Quantum dynamics of a domain wall in a quasi one-dimensional $XXZ$ ferromagnet

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by  Pavel Tikhonov

Quantum dynamics of a domain wall in a quasi one-dimensional XXZ ferromagnet
Pavel Tikhonov1 and Efrat Shimshoni1
1Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
We derive an effective low-energy theory for a ferromagnetic (2N +1)-leg spin-1 ladder with
strong XXZ anisotropy (cid:12)(cid:12)J(cid:107)z(cid:12)(cid:12) (cid:28) (cid:12)(cid:12)(cid:12)J(cid:107)xy(cid:12)(cid:12)(cid:12), subject to a kink-like non-uniform magneti2c field Bz(X)
which induces a domain wall (DW). Using Bosonization of the quantum spin operators, we show
that the quantum dynamics is dominated by a single one-dimensional mode, and is described by
a sine-Gordon model. The parameters of the effective model are explored as functions of N, the
easy-plane anisotropy ∆ = −Jz/Jxy, and the strength and profile of the transverse field B (X).
(cid:107) (cid:107) z
We find that at sufficiently strong and asymmetric field, this mode may exhibit a quantum phase
7 transition from a Luttinger liquid to a spin-density-wave (SDW) ordered phase. As the effective
1 Luttingerparametergrowswiththenumberoflegsintheladder(N),theSDWphaseprogressively
0 shrinks in size, recovering the gapless dynamics expected in the two-dimensional limit N →∞.
2

n
I. INTRODUCTION interactions. In particular, in the recent years there has
a
J been growing interest in systems based on heavy ele-
ments with strong spin-orbit coupling, which pave the
0 Low dimensional quantum systems attract much ex-
3 way to realizing a variety of unconventional spin models.
perimental and theoretical attention, due to the rich
A prominent example is the proposal10 to realize spin
physics arising from their enhanced quantum fluctua-
] interaction terms in Iridate crystals that are effectively
l tions. Mostprominently,quantumeffectsaremanifested
e consistent with the highly anisotropic Kitaev model11,
- by quasi-one dimensional (1D) spin systems, namely, and hence support a quantum spin-liquid ground state
r spin ladders1. Theoretical studies primarily focused on
t in a genuinely two-dimensional (2D) system.
s models of coupled spin-1 chains with anti-ferromagnetic
. 2 An alternative route to the formation of anisotropic
t (AFM) exchange interactions. In addition to being mo-
a exchange interactions naturally arises in systems with
tivated by the existence of real materials, these were in-
m
iso-spin degrees of freedom, such as the layer or valley
spired by the seminal work of Haldane2, which pointed
- indexinbi-layersorbipartitelattices. Afascinatingplay-
d out the crucial distinction between odd and even spin S.
ground for such realizations of quantum spin models is
n Thesamephysicscarriesthroughtomulti-legspin-1 lad-
2 provided by quantum Hall ferromagnetism (QHFM)12,
o der systems3–5, where the number of legs N replaces the
c generalspinS = N intheHaldanechain. Aparticularly established in 2D electron systems subject to a strong
[ 2 magnetic field. Most notable is its manifestation in
striking signature of quantum fluctuations arises in the
graphene: the multi-component nature of the spin/iso-
1 caseofoddN,characterizedbyamagneticallydisordered
spin manifold leads to a plethora of exchange-induced
v ground-statewithpower-lawspin-spincorrelations. This
6 broken symmetry phases, where anisotropy plays a cru-
behavior, which reflects the presence of gapless spin-flip
5 cial role13. Yet another type of system effectively de-
excitations (spinons), is the simplest realization of a so-
5 scribed by inherently anisotropic quantum spin models
called ”spin-liquid” phase6 beyond 1D.
8 is the superconducting (SC) ladder14, which implements
0 Notably, the above described quantum features are a mapping of the complex SC order parameter field to a
1. characteristictoAFMspinladders. Ferromagnetic(FM) Bosonic representation of local spin operators.
