CERNPHTH/2005025
February 10, 2005
Supersymmetric Models for Fermions on a Lattice
[50pt]
Nevena Ilieva , Heide Narnhofer and Walter Thirring
[20pt]
Physics Dept., Theory Division, CERN, CH1211 Geneva 23, Switzerland University of Vienna, Institute for Theoretical Physics Boltzmanngasse 5, A1090 Vienna, Austria, and Erwin Schrödinger International Institute for Mathematical Physics Boltzmanngasse 9, A1090 Vienna, Austria
We dedicate this article to Julius Wess on the occasion of his birthday.
We investigate the large behaviour of simple examples of
supersymmetric interactions for fermions on a lattice. Witten’s
supersymmetric quantum mechanics and the BCS model appear just as
two different aspects of one and the same model. For the BCS
model, supersymmetry is only respected in a coherent superposition
of Bogoliubov states. In this coherent superposition mesoscopic
observables show better stability properties than in a Bogoliubov
state.
PACS codes: 12.60Pb, 74.20, 74.50.+r
On leave from Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko Chaussee 72, 1784 Sofia, Bulgaria
E–mail address: ,
1 Introduction
We realize the simplest supersymmetric system by a finite fermion lattice. The supersymmetric structure is determined by a nonhermitian supercharge . This is an odd nilpotent element of the Fermi algebra . The sum and the anticommutator of with its adjoint give two hermitian elements, and . They generate the supertransformation and the time evolution, two commuting automorphism groups of . Already at this level of generality the Hilbert space assumes a structure. It is the sum of the nullspace of these operators plus the tensor product of and a rest In has strictly positive eigenvalues, which are twofold degenerate, while has the eigenvalues and leaves these spaces invariant. Thus the supertransformation and the time evolution are linked. However this closeness is lost once we go to the limit where , the number of lattice points, goes to infinity. There it can even happen that on local operators the supertransformation is not well defined whereas the time evolution is.
To restrict the many possibilities for , we impose on it a locality condition: products of operators may only contain operators of the same lattice site. If there is only one fermion per site, and are essentially unique. The time evolution becomes trivial, but the supertransformation remains and is in fact nonlocal. If there are two fermions per site, the time evolution becomes the free time evolution and there is a nonlocal supertransformation associated to it.
Since even for a local the supertransformation is nonlocal, we drop the locality requirement. We construct a Clifford variable , which is usually associated with supersymmetry. Note that a Grassman would mean , which is not possible in a algebra, but our is nilpotent and anticommutes with the other odd elements. With three fermions per site we can construct a supersymmmetric version of the BCS model. This is mathematically well explored and we can behold the many vistas that the limit offers. The limit depends on the state on which the representation is based and we shall study it for three different states: the ground state of , its ceiling state, and the Bogoliubov state, which in the limit has the same energy per particle as the ceiling state but remains pure on the quasilocal algebra.
We shall study the following three types of limiting observables:
a) Local operators, i.e. polynomials in the operators , the elements of , which are localized at the site ;
b) Mesoscopic observables, i.e. limits of ;
c) Macroscopic observables, i.e. limits of .
Our automorphisms turn out to be different in all cases. It is not even true that the microscopic time evolution determines the mesoscopic one. The supertransformation is finite and nontrivial only for the ground state, where Witten’s supersymmetric quantum mechanics emerges in the limiting procedure. It seems remarkable that the difference between a statistical mixture of the Bogoliubov states and the ceiling state, which can be considered as a coherent mixture of the Bogoliubov states, can only be observed on the mesoscopic and macroscopic level respectively. Especially the mesoscopic algebra is stable in the ceiling state under the emerging time evolution, whereas it is not so in the Bogoliubov state, which also breaks supersymmetry.
2 Algebraic framework
The basic structural elements of supersymmetry are a algebra and an odd nilpotent element (the supercharge):
(2.1) 
is supposed to be the generator of the time evolution and by Eq.(2.1) has the properties
(2.2) 
For (iii), note that either or must be different from zero and also belongs to the eigenvalue . Also, cannot be , would be in contradiction with the assumption .
Equation (2.2) implies that the Hilbert space can be written as a sum of a zerospace (projection ) and a tensor product of and the rest, : . Defining we have In this decomposition we can write in a matrix representation
