Dynamics of vortices in chiral media: the chiral propulsion effect
DDynamics of vortices in chiral media: the chiral propulsion effect
Yuji Hirono, ∗ Dmitri E. Kharzeev,
2, 3 and Andrey V. Sadofyev Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA Department of Physics and RIKEN-BNL Research Center,Brookhaven National Laboratory, Upton, New York 11973-5000, USA Theoretical Division, MS B283, Los Alamos National Laboratory, Los Alamos, NM 87545 (Dated: March 2, 2018)We study the motion of vortex filaments in chiral media, and find a semi-classical analog of theanomaly-induced chiral magnetic effect. The helical solitonic excitations on vortices in a parity-breaking medium are found to carry an additional energy flow along the vortex in the directiondictated by the sign of chirality imbalance; we call this new transport phenomenon the ChiralPropulsion Effect (CPE). The dynamics of the filament is described by a modified version of thelocalized induction equation in the parity-breaking background. We analyze the linear stabilityof simple vortex configurations, and study the effects of chiral media on the excitation spectrumand the growth rate of the unstable modes. It is also shown that, if the equation of motion ofthe filament is symmetric under the simultaneous reversal of parity and time, the resulting planarsolution cannot transport energy.
PACS numbers:
INTRODUCTION
The physics of chiral media has attracted a significant attention recently. Remarkably, it appears that the quantumchiral anomaly [1, 2] significantly affects the macroscopic behavior of chiral media and induces new transport phe-nomena, such as the Chiral Magnetic [3–7] and Chiral Vortical Effects [8–12] (CME and CVE, respectively). CMEand CVE refer to the generation of electric currents along an external magnetic field or vorticity in the presence ofa chirality imbalance. The resulting currents are non-dissipative due to the protection by the global topology of thegauge field. These chiral effects are expected to occur in a variety of systems: the quark-gluon plasma, Dirac andWeyl semimetals, primordial electroweak plasma, and cold atoms. In quark-gluon plasma, the chirality imbalance canbe produced by topological fluctuations of QCD, or by the combination of electric and magnetic fields that accompanyheavy-ion collisions. The parallel electric and magnetic fields can also be used to create the chirality imbalance incondensed matter systems, see e.g. [13]. In addition, CME and CVE lead to a new class of instabilities in thesesystems [14–20].The CME has been observed experimentally in Dirac [13, 21, 22] and Weyl semimetals [23–25]. There is an ongoingsearch for CME and the local parity violation [3, 4] induced by the topological fluctuations in the quark-gluon plasmain heavy-ion collisions at RHIC and LHC, see Ref. [6] for a review. In particular, the forthcoming isobar run in theSpring of 2018 at RHIC is expected to provide a conclusive result on the occurrence of CME in heavy-ion collisions[26].Recently, the STAR collaboration reported the experimental observation of Λ hyperon polarization along the normalto the reaction plane of the heavy-ion collision, pointing towards the existence of large vorticity in the produced quark-gluon fluid [27]. The role of vortical flows in heavy-ion collisions has been discussed e.g. in Refs. [28–39]. It is naturalto ask how the dynamics of vortices is influenced by the chiral anomaly.In this paper, we consider the dynamics of vortices in a fluid with broken parity; the electromagnetic fields aretreated as fully dynamical. We find a new chiral transport effect – an additional, asymmetric energy flow along thevortex filament in the direction determined by the sign of chirality imbalance, the Chiral Propulsion Effect (CPE). a r X i v : . [ h e p - t h ] J a n EXCITATIONS ON VORTICES IN CHIRALLY IMBALANCED MEDIA
Consider the motion of a vortex filament in a fluid. It can be described by the localized induction equation (LIE) ,˙ X = C X (cid:48) × X (cid:48)(cid:48) , (1)where X = X ( t, s ) denotes the position of a vortex, t is the time, s is the arc-length parameter, the dot and theprime indicate the derivatives with respect to t and s respectively, and C is a parameter dependent on the propertiesof the fluid. Interestingly, the LIE (1) can be mapped to the non-linear Schr¨odinger equation (NLSE) by the so-calledHasimoto transformation [45], ψ ( t, s ) = κ ( t, s ) exp (cid:20) i (cid:90) s τ ( t, s (cid:48) ) ds (cid:48) (cid:21) , (2)where κ ( t, s ) is the curvature and τ ( t, s ) is the torsion of a vortex. NLSE is known to be a completely integrable systemwhich has solitonic solutions and an infinite sequence of commuting conserved charges. The Hasimoto transformationhas been shown to be a Poisson map that preserves the Poisson structures [46]. The LIE thus describes a completelyintegrable system. The LIE possesses solutions that represent helical excitations propagating along the vortex; theyare known as Hasimoto solitons.Let us now consider a system in which parity is broken by the presence of magnetic helicity; the correspondingterm in the action is S χ = (cid:90) dt µ H , (3)where µ is the “chiral” chemical potential , H is the magnetic helicity given by H = e π (cid:82) d x A · B , where A isthe vector potential and B is the magnetic field. It is worth mentioning that taking the derivative of this actionwith respect to the vector potential, one readily finds the CME current: J CME = δS χ /δ A ∝ B . Supplementingthe non-relativistic Abelian Higgs model with the term given by Eq. (3), one can find the equation of motion for aquantized magnetic vortex at finite µ , as derived by Kozhevnikov [48, 49] :˙ X = C X (cid:48) × X (cid:48)(cid:48) + µ (cid:20) X (cid:48)(cid:48)(cid:48) + 32 ( X (cid:48)(cid:48) ) X (cid:48) (cid:21) , (4)where a tangential term µ ( X (cid:48)(cid:48) ) X (cid:48) is added to keep the arc-length-preserving property . Note that the tangentialmotion does not change the shape of the vortex. Hereafter, we set the constant C in Eq. (4) to unity by a correspondingtime rescaling. Let us note that Eq.(4) has previously emerged in a different context: it describes the motion of avortex tube containing an axial flow, and is known as the Fukumoto-Miyazaki equation (FME) [51, 52]. Remarkably,through the Hasimoto transformation, the FME can be mapped to the integrable Hirota equation [53], i ˙ ψ + ψ (cid:48)(cid:48) + 12 | ψ | ψ − iµ (cid:18) ψ (cid:48)(cid:48)(cid:48) + 32 | ψ | ψ (cid:48) (cid:19) = 0; (5)this map can be utilized to obtain the solitons of the FME.In this paper we are interested in the behavior of chiral solutions. We can find a simple explicit solution of theFME (4) having the form of a helix, X helix ( t, s ) = 1 A κ cos[ A ( s − v p t )] κ sin[ A ( s − v p t )] τ A ( s − v g t ) , (6) Although originally the LIE has been introduced for thin vortices in classical fluids [40, 41], it can also describe the dynamics of quantizedvortices in superfluids and superconductors [42–44]. Usually it is denoted by µ , but as this will be the only chemical potential that will appear in this paper, we simplify this notation to µ . Please note that the chiral chemical potential does not correspond to a conserved quantity, as the chiral charge is not conserved dueto the chiral anomaly. The state with µ (cid:54) = 0 therefore does not correspond to a true ground state of the system; for discussion, see e.g.[47]. The term X (cid:48)(cid:48)(cid:48) can also be derived in a fluid-dynamical system using the kinetic helicity as the Hamiltonian [50]. where the constants κ and τ give the curvature and the torsion of the helix, A = (cid:112) κ + τ , and the phase and groupvelocities are given by v p = τ + µ ( τ − κ ) , v g = − κ τ − κ µ. Note that the sign of τ determines the handedness ofthe helix. The radius R and the pitch (cid:96) of the helix are given by R = κ / ( κ + τ ) , (cid:96) = 2 πτ / ( κ + τ ). The solutionis reduced to a circular loop in the limit τ = 0 in Eq. (6).Using the map between the FME and the Hirota equation, we find a propagating solitonic solution of the FME, X sol ( t, s ) = − (cid:15)(cid:15) + τ sech[ (cid:15)ξ ] cos [ η ] − (cid:15)(cid:15) + τ sech[ (cid:15)ξ ] sin [ η ] s − (cid:15)(cid:15) + τ tanh[ (cid:15)ξ ] . (7)where η ≡ τ s + ( (cid:15) − τ ) t + µτ (3 (cid:15) − τ ) , ξ ≡ s − (2 τ + µ (cid:0) τ − (cid:15) (cid:1) ) t, and (cid:15) and τ are constants. This solitonhas a constant torsion given by τ and propagates in the z direction. Its speed is modified by µ and reduces to theoriginal Hasimoto soliton at µ = 0.Let us discuss the kinetic properties of these solutions. The kinetic energy of a soliton can be found as E = (cid:82) ds κ = 4 (cid:15). In addition, the helical nature of the configuration [54] can be characterized by the quantity H = (cid:82) ds κ τ = 8 (cid:15)τ . This is the second conserved quantity in the NLS hierarchy [46, 54]. In the case of a planarconfiguration, namely if the torsion is vanishing, H = 0. Note that these quantities do not depend on µ .Let us now turn to the momentum carried by these solutions. In the thin vortex limit, the electromagnetic fieldscan be expressed in terms of the vortex coordinates X ( t, s ) as B ( t, x ) = ϕ (cid:90) ds X (cid:48) δ ( x − X ) , (8) E ( t, x ) = − ϕ (cid:90) ds ˙ X × X (cid:48) δ ( x − X ) , (9)where ϕ is the magnetic flux and the electric field locally has the structure “ v × B .” The momentum of the magneticflux is given by the Poynting vector, P = (cid:90) d x E × B = ϕ M (cid:90) ds ˙ X , (10)where M is the inverse of the core size of the vortex. The helix solution moves in the z -direction. The z -componentof momentum per unit length of the coil is evaluated as( ¯ P helix ) z = κ τ (cid:18) µτ (cid:19) ϕ M . (11)The z component of the momentum of the soliton solution can be calculated using Eq. (7):( P sol ) z = 4 (cid:15) (2 τ + µ (3 τ − (cid:15) )) (cid:15) + τ ϕ M . (12)In both cases, there are contributions proportional to µ . Therefore the chiral medium provides a thrust to the solitons,propelling them along the vortex - we will call this the Chiral Propulsion Effect (CPE).In the case of the solitons (12), at µ = 0 the velocity is proportional to the torsion τ – this means that for thewave to have a finite momentum in a chirally symmetric medium, the vortex has to deform in a parity-breaking way.On the other hand, even if the solution is planar, due to the circulation in the vortex it can still experience the thrustif parity is broken in the medium. Indeed, Eq (12) shows that for µ (cid:54) = 0 the thrust remains even in the τ → H = 0. As we will discuss later, a planar solution is forbidden to have a finiteenergy flow in a PT symmetric theory. The LIE has the PT symmetry, while in the FME case it is broken. PROPERTIES OF FLUCTUATIONS
Let us now examine the effect of the chiral medium on the fluctuations around the circle and helix solutions. We usethe local coordinate system called the Frenet–Serret (FS) frame, which is commonly used to parametrize the shapeof a curve. There is an ambiguity in the parametrization in s , and we fix this by requiring | X (cid:48) | = 1. Then, the unittangent vector is written as t = X (cid:48) . Given a curvature κ ( t, s ) and a torsion τ ( t, s ), the shape of a curve is determined,up to a trivial translation and rotation, by the FS formulas, ∂ s tnb = κ − κ τ − τ tnb , (13)where n ∝ t (cid:48) is the unit normal vector, and b ≡ t × n is the unit binomal vector. The time evolution of a curve isdescribed by ∂ t tnb = α β − α γ − β − γ tnb , (14)where α, β, γ are functions of κ and τ and their functional forms are determined from Eq. (4). The FS basis has tosatisfy the compatibility conditions, ∂ s ∂ t t = ∂ t ∂ s t , ∂ s ∂ t n = ∂ t ∂ s n , ∂ s ∂ t b = ∂ t ∂ s b . Using these conditions, we find(see the Supplementary Material) the time-evolution equations for τ and κ :˙ κ = − τ κ (cid:48) − κτ (cid:48) + µ (cid:0) κ (cid:48)(cid:48)(cid:48) − τ κ (cid:48) + 3 κ κ (cid:48) − κτ τ (cid:48) (cid:1) , (15)˙ τ = κ (cid:48)(cid:48)(cid:48) κ − κ τ τ (cid:48) + κ κ (cid:48) − κ (cid:48) κ (cid:48)(cid:48) κ + µ κ (cid:16) κ τ (cid:48) + 6 κ (cid:48)(cid:48)(cid:48) κτ + 2 κ τ (cid:48)(cid:48)(cid:48) + 12 κκ (cid:48)(cid:48) τ (cid:48) − κ τ τ (cid:48) + 6 κκ (cid:48) τ (cid:48)(cid:48) − κ (cid:48) ) τ (cid:48) + 6 κ τ κ (cid:48) − τ κ (cid:48) κ (cid:48)(cid:48) (cid:17) . (16)If we take µ = 0 in Eqs. (15) and (16) the Da Rios equations are reproduced [55].We consider linear fluctuations, δκ and δτ , around constant κ and τ . By taking δκ, δτ ∝ e − iωt + ips , the followingdispersion relation is obtained from Eqs. (15) and (16), ω = 2 pτ + µp (cid:18) p − κ + 3 τ (cid:19) ± (cid:112) p ( p − κ )(1 + 3 µτ ) . (17)Equation (17) compactly encodes the information of the fluctuations around three different configurations: a circle,a helix and a straight line. Let us first discuss a circle, in which case the torsion is zero. The periodicity of a circlerequires p = nκ with an integer n , then the frequency ω simplifies to ω = ± κ (cid:112) n ( n −
1) + µκ n (cid:18) n − (cid:19) . (18)If we take µ = 0, Eq. (18) coincides with the result of previous studies [56–59]. The mode with n = 0 corresponds tothe change of radius, n = ± µ = 0, these modes are the zero modes of the soliton. At µ (cid:54) = 0, because of the chirality imbalance, n = ± n ↔ − n is also lifted. The frequency is alwaysreal, which means that a circle is stable.Let us now consider the helix-shape solution. The lowest value of p is determined by the length L of a helix as2 π/L . At µ = 0, the imaginary part appears if p < κ , which means that these long-wavelength modes are unstable.Since the factor (1 + 3 µτ ) is always nonnegative, this condition is unchanged, except for a very special choice of thechiral chemical potential µτ = − /
3. However, a finite µ changes the growth rate of unstable modes. In the small p limit, the growth rate is given by Im ω = ± (cid:112) κ (1 + 3 µτ ) p + O ( p ) , (19)which is different for the right-handed ( τ >
0) and left-handed ( τ <
0) helices. The real part of ω to the first orderin p is given by Re ω = (cid:20) τ + 32 µ (2 τ − κ ) (cid:21) p + O ( p ) . (20)Hence, the chirality imbalance also modifies the velocity of the wave propagating along the helix.In the limit κ → , τ →
0, the helix approaches a straight line [60], and the dispersion for the fluctuations arounda straight vortex, ω = ± p + µp , is obtained. The leading p behavior corresponds to the famous Kelvin waves [61],and the second term represents the modification due to a chirality imbalance. FIG. 1: Non-planar (left, ( (cid:15), τ , t ) = (1 , . , (cid:15), τ , t ) = (1 , , ABSENCE OF PROPAGATION OF PLANAR SOLUTIONS IN THE PT-SYMMETRIC CASE
In the case of the LIE, the velocity of a Hasimoto soliton is given by v = 2 τ (take µ = 0 in Eq. (7)). At τ (cid:54) = 0,the solutions are “chiral,” in the sense that their handedness is correlated with the direction of the propagation. Amirror image of a solution propagates in the opposite direction from the original one. If we look at a planar ( τ = 0)solution at µ = 0 (see the right figure of Fig. 1), it just rotates around its axis and cannot convey energy along thevortex. In fact, this is a generic feature. Here, we consider a class of solutions that are asymptotically straight lines,like the Hasimoto solitons. We will now show that these solutions cannot propagate if the equation of motion (EOM)has the PT symmetry. The LIE has this symmetry, while the FME does not.It suffices to show that, when the solution is planar, the velocity ˙ X is restricted to the direction of b , since abinomal motion cannot make the soliton propagate along the vortex. Consider a current written in the form J = (cid:90) ds f ( κ, τ ) ˙ X , (21)where f ( κ, τ ) is a function of κ and τ . The energy current is written in this form. Let us denote the unit vector in thedirection of the asymptotic line by (cid:96) . Since (cid:96) is within the plane spanned by { t , n } for a planar solution, it is alwaysorthogonal to b , (cid:96) · b = 0. Thus, if ˙ X ∝ b , then (cid:96) · J = 0 holds and there is no energy flow in the direction of (cid:96) .Let us examine the transformation property of the EOMs. The parity reflection, X → − X , acts on the FS systemas { t , n , b , κ, τ } −→ {− t , − n , b , κ, − τ } . (22)The binomal vector is parity even, because b = t × n , while the torsion τ = ( n (cid:48) · b ) is parity odd. The RHS of theLIE, X (cid:48) × X (cid:48)(cid:48) , is P-even, while the modification to the LIE in the FME, X (cid:48)(cid:48)(cid:48) + ( X (cid:48)(cid:48) ) X (cid:48) , is P-odd.A general EOM can be written in the form˙ X = a ( κ, τ ) t + b ( κ, τ ) n + c ( κ, τ ) b . (23)From our assumption, the theory has the PT symmetry. The LHS is even under PT. The RHS is T-even, so it hasto be P-even. For a planar solution, τ = 0, the coefficients in Eq. (23) are all P-even, because κ is a P-even quantity.Thus, the coefficients of t and n have to vanish, a ( κ, τ = 0) = b ( κ, τ = 0) = 0, and ˙ X ∝ b .The results above can be further generalized. The LIE can be mapped to NLSE, which has an infinite sequence ofcommuting invariants. Those invariants are the generators of the Hamiltonian flows. Correspondingly, the LIE alsohas infinitely many commuting Hamiltonian flows [46], which are called the LIE hierarchy. The first and second termof the RHS of the FME (4) are the first two Hamiltonian flows, V = κ b , V = κ t + κ (cid:48) n + κτ b , · · · (24)In Ref. [46], a recursion operator that successively generates the next flow is constructed, R V ≡ −P [ t × ∂ s V ] , (25)where P denotes the reparametrization procedure to keep the arc-length-preserving nature, which is done by addinga tangential term (see also Ref. [50]). Once we know V n , we can obtain the next flow by V n +1 = R V n . One can show(see the Supplementary Material) that V n is P-even(odd) if n is an even(odd) number. Thus, every EOM with aneven n has the PT symmetry, and