Remarks on the notion of quantum integrability

Abstract

We discuss the notion of integrability in quantum mechanics. Starting from a
review of some definitions commonly used in the literature, we propose a
different set of criteria, leading to a classification of models in terms of
different integrability classes. We end by highlighting some of the expected
physical properties associated to models fulfilling the proposed criteria.

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arXiv:1012.3587v1 [cond-mat.str-el] 16 Dec 2010
Remarks on the notion of quantum integrability
Jean-S´ ebastien Caux and Jorn Mossel
Institute for Theoretical Physics, Universiteit van Amsterdam,
Science Park 904, Postbus 94485,
1090 GL Amsterdam, The Netherlands
December 17, 2010
Abstract
We discuss the notion of integrability in quantum mechanics. Starting
from a review of some definitions commonly used in the literature, we
propose a different set of criteria, leading to a classification of models in
terms of different integrability classes. We end by highlighting some of the
expected physical properties associated to models fulfilling the proposed
criteria.
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1 Introduction
Classical mechanics is a subject with a unique level of maturity: it is one of
the most enjoyable to learn, and through the beauty and powerfulness of its
formalism is often considered as a prototypical example of ‘how things should
ideally be done’ in physics. One of the most powerful concepts in the study
of the dynamics of classical systems is the notion of integrability (in the sense
of Liouville, see e.g. [1]), namely that if a system with n degrees of freedom
(i.e. with 2n-dimensional phase space) possesses n independent first integrals
of motion in involution (i.e. Poisson-commuting), then the system is integrable
by quadratures. The meaning of ‘integrable’ here is thus transparent, namely
that the differential equations describing the time evolution can be explicitly
integrated using action-angle variables. The solutions of the equations of motion
thus display periodic motion on tori in phase space, and ergodicity is absent,
in contrast to non-integrable models which explore phase space densely in the
course of time. Besides providing explicit solutions to the time evolution, the
classical notion of integrability thus partitions classical models into separate
classes of integrable and non-integrable models with manifestly different physical
behaviour.
It thus comes as a surprising (and insufficiently known) fact that trans-
lating the notion of integrability to the quantum context has faced numerous
pitfalls, and remains to this day a subject of debate. This leads to some un-
fortunate widespread confusion, since integrability is mentioned very often in
contemporary discussions and publications concerning among other themes in-
and out-of-equilibrium dynamics, relaxation and thermalization of many-body
quantum systems under current theoretical and experimental investigation. If
quantum integrability is ill-defined, how can we thus invoke it at all?
Questioning the precise meaning of ‘quantum integrability’ has been done
on many occasions. Nearly two decades ago, in an eminently readable article,
Weigert [2] summarized some fundamental issues and discussed the shortcom-
ings of commonly used definitions. Delving further into the details is however
not usually done in research articles, but rather in private discussions or pro-
ceedings of lectures given by eminent researchers in the field, e.g. [3]. It is also
a subject of ongoing work (see for a recent example [4]). Since quantum integra-
bility was very often mentioned and discussed by many participants during the
StatPhys 24 conference, we found it appropriate to use the occasion offered by
these proceedings to share a few hopefully worthwhile thoughts, observations,
suggestions and conclusions on this important theme.
The paper is organized as follows. We first put the problem in context,
highlighting precisely what the problem is, and what kind of solution would
ideally be required. In Section 3, we review many definitions commonly found
in the literature, and collect our thoughts and comments on each of them.
After summarizing the conclusions reached, we propose a new categorization in
Section 4, and provide examples of where known models fit within our scheme in
Section 5. The physical consequences of our definition are discussed in Section
6, which is followed by our conclusion.
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2 Motivations
We can begin with the simplest amd most important questions: Don’t we have
a proper definition already? Why is this question interesting and important?
The first question will be answered in the negative in the following section. The
second question is best answered by reflecting on the classical case: since the
presence or absence of integrability in a classical system is associated to such
drastic differences in physical behaviour, the lack of a proper understanding
of the quantum equivalent inevitably means that we must be ‘missing out’ on
some important properties and features; the lack of a quantum equivalent to
the KAM theorem [5, 6, 7] (on the stability of quasi-periodic motion in the
presence of small perturbations) is possibly the most striking illustration of this
point, and makes it difficult to extract hard statements on the equilibration and
thermalization of many-body quantum systems.
The lack of correspondence between classical and quantum integrability,
which we will discuss further below, leads us to ask more basic questions about
the differences between classical and quantum systems [8]. A first point worth re-
membering is that quantum mechanics differs markedly from classical mechanics
in the way it counts degrees of freedom. In quantum mechanics, discretization
of levels means that we can comfortably work with finite-dimensional Hilbert
spaces: spins, bound atomic levels, etc. have eigenstates which we can label with
a discrete quantum number taking a finite set of values, and we typically say
that the number of degrees of freedom of a quantum system is the dimension-
ality of its Hilbert space. By contrast, in classical mechanics, we count degrees
of freedom by specifying how many pairs of conjugate phase space variables
are necessary to specify the configuration of a system. Each variable can take
on a continuum of values. In any quantum-classical correspondence, we would
thus associate the number of classical degrees of freedom to the multiplicity of
infinities of the dimension of the Hilbert space. There thus cannot be a clas-
sical equivalent to a quantum system with a finite-dimensional Hilbert space,
and this already means that classical integrability is insufficient as a basis for
defining quantum integrability in general.
