DMRG applied to quantum Hall systems

In this chapter we will use the above-mentioned DMRG algorithm to study a quantum Hall system. One ob- vious problem is that quantum Hall systems are two-dimensional, whereas in the previous subsection DMRG was defined for a one-dimensional system. DMRG is commonly used to study two-dimensional systems by defining such systems in a quasi-one-dimensional setup such as on a strip or cylinder; in this chapter we use a cylinder. In a one-dimensional system the bond dimensionχneeded to accurately approximate the system increased polynomially in system size, and this is why the problem is numerically tractable. In a quasi-1D setup the requiredχ∼eLx_{, whereL}

xis the diameter of the cylinder. Therefore we will only be able to deal with fairly small cylinders. This is still reasonable. The main competing numerical method used to study quantum Hall systems, exact diagonalization, has an exponential cost in both the diameter of the cylinder and its length, so DMRG can study significantly larger sizes.

To put the quantum Hall system on a cylinder, we use the Landau gauge(Ax, Ay) = (By,0). The Hamiltonian in this gauge, in the absence of interactions between electrons, is2

H0= ~2 2mp 2 y+ ~2 2m p2x− eB c y 2 . (5.8)

WhereBis the magnetic field,xis the direction around the circumference of the cylinder andyis the direction along the cylinder. This Hamiltonian has a number of ground states, which are given by the following wave functions: φn(x, y) =e −iknx_e−(y−knℓ2B)2 p 2Lxπ1/2 kn= 2πn Lx . (5.9)

1_{More accurately, we will represent only a few sites and use periodic boundary conditions to get an ‘infinite’ system.}

2_{This is the Hamiltonian for a single species, which will be discussed for most of this section. The generalization to more species is} fairly straightforward[43].

Here ℓB ≡

q

~

eB is the magnetic length, which we will soon set to be1. These wave functions are a combination of a plane wave in thexdirection and the solution to a harmonic oscillator in theydirection. The above equation is for the lowest energy solution to the harmonic oscillator problem, which corresponds to the lowest Landau level. Electrons in higher Landau levels have wave functions corresponding to higher energy eigenstates of the harmonic oscillator problem.

We will perform numerics in the orbital basis, indexed bynand defined by:

c(r) =X

n

φn(x, y)cn, (5.10)

wherecnannihilates an electron in orbitalnandc(r)annihilates an electron at positionr = (x, y). Each orbital carriesx-momentumknand is localized at positionknℓ2B. The occupation for each orbital is either0 or1. Therefore in the orbital basis the quantum Hall problem reduces to a one-dimensional Hubbard model (though with some unusual interactions) and this is a problem that can be solved with DMRG.

We want to find ground states of the above Hamiltonian, including interactions

δH=X(r, r′)V(r−r′)ρ(r)ρ(r′). (5.11)

V(r)will typically be a Coulomb interaction, andρ(r) ≡ c†_{(r)c(r) is the electron density. Naturally the}

eigenstates of the full Hamiltonian may not be spanned by the basisφn, but we restrict ourselves to this basis anyways, this is projection to the lowest Landau level. In the systems considered in this chapter this is a reasonable approximation to make. The energy separation between Landau levels fromH0is proportional to the magnetic fieldB, whileδH ∼ √B. Therefore at large enough fields higher Landau levels can be neglected. It has also been shown that expansions in the Landau level mixing parameterκ(which is essentially

δH/H0) are stable even forκ≈1. Though neglecting higher Landau levels is a reasonable approximation in this work, in other systems such as graphene, where the Landau level spacing is∼√B, or in cases where Landau level mixing breaks a particle-hole symmetry, it cannot be neglected.

Eq. (5.11) can be written in the orbital basis as:

δH= X

n,m,k

Vmkc†n+mc†n+kcn+m+kcn. (5.12)

We see that the Coulomb Hamiltonian induces both hopping between orbitals as well as diagonal terms. Note that all of these hoppings preserve momentum in thex-direction. IfVmk is a Coulomb interaction then it is proportional to_|r−r′_|−1_{. In the orbital basis, the orbitals are at positions2πnℓ}2

B/Lx, i.e., they get closer together as the diameter of the cylinder is increased, and in orbital space the Coulomb interaction gets longer- ranged asLxis increased. The longer-ranged interaction induces more entanglement between the different orbitals, and so a largerχis required to represent the matrix product state.

In the previous subsection we saw how to represent both on-site interactions and exponentially decaying interactions using matrix product operators. In the quantum Hall problem we have a power-law decaying operator. This can be approximated by a linear combination of exponentials, and that is the approach taken in this chapter[42].