A Geometric Interpretation of the Second-Order Structure Function
Arising in Turbulence
Vladimir N. Grebenev · Martin Oberlack
Received: 25 February 2008 / Accepted: 22 October 2008 / Published
online: 20 November 2008 © Springer Science + Business Media B.V.
2008
Abstract We primarily deal with homogeneous isotropic turbulence
and use a closure model for the von Kármán-Howarth equation to
study several geo- metric properties of turbulent fluid dynamics.
We focus our attention on the application of Riemannian geometry
methods in turbulence. Some advantage of this approach consists in
exploring the specific form of a closure model for the von
Kármán-Howarth equation that enables to equip a model manifold (a
cylindrical domain in the correlation space) by a family of inner
metrics (length scales of turbulent motion) which depends on time.
We show that for large Reynolds numbers (in the limit of large
Reynolds numbers) the radius of this manifold can be evaluated in
terms of the second-order structure function and the correlation
distance. This model manifold presents a shrinking cylindrical
domain as time evolves. This result is derived by using a
selfsimilar solution of the closure model for the von
Kármán-Howarth equation under consideration. We demonstrate that in
the new variables the selfsimilar solution obtained coincides with
the element of Beltrami surface (or pseudo-sphere): a canonical
surface of the constant sectional curvature equals −1.
V. N. Grebenev (B) Institute of Computational Technologies, Russian
Academy of Science, Lavrentjev ave. 6, Novosibirsk 630090, Russia
e-mail:
[email protected]
M. Oberlack Fluid Dynamics, Technische Universität Darmstadt,
Hochschulstrasse 1, Darmstadt 64289, Germany e-mail:
[email protected]
2 V.N. Grebenev, M. Oberlack
Keywords Beltrami surface · Closure model for the von
Kármán-Howarth equation · Homogeneous isotropic turbulence ·
Riemannian metric · Two-point correlation tensor · Length scales of
turbulent motion
Mathematics Subject Classifications (2000) 76F05 · 76F55 · 53B21 ·
53B50 · 58J70
1 Introduction
Turbulent fluid dynamics is characterized by ranking turbulent
motions in size from scales ∼ l of the flow under consideration to
much smaller scales which become progressively smaller as the
Reynolds number increases. One of a fundamental problem of
turbulent fluid dynamics consist of studying the shape dynamics of
a fluid volume. The first concept in Richardson point of view is
that the turbulence can be considered to compose eddies (a
turbulent motion localized within a region of size l) of different
sizes. Richardson’s notion is that the eddies are evolved in time,
transferring their energy to smaller scale motions. These smaller
eddies undergo a similar cascade process, and transfer their energy
to yet smaller eddies in the inertial range and so on—continuous
until the Reynolds number is sufficiently small that molec- ular
viscosity is effective in dissipating the kinetic energy. The
characteristic features of turbulence—its distribution of eddy
sizes, shapes, speeds, vorticity, circulation, and viscous
dissipation—may all be captured within the statistical approach to
fully developed turbulence, and several questions can be posed.
What are the sizes of the eddies which are generated in Richardson
scenario? As time increases, how the shape of eddies is deformed?
While there are many efforts in this direction, the aim of this
paper is to present an approach that is based on the use of methods
of Riemannian geometry for studying the shape dynamics of eddies,
in particular, on the interaction between the deformation of
geometric quantities (shape form, curvature and other) of a
manifold (a singled out fluid volume) equipped with a family of
Riemannian metrics (length scales of turbulent motion) and the
deformation of these Riemannian metrics in time t. Our approach is
conceptually similar to the Ricci flow ideas [1]. The Ricci flow is
an evolution differential equation on the space of Riemannian
metrics, the behavior of smooth Riemannian metrics which evolves
under the flow may serve as a model to tell us something about the
geometry of an underlying manifold. The advantage of this approach
is that we can control the deformation of geometric quantities of
the manifold under consideration in time, and often a Ricci flow
deforms an initial metric to a canonical metric and a key point is
to control the so-called injectivity radius of the metrics.
A well-known example of the above is a Ricci flow that is starting
from a round sphere SN with an initial metric gmn(x, 0) = g(0) such
that Rmn = λgmn(x, 0), λ ∈ R where Rmn is the Ricci tensor. This
metric is known as
Interpretation of the Second-Order Structure Function 3
∂
where the evolving metrics are given by the formula
gmn(x, t) = ρ2(t)gmn(0) ≡ (1 − 2λt)gmn(0), λ = N − 1,
and the sphere shrinks homothetically to a point as t → T = 1/2(N −
1). Another example of this type would be if g0 is a hyperbolic
metric or an
Einstein metric of negative scalar curvature. Then the manifold
will expand homothetically for all times. Indeed if Rmn = −λgmn(x,
0) then ρ(t) satisfies
dρ dt
with the solution
ρ2(t) = 1 + 2λt.
Hence the evolving metrics gmn(x, t) = ρ2(t)gmn(x, 0) exists and
expands ho- mothetically for all times.
These illustrative examples give us a feeling how the Ricci flow
can deform a manifold equipped with an initial Riemannian metric
g(0). In the general case, the Ricci flow behaves more
wildly.
In this paper, we deal with homogeneous isotropic turbulence and
emphasis is placed on the use of the specific form of a closure
model [2, 3] for the von Kármán-Howarth equation [4] to introduce
into consideration a family of Riemannian metrics. Inspired by the
Ricci flow idea, we study the behavior of Riemannian metrics
constructed and as a consequence, the deformation of some metric
quantities of an underlying Riemannian manifold can be deter-
mined. In order to equip a model manifold (a singled out fluid
volume within turbulent flow) by a family of Riemannian metrics
(length scales of turbulent motion), we rewrite this model in the
form of an evolution equation and show that the right-hand side of
this evolution equation coincides with the so-called radial part of
a Laplace-Beltrami type operator. This enables to construct
Riemannian metrics (length scales of turbulent motion) compatible
with the specific form of this closure model. We recall that the
Laplace-Beltrami oper- ator contains a metric tensor of a
Riemannian manifold where this operator is defined on. This is a
crucial peculiarity of this operator that makes its possible to
investigate geometric characteristics of an underlying Riemannian
manifold. Using the selfsimilar solution obtained of the closure
model for the von Kármán-Howarth equation under consideration, we
calculate explicitly the deformation of this family of metrics in
time. As a remarkable fact, we note that the above-mentioned
selfsimilar solution coincides in the new variables with the
element of Beltrami surface (or pseudo-sphere). This is a canonical
surface of the constant (sectional) curvature equals −1 [5].
Examining length scales of turbulence motion, we can see that some
scales analyzed are based on the use of Euclidian metric to measure
a distance.
4 V.N. Grebenev, M. Oberlack
However, it is not so clear why we use Euclidian metric in
turbulence to define a length scale of turbulent motion without
taking into account the geometry of turbulent pattern. The
well-known example, where we need a correction of (linear) length
scale, is the use of Prandtl’s mixing-length scale lm [6] in the
problem of decaying fluid oscillations near a wall. In this
problem, a modifica- tion of Prandtl’s mixing-length scale is taken
in the following (nonlinear) form: lm = κr(1 − exp(−r/A)) [6]. The
length scale lm plays the role of a measure of the transversal
displacement of fluid particles under turbulent fluctuations.
Although the above example comes from the theory of wall turbulent
flows, nevertheless this fact reflects understanding to make a
correction of some (linear) length scales.
We note that even in the case of homogeneous isotropic turbulence
there is a relatively small number of publications devoted to
numerical modeling isotropic homogeneous turbulence [7] and there
are very few results devoted to mathematical analysis of the von
Kármán-Howarth equation for the isotropic two-point correlation
function. We only mention here the paper [8] wherein this unclosed
equation was studied in the framework of the group classification
problem of differential equations [9]. We do not discuss the
details of the Kolmogorov theory (which tell us that the
statistical properties of small scales depend only on the mean rate
of energy dissipation ε and the correlation distance r) but remark,
however, that still are many discussions on whether small scale
fluctuations are isotropic or not and that the Richardson scenario
may not be valid. Consequently, the velocity statistics in the
inertial sub-range may have nonuniversal features. The notion of
intermittency is attributed to the violation of local homogeneity
of turbulence. This phenomenon leads to the anomalous scaling and
reflects a symmetry breaking in the case of ν → 0. From a physical
point of view as the viscosity tends to zero turbulence become
highly intermittent, and vorticity is concentrated on sets of a
small measure and scenario of turbulent motion is complicated
significantly.
Here we do not review the papers based on the methods of Lagrangian
formalism (i.e. the description of turbulent motion of fluids
particles) for the stochastic description of turbulence since our
approach lies in another field of mathematical investigations of
this phenomenon. The difference between the application of
Lagrangian formalism method for turbulence (exhaustive reviews on
this topic can be found in [10, 11]) and the approach presented
here is the same as using Lagrangian and Euler variables in
hydrodynamics. We do not look at how a marked fluid particle or an
ensemble of marked fluid particles (the separation distance between
marked particles) is traveled in turbulent flow but we prefer to
observe entirely the deformation of length scales of turbulent
motion localized within a singled out fluid volume of this flow in
time.