(2.3) 
Here we identify ; denotes the positive square root of , but there are others: if then . The gauge transformation is effected by , which has the familiar matrix representation
(2.4) 
so that .
In every element can be written as . This gives the algebra a grading by calling the first two terms even and the others odd. All this emerges from a single nilpotent operator, namely . The operators generate a supertransformation
(2.5) 
which mixes even and odd elements of . To isolate the various aspects, we start with a finitedimensional , and later investigate the limit .
3 Fermions on a lattice
The supertransformation is a nonlinear transformation of the ’s that preserves their algebraic relations. For warming up we start with the simplest case:
The supertransformation is
(3.1) 
In this special case has to be trivial since it is twofold degenerate, (ii) cannot appear since create all of . The supertransformation is a twodimensional generating subset (but not subgroup) of the automorphism group. The latter is threedimensional and isomorphic to . Equation (2.3) tells us that in its embryonic form the supertransformation is a Bogoliubov transformation plus a quadratic term.
To get such an explicit expression also for higher we restrict the systems to be considered first by a locality condition. We think in terms of a lattice and assume to be the sum of charges situated at the lattice sites:
(3.2) 
Equation (2.1) requires , in addition to .
I. One kind of fermions at each lattice site
The most general is of the form
(3.3) 
Since the third power of a fermion operator at a point vanishes, must be linear in and , and the nilpotence leaves only or (which one does not matter). The phase of the is arbitrary, so we may take , that is . Thus
(3.4) 
so with being a number the time evolution is trivial. The introduced in Section 2 is empty, a vector with would be annihilated by all and . Since they span all of such a vector must be zero. The of Section 2 is and is a collective fermion coordinate. Since it obeys the CARrelations we have although it is a sum of operators with of a norm .
However, the supertransformation is not trivial; also it is not just a tensor product of the unitaries of the baby model since the and in (3.2) at different points anticommute. We find rather
so that
(3.5) 
where we use the notation One readily verifies that this is an automorphism of ,
but it is not a local transformation — depend on all the other ’s and ’s.
II. Two fermions at each lattice site
We think of it as of electrons with spin up and down. Thus is generated by and . So now we afford at each site a product of three operators, of which there are four types:
Each of them is nilpotent. A typical local supercharge is
(3.6) 
and
(3.7) 
A priory, seems to be of 6th power in the ’s but by locality it can be only quartic and reduces it to something quadratic:
(3.8) 
Thus a free time evolution where half of the fermions are quiet is supersymmetric with a local supercharge. In this case the vacuum satisfies
irrespective of the down spins. All these vectors belong to the eigenvalue zero of and span In fact the eigenvalues are at least fold degenerate.
We determine the automorphism of generated by the supertransformation by the same method as before:
leading to
(3.9) 
Similarly,
leading to
(3.10) 
Of course there is an alternative form where is pulled out to the left and with which one verifies that this strange transformation is actually an automorphism group of that mixes spin up and down as well as even and odd; however, the time evolution leaves invariant up to a phase and does not mix between even and odd elements or between elements at different sites.
Remark
Instead of an opposite spin one might use the next neighbour and try
(3.11) 
but this is not nilpotent. To meet this condition more refined constructions are necessary (see, e.g. [1]).
III. Three fermions at each lattice site
In the case of three fermions at each lattice site, we start again with the CARalgebra generated by :
(3.12) 
However even with strictly local the charge in (3.2) creates a nonlocal supertransformation, so we drop locality. Instead we impose translation invariance of the ’s in such a way that becomes translation and even permutation invariant. Furthermore we think of the and as Cooper pairs, and thus consider the subalgebra of generated by and , . Although the ’s commute for different sites they do not form a bona fide Bose field since there is at most one pair per site, . However in the anticommutator is a projection of the centre and therefore in an irreducible representation it equals unity. These are the representations we are interested in and therefore we can think of the ’s as of spin variables:
Thus our algebra is defined by
(3.13) 
The supertransformation, and therefore the dynamics, will be defined by fluctuation variables.
Definition
(3.14) 
Proposition