0 ladders, on the other hand, are more ”classical” in na- A particularly appealing aspect of the latter two re-
7 tureandintheHeisenberg(SU(2)-symmetric)case,long
alizations is that the control of parameters, as well as
1 rangeFMorderisestablished. Nevertheless,quantumef-
the measurement of physical properties, are accessible
v: fects were observed in certain spin-ladder compounds7, by electric means. Specifically in graphene QHFM,
i and were shown theoretically to yield a rich phase
X electric conduction distinguishes a FM order in the
diagram8,9. A key ingredient promoting quantum fluc- bulk from other broken-symmetry phases15,16 as it sup-
r tuations in such systems is anisotropy in the exchange
a ports a gapless conducting mode. This mode is associ-
interactions of the XXZ type. As a result, the spin sys-
ated with quantum fluctuations of a domain wall (DW)
tem becomes either an easy-plane or easy axis FM, and configuration17,whichformsattheedgeofthesamplein
canundergoaquantumphasetransition,e.g. inthepres-
the FM state and can be modeled as a Luttinger liquid
ence of a transverse field. (LL)18. Due to the spin-charge coupling characteristic
Realistic magnetic materials quite often possess to quantum Hall systems, this mode is relatively pro-
Heisenbergexchangeinteractions,hencemuchoftheear- tected from backscattering and exhibits a nearly perfect
lier literature on quantum magnetism did not regard electrical conductivity19. Similar conducting DW chan-
anisotropy as a significant parameter. However, the cou- nels can also form in the bulk, e.g. in bilayer graphene
pling of real spin to additional degrees of freedom may whereanon-uniformeffectiveZeemanfieldisinducedby
introduce appreciable (and possibly tunable) anisotropic a spatially dependent gate voltage20,21. Derivation of an
2
B
amodelofcoupledferromagneticspinchainsandidentify
z
the U(1) mode that dominates the low-energy behavior.
In Sec. III we detail the derivation of its effective low
energy theory, and the implied phase diagram. Finally,
X we present concluding remarks in Sec. IV.
... ...
B B B =0B B
-2 -1 0 1 2
II. THE MODEL
We consider a quasi-1D model for a FM stripe as a
(2N + 1)-leg ladder of coupled XXZ spin-1 chains in
2
a non-uniform magnetic field oriented along the z-axis,
that exhibits a sign reversal (see Fig. 1). The Hamilto-
nian describing this system is
(cid:88)
H = H +Hint (2.1)
i i
i
FwIaGll.c1o:nfi(Cguorloartioonnliinnea.)sSycshteemmadteisccrreibperedsebnytaEtqio.n(2o.f1)a;dhoemreaXin Hi =(cid:88)J2|x|y (cid:0)Sj+,iSj−+1,i+h.c.(cid:1)+J|z|Sjz,iSjz+1,i−BiSjz,
is a continuum representation of the chain index i. j
Hint =(cid:88)J⊥xy (cid:0)S+S− +h.c.(cid:1)+JzSz Sz ,
i 2 j,i j,i+1 ⊥ j,i j,i+1
effective low energy theory describing the dynamics of j
these modes (and hence their transport properties) re-
quires the quantum analysis of a spin system subject to
non-uniform fields19,22. where all the coupling constants are ferromagnetic
In this paper we address the problem of a generic spin (Jz,Jz,Jxy < 0, while for convenience Jxy > 0 [25]),
(cid:107) ⊥ ⊥ (cid:107)
configuration in the quantum regime, where spin fluc-
i∈−N,...,−1,0,1,...,N isthelegindex,j isasitein-
tuations can not be treated in the framework of spin
dex along the chain and B is a magnetic field acting on
wave approach23. To this end, we analyze the low en- i
chain i. The natural degrees of freedom of unpolarized
ergy dynamics of a smooth DW of an arbitrary finite
spins are Euler angles, which can be easily introduced