The paper is organized as follows. Section 2 is devoted to a
closure model for the von Kármán-Howarth equation. Observe that
this model holds (see [2]) for a wide range of well accepted
turbulence theories for homogeneous isotropic turbulence as there
is Kolmogorov first and second similarity hypothesis. In Section 3,
we show how to equip a model manifold (a singled out fluid
Interpretation of the Second-Order Structure Function 5
volume) by a family of Riemannian metrics (length scales of
turbulent motion) exploring the specific form of the
above-mentioned closure model for the von Kármán-Howarth equation
limited to sufficiently large Reynolds numbers. Moreover, we give a
geometric interpretation of the second-order structure function
DLL. At the end of Section 3 we present the results [12] of group
analysis of the von Kármán-Howarth equation (in its inviscid form)
and indicate two scaling symmetries admitted by this equation that
enable us to find a whole class of selfsimilar solutions. We show
that one implicit self- similar solution, which corresponds to
Loitsyansky decay low [13], coincides (in the new variables) with
the element of Beltrami surface (or pseudo- sphere). Negativity of
the curvature of Beltarmi surface means a stochastic behavior of
geodesic curves located on this surface [14]. As was noted by
Arnold [14], this property leads to the so-called exponential
instability of the geodesic flow. Here we do not develop this
topic. Appendix includes a formal derivation of the closure
relationship [2] (the algebraic approximation for the triple
correlation function) for the von Kármán-Howarth equation limited
to sufficiently large Reynolds numbers in the framework of the
method of differential constraints [15]. In concluding remarks, we
provide the results obtained by physical comments to some
extent.
2 Closed Model for the Von Kármán-Howarth Equation
We begin with basic notions of homogeneous isotropic
turbulence.
2.1 Two-Point Velocity Correlation Tensor
Traditional Eulerian turbulence models employ the Reynolds
decomposition to separate the fluid velocity u at a point x into
its mean and fluctuating components as u = u + u′ where the symbol
(·) denotes the Eulerian mean sometimes also called Reynolds
averaging.
In particular, the concept of two- and multi-point correlation
functions was born out of the necessity to obtain length-scale
information on turbulent flows. At the same time the resulting
correlation equations have considerably less unknown terms at the
expense of additional dimensions in the equations. In each of the
correlation equations of tensor order n an additional tensor of the
order n + 1 appears as unknown term, see for details [16]. The
first of the infinite sequence of correlation functions is the
two-point correlation tensor defined as
Bij(x, x′; tc) = (u′ i(x; tc) − u′
i(x′; tc))(u′ j(x; tc) − u′
j(x′; tc)), (2.1)
where u′(x; tc) and u′(x′; tc) are fluctuating velocities at the
points (x; tc) and (x′; tc) for each fixed tc ∈ R+. Therefore,
Bij(x, x′; t) defines a tensor field of the independent variables
x, x′ and t on a domain D of the Euclidian space R+ × R
6.
6 V.N. Grebenev, M. Oberlack
The assumption of isotropy and homogeneity of a turbulent flow
(invariance with respect to rotation, reflection and translation)
implies that this tensor may be written in the form [13]
Bij(r, tc) = u′ i(x; tc)u′
j(x + r; tc), (2.2)
which acts in the so-called correlation space K3 ≡ {r = (r1, r2,
r3)}, K3 R
3
for each tc, where r = x − x′. Moreover, for isotropic turbulence
Bij(r, tc) is a symmetric tensor which depends only on the length
|r| of the vector r = r(x, x′, tc), (x, x′) ∈ R
6, and the correlations Bij can be expressed by using only the
longitudinal correlational function BLL(|r|, tc) and the
transversal correlation function BNN(|r|, tc).
2.2 Closure of the Von Kárman-Howarth Equation
The correlation functions directly connect the concept of length
scales with the result of an actual flow measurement. However, the
two-point correlation functions yield no information on the energy,
that is contained in a given interval of separation r. The
third-order correlations function BLL,L pro- vides information
about the energy fluxes between scales. The von Kármán- Howarth
equation relates the time derivative of the component BLL of the
two-point correlation tensor to the divergences of the third-order
correlation function BLL,L and has the following form
∂ BLL
∂ BLL
∂r
) , (2.3)
where ν is the kinematic viscosity coefficient, r = |r|. This
equation directly follows from the Navier-Stokes equations
[13].
Originally, the invariance theory of isotropic turbulence was
introduced by von Kármán and Howarth [4] and refined by Robertson
[17], who reviewed this equation in the light of classical tensor
invariant theory. Arad, L’vov and Procaccia [18] extended these
fundamental results by considering projections of the fluid
velocity correlation dynamics onto irreducible representation of
the SO(N) symmetry group.
Equation 2.3 is not closed since it contains two unknowns BLL and
BLL,L
which cannot be defined from (2.3) alone without the use of
additional hypotheses. The simplest assumption is the
Kármán–Howarth hypothesis on the similarity of the correlation
functions BLL and BLL,L which is
BLL(r, t) = u′2(t) f (η), BLL,L(r, t) = (u′2(t))3/2h(η), η =
r/L(t), (2.4)
where u′2(t) is the velocity scale for the turbulent kinetic
energy, (u′2(t))3/2 is the scale for the turbulent transfer and
L(t) is a single global length scale of the turbulence.
Substituting these hypothesized expressions into Eq. 2.3, it
is
Interpretation of the Second-Order Structure Function 7
straightforward to demonstrate that this equation admits a complete
similarity solution of type (2.4) only when the Reynolds number Re
= u′2(t)1/2 L(t)/ν is a finite constant. In fact, this directly
relates u′2(t) to L(t). It is known that this condition is normally
not satisfied in experimental measurements of decaying isotropic
turbulence at finite Reynolds numbers.
Batchelor and Townsend [19] carried out a similarity analysis of
this problem in Fourier space and showed that a similarity solution
under this constraint could be found during the final period of
decay when the nonlinear terms become negligible. Millionshchikov
in [20] outlined a more general hypotheses which produces
parametric models of isotropic turbulence based on a closure
procedure for von the Kármán–Howarth equation. The essence of these
hypotheses is that BLL,L is given by the following relation of
gradient-type
BLL,L = 2K ∂ BLL
∂r , (2.5)
where K has the dimension of the turbulent kinematic viscosity
which is characterized by a single length and velocity scale.
Millionshchikov’s hypothe- ses [20] assumes that
K = κ1u′21/2 r, u′2 = BLL(0, t), (2.6)
where κ1 denotes an empirical constant. An initial-boundary value
problem for the Millionshtchikov closure model has been studied in
[21] wherein the theory of contractive semigroups was applied to
find a solution to the problem by the use of a Chorin-type
formula.
A way of closing the von Kármán–Howarth equation was suggested by
Oberlack in [2] which connects the two-point correlation functions
of the third- order BLL,L and the second order BLL by using the
gradient type hypothesis, that according to [2, 3] takes the
form
K = κ2rD1/2 LL, DLL = 2[u′2 − κ0 BLL(r, t)], κ0 = 1, κ2 =
√ 2
5C3/2 , (2.7)
where C is the Kolmogorov constant. The Millionshchikov hypotheses
is a consequence of the above formula in the case of κ0 = 0.
Comparison with experimental data was done calculating the triple
corre- lation h (the normalized triple-correlation function) out of
measured values of the normalized double correlation function f
using the model (2.5), (2.7). The normalized double correlation
function f was recovered simultaneously with the triple correlation
h in Stewart/Taunsend experiments [22]. Good agreement between
measured and computed values of h was achieved within the range of
the reliable data [2].
In [23], isotropic homogeneous turbulence dynamics was described by
a closure system of partial differential equations for the
two-point double- and
8 V.N. Grebenev, M. Oberlack
triple correlation functions coming from using the
finite-dimensional probabil- ity density equation. The following
system of equations was written:
∂ BLL
τ BLL,L, (2.9)
where the first equation coincides with the von Kármán-Howarth
equation, τ is the quantity which characterizes the correlation
time. Applying the so- called local equilibrium approximation to
the second equation, the closure relationship (2.5), (2.7) can be
obtained but as it was noted by Chorin [24], such approach is based
only on a physical hypothesis. In the Appendix to this paper, we
give a formal derivation of this formula based on studying the
Riemannian invariants of characteristics of system (2.8),(2.9).
This enables to find an invariant manifold admitted by (2.8),(2.9)
and to construct a reduced system. Conceptually, this procedure is
a similar to the approach suggested in [25]. The principle
difference is that we apply the method of differential constraints
[15] worked out by Cartan and Yanenko to study overdetermined
systems.
Finally, we note that it was in fact Hasselman [26] who was the
first to hypothesize a connection between the correlation functions
of the second- and third-order. His model for isotropic turbulence
contains one empirical constant and a rather complicated expression
for the turbulent viscosity coefficient.
3 A Model Manifold Defined by Closure of the Von Kármán–Howarth
Equation
First we review certain definitions and statements from Riemannian
geometry. Then we construct the so-called model manifold by
exploring the closure model (2.5), (2.7) for the von Kármán-Howarth
equation and give a geometric interpretation of the second-order
structure function DLL. To study explicitly the deformation of a
family of Riemannian metrics constructed in time, we use a
selfsimilar solution of the closure model for the von
Kármán–Howarth equation.
3.1 Laplace-Beltrami Operator
We recall the definition of some operators on a Riemannian manifold
U . Consider a vector field F = Fn∂/∂xn on U . The operator div is
determined by the formula
div F = 1√ g
Interpretation of the Second-Order Structure Function 9
where g = det gmn, and the mth component of the operator ∇ is
defined according to the formula
(∇ f )m = N∑
,
here gnm are elements of the matrix gnm−1. Further we denote
= div ∇ the Laplace-Beltrami operator. The Laplace-Beltrami
operator with a positive smooth weighted function σ(x) is defined
in a similar way using the following formula
div F = 1
√ gFn).
Here σ(x) presents the density of a Borel measure μ on U . If μ is
the Riemannian volume, then σ(x) ≡ 1.
Let Z be a Riemannian manifold which is isometric to
Z X × Y,
where X is an arbitrary manifold of dim X = N1 and Y is a compact
N2- dimensional manifold. Then a metric dz2 on Z is determined
by
dz2 = dx2 + γ 2(x)dy2, (3.10)
where γ (x) is a positive smooth function and dx2, dy2 are metrics
on X, Y correspondingly.
