Numerical Renormalization Group Calculations for Impurity ...

2.2. Quantum phase transitions and quantum critical points 7
with which the Hamiltonian in Eq. (2.18) is written in a diagonal form of
H I (g) = ∑ k
ε k (g)(γ † k γ k − 1/2). (2.20)
The diagonal form in Eq. (2.20) is obtained using the two consequent transformations
of the Jordan-Wigner transformation 3 and the Bogolioubov transformation.
4
The ground state, |0〉, of H I (g) has no γ fermions and therefore satisfies
γ k |0〉 = 0 for all k. The excited states are created by occupying the singleparticle
states; they can clearly be classified by the total number of occupied
states and a n-particle state has the form γ † k 1
γ † k 2
...γ † k n
|0〉, with all the k i distinct.
The energy gap between the ground state and the first excited one occurs at
k = 0 and equals
∆(g) = 2J(1 − g). (2.21)
Therefore the model H I (g) exhibits a quantum phase transition at the critical
coupling g = 1, which separates an ordered state with Z 2 symmetry broken (g ≪
1) from a quantum paramagnetic state where the symmetry remains unbroken
(g ≫ 1). The state at g = 1 is critical and there is a universal continuum quantum
field theory that describes the critical properties in its vicinity.
We shall now obtain the critical theory for the model in Eq. (2.18). We define
the continuum Fermi field
Ψ(x) = 1 √ a
c i , (2.22)
that satisfies
{Ψ(x), Ψ † (x ′ )} = δ(x − x ′ ). (2.23)
To express H I (g) in terms of Ψ and the expansions in spatial gradients yields the
continuum H F ,
∫
H F = E 0 +
[ ]
c
dx
2 (Ψ†∂Ψ† ∂x − Ψ∂Ψ ∂x ) + ∆Ψ† Ψ + ..., (2.24)
where the ellipses represent terms with higher gradients, and E 0 is an uninteresting
additive constant. The coupling constant in H F are
∆ = 2J(1 − g), c = 2Ja. (2.25)
Notice that at the critical point g = 1, we have ∆ = 0, and we have ∆ > 0 in the
magnetically ordered phase and ∆ < 0 in the quantum paramagnet.
3 To map the Hamiltonian H I (g) with spin-1/2 degrees of freedom into a quadratic ones with
the spinless Fermi operators
4 To transform the quadratic Hamiltonian into a form whose number is conserved.