width and shape, in a FM with anisotropic exchange in- bybosonization27. Takingthecontinuumlimitalongthe
teractions. This generalizes an earlier study of a sharp chains (Sν →Sν(x)),
DW configuration24, which has been predicted to pos- j,i i
sessaphasetransitiontoaspin-orderedconfigurationas
e∓iθi(x)
a function of anisotropy strength. In the present work S±(x)= √ [cos(2ϕ (x))+cos(2k x)] (2.2)
we model the DW as a multi-leg ladder of ferromagnetic i 2πα i F
XXZ spin-1 chains subject to a kink-like magnetic field 1 1
2 Sz(x)=− ∂ ϕ (x)+ cos(2ϕ (x)−2k x) (2.3)
Bz(X),andderivea1Dlow-energyeffectivetheorywhich i π x i πα i F
allows to explore the dependence of its quantum dynam-
icsonfieldstrengthandshape. Inparticular,weidentify where (cid:2)ϕ (x), 1∂ θ (x(cid:48))(cid:3) = iδ(x−x(cid:48))δ , α is a short
i π x j ij
the regime of parameters where a quantum phase transi- distance cut-off (lattice spacing) and k = π . In terms
F 2α
tion from a LL to spin density wave (SDW) can occur. of the new fields, the original Hamiltonian has the fol-
Thepaperisorganizedasfollows: InSec. IIwepresent lowing form:
3
N ±N
(cid:88) (cid:88)
H = H + Hint, (2.4)
i i
i=−N i=±1...
1 (cid:90) (cid:104) u (cid:105) 2g (cid:90) 1 B
H = dx uK(∂ θ )2+ (∂ ϕ )2 + 3 dxcos(4ϕ −4k x)+ B ∂ ϕ − i cos(2ϕ −2k x)
i 2π x i K x i (2πα)2 i F π i x i πα i F
(cid:34)
Hint =(cid:90) dx 2g1 cos(cid:0)θ −θ (cid:1)+ J⊥zα∂ ϕ ∂ ϕ
±i (2πα)2 ±i ±(i−1) π2 x ±i x ±(i−1)
(cid:35)
+ 2g2 cos(cid:0)2ϕ −2ϕ (cid:1)+ 2g2 cos(cid:0)2ϕ +2ϕ −4k x(cid:1) ,
(2πα)2 ±i ±(i−1) (2πα)2 ±i ±(i−1) F
and g = πJxyα, g = Jzα, g = Jzα; The Luttinger Therefore we include these leading operators in the low-
1 ⊥ 2 ⊥ 3 (cid:107)
liquid (LL) parameters K and u are obtained exactly energy theory and account for all other non-quadratic
(cid:12) (cid:12) terms in Eq. (2.4) as perturbations. We then perform
for an XXZ-chain with (cid:12)Jz(cid:12) < Jxy by a Bethe-ansatz
(cid:12) (cid:107)(cid:12) (cid:107) a transformation D which simplifies the unperturbed
calculation26, yielding for zero magnetic field Hamiltonian. ThematrixDcanberelatedtoacanonical
transformation U on the K-matrix of coupled Luttinger
π
K = , (2.5) liquids. Here we keep the ϕ and θ fields separate as in
2arccos(∆)
the original Hamiltonian (2.4), and define
1 (cid:18) (cid:18) 1 (cid:19)(cid:19)Jxy
(cid:107)
u= sin π 1−
1− 1 2K 2 ϕ¯ =(cid:0)DT(cid:1)−1ϕ , θ¯ =D θ ,
2K i ij j i ij j
Jz
∆≡− (cid:107) , Dn,m =δn,m+δn,0−δn−sg(n),m, (2.10)
Jxy
(cid:107)
where D is the (2N +1)×(2N +1) matrix, n is a row
sothatforferromagneticinteractionsK >1. Itisimpor- number, m is a column number, −N <n,m<N and
tanttopointoutthatthisresultremainsagoodapprox-
(cid:12) (cid:12) 
imation even in a finite magnetic field for (cid:12)(cid:12)J(cid:107)z(cid:12)(cid:12) (cid:28) J(cid:107)xy. 1 m>0
The Gaussian part of the Hamiltonian (2.4) can be con- sg(m)= 0 m=0. (2.11)
veniently written in a matrix form −1 m<0
1 (cid:90) (cid:104) u (cid:105)
H = dx uK∂ θ δ ∂ θ + ∂ ϕ δ ∂ ϕ Note that the transformation Eq. (2.10) preserves the
Gauss 2π x i ij x j K x i ij x j canonical commutators:
Jzα(cid:90)
+ 2⊥π2 dx∂xϕiCij∂xϕj, (2.6) (cid:2)ϕ¯i(x),∂x(cid:48)θ¯k(x(cid:48))(cid:3)=iπδ(x(cid:48)−x)δik (2.12)
where the matrix C couples adjacent modes and is de- (See App. A). The central mode θ¯ ,ϕ¯ corresponds to
0 0
fined as
the symmetric combination of the original fields
C ≡δ +δ (2.7)
ij i,j−1 i−1,j θ¯ =(cid:88)θ , ϕ¯ = 1 (cid:88)ϕ . (2.13)
0 i 0 2N +1 i
in which Einsteinian summation over i,j is implied.
