Dissipative dynamics of topological defects in frustrated Heisenberg spin systems

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Dissipative dynamics of topological defects in frustrated Heisenberg spin systems

V. Juricic,^1,2L. Benfatto,¹A. O. Caldeira,³ and C. Morais Smith^1,2

1Département de Physique, Université de Fribourg, Pérolles, CH-1700 Fribourg, Switzerland

2Institute for Theoretical Physics, University of Utrecht, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands

3Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-970 Campinas, São Paulo, Brazil 共Received 7 October 2004; published 28 February 2005兲

We study the dynamics of topological defects of a frustrated spin system displaying spiral order. As a starting point we consider the SO共3兲nonlinear sigma model to describe long-wavelength fluctuations around the noncollinear spiral state. Besides the usual spin-wave magnetic excitations, the model allows for topologi- cally nontrivial static solutions of the equations of motion, associated with the change of chirality共clockwise or counterclockwise兲 of the spiral. We consider two types of these topological defects, single vortices and vortex-antivortex pairs, and quantize the corresponding solutions by generalizing the semiclassical approach to a non-Abelian field theory. The use of the collective coordinates allows us to represent the defect as a particle coupled to a bath of harmonic oscillators, which can be integrated out employing the Feynman-Vernon path- integral formalism. The resulting effective action for the defect indicates that its motion is damped due to the scattering by the magnons. We derive a general expression for the damping coefficient of the defect, and evaluate its temperature dependence in both cases, for a single vortex and for a vortex-antivortex pair. Finally, we consider an application of the model for cuprates, where a spiral state has been argued to be realized in the spin-glass regime. By assuming that the defect motion contributes to the dissipative dynamics of the charges, we can compare our results with the measured inverse mobility in a wide range of temperature. The relatively good agreement between our calculations and the experiments confirms the possible relevance of an incom- mensurate spiral order for lightly doped cuprates.

DOI: 10.1103/PhysRevB.71.064421 PACS number共s兲: 75.10.Nr, 74.25.Fy, 74.72.Dn

I. INTRODUCTION

Two-dimensional frustrated Heisenberg spin systems with noncollinear or canted order have attracted much attention recently. Noncollinear order arises due to frustration, which may originate from different sources. The most common kind of frustration is realized in antiferromagnets on a two- dimensional共or three-dimensional stacked兲triangular lattice.

Prototypes of these geometrically frustrated magnets are pyrochlores.^1–3A second source of frustration may be a com- petition between nearest-neighbor and further-neighbor ex- change interactions between spins. Typical examples are helimagnets, where a magnetic spiral is formed along a cer- tain direction of the lattice.¹A third kind of frustration may occur by chemical doping of a magnetically ordered system.

In this case, the spin current of the itinerant doped charges couples to the local magnetic moment of the magnetic host, leading to the formation of a noncollinear magnetic state.

This situation may be realized in lightly doped cuprate superconductors.^4–12

The main characteristic of the noncollinear state is that the spin configuration must be described by a set of three orthonormal vectors or, alternatively, by a rotational matrix which defines the orientation of this set with respect to some fixed reference frame. As a consequence, the order-parameter space is isomorphic to the three-dimensional rotational group SO共3兲, and in the low-temperature phase, when the rotational symmetry is fully broken, three spin-wave modes are present in the system, instead of two, as in the nonfrustrated case.

Moreover, topological defects may arise in the system, asso- ciated to the chiral degeneracy of the spiral, which can rotate

clockwise or counterclockwise. Because the order-parameter space has a nontrivial first homotopy group,␲1关SO共3兲兴=Z2, the topological excitations are vortex like. On the other hand, skyrmions are not present because the second homotopy group of the SO共3兲is trivial,␲2关SO共3兲兴= 0.¹³

A convenient field-theoretical description of frustrated Heisenberg systems in the long-wavelength limit is provided by the SO共3兲 nonlinear sigma 共NL␴兲 model.^13–17Its critical behavior in two dimensions has been extensively investi- gated, both in the absence and in the presence of topological excitations. Studies in the former case have revealed a dy- namical enhancement of the symmetry from O共3兲丢O共2兲to O共4兲 under renormalization group flow in d = 2 +⑀^{, which} means that in the critical region all three spin-wave modes have the same velocity.^14,17When topological excitations are included, a complex finite-temperature behavior is found.¹⁸ Numerical studies, as well as analysis involving entropy and free-energy arguments, indicate the occurrence of a transition driven by vortex-antivortex pairs unbinding at a finite tem- perature T_v.^19–22In contrast to the XY case, here vortices and spin waves are coupled already in the harmonic approxima- tion, and anharmonic spin-wave interactions yield a finite correlation length for arbitrarily low temperatures.^17,23 Therefore, the transition mediated by vortices is a crossover rather than a true Kosterlitz-Thouless共KT兲transition.²⁴Free vortices start to proliferate at the temperature T_v, similarly as vortices in the XY model do above the KT-transition tempera- ture.