We assume that the density σ(z) of a Borel measure μ on Z can be
written as σ(z) = τ(x)η(y). Then the Laplace-Beltrami operator
given on Z takes the form
Z = A + γ −2 B, (3.11)
where A is the Laplace-Beltrami operator on X with the weighted
function γ N2τ and B denotes the Laplace-Beltrami operator defined
on Y with the weighted function η [27].
As an elementary example which illustrates the above construction
we consider the Laplace operator
= ∂2
∂z2
written in the spherical coordinates r, , ψ (x = r sin cos ψ , y =
r sin sin ψ , z = r cos )
= 1
r2
where
∂2
∂2
denotes the inner Laplacian on the unite sphere S2. Then a
Riemannian metric dz2 of Z = R+ × S2 is defined by the
formula
dz2 1 = dr2 + r2dθ2, dθ2 = d2 + sin2 dψ2, (3.12)
where dθ2 is the standard inner metric of S2 and dz2 1 is
equivalent to the usual
Euclidian metric. If we substitute a function g2(r), g(0) = 0, g(r)
0 instead of r2 into the second term of (3.12), then the
corresponding Laplace-Beltrami operator given on Z which is
equipped by the metric
dz2 2 = dr2 + g2(r)dθ2
takes the form
g2(r) ∂
∂r + 1
g2(r) 2.
Here Z is a cylindrical domain of the radius γ = g(r) and dz2 2
determines
another inner metric on Z . Therefore the definition of the
Laplace-Beltrami operator on the cross product of Riemannian
manifolds and the above exam- ples show us that the so-called
radial part A of the operator Z completely defines the form of a
Riemannian metric dz given on Z X × Y.
3.2 Metric Properties of a Model Manifold Determined by the Model
Limited to Sufficiently Large Reynolds Number
Let us consider a cylindrical domain Z = R+ × S2 in the correlation
space K3( R
3). In order to equip this cylindrical domain by an inner metric,
we explore the specific form of the right-hand side of the closure
model for the von Kármán–Howarth equation.
So, assuming the Reynolds number to be large, the first order O(1)
of Eq. 2.3 with the closure relationships (2.5), (2.7) reduces to
its inviscid form
∂ BLL
∂r BLL, r = |r|, r ∈ K3. (3.13)
Let q = 2r1/2, BLL(q, t) ≡ BLL(r, t) and DLL = 2[u′2 − BLL]. Then
Eq. 3.13 can be rewritten in the form
∂ BLL
∂q BLL. (3.14)
Further, let Z = R+ × S2 be a manifold with the metric
dz2 = dq2 + γ 2(q, tc)dθ2 (3.15)
Interpretation of the Second-Order Structure Function 11
where γ = qβ Dα LL, α = 1/4 and β = 9/2. This manifold represents a
cylindri-
cal domain such that γ = qβ Dα LL is the radius of the hypersurface
{q} × S2
∂
∂t − κ2 D1/2
LL Z (3.16)
∂
) BLL(q, t) = 0, q = q(r) (3.17)
due to Eq. 3.14. The direct calculations show that Eqs. 3.17 and
3.14 coincide with each other. Therefore the domain of definition
of operator (3.16) evolves in time and the radius of the
hypersurface {q} × S2 is determined by the formula
γ = qβ Dα LL, DLL = 2[u′2 − BLL]. (3.18)
It means that by solutions of Eq. 3.13 we can control a deformation
of the metric (3.15).
Therefore if we single out a fluid volume (of spherical form) in
(infinite) homogeneous isotropic flow embedded into the correlation
space K3 (i.e. we introduce the correlation variables instead of
physical ones), then a length scale of turbulent motion localized
within this volume can be defined according to the formula (3.15)
(written in the spherical coordinates r, , ψ) where γ (the
injectivity radius of the metric dz2) is determined by (3.18). We
note that this spherical domain (with the punctured point r = 0) is
isometric to Z with the same metric. The length scale of turbulent
motion constructed by this way is a family of scales parametrized
by the time t. The formula (3.18) tell us how the length scale of
possible transverse displacements of fluid particles depends on the
second-order structure function DLL and the correlation distance r.
This kind of argument may be also used to describe the shape
dynamics of this fluid volume in terms of the deformation of length
scales of turbulent motion in the transverse direction. We need
only to control the deformation of a measure ( length scale) of
transversal displacement of fluid particles. The function γ defines
a measure (length scale) of these transverse displacements. The
Ricci flow which shrinks homothetically a round sphere to a point
serves as an illustrative example of similar phenomenon.
The injectivity radius γ of the metric dz2 determines the geometric
structure on Z . In particular, a cylindrical domain Z is isometric
to the hyperbolic space H
3 (or a domain of this space) when γ = sinh q. It means that the
metric constructed is nonequivalent to the usual (Euclidian) metric
in general.
In order to study explicitly the behavior of the function γ (q, t)
which deter- mines the radius of the hypersurface {q} × S2, we use
the inviscid form of the von Kármán-Howarth equation which admits
the two-parameter Lie scaling subgroup and one-parameter Lie
subgroup of translation transformation in
12 V.N. Grebenev, M. Oberlack
time [12]. Therein this factum was applied to introduce a
selfsimilar ansatz and to find a whole class of selfsimilar
solutions. Other symmetries of fundamental fluid mechanics such as
rotation invariance, translation invariance in time, Galilean
invariance are implicitly met due to the a priori constraint of
isotropic turbulence.
Let us write the inviscid form of the von Kármán-Howarth equation
in the following normalized form
∂u′2(t) f (r, t) ∂t
= 1
r4
3/2 h(r, t), (3.19)
where f and h are respectively the normalized two-point double and
triple velocity correlation. The unclosed Eq. 3.19 admits the
following two scaling groups
Ga1 : t∗ = t, r∗ = ea1r, u′2∗ = e2a1 u′2, f ∗ = f, h∗ = h,
Ga2 : t∗ = ea2 t, r∗ = r, u′2∗ = e−2a2 u′2, f ∗ = f, h∗ = h,
or in the infinitesimal form
Xa1 = r ∂
∂u′2 .
The operators Xa1 and Xa2 generate the two-parametric Lie scaling
group
Ga1,a2 : t∗ = ea2 t, r∗ = ea1r, u′2∗ = e2(a1−a2)u′2, f ∗ = f, h∗ =
h.
It is easy to check that Eq. 3.13 is invariant under the
two-parametric group Ga1,a2 .
We note that
t−3(σ+1)/(σ+3) .
is a differential invariant of Ga1,a2 and the closure relationships
sug- gested (2.5), (2.7) are in agreement with group properties of
the inviscid form of the von Kármán-Howarth equation. Other
invariants of Ga1,a2 are
ξ = r t2/(σ+3)
, h= u′23/2 h
a1 .
where ai, i = 1, 2 are given above. Here r is scaled by the
integral length scale lt ∝ t2/(σ+3). The invariants above enable us
to reduce the number of variables in Eq. 3.13 and as a result, we
have the following ordinary differential equation
2κ2
dξ
] + δξ
Interpretation of the Second-Order Structure Function 13
where γ = 2(σ + 1)/(σ + 3), δ = 2/(σ + 3) and σ is undetermined at
this point. Eq. 3.20 can be integrated by Loitsyansky
invariant
= u′2 ∫ ∞
0 r4 f (r, t)dr, ≡ const. (3.21)
To determine the value of the parameter σ , we rewrite (3.21) in
the form (using the invariants above)
= u′2 ∫ ∞
∫ ∞
0 ξ 4 f (ξ)dξ
which determines σ = 4 and hence γ = 10/7, δ = 2/7. To find a
closed form solution of (3.20), we first multiply (3.20) on ξ 4 and
then integrate the equation obtained by parts using the computed
values α and β. As a result, we obtain the formula (see,
[12])
ξ = 7κ2
) (3.22)
which defines a solution of Eq. 3.20 in implicit form. It follows
from the formula (3.22) that
f (ξ) ≈ e−2ξ/3, ξ 1, f (ξ) ≈ 1 − ξ 2/3, ξ 1.
The computed evolution of u′2(t) and the integral length scale lt
read as follows
u′2(t) ∝ (t + a)−10/7, lt ∝ (t + a)2/7, a ∈ R. (3.23)
Therefore there exists a selfsimilar solution of Eq. 3.13 in the
following form
BLL(r, t) = u′2(t) f (ξ) ≡ (t + a)−10/7 f (ξ). (3.24)
The exact form of the function u′2(t) makes it possible to
calculate exactly the evolution of the radius γ of {q} × S2 in
time
γ f = qβ Dα LL ≡ Ar9/4(t + a)−5/14(1 − f (rt−2/7))1/4, (3.25)
where A is a positive constant equals 29/2.
Remark 3.1 To the best of our knowledge, there are no published
results establishing the existence and uniqueness of solutions to
initial-boundary value problems for Eq. 3.13. Equation 3.13 is a
nonlocal degenerate parabolic equa- tion. This makes very delicate
the proof of solvability of initial-boundary value problems for
this equation. Moreover, the large time behavior of solutions and
accompanying qualitative properties are not studied yet. To
overcome this gap, we use the results of numerical modeling. As
have been shown in [2], numerical calculations indicate some
important constraints of the original closed model of the von
Kármán-Howarth equation: if the Reynolds number is sufficiently
large, solutions from arbitrary initial conditions relax after a
small amount of time to a selfsimilar state, controlled by the
large scale structure. Moreover,
14 V.N. Grebenev, M. Oberlack
the self-similar solution obtained demonstrates a good agreement
between measured and analytic calculated values of f and h if the
Reynolds number to be sufficient large. Thus we can conclude, based
on numerical experiments, that the function γ behaves approximately
as γ f for large time.
We give a geometric interpretation of the solution obtained (3.22).
