It features a lattice with nearest neighbour interactions determined by the alignment or anti-alignment of spin projections along the
axis, as well as an external magnetic field perpendicular to the
axis) which creates an energetic bias for one x-axis spin direction over the other.
An important feature of this setup is that, in a quantum sense, the spin projection along the
This means classical statistical mechanics cannot describe this model, and a quantum treatment is needed.
Specifically, the model has the following quantum Hamiltonian: Here, the subscripts refer to lattice sites, and the sum
is another coupling coefficient that determines the relative strength of the external field compared to the nearest neighbour interaction.
Below the discussion is restricted to the one dimensional case where each lattice site is a two-dimensional complex Hilbert space (i.e., it represents a spin 1/2 particle).
symmetry group, as it is invariant under the unitary operation of flipping all of the spins in the
The model can be exactly solved for all coupling constants.
It is more convenient to write the solution explicitly in terms of fermionic variables defined by Jordan-Wigner transformation, in which case the excited states have a simple quasiparticle or quasihole description.
In this phase the ground state breaks the spin-flip symmetry.
The ground state preserves the spin-flip symmetry, and is nondegenerate.
, the system has gapless excitations and its low-energy behaviour is described by the two-dimensional Ising conformal field theory.
, and is the simplest of the unitary minimal models with central charge less than 1.
Besides the identity operator, the theory has two primary fields, one with conformal weights
Then the transverse field Ising Hamiltonian (assuming an infinite chain and ignoring boundary effects) can be expressed entirely as a sum of local quadratic terms containing Creation and annihilation operators.
This Hamiltonian fails to conserve total fermion number and does not have the associated
That is, the Hamiltonian commutes with the quantum operator that indicates whether the total number of fermions is even or odd, and this parity does not change under time evolution of the system.
The Hamiltonian is mathematically identical to that of a superconductor in the mean field Bogoliubov-de Gennes formalism and can be completely understood in the same standard way.
The exact excitation spectrum and eigenvalues can be determined by Fourier transforming into momentum space and diagonalising the Hamiltonian.
, the Hamiltonian takes on an even simpler form (up to an additive constant):
A nonlocal mapping of Pauli matrices known as the Kramers–Wannier duality transformation can be done as follows:[4]
Then, in terms of the newly defined Pauli matrices with tildes, which obey the same algebraic relations as the original Pauli matrices, the Hamiltonian is simply
In terms of the Majorana fermions mentioned above, this duality is more obviously manifested in the trivial relabeling
Note that there are some subtle considerations at the boundaries of the Ising chain; as a result of these, the degeneracy and
symmetry properties of the ordered and disordered phases are changed under the Kramers-Wannier duality.
The transverse field Ising model represents the case where
The quantum transverse field Ising model in
dimensions is dual to an anisotropic classical Ising model in