# Krylov complexity in a natural algebra basis

Krylov complexity in a natural basis
for the Schrödinger algebra

Dimitrios Patramanis1 and Watse Sybesma2

1 Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw,
Poland

2 Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland
<EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

We investigate operator growth in quantum systems with two-dimensional
Schrödinger group symmetry by studying the Krylov complexity. While feasible
for semisimple Lie algebras, cases such as the Schrödinger algebra which is
characterized by a semi-direct sum structure are complicated. We propose to
compute Krylov complexity for this algebra in a natural orthonormal basis,
which produces a pentadiagonal structure of the time evolution operator,
contrasting the usual tridiagonal Lanczos algorithm outcome. The resulting
complexity behaves as expected. We advocate that this approach can provide
insights to other non-semisimple algebras.

###### Contents

  1. 1 Introduction
  2. 2 Krylov complexity
    1. 2.1 Krylov basis
    2. 2.2 Heisenberg-Weyl algebra
    3. 2.3 $SL_{2}(\mathbb{R})$ algebra
  3. 3 Two-dimensional Schrödinger algebra
    1. 3.1 Schrödinger symmetries
    2. 3.2 Computation of coherent states
    3. 3.3 Krylov complexity in a natural basis
    4. 3.4 Interpreting complexity
  4. 4 Discussion

##  1 Introduction

Krylov complexity [1] is a measure introduced in order to quantify how quantum
operators change under time evolution. In complex quantum systems one
generically expects that, due to the non-trivial dynamics, a typical operator
undergoing time evolution will tend to “spread” such that it affects a larger
number of the degrees of freedom of the system thereby increasing in
complexity. In [2] it was shown that symmetry plays an important role in
determining the features of this growth and following similar methods in this
work we will be proposing a more general framework that includes symmetries
described by non-semisimple Lie groups.

Krylov complexity is conjectured to grow at a maximal rate for chaotic
systems, although there are subtleties that need to be taken into account,
especially for quantum field theories due to their infinite degrees of freedom
in contrast to ordinary quantum mechanical systems, see discussions in [3, 4,
5, 6, 7, 8, 9, 10]. When one considers quantum many body systems such as spin
chains it is more straightforward to produce evidence that Krylov complexity
can for example distinguish between integrable and chaotic dynamics as was
argued in [11, 12, 13].

However, despite its apparent relevance in the discourse regarding quantum
chaos, one might ask why is it worth studying Krylov complexity over all the
different available measures of which there has been a profound proliferation
in recent years? We believe that there are two main reasons. First, Krylov
complexity can be applied to any quantum system making it computationally
available, at least in principle, for a plethora of different cases including
but not limited to condensed matter and many-body systems [14, 13, 15, 16,
17], quantum and conformal field theories [3, 4, 18, 5, 19, 20], open systems
[21, 22, 23, 24, 25], topological phases of matter [26, 27] and many other
topics related to aspects of the above and not only [28, 29, 30, 31]. Second,
it is related by its construction to inherent properties and characteristic
parameters of the system, namely the Hamiltonian and the Hilbert space that it
defines. The reason why this is important is because it removes the
arbitrariness which is to a large extent present in other definitions of
complexity. Namely, one usually needs to define a set of elementary operations
that can be performed on the system and assign a cost to them according to
some justifiable albeit arbitrary criteria. For a review on this topic see for
example [32, 33]. In the context of Krylov complexity these choices are
reduced to the choice of an inner product using which the Lanczos algorithm
can map the dynamics of a quantum mechanical system onto a semi-infinite one-
dimensional chain with nearest neighbor hopping. Krylov complexity is then
simply defined as the average position on the chain as a function of time. In
the current note we review some important examples of semisimple algebras for
which one can analytically compute Krylov complexity: the case of the
Heisenberg-Weyl algebra, corresponding to states expressible in the one-
dimensional harmonic oscillator basis, and $SL_{2}(\mathbb{R})$, which for
example represents the two-dimensional conformal algebra.

Subsequently, we will go beyond studying the Krylov complexity of semisimple
algebras, which as we will discuss poses certain analytic challenges. We focus
on the specific example of the two-dimensional Schrödinger algebra. This
algebra is the maximal symmetry algebra for a two-dimensional Schrödinger
equation with a quadratic potential or no potential at all and appears as a
subalgebra of the three-dimensional conformal algebra $SO(3,2)$, see e.g.
[34]. As the two-dimensional Schrödinger equation is relevant for, e.g., cold
atom traps and optical systems [35], having a prediction for Krylov complexity
can yield insights in both directions. Moreover, having analytic results for
systems with such a symmetry structure can potentially set the groundwork to
study more complex systems that are relevant in high energy physics.

However, the usual Krylov approach for the two-dimensional Schrödinger group
is problematic from an analytic point of view and only numerical
approximations are available [36], making it difficult to work with. In this
note we propose an alternative method to probe the Krylov subspace that relies
on making use of the semi-direct sum structure of the symmetry algebra and
methods used in the study of coherent states. We find that the semi-direct sum
structure presents us with an orthogonal basis that translates to a picture of
hopping on a semi-infinite chain where next-to-nearest neighbor interactions
are allowed, contrasting conventional Krylov complexity approaches. We
advocate that generically in the case where symmetry algebras are found to
allow for a semi-direct sum structure, it can be more fruitful to use a
“natural” orthogonal basis that as a result induces more interactions from the
chain point of view, but can allow for direct evaluation.