Let us rewrite (3.22) in the form
1
[ 1 + (1 − f )1/2
1 − (1 − f )1/2
x = ξ/14κ2, y = f 1/2. (3.27)
Then (3.22) is transformed to the well-known tractrix equation
[28]
x = x(y) = −(a2 − y2)1/2 + a 2
ln
[ a + (a2 − y2)1/2
a − (a2 − y2)1/2
] , a = 1 (3.28)
arising in differential geometry. The curve x = x(y) coincides with
the element of Beltarmi surface. This is a remarkable fact since
Beltrami surface is a canonical surface of the constant (sectional)
negative curvature equals −1. Reflecting this surface with respect
to the plane yOz of the Cartesian space R
3, we obtain the so-called pseudo-sphere: a hyperbolic manifold of
the constant negative curvature. This manifold has singular points
at x = 0 which forms the so-called break circle where the manifold
loses smoothness. We note that according to our construction the
longitudinal correlation function BLL(r, t) for each fixed time
takes a constant value on the hypersurface {r} × S2 or, in another
words, BLL(r, t) is a radially-symmetric function. Fixing the angle
coordinate of the sphere S2, for example ψ = ψc (or considering a
cross-section of the sphere along a latitude), we can construct in
the Cartesian coordinates (x, y, z) a surface of revolution
generated by the curve x = x(y) (or the graphic of f ) which
coincides with Beltrami surface. The parametric equations of the
curve x = x(y) (or the graphic of f ) read
x = ln cot 1
π
2 .
Revolving this curve about Ox-axis, we obtain the following family
of curves
xω ≡ x, yω = sin θ cos ω, zω = sin θ sin ω, −∞ < ω < ∞
(3.29)
where ω is an angle of rotation of the plane XY and Eq. 3.29
present the so- called universal covering of Beltrami surface. Here
the value of angle θ = π/2 corresponds the singular points of the
pseudo-sphere. Parametric equations of
Interpretation of the Second-Order Structure Function 15
the surface of revolution generated by (3.24) for each fixe time tc
for a cross- section of the sphere along a latitude can be written
in the form
x = ln cot 1
(tc + a)10/7 sin2 θ cos2 ω, (3.30)
z = sin θ sin ω, 0 < θ < π
2 , 0 ω 2π. (3.31)
Equations 3.30 and 3.31 describe the evolution of this surface in
time.
4 Concluding Remarks
The physical conclusions of this works read as: the Riemannian met-
ric (3.15), (3.18) enables us to introduce into consideration the
family of length scales (parametrized by the time t) of turbulent
motion exploring the closure model for the von Kármán-Howarth
equation in the limit of large Reynolds numbers; the length scales
constructed by this way are compatible with the form of this
closure model; the formula (3.18) tell us how the length scale of
possible transverse displacements of fluid particles depends on the
second- order structure function and the correlation distance;
using the selfsimilar solution (3.24) and the formula (3.25), we
can estimate asymptotically de- creasing the length scales of
turbulent motion in the transverse direction. The frame of this
consideration is limited by the case of homogeneous isotropic
turbulence under the assumption of large Reynolds numbers (in the
limit of large Reynolds numbers).
In this paper we did not consider such questions as solvability of
initial- boundary value problems for Eq. 3.13, large time behavior
of solutions and other accompanying topics.
Acknowledgements This work was supported by DFG foundation and was
partially supported by RFBR (grant No 07-01-0036). The first author
grateful to Prof. V.V. Pukhnachev and G.G. Chernykh for helpful
discussions. Also the authors thank Dr. M. Frewer for the useful
comments and an anonymous reviewer for the reference [25].
Appendix
We show that the approximation (2.5) and (2.7) can be obtained by
using invariants of the characteristics of system (2.8),
(2.9).
So, assuming the Reynolds number to be large, the first order O(1)
of Eqs. 2.8 are reduced to the system of first-order partial
differential equations
∂ BLL
16 V.N. Grebenev, M. Oberlack
In order to find the Riemannian invariant I of this system we
consider the operator
L = Dt + λDr,
where Dt and Dr are the total derivatives with respect to the
variables t and r. Here λ is determined by the equation
det
) = 0, (4.34)
where E denotes the unit matrix. Then the Riemannian invariant I of
the characteristic equation
dx dt
= λ (4.35)
is determined by the equation L(I) = 0 due to (4.32), (4.33).
Applying this definition to (4.32) and (4.33), we can rewrite this
system in the form
L(I)|[(4.32), (4.33)] = 0. (4.36)
Solving Eq. 4.34, we find that this equation has the roots
λ1 = 1/2DLL, λ2 = −1/2DLL.
In the case of λ = λ1 our equation for the invariants takes the
form
Dt I(r, t, w, v) + 1/2D1/2 LL Dr I(r, t, w, v)
(4.32), (4.33)]
= 0, w = BLL,
v = BLL,L. (4.37)
Equation 4.37 is a first-degree polynomial with respect to the
variables v, wr
and vr. As a result, we obtain the following system
4
LL Iw + 1/2DLL Iv = 0, Iw + 1/2D1/2 LL Iv = 0,
It + 1/2D1/2 LL Ir = 0. (4.38)
Suppose that DLL ≡ 0 then we find
τ = 3/2 a1r
D1/2 LL
that coincides with the formula suggested in [23] which have been
obtained by the dimensional analysis. Then from the Eqs. 4.38 it
follows that the Riemannian invariant I1 (which corresponds to the
root λ1 of Eq. 4.34) is defined by the formula
I1 = v − ∫
Interpretation of the Second-Order Structure Function 17
We recall that invariants of the so-called characteristic
differential Eq. (4.34) generate invariant manifolds of the
corresponding system of differential equa- tions i.e. according to
[15] the following equality holds
VF(Ii)|[Ii]r = 0, F = (F1, F2) (4.41)
where VF is the vector field
VF = ∂
∂t +
, g = (g1, g2) ≡ (w, v)
and [Ii]r denotes the equation Ii = 0 and differential
prolongations of this equation with respect to r. Indeed, the
condition of invariance
L(Ii)|(4.32), (4.33) = 0
Dt Ii|(4.32), (4.33) |[Ii]r = 0,
which is equivalent to (4.41). Therefore if we consider the system
(4.32), (4.33) on the invariant manifold I1 = 0 then Eq. 4.32 is
reduced to
∂ BLL
LLdw. (4.42)
Thus we can construct a von Kármán-Howarth’s type differential
model by Eq. (4.42) and using the Taylor series for the
function
1/2 ∫
D1/2 LLdw.
Taking into account that v(0) = 0, we write the invariant manifold
in the form
0 = I1 = v(r) − 1/2 ∫
(4.43) Denoting
LLwr,
as the first-order approximation of I1, we can reduce Eq. 4.42
to
∂ BLL
LL ∂ BLL
∂r (4.44)
on the set Iappr 1 = 0 which is similar to the von Kármán-Howarth
equation but
with another model constant κ3 = 0.5.
Remark 4.1 Using the recommended value of the Kolmogorov constant C
≈ 1.9, and calculating 2κ2 in (2.7) which approximately equals 0.2,
we find that κ3
is close to 2κ2 with respect to the order of these
quantities.
18 V.N. Grebenev, M. Oberlack
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Heisenberg-Integrable Spin Systems
Robin Steinigeweg · Heinz-Jürgen Schmidt
Received: 20 June 2007 / Accepted: 24 October 2008 / Published
online: 21 November 2008 © Springer Science + Business Media B.V.
2008
Abstract We investigate certain classes of integrable classical or
quantum spin systems. The first class is characterized by the
recursively defined property P saying that the spin system consists
of a single spin or can be decomposed into two uniformly coupled or
disjoint subsystems with property P. For these systems the time
evolution can be explicitly calculated. The second class consists
of spin systems where all non-zero coupling constants have the same
strength (spin graphs) possessing N − 1 independent, commuting
constants of motion of Heisenberg type. These systems are shown to
have the above property P and can be characterized as spin graphs
not containing chains of length four as vertex-induced sub-graphs.
We completely enumerate and characterize all spin graphs up to N =
5 spins. Applications to the construction of symplectic numerical
integrators for non-integrable spin systems are briefly
discussed.
Keywords Completely integrable systems · Heisenberg spin
systems
Mathematics Subject Classifications (2000) 70 H 06 · 37 J 35 · 81 Q
05 · 94 C 15 · 82 D 40
R. Steinigeweg · H.-J. Schmidt (B) Fachbereich Physik, Universität
Osnabrück, Barbarastr. 7, 49069 Osnabrück, Germany e-mail:
[email protected]
20 R. Steinigeweg, H.-J. Schmidt
1 Introduction
Integrable physical systems are rare; nevertheless there exists a
vast literature about these systems, let it be classical or quantum
ones. They are interesting in their own right but also in
connection with properties like chaotic behavior [1, 2], level
statistics [3, 4] or transport properties [5–7]. In this paper we
deal with a special class of integrable spin systems, called
Heisenberg-integrable systems. This is motivated by investigations
on magnetic molecules.
The synthesis of these molecules has undergone rapid progress in
recent years building on successes in coordination and
polyoxometalate chemistry [8–11]. Each of the identical molecular
units can contain as few as two and up to several dozen
paramagnetic ions. It appears that in the majority of these
molecules the localized single-particle magnetic moments (“spins”)
couple antiferromagnetically and the spectrum is rather well
described by the Heisenberg model [12]. The synthesis of these
magnetic molecules has addi- tionally aroused the interest in small
spin systems, as opposed to the traditional topic of infinitely
large spin lattices. Nevertheless, small spin systems may have
large Hilbert spaces: Since the dimension of the Hilbert space for
N spins of spin quantum number s, given by (2s + 1)N , grows
rapidly with N and s, the numerical evaluation of all energy
eigenvalues may be practically impossible. Analogous remarks apply
to the study of classical spin systems, where s → ∞. Hence it is
useful to know as much as possible about integrable spin sys- tems
where the Hamiltonian can be analytically diagonalized. Integrable
spin systems may additionally be used to test numerical procedures
or even to construct numerical integrators [13]. Here we want to
systematize the scattered knowledge on integrable spin systems of a
particular kind, the Heisenberg- integrable systems.
