Comment on “Aharonov-Casher and ScalarAharonov-Bohm Topological Effects”

In a recent Letter [1], Dulat and Ma (DM) derive (i) therelativistic Hamiltonian of a fixed spin from the spin-stateprojected Lagrangian Lþ with the Uð1Þmm gauge structurefor a neutral spin half particle with an anomalous magneticmoment μ, discuss (ii) the conditions for a topologicalAharonov-Casher (AC) and scalar Aharonov-Bohm (SAB)effects, and then make (iii) the conclusion that the argu-ments of Peshkin and Lipkin (PL) [2], which state that theAC and SAB effects are not topological, are incorrect, bystating that “they (PL) are based on the wrong Hamiltonianwhich yields their incorrect conclusion.” based on theobservation that there is no Uð1Þmm gauge structure inPL’s Hamiltonian. In this Comment we point out (i) that theHamiltonian, Eq. (17), is not physical, (ii) then that theconditions are irrelevant for topological AC and SABeffects, and (iii) conclusively that the nonrelativisticHamiltonian employed by PL [2] has the same Uð1Þmmgauge structure for fixed spins and then is not wrong, buttheir incorrect interpretation of the spin autocorrelations ledto the incorrect conclusion.In Ref. [1], in order to obtain the Uð1Þmm gauge

symmetry in the Lagrangian L, DM use the spin projectionoperator such that the polarization direction of the neutralparticle does not vary. However, the spin-state projectedLagrangian Lþ cannot preserve the relativistic invariancebecause the spin should undergo a Wigner rotation, whichis not covariant under a general Lorentz transformation. Infact, one can prove that the equality in Eq. (4) does not holdfor sμ ¼ ð0; 0; 0; 1Þ in the particle rest frame. This impliesthat a fixed spin is possible only in a certain referenceframe. Consequently, the Hamiltonian, Eq. (17), derivedfrom the Lþ and its nonrelativistic approximatedHamiltonian, Eq. (18), are not physical. Thus, the con-ditions, Eqs. (23)–(25), in order to recover the original ACand SAB Hamiltonian in Eq. (18) are not physicallymeaningful for the necessary conditions of the AC andSAB setups.The nonrelativistic Hamiltonian of PL [2] including both

the AC and SAB effects can be written as

H ¼ 1

2mðp − μσ ×EÞ2 − μσ · B (1)

for a neutral particle in electric and magnetic fields E andB. This Hamiltonian has the SUð2Þspin gauge symmetry.However, for fixed spins, Eq. (1) has the same Uð1Þmmgauge structure with A ¼ s ×E and A0 ¼ s ·B asintroduced in Ref. [1], where new electric and magneticfields can be defined as E ¼ −∇A0 and B ¼ ∇ ×A,respectively. Then, A and A0 can induce the topologicalAC and SAB effects for fixed spins, respectively. To

generate topological AC and SAB phases, as is known,the prerequisite conditions of being force-free and torque-free are required for the AC and SAB setups. For a givenH, the force ðdp=dtÞ ¼ ½p;H� and the spin torqueðdσ=dtÞ ¼ ½σ;H� on the neutral particle should be zerofor the topological AC and SAB setups. For the SABsetup, as an example, the force and the spin torque aregiven by ðdp=dtÞ ¼ iℏμ∇ðσ ·BÞ and ðdσ=dtÞ ¼ μσ × B(introduced as Eq. (8) in Ref. [2] and Eq. (27) inRef. [1]), respectively. Let us consider that the spin isfixed in the j þ zi state and þz is the direction of themagnetic field B ¼ Bzz. Then, the force becomes zero,∇ðs ·BÞ ¼ 0 ¼ E. The spin torque also becomes zero, asshown as follows: σx and σy are actually the spin flipoperators so that the expectation value of the spin torque,hþzðtÞjσx=yBzj þ zðtÞi, should be zero for fixed spins.Thus, no Larmor-type precession occurs, which cannotgive the frequency ω ¼ 2μB=ℏ in the spin correlations ofEq. (11) in Ref. [2]. Resultantly, PL have missed thetrivial solution of the spin correlation equations, CðtÞ ¼ 0and SðtÞ ¼ 0 in Eq. (12) in Ref. [2]. Indeed, DM havealso noticed [1] that the effects of the quantum fluctua-tions in Eq. (12) in Ref. [2] can exist only at a very shorttime, Δt ≈ 0. Consequently, the SAB and AC effects canbe topological for fixed spins. The reason why PL havemade the incorrect conclusion in Ref. [2] is not becauseof using a wrong Hamiltonian as DM have claimed inRef. [1], but because of their incorrect interpretation ofthe spin torque equation.

We acknowledge support from the National ResearchFoundation of Korea Grant funded by the Korean Gov-ernment(2012-0003786) (T. C.) and the National NaturalScience Foundation of China under the GrantNo. 11374379 (S. Y. C.).

T. Choi1,* and S. Y. Cho2,†1Division of General EducationSeoul Women’s UniversitySeoul 139-774, Korea2Center for Modern Physics and Department of PhysicsChongqing UniversityChongqing 400030, China

Received 20 November 2013; published 14 April 2014DOI: 10.1103/PhysRevLett.112.158901PACS numbers: 03.65.Vf, 11.15.-q, 11.30.-j,

*tschoi@swu.ac.kr†sycho@cqu.edu.cn

[1] S. Dulat and K. Ma, Phys. Rev. Lett. 108, 070405(2012).

[2] M. Peshkin and H. J. Lipkin, Phys. Rev. Lett. 74, 2847(1995).

PRL 112, 158901 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

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