While this approach in itself is novel, there have been several works already
where an orthonormal basis for the Krylov subspace is not obtained through the
Lanczos algorithm and as such the Liouvillian (Hamiltonian) is not
tridiagonal. These include [18, 37], where (similarly to this work) the
symmetry algebra is considered for the full Virasoro group and
$SL_{n}(\mathbb{R})$ respectively. For the former this leads to a block
diagonal structure of the Liouvillian and for the latter to a “trivially” non-
tridigonal structure along the same lines with $SL_{2}(\mathbb{R})$ which we
examine in more detail in the following sections. Importantly, in both cases,
this leads to analytically tractable results, although there does not appear
to be a complete understanding of the precise difference between the
implementation of this approach as compared to the Lanczos algorithm. Another
instance where an explicitly non-tridiagonal form of the Liouvillian is
required in order to make progress arises in open quantum systems (see e.g.
[23]). The reason is that the Liouvillian for these systems is not a Hermitian
operator anymore and as such the Lanczos algorithm is insufficient, so a more
general orthogonalization scheme has to be implemented instead. In particular
one can use the Arnoldi iteration which puts the Liouvillian in Hessenberg
form (triangular matrix with non-zero entries in the first subdiagonal). The
common element in all of these considerations is that while it is possible to
obtain an orthonormal basis for the Krylov subspace in different ways, the
differences between these bases and the associated complexities are unclear.
We comment further on this point from the perspective of our results in the
discussion section.

The remainder of this note is organized in the following way. In Section 2 we
review Krylov complexity and establish notation. In Section 3 we introduce the
Schrödinger algebra and present our results. Finally, in section 4 we present
a discussion on our obtained results.

##  2 Krylov complexity

In this section we introduce Krylov complexity and establish notation through
a series of examples.

###  2.1 Krylov basis

There are many works that include a pedagogical introduction to the notion of
the Krylov basis, to which we refer the reader for a detailed derivation [1,
38, 2, 39, 40]. The central concept is the expansion of a time evolved
operator
$\mathcal{O}(t)=e^{iHt}\mathcal{O}e^{-iHt}=e^{i\mathcal{L}t}\mathcal{O}$ in
the Krylov subspace $\mathcal{H}_{\mathcal{O}}$ defined as the linear span of
the action of the Liouvillian on the initial operator

$\mathcal{H}_{\mathcal{O}}=\text{span}\\{\mathcal{L}^{n}\mathcal{O}\\}_{n=0}^{+\infty}\leavevmode\nobreak\
.$ (1)

In other words one seeks to decompose an operator at some arbitrary time as

$|\mathcal{O}(t))=\sum_{n}\phi_{n}|K_{n})\leavevmode\nobreak\ .$ (2)

Here we adopt the curly bracket notation to denote a state in the operator
Hilbert space. Therefore, $|K_{n})$ denotes an orthonormal basis for the
Krylov subspace and $\phi_{n}$ the appropriate coefficients satisfying
$\sum_{n}|\phi_{n}|^{2}=1$. The most common way of obtaining such a basis is
through the Lanczos algorithm which applies the Gram-Schmidt orthogonalization
scheme to the elements $|\mathcal{O}_{n})=\mathcal{L}^{n}|\mathcal{O})$ [41].
This has the effect of tridiagonalizing the Liouvillian such that

$\mathcal{L}|\mathcal{O}_{n})=b_{n+1}|\mathcal{O}_{n+1})+b_{n}|\mathcal{O}_{n-1}).$
(3)

This tridiagonal structure manifests explicitly when the elements of the
Liouvillian are represented in matrix form.

$\mathcal{L}=\begin{pmatrix}0&b_{1}&0&\cdots&0\\\
b_{1}&0&b_{2}&\ddots&\vdots\\\ 0&b_{2}&0&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{n}\\\ 0&\cdots&0&b_{n}&0\end{pmatrix}\,.$ (4)

Note that the diagonal elements are zero due to the properties of the operator
inner product, which we assume (as is standard) to involve a trace. Having
obtained the basis one can then study different aspects of the probability
distribution provided by $\phi_{n}$ to probe the growth of the operator in the
Krylov subspace and assess its resulting complexity. In particular the average
of that distribution $\sum_{n}n|\phi_{n}|^{2}$ is dubbed Krylov complexity and
quantifies, albeit in a somewhat crude manner, the growth of the operator. A
natural question that arises is why should one consider the average as a
measure of complexity rather than any other moment of the distribution. There
are some formal arguments to be made in favour of that choice as is done for
example in [42], but here we will take a more heuristic approach that will
prove useful in our subsequent considerations. The so-called “Krylov chain”
picture was pointed out in the seminal work of [1] and it has since been
explored by several others [14, 13, 43]. This picture arises from the
realization that the dynamics of the Krylov subspace can be mapped to that of
a particle hopping on a semi-infinite, one-dimensional chain. This
identification is made by the discrete Schrödinger equation that relates the
Lanczos coefficients $b_{n}$ with the coefficients $\phi_{n}$ as

$-i\frac{d\phi_{n}}{dt}=b_{n+1}\phi_{n+1}+b_{n}\phi_{n-1}\leavevmode\nobreak\
.$ (5)

In this picture the $b_{n}$ play the role of the hopping coefficients between
two adjacent sites and $\phi_{n}$ the amplitude of finding the particle at
each site. Krylov complexity can be interpreted as the average position on the
chain at a particular time. This is indeed a very useful piece of information,
especially in cases where access to other aspects of the probability
distribution defined by the $\phi_{n}$ is limited. However, there is much more
refined information that one can extract from said probability distribution if
it can be obtained analytically. For instance one can study the Krylov
variance [18], entropy [39, 44, 45], logarithmic negativity and capacity of
entanglement [43].