Classical spin systems are examples of Hamiltonian mechanical
systems. Hence the term “integrable” has a precise meaning in the
context of the Liouville-Arnold theorem [14]. It requires that
there exist N independent, commuting constants of motion, where N
denotes the number of spins and “commutation” is understood w. r.
t. the Poisson bracket. For integrable systems one can find
so-called action-angle variables In, n such that
In = − ∂ H ∂n
= ωn = const. , n = 1, . . . , N . (1)
Hence the equations of motion can be solved explicitly if the
integrations involved in the definitions of the In, n can be
performed, see [14]. The task to independently characterize the
class of all integrable spin systems IS seems extremely difficult.
Most published work on integrable spin systems deals with special
examples and numerical case studies [1, 2]. Also in this article we
will not characterize IS itself, but certain subclasses of
IS.
For quantum systems the corresponding notion of “integrability” is
less precise. Although some prominent examples of integrable
classical systems
Integrable Spin Systems 21
have solvable quantum mechanical counterparts, including the
harmonic os- cillator, the Kepler problem and the two center Kepler
problem, there is no comparable general theory of integrable
quantum systems. For example, the general Heisenberg spin triangle
with different coupling constants is an integrable classical
system, but we do not know of any procedure to analytically
calculating the eigenvectors and eigenvalues of the general quantum
spin trian- gle for arbitrary individual spin quantum number s.
Moreover, some quantum systems like the s = 1
2 Heisenberg spin chain of length N are considered as completely
integrable, for example by quantum inverse scattering methods [15],
but possess non-integrable classical counter-parts for N ≥ 5.
However, for the subclasses of integrable spin systems to be
considered below the eigenvalue problem of the corresponding
quantum spin Hamiltonian can be analytically solved.
In this article we investigate a subclass HIS ⊂ IS of the class of
integrable spin systems, called Heisenberg integrable systems, or,
shortly, H-integrable systems. They are defined by the extra
condition that N − 1 of the N constants of motion, as well as the
Hamiltonian H itself, are of Heisenberg type, i. e. consist of
linear combinations of scalar products of spin vectors:
E(n) = ∑
E(n) μν sμ · sν, n = 1, . . . , N − 1. (2)
The remaining N-th constant of motion is chosen as the 3−component
of the total spin S(3). We conjecture that HIS = IS if H is of
Heisenberg type, but we have not proven this although we have some
evidence from numerical studies of Ljapunov exponents for small
spin systems, see also [16]. We did not obtain an independent
characterization of HIS itself, but only for two subclasses HIG ⊂
BS ⊂ HIS called “Heisenberg integrable spin graphs” and
“B-partitioned spin systems”. “Spin graphs” are systems with
Heisenberg Hamiltonians
H = ∑
Jμνsμ · sν , (3)
satisfying Jμν ∈ {0, 1}. Obviously, the coupling scheme of such a
system can be represented by an undirected graph, the N vertices v
∈ V of which correspond to the N spins and the edges (μ, ν) ∈ E to
those pairs of spins where Jμν = 1.
For our purposes it is convenient to consider a special notion of a
“sub- system”, see Section 2. In the case of a spin graph G = (V,
E) a subsystem will consist of a subset V ′ ⊂ V of vertices
together with all edges e ∈ E which connect vertices of V ′. In
other words, the sub-graph G ′ = (V ′, E ′) is obtained by removing
a number of vertices along with any edges which contain a removed
vertex. This notion of a subsystem is equivalent to what graph
theorists call a “vertex-induced subgraph”, see [25].
22 R. Steinigeweg, H.-J. Schmidt
All spin graphs with N ≤ 4 turn out to be H-integrable, with the
excep- tion of the 4-chain. One main result of this article is that
a spin graph is H-integrable iff it contains no 4-chain as a
subsystem (in the sense explained above) iff it is the union of two
uniformly coupled or disjoint H-integrable subsystems (“uniform or
disjoint union”). By recursively applying the uniform union
property we obtain a partition of the whole spin graph into smaller
and smaller H-integrable subsystems with uniform or vanishing
coupling. This sequence of partitions can be encoded in a binary
“partition tree” B. Removing the condition Jμν ∈ {0, 1} we arrive
at the slightly more general notion of B-partitioned spin systems
for which the time evolution can be analytically calculated. The
observation that the uniform union of two integrable systems is
again integrable is certainly not new, see e. g. [17, 18] for a
special case concerning quantum spin systems or [19] for classical
systems. However, the use of partition trees in order to obtain the
time evolution or the eigenvectors in closed form seems to be
novel.
Moreover, our B-partitioned spin systems are closely related to
integrable Hamiltonians constructed from co-algebras, see [20],
Theorem 2. More pre- cisely, if we specialize B to be a monotone
partition (for the definition see Section 5.1) then our N − 1
quadratic constants of motion can be expressed by the Casimir
elements C(m), 2 ≤ m ≤ N defined in [20] in the case of the Lie
algebra g = so(3). Recall that the construction in [20] includes
the Gaudin- Calogero systems [21, 22] for the choice of g = so(2,
1). It seems that the results of [20] also hold for the case of a
general partition tree B as considered in this paper.
Our article is organized as follows. In Section 2 we present the
pertinent definitions and first results on classical integrable or
H-integrable spin systems. Further results on subsystems and
uniform unions are contained in Section 3. Among these is Theorem 1
saying that any subsystem of an H-integrable spin system is again
H-integrable.
Section 4 contains our results on spin graphs. For example, Theorem
2 states that each H-integrable spin graph is the uniform union of
two H-integrable subsystems. Finally we will prove that a spin
graph is H-integrable iff it does not contain any 4-chain, see
Theorem 3. As an application, we enumerate all connected spin
graphs up to N = 5 in the Appendix. If they are H-integrable we
indicate the uniform decomposition as well as the N commuting
constants of motion; if they are not we display some 4-sub-chain.
The next Section 5 is devoted to the explicit form of the time
evolution for B-partitioned spin systems, see Theorem 4. It turns
out that their time evolution can be described by a suitable
sequence of rotations about constant axes. This is closely related
to the definition of action-angle variables satisfying (1), as we
will show in Section 5.2.
In Section 5.3 we will sketch how to calculate the eigenvectors and
eigenval- ues of the Hamiltonian in the quantum version of
B-partitioned spin systems. Section 6 contains a summary and an
outlook.
Integrable Spin Systems 23
2 Definitions and First Results
Classical spin configurations are most conveniently represented by
N-tuples of unit vectors s = (s1, . . . , sN), |sμ|2 = 1 for μ = 1,
. . . , N. The compact man- ifold of all such configurations is the
phase space of the spin system
P = PN = { (s1, . . . , sN)
. (4)
The three components si μ (i = 1, 2, 3) of the μ-th spin vector can
be viewed as
functions on P
si μ : P −→ R . (5)
In order to formulate Hamilton’s equation of motion we need the
Poisson bracket between two arbitrary smooth functions
f, g : P −→ R . (6)
The Poisson bracket has to satisfy a couple of general properties,
namely bilin- earity, antisymmetry, Jacobi identity and Leibniz’
rule, see e. g. [23] 10.1. Hence it suffices to define the Poisson
bracket between functions of the form (5):
{ si μ, s j
δμνεijksk μ , (7)
where δμν denotes the Kronecker symbol and εijk the components of
the totally antisymmetric Levi-Civita tensor. This definition turns
P into a Poisson manifold.
We will sketch a more abstract way to endow PN with a Hamiltonian
structure: Let G be a Lie group and g∗ the dual of its Lie algebra.
g∗ is endowed with a canonical Poisson bracket and, moreover, is a
disjoint union of the orbits of the co-adjoint action of G upon g∗.
Every such orbit is a natural symplectic manifold, see [23],
Chapter 14. The phase space PN of a classical spin system results
if G is taken as the N-fold direct product of the rotation group
SO(3). In this case g∗ ∼= (R3)N can be given the structure of a
product of Euclidean spaces, unique up to an arbitrary positive
factor, such that G operates isometrically on g∗. Then PN is the
co-adjoint orbit consisting only of unit vector configurations.
Hence the εijk in (7) has its origin in the Lie bracket of the Lie
algebra of SO(3), which is also the origin of the commutation
relations of angular momenta in quantum mechanics. In the sequel we
will, however, not make use of this abstract approach.
Having defined the Poisson bracket, we can write down the
differential equation corresponding to a given smooth function H :
P −→ R:
d dt
si μ = {
24 R. Steinigeweg, H.-J. Schmidt
The r. h. s. of (8) can be viewed as the vector field XH on P
generated by the function H. If H is the Hamiltonian of the spin
system, (8) is Hamilton’s equation of motion and XH is the
corresponding Hamiltonian vector field. It is complete since P is
compact, see [24] Cor. 4.1.20. Generally, we define
Ft(H)s(0) = s(t), t ∈ R , (9)
where s(t) = (s1(t), . . . , sN(t)) is the solution of (8) with
initial value s(0). Ft(H) : P −→ P is called the flow of H and is
defined for all t ∈ R due to the completeness of the Hamiltonian
vector field.
Lemma 1 Let H, K : P −→ R be smooth functions. Then the flows Ft(H)
and Ft(K) commute iff {H, K} = 0.
