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arXiv:2307.01699 · Integral Equations and Operator Theory

Logarithmically Enhanced Area-Laws for Fermions in Vanishing Magnetic Fields in Dimension Two

Paul E. Pfeiffer, Wolfgang Spitzer

Published Sep 14, 2024• 1 citation
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Abstract

Abstract We consider fermionic ground states of the Landau Hamiltonian, $$H_B$$ H B , in a constant magnetic field of strength $$B>0$$ B > 0 in $${\mathbb {R}}^2$$ R 2 at some fixed Fermi energy $$\mu >0$$ μ > 0 , described by the Fermi projection $$P_B:=1(H_B\le \mu )$$ P B : = 1 ( H B ≤ μ ) . For some fixed bounded domain $$\Lambda \subset {\mathbb {R}}^2$$ Λ ⊂ R 2 with boundary set $$\partial \Lambda $$ ∂ Λ and an $$L>0$$ L > 0 we restrict these ground states spatially to the scaled domain $$L \Lambda $$ L Λ and denote the corresponding localised Fermi projection by $$P_B(L\Lambda )$$ P B ( L Λ ) . Then we study the scaling of the Hilbert-space trace, $$\textrm{tr} f(P_B(L\Lambda ))$$ tr f ( P B ( L Λ ) ) , for polynomials f with $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 of these localised ground states in the joint limit $$L\rightarrow \infty $$ L → ∞ and $$B\rightarrow 0$$ B → 0 . We obtain to leading order logarithmically enhanced area-laws depending on the size of LB. Roughly speaking, if 1/B tends to infinity faster than L, then we obtain the known enhanced area-law (by the Widom–Sobolev formula) of the form $$L \ln (L) a(f,\mu ) |\partial \Lambda |$$ L ln ( L ) a ( f , μ ) | ∂ Λ | as $$L\rightarrow \infty $$ L → ∞ for the (two-dimensional) Laplacian with Fermi projection $$1(H_0\le \mu )$$ 1 ( H 0 ≤ μ ) . On the other hand, if L tend

DOI
10.1007/s00020-024-02778-3

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Abstract We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane R2{{\mathbb {R}}}^2R2 R 2 perpendicular to an external constant magnetic field of strength B>0B>0B>0 B > 0 . We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential μ≥B\mu \ge Bμ≥B μ ≥ B (in suitable physical units). For this (pure) state we define its local entropy S(Λ)S(\Lambda )S(Λ) S ( Λ ) associated with a bounded (sub)region Λ⊂R2\Lambda \subset {{\mathbb {R}}}^2Λ⊂R2 Λ ⊂ R 2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region Λ\Lambda Λ Λ of finite area ∣Λ∣|\Lambda |∣Λ∣ | Λ | . In this setting we prove that the leading asymptotic growth of S(LΛ)S(L\Lambda )S(LΛ) S ( L Λ ) , as the dimensionless scaling parameter L>0L>0L>0 L > 0 tends to infinity, has the form LB∣∂Λ∣L\sqrt{B}|\partial \Lambda |LB​∣∂Λ∣ L B | ∂ Λ | up to a precisely given (positive multiplicative) coefficient which is independent of Λ\Lambda Λ Λ and dependent on B and μ\mu μ μ only through the integer part of (μ/B−1)/2(\mu /B-1)/2(μ/B−1)/2 ( μ / B - 1 ) / 2 . Here we have assumed the boundary curve ∂Λ\partial \Lambda ∂Λ ∂ Λ of Λ\Lambda Λ Λ to be sufficiently smooth which, in particular, ensures that its arc length ∣∂Λ∣|\partial \Lambda |∣∂Λ∣ | ∂ Λ | is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case B=0B=0B=0 B = 0 , where an additional logarithmic factor ln⁡(L)\ln (L)ln(L) ln ( L ) is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space  L2(R2)\text{ L}^2({{\mathbb {R}}}^2) L2(R2) L 2 ( R 2 ) to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies. As opposed to the case B=0B=0B=0 <m
Abstract We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R3{{\mathbb {R}}}^3R3 R 3 subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L2ln⁡(L)L^2\ln (L)L2ln(L) L 2 ln ( L ) , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region LΛ⊂R3L\Lambda \subset {{\mathbb {R}}}^3LΛ⊂R3 L Λ ⊂ R 3 as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln⁡(L)\ln (L)ln(L) ln ( L ) factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂Λ\partial \Lambda ∂Λ ∂ Λ , is not solely on its area but on the “surface perpendicular to the direction of the magnetic field”. We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L2L^2L2 L 2 of the relevant trace for piecewise C1,α\textsf{C}^{1,\alpha }C1,α C 1 , α smooth surfaces ∂Λ\partial \Lambda ∂Λ ∂ Λ .
We prove a logarithmically enhanced area law for all Rényi entanglement entropies of the ground state of a free gas of relativistic Dirac fermions. Such asymptotics occur in any dimension if the modulus of the Fermi energy is larger than the mass of the particles and in the massless case at Fermi energy zero in one space dimension. In all other cases of mass, Fermi energy and dimension, the entanglement entropy grows no faster than the area of the involved spatial region. The result is established for a general class of test functions which includes the ones corresponding to Rényi entropies and relies on a recently proved extension of the Widom-Sobolev formula to matrix-valued symbols by the authors.
In many-body systems the extent and range of spatial quantum correlations induced by entanglement provide a great deal of information about several qualitative physical properties. One way of studying this information is to examine the scaling behaviour of the ground state entanglement entropy with respect to a scaled version of a distinguished spatial subregion. In various systems the entanglement entropy grows proportionally to the surface area of the subregion which is referred to as an area law. In this thesis we examine the connection between the scaling behaviour of the entanglement entropy and many-body localisation. In recent years it was show that a number of systems, which are known to be in the localised phase, exhibit area laws of the entanglement entropy. It is commonly expected that the entanglement entropies of delocalised ground states do not satisfy area laws, though not many examples of different scaling behaviours have been shown, yet. The aim of this thesis is to provide further examples of violations of area laws in the context of delocalised systems. In three different models we show that the entanglement entropy of the ground states grows at least like a logarithmically enhanced area law. The first part of this thesis, based on joint work with P. Müller and L. Pastur [MPS20], considers the random dimer model. Even though this non-interacting, one-dimensional model is spectrally localised, there exist critical points in its spectrum at which the localisation length diverges. We consider the ground state corresponding to a Fermi energy positioned at one of these critical energies. In the case of small disorder we show a logarithmic lower bound to the expectation of the entanglement entropy. Moreover, we proof a logarithmic lower bound to the finite-volume entanglement en- tropy at these critical points for any disorder strength. In the second part of this thesis, which is based on joint work with P. Müller [MS20], we consider a multi-dimensional continuum Schrödinger operator, which is given by a perturbation of a negative Laplacian by a compactly supported, bounded potential. We establish both an upper and a lower bound to the entanglement entropy corresponding to a positive Fermi energy. These bounds prove that the scaling behaviour of the entanglement entropy is a logarithmically enhanced area law. This is the same scaling behaviour as the one occurring in the case of free fermions, one of the few delocalised systems for which an asymptotic expansion of the entanglement entropy is known. Finally, in the third and last part, based on joint work with C. Fischbacher [FS20], we consider the finite XXZ spin chain with periodic boundary conditions in the Ising phase. We show that for each eigenvalue in the droplet band there exists at least one eigenvector such that the corresponding entanglement entropy grows at least logarithmically, provided the anisotropy parameter ∆ is sufficiently large. In addition, we show a Combes–Thomas estimate for this model, which may be of independent interest.
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