Our main motivation is to study the Krylov subspace as it arises from the
symmetry of the quantum system of interest in the spirit of [2]. Assuming that
this symmetry is described by a Lie group and that the system is closed, then
the Liouvillian can be written as a linear combination of the algebra
generators. For three-dimensional Lie algebras that admit a representation in
terms of some generalized ladder operators one can write the Liouvillian in
the following form

$\mathcal{L}=\xi L_{+}-\overline{\xi}L_{-}\leavevmode\nobreak\ .$ (6)

The action of such a Liouvillian on a general Fock state $\ket{n}$ is then by
construction identical to how the Liouvillian acts on the Krylov basis and by
an appropriate choice of $\xi$ one can identify the two in a precise
mathematical manner. This enables us to identify the operator
$e^{i\mathcal{L}t}$ as a group element, thus leading to an interpretation of
the time evolved operator

$|\mathcal{O}(t))=e^{i\mathcal{L}t}|\mathcal{O}(0))=\sum_{n}\phi_{n}\ket{n}\leavevmode\nobreak\
,$ (7)

as a coherent state. The latter have been studied extensively [46] and are
very well known for a number of different symmetry groups. This is
particularly useful given that one can readily obtain the coefficients
$\phi_{n}$ from which the Krylov complexity can be extracted, but it
additionally endows these results with a certain degree of universality. More
specifically, this construction implies that any system with the same type of
symmetry will exhibit the same behaviour in terms of its Krylov complexity. As
mentioned in the introduction our aim here is to study a group with more
general structure for which the Liouvillian is not automatically tridiagonal.
Our vehicle in that endeavour is the Schrödinger group which in 1+1 dimensions
is the semi-direct product of the Heisenberg-Weyl and $SL_{2}(\mathbb{R})$
groups. For that reason, we review the methods and results for these two
groups in the following subsections.

###  2.2 Heisenberg-Weyl algebra

The Heisenberg-Weyl algebra ($HW$) is defined by (we borrow the notation
employed in [2])

$[a,a^{\dagger}]=1\,,\quad[\hat{n},a^{\dagger}]=a^{\dagger}\,,\quad[\hat{n},a^{\dagger}]=-a\,,$
(8)

with all other commutators vanishing, $\hat{n}=a^{\dagger}a$ and the usual
creation and annihilation operators $a^{\dagger}$ and $a$. Defining a vacuum
state $|0\rangle$ via $a|0\rangle=0$ we can consider the following orthonormal
basis for the corresponding Hilbert space:

$|n\rangle=\frac{1}{\sqrt{n!}}(a^{\dagger})^{n}|0\rangle\,,$ (9)

such that

$a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle\,,\quad
a|n\rangle=\sqrt{n}|n-1\rangle\,.$ (10)

To compute the associated Krylov complexity it was shown [2] that we can
simply make the following identifications

$\mathcal{L}=\alpha(a^{\dagger}+a)\,,\quad|\mathcal{O}_{n})=|n\rangle\,.$ (11)

This allows us the write the Heisenberg operator state in Krylov space as

$|\mathcal{O}(t))=e^{i\alpha
t(a^{\dagger}+a)}|0\rangle=\sum_{n=0}^{\infty}(it)^{n}\phi_{n}(t)|n\rangle\,,\quad\phi_{n}=e^{-\alpha^{2}t^{2}}\frac{\alpha^{n}t^{n}}{\sqrt{n!}}\,,\quad\sum_{n=0}^{\infty}|\phi_{n}|^{2}=1\,,$
(12)

where at the second equality sign we simply wrote the exponent in its series
representation and worked out that
$(a^{\dagger})^{n}|0\rangle=\frac{1}{\sqrt{n!}}|n\rangle$. This allows us to
compute Krylov complexity $K_{\mathcal{O}}$:

$K_{\mathcal{O}}=\sum^{\infty}_{n=0}n|\phi_{n}(t)|^{2}=\alpha^{2}t^{2}\,.$
(13)

While the $HW$ group appears elementary it actually arises in cases much more
sophisticated than that of the harmonic oscillator with which it is usually
associated. For example in [47] the authors discuss how the symmetry of SYK in
the triple scaling limit is described by $HW$ and how that leads to the
universal Krylov complexity result presented above.

###  2.3 $SL_{2}(\mathbb{R})$ algebra

We will now consider the $SL_{2}(\mathbb{R})$ (or the isomorphic $SU(1,1)$)
algebra and compute complexity in two different bases: the Heisenberg-Weyl
basis and the ‘natural’ $SL_{2}(\mathbb{R})$ basis. The algebra is defined as

$[L_{0},L_{\pm 1}]=\mp L_{\pm 1}\,,\quad[L_{0},L_{\pm 1}]=2L_{0}\,,$ (14)

where we define a vacuum $|h\rangle$ via $L_{1}|h\rangle=0$ and
$L_{0}|h\rangle=h|h\rangle$ where positive integer $h$ labels the different
states that are allowed. An orthonormal basis is given by

$|h,n\rangle=\sqrt{\frac{\Gamma(2h)}{n!\Gamma(2h+n)}}L^{n}_{-1}|h\rangle\,,$
(15)

such that

$L_{0}|h,n\rangle=(h+n)|h,n\rangle\,,\quad
L_{\pm}|h,n\rangle=\sqrt{\left(n+\frac{1\mp 1}{2}\right)\left(2h+n-\frac{1\pm
1}{2}\right)}|h,n\rangle\,.$ (16)

To compute the associated Krylov complexity $K_{\mathcal{O}}$ we can make the
following identifications [2]

$\mathcal{L}=\beta(L_{-1}+L_{1})\,,\quad|\mathcal{O}_{n})=|h,n\rangle\,.$ (17)

It then follows that

$\phi_{n}(t)=\sqrt{\frac{\Gamma(2h+n)}{n!\Gamma(2h)}}\frac{\tanh^{n}(\beta
t)}{\cosh^{2h}(\beta t)}\,,\quad K_{\mathcal{O}}=2h\sinh^{2}(\beta t)\,.$ (18)

What happens when we express the above in the $HW$ basis instead? After some
algebra we establish the mapping

$L_{0}=\frac{1}{4}(a^{\dagger}a+aa^{\dagger})\,,\quad
L_{+1}=\frac{1}{2}a^{2}\,,\quad L_{-1}=\frac{1}{2}(a^{\dagger})^{2}\,,$ (19)

with value of $h=1/4$, which we treat as a continuation from the usual integer
values. As a result the exponentiated Liouvillian takes the form of a so-
called squeeze operator which one encounters quite frequently in quantum
optics [48]. In that case it is known that [46]

$|\mathcal{O}(t))=\sum^{\infty}_{n=0}i^{n}\frac{\sqrt{(2n)!}}{2^{m}m!}\frac{\tanh^{n}(\beta
t)}{\sqrt{\cosh(\beta t)}}|2n\rangle\,,$ (20)

which leads to the same complexity as previously computed, but restricted to
$h=1/4$.