i i
Being in the ferromagnetic region (K >1), the terms
of the form cos(cid:0)θ −θ (cid:1) (parametrized by the Werescaleitinawaythatallowsustointerpretθ¯ asthe
±µ ±(µ−1) 0
coupling constant g ) are the most relevant among the azimuthal angle of an effective spin operator. Namely, it
1
non-Gaussian contributions to H, namely, with the low- encodes a U(1) mode associated with a global rotation
est scaling dimension: angle in the XY-plane. The remaining fields (θ¯ with
i
i (cid:54)= 0) are rescaled in order for the most relevant terms
β2 to be of the form cos2θ˜:
d(cosβθ )(cid:39) (2.8) i
i 4K
1
and hence θ˜i = 2θ¯i, ϕ˜i =2ϕ¯i, ∀i(cid:54)=0, (2.14)
1 1 1
d = < . (2.9) θ˜ = θ¯ , ϕ˜ =(2N +1)ϕ¯ .
g1 2K 2 0 2N +1 0 0 0
4
Intermsofthenewfields,theHamiltonian(2.4)acquires where
the form
H =H +H +Hint
0 b
(cid:88)
H = Hi (2.15)
b b
i(cid:54)=0
Hint =Hint+Hint,
b,b b,0
1 (cid:90) (cid:20) (cid:16) (cid:17)2 u (cid:21) 1 (cid:90)
H = dx u K ∂ θ˜ + 0 (∂ ϕ˜ )2 + B˜ dx∂ ϕ˜ (2.16)
0 2π 0 0 x 0 K x 0 π 0 x 0
0
1 (cid:90) (cid:20) (cid:16) (cid:17)2 u (cid:21) 2g (cid:90) 1 (cid:90)
Hi = dx u K ∂ θ˜ + i (∂ ϕ˜ )2 + 1 dxcos2θ˜ + B˜ dx∂ ϕ˜
b 2π i i x i Ki x i (2πα)2 i π i x i
Hint = (cid:88) 1 (cid:90) dx(cid:20)u4K∂ θ˜ (cid:0)DDT(cid:1)−1∂ θ˜ +∂ ϕ˜ (cid:26) u (cid:0)DDT(cid:1) + J⊥zα(cid:0)DCDT(cid:1) (cid:27)∂ ϕ˜ (cid:21)+(cid:88)(cid:90) dx 2g2 Oˆc
b,b 2π x i i,j x j x i 4K i,j 2π2 i,j x j (2πα)2 i
i(cid:54)=j(cid:54)=0 i(cid:54)=0
(cid:34) (cid:35)
Hint = J⊥zα 1 (cid:90) dx∂ ϕ˜ (∂ ϕ˜ +∂ ϕ˜ )+(cid:88)(cid:90) dx −BiOˆa+ 2g2 Oˆb+ 2g3 Oˆd .