In the present paper we study the physical properties of frustrated Heisenberg spin system, which are sensitive to the dynamics of the above-mentioned topological defects. The

Published in "Physical Review B 71: 064421, 2005"

which should be cited to refer to this work.

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approach we use has been employed to describe the dynam- ics of excitations in a very broad class of one- or two- dimensional systems.²⁵The central idea is the application of the collective-coordinate method²⁶ to quantize a nontrivial static solution of the classical equation of motion of the field-theoretical model in question. In our case, we find that single vortexlike excitations or vortex-antivortex pairs are the localized static solution of the SO共3兲NL␴ ^{model. A} proper description of the quantum levels associated with these solutions is provided, on the semiclassical level, by a theory in which the topological excitation is represented by a single quantum-mechanical variable coupled to a bath of quantum harmonic oscillators, which are the fluctuations about the classical solution itself. Thus, the resulting effec- tive model represents a particle共the topological defect兲scat- tered by the linearized excitations of the system. The latter can be integrated out using the standard system-plus- reservoir approach,²⁵leading to a dissipative equation of mo- tion for the topological excitation. As a consequence, any physical property of these systems that depends on the mo- tion of the topological excitations may be expressed in terms of transport coefficients—such as mobility and diffusion—of these damped defects. Since we do neglect any interaction between the defects, our results are only valid for a diluted gas of topological excitations. Part of our results concerning the mobility of a vortex-antivortex defect has been recently published in Ref. 12. Here, besides a complete presenta- tion of the technical details, we discuss also the transport at finite frequencies and we compare—qualitatively and quantitatively—the cases where the defect is represented by a single vortex or by a vortex-antivortex pair. Moreover, in the light of new experimental results by Ando et al.,²⁷ we discuss the relevance of the single vortices for the transport in the spin-glass phase of cuprates.

The structure of the paper is the following. Starting from the SO共3兲NL␴ model, we derive in Sec. II the quantum Hamiltonian describing the dynamics of the topological de- fect coupled to a bath of magnetic excitations. In Sec. III the equation governing the evolution of the reduced density ma- trix for the topological defect is obtained and the influence functional, which describes the effect of the magnon bath on the dynamics of the vortices, is evaluated. Section IV is de- voted to the derivation of the effective action for the defect after the magnons have been integrated out, and in Sec. V the inverse mobility is calculated. In Sec. VI we discuss how a spiral state may be realized in cuprates, and we then apply our results to this specific case. Section VII contains our conclusions. Details of the calculations are given in the Ap- pendixes.

II. THE MODEL

In the spiral state the spin configuration S at each site r is described by means of a Dreibein order parameter n_k^a僆SO共3兲 with k = 1, 2, 3 and n_k^an_q^a=␦kq,¹⁴so that

S

S = n = n₁cos共k_s· r兲− n₂sin共k_s· r兲, 共1兲 where S =兩S兩and the wave vector k_s=共␲^{/ a ,}␲^{/ a}兲+ Q, with a denoting the lattice constant. Here, Q =共2␲^{/ m}xa , 2␲^{/ m}ya兲

measures the incommensurate spin correlations. Indeed, the magnetic susceptibility corresponding to the spin modulation 共1兲 has two peaks at k_s and −k_s 关equivalent to 共␲^{/ a ,}␲^{/ a}兲

− Q兴, as represented in Fig. 1 in the case of m_x= −m_y. The resulting spin order for n₁and n₂in the plane is represented in Fig. 2, where m_x= −m_y= 20. Observe that the periodicity of the spin texture is 2␲/ Q for even values of m_x, m_y, and twice it for odd values.

As discussed in Refs. 14–16, a proper continuum field theory for the spiral state is provided by the SO共3兲quantum NL␴^model

S=

冕

^{dt d2x关␬k共⳵tnk兲2− pk␣共⳵␣nk兲2兴.}

Here, the index␣stands for the spatial coordinates and sum- mation over repeated indices is understood. The spatial an- isotropy of the spin stiffness p_k␣depends on the components Q_␣ of the incommensurate wave vector. Since at the fixed point all the spin-wave velocities are equal,^14,17we will con- sider the case␬k⬅␬^{, p}k␣⬅p_␣ and we will choose a system of coordinates parallel共x_储兲and perpendicular共x_⬜兲to the spi- ral axis, respectively,

S=␬

冕

^{dtdx⬜dx储关共⳵tnk兲2− c⬜2共⳵⬜nk兲2− c储2共⳵储nk兲2兴, 共2兲}

where c_⬜⬅

冑

p_⬜/␬ ^{and c}储⬅

冑

p_储/␬ are the spin-wave veloci- ties perpendicular and parallel to the spiral axis. Even though FIG. 1. Incommensurate magnetic response for the spiral spin modulation 共1兲. The magnetic susceptibility corresponding to the spiral order with the wave vector k_sexhibits two peaks at the points 共␲/ a ,␲/ a兲± Q marked by a circle in the figure. In this case Q has finite components in both the x and y directions, and the distance between the peaks is twice the modulus of Q.