##  3 Two-dimensional Schrödinger algebra

###  3.1 Schrödinger symmetries

The Schrödinger group is the maximal symmetry group corresponding to the free
Schrödinger equation [49] and is isomorphic to the maximal symmetry group of
the Schrödinger equation with a harmonic potential [50]. In general, from a
group perspective the Schrödinger group can be viewed as the extension of the
Galilean group. The generators of the Galilean group consist of time ($H$) and
spatial translations ($P_{i}$), Galilean boost ($G_{i}$) and spatial
rotations. The Galilei algebra admits a central extension $M$ of the
commutator of the boost generator $G_{i}$ and the spatial translation
generator $P_{i}$. Adding this extension yields the Bargmann algebra. In order
to reach the Schrödinger algebra we take into account a dilatation operator
$D$ under which time scales twice as fast as space and a special conformal
symmetry generator $K$. We tailored the names of these generators to the case
of the free Schrödinger equation. In two dimensions the only non-vanishing
commutators of this algebra are given by (where we dropped the index $i$ as it
runs over the spatial indices)

$[P,G]=-M\,,$ (21) $[D,H]=-2H\,,\quad[D,K]=2K\,,\quad[H,K]=D\,,$ (22)
$[D,P]=-P\,,\quad[D,G]=G\,.$ (23)

In (21) we recognize a Heisenberg-Weyl sub-algebra and in (22) we recognize an
$SL_{2}(\mathbb{R})$ sub-algebra. The remaining brackets in (23) are between
elements of either sub-algebra. We conclude that the two-dimensional
Schrödinger algebra is a semi-direct sum $sl_{2}(\mathbb{R})\ltimes hw$.

Using the insight from Section 2 in which we showed that $sl_{2}(\mathbb{R})$
can be written in terms of $hw$ generators we cast all the algebra elements of
the two-dimensional Schrödinger in terms of ladder operators:

$P=\frac{1}{\sqrt{2}}(a^{\dagger}-a)\,,\quad
G=\frac{1}{\sqrt{2}}(a^{\dagger}+a)\,,\quad M=aa^{\dagger}-a^{\dagger}a\,,$
(24) $H=-\frac{1}{4}(a+a^{\dagger})^{2}\,,\quad
K=-\frac{1}{4}(a-a^{\dagger})^{2}\,,\quad
D=\frac{1}{2}(a^{2}+(a^{\dagger})^{2})\,.$ (25)

Therefore, the Liouvillian expressed as a general element of the Lie algebra
can be cast in the form

$\mathcal{L}=\alpha(a^{\dagger}+a)+\frac{\beta}{2}((a^{\dagger})^{2}+a^{2})\,,\quad\alpha,\beta\in\mathbb{R}\,,$
(26)

and accordingly the element of the Schrödinger group that results from its
exponentiation can be parametrized as

$e^{i\mathcal{L}t}=S(v,w)=e^{(va-\overline{v}a^{\dagger})}e^{(\frac{w}{2}a^{2}-\frac{\overline{w}}{2}(a^{\dagger})^{2})}\leavevmode\nobreak\
,$ (27)

where we have not taken into account bilinears of $a$ and $a^{\dagger}$ that
act diagonally on the Fock states in either of the two expressions involving
the Liouvillian, since they just produce a phase that can be included in the
normalization. We provide an explicit mapping between the parameters
$\alpha,\beta$ and $v,w$ in subsection 3.4.

The Schrödinger group, apart from describing the symmetries of a non-
relativistic CFT [51] and thus being relevant for a non-relativistic limit of
AdS/CFT [52, 34], finds applications in systems with experimental
realizations. In particular, it is used to describe specific processes in
molecular dynamics and two-photon processes [35, 53]. Namely, within the study
of coherent states, squeezing refers to saturating the uncertainty relation
and in quantum optics, to reach such a state, one needs to take into account
pairs of photons on top of single photon states. Two-photon processes and the
collision of molecules can be modelled with a Hamiltonian expanded in a power
series and truncated at quadratic order thus producing an effective
Hamiltonian of the form

$H=\hbar\omega(aa^{\dagger}+\frac{1}{2})+f_{2}(t)(a^{\dagger})^{2}+f_{2}^{*}(t)a^{2}+f_{1}(t)a^{\dagger}+f_{1}^{*}(t)a\leavevmode\nobreak\
.$ (28)

In the Liouvillian presented above the coefficients $\alpha,\beta$ are time
independent, but generally they can be promoted to time-dependent as in this
Hamiltonian to capture squeezing that is not necessarily linear in time.

