Ph.D. Thesis - Physics

superconductors exhibit the Meissner effect, meaning that in the superconducting state the
bulk material excludes magnetic flux. Type II superconductors, on the other hand, permit
magnetic flux penetration, for a range of magnetic fields, in quantized units of hc/(2ec)
(written in gaussian units, where ec is the positive fundamental charge). The behavior of
Type II superconductors, unlike that of Type I, still lacks a complete microscopic theory.
Well-explained as Type-I superconductors are, it is still not trivial to calculate specific
properties of such a system. Indeed, the Hilbert space has the dimension of the number
of lattice sites (which we call N). Thus calculating the energy spectrum in general is
intractable on a classical computer, with in general 2 N steps being required to diagonalize
the Hamiltonian. This Hamiltonian, which governs the electrons in the superconducting
state, is given by
HBCS =
N
m=1
ǫm
2 (nm + n−m) +
N
m,l=1
Vmlc † mc †
−m clc−l , (3.1)
where ǫm is the onsite energy for a pair with quantum number m, n± = c †
±m c±m is the number
operator for an electron in mode ±m, and the matrix elements Vml = | 〈m, −m|V |l, −l〉 |
can be calculated using methods such as those found in Ref. [Mah00]. In a Cooper pair,
the two electrons have opposite momenta and spins; here, the quantum numbers m signify
the motional and spin states of the electrons: m ↦→ (p, ↑), −m ↦→ (−p, ↓). We shall refer
to these distinct quantum numbers m as modes.
Physically, the onsite energies signify the energy of one pair, while the hopping terms
specify the energy needed to transition from a mode l to a mode m. BCS dynamics is
regulated by a competition between these two energies, similar to the physics of the Hubbard
model. In general, the BCS ground state is a superposition of modes of different m.
Although the number of states m is clearly staggering in a superconductor of macro-
scopic size, superconductivity has also been observed in ultrasmall (O(nm)) superconducting
metallic grains. The number of states in these systems within the Debye cutoff from the
Fermi energy is estimated to be ≈100 [WBL02]. Although the BCS ansatz is expected to
hold in the thermodynamic limit, it would be desirable to test its predictions for these small
systems.
To test the predictions of the BCS Hamiltonian, we require solutions of its energy spec-
trum. The BCS Hamiltonian is a member of a class of pairing Hamiltonians, which are
of interest in both condensed matter and nuclear physics. There are certain methods of
classical simulation that can provide exact solutions for certain instances of pairing Hamil-
tonians. Exact solutions are possible for an integrable subset of general pairing models.
The number of free parameters that define an integrable model, for a given number of
modes N, is 6N+3, whereas the set of free parameters of general Hamiltonians is equal to
2N 2 − N [DRS03]. Therefore, the fraction of models that are integrable approaches zero as
the system size increases. The most successful approximate method is the Density Matrix
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