Proof The if-part is a standard result, since the commutation of
the flows is equivalent to 0 = [XH, XK], see [24] 4.2.27, and [XH,
XK] = −X{H,K}, see [23] 5.5.4. For the only-if-part we conclude 0 =
[XH, XK] = −X{H,K}, hence {H, K} = c = const.. For c = 0 the
differential equation
d dt
K(s(t)) = {K, H} = −c (10)
has unbounded solutions, which is impossible for compact P . Hence
{H, K} = 0.
For the rest of this section we consider a fixed Heisenberg
Hamiltonian H : P −→ R. It will be convenient to identify the spin
system with its Hamiltonian.
A constant of motion is a smooth function f : P −→ R which commutes
with the Hamiltonian: { f, H} = 0. H is said to be of Heisenberg
type, or, short, a Heisenberg Hamiltonian, if it is of the
form
H(s) = ∑
Jμνsμ · sν . (11)
The real numbers Jμν are called coupling constants. It will be
convenient to set Jνμ = Jμν for μ < ν. Define the total spin
vector
S ≡ N∑
μ=1
sμ (12)
with components S(i), i = 1, 2, 3 and square S2 ≡ S · S. A
Heisenberg Hamiltonian commutes with all components of the total
spin and its square:
0 = { H, S2
Integrable Spin Systems 25
Let A = ∅ be any subset of {1, . . . , N} and a = |A|. Then PA
denotes the phase space of the subsystem A, i. e. the manifold of
all spin configurations of the form sA = (sμ1 , . . . , sμa) such
that μ1 < μ2 < . . . < μa and |sμi |2 = 1 for all μi ∈ A.
The Hamiltonian HA of the subsystem A will be defined by
HA(sA) = ∑
Jμνsμ · sν . (14)
Similarly, we define SA = ∑ μ∈A sμ together with its components
S(i)
A and its square S2
A. Next we consider a decomposition of {1, . . . , N} into two
disjoint subsets,
{1, . . . , N} = A∪B such that A, B = ∅. Let further NA ≡ |A| and
NB ≡ |B|. The Heisenberg Hamiltonian H is accordingly decomposed
into three parts:
H(s) = ∑
Jμνsμ · sν (15)
≡ HA + HB + HAB . (16)
Here and in the sequel we identify functions of Heisenberg type
defined on the phase space of a subsystem PA with their unique
extension to the total phase space P , defined by Jμν = 0 for all
μ, ν with μ /∈ A or ν /∈ A.
If the coupling constants Jμν, μ ∈ A, ν ∈ B occurring in HAB have
all the same non-zero value, say, Jμν = cAB ∈ R, cAB = 0, we will
call the system H the uniform union of the subsystems HA and HB. If
the analogous condition holds with cAB = 0 we call the system H the
disjoint union of the subsystems HA and HB. A Heisenberg system is
called connected if it is not the disjoint union of two subsystems.
A Heisenberg system where all coupling constants are non-zero and
have the same value will be called a pantahedron.
Definition 1 A Heisenberg spin system H is called Heisenberg
integrable, or, short, H-integrable, if there exist N − 1
independent constants of motion E(n)
of Heisenberg type which commute pairwise:
{ E(n), E(m)
} = 0 for all n, m = 1, . . . , N − 1 . (17)
The Heisenberg Hamiltonian H, which commutes with all E(n), will in
general be a linear combination of the E(n) and hence need not be
explicitly included in the set of the independent constants of
motion.
26 R. Steinigeweg, H.-J. Schmidt
The above condition of independence means that there exists some s
∈ P such that the set of covectors {dE(1)(s), . . . , dE(N−1)(s)}
is linearly independent. It follows that this condition is also
satisfied in some neighborhood of s ∈ P . But it cannot hold
globally: If you take some F in the linear span of the E(n)
such that s ∈ P is a critical point of F, i. e. dF(s) = 0, then it
is obviously violated. Later we will derive a simple criterion for
the independence of a number of Heisenberg constants of motion, see
Proposition 1.
If a connected spin system is H-integrable, it can be easily shown
that {E(1), . . . , E(N−1), E(N) = S(3)} will be a set of N
independent, commuting con- stants of motion. Hence any
H-integrable system is also integrable in the sense of the
Liouville-Arnold theorem. We conjecture that the converse is also
true.
For an integrable spin system the N constants of motion E(n) are
not uniquely determined. First, one can consider linear
transformations
F(n) = N∑
m=1
Anm E(m), n = 1, . . . , N , (18)
where the Anm are the entries of an invertible matrix. These
transformations leave invariant the space E of functions spanned by
the E(n) n = 1, . . . , N. But also the space E need not be
uniquely determined by the Hamiltonian H: Consider the disjoint
union H of two H-integrable subsystems HA and HB. Then one could
either consider the union of the two sets of independent, commuting
constants of motion for the subsystems, including S(3)
A and S(3)
B , or, alternatively, the union of the Heisenberg constants of
motion of the subsystems together with S2 − S2
A − S2 B and S(3). Since the second choice of
the E(n) is always possible, our Definition 1 entails that the
disjoint union of H-integrable spin systems will be again
H-integrable, see Proposition 3.
For later reference we mention the following well-known fact
without proof:
Lemma 2 For any spin system with N spins there exist at most N
independent, commuting constants of motion.
A spin graph is a Heisenberg spin system where all non-zero
coupling con- stants have the same strength J = 0. Without loss of
generality we may assume Jμν ∈ {0, 1}. As explained in the
introduction, the system can be represented as an undirected graph
with N vertices. Of course, the above definition of a connected
Heisenberg spin system coincides with the graph-theoretic notion of
connectedness if the system is a spin graph. Recall that subsystems
of spin graphs are understood as vertex-induced subgraphs, see
above.
The following fact is well-known, see [25], Theorem 1.6:
Lemma 3 In a connected spin graph with N ≥ 2 vertices one can
remove two suitable vertices such that the remaining subsystem is
still connected.
Integrable Spin Systems 27
In order to evaluate the Poisson bracket between functions of
Heisenberg type and to argue with the resulting equations we need
the following lemmas which are easily proven:
Lemma 4
} = sμ · (sλ × sν) = det (sμ, sλ, sν
)
(ii) {sμ · sν, k · sν
} = det (sμ, k, sν
N∑
N∑
) = 0 (22)
holds for all (s1, . . . , sN) ∈ P then all coefficients of the
corresponding equation must vanish.
Proof By induction over N using the replacement sN+1 → −sN+1.
Now we can formulate a criterion for the commutation of two
functions of Heisenberg type:
Lemma 6 Let H be a Heisenberg system and E a function of Heisenberg
type. E commutes with H iff
Eμ ν(Jμ λ − Jν λ) + Eμ λ(Jν λ − Jμ ν) + Eν λ(Jμ ν − Jμ λ) = 0
(23)
for all μ < ν < λ ≤ N.
Proof The lemma is proven in a straightforward manner by using
Lemma 4(i), cyclic permutations of triple products and (22).
28 R. Steinigeweg, H.-J. Schmidt
Next we will show that the independence of M functions of
Heisenberg type is equivalent to the linear independence of the
corresponding symmetric matrices.
Proposition 1 Let E(n) : P −→ R be M functions of Heisenberg type,
i. e.
E(n)(s) = ∑
E(n) μν sμ · sν, n = 1, . . . , M . (24)
and denote by E (n) the symmetric N × N-matrix with entries E
(n) μν = E(n)
μν and E
(n) μμ = 0 for μ, ν = 1, . . . , N − 1. Then the following
conditions are equivalent:
(i) There exists an s ∈ P such that the set of covectors
{dE(n)(s)|n = 1, . . . , M} is linearly independent.
(ii) The set of matrices {E(n)|n = 1, . . . , M} is linearly
independent.
Proof We will prove the equivalence of the negations of (i) and
(ii). (i) ⇒ (ii). Assume that {E(n)|n = 1, . . . , M} is linearly
dependent. That is,
there exists a non-vanishing real vector (λ1, . . . , λM) such that
∑
n λnE (n) =0.
It follows that E(s) ≡ ∑ n λn E(n)(s) = 0 for all s ∈ P and hence 0
=
d (∑
n λn E(n)(s) ) = ∑
n λndE(n)(s). Hence the set of covectors {dE(n)(s)|n = 1, . . . ,
M} is linearly dependent for all s ∈ P .
(ii) ⇒ (i). It is possible to invert the sequence of arguments of
the first part of the proof, except at the step dE = 0 ?⇒ E = 0.
Here we can only conclude E = c = const. and obtain the apparently
weaker condition
c = E(s) = ∑
λn E(n) μν sμ · sν (25)
for all s ∈ P . Replacing sν by −sν for fixed ν yields ∑
μ Eμνsμ · sν = 0. By sum- ming over ν we obtain
∑ μ<ν Eμνsμ · sν = 0 and, by (20), Eμν = 0 for all μ <
ν.
Hence ∑
n λnE (n) = 0 and the set {E(n)|n = 1, . . . , M} is linearly
dependent.
We note that (23) can be viewed as a system of (N
3
2
) unknowns Eμν . An H-integrable system admits at least N − 1
linearly
independent solutions, according to Proposition 1. In general, it
will admit more solutions, but only N − 1 of these will lead to
commuting constants of motion. Hence, in the H-integrable case, the
matrix M of the system of linear equations (23) has the rank r ≤
(N
2
) − (N − 1). For N = 4 the rank condition r = (N
2
) − (N − 1) = 3 is even sufficient since it implies the existence
of a
Integrable Spin Systems 29
constant of motion E which is not of the form λH + μS2. After some
algebra we obtain:
Proposition 2 A Heisenberg system with N = 4 spins is H-integrable
iff
J13 − J23 J23 − J12 0 J14 − J24 0 J24 − J12
0 J14 − J34 J34 − J13
= 0 . (26)
This criterion can be used to independently check the results on
spin graphs with N = 4, see Appendix.