###  3.2 Computation of coherent states

Acting with this general element on a quantum state of a system with
Schrödinger symmetry provides a family of generalized coherent states in line
with Perelomov [46]. Let us discuss the construction of those coherent states
following [54]. We first consider the Baker-Campbell-Hausdorff (BCH)
decomposition of the displacement operator combined with a complex phase
factor $\theta$

$\begin{split}S(v,w)&=\theta
e^{(va-\overline{v}a^{\dagger})}e^{(\frac{w}{2}a^{2}-\frac{\overline{w}}{2}(a^{\dagger})^{2})}\\\
&=\theta
e^{-\frac{|v|^{2}}{2}}e^{-\overline{v}a^{\dagger}}e^{va}e^{-\frac{1}{2}\frac{\overline{w}}{|w|}\tanh{(|w|)}(a^{\dagger})^{2}}e^{-\ln{(\cosh{(|w|)})}(a^{\dagger}a+\frac{1}{2})}e^{\frac{1}{2}\frac{w}{|w|}\tanh{(|w|)}a^{2}}\leavevmode\nobreak\
,\end{split}$ (29)

where we have used the standard results for the BCH decomposition of the two
exponential terms, see e.g. [35]. One might easily guess that expanding the
action of that operator on a general quantum state is impractical. Instead one
can obtain a recurrence formula for the coefficients
$\phi_{k}=\bra{k}S\ket{0}$. However, before that we need to calculate
$\phi_{0}$ which will serve as normalization.

$\begin{split}\phi_{0}=\bra{0}S\ket{0}&=\theta\frac{e^{-\frac{|v|^{2}}{2}}}{\cosh{(|w|)}^{\frac{1}{2}}}\bra{0}e^{va}e^{-\frac{1}{2}\frac{\overline{w}}{|w|}\tanh{|w|}(a^{\dagger})^{2}}\ket{0}\\\
&=\theta\frac{e^{-\frac{|v|^{2}}{2}}}{\cosh{(|w|})^{\frac{1}{2}}}e^{-\frac{1}{2}v^{2}\frac{\overline{w}}{|w|}\tanh{|w|}}\leavevmode\nobreak\
.\end{split}$ (30)

Subsequently one can compute $\phi_{k}$ by observing that

$\bra{k}Sa\ket{0}=\bra{k}SaS^{-1}S\ket{0}=0\leavevmode\nobreak\ .$ (31)

Inserting above the following Bogoliubov transformation (i.e., the result of
commuting $a$ with $S^{-1}$)

$SaS^{-1}=\cosh{|w|}a+\frac{\overline{w}}{|w|}\sinh{|w|}a^{\dagger}+\overline{v}\cosh{|w|}+v\frac{\overline{w}}{|w|}\sinh{|w|}\leavevmode\nobreak\
,$ (32)

we obtain the following relation

$\sqrt{k+1}\cosh{|w|}\phi_{k+1}+\sqrt{k}\frac{\overline{w}}{|w|}\sinh{|w|}\phi_{k-1}+(\overline{v}\cosh{|w|}+v\frac{\overline{w}}{|w|}\sinh{|w|})\phi_{k}=0\leavevmode\nobreak\
.$ (33)

This three-term recurrence relation can be solved in terms of Hermite
polynomials, thus producing an analytical expression for all $\phi_{k}$ [54]

$\phi_{k}=\frac{1}{\sqrt{k!}}\left(\frac{1}{2}\frac{\overline{w}}{|w|}\tanh{|w|}\right)^{\frac{k}{2}}H_{k}(s)\phi_{0}\leavevmode\nobreak\
,$ (34)

where

$s=-\frac{1}{\sqrt{2}}\left(v\left(\frac{\overline{w}}{|w|}\right)^{\frac{1}{2}}\sqrt{\tanh{|w|}}+\overline{v}\left(\frac{w}{|w|}\right)^{\frac{1}{2}}\frac{1}{\sqrt{\tanh{|w|}}}\right)\,.$
(35)

Here, the appearance of orthogonal polynomials is reminiscent of [55], where
the authors investigate the relationship of Krylov complexity and orthogonal
polynomials. While this hints at a possible connection, the orthogonal
polynomials here enter the expression for $\phi_{k}$ whereas in [55] they play
the role of the operator states themselves. It is unclear at this time whether
there is an overarching description of this problem in terms of orthogonal
polynomials or this is simply a coincidence. Moving past this digression, we
have achieved our initial goal of obtaining a distribution over the Fock basis
of the Perelomov coherent states for the two-dimensional Schrödinger group in
the form

$S\ket{0}=\sum_{k=0}^{\infty}\phi_{k}\ket{k}\leavevmode\nobreak\ ,$ (36)

with the coefficients $\phi_{k}$ specified above.

###  3.3 Krylov complexity in a natural basis

As we showed previously, the probability distribution over the Fock basis is
given by (34). First, let us show that indeed the condition
$\sum_{k}|\phi_{k}|^{2}=1$ is satisfied. For that we will need to use Mehler’s
formula

$\sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{n!}\left(\frac{z}{2}\right)^{n}=(1-z^{2})^{-\frac{1}{2}}e^{\frac{2xyz-(x^{2}+y^{2})z^{2}}{1-z^{2}}}\leavevmode\nobreak\
,$ (37)

which for $x=s$ and $y=\overline{s}$ simplifies to

$\sum_{n=0}^{\infty}\frac{H_{n}(s)H_{n}(\overline{s})}{n!}\left(\frac{z}{2}\right)^{n}=(1-z^{2})^{-\frac{1}{2}}e^{\frac{2|s|^{2}z-(s^{2}+\overline{s}^{2})z^{2}}{1-z^{2}}}\leavevmode\nobreak\
.$ (38)

The sum expression $|\phi_{n}|^{2}$ in terms of (34), using that
$\overline{H}_{n}(z)=H_{n}(\overline{z})$, simply becomes

$\sum_{k=0}^{\infty}\frac{H_{k}(s)H_{k}(\overline{s})}{k!}\left(\frac{\tanh{|w|}}{2}\right)^{k}|\phi_{0}|^{2}=\cosh|w|e^{\frac{2|s|^{2}\tanh|w|-(s^{2}+\overline{s}^{2})\tanh^{2}|w|}{1-\tanh^{2}|w|}}|\phi_{0}|^{2}=1\,,$
(39)

where we used the definition of $s$ from (35) to show the last step. In
computing Krylov complexity we consider rewriting the definition as follows

$\displaystyle K_{\mathcal{O}}=$ $\displaystyle\leavevmode\nobreak\
\sum_{k=0}^{\infty}k|\phi_{k}|^{2}$ (40) $\displaystyle=$
$\displaystyle\leavevmode\nobreak\
|\phi_{0}|^{2}z\partial_{z}\sum_{k=0}^{\infty}\left(\frac{z}{2}\right)^{k}H_{k}(s)\overline{H}_{k}(s)-|\phi_{0}|^{2}z\sum_{k=0}^{\infty}\left(\frac{z}{2}\right)^{k}\partial_{z}\left[H_{k}(s)\overline{H}_{k}(s)\right]$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
|\phi_{0}|^{2}z\partial_{z}|\phi_{0}|^{-2}-|\phi_{0}|^{2}y\partial_{y}|\phi_{0}|^{-2}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\ |v|^{2}+\sinh^{2}|w|\,,$

where $z=\tanh|w|$ and $y=\left(v/|v|\right)^{2}$ and in the third equality we
used the identity

$z\partial_{z}s=y\partial_{y}s\,.$ (41)

In fact, using induction one can easily arive at the conclusion

$K_{(n)}=\sum^{\infty}_{k=0}k^{n}|\phi_{k}|^{2}=\left.|\phi_{0}|^{2}\left(\frac{\sinh(2|w|)}{2}\partial_{|w|}\right)^{n}|\phi_{0}|^{-2}\right|_{s}\,,$
(42)

where $n$ is any integer and $|_{s}$ indicates that when taking the derivative
with respect to $|w|$, $s$ is kept fixed.

The variance $\sigma^{2}=\langle k^{2}\rangle-\langle k\rangle^{2}$ can be
computed using the identity in (42) and reads

$\sigma^{2}=|v|\cosh 2|w|+\sinh|w|\cosh|w|\left(\sinh
2|w|-\frac{\overline{v}^{2}w+v^{2}\overline{w}}{|w|}\right)\,.$ (43)

To get some more feeling for expression (40), it is instructive to reproduce
the limit in which either the Heisenberg-Weyl part or the $SL_{2}(\mathbb{R})$
part are turned off by hand. If we take $v=-\overline{v}=i\alpha t$ and
$w=\overline{w}=0$ we find

$K_{\mathcal{O}}=\alpha^{2}t^{2}$ (44)

where we assumed $\alpha$ to be real and we indeed reproduce the correct
Krylov complexity from (13). In the opposite case we consider
$v=\overline{v}=0$ and $w=-\overline{w}=i\beta t$, where $\beta$ is real, and
we find

$K_{\mathcal{O}}=\sinh^{2}\beta t$ (45)

which indeed reproduces (18) (with $h=1/4$) up to a factor of 2. This can be
attributed to the fact that the chain picture that we obtain for the
Schrödinger group has double the number of sites compared to
$SL_{2}(\mathbb{R})$. Setting $v=0$ has the effect of the probability
vanishing on odd sites, which however are still counted by the complexity. On
the contrary the way that the Krylov basis is built for $SL_{2}(\mathbb{R})$
(using the harmonic oscillator realization of the algebra presented
previously) is such that only the even sites are taken into account to begin
with and this subtle difference leads to this factor of two discrepancy. This
effect is captured graphically in figure 1.

Figure 1: In this figure we compare the probabilities for the Schrödinger
group (yellow dots) and $SL_{2}(\mathbb{R})$ (blue dots) as a function of
their index $k$. Notice that for odd $k$ the probabilities for the Schrödinger
group vanish, while for even $k$ they are pairwise equal to
$SL_{2}(\mathbb{R})$, but shifted.

###  3.4 Interpreting complexity

The displacement operator $S(v,w)$ introduced in (29) leads to Perelomov
coherent states for the two-dimensional Schödinger group. In order to
interpret the result for complexity we need to relate the displacement
operator $S(v,w)$ to the Liouvillian

$\mathcal{L}=\alpha(a^{\dagger}+a)+\frac{\beta}{2}((a^{\dagger})^{2}+a^{2})\,,\quad\alpha,\beta\in\mathbb{R}\,,$
(46)

such that complex parameters $v$ and $w$ inherit the appropriate time
dependence. To relate these quantities we will use the identity

$e^{i[\alpha(a^{\dagger}+a)+\frac{\beta}{2}((a^{\dagger})^{2}+a^{2})]t}=\theta
e^{(va-\overline{v}a^{\dagger})}e^{(\frac{w}{2}a^{2}-\frac{\overline{w}}{2}(a^{\dagger})^{2})}\,,$
(47) $v=\frac{\alpha}{\beta}(1-\cosh\beta t)+i\frac{\alpha}{\beta}\sinh\beta
t\,,\quad w=i\beta t\,,$ (48)

and we pick up a phase factor
$\theta=\exp\left[i\frac{\alpha^{2}}{\beta^{2}}\left(\sinh\beta t-\beta
t\right)\right]$. To obtain the above identity we circumvented using the BCH
formula by adopting an explicit matrix representation [35] that allows us to
perform ordinary matrix exponentiation:

$\eta\left(a^{\dagger}a+\frac{1}{2}\right)+\delta+R(a^{\dagger})^{2}+La^{2}+ra^{\dagger}+la\quad\mapsto\quad\begin{pmatrix}0&0&0&0\\\
r&\eta&2R&0\\\ -l&-2L&-\eta&0\\\ -2\delta&-l&-r&0\end{pmatrix}\,,$ (49)

where $\eta$, $\delta$, $R$, $L$, $r$, $l$ are complex numbers.