FIG. 2. Spin background corresponding to Eq. 共1兲 and to the case depicted in Fig. 1. Here, a value of Q =共2␲/ 20a , −2␲/ 20a兲 has been chosen.

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for the moment we will keep our derivation on general grounds, in Sec. VI we will specify the values of the param- eters ␬ and c for the case of cuprates, where they can be related to measurable quantities.

Given the action共2兲as our starting model, our first aim is to analyze whether the equations of motion admit topologically nontrivial solutions. For that purpose, it is con- venient to introduce an equivalent representation of the n_k^a order parameter through an element g僆SU共2兲 as n_k^a=共1 / 2兲tr关␴^ag␴^kg−1兴, where ␴^a are Pauli matrices, and to introduce the fields

A_␮^a= 1

2itr关␴^ag−1⳵_␮^g兴, 共3兲 which are related to the first derivatives of the order parameter through⳵_␮ⁿk

a= 2⑀ijkA_␮ⁱn^a_j.²³ Here, ⳵_␮⬅共⳵t,ⵜ兲 and

⑀123=⑀¹²³= 1. Using that 共⳵␮n兲²= 8A_␮² 共no summation over index␮is imposed here兲, the action共2兲reads

S= 8␬

冕

^{dt dx储dx⬜}

^共

^{A02− c⬜2A⬜2 − c储2A储2}

^兲

^{. 共4兲}

The above action may be mapped to an isotropic form by introducing the coordinates r =共x , y兲, with

x =

冑

c^c_⬜^储x_⬜, y =

冑

^cc^⬜_储x_储. We then find

S=N

冕

^{dt d2rA␮2⬅N}

冕

^{dt d2r共A02− c2A␣2兲, 共5兲}

with the isotropic spin-wave velocity c =

冑

c_储c_⬜and the con- stantN= 8␬. The most generic expression for the element g is given by

g关␣ជ兴= exp

冉

^{2i␣ជ共r,t兲·␴ជ}

冊

^{, 共6兲}

and the corresponding Lagrangian obtained from the action 共5兲reads

L₀=1

4N

冕

^{d2r共⳵␮␣ជ兲2, 共7兲}

with ⳵␮A⳵␮B⬅⳵tA⳵tB − c²ⵜAⵜB. By making the ansatz¹⁸

␣ជ共r , t兲= mជ⌿共r , t兲, where mជ is a constant unit vector and⌿a scalar function, the Lagrangian共7兲reduces to

L₀=¹₄N

冕

^{d2r共⳵␮⌿兲2. 共8兲}

The equation of motion for the field⌿

⳵t

2⌿− c²ⵜ²⌿= 0, 共9兲

possesses static topologically nontrivial solutions in the form of a single-vortex defect at R =共X , Y兲

⌿1v= arctan

冉

^{x − Xy − Y}

冊

^{, 共10兲}

and a vortex-antivortex pair

⌿2v= arctan

冉

^{x − Xy − Y1}1

冊

^{− arctan}

冉

^{x − Xy − Y2}2

冊

= arctan

再

^{关共关r − Rd⫻共兲r − R2− d2兲兴/4z兴}

冎

^{, 共11兲}

where now R =共R₁+ R₂兲/ 2 is the center of mass and d = R₂− R₁ the relative coordinate of the defect pair. If in Eqs. 共10兲 and共11兲 the role of x and y coordinates is inter- changed, one only changes the vorticity. Without loss of gen- erality, we may assume the unity vector to be in the z direc- tion, m = eˆ_z. Thus, the n_k fields which define the spin configuration according to Eq. 共1兲 are given by n₁=共cos⌿, −sin⌿, 0兲, n₂=共sin⌿, cos⌿, 0兲, n₃=共0 , 0 , 1兲. The spin patterns corresponding to a single vortex关⌿ from Eq.共10兲兴or to a vortex-antivortex pair关⌿from Eq.共11兲兴are represented in Figs. 3 and 4, respectively.

The main difference between the two possible static solu- tions共10兲and共11兲is their energy. As shown in Appendix A, the energy of a single-vortex diverges with the logarithm of the system size ᐉ, E关⌿1v兴⬀lnᐉ. On the other hand, the vortex-antivortex pairs have finite energy, depending on the distance d between defects, E关⌿2v兴⬀ln d. A similar situation is realized in the standard XY model, where indeed the pres- ence of single defects below the KT transition is not ener- getically favorable in the thermodynamic limit.²⁴ However,

FIG. 4. Spin background corresponding to the vortex-antivortex solution of Eq. 共11兲. The centers of the vortices are marked by a circle. The spiral incommensurability Q is the same used in Fig. 2.

FIG. 3. Spin background corresponding to the one-vortex solution of Eq. 共10兲. The center of the vortex is marked by the circle. The spiral incommensurability Q is the same used in Fig. 2.

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in the case of the model 共2兲, which possesses asymptotic freedom, the correlation length ␰ is finite at any finite temperature,^17,23 so that the logarithmic divergence of the single-vortex energy should be understood up to the length scale␰^{, E}关⌿1v兴⬀ln␰. In addition, the energy of the vortex- antivortex pair should be bounded below at distances of the order of few lattice spacings, which is the intrinsic cutoff of the theory. Because the procedure that we describe in the following does not depend on the exact form of the static solution, we will refer to a static topological defect solution

⌿v specifying only at the end of the calculations the differ- ences between the cases共10兲and共11兲.

Following a procedure analogous to the one described in Ref. 26 to quantize the kink solution of the scalar field theory, we analyze now the effect of the fluctuations around the static topologically nontrivial configuration, which is a saddle point of the action that corresponds to the Lagrangian 共8兲. In order to reach this aim, we write the generic field g僆SU共2兲of Eq.共6兲in the form of a product of the field g_s corresponding to a static solution mជ⌿v共r兲 and the field g_␧ corresponding to the fluctuations around it

g共r,t兲= g_s关⌿v共r兲兴g_␧关␧ជ共r,t兲兴, 共12兲 where

g_␧关␧ជ兴= exp

冉

^{2i␧ជ·␴ជ}

冊

^.

Observe that the description of the fluctuations via Eq.共12兲 differs from the standard approach used for a scalar field theory,²⁶ and it is related to the symmetry properties of the order parameter. Indeed, since the full g has to be an element of the SU共2兲group, and both g_s, g僆SU共2兲, then the fluctua- tions g_␧ around g_s have to belong to SU共2兲 as well. If, in- stead, we had used the expansion␣ជ^{= m}ជ⌿v+␧ជ, the equations of motion for the␧ជfield would have been independent of the static solution ⌿v, leading to a failure of the semiclassical expansion. Using Eqs.共3兲and共12兲, we can express the ac- tion共5兲in terms of the fields⌿vand␧ជ. Retaining only terms up to second order in␧ជ^{, we find}共Appendix B兲

A_␮^a=1

2m^a⳵_␮⌿v

冉

^{1 −␧2+2␧z2}

冊

^{+14␧a␧bmb⳵␮⌿v}

+1

2⑀^abc␧^bm^c⳵␮⌿v+1

2⳵␮␧^a+ 1

4⑀^abc␧^b⳵␮␧^c, where␧²=␧x

2+␧y

2. The corresponding Lagrangian then reads 共Appendix B兲

L = L₀+N

冕

^{d2rL1, 共13兲}

with L₀given by Eq.共8兲and

L1关⌿v兴=¹₄共⳵␮␧ជ兲²+₂¹共mជ^·⳵␮␧ជ兲⳵␮⌿v+¹₄⑀^abc⳵␮␧^a␧^bm^c⳵␮⌿v

=¹₄共⳵␮␧兲²+₄¹␧²共⳵␮␪兲²+¹₄共⳵␮␧^z兲²+¹₂⳵␮␧^z⳵␮⌿v

−¹₄␧²⳵␮⌿v⳵␮␪^. 共14兲 Here, we used the fact that m = eˆ_zand introduced polar coor-

dinates ␧ជ⁼共␧cos␪^,␧sin␪^,␧z兲. Since the Lagrangian L1 is evaluated at the vortex-like solution⌿vof Eq.共9兲, the equa- tions of motion for the fluctuations around the topological defect also depend on⌿v

␧: 共⳵t

2− c²ⵜ²兲␧−␧共⳵_␮␪兲²− c²␧ⵜ⌿vⵜ␪^{= 0,} 共15兲

␪^: ⳵_␮共␧²⳵_␮␪兲+c²

2 ⵜ共␧²ⵜ⌿v兲= 0, 共16兲

␧z: 共⳵t

2− c²ⵜ²兲␧^z= 0. 共17兲 Equation共16兲admits the solution␪⁼⌿v/ 2, whereas Eq.共17兲 indicates that the field␧zis free. By using these two condi- tions, we can rewrite the total Lagrangian L in Eq.共13兲as

L =N

4

冕

^d2r

冋

^{共⳵␮⌿v兲2+共⳵␮␧兲2−14␧2共⳵␮⌿v兲2}

册

^{, 共18兲}

and the equation of motion共15兲as

关

^{⳵t2− c2ⵜ2−1}4共ⵜ⌿v兲²

兴

^{␧= 0.}

Since the field⌿v in the previous equation does not depend on time, we decompose the field␧into its time- and space- dependent parts

␧共r,t兲=

兺

_nm^{qnm共t兲␩nm共r兲, 共19兲}

and identify the normal modes ␩nm with the eigenfunctions of the operator

c²关ⵜ²+ V共r兲兴␩nm= −␻nm

2 ␩nm. 共20兲

This equation has the typical form of a Schrödinger-type equation for a particle scattered by a potential V共r兲=共ⵜ⌿v兲²/ 4. The two indices n and m refer, respectively, to the radial and angular part of the wave function. By using a standard approach to scattering problems in two dimen- sions, one may express the wave functions␩nmin terms of the eigenfunctions of the free problem共V = 0兲, corrected by a phase shift␦mdue to the scattering by the potential V共r兲²⁸

␩nm=1

2

冑

^k2^nmᐉe^im␽

关

^{H兩共m1兲兩共knmr兲+ e−2i␦mH兩共m2兲兩共knmr兲}

兴

^.

共21兲 Here, H_兩^共_m^1,2_兩^兲 are Hankel functions of the first and second kinds, m is an integer, and␽is a polar angle. The k_nmvalues are determined by requiring the vanishing of the wave func- tion共21兲at the boundary r =ᐉ. By using the asymptotic form of the Hankel functions, we obtain k_nmᐉ=共2n + 1兲␲^{/ 2} +共2兩m兩+ 1兲␲^{/ 4 −}␦m, where n is a positive integer. Since the field␧is real, we may rewrite the expansion共19兲in the form

␧共r,t兲=_n,m

兺

_艌0^{关qnm共t兲␩nm共r兲+ qnm* 共t兲␩nm* 共r兲兴, 共22兲}

where we used the identities␩n,m= e^−2i␦m␩n,−m

* and␦m=␦−m. Note that the sum in Eq. 共22兲 is over the positive angular momenta, as will be the case in what follows.

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The static defect solution ⌿v共r兲 of Eq. 共9兲 is invariant under translation of the center of the defect共i.e., the position of the vortex or the center of mass of the vortex-antivortex pair兲. A consequence of this invariance²⁶ is that Eq. 共20兲 admits zero-frequency modes. A consistent treatment of them requires the use of the collective coordinate method.^25,26The center of mass of the defect is then promoted to a dynamical variable, yielding

⌿v共r兲→⌿v关r − R共t兲兴, 共23兲 and

␧共r,t兲→␧关r − R共t兲,t兴 ⬅

兺

_nm^{共qnm共t兲␩nm„r − R共t兲) + c.c.兲,}

共24兲 where the last sum is over all nonzero-frequency modes. By inserting these expressions into the full Lagrangian 共18兲 evaluated at the saddle-point solution, we obtain

N

4

冕

^{dt d2r共⳵t⌿v兲2=N4}

冕

^{dt d2rR˙␣R˙␤⳵␣⌿v⳵␤⌿v}

=M

2

冕

^{dt R˙2共t兲, 共25兲}

where

M =N

2

冕

^{d2r共ⵜ⌿v兲2,}

is the mass of the topological defect, which is proportional to its energy共see Appendix A兲. The time derivative of the field

␧yields N

4

冕

^{dt d2r共⳵t␧兲2=N4}nm,kl

^兺 冕

^{dt d2r关q˙nm* 共t兲␩nm*}

− q_nm^* ⳵_␣␩nm

* R˙

␣共t兲+ c.c.兴

⫻关q˙_kl共t兲␩kl− q_kl⳵␤␩klR˙

␤共t兲+ c.c.兴

=N 2

兺

nm

冕

^{dt关兩q˙nm兩2+}

^兺

kl

R˙共t兲共q˙_nmq_kl^*G_nm,kl^* + q˙_nm^* q_klG_nm,kl兲兴, 共26兲 where the coupling constants G are related to the eigenfunc- tions␩^via

G_nm,kl=

冕

^{d2r␩klⵜ␩nm* ,}

and we neglected terms of order q²R˙². Here, we used that 兰d²r␩nm␩kl= 0 and兰d²r␩nmⵜ␩kl= 0 for m and l positive. By substituting Eqs.共25兲and共26兲into the Lagrangian共18兲, we obtain

L =¹₂MR˙²+

兺

nm

共兩q˙_nm兩²−␻nm

2 兩q_nm兩²兲+

兺

nm,kl

R˙共t兲共q˙_nmq_kl^*G_nm,kl^* + q˙_nm^* q_klG_nm,kl兲,

where we rescaled q→q

冑

N/ 2. Using that G_nm,kl^* = −G_kl,nm,

and neglecting terms which are quadratic in G, the corre- sponding Hamiltonian reads

H = 1

2M共P − P_E兲²+

兺

_nm^{共兩pnm兩2+␻nm2 兩qnm兩2兲, 共27兲}

with

P_E=

兺

nm,kl

共p_nmG_nm,klq_kl+ p_nm^* G_nm,kl^* q_kl^*兲.

Here, P is the momentum canonically conjugate to the center of the defect R, and q_nm and p_nm are the coordinates and momenta of the magnons. The classical Hamiltonian 共27兲 can be promptly quantized by introducing two sets of inde- pendent creation and annihilation operators, aˆ^†,aˆ, and bˆ^†,bˆ.

The quantum Hamiltonian reads Hˆ = Hˆ_v+ Hˆ

B+ Hˆ

I, 共28兲

where

Hˆ

v= Pˆ²

2M, 共29兲

is the Hamiltonian of a free defect, and Hˆ

B=

兺

nm

ប␻nm共aˆ_nm^† aˆ_nm+ bˆ_nm^† bˆ_nm兲, 共30兲 is the Hamiltonian of the bath of magnons which consists of two independent sets of noninteracting harmonic oscillators described by the operators aˆ,aˆ^†and bˆ,bˆ^†, as it is expected in two dimensions. The interaction between the bath and the topological defect is described by the Hamiltonian

Hˆ

I= −បPˆ

M _nm,kl

兺

^{关Dnm,klaˆnmaˆkl† − Dkl,nmbˆnmbˆkl† + Cnm,kl}

⫻共aˆ_nm^† bˆ_kl^† − aˆ_klbˆ_nm兲兴, 共31兲 with the coupling constants given by

D_nm,kl= i 2

␻nm+␻kl

冑

_␻nm␻kl

G_kl,nm,

共32兲 C_nm,kl= i

2

␻nm−␻kl

冑

_␻nm␻kl

G_nm,kl.

The terms with the coupling constants D / C describe the scattering/creation共annihilation兲of magnetic excitations by the defect. Since we consider here only the low-energy dy- namics of the topological defect, we neglect the off-diagonal terms in the interaction Hamiltonian, i.e., we set C = 0. In the following we shall integrate out the bath degrees of freedom in order to study the effective dynamics of the defects. For that purpose we employ the Feynman-Vernon formalism.

III. REDUCED DENSITY MATRIX

In this section we derive the reduced density matrix for the defect. The system under consideration consists of two

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subsystems: the topological defect and the bath of magnons.

Thus, the Hilbert space of the full system, H, is a direct product of the subsystem Hilbert spaces H=Hv丢HB

⬅Hv丢H_B^共a兲丢H_B^共b兲, and the state of the full system is also a direct product, 兩x ,␣ជ典⬅兩x典丢兩␣ជ典⬅兩x典丢兩␣ជa典丢兩␣ជb典. We use the coordinate representation for the defect共x are the eigen- values of its center-of-mass position operator兲, and the co- herent state representation for the bath, aˆ_nm兩␣ជa典=␣nm,a兩␣ជa典 and bˆ_nm兩␣ជb典=␣nm,b兩␣ជb典. The reduced density matrix is de- fined as^˜ˆ␳v共t兲= tr_B关␳^ˆ共t兲兴, where tr_B denotes the trace over the bath degrees of freedom, and␳^ˆ共t兲is the density matrix of the full system, whose evolution is described by

␳^ˆ共t兲= e^{−iHˆ t/ប}␳^ˆ共0兲e^{iHˆ t/ប}. 共33兲 Here, Hˆ is the Hamiltonian of the full system given by Eqs.

共28兲–共32兲. The matrix elements of the density operator in the basis introduced before are

␳^ˆ共x,␣ជ^;y,␤ជ;t兲=具x,␣ជ兩␳^ˆ共t兲兩y,␤ជ_典, and the reduced density matrix of the vortex reads

␳

˜ˆ_v共x,y,t兲=

冕

^d␲^{22N␣ជ具x,}␣ជ兩␳^ˆ共t兲兩y,␣ជ典

=

冕

^d␲^{22N␣ជ具x,}␣ជ兩e^{−iHˆ t/ប}␳^ˆ共0兲e^{iHˆ t/ប}兩y,␣ជ典. 共34兲 After insertion of the unity operator on both sides of␳^ˆ共0兲in Eq.共34兲, the reduced density matrix acquires the form

␳

˜ˆ_v=

冕

^d␲^22N␣ជ

冕

^d2x⬘

冕

^d2y⬘

冕

^d␲^22N␤ជ

冕

^d␲^{22N␤ជ⬘具x,}␣ជ兩e^{−iHˆ t/ប}兩x⬘^,␤ជ_典

⫻具x⬘^,␤ជ_兩␳^ˆ共0兲兩y⬘^,␤ជ⬘^典具y⬘^,␤ជ⬘^{兩eiHˆ t/ប兩y,}␣ជ典. 共35兲 In order to calculate the time evolution of the reduced den- sity matrix, we have to define the initial condition for the density matrix of the full system. For the sake of simplicity, we choose the factorizable one

␳^ˆ共0兲=␳^˜ˆv共0兲^˜ˆ␳B共0兲, 共36兲 which implies that the bath and the topological defect are decoupled at t = 0. The bath is assumed to be initially in thermal equilibrium at temperature T

␳

˜ˆ_B= e^−UHˆB

tr关e^−UHˆB兴=^˜ˆ␳B,a␳^˜ˆB,b, 共37兲 where U⬅ប/共k_BT兲. Here, we used the fact that the baths do not interact, so the density matrix of the full bath is the product of the density matrices for the separate baths. By substituting Eqs.共36兲and共37兲into Eq.共35兲, we obtain

␳

˜ˆ_v共x,y,t兲=

冕

^d2x⬘

冕

^{d2y⬘J共x,y,t;x⬘,y⬘,0兲˜ˆ␳v共x⬘,y⬘,0兲,}

共38兲 with the superpropagator J given by

J共x,y,t;x⬘^,y⬘^,0兲

=

冕

^d␲^22N␣ជ

冕

^d␲^22N␤ជ

冕

^d␲^{22N␤ជ⬘˜ˆ}␳B共␤ជ^*_,␤ជ⬘^,0兲K共x,␣ជ^*;x⬘^,␤ជ_;t兲

⫻K^*共y,␣ជ^*;y⬘^,␤ជ⬘^;t兲. 共39兲

A. Superpropagator

We consider the superpropagator共39兲. The kernel K共x,␣ជ^*;y,␤ជ_{;t兲 ⬅ 具x,}␣ជ兩e^{−iHˆ t/ប}兩y,␤ជ_{典, 共40兲} can be expressed in the path-integral formalism as共Appen- dix C兲

K共x,␣ជ^*;y,␤ជ_;t兲=

冕

y x

Dx exp

再

^{−兩␣ជ2兩2−兩␤ជ2兩2}

冎 ^冕

^{␤ជ␣ជ*D␨ជ}

⫻exp

再

^{12关␨ជ*共0兲·␤ជ+␨ជ共t兲·␣ជ*兴}

冎

⫻exp

再

^{បiS0关x兴}

冎

^exp

^兵

^{SI关x,␨ជ兴}

^其

^{, 共41兲}

where S₀关x兴=兰0

tdt⬘^{共M / 2兲x˙2} stands for the free action and

S_I关x,␨ជ_兴=

冕

0 t

dt⬘

再

^{12共␨ជ·␨ជ˙*−␨ជ*·␨ជ˙兲−បi}

^关

^{hB共␨ជ*,␨ជ兲}

− x˙ · h_I共␨ជ^*_,␨ជ_兲

兴 冎

^{⬅SI,a关x,␨ជa兴+ SI,b关x,␨ជb兴 共42兲}

denotes the interaction. Here h_B=

兺

nm,i=a,b

ប␻nm␨nm,i

* ␨nm,i⬅h_B,a+ h_B,b,

h_I=ប_nm,kl

兺

^{Dnm,kl␨kl,a* ␨nm,a− Dkl,nm␨nm,b␨kl,b* ⬅hI,a+ hI,b.}

By inserting Eq. 共41兲 into Eq. 共39兲 and using the reduced density matrix of the bath in the coherent state representation 共Appendix D兲, we obtain a simpler expression for the super- propagator,

J共x,y,t;x⬘^,y⬘^,0兲=

冕

x⬘

x

Dx

冕

y⬘

y

Dye^{i/ប关S0共x兲−S0共y兲兴}F关x,y兴, 共43兲 where F=FaFb is the total influence functional, and Fi共i = a , b兲is the influence functional for the bath i given by 共to simplify notation we omit the index i in the integration variables兲

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Fi关x,y兴=

冕

^d␲^2N␣ជ

冕

^d␲^2␤Nជ

冕

^d␲^2␤ជN⬘␳B共␤ជ^*_,␤ជ⬘^,0兲

⫻exp

再

^{−兩␣ជ兩2−兩␤ជ2兩2−兩␤ជ2⬘兩2}

冎

⫻

冕

_␤^{ជ␣ជ*D␨ជ}

冕

␤ជ

⬘^*

␣ជ

D␥ជ^exp

再

¹²

^关

^{␨ជ*共0兲·␤ជ+␨ជ共t兲·␣ជ*}

+␥ជ共0兲·␤ជ⬘^*+␥ជ^*共t兲·␣ជ

兴 冎

^{exp兵SI,i关x,␨ជ兴+ SI,i*关y,␥ជ兴其,}

共44兲 with the initial conditions

␨ជ_共0兲=␤ជ_, ␨ជ^*_共t兲=␣ជ^*, 共45兲

␥ជ^*共0兲=␤ជ⬘^* ␥ជ共t兲=␣ជ^. 共46兲

B. Influence functional

We now evaluate the influence functional, which de- scribes the influence of the bath on the effective dynamics of the defect. The only difference between the functionalsFa

andFbis in the form of the interaction S_I; see Eqs.共42兲and 共44兲. Note that the actions S_I,a and S_I,b are related by the substitution D_nm,kl→−D_kl,nm. Thus, it is enough to calculate the functionalFa, and consequentlyFbis obtained using the latter transformation. In order to simplify notation, in what follows we write the integration variables without the index a. First, we calculate the path integrals in Eq.共44兲using the stationary phase approximation共SPA兲. In order to apply the SPA, we have to solve the equations of motion correspond- ing to S_Iand S_I^*. Because S_I^*共x ,␨ជ_{兲= −SI共x ,}␨ជ_兲, we need to con- sider only S_I. The equations of motion are promptly obtained from␦^SI/␦␨nm

* = 0,␦^SI/␦␨nm= 0, and they read

␨^˙nm+ i␻nm␨nm− ix˙

兺

_kl ^{Dkl,nm␨kl= 0,}

共47兲

␨^˙nm

* − i␻nm␨nm

* + ix˙

兺

kl

D_nm,kl␨kl

* = 0.

Notice that the two equations are identical; one is the com- plex conjugate of the other共recall that D_nm,kl= D_kl,nm^* 兲.

The SPA requires the evaluation of the action S_I on the classical trajectory, which is the solution of the above equa- tions of motion. Straightforward calculations show that the value of S_I at the stationary point is zero. If we define

␨ជ₌␨ជ_cl+⌬ជ, then the functional integral over ␨ជ becomes the functional integral over the fluctuations⌬ជ around the saddle point. Expanding the action around its saddle point, we find that the relevant contribution comes from the second deriva- tive of the action at the stationary point, because both the value and the first derivative of the action are zero at the saddle point. The second derivative of the action evaluated at the stationary point is a constant operator, so the integration over the fluctuations⌬ yields, up to an irrelevant constant

Fa关x,y兴=

冕

^d␲^2N␣ជ

冕

^d␲^2␤Nជ

冕

^d␲^2␤ជN⬘

⫻␳B,a共␤ជ^*_,␤ជ⬘兲e^{−兩␣ជ兩2−共兩␤ជ兩2/2兲−共兩␤ជ⬘兩2/2兲}

⫻exp

再

¹²

^关

^{␨ជ*共0兲·␤ជ+␨ជ共t兲·␣ជ*}

+␥ជ共0兲·␤ជ⬘^*+␥ជ^*共t兲·␣ជ

兴 冎

^{. 共48兲}

Therefore, in the SPA, S_Iand S_I^*only contribute to the influ- ence functional through the boundary terms, which may be determined using the solutions of the equations of motion 共see Appendix E兲.

After inserting Eqs.共D2兲and共E8兲into Eq.共48兲, and per- forming the Gaussian integrals over␣^,␤^{, and}␤⬘, the influ- ence functional acquires the form

Fa关x,y兴= 1

det共1 − n⌫^a兲, 共49兲 where the matrix⌫^ais given by

⌫nm,kl

a 关x,y兴=¹₂关W_kl,nm共x,t兲+ W˜

nm,kl共x,0兲+ W_kl,nm共y,0兲 + W˜

nm,kl共y,t兲兴+1

4

兺

_pq^{关W˜nm,kl共x,0兲}

+ W_pq,nm共x,t兲兴关W_kl,pq共y,0兲+ W˜

pq,kl共y,t兲兴, 共50兲 and n_pq= 1 /关exp共U␻pq兲− 1兴 is the bosonic occupation num- ber. Equation共E7兲enables us to express the matrix⌫^aonly in terms of the functionals W and W˜

⌫nm,kl

a 关x,y兴=¹₂关W_kl,nm共x,t兲+ W˜

nm,kl共x,0兲+ W˜

kl,nm

* 共y,0兲 + W_nm,kl^* 共y,t兲兴+¹₄

兺

pq

关W˜

nm,pq共x,0兲 + W_pq,nm共x,t兲兴关W˜

kl,pq

* 共y,0兲+ W_pq,kl^* 共y,t兲兴. 共51兲 Using the formula ln detA= tr lnA for the matrix A=共1 − n⌫^a兲⁻¹, we find

Fa关x,y兴= exp关tr共n⌫^a兲兴= exp

冋 ^兺

^{pqnpq⌫pq,pqa}

册

^{, 共52兲}

and the total influence functional reads

F=FaFb= exp

冋 ^兺

^{pqnpq⌫pq,pq}

册

^{, 共53兲}

in the lowest order in n⌫, where ⌫⬅⌫^a+⌫^b. The diagonal elements of the matrix ⌫^a are obtained from Eq. 共51兲, while the matrix ⌫^b is obtained from ⌫^a by the substitution D_nm,kl→−D_kl,nm= −D_nm,kl^* . The functionals W and W˜ are given implicitly by Eqs.共E3兲. From their form we see that they actually represent the amplitude of scattering of the mode nm to the mode kl through virtual intermediate states.

These functionals can be determined iteratively from these