3 Subsystems and Uniform Union
In this section we will collect some general results on
H-integrable systems in connection with subsystems and uniform
unions.
Lemma 7 Let E be a Heisenberg constant of motion of a Heisenberg
system H and ∅ = A ⊂ {1, . . . , N}. Then the restriction EA is a
Heisenberg constant of motion of the subsystem HA.
Proof The lemma follows from Lemma 6 since the restricted functions
EA and HA commute iff the equations (23) hold for μ < ν < λ
with μ, ν, λ ∈ A.
Proposition 3 If a Heisenberg system H is the uniform or disjoint
union of two H-integrable subsystems HA and HB, then H itself is
H-integrable.
Corollary 1 Each pantahedron is H-integrable.
Proof Let EA be one of the NA − 1 independent, commuting Heisenberg
constants of motion of HA. In particular, EA commutes with HA and
also with HB since B ∩ A = ∅. Since EA is a function of Heisenberg
type it commutes with S2, S2
A and S2 B, hence also with HAB since
HAB = 1
2 cAB
( S2 − S2
) . (27)
It follows that EA commutes with H. The same holds for a
corresponding constant of motion EB of the second subsystem. Hence
the NA − 1 functions EA together with the NB − 1 functions EB and
S2 − S2
A − S2 B form a set of
N − 1 independent, commuting constants of motion of Heisenberg
type. This means that H is H-integrable.
The converse of Proposition 3 is not true: The general spin
triangle is a Heisenberg spin system with N = 3 and three different
coupling constants. It
30 R. Steinigeweg, H.-J. Schmidt
has H and S2 as independent, commuting constants of motion and is
hence H-integrable, but it is not the uniform or disjoint union of
two H-integrable subsystems.
Next we will show that H-integrability is heritable to
subsystems.
Theorem 1 Any subsystem HA of an H-integrable system is itself
H-integrable.
Proof Consider N − 1 independent, commuting constants of motion E1,
. . . , EN−1 of the form
Ei = ∑
Ei, μ ν sμ · sν . (28)
These constants of motions span a linear space F . According to the
assumptions of the theorem {1, . . . , N} is the disjoint
union
of two nonempty subsets A and B such that N = NA + NB with NA = |A|
and NB = |B|. We arrange the coefficients Ei, μ ν in the form of a
matrix E with(N
2
(NA 2
μ < ν and μ, ν ∈ A; the next (NB
2
) rows contain the coefficients with μ < ν and
μ, ν ∈ B and finally the remaining NA NB rows those with μ ∈ A, ν ∈
B. In this way the matrix is divided into three blocks, see the
following figure.
E E1 2 EN–1. . .
(N 2 )A
. . . N – 1
Next this matrix will be transformed into a lower triangular form
by ele- mentary Gauss transformations. We allow arbitrary
permutations of columns
Integrable Spin Systems 31
and arbitrary permutations of rows within the three blocks, see the
following figure.
1 2 N–1. . .
N – 1
The resulting matrix F begins with dA linearly independent columns
span- ning a linear space FA. dA is the maximal number of
independent constants of motions of the subsystem A obtained as
restrictions to A of functions from F . The next dB columns of F
span the linear space FB. dB is the maximal number of independent
constants of motions of the subsystem B obtained as restrictions to
B of functions from F which vanish on A. The remaining dAB
columns span the linear space FAB of functions from F vanishing on
A and on B. Since elementary Gauss transformations do not change
the rank of the matrix, we have
dA + dB + dAB = N − 1 . (29)
For sake of simplicity we identify the columns of F with the
corresponding constants of motion. By Lemma 7, the restrictions to
A of F1, . . . , FdA will be constants of motion of HA. According
to Lemma 2 there are at most NA − 1 independent constants of
motion. The analogous argument holds for FdA+1, . . . , FdA+dB and
the subsystem HB. Hence
dA ≤ NA − 1 and dB ≤ NB − 1 . (30)
It follows that N − 1 = dA + dB + dAB ≤ NA − 1 + NB − 1 + dAB = N −
2 + dAB, hence
dAB ≥ 1 . (31)
32 R. Steinigeweg, H.-J. Schmidt
Next we want to show that dAB ≤ 1. In this case we are done: dA
< NA − 1 would imply N − 1 = dA + dB + dAB < NA − 1 + NB − 1
+ 1 = N − 1 which is a contradiction. Hence dA = NA − 1 and the
subsystem HA would be H-integrable.
Proving dAB ≤ 1 is equivalent to show that FAB is at most
one-dimensional, i.e. that the ratios of the coefficients Fμλ/Fνκ
with μ, ν ∈ A and λ, κ ∈ B are uniquely determined. Hence consider
some F ∈ {FdA+dB+1, . . . , FN−1}. Its restrictions are FA = FB =
0. For μ, ν ∈ A and λ ∈ B Lemma 6 yields
Fμ λ(Jν λ − Jμ ν) + Fν λ(Jμ ν − Jμ λ) = 0. (32)
Then the following ratio of coefficients
Fμ λ
Fν λ
(33)
is uniquely determined, except in the case where the nominator Jμ ν
− Jμ λ and the denominator Jν λ − Jμ ν vanish simultaneously:
Jμ ν = Jμ λ = Jν λ . (34)
Define, for fixed μ ∈ A and λ ∈ B, M(μ, λ) to be the set of ν ∈ A
satisfying (34). If M(μ, λ) = A then HA is a pantahedron and hence
H-integrable. If M(μ, λ) = A then there exists a κ ∈ A such that κ
/∈ M(μ, λ) and the ratios Fμλ / Fκλ and Fνλ / Fκλ are uniquely
determined. Hence also the ratio
Fμ λ
Fν λ
= Fμλ
Fκλ
· Fκλ
Fνλ
(35)
will be uniquely determined. By analogous reasoning, also the ratio
Fνλ / Fνκ
with ν ∈ A and λ, κ ∈ B, and hence Fμλ/Fνκ with μ, ν ∈ A and λ, κ ∈
B will be uniquely determined. This completes the proof.
The preceding proof also shows:
Corollary 2 Let H be an H-integrable spin system with two
subsystems HA and HB such that {1, . . . , N} = A∪B. According to
Theorem 1, HA and HB are also H-integrable. Then the N − 1
independent, commuting constants of motion F1, . . . , FN−1 can be
chosen such that
(i) F1, . . . , FNA−1 are independent, commuting constants of
motion of HA
and vanish on B, (ii) FNA , . . . , FN−2 are independent, commuting
constants of motion of HB
and vanish on A, (iii) FN−1 vanishes on A and on B.
Integrable Spin Systems 33
Proof It remains to show that the F1, . . . , FNA−1 of (i) vanish
on B. This can be done by further Gauss transformations which add
suitable multiples of columns of (ii) to the columns of (i).
4 Spin Graphs
As explained in Section 2, spin graphs are Heisenberg spin systems
such that the coupling constants satisfy Jμ ν ∈ {0, 1}. For these
systems, H-integrability can be completely analyzed. According to
Proposition 2 the uniform or disjoint union of H-integrable systems
is again H-integrable, but not all H-integrable systems are
obtained in this way. However, all H-integrable spin graphs are the
uniform or disjoint unions of H-integrable subsystems, as we will
show. This means that there is a construction procedure by which
one can compose all H-integrable spin graphs from small
constituents. Starting with two single spins, which are trivially
H-integrable, we can either form a disjoint union or a spin dimer.
The uniform union of a dimer and a single spin yields a uniform
triangle; the uniform union of a single spin with a pair of
disjoint spins is a 3-chain, etc. Remarkably, the 4-chain cannot be
obtained in this way and is hence not H-integrable.
Our first result is:
Lemma 8 Each connected H-integrable spin graph with N > 1
vertices is the uniform union of two subsystems.
Proof The proof will be performed by induction over N. For N = 2
the theorem holds since the dimer is the uniform union of two
single spins.
Next we assume the theorem to hold for all spin graphs with N or
less ver- tices and consider a connected spin graph with N + 1
vertices and Hamiltonian H. According to Lemma 3 we may assume that
the subsystem HN with vertices {1, . . . , N} is connected and, by
Theorem 1, H-integrable. We denote by HN+1
the single spin system with vertex N + 1. By the induction
hypothesis and since N > 1, HN is the uniform union of two
H-integrable subsystems HA and HB, where {1, . . . , N} = A∪B and
A, B = ∅.
HA is further decomposed into subsystems H0 A and H1
A with vertex sets A0
and A1, respectively. A0 consists of those spins in A which do not
couple to N + 1; A1 consists of the remaining spins in A which thus
uniformly couple to N + 1. The analogous decomposition is performed
w. r. t. HB.
Since A, B = ∅ we must have A0 = ∅ or A1 = ∅, and, similarly, B0 =
∅ or B1 = ∅. In the case A0 = B0 = ∅ the proof is done since this
means that HN
couples uniformly with HN+1. The case A1 = B1 = ∅ can be excluded
since it implies that H is disconnected. Hence it suffices to
consider the case A1 = ∅ and B0 = ∅ in what follows. The other
remaining case A0 = ∅ and B1 = ∅ can
34 R. Steinigeweg, H.-J. Schmidt
be treated completely analogously. The situation is illustrated in
the following figure.
H1 0
B BH
If A0 = ∅, the total system H is a uniform union of the two
subsystems with vertex sets A and {N + 1} ∪ B and the proof is
done. Hence we may assume A0 = ∅. If the coupling between the
subsystems H0
A and H1 A is uniform we may
rearrange the decomposition by setting A′ 0 = ∅ and B′
0 = A0 ∪ B0, leaving A1
and B1 unchanged. But this case has already be considered above.
Hence we may assume that the coupling between H0
A and H1 A is non-uniform.
