Fermions and the Jordan-Wigner transform V: Diagonalization Continued
In today’s post we continue with the train of thought begun in the last post, learning how to find the energy spectrum of any Hamiltonian quadratic in Fermi operators. Although we don’t dwell in this post on the connection to specific physical models, this class of Hamiltonians covers an enormous number of models of relevance in condensed matter physics. In later posts we’ll apply these results (together with the Jordan-Wigner transform) to understand the energy spectrum of some models of interacting spins in one dimension, which are prototypes for an understanding of quantum magnets and many, many other phenomena, including many systems which undergo quantum phase transitions.
Update: Thanks to Steve Mayer for fixing a bug in the way matrices are displayed.
Note: This post is one in a series describing fermi algebras, and a powerful tool known as the Jordan-Wigner transform, which allows one to move back and forth between describing a system as a collection of qubits, and as a collection of fermions. The posts assume familiarity with elementary quantum mechanics, comfort with elementary linear algebra (but not advanced techniques), and a little familiarity with the basic nomenclature of quantum information science (qubits, the Pauli matrices).
The Hamiltonian we diagonalized in the last post can be generalized to any Hamiltonian which is quadratic in Fermi operators, by which we mean it may contain terms of the form and . We will not allow linear terms like . Additive constant terms are easily incorporated, since they simply displace all elements of the spectrum by an amount . There are several ways one can write such a Hamiltonian, but the following form turns out to be especially convenient for our purposes:
The reader should spend a little time convincing themselves that for the class of Hamiltonians we have described, it is always possible to write the Hamiltonian in this form, up to an additive constant , and with hermitian and antisymmetric.
This class of Hamiltonian appears to have first been diagonalized in an appendix to a famous Annals of Physics paper by Lieb, Schultz and Mattis, dating to 1961 (volume 16, pages 407-466), and the procedure we follow is inspired by theirs. We begin by defining operators :
We will try to choose the complex numbers and to ensure that: (1) the operators satisfy Fermionic CCRs; and (2) when expressed in terms of the , has the same form as , and so can be diagonalized.
A calculation shows that the condition is equivalent to the condition
while the condition is equivalent to the condition
These are straightforward enough equations, but their meaning is perhaps a little mysterious. More insight into their structure is obtained by rewriting the connection between the s and the s in an equivalent form using vectors whose individual entries are not numbers, but rather are operators such as and , and using a block matrix with blocks and :
The conditions derived above for the s to satisfy the CCRs are equivalent to the condition that the transformation matrix
is unitary, which is perhaps a somewhat less mysterious condition than the earlier equations involving and . One advantage of this representation is that it makes it easy to find an expression for the in terms of the , simply by inverting this unitary transformation, to obtain:
The next step is to rewrite the Hamiltonian in terms of the operators. To do this, observe that:
It is actually this expression for which motivated the original special form which we chose for . The expression is convenient, for it allows us to easily transform back and forth between expressed in terms of the and in terms of the . We already have an expression in terms of the operators for the column vector containing the and terms. With a little algebra this gives rise to a corresponding expression for the row vector containing the and terms:
This allows us to rewrite the Hamiltonian as
where we have used the shorthand to denote the vector with entries , and
Supposing we can choose such that is diagonal, we see that the Hamiltonian can be expressed in the form of , and the energy spectrum found, following our earlier methods.
Since is hermitian and antisymmetric it follows that also is hermitian, and so can be diagonalized for some choice of unitary . However, the fact that the s must satisfy the CCRs constrains the class of s available to us. We need to show that such a can be used to do the diagonalization.
We will give a heuristic and somewhat incomplete proof that this is possible, before making some brief remarks about what is required for a rigorous proof. I’ve omitted the rigorous proof, since the way I understand it is uses a result from linear algebra that, while beautiful, I don’t want to explain in full detail here.
Suppose is any unitary such that
where is diagonal, and we used the special form of to deduce that the eigenvalues are real and appear in matched pairs . We’d like to show that can be chosen to be of the desired special form. To see that this is plausible, consider the map , where is a block matrix:
Applying this map to both sides of the earlier equation we obtain
But , and so we obtain:
It is at least plausible that we can choose such that , which would imply that has the required form. What this actually shows is, of course, somewhat weaker, namely that commutes with .
One way of obtaining a rigorous proof is to find a satisfying
and then to apply the cosine-sine (or CS) decomposition from linear algebra, which provides a beautiful way of representing block unitary matrices, and which, in this instance, allows us to obtain a of the desired form with just a little more work. The CS decomposition may be found, for example, as Theorem VII.1.6 on page 196 of Bhatia’s book “Matrix Analysis” (Springer-Verlag, New York, 1997).
Problem: Can we extend these results to allow terms in the Hamiltonian which are linear in the Fermi operators?
In this post we’ve seen how to diagonalize a general Fermi quadratic Hamiltonian. We’ve treated this as a purely mathematical problem, although most physicists will probably have little trouble believing that these techniques are useful in a wide range of physical problems. In the next post we’ll explain a surprising connection between these ideas and one-dimensional spin systems: a tool known as the Jordan-Wigner transform can be used to establish an equivalence between a large class of one-dimensional spin systems and the type of Fermi systems we have been considering. This is interesting because included in this class of one-dimensional spin systems are important models such as the transverse Ising model, which serve as a general prototype for quantum magnetism, are a good basis for understanding some naturally occurring physical systems, and which also serve as prototypes for the understanding of quantum phase transitions.