Now that we know $v(t)$ and $w(t)$ we can interpret complexity (40) as a
function $t$:

$K_{\mathcal{O}}=\alpha^{2}t^{2}+\sinh^{2}\beta
t+\alpha^{2}\left[\frac{4\cosh\beta t\sinh^{2}\frac{\beta
t}{2}}{\beta^{2}}-t^{2}\right]\,,$ (50)

where the first term reproduces the $HW$ complexity ($\beta=0$) and the second
term reproduces the $SL_{2}(\mathbb{R})$ complexity ($\alpha=0$), for a
comparison see Section 2. The third term in square brackets, which we dub the
interaction term, vanishes in either cases and we interpret it as arising from
the fact that elements of either sub-algebras do not commute. When expanding
the hyperbolic functions in the interaction term we get an infinite series in
even power that cancels the single $t^{2}$, showing that the interaction term
is always positive. This means that the Schrödinger complexity is more than
the sum of the separate complexities of $HW$ and $SL_{2}(\mathbb{R})$. While
for non-zero $\alpha,\beta$ the interaction term is always present, the
relative size of these parameters determines the character of the system. In
other words if $\frac{\alpha}{\beta}$ is small then the system behaves closer
to pure $SL_{2}(\mathbb{R})$ and conversely when $\frac{\alpha}{\beta}$ is
large then the behavior resembles pure HW, as seen in figure 2.

Figure 2: This figure contains the plots of complexity as a function of time
for the cases where $\alpha=0.01,\beta=1$ (red curve) and
$\alpha=1,\beta=0.01$ (black curve). We observe that the red curve clearly
exhibits the exponential behavior characteristic of $SL_{2}(\mathbb{R})$
whereas the black curve is closer to the quadratic behavior of HW.

Let us consider early and late time limits to explore the effects of the
interaction term. For early time we find

$K_{\mathcal{O}}=(\alpha^{2}+\beta^{2})t^{2}+\mathcal{O}(t^{3})\,,$ (51)

where the $\beta^{2}$ comes from the $\sinh^{2}$ and the interaction term does
not contribute. The effects of the interaction term becomes apparent at late
times

$K_{\mathcal{O}}=\left(\frac{1}{4}+\frac{1}{2}\frac{\alpha^{2}}{\beta^{2}}\right)e^{\beta
t}=e^{\beta(t-t_{s})}\,,$ (52)

which implies that the Lyapunov exponent remains unaltered, but the scrambling
time $t_{s}$, which can be a probe for operator growth, does receive a
correction:

$t_{s}=\frac{1}{\beta}\log\frac{4\beta^{2}}{\beta^{2}+2\alpha^{2}}\,.$ (53)

For $\alpha>0$, the scrambling time decreases as compared to the
$SL_{2}(\mathbb{R})$ case and for $\alpha>\beta\sqrt{3/2}$ the scrambling time
becomes negative, as can also happen for the pure $SL_{2}(\mathbb{R})$ case
for different representations, i.e., values of $h$ in (18). This is to be
expected as it signals that one needs more information than just the leading
order term for such cases. We furthermore point out that the first and second
derivative of the complexity with respect to time are positive for $t\geq 0$.

The variance in (43) combined with $v(t)$ and $w(t)$ and normalized with
$K_{\mathcal{O}}^{2}$ yields

$\displaystyle\sigma^{2}=$ $\displaystyle\leavevmode\nobreak\
\left\\{\alpha^{2}t^{2}+\frac{1}{2}\sinh^{2}2\beta
t+\frac{\alpha^{2}}{2\beta^{2}}\left[4\left(\cosh\beta t+\cosh 3\beta
t\right)\sinh^{2}\frac{\beta t}{2}-2\beta^{2}t^{2}\right.\right.$ (54)
$\displaystyle\leavevmode\nobreak\ \left.\left.+32\cosh\frac{\beta
t}{2}\cosh^{3/2}\beta t\sinh^{4}\frac{\beta
t}{2}\right]\right\\}/\left[\sinh^{2}\beta
t+4\frac{\alpha^{2}}{\beta^{2}}\cosh\beta t\sinh^{2}\frac{\beta
t}{2}\right]^{2}\,,$

where the first two terms correspond to the variance of purely $HW$ and
$SL_{2}(\mathbb{R})$ respectively. The term in the numerator within brackets
is positive for $t>0$. In order to appreciate the effect of the interaction
term it is useful to investigate the plots of the probabilities $\phi_{k}^{2}$
which appear in figure 3.

Figure 3: Probabilities for the Schrödinger group as functions of their index
k (made into a continuous variable for illustration purposes) for $\alpha=0$
(blue curve), $\alpha=0.25$ (yellow curve) and $\alpha=0.5$ (green curve). We
notice that the magnitude of the probability drops for larger values of
$\alpha$. Since the probability functions are normalized. This illustrates the
faster spreading that occurs due to the interaction term in (54).

It is also worth examining in more detail the form of $|\phi_{0}|^{2}$, which
is the autocorrelator (survival amplitude) of the system

$|(\mathcal{O}(0)|\mathcal{O}(t))|^{2}=|\phi_{0}(t)|^{2}=\frac{e^{-\frac{\alpha^{2}(e^{2\beta
t}-1)^{2}}{8\beta^{2}}+2\beta t}}{\cosh{(2\beta t)}}\leavevmode\nobreak\ ,$
(55)

where we have restored the time dependence using (47). We remind the reader
that the information contained in the autocorrelation function is equivalent
to the Lanczos coefficients, or in our case the hopping coefficients. We
observe that the leading contribution to the decay of the autocorrelation
function is of the form $e^{-\alpha^{2}e^{2\beta t}}$, that is doubly
exponential. This is unlike any of the semisimple groups that were studied so
far, although it corroborates the enhanced spreading of the wavefunction as
argued previously. We illustrate this result by comparing to the results for
pure $HW$ and $SL_{2}(\mathbb{R})$ in figure 4.

