This is actually a very tricky question, mathematically. Physicists may think this question to be trivial. But it takes me one hour in a math summer school to explain the notion of gapped Hamiltonian.

To see why it is tricky, let us consider the following statements. Any physical
system have a finite number of degrees of freedom (assuming the universe is finite). Such physical
system is described by a Hamiltonian matrix with a finite dimension.
Any Hamiltonian matrix with a finite dimension has a discrete spectrum.
So all the physical systems (or all the Hamiltonian) are gapped.

Certainly, the above is not what we mean by "gapped Hamiltonian" in physics.
But what does it mean for a Hamiltonian to be gapped?

Since a gapped system may have gapless excitations at boundary, so
to define gapped Hamiltonian, we need to put the Hamiltonian on a space with no boundary. Also, system with certain sizes may contain non-trivial excitations
(such as spin liquid state of spin-1/2 spins on a lattice with an ODD number of sites), so we have to specify that the system have a certain sequence of sizes as we take the thermodynamic limit.

So here is a definition of "gapped Hamiltonian" in physics:
Consider a system on a closed space, if there is a sequence of sizes
of the system $L_i$, $L_i\to\infty$ as $i \to \infty$,
such that the size-$L_i$ system on closed space has the following "gap property", then the system is said to be gapped.
Note that the notion of "gapped Hamiltonian" cannot be even defined for a single Hamiltonian. It is a properties of a sequence of Hamiltonian
in the large size limit.

Here is the definition of the "gap property":
There is a fixed $\Delta$ (ie independent of $L_i$) such that the
size-$L_i$ Hamiltonian has no eigenvalue in an energy window of size $\Delta$.
The number of eigenstates below the energy window does not depend on
$L_i$, the energy splitting of those eigenstates below the energy window
approaches zero as $L_i\to \infty$.

The number eigenstates below the energy window becomes the ground state degeneracy of the gapped system.
This is how the ground state degeneracy of a topological ordered state
is defined.
I wonder, if some one had consider the definition of gapped many-body system very carefully, he/she might discovered the notion on topological order mathematically.

This post imported from StackExchange Physics at 2014-04-04 16:13 (UCT), posted by SE-user Xiao-Gang Wen