishahiha, A. Faraji Astaneh, M. R. Mohammadi Mozaffar and A. Mollabashi, “Complexity Growth with Lifshitz Scaling and Hyperscaling Violation,” JHEP 07, 042 (2018) doi:10.1007/JHEP07(2018)042 [arXiv:1802.06740 [hep-th]].
  * [63] P. Caputa and S. Datta, “Operator growth in 2d CFT,” JHEP 12, 188 (2021) [erratum: JHEP 09, 113 (2022)] doi:10.1007/JHEP12(2021)188 [arXiv:2110.10519 [hep-th]].
  * [64] P. Caputa, J. M. Magan and D. Patramanis, “Geometry of Krylov complexity,” Phys. Rev. Res. 4 (2022) no.1, 013041 doi:10.1103/PhysRevResearch.4.013041 [arXiv:2109.03824 [hep-th]].