Figure 4: Comparison for the autocorrelator functions for $\alpha=1,\beta=0$
(yellow curve), $\alpha=0,\beta=1$ (green curve), $\alpha=1,\beta=1$ (blue
curve). We observe that the autocorrelator for the Schrödinger group decays
faster than both the $HW$ and $SL_{2}(\mathbb{R})$ groups.

##  4 Discussion

In this paper we proposed an approach to computing Krylov complexity for the
two-dimensional Schrödinger group in what we dub to be a natural orthonormal
basis, which does not involve the usual tridiagonal Liouvillian, but a
pentadiagonal one instead. We argue that regardless the same Krylov subspace
is probed. The naturalness of the basis arises from the fact that the two-
dimensional Schrödinger group is generated by a non-semisimple algebra that
can be written as the semi-direct sum of the Heisenberg-Weyl algebra and
$SL_{2}(\mathbb{R})$, which can both be naturally expressed in the Heisenberg-
Weyl basis. We advocate that this approach might provide insights to other
non-semisimple algebras for which the usual Krylov basis is technically non-
attainable.

We find that the Krylov complexity in its natural basis is greater than the
sum of the Krylov complexity of the separate sub-algebras and the same holds
for its variance. At late time we recover the same Lyapunov exponent as for
$SL_{2}(\mathbb{R})$, but we do find a naive scrambling time of smaller value.
This is an intriguing aspect of our results as it creates the following
picture. At late times the dynamics of the system are dominated by the
$SL_{2}(\mathbb{R})$ degrees of freedom, which is why we observe the
exponential growth of Krylov complexity with the same Lyapunov exponent.
However, the spreading of the wave function, or in other words the
distribution of the state over the Krylov subspace happens at a faster rate as
shown by our variance calculations and the smaller scrambling time. While a
rigorous interpretation of this phenomenon is outside of our reach for the
moment, this highlights the importance of probing multiple aspects of the
probability distribution rather than just the average. We believe this
behavior makes sense from an intuitive point of view: due to its semi-direct
sum structure, elements in different sub-algebras have non-trivial commutation
relations and as such there should be an increase of the total complexity
compared to the naive sum of the sub-algebras.

Our complexity grows convexly, which contrasts the numerical findings of [36],
who considered Krylov complexity in an approximate tridiagonal basis. They
furthermore explicitly find that the complexity of the two-dimensional
Schrödinger algebra is less than its separate sub-algebras. It would be
interesting to understand this discrepancy, especially since in the end both
methods probe the same Krylov subspace.

Doing computations in the natural basis, from the perspective of the harmonic
oscillators basis $a^{\dagger}$ and $a$, amounts to taking into account
generators $(a^{\dagger})^{2}$ and $a^{2}$ in the Liouvillian. One might
wonder: why stop at quadratic order? It turns out that higher orders, say,
$a^{\dagger 3}$ and $a^{3}$ do not close the algebra. Commuting these
generators yields fourth order generators in terms of the oscillator basis and
one is led to conclude that the algebra does not close for a finite number of
generators.

The autocorrelator computed using the Krylov basis for $SL_{2}(\mathbb{R})$
can be reproduced using a thermofield double setup in the context of
holography. The Schrödinger algebra is non-Lorentzian and its holographic
manifestation is less well understood [56, 34]. Perhaps the here presented
results, the autocorrelator that exhibits doubly-exponential scrambling
specifically, can provide a bench mark for further developing Schrödinger
holography.

Finally, it is important to ask what our results imply for the bigger picture
and how do they fit into our understanding of Krylov complexity and its
relation to other complexity measures. Here we have established that one can
obtain analytic results for systems with symmetry that go beyond the case of
semisimple Lie algebras. The key ingredient in this endeavour is the use of
generalized coherent states, which appear to play a prevalent role not only
when it comes to Krylov complexity, but also in approaches to complexity that
rely on geometry as is the case in [57, 58, 59, 60, 61] for example. This is
due to the natural organization of coherent states in metric spaces as
explained in detail in [46]. It would be interesting to think how our results
fit into that framework as the Schrödinger group provides an incremental step
in tackling more complicated problems like the Virasoro group which, being
centrally extended, also falls into the category of non-semisimple groups. Due
to its relevance in AdS/CFT this group has been studied extensively and there
are some results with regard to complexity that take advantage of the
associated geometry and are possibly implicitly related to certain classes of
coherent states [59]. While there have been recent advances in understanding
Krylov complexity from a holographic perspective [62, 37, 47] a rigorous
holographic prescription is not yet available. Thus, we believe that our work
is not only important for a non-relativistic limit of AdS/CFT, but also
provides the groundwork for more general considerations.

### Acknowledgements

It is a pleasure to acknowledge the scientific program “Reconstructing the
Gravitational Hologram with Quantum Information”, which took place at the INFN
Galileo Galilei Institute (GGI) for Theoretical Physics, National Centre for
Advanced Studies in the summer of 2022, in Florence, where this work was
initiated. We furthermore wish to thank Paweł Caputa for insightful
discussions and support and Jose Barbon, Jan Boruch, Sinong Liu and Javier
Magan for helpful comments. DP is supported by “Polish Returns 2019” grant of
the National Agency for Academic Exchange (NAWA) PPN/PPO/2019/1/00010/U/0001
and Sonata Bis 9 2019/34/E/ST2/00123 grant from the National Science Center,
NCN. WS is supported by the Icelandic Research Fund via the Grant of
Excellence titled “Quantum Fields and Quantum Geometry” and by the University
of Iceland Research Fund. WS furthermore thanks Paweł and Dimitris for their
unrivalled hospitality in Warsaw.

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