On Entanglement Entropy for magnetic Hamiltonians

This doctoral thesis concerns the entanglement entropy of ideal Fermi gases subject to some simple magnetic Hamiltonians in the asymptotic for large domains.

To mathematically formalize this, we first introduce a Hamiltonian operator H acting on a dense subset of L²(ℝ^d) for d≤3. It describes the kinetic energy of an electron and all external forces acting on our ideal Fermi gas. This Hamiltonian fully describes the behaviour of our idealised Fermions, as they do not have pairwise interactions. We mostly work with the Landau Hamiltonian in two or three spatial dimensions, which stems from a constant magnetic field of strength B>0. In two dimensions, it is given by

H_B≔(-i∂₁-x₂)²+(-i∂₂+x₁)²,

while in three dimensions, it is given by

H_B≔(-i∂₁-x₂)²+(-i∂₂+x₁)²-∂₃² .

In both cases xj are the euclidean spatial coordinates and ∂j are the partial differentials with respect to these coordinates.

Furthermore, we introduce a domain Λ⊂ℝ^d, which is an open set such that the boundary ∂ Λ satisfies some regularity assumptions, which vary throughout this thesis. Let μ>0 be the so-called chemical potential. We work with the ground state projector, which agrees with the thermal equilibrium at zero temperature. It is written as 1_(≤μ)(H) and is the spectral projector associated to the self-adjoint operator H and the part of the spectrum which lies inside the interval (-∞,μ]. The final piece of notation we need is the spatial projector 1_Λ, which is the multiplication operator with the characteristic function of the domain  Λ.

For some test functions h:[0,1]→R vanishing at 0, we study the asymptotic expansion as L→∞ of the expression

tr h(1_(LΛ) 1_(≤μ)(H)1_(LΛ)) ,

where tr refers to the usual Hilbert space trace of the trace class operator inside. When we insert the entropy function h(t)≔-t ln(t)-(1-t)ln(1-t), we arrive at the bipartite (or local) entanglement entropy of the system. This is of some physical interest and the motivation for this study.

Our first main result is the following theorem.

Suppose that  Λ⊂ℝ³ is a piecewise Lipschitz region and let μ>B>0. Let ν≔⌈(μ/B-1)/2⌉ and let h:[0,1]→R be a continuous function, which is β-Hölder continuous at 0 and 1 for some 1≥β>0, and assume that h(0)=h(1)=0. Then, we have the asymptotic expansion

trh(1_(LΛ)1≤μ(HB)1_(LΛ))=L²ln(L)νB/π I(h) ∫_(∂Λ) ​dH²(v) |n(v)⋅e₃|+o(L²ln(L)) .

In particular, as the γ-Rényi entropy function h_γ is β-Hölder continuous for any βγ,1), the γ-Rényi entanglement entropy, S_γ(LΛ), of the ground state at Fermi energy μ localized to LΛ, satisfies the asymptotic expansion

S_γ(LΛ)=L²ln(L)νB (1+ γ)/(24γπ)  ∫_(∂Λ​) dH²(v) |n(v)⋅e₃|+o(L²ln(L))

as L→∞.

Essentially, this theorem shows an enhanced area law for HB, the Landau Hamiltonian in three dimensions. The noteworthy point here is that, in contrast to the free Hamiltonian H = -Δ, the dependence on the domain is not rotationally invariant, but it depends strongly on the direction of the magnetic field, which is oriented in e₃ direction.

The proof of this theorem takes about a quarter of this thesis. The first step is to work with polynomial test functions h and find appropriate arguments to reduce the leading order asymptotic to the known asymptotic of the one-dimensional free case (with the Laplace operator on R as the Hamiltonian) and the two-dimensional Landau operator H_B. Then, we need some p-Schatten quasi-norm estimates to go from polynomials to merely Hölder continuous test functions h.

The remaining part of this thesis is dedicated to the study of the two-dimensional case with the Hamiltonian H_B. For fixed magnetic field strength and chemical potential, this has essentially already been fully understood before the work on this thesis began. This thesis focuses on the joint limit B→0 and L→∞. Here, depending on which one approaches its limit faster, we get different, interesting behaviours. We prove the following conjecture in a variety of circumstances.

 For any Hölder-continuous function f with Hölder exponent strictly bigger than 0, which satisfies f(0)=f(1)=0, any bounded Lipschitz domain  Λ, and any μ>0 we have the asymptotic expansion

tr f (1_(LΛ)1_(H_B≤μ)1_(LΛ))=

• |∂Λ| 2√μ /π I(f)Lln(μL)+o(μLln(μL))      if BL<μ ,
• |∂Λ| 2√μ /π I(f)Lln(μ/B)+o(μLln(μ/B))   if BL≥μ ,

as L→∞ and B→0, where we defined the functional

f↦I(f)≔1/(4π²) ∫₀¹f(t) /(t(1-t)) dt .

The proofs of this conjecture under varied assumptions make up the bulk of the remaining three quarters of this thesis.

For this study, due to scaling invariance, we assume μ≔2. After we tackle the case f(t)=t(1-t), where we only need to assume that  Λ has C²-boundary and can handle the full range of B,L, we continue with the study of polynomial test functions f. Here, we prove the conjecture in the case BL^2.5≪1 using that the projection 1≤μ(HB) converges pointwise to 1≤μ(-Δ), as B tends to zero. Furthermore, we prove it in the case B²L≫1, assuming that ∂ Λ is C²-smooth. Lastly, we prove it under the assumption BL/ln(L)≫1, assuming that  Λ is a polygon.

This part of the proof relies on a very good understanding of the sine kernel asymptotics of the Hermite kernel, which are discussed along the way. It also relies on the previously established results for a fixed constant magnetic field.

After that, we try to get from the polynomial test functions f to the Hölder continuous functions. In the case BL≫1, we recover the same results as for polynomial test functions. This thesis does not study the case BL≪1, as the author has not been able to find a satisfactory approach to this case. The author still considers the case BL≫1 more interesting, as it results in a new formula, while the case BL≪1 yields the same result as B=0, which can thus be regarded as a slight perturbation of the free case H=-Δ. For the purpose of proving this, it is, however, not quite that simple.

To get the required bounds, we estimate some p-Schatten quasi-norms. As the methods the author had been aware of prior to working on this thesis were not applicable, the author utilized a method, which relies on “guessing” an orthonormal basis, which is sufficiently close to the eigenbasis of an operator. We discuss this in a lot of detail and provide two other motivational examples, where this method can be applied. This also leads to a (as far as the author is aware) new such quasi-norm estimate for the Hermite kernel associated to the ground state of the harmonic oscillator.

The doctoral thesis closes with a brief outlook of potential future research which might be inspired by this work.