By Corollary 2 (iii) and since HN is H-integrable, the total system
H possesses a non-zero constant of motion of the form
E = ∑
Lemma 6 implies
Eμ N+1(Jμ λ − Jλ N+1) + Eλ N+1(Jμ N+1 − Jμ λ) = 0, (37)
for all μ<λ< N + 1 since Eμ λ = 0. We will show that the case
A0, A1, B0 = ∅ is in contradiction to the above-mentioned fact that
E is non-zero. To this end we consider (37) in the following four
cases:
– μ0 ∈ B0 and λ1 ∈ A1 (Jμ0 λ1 = Jλ1 N+1 = 1 and Jμ0 N+1 = 0)
⇒ Eλ1 N+1 = 0 . (38)
– μ1 ∈ B1 and λ0 ∈ A0 (Jμ1 λ0 = Jμ1 N+1 = 1 and Jλ0 N+1 = 0)
⇒ Eμ1 N+1 = 0 . (39)
– μ0 ∈ B0 and λ0 ∈ A0 (Jμ0 N+1 = Jλ0 N+1 = 0 and Jμ0 λ0 = 1)
⇒ Eμ0 N+1 = Eλ0 N+1 (40)
– λ0 ∈ A0 , λ1 ∈ A1 and Jλ0 λ1 = 0 (Jλ0 N+1 = 0 and Jλ1 N+1 =
1)
⇒ Eλ0 N+1 = 0 . (41)
Integrable Spin Systems 35
and λ′ 1 ∈ A1 such that Jλ′
0 λ′ 1 = 0. For this λ′
0 the coefficient Eλ′ 0 N+1 vanishes
by (41). Equation (40) then yields Eμ0 N+1 = 0 for all μ0 ∈ B0 and,
further, Eλ0 N+1 = 0 for all λ0 ∈ A0. By the equations (38) and
(39) the remaining coefficients of E vanish, which leads to a
contradiction.
Lemma 8, Theorem 1 and Proposition 3 together yield:
Theorem 2 Each H-integrable spin graph is the uniform or disjoint
union of two H-integrable subgraphs.
It follows from Theorem 2 that all spin graphs with N ≤ 3 are
H-integrable, but that the chain with N = 4 is not H-integrable
since it is not the uniform union of smaller systems. By virtue of
Theorem 1 every spin graph containing a 4-chain will not be
H-integrable. The converse is also true and yields the following
graph-theoretic characterization of H-integrable spin graphs.
Theorem 3 A spin graph is not H-integrable iff it contains a chain
of length four as a vertex-induced sub-graph.
Proof It remains to show the only-if-part. This will be done by
induction over N.
For N = 4 the theorem is proven by a complete classification of all
con- nected spin graphs, see the Appendix. Next we assume that the
theorem holds for all spin graphs with N or less spins and consider
a spin graph H with N + 1 vertices which is not H-integrable.
If H is the union of two disjoint subsystems, at least one of them
is not H-integrable by Proposition 3 and hence contains a 4-chain
by the induc- tion assumption. Thus we may assume that H is
connected. By virtue of Lemma 3 we can remove a suitable vertex
with number, say, N + 1, such that the remaining subsystem HN is
still connected. Further we may assume that HN is H-integrable,
since otherwise it would contain a 4-chain by the induction
assumption. The coupling between HN and HN+1 is neither uniform nor
zero, since then H would be integrable by Proposition 3 or
disconnected. Hence we may decompose HN into a maximal subsystem
H0
N which is not coupled with HN+1 and the remainder H1
N which is uniformly coupled with HN+1. Both subsystems are
non-empty and H-integrable by Theorem 1.
H
36 R. Steinigeweg, H.-J. Schmidt
HN is connected and H-integrable and hence, by Theorem 2, the
uniform union of two non-empty subsystems HA and HB. Both
subsystems are further decomposed into H0
A, H1 A and H0
B, H1 B according to their coupling with
HN+1, similarly as HN above. Let A0, A1, B0, B1 be the
corresponding subsets of {1, . . . , N}. A0 ∪ B0 = ∅ and A1 ∪ B1 =
∅, see above.
H1 0
B BH
We consider the case A0 = ∅. This means that HN+1 as well as HB is
uniformly coupled to HA. The restriction of H to {N + 1} ∪ B cannot
be H-integrable, since then H would be H-integrable by Proposition
3. Hence {N + 1} ∪ B contains a 4-chain by the induction
assumption. Analogously we can argue in the case B0 = ∅. Hence we
may assume A0 = ∅ and B0 = ∅.
Assume that HN is a pantahedron. Then H would be the uniform union
of H1
N and the disjoint union of HN+1 and H0 N and hence H-integrable
by
Proposition 3, contrary to previous assumptions.
HN+1
H N 0
H N 1
Thus H is not a pantahedron and possesses at least one uncoupled
pair of spins, say (μ, ν) such that Jμν = 0. We have μ ∈ A0, ν ∈ A1
or μ ∈ B0,
Integrable Spin Systems 37
ν ∈ B1, since A and B are uniformly coupled. Because of A0 = ∅ and
B0 = ∅ we can choose any λ ∈ A0 or λ ∈ B0 and obtain a 4-chain (N +
1, ν, λ, μ).
H1 0
5.1 Explicit Form
In this section we consider spin systems which are the uniform or
disjoint union of subsystems A and B in such a way that these
subsystems enjoy the same property, and so on, until one or both
subsystems consist of single spins. Thus we obtain a nested system
of partitions which can be encoded in a binary partition tree B.
Examples are H-integrable spin graphs which are special
B-partitioned systems by virtue of Theorem 2.
The time evolution of such systems can be exactly described by
means of a recursive procedure, see [19]. In this section we will,
additionally, provide an explicit formula for the general solution
of the equations of motion which was assumed to be “too cumbersome”
in [19], using the notion of a “partition tree”.
Definition 2 A partition tree B over a finite set {1, . . . , N} is
a set of subsets of {1, . . . , N} satisfying
1. ∅ /∈ B and {1, . . . , N} ∈ B, 2. for all M, M′ ∈ B either M ∩
M′ = ∅ or M ⊂ M′ or M′ ⊂ M, 3. for all M ∈ B with |M| > 1 there
exist M1, M2 ∈ B such that M = M1∪M2.
It follows from 2(ii) that the subsets M1, M2 satisfying M = M1∪M2
in definition 2(iii) are unique, up to their order. M1, M2 are
hence defined for all M ∈ B with |M| > 1. M1 and M2 denote the
two uniquely determined “branches” starting from M. It follows that
B is a binary tree with the root {1, . . . , N} and singletons {μ}
as leaves. More general partitions into k disjoint
38 R. Steinigeweg, H.-J. Schmidt
subsets can be reduced to subsequent binary partitions and hence
need not be considered. For all M ∈ B there is a unique path
PM(B) ≡ {M′ ∈ B | M ⊂ M′} (42)
joining M with the root of B. It is linearly ordered since M ⊂ M′
and M ⊂ M′′ imply M′ ⊂ M′′ or M′′ ⊂ M′ by Definition 2 (ii).
Especially, every element μ ∈ {1, . . . , N} belongs to a unique,
linearly ordered construction path
Pμ(B) ≡ {M ∈ B | μ ∈ M} . (43)
A partition tree B is called monotone, iff it contains, besides the
singletons {μ}, all ordered subsets of {1, 2, . . . , N} of the
form {1, 2, . . . , n}, n ≤ N.
Although there are many different partition trees over a fixed
finite set, all have the same size which can be easily determined
by induction over N:
Proposition 4 |B| = 2N − 1 .
For M = {1, . . . , N} we will denote by M the “successor” of M,
that is, the smallest element of PM(B) except M itself. To simplify
later definitions we set {1, . . . , N} ≡ 0 and denote by B the
class of all successors, i. e.
B ≡ {M | M ∈ B} . (44)
It follows that |B| = N. Later we will use B as an index set for N
action variables. For μ = ν ∈ {1, . . . , N} let Mμν ∈ B denote the
smallest set of B such that μ, ν ∈ Mμν , i. e. Mμν ∈ B is the set
where both construction paths of μ and ν meet the first time.
Consider real functions J defined on a partition tree
J : B −→ R . (45)
J(Mμν)sμ · sν (46)
defines a Heisenberg Hamiltonian. The corresponding spin system
will be called a B-partitioned system or sometimes, more precisely,
a (B, J)− system. The N-pantahedron is a (B, J)-system where B is
arbitrary and J is a constant function. As mentioned before, all
H-integrable spin graphs are (B, J)-systems. For example, the spin
square is obtained by the partition tree
B = {{1, 2, 3, 4}, {1, 3}, {2, 4}, {1}, {2}, {3}, {4}} (47)
and the function J with J({1, 2, 3, 4}) = 1 and J(M) = 0 else.
Recall that SM denotes the total spin vector of the subsystem M
⊂
{1, . . . , N} with length SM and that Ft(H) denotes the flow map
describing
Integrable Spin Systems 39
time evolution of spins according to a Hamiltonian H. It follows by
Lemma 6 that the squares of the total spins corresponding to a
partition tree commute:
{ S2
} = 0 for all M, M′ ∈ B . (48)
We consider rotations in 3-dimensional spin space:
Definition 3 D( ω, t) will denote the rotation matrix with axis ω
and angle | ω| t. D(0, t) equals the identity matrix I.
The proof of the following proposition is easy and will be
omitted:
Proposition 5 Let M, M′ ⊂ {1, . . . , N}. Then
1. Ft (
) = D(SM, t) ,
2. D(SM′ , t)SM = SM if M′ ⊂ M or M′ ∩ M = ∅ .
A short calculation shows that the Hamiltonian (46) of a (B,
J)-system can also be written as
H = 1
M , (49)
if we set J({μ}) = J(0) = 0 for all μ ∈ {1, . . . , N}. It follows
that
Ft(H) = ∏
,
(50) where t