mathematical physics, analysis and geometry - volume 12

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 geometric 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 molecular 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 homothetically 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 determined. 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 operator 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.
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
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 selfsimilar 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.
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
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 hypotheses [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 correlation 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
triple correlation functions coming from using the finite-dimensional probability 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
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 examples 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)
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 determines 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
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 suggested (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
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 equation. 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,
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
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 metric (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
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 − ∫
We recall that invariants of the so-called characteristic differential Eq. (4.34) generate invariant manifolds of the corresponding system of differential equations 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.
References
1. Topping, P.: Lectures on the Ricci Flow. London Mathematical Society. Notes Series. Cambridge University Press, Cambridge (2006)
2. Oberlack, M., Peters, N.: Closure of the two-point correlation equation as a basis for Reynolds stress models. Appl. Sci. Res. 55, 533–538 (1993)
3. Lytkin, Yu.M., Chernykh, G.G.: On a closure of the von Kármán–Howarth equation. Dyn. Continuum Mech. 27, 124–130 (1976)
4. von Kármán, Th., Howarth, L.: On the statistical theory of isotropic turbulence. Proc. R. Soc. A 164, 192–215 (1938)
5. Lee, J.M.: Riemannian Manifolds. Graduate Text in Math. 176. Springer, New York (1997) 6. Piquet, J.: Turbulent Flows. Models and Physics. Springer, New York (1999) 7. Chernykh, G.G., Korobitsina, Z.L., Kostomakha, V.A.: Numerical simulation of isitropic
dynamics. IJCFD 10, 173–182 (1998) 8. Khabirov, S.V., Ünal, G.: Group analysis of the von Kármán-Howarth equation. I : Submod-
els. Commun. Non. Sci. Numer. Simul. 7(1–2), 3–18 (2002) 9. Ovsyannikov, L.V.: Group Methods of Differential Equations. Academic, New York (1982)
10. Pope, S.B.: Lagrangian PDF methods for turbulent flows. Annu. Rev. Fluid Mech. 26, 23–63 (1994)
11. Pope, S.B.: On the relationship between stochastic Lagrangian models of turbulence and secon-moment closure. Phys. Fluids 6, 973–985 (1994)
12. Oberlack, M.: On the decay exponent of isotropic turbulence. In: Proceedings in Applied Mathematics and Mechanics, 1. http://www.wiley-vch.de/publish/en/journals/ alphabeticIndex/2130/?sID= (2000)
13. Monin, A.S., Yaglom, A.M.: Statistical Hydromechanics. Gidrometeoizdat, St.-Petersburg (1994)
14. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1986) 15. Sidorov, A.F., Shapeev, V.P., Yanenko, N.N.: The Differential Constraints Method and its
Application to Gas Dynamics. Nauka, Novosibirsk (1984) 16. Oberlack, M., Busse, F.H. (eds.): Theories of Turbulence. Lecture Notes in Computer Science.
442. Springer, New York (2002) 17. Robertson, H.P.: The invariance theory of isotropic turbulence. Proc. Camb. Philol. Soc. 36,
209–223 (1940) 18. Arad, I., L’vov, S.L., Procaccia, I.: Correlation functions in isotropic turbullence: the role of
symmetry group. Phys. Rev. E. 59, 6753–6765 (1999) 19. Batchelor, G.K., Townsend, A.A.: Decay of turbulence in the final period. Proc. R. Soc. A 194,
527–543 (1948) 20. Millionshtchikov, M.: Isotropic turbulence in the field of turbulente viscosity. Lett. JETP. 8,
406–411 (1969) 21. Grebenev, V.N., Oberlack, M.: A Chorin-type formula for solutions to a closure model of the
Kármán–Howarth equation. J. Nonlinear Math. Phys. 12, 1–9 (2005) 22. Steward, R.W., Townsend, A.A.: Similarity and self-preservation in isitropic turbulence.
Philos. Trans. R. Soc. A 243(827), 359–386 (1951) 23. Onufriev, A.: On a model equation for probability density in semi-empirical turbulence trans-
fer theory. In: The Notes on Turbulence. Nauka, Moscow (1994) 24. Chorin, A.: Lecture Notices in Math, 615. Springer, Berlin (1977) 25. Chen, G.Q., Levermpre, C.D., Liu, T.-P.: Hyperboloic conservation kaws with stiff relaxation
terms and entropy. Comm. Pure Appl. Math. 47(6), 787–830 (1994) 26. Hasselmann, K.: Zur Deutung der dreifachen Geschwindigkeits korrelationen der isotropen
Turbulenz. Dtsh. Hydrogr. Zs. 11, 207–211 (1958) 27. Svetlov, A.V.: A discreteness criterion for the spectrum of the Laplace–Beltrami operator on
quasimodel manifolds. Siberian Math. J. 43, 1103–1111 (2002) 28. Michenko, A.S., Fomenko, A.T.: Lectures on Differential Geometry and Topology. Factorial,
Moscow (2000)
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 additionally 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 systems 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 triangle 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 “subsystem”, 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].
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 precisely, 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 eigenvalues of the Hamiltonian in the quantum version of B-partitioned spin systems. Section 6 contains a summary and an outlook.
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 manifold 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 = {
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
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.
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 constants 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 constants 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.
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).
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
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
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 elementary Gauss transformations. We allow arbitrary permutations of columns
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)
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.
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 vertices 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
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)
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 connected 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 induction 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
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,
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
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
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

mathematical physics, analysis and geometry - volume 12

Documents