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arXiv:2007.06316 · Communications in Mathematical Physics

Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field

Hajo Leschke, Alexander V. Sobolev, Wolfgang Spitzer

Published Dec 21, 2020• 9 citations
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Abstract

Abstract We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane $${{\mathbb {R}}}^2$$ R 2 perpendicular to an external constant magnetic field of strength $$B>0$$ 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 $$\mu \ge B$$ μ ≥ B (in suitable physical units). For this (pure) state we define its local entropy $$S(\Lambda )$$ S ( Λ ) associated with a bounded (sub)region $$\Lambda \subset {{\mathbb {R}}}^2$$ Λ ⊂ R 2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $$\Lambda $$ Λ of finite area $$|\Lambda |$$ | Λ | . In this setting we prove that the leading asymptotic growth of $$S(L\Lambda )$$ S ( L Λ ) , as the dimensionless scaling parameter $$L>0$$ L > 0 tends to infinity, has the form $$L\sqrt{B}|\partial \Lambda |$$ L B | ∂ Λ | up to a precisely given (positive multiplicative) coefficient which is independent of $$\Lambda $$ Λ and dependent on B and $$\mu $$ μ only through the integer part of $$(\mu /B-1)/2$$ ( μ / B - 1 ) / 2 . Here we have assumed the boundary curve $$\partial \Lambda $$ ∂ Λ of $$\Lambda $$ Λ to be sufficiently smooth which, in particular, ensures that its arc length $$|\partial \Lambda |$$ | ∂ 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 $$B=0$$ B = 0 , where an additional logarithmic factor $$\ln (L)$$ 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 $$\text{ L}^2({{\mathbb {R}}}^2)$$ 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 $$B=0$$ <m

DOI
10.1007/s00220-020-03907-w

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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
We consider the two-dimensional ideal Fermi gas subject to a magnetic field which is perpendicular to the Euclidean plane R2\mathbb R^2R2 and whose strength B(x)B(x)B(x) at x∈R2x\in\mathbb R^2x∈R2 converges to some B0>0B_0>0B0​>0 as ∥x∥→∞\|x\|\to\infty∥x∥→∞. Furthermore, we allow for an electric potential VεV_\varepsilonVε​ which vanishes at infinity. They define the single-particle Landau Hamiltonian of our Fermi gas (up to gauge fixing). Starting from the ground state of this Fermi gas with chemical potential μ≥B0μ\ge B_0μ≥B0​ we study the asymptotic growth of its bipartite entanglement entropy associated to LΛLΛLΛ as L→∞L\to\inftyL→∞ for some fixed bounded region Λ⊂R2Λ\subset\mathbb R^2Λ⊂R2. We show that its leading order in LLL does not depend on the perturbations Bε:=B0−BB_\varepsilon := B_0 - BBε​:=B0​−B and VεV_\varepsilonVε​ if they satisfy some mild decay assumptions. Our result holds for all ααα-R\' enyi entropies α>1/3α>1/3α>1/3; for α≤1/3α\le 1/3α≤1/3, we have to assume in addition some differentiability of the perturbations BεB_\varepsilonBε​ and VεV_\varepsilonVε​. The case of a constant magnetic field Bε=0B_\varepsilon = 0Bε​=0 and with Vε=0V_\varepsilon= 0Vε​=0 was treated recently for general μμμ by Leschke, Sobolev and Spitzer. Our result thus proves the stability of that area law under the same regularity assumptions on the boundary ∂Λ\partial Λ∂Λ.
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 .
Abstract We consider fermionic ground states of the Landau Hamiltonian, HBH_BHB​ H B , in a constant magnetic field of strength B>0B>0B>0 B > 0 in R2{\mathbb {R}}^2R2 R 2 at some fixed Fermi energy μ>0\mu >0μ>0 μ > 0 , described by the Fermi projection PB:=1(HB≤μ)P_B:=1(H_B\le \mu )PB​:=1(HB​≤μ) P B : = 1 ( H B ≤ μ ) . For some fixed bounded domain Λ⊂R2\Lambda \subset {\mathbb {R}}^2Λ⊂R2 Λ ⊂ R 2 with boundary set ∂Λ\partial \Lambda ∂Λ ∂ Λ and an L>0L>0L>0 L > 0 we restrict these ground states spatially to the scaled domain LΛL \Lambda LΛ L Λ and denote the corresponding localised Fermi projection by PB(LΛ)P_B(L\Lambda )PB​(LΛ) P B ( L Λ ) . Then we study the scaling of the Hilbert-space trace, trf(PB(LΛ))\textrm{tr} f(P_B(L\Lambda ))trf(PB​(LΛ)) tr f ( P B ( L Λ ) ) , for polynomials f with f(0)=f(1)=0f(0)=f(1)=0f(0)=f(1)=0 f ( 0 ) = f ( 1 ) = 0 of these localised ground states in the joint limit L→∞L\rightarrow \infty L→∞ L → ∞ and B→0B\rightarrow 0B→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 Lln⁡(L)a(f,μ)∣∂Λ∣L \ln (L) a(f,\mu ) |\partial \Lambda |Lln(L)a(f,μ)∣∂Λ∣ L ln ( L ) a ( f , μ ) | ∂ Λ | as L→∞L\rightarrow \infty L→∞ L → ∞ for the (two-dimensional) Laplacian with Fermi projection 1(H0≤μ)1(H_0\le \mu )1(H0​≤μ) 1 ( H 0 ≤ μ ) . On the other hand, if L tend
The leading asymptotic large-scale behaviour of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensional Euclidean space at zero absolute temperature, T = 0, is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at Tgt0T\\gt 0Tgt0. The leading large-scale term of this thermal EE turns out to be twice the first-order finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the interface of the bipartition. However, it is given by a rather complicated integral derived from a semiclassical trace formula for a certain operator on the underlying one-particle Hilbert space. But in the zero-temperature limit Tdownarrow0T\\downarrow 0Tdownarrow0, the leading large-scale term of the thermal EE considerably simplifies and displays a mathrmln(1/T)\\mathrm{ln}(1/T)mathrmln(1/T)-singularity which one may identify with the known logarithmic enhancement at T = 0 of the so-called area-law scaling.
Abstract The leading asymptotic large-scale behaviour of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensional Euclidean space at zero absolute temperature, T = 0, is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T > 0 . The leading large-scale term of this thermal EE turns out to be twice the first-order finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the interface of the bipartition. However, it is given by a rather complicated integral derived from a semiclassical trace formula for a certain operator on the underlying one-particle Hilbert space. But in the zero-temperature limit T ↓ 0 , the leading large-scale term of the thermal EE considerably simplifies and displays a ln ( 1 / T ) -singularity which one may identify with the known logarithmic enhancement at T = 0 of the so-called area-law scaling.
The leading asymptotic large-scale behavior of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensionsal Euclidean space at zero absolute temperature, T=0, is by now well understood. Here, we announce and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T>0. The leading large-scale term of this thermal EE turns out to be twice the leading finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the boundary surface of the bipartition. However, it is given by a rather complicated analytical expression derived from semiclassical functional calculus and differs, at least at high temperature, from simpler expressions previously obtained by arguments based on conformal field theory. In the zero-temperature limit, the leading large-scale term of the thermal EE considerably simplifies and displays a ln(1/T)-singularity which one may identify with the known logarithmic correction at T=0 to the so-called area-law scaling. Our results extend to the whole one-parameter family of (quantum) R\'enyi entropies.