When thinking about conserved charges, the notion of Liouville integrability
includes a specification of how many independent charges we need, namely a
number identical to the number of degrees of freedom n (in which case it is said
that the system possesses a complete set of charges, or is completely integrable).
If we can provide more than n charges, the system is said to be superintegrable;
if we can produce 2n charges, the system is maximally superintegrable (see
e.g. [9] and references therein). One semantic pitfall is associated to the word
‘complete’. Namely, one fundamental notion in quantum mechanics is that of
a complete set of commuting observables (CSCO), namely a set of commuting
quantum operators whose eigenvalues are sufficient to uniquely specify a state
in Hilbert space. In the context of integrability, the word ‘complete’ takes
on a different meaning: for a fully nondegenerate system, a single operator (the
Hamiltonian) already forms a CSCO. The cardinality of a CSCO is thus patently
not the number of conserved charges we should be looking for in the quantum
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case.
quantum operators in Hilbert space, meaning that we should be able to display
a number of commuting operators coinciding with the dimensionality of the
Hilbert space in order to call our set of charges ‘complete’. To avoid this pitfall,
we will thus rather talk about ‘maximal’ sets than ‘complete’ ones.
Before going further with our discussion of quantum integrability, it is worth-
while to follow the example of [2] and formulate a number of requirements for
a meaningful and useful definition of this concept. Most importantly,
We should of course be looking for a maximal abelian subalgebra of
1. it should be unambiguous;
2. it should partition the set of all possible quantum models into distinct
classes;
3. different classes of models should display distinguishable physical behaviour.
In addition to these, we could formulate a number of extra requirements, namely:
a) the contact with the classical limit should be natural; b) the contact with
integrable field theory should be natural; c) the different classes should be (algo-
rithmically) distinguishable, i.e. it should be easily feasible to determine which
class a model belongs to, etc. These are however less crucial criteria than the
ones we have selected.
3 Common definitions used in the literature
In this section, we initiate our discussion by summarizing a number of definitions
of quantum integrability encountered in the literature. We briefly comment each
one.
QI:N
independent commuting quantum operators Qα, α = 1,...,dim(H).
A system is quantum integrable QI:N if it possesses a maximal set of
Allowing for a bit of flexibility in the precise terms used, this is (at least
in spirit) overwhelmingly the most common definition of quantum integrability
encountered in the literature. It has the appeal of being directly related to the
classical notion of integrability, in the sense of being essentially a word-for-word
translation after replacing Poisson brackets with commutators.
Definition QI:N is given the label N for a simple reason: it is too naive.
Its fatal flaw is absolutely trivial: all quantum models associated to (limits
of) finite-dimensional Hilbert spaces fall under the label QI:N. By the spectral
theorem, all Hermitian Hamiltonians are readily diagonalizable; one thus obtains
dim(H) orthogonal state vectors |Ψα? from which one can build projectors Qα=
|Ψα??Ψα|, the set of which constitutes a maximal independent commuting set.
So is every quantum system we can think of to be called integrable? Well, a
court jester would amuse himself doing precisely this, but we have to reject this
pathway by invoking one of the requirements we had about a proper definition,
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namely that it should separate models into distinct classes. Definiton QI:N
blatantly fails to do so, and is thus to be rejected as being formally useless.
Loopholes of definition QI:N were also extensively discussed in [2]. Loop-
hole (A) there corresponds to the flaw mentioned above. A second loophole
mentioned is associated to the notions of ‘maximal’ and ‘independent’ quan-
tum operators: a theorem of von Neumann [10] states that it is possible to
encode any number of commuting Hermitian operators into a single Hermitian
operator Q (in other words, any operator Qαcan then be viewed as a function
Qα= fα(Q)), so the very basic notion of the number of independent operators
actually seems ill-defined. Going further, since we allegedly can’t even prop-
erly count the number of charges we have, we are then prevented from honestly
declaring that a set be maximal.
While we cannot repair the first fatal flaw of definition QI:N, let us address
the second point in more detail. We do not dispute the validity of von Neu-
mann’s theorem, however we do not agree that it is of relevance here. First
of all, thinking about the counting of conserved charges which can be defined
as how one would represent them in terms of matrices in the eigenbasis, there
is no doubt that the number required to call a set maximal coincides with the
dimensionality of Hilbert space dim(H), since this is the number of indepen-
dent diagonal entries. As far as counting is concerned, the notion of algebraic
independence (that is, the set of charges does not obey any nontrivial polyno-
mial equation) is sufficient to make it well-defined, and is already in use in the
literature (see for example [9] and references therein).
We thus have to look for something beyond the naive definition. Of course,
one of the main assumptions was that in practice, we could actually diagonalize
the Hamiltonian to obtain the charges as projectors. This suggests a more
pragmatic definition:
QI:ES
words if we can construct its full set of eigenstates explicitly.
A system is quantum integrable QI:ES if it is exactly solvable, in other
While this reminds us of the action-angle variables in the classical case, the
reader will probably agree that this washed-down definition does not take us
very far. This definition could be further categorized according to which method
is employed to obtain the eigenstates: Fourier transform for free theories, Bethe
Ansatz for specific models (although the completeness of the set of Bethe eigen-
states is not formally proven for all models), etc. We also reject this definition
on the grounds that it does not fulfill all our criteria: the third, in particular,
is hard to relate to.
QI:HO
monic oscillators.
A system is quantum integrable QI:HO if it can be mapped to har-
This is not really practical: such a mapping is hard to construct explicitly,
even for models we know how to solve exactly. Anyway, the existence of such a
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