;

;

.
Remarks

represents a collective Fermi mode and will serve as a Clifford variable. The fact that the number of single fermion modes equals the number of pairs is not essential;

is a collective Bose mode and, in a representation based on the “vacuum” (all spins down), it assumes for the properties of in quantum mechanics and we will arrive at Witten’s supersymmetric quantum mechanics [2];

By anticommutativity the are so correlated that . On the contrary, since we can think of as of , with , .
In agreement with our desideratum , , we take and thus obtain
(3.15) 
Remarks

In the BCS model (in the degenerated case) and we see that it differs from only by . Thus the energies per particle coincide for ;

In the situation, where with Pauli matrices for , we have .
Next we shall briefly comment on the ground state and the ceiling state of For their discussion it has to be kept in mind that, for , stays bounded and only can grow big; is then essentially . The smallest requires as big as . But the lowest wants close to zero and maximal . The Fock vacuum gives if the single fermions are anticorrelated to the pairs, . Then and thus . For even there are more ground states if all pairs are pairwise anticorrelated. By this we mean that there is a permutation , such that and . Then and again , however acts on . In this case
For the ground state of we want , . For even this is possible by applying times onto . For big this becomes awkward and here it is expedient to make a Bogoliubov transformation. However the standard form
does not leave the pair algebra invariant and we have to use the transformation (2.1) with , , i.e.
such that becomes
If denotes the new vacuum we note We conclude
(3.16) 
4 The limit
In the limit , new features appear. We have to distinguish between local, mesoscopic and macroscopic observables: typically Which limits exist and how they behave under the time evolution (TE) and the supertransformation (ST) will depend very much on the state.
For the limiting procedure we impose only minimal requirements. We assume that a state for arbitrary is given and we check whether the expectation values converge. In addition we demand that these limits can be interpreted as the expectation values of a limiting algebra. We want the latter to be as big as necessary for the mesoscopic observables to still reflect some quantum features.
Let us first turn to the global unscaled quantities
(4.1) 
In terms of these operators together with from (3.14) in the model of Section 3:
(4.2) 
is independent of the gauge transformation . For the time evolution (with ) this leads to
(4.3) 
whereas for the supertransformation () we obtain
(4.4) 
For evidently . For the time evolution we can use the fact that is a norm bounded sequence. Therefore if the time derivatives will have weak accumulation points. To be able to construct a corresponding automorphism group, however, we need strong convergence that will only hold in favourable representations. Especially will be in the centre of the representation, and supersymmetry becomes trivial in local and global operators and .
The ground state of
The ground state is given as the expectation value with the “vacuum vector” with all spins down: , . This means for quasilocal operators:
(4.5) 
The expectation value factorizes and is the same for all . Following [3] we can extend the set of observables and introduce the fluctuation operators
(4.6) 
Hence
where the vector is now the ground state of a harmonic oscillator over a Weyl algebra generated canonically by and . If in addition to we have a local polynomial , then for fixed and the expectation value factorizes since the interference between the two factors vanishes as .
Furthermore
so the Weyl relations hold in the limit , which allows us to call
Thus maps our operators into the factorizable, hence commuting product such that , . The evolution equations (4.3),(4.4) become (with ):
(4.7) 
(4.8) 
They are indeed implied by the limiting generators
(4.9)  
(4.10) 
Notice, however, that the evolution of the global operators is not the limit of the (norm) evolution of the local ones [4]. The local ’s remain constant, but the mesoscopic and move.
Remark
In the matrix representation
the ground state of , , is given by the vector
It is thus the same for all , so there is no symmetry breaking. From the well known relation for the eigenvectors of a total spin and , expanding around ,
(4.11) 
it follows that for all spins in the direction the macroscopic and the mesoscopic converge.
The ceiling state of
Next we want to examine the ground state of , which is essentially . Thus should be as big as possible and . Since for each eigenvalue of there is an eigenvector to with eigenvalue , we have to find a maximal eigenvector to . This vector will depend on but all are equally suitable for . Therefore the ground state of is degenerate.
To get the limiting state we have to find a sequence such that
(4.12) 
With the notation
we can write the operators and in matrix form as follows:
(4.13) 
An eigenvector has to satisfy
(4.14) 
Evidently, we can choose , and all lead to
With an appropriate symmetrization we need
Then
Thus for finite we have
This form of does not lend itself easily to , but we can write as an integral over vectors with all spins pointing in the direction.
Proposition
(4.15) 
Proof
The integrand is periodic with period ; thus, changing to does not alter the integral. Hence, and . Now
Coherent superpositions of eigenvectors of the supersymmetry operator lead to the same state in the quasilocal algebra. This state is an integral over product states. The corresponding von Neumann algebra in the GNS representation has a nontrivial centre. The elements of the centre correspond to the orientation in the – plane of the pure product states.
Note that although we started with a coherent superposition the limiting state over the local algebra appears to be a mixed state: