E. W. Carlson, D. X. Yao and D. K. Campbell- Spin waves in striped phases

8/3/2019 E. W. Carlson, D. X. Yao and D. K. Campbell- Spin waves in striped phases

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Spin waves in striped phases

E. W. Carlson,1 D. X. Yao,2 and D. K. Campbell21Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA

2Department of Physics and Department of Electrical and Computer Engineering, Boston University, Boston, Massachusetts 02215, USA

(Received 3 February 2004; published 11 August 2004)

In many antiferromagnetic, quasi-two-dimensional materials, doping with holes leads to stripe phases, inwhich the holes congregate along antiphase domain walls in the otherwise antiferromagnetic texture. Using a

suitably parametrized two-dimensional Heisenberg model on a square lattice, we study the spin wave spectra

of well-ordered spin stripes, comparing bond-centered antiphase domain walls to site-centered antiphase do-

main walls for a range of spacings between the stripes and for stripes both aligned with the lattice (vertical)

and oriented along the diagonals of the lattice (diagonal). Our results establish that there are qualitative

differences between the expected neutron scattering responses for the bond-centered and site-centered cases. In

particular, bond-centered stripes of odd spacing generically exhibit more elastic peaks than their site-centered

counterparts. For inelastic scattering, we find that bond-centered stripes produce more spin wave bands than

site-centered stripes of the same spacing and that bond-centered stripes produce rather isotropic low energy

spin wave cones for a large range of parameters, despite local microscopic anisotropy. We find that extra

scattering intensity due to the crossing of spin wave modes (which may be linked to the resonance peak in

the cuprates) is more likely for diagonal stripes, whether site- or bond-centered, whereas spin wave bands

generically repel, rather than cross, when stripes are vertical.

DOI: 10.1103/PhysRevB.70.064505 PACS number(s): 74.72.h, 75.30.Ds, 76.50.g, 75.10.Jm

I. INTRODUCTION

Many doped strongly correlated materials exhibit evi-dence for an emergent length scale in the form of stripes,i.e., regular antpihase domain walls in an otherwise antifer-romagnetic texture. The strongest evidence for striped struc-tures in nickelate perovskites and some related cuprates hascome from neutron scattering,15 which is capable of detect-ing the spin texture directly through diffraction. Since sev-

eral theories of high temperature superconductivity makecontact with such structures,612 it is important to improveour microscopic picture of them. In particular, it is not yetknown from experiment whether the antiphase domain wallssit primarily on nickel (copper) sites, or rather sit primarilyon oxygen sites.

When undoped, the nickel-oxygen (and copper-oxygen)planes in these materials are antiferromagnetic, with spinmoments localized on the Ni (Cu) sites, as evidenced by apeak in elastic neutron scattering at ,.13 Upon hole dop-ing, this peak is observed to split into four (or perhapstwo14,15) incommensurate peaks,16 indicating an extramodulation on top of the antiferromagnetic wavelength. Forthe case of collinear spins, this is consistent with the forma-tion of periodic antiphase domain walls in the antiferromag-netic texture (i.e., stripes).

On a two-dimensional square lattice, these domains con-sist of a strip of antiferromagnet with spin up on, say, theA sublattice, separated by a domain wall from a strip ofantiferromagnet with spin up on the B sublattice, and soon, as in Fig. 1. The figures necessarily depict a certain widthfor each antiphase domain wall, but the widths are notknown and are in reality likely less sharp than shown in thefigure. In both cases, neighboring antiferromagnetic patcheshave spin up on opposite sublattices, which washes out any

signal at the antiferromagnetic peak ,. Rather, satellitepeaks are observed around ,, at a distance determinedby the spacing between domain walls. When the domainwalls are site-centered, all couplings are antiferromagnetic,including couplings across the domain walls. Bond-centereddomain walls, however, have some ferromagneticcouplings.17 That is, bond-centered configurations consist ofantiferromagnetic patches which are ferromagneticallycoupled across the domain wall. As shown in Fig. 1, we

consider stripes aligned with the lattice direction (calledvertical stripes) or aligned along the lattice diagonals(called diagonal stripes).

In this article we focus on the spin wave spectra and ex-pected magnetic scattering intensities of bond-centered andsite-centered stripe phases of various spacings and orienta-tions. Other stripe phases are certainly possible, such asphases which mix site- and bond-centered domain walls, orphases in which the spacing of the antiphase domain walls isnot commensurate with the underlying lattice, or dynamicstripes,18 which fluctuate in time. We will not consider thesecases here, but focus on well-ordered spin stripes which havepurely site- or bond-centered domain walls. As we will showbelow, there are qualitative differences between the spinwave spectra of bond- and site-centered domain walls, indi-cating that in some cases inelastic neutron scattering may beable to distinguish between the two. In addition, there is adifference in the number of peaks in the elastic spin structurefactor for odd stripe spacings, indicating that elastic neutronscattering alone may be able to distinguish as well.

II. MODEL

We consider static, ordered arrays of antiphase domainwalls in an otherwise antiferromagnetic texture. Although the

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domain walls collect charge,1922 we neglect this chargecomponent, as we are interested solely in the response of thespin degrees of freedom. We use a Heisenberg model on atwo-dimensional square lattice:

H=1

2

r,r

Jr,rSrSr, 1

where r , r runs over all spin sites, and the exchange cou-pling is Jr,r. Within an antiferromagnetic patch, nearestneighbor couplings are antiferromagnetic with Jr,r=Ja0.Couplings across a domain wall depend upon the configura-tion and are enumerated below. All other couplings are ne-glected. When comparing to the nickel oxides (copper ox-ides), our lattice corresponds to the nickel (copper) siteswithin the nickel-oxygen (copper-oxygen) planes.

A. Vertical stripes

We consider first the case where stripes run parallel to theNi-O (Cu-O) bond direction; we call these vertical stripes.As illustrated in Fig. 1, when the domain wall is centered ona lattice site, we may describe the system as having no netspin on the domain wall.23 In this case, spins from the edgesof neighboring antiferromagnetic patches are coupled acrossthe domain wall antiferromagnetically, Jr,r=Jb0 with Sr=0 on the domain wall, as illustrated in Fig. 1(a). Within theantiferromagnetic patches, nearest neighbor spins are ofcourse also antiferromagnetically coupled, Jr,r=Ja0.When, however, the domain wall is bond-centeredthat is,situated between two sites as in Fig. 1(b)spins from theedges of neighboring antiferromagnetic patches are ferro-

magnetically coupled, and we have Jr,r=Jb0 across thedomain wall. Nearest neighbor exchange couplings withineach antiferromagnetic patch remain antiferromagnetic,Jr,r=Ja0. We shall see that this ferromagnetic coupling Jbof spins across the domain wall leads to distinctive featuresfor the spin waves in the bond-centered case.

We define the magnetic Bravais lattice as follows.24 Let p

denote the distance between domain walls. We will hence-forth work in units where the square lattice spacing a =1. Forp = odd, we choose the basis vectors A1 =p , 0 and A2= 0 , 2, and for p = even, we use A1 = p , 1 and A2 = 0 , 2.For site-centered configurations, there are N= 2p sites withineach unit cell which include 2p 1 spins and 2 sites with nostatic spin component. For bond-centered domain walls,there are N= 2p spins in each unit cell. (See Fig. 2.)

We use the notation VSp and VBp to refer to verticalstripes of spacing p in a site (S)- or bond (B)-centered con-figuration, respectively. For example, VS3 refers to a verticalsite-centered configuration with spacing p =3 between do-main walls.

B. Diagonal stripes

For diagonal stripes, the antiphase domain walls are ori-ented along the 1 , 1 direction in a square lattice (recallwe have set the lattice spacing a = 1). For the same micro-scopic interaction strengths (deriving Jr,r from, e.g., a Hub-bard model), spins are more strongly coupled across the do-main wall than in the vertical case. For example, withdiagonal bond-centered stripes, each spin neighboring thedomain wall interacts with two nearest neighbor (ferromag-netically coupled) spins across the domain wall, as shown in

FIG. 1. (Color online) (a) Site-centered vertical stripe pattern with p =4 lattice constants between domain walls. In this configuration,

exchange couplings Ja0 and Jb0 are all antiferromagnetic. (b) Bond-centered vertical stripe pattern with spacing p =4. The exchange

coupling Ja0 is antiferromagnetic, while Jb0 is ferromagnetic. (c) Diagonal site-centered domain walls have coupling Jb0 for nextnearest neighbor spins coupled across the domain wall along the vectors (2,0) and (0,2), and coupling Jc0 diagonally to Manhattan

second neighbors across the domain walls along the vector (1,1), in units where the square lattice spacing a = 1. (d) Diagonal bond-centered

domain walls have nearest neighbor ferromagnetic coupling Jb0 across the domain wall. The size of each figure has been chosen for visual

clarity.

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Fig. 1(d

). Contrast this with the vertical stripes of Figs. 1

(a

)

and 1(b), where each spin neighboring a domain wall inter-acts with only one spin across the domain wall. Diagonalsite-centered stripes are even more strongly coupled, withtwo different types of interactions across the domain wall,one of which we label Jb because it connects spins along abond direction [connecting spins along the vectors (2,0) and(0,2) across the domain wall], and the other we label Jc [con-necting spins along the vector (1,1) across the domain wall],as shown in Fig. 1(c).

For diagonal stripes, the magnetic Bravais lattice differsfrom the vertical case. For p = odd spacing between domainwalls, we choose the basis vectors A1 =p , 0 and A2=1,1, and for p = even, we use A1 = 2p , 0and A2 = 1,1. For site-centered configurations, when p iseven there are N= 2p sites within each unit cell which in-cludes 2p 1 spins and 2 sites with no static spin compo-nent, and when p is odd, there are N=p sites within each unitcell, which includes p 1 spins, and one empty site. Forbond-centered domain walls, there are N= 2p spins in eachunit cell when p is even, and there are N=p spins in the unitcell when p is odd. (See Fig. 7.)

We use the notation DSp and DBp to refer to diagonalstripes of spacing p in a site (S)- or bond (B)-centered con-figuration, respectively.

III. SPIN WAVE THEORY

The elementary excitations of ordered spin textures maybe studied using the well-known technique of Holstein-Primakoff bosons. The same dispersion is obtained by quan-tizing the classical spin waves, and the methods are equiva-lent as S. We use each description when convenient. Asit is physically more transparent, we review here the lattermethod,25 discussing the former in Appendix A.

In the classical spin wave approach, each spin is treated asprecessing in the effective field produced by its coupledneighbors, via the torque equations of a spin in a magnetic

field.

25

The rate of change of the spin at positionr

is de-scribed by

dSr

dt= rHr

ef f, 2

where r and Href f are, respectively, the corresponding mag-

netic moment and effective magnetic field at position r, de-fined by

r = gBSr

Href f =

1

gBr

Jr,rSr. 3

Within our model, Eq. (1), the torque equations become

dSrx

dt=

1

Sryr

Jr,rSrz

Srz

r

Jr,rSry

dSry

dt=

1

Srzr

Jr,rSrx

Srx

r

Jr,rSrz

dSrz

dt 0, 4

where we have assumed large S and small oscillations, so

that changes in Sz can be neglected. We seek solutions of theform

Srx = Si

x expik r t ,

Sry = Si

y expik r t , 5

where i labels spins within the unit cell, i.e., i = 1 , 2 , . . . ,N; Nis the total number of spins in the unit cell; k=kx , ky, andr=rx , ry. Setting the determinant of the coefficients of Si

x

and Siy to zero yields the dispersion relations for the spin

wave.

FIG. 2. (Color online) Vertical

site- and bond-centered configura-

tions, showing even and odd spac-

ing. S refers to site-centered

configurations, and B refers to

bond-centered configurations. Thenumber label is the spacing p be-

tween domain walls. Dotted verti-

cal lines mark antiphase domain

walls. The solid boxes denote unit

cells. The height of the arrows

represents the net spin on a site,

which is expected to peak be-

tween domain walls.

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We calculate the zero-temperature dynamic structure fac-tor using Holstein-Primakoff bosons:

Sk, = f

i=x,y,z

fSik02 f. 6

Here 0 is the magnon vacuum state and f denotes the finalstate of the spin system with excitation energy f. Since S

z

does not change the number of magnons, it leads to the elas-tic part of the structure factor. Single magnon excitationscontribute to the inelastic response through Sxk and Syk.

IV. RESULTS FOR VERTICAL STRIPES

We begin with our results for ordered, vertical stripephases. We discuss magnon excitation energies as functionsof momentum, the dynamic spin structure factors, the elasticresponse, the velocities of the acoustic bands, and analyticresults for dispersion relations for small unit cell sizes. Fig-

ure 2 shows schematic representations of vertical stripes thatare site- and bond-centered, with both even and odd spacing.In this figure (in contrast to Fig. 1) we have used the lengthof the arrow to represent the net spin on a site. The net spinis expected to be smaller near domain walls (as it is alwayszero on a domain wall). Our zero frequency results incorpo-rate this general form factor. For the finite results, we usea form factor with the same net spin on each occupied site.

A. Elastic peak at 0 ,

Elastic neutron scattering can in principle detect one im-portant qualitative difference between bond- and site-

centered stripes. For odd stripe spacings, both bond- andsite-centered stripes have magnetic reciprocal lattice vectorsat 0 ,. However, site-centered stripes are forbidden fromproducing weight at 0 ,, whereas bond-centered stripesgenerically show weight at this point. This is related to thediscrete Fourier transform of the spin structure. Taking ad-vantage of the antiferromagnetic long range order in one di-rection and the finite spacing between stripes in the other, wecan describe the spin structure in real space by a function

Szn,m = cosmj=0

j

Aj ei2/pjn 7

=fngm , 8

where m is the discrete y coordinate parallel to the stripes, nis the discrete x coordinate perpendicular to them, and wherej=p 1 for p odd, with j= 2p 1 for p even. The functionsfn and gm are shown schematically in Fig. 3. The elasticscattering cross section is proportional to the Fourier trans-form of Szn , m:26

ddel m,n eikmm+knnSzm,nSz0, 0

= m

eikm m cosmj=0

j

Ajn

eiknnei2j/pn

= Nmkm, + km,j=0

j

Ajn

Nnkn,2j/p. 9

We emphasize that this expression allows for any form factor

and is not restricted to configurations where each occupied

site has a full quantum of spin. In the case where each occu-

pied site has the same net spin, the ratio of intensity at themain peaks /p , to that at 0 , is 2 in the VB3case, and 2.6 in the VB5 case. Site-centered stripes always

have A0 =0, while A0 is generically nonzero for bond-

centered stripes (although it can be fine-tuned to zero). A

finite j = 0 term produces elastic weight at 0 ,. This can beunderstood heuristically from considering the function fn,shown schematically for the VB5 case in Fig. 3. Odd-spacingbond-centered stripes generically have a net magnetization inthe function fn, while symmetry forbids this for site-centered stripes.

FIG. 3. (Color online) Schematic representation of vertical

stripes with p = odd widths, indicating the pattern of the functions

gm and fn. Note that for the bond-centered case with odd stripespacings, the function fn can have a net magnetization, producingelastic weight at the peak 0 ,.


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B. Analytic results for small p

For small p, which corresponds to small unit cell sizes,we can obtain analytic results for the dispersion relation ofthe acoustic mode. For the case VS3, we find

VS3Ja S

2 = 4 + 1 + C 4 + 1D , 10where

= JbJa ,

C= 2f3kx + 8fky 4f2ky,

D = 1 fky1 2f3kx + 12 11and the function f is defined as

fx = 1 cosx. 12

The acoustic spin wave velocity parallel to the stripe di-

rection k y may be obtained by setting kx =0 above, andtaking ky1. In this case, fkx = 0, fky

1

2 ky2, and VB2

vky, where

v =1

2 + 3 vAF, 13

and vAF= 22JaS is the velocity of the pure antiferromagnetwith coupling Ja and no antiphase domain walls. The spinwave velocity perpendicular to the stripe direction may besimilarly obtained:

v =32

4 + 3

+ 1vAF. 14

For

1, these approachv 3 /2

vAF and

v

1 / 2vAF.For the case VB2, the problem reduces to diagonalizing a

44 matrix, with the result

VB2Ja S

2 = 22 + 3 + 2 + A 22 + 3 + 22 + B15

where

A = 2f2ky,

B = 12

2f4kx 4fky 42 + 3f2kx

4fky1 fky + 2 + 31 f2kx . 16

The spin wave velocities in the case VB2 are

v =32vAF, 17

independent of, and

v = 32 + 1

vAF. 18

For 1, we note that v saturates at

v32vAF. 19

That v saturates with large is in contrast to the behaviorof site-centered cases and can lead to rather isotropic spinwave cones for the bond-centered case,3,4 despite local mi-croscopic anisotropy. As discussed in the next section, forbond-centered stripes with any spacing p, v is independent

of and v saturates with large .

C. Numerical results

For most values of the stripe spacing p, the spin wavematrices are sufficiently large that one must use numericaldiagonalizations to obtain the dispersion relations of the vari-ous modes. From the corresponding eigenfunctions we canthen also calculate the spectral intensity (proportional to thedynamic structure factor) that these magnon states wouldcontribute to the inelastic neutron scattering. Figures 4 and 5show the calculated dispersion and scattering intensities forsite- and bond-centered vertical stripes of various spacings.Our results for site-centered stripes are consistent with those

of Ref. 24. For the site-centered case, bands never cross for1. At the critical couplings =1 and =2.5, site-centeredbands appear to cross. Away from these couplings, verticalsite-centered bands generally repel rather than cross. For =1, the dispersion is very similar to that of a pure antiferro-magnet, albeit with different magnetic reciprocal lattice vec-tors. For any coupling , as p, the result for a puretwo-dimensional antiferromagnet is recovered. For p increas-ing but finite, the number of bands as well as the number ofreciprocal lattice vectors increases. However as p, allspectral weight is transferred to the response of a pure anti-ferromagnet.

Figure 5 shows representative results for vertical bond-

centered antiphase domain walls with spacings p =2, 3, and4. In this case, the critical point where bands appear to touchis at c 0.56 and is at most only weakly dependent on p.Away from the critical coupling, bands never appear to cross,but rather level repulsion is observed. There are other notabledifferences between the site- and bond-centered cases. Forone thing, for the same spacing p, bond-centered configura-tions yield one more band: site-centered configurations havep 1 bands, whereas there are p bands for bond-centeredconfigurations.

A qualitative difference between the two cases is the scal-ing of the band energies with coupling . For site-centeredconfigurations, all bands increase their energy monotonicallywith the coupling ratio . This is in contrast with the bond-centered case, where for large Jb, only the top band is af-fected by the ferromagnetic coupling (that is, it increaseslinearly with ), but all other bands saturate as is in-creased. The behavior of the top band can be understood byconsidering the spins that are ferromagnetically coupledacross the domain wall. In the top band, these spins precess out of phase with each other, and the dispersion is domi-nated by the behavior of the effective ferromagnetic dimers,yielding 2JbS/ as Jb, as shown in Appendix B.

An important consequence of the saturation of the lowerbands as gets large in the bond-centered cases is that the


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low-energy spin wave velocities alone, v and v, cannotreadily be used to extract the relation between the bare ex-change couplings Ja and Jb. We explore this point in moredetail in the next section.

In Fig. 6 we present the spin wave velocities perpendicu-lar v and parallel v to the stripe orientation for theacoustic (lowest) bands as functions of the coupling constantratio . These are compared to the reference velocity, vAF ofthe pure antiferromagnet, which is independent of the cou-pling and equivalent to p.

While in both the site- and bond-centered cases [Figs. 6(a)and 6(b), respectively] the perpendicular velocity depends onthe coupling ratio, in the bond-centered case v rapidly satu-rates to a value close to v for large . As a consequence, thevalue of the coupling ratio =Jb/Ja cannot be determinedsolely by the ratio of the acoustic velocities but requires ad-ditional information, such as vAF.

The curves of v and v cross at =1 for the bond-centered case, apparently independent of p for the widths wehave studied. The crossing is at most weakly dependent on pin the site-centered case, occurring at =2/7 for DS3, and at=0.3 for DS4. For all spacings studied, we find that in thebond-centered case, v is independent of the coupling andthat v rapidly saturates with large . As p gets larger, bothof these velocities approach vAF. For the VB3 configuration,v=0.9vAF, independent of . Notice that the independenceof v upon and the rapid saturation of v as becomeslarger than 1 means that bond-centered configurations canproduce rather isotropic spin wave cones.3,4

V. RESULTS FOR DIAGONAL STRIPES

Figure 7 depicts representative diagonal configurations,for site- and bond-centered domain walls and with even andodd spacing. As mentioned in Sec II B, for a given micro-

scopic model, diagonal stripes are more strongly coupledacross the domain wall than vertical stripes. In addition,there are more parameters to consider for site-centered diag-onal stripes: we must include Jc as well as Jb (see Fig. 1),since both couplings appear to the same order if derivedfrom, e.g., a Hubbard-like model.

A. Elastic peak at (0,0)

Like their vertical counterparts, bond-centered diagonalstripes can produce new peaks in the elastic response. Withdiagonal stripes the new weight is physically transparent. Forall bond-centered domain walls, nearest-neighbor spins areferromagnetically coupled across the wall, and in the diago-nal case, nearest neighbor pairs along a single domain wallall have their moments pointing in the same direction, lead-ing to a domain wall magnetization. As Fig. 7 illustrates, fordiagonal stripes with even p, adjacent domain walls havealternating signs of the magnetization. But diagonal stripeswith odd spacing have the same magnetization direction oneach domain wall. This generically leads to net ferromag-netism and a peak at (0,0), unless parameters are fine-tuned.In a three-dimensional antiferromagnet (as may happen,17

e.g., with weakly coupled planes) domain walls are two di-mensional (planar), and this peak appears either at 0 , 0 , if

FIG. 4. Spin wave spectra and intensities for vertical, site-centered stripes. All spectra are reported at ky = as a function of the transverse

momentum kx. The frequency is in units of JaS. Apparent crossings only occur at =1 and =2.5.


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the diagonal in-plane stripes lie directly on top of each otherfrom plane to plane (meaning there is also no net magneti-zation on a domain wall), or at (0,0,0) if the stripes are di-agonal within a plane and in their correlation from plane toplane.

B. Analytic results for small p

As for the case of vertical stripes discussed in Sec. IV B,for small p, it is possible to obtain analytic forms for theacoustic dispersion relations for diagonal stripes in both the

site- and bond-centered cases.For the case DS3, the analytic dispersion is

DS3JaS

2 2 = fkx ky + 2f2kx ky + 2cfkx ky+ + cf2kx + ky + fkx + 2ky 20

+ f3kx + f3ky , 21

where c = Jc/Ja, and where the function f is defined as

fx = 1 cosx , 22

as in Sec. IV B.The dispersion perpendicular to the stripes, along the k

= kx , kx direction, is then

kx,kx

JaS= 82 + csin3kx

2 , 23

which yields for the velocity in that direction

v =32 + c

22vAF, 24

which approaches v3

22 vAF for large , and v

3

2c vAF for large c.

FIG. 5. Spin wave spectra and intensities for vertical, bond-centered stripes. All spectra are reported at ky = as a function of the

transverse momentum kx. The frequency is in units of JaS.


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In the parallel direction kx , kx, the dispersion becomes

1

8kx, kx

JaS2 = 1 + + c + cos kx + cos2kx 25

1 + 2 + 2 cos kxsin2 kx

2, 26

which gives

v =1 + 42 + 2 + c

22vAF. 27

This approaches vvAF for large , and vc/ 8 vAFfor large c.

For the case DB2, the analytic dispersion is

DB2

JaS 2

2 = 4

1 +

+ A

161 +

2

+ B

28where

A = 1 2fkx ky,

B = 81 + 2fkx ky + 2 fkx ky2fkx ky

2f2kx 2ky f3kx + ky fkx + 3ky . 29

Perpendicular to the stripes, along k = kx , kx, the velocity is

v = + 1

vAF, 30

saturating to vvAF as 1. In the direction k = kx , kx,parallel to the stripes, the velocity is

v =1

2 + 1vAF. 31

C. Numerical results

In Fig. 8, we plot the dispersion and intensities for DS3and DS4 along kx , kx for various values of the couplingratio = Jb/Ja, setting Jc = 0. (See Fig. 1 for the definitionsofJb and Jc.) Similar results using a Jc only model (i.e., withJb = 0) are reported in Ref. 24. Our results show similar bandstructures but with critical coupling =1, which is only half

of the Jc only model.However, the effects of Jb and Jc depend upon the direc-

tion in k space. In Fig. 9 we show the effects of varying thecouplings Jb and Jc for two cuts in momentum space forDS3. For a cut perpendicular to the stripe direction, Jb and Jchave more or less the same effect, although since Jb couplesmore spins than Jc, it has a more dramatic effect. Increasingeither coupling broadens the bandwidth in a roughly linearmanner with negligible effect on the shape. However, for the

FIG. 6. Spin wave velocities for (a) VS3 and (b) VB3 as a

function of the coupling ratio . The solid curves in panel (a) are

analytic results for VS3 calculated in Sec. IV B. Symbols in both

(a) and (b) are numerical results. The velocities parallel and perpen-

dicular to the stripe direction are equal to each other for =2 /7 and

=1 in the site- and bond-centered cases, respectively. Qualitatively

similar behavior is found for other stripe spacings.

FIG. 7. (Color online) Diago-

nal site- and bond-centered con-

figurations, showing even and odd

spacing. Dotted lines denote do-

main walls. Solid parallelograms

denote unit cells.


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cut kx , 2kx, we see that the presence of Jb produces inflec-tion points when Jc =0, and can produce flat-topped disper-sions if Jc is included as well.

We show in Fig. 10 the calculated dispersion relations andintensities for the bond-centered diagonal case, for spacingsp =2, 3, and 4. As in the vertical case, the number of bands isequal to p. A striking difference in the spectra of odd spac-ings is seen, as the net ferromagnetism in the system changesthe low-energy character of the spin waves from a linear(antiferromagnetic-like) to a quadratic (ferromagnetic-like)

dispersion. Rather than the band repulsion observed in thevertical case (except at finely tuned values of the coupling),crossing of optical bands is generic in the bond-centered di-agonal case. Note the ability of optical bands to cross, indi-cating a difference in symmetry for the crossing bands. Alsoevident in the dispersion of DB4 is the downturn of theacoustic band at 2p magnetic reciprocal lattice vectors, twiceas many as in the odd case. (See Sec. II B.) This is expectedbecause of the doubling of the unit cell necessary to accom-modate even spacing. Note, however, that spectral weight is

FIG. 8. Dispersion and intensities for DS3 and DS4 along kx , kx, direction with Jb only. The frequency is in units ofJa S. For all plots,Jc = 0.

FIG. 9. Dispersion and intensities for DS3

along kx , kx and kx , 2kx, directions, compar-ing the effects of Jb and Jc. The frequency is in

units of JaS.


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forbidden at these extra reciprocal lattice vectors, includingthe , point.

In Fig. 11, we plot the spin wave velocities for DS3.When Jb and Jc are both finite, there is a wide range of

couplings for which the spin wave velocities parallel and

perpendicular to the stripes are nearly equal, while this ap-proximate isotropy is confined to a narrow range of ifeither Jb or Jc is zero. Figure 12, which presents v and v

FIG. 11. Velocities parallel v and perpendendicular v to the stripe direction, as compared to vAF, for DS3. In the first panel, Jc=0, and the velocities are plotted as a function of b =Jb/Ja. In the second panel, Jb =0, and the velocities are plotted as a function of c=Jc/Ja. In the third panel, Jb =Jc, and the velocities are plotted as a function ofb =c.

FIG. 10. Dispersion and intensities for diagonal bond-centered domain walls along kx , kx direction, at =0.1,0.5,1.0, for p =2, 3, and4. The frequency is in units of JaS.


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for the case DB4, shows the characteristic saturation of vwith large for bond-centered stripes.

VI. EXPERIMENTAL IMPLICATIONS

We have shown that for a certain class of nontrivial spinorderings on a lattice, the spin wave response is sensitive tothe microscopic placement of the antiphase domain walls.Furthermore, even elastic neutron scattering can in principledistinguish site- from bond-centered for odd stripe spacings,whether vertical or diagonal.

While both site- and bond-centered odd width verticalstripe configurations will produce elastic weight at/p ,, only configurations that are phase-shiftedfrom the site-centered configuration (e.g., a bond-centeredconfiguration) are capable of producing weight at 0 ,, andthe observation of this peak along with peaks at

/p ,

would rule out a site-centered vertical con-figuration. A similar ferromagnetic peak, i.e., at (0,0), wouldrule out site-centered diagonal stripes.17

Figures 6 and 12 illustrate another important implicationfor experiments: The transverse spin wave velocity v in theacoustic band saturates for large in the bond-centered casefor both vertical and diagonal stripes. In fact, all but the topband in the bond-centered case saturate and become indepen-dent of Jb for large Jb. As noted above, this unfortunatelymeans that an estimate of =Jb/Ja cannot necessarily bediscerned directly from the ratio v/v but requires eitherindependent knowledge of whether the stripes are site- orbond-centered, or an appropriate estimate of the bare cou-pling (from, e.g., vAF).

A prominent piece of phenomenology in the cuprates isthe resonance peak observed in neutron scattering,2729

which is the presence of extra scattering weight appearing at, at finite frequency, typically of order 40 meV. Oneproposal is that this may be due to spin waves crossing.30,31

We note that for vertical stripes, spin waves generically repeland appear to cross only at finely tuned values of the cou-pling. For site-centered configurations, this corresponds to=1 and 2.5, while for bond-centered configurations, thecritical coupling is near 0.56. However, a finite energyresolution measurement would not be able to distinguish ac-

tual crossings from near crossings. In the bond-centered casewith large , the first optical mode has more weight at ,than the acoustic band, which would tend to leave the weightnear the elastic incommensurate peaks /p , ratherdisconnected from what might be called a resonance peakin this configuration. We also note that our calculations showthat band crossings are more generic in the presence of di-agonal stripes than vertical stripes.

The nickelate compound La1.69Sr0.31NiO4 shows evidencefrom neutron scattering of diagonal stripes with spacing p= 3.3,4 As Sr is substituted for La, holes are doped into theNiO2 planes. Neutron scattering has been used to map outthe acoustic spin wave dispersion for this material. The datareveal rather isotropic spin wave cones, i.e., that v and vare rather similar, with v1.030.06vAF and v0.860.06vAF,3 where vAF is the acoustic spin wave ve-locity of the undoped antiferromagnet. For the DS3 state, ifwe include only Jb or only Jc, we find no coupling strength for which these two relations can be simultaneously satisifed.The presence of the two couplings together, as shown in Fig.11 with Jb =Jc, can account for the proper relationship among

the velocities, but only for a small range of rather smallcoupling ratio. As a general trend, we find approximate isot-ropy of the spin wave cones to be more robust for bond-centered stripes (in both vertical and diagonal cases), and soone might suspect bond-centered stripes could be responsiblefor the near isotropy of the spin waves in this material. How-ever, as we have shown, the DB3 configuration yields a fer-romagnetic spin wave dispersion, which is certainly not sup-ported by the data.

In the related compound La2NiO4.133,17 signatures of spin

stripes have been detected in neutron scattering. Incommen-surate peaks are observed to persist up to a temperature Tm,above which magnetic peaks indicative of stripe structure

can be regained by application of a 6T magnetic field. Thefield-induced stripe spacing (both above and slightly belowTm) is smaller than the zero-field stripe spacing observedbelow Tm. As noted by the authors,

17 the ferrimagnetic re-sponse is naturally explained by bond-centered stripes. In thehigh temperature field-induced stripe phase, the diagonalstripes have spacing p =3. Our results in Fig. 10 suggest thatthis field-induced transition should be accompanied by a dra-matic change in the low-energy spin wave dispersion, fromlinear to quadratic.

We have also shown that (as in the site-centered case24)the number of bands in a bond-centered configuration is setby the number of spins in the unit cell, rather than by thespacing p. Generally, for both vertical and diagonal stripes,

site-centered stripes have p 1 spin wave bands, and bond-centered stripes have p bands. The exception is the case ofdiagonal site-centered stripes with odd spacing p, which has1

2p 1 bands. An experimental consequence of this is that

for a given value of p, bond-centered stripes have p spinwave bands, whereas site-centered stripes have at most p 1 bands. Although not yet observed experimentally, thismeans that the upper bands can also be used to distinguishsite- from bond-centered stripes. Finding p bands along withincommensurate peaks indicative of spacing p would rule outsite-centered stripes. For diagonal odd width stripes, the

FIG. 12. Velocities parallel v and perpendendicular v tothe stripe direction, as compared to vAF, for DB4.


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threshold is even lower. For, e.g., DS3, only one spin waveband is expected, whereas for DB3, we expect to find threebands. The observance of a second band (or eqivalently aspin wave crossing) for diagonal p =3 stripes would rule outa site-centered configuration. Of course, negative evidence isdicier, and the observance of the smaller number of bandscannot distinguish the two, as it cannot rule out the possibil-ity that the top band is too faint to be observed.

VII. CONCLUSIONS

In conclusion, we have studied regular arrays of antiphasedomain walls in two-dimensional Heisenberg antiferromag-nets and find that their location relative to the lattice, i.e.,whether they are site-centered or bond-centered, producesdistinct effects which may be measurable in a diffractionprobe such as neutron scattering. In particular, arrays of odd-width, bond-centered antiphase domain walls genericallyproduce more elastic peaks than site-centered stripes. In ad-dition, bond-centered stripes generically produce more bandsthan site-centered stripes. We further find that low-energyspin wave velocities are not always directly related to the

exchange couplings in the model, and in particular for bond-centered configurations, rather isotropic spin wave cones arepredicted for a wide range of parameters.

ACKNOWLEDGMENTS

It is a pleasure to thank I. Affleck, A. Castro Neto, S.Kivelson, B. Lake, Y. S. Lee, S. Rosenkranz, A. Sandvik, andJ. Tranquada for helpful discussions. This work was sup-ported by the U.S. Government and by Boston University.

APPENDIX A: SPIN-WAVE METHODS

We rewrite the Hamiltonian equation (1) using the ladderoperators:

H=1

2

r,r

Jr,rSrzSrz + 12 Sr+Sr + SrSr+ . A1We now replace the spin operators by Holstein-Primakoff(HP) bosons32

Si+ = 2S ai , Si = 2S ai+, Siz = S ai+ai , A2

for odd sites i occupied by a spin up, and

Si+ = 2S ai+ , Si = 2S ai, Siz = S + ai+ai, A3

for even sites i occupied by a spin down. Here, i labels eachspin within a unit cell, i.e., i = 1 , 2 , . . . ,N, where N is thenumber of spins in the unit cell. We use odd i to representSz = spins and even i for Sz = spins. We Fourier transformthe bosonic operators via

aik =1

n rodd iare

ikr ,

ajk =1

n reven iare

ikr . A4

Finally, we get the Hamiltonian in momentum space

H= ij

Ai,j ai+kajk +

1

2

ij

Bij ai+kaj

+k

+ Bij*

ajkaik , A5

where A =A+ and BT=B.The quadratic Hamiltonian (A5) can be diagonalized via a

canonical symplectic transformation33 T, b = Ta, using the

bosonic metric

= I 00 I

, A6where I is the NN indentity matrix. This leads to

Hk =

b+kkbk + 12k . A7We now consider the structure factor. Only Sx and Sy con-

tribute to the inelastic part of the structure factor. In terms ofHP bosons,

Skx

=

1

2 Sk+

+ Sk

= S

2 ioddai+

k + aik

+ ieven

ai+k + ai k . A8

We then substitute Eq. (A8) into the structure factor andkeep only the creation operators b1

+k , b2+k , . . . , which

connect the vacuum to singly excited states. This gives

Sink, = 2f

fSkx 02 f

= S1i

ibi+02 = S

i

i2 , A9

where i is the ith component of the (orthonormalized) ei-genvector of the Hamiltonian using the bosonic metric,corresponding to eigenvalue .

APPENDIX B: DIMERIZED SPIN MODEL

We consider an isolated system of two spins with ferro-magnetic coupling Jb. In the ground state, the two spins arealigned. When the spins tilt a bit, each produces an effectivefield acting on the other. Using the classical spin wavemethod, we have

dS1

dt=

Jb

S1 S2,

dS2

dt=

Jb

S2 S1. B1

Ignoring the change in Sz, the x, y components of the twospins satisfy

dS1x

dt=

dS2x

dt=

JbS

S1

y S2y,

dS1y

dt=

dS2y

dt=

JbS

S1

x S2x. B2

Integrating yields S1x,y = S2

x,y + c, where c is a constant of in-tegration. Since we allow only Sz to have a constant compo-


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nent, c =0. Taking the second derivative of S1x, we find

d2S1x

dt2=

JbS

dS1y

dt

dS2y

dt

= JbS

2

JbSS1x S2

x + JbSS1x S2

x

= 2J

b

2S2

2 S1x S2x = 4

Jb

2S2

2 S1x , B3

which is a harmonic oscillator equation. If we set S1xx , t

= uxeit, we see that the oscillation frequency is

= 2Jb

S, B4

which recovers the large Jb limit of Eq. (15).

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E. W. Carlson, D. X. Yao and D. K. Campbell- Spin waves in striped phases

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Transcript of E. W. Carlson, D. X. Yao and D. K. Campbell- Spin waves in striped phases