the solution of the EOM cannot propagate if its planar.To summarize, we have found a new phenomenon affecting the dynamics of vortex solitons in chirally imbalancedmedia - the Chiral Propulsion Effect. The CPE refers to an asymmetric energy flow along the vortex filament inthe direction determined by the sign of the chirality imbalance. The energy is carried along the vortex by helicalexcitations analogous to the Hasimoto solitons. We have also found that the growth rate of unstable modes on thehelical soliton solution is modified by the chirality of the medium. It is shown that, if the equation of motion respectsthe PT symmetry, a planar solution cannot transfer energy – this indicates that the existence of the CPE is entirelydue to the breaking of parity in the medium.This work was supported in part by the U.S. Department of Energy under contracts No. DE-FG-88ER40388 andDE-SC-0017662 (D.K.), DE-AC02-98CH10886 (Y.H. and D.K.), and within the framework of the Beam Energy ScanTheory (BEST) Topical Collaboration. The work of A.S. is partially supported through the LANL LDRD Program. SUPPLEMENTARY MATERIALDerivation of Eqs. (15) and (16)
The compatibility conditions for the Frenet-Serret frame, ∂ s ∂ t t = ∂ t ∂ s t , ∂ s ∂ t n = ∂ t ∂ s n , ∂ s ∂ t b = ∂ t ∂ s b , result inthe following relations, ∂ s β = − ατ + γκ, (26) ∂ t (cid:18) κτ (cid:19) = (cid:18) − ββ (cid:19) (cid:18) κτ (cid:19) + ∂ s (cid:18) αγ (cid:19) . (27)A generic equation of motion (EOM) of a curve is written in the form˙ X = a ( κ, τ ) t + b ( κ, τ ) n + c ( κ, τ ) b , (28)where a ( κ, τ ) , b ( κ, τ ) , c ( κ, τ ) are functions of κ and τ . They are related to α, β, γ in Eq. (14) as α = aκ + b (cid:48) − cτ, β = bτ + c (cid:48) , γ = β (cid:48) + τ ακ , (29)which can be checked by using the FS formulas. The FME (4) can be written in terms of the FS basis as˙ X = µ κ t + µκ (cid:48) n + κ (1 + µτ ) b . (30)From this expression we can read off a , b and c in Eq. (28). By plugging them into Eqs. (29), we obtain the expressionsfor α, β, γ . Substituting them into Eq. (27), we find the time-evolution equations for τ and κ given by Eq.(15) andEq.(16). The parity symmetry of the higher-order flows of the LIE hierarchy
Here we determine the parity symmetry of the higher-oder flows of the LIE hierarchy, using the recursion operator R . It is shown that V n is P-even(odd) if n is an even(odd) number. Suppose V n is a flow of a particular parity (evenor odd). The n + 1-th flow can be generated by the operation, V n +1 = −R ( X (cid:48) × ∂ s V n )= − X (cid:48) × ∂ s V n + ¯ a ( κ, τ ) t , (31)where ¯ a ( κ, τ ) t is the added term to keep the arc-length unchanged. The multiplication of X (cid:48) × ∂ s changes the parity,because of a factor of X (cid:48) , and the first term has the opposite parity from V n . One can see that the reparametrizationoperation does not change the parity of the flow, as follows. V n +1 can be written in the form, V n +1 = b ( κ, τ ) n + c ( κ, τ ) b + ¯ a ( κ, τ ) t . The arc-length preserving condition, t · V (cid:48) n +1 = 0, implies that the newly added term has to satisfy¯ a (cid:48) = κb. Since κ is P-even, ¯ a has the same parity as b , and ¯ a t has the same parity as b n . Therefore, V n is P-even(odd)if n is an even(odd) number. ∗ Electronic address: [email protected][1] S. L. Adler, Physical Review , 2426 (1969).[2] J. S. Bell and R. Jackiw, Il Nuovo Cimento A (1965-1970) , 47 (1969).[3] D. Kharzeev, Phys. Lett. B633 , 260 (2006), hep-ph/0406125.[4] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl.Phys.
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