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arXiv:1409.3269 · Journal of Physics A Mathematical and Theoretical

Entanglement entropy as a witness of the Aharonov–Bohm effect in QFT

Raúl E Arias, David D Blanco, Horacio Casini

Published Mar 18, 2015• 6 citations
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Abstract

We study the dependence of the entanglement entropy with a magnetic flux and show that the former quantity witnesses an Aharonov–Bohm-like effect. In particular, we consider free charged scalar and Dirac fields living on a two-dimensional cylinder and study how the entanglement entropy for a strip-like region on the surface of the cylinder is affected by a magnetic field enclosed by it.

DOI
10.1088/1751-8113/48/14/145401

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arXiv:1604.02110•Physics Letters B•Jun 3, 2016•28 citations
We study the dependence of the entanglement entropy with a magnetic flux and show that the former quantity witnesses an Aharonov–Bohm-like effect. In particular, we consider free charged scalar and Dirac fields living on a two-dimensional cylinder and study how the entanglement entropy for a strip-like region on the surface of the cylinder is affected by a magnetic field enclosed by it.
We consider the Aharonov-Bohm effect on entanglement entropy for one interval in (1+11+11+1) dimensional conformal field theory on a one dimensional ring. The magnetic field is confined inside the ring, i.e., there is a Wilson loop on the ring. The Aharonov-Bohm phase factor which is proportional to the Wilson loop is represented as insertion of twist operators. We compute exactly the R\'enyi entropy from a four point function of twist operators in a free charged scalar field.
The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, ΦΦΦ, due to a (classical) magnetic field, B{\mathbf B}B, which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, R2\mathbb{R}^2R2, of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, R2−{0}\mathbb{R}^2-\{0\}R2−{0}, corresponding to the charge.
The Aharonov-Bohm effect is a fundamental topological phenomenon with a wide range of applications. It consists of a charge encircling a region with a magnetic flux in a superposition of wave packets having their relative phase affected by the flux. In this work we analyze this effect using an entropic measure known as realism, originally introduced as a quantifier of a system's degree of reality and mathematically related to notions of global and local quantum coherence. More precisely, we look for observables that lead to gauge-invariant realism associated with the charge before it completes its loop. We find that the realism of these operators has a sudden change when the line connecting the center of both wave packets crosses the solenoid. Moreover, we consider the case of a quantized magnetic-field source, pointing out similarities and differences between the two cases. Finally, we discuss some consequences of these results.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential A\mathbf{A}A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics A\mathbf{A}A is in some sense more ``fundamental'' than B\mathbf{B}B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant B\mathbf{B}B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the B\mathbf{B}B field in the region inaccessible to the particle.
We study the time-dependent Aharonov–Bohm effect on the noncommutative space. Because there is no net Aharonov–Bohm phase shift in the time-dependent case on the commutative space, therefore, a tiny deviation from zero indicates new physics . Based on the Seiberg–Witten map we obtain the gauge invariant and Lorentz covariant Aharonov–Bohm phase shift in general case on noncommutative space. We find there are two kinds of contribution: momentum-dependent and momentum-independent corrections. For the momentum-dependent correction, there is a cancellation between the magnetic and electric phase shifts, just like the case on the commutative space. However, there is a non-trivial contribution in the momentum-independent correction. This is true for both the time-independent and time-dependent Aharonov–Bohm effects on the noncommutative space. However, for the time-dependent Aharonov–Bohm effect, there is no overwhelming background which exists in the time-independent Aharonov–Bohm effect on both commutative and noncommutative space. Therefore, the time-dependent Aharonov–Bohm can be sensitive to the spatial noncommutativity. The net correction is proportional to the product of the magnetic fluxes through the fundamental area represented by the noncommutative parameter θ , and through the surface enclosed by the trajectory of charged particle. More interestingly, there is an anti-collinear relation between the logarithms of the magnetic field B and the averaged flux Φ / N ( N is the number of fringes shifted). This nontrivial relation can also provide a way to test the spatial noncommutativity. For B Φ / N ∼ 1 , our estimation on the experimental sensitivity shows that it can reach the 10 GeV scale. This sensitivity can be enhanced by using stronger magnetic field strength , larger magnetic flux, as well as higher experimental precision on the phase shift.