b,0 2π2 2N +1 x 0 x N x −N πα i (2πα)2 i (2πα)2 i
i(cid:54)=0
Here H is the Hamiltonian of the symmetric mode with Oˆb =cos(cid:0)2ϕ +2ϕ (cid:1) (2.20)
0 ±i ±i ±(i−1)
 (cid:16) (cid:17)
(cid:18) (cid:19)1 cos(cid:16)2N4+1ϕ˜0+ϕ˜±(i−1) (cid:17) ;i=N
u0 =u 1+ gu22N(cid:32)N+1π2K 2 (cid:33)1 (2.17) =ccooss(cid:16)22NN44++11ϕϕ˜˜00+−ϕϕ˜˜∓±i(i−−1ϕ)˜−±(iϕ˜+±1)(i(cid:17)+1) ;;i1=<1i<N
1 2
K =(2N +1)K ,
0 1+ gu22NN+1π2K Oˆc =cos(cid:0)2ϕ −2ϕ (cid:1) (2.21)
±i ±i ±(i−1)
 (cid:0) (cid:1)
cos 2ϕ˜ −ϕ˜ ;i=N
 ±i ±(i−1)
and Hb describes a bath of 2N modes; Hbi is the Hamil- = cos(cid:0)2ϕ˜±i−ϕ˜±(i−1)−ϕ˜±(i+1)(cid:1) ;1<i<N
tonian of the i’th mode, which has the form of a sine- cos(cid:0)2ϕ˜ +ϕ˜ −ϕ˜ (cid:1) ;i=1
±i ∓i ±(i+1)
Gordon model where the LL parameters of mode i is
Oˆd =cos(4ϕ ) (2.22)
(cid:18) (cid:19)1 ±i ±i
2(N +|i|)(N −|i|+1) 2  (cid:16) (cid:17)
ui =u 2N +1 (2.18) cos(cid:16)2N4+1ϕ˜0+2ϕ˜±i (cid:17) ;i=N
(cid:18)8(N +|i|)(N −|i|+1)(cid:19)12 = cos 2N4+1ϕ˜0+2ϕ˜±i−2ϕ˜±(i+1) ;1(cid:54)i<N
Ki =K 2N +1 . cos(cid:16)2N4+1ϕ˜0−2ϕ˜1−2ϕ˜−1(cid:17) ;i=0.
Finally,allmodescouplelinearlytoaneffectivemagnetic
Hint and Hint describe interactions within the bath field B˜ where
b,b b,0 i
modes and between the bath and the symmetric 0’th
mode, respectively, in which (cid:40) 1 (cid:80) B i=0
B˜ = 2N+1 j j . (2.23)
i 1(cid:0)B −B (cid:1) i(cid:54)=0
2 i i−sg(i)
Oˆ±ai =cos(2ϕ±i−2kFx) (2.19) This leads to a shift of the Fermi momenta k → ki =
=cccooosss(cid:16)(cid:16)(cid:16)222NNN222+++111ϕϕϕ˜˜˜000++−ϕϕϕ˜˜˜±±1ii−−−ϕ˜2ϕ−˜k±1F(−ix+(cid:17)21)kF−x2(cid:17)kFx(cid:17) ;;;ii1==(cid:54)N0i<N kF2k−Bi k≡Bi ,−w(cid:104)h∂exrϕ˜ei(cid:105)=(cid:40)BB˜˜uu0iKKi0i0=∼BK˜uuiK(cid:80)jBj ii(cid:54)==00F . (2F.24)
5
The Hamiltonian in the form Eqs. (2.15), (2.16) is mode. We start with the already mentioned first part
still rather complicated, as Hint includes many coupling of Hint. In addition to the effective magnetic field, it
b,0
terms between the modes. However, since the cosine induces corrections to the LL parameters (see App. B):
terms cos2θ˜ are highly relevant, the θ˜-fields in the
i i
bath are ordered and their fluctuations are gapped. The 1 (cid:18) Jzα (cid:19)2(cid:90) (cid:90)
masses of the fluctuation fields δθ˜i are given to a good δSiGnat,u2ss (cid:39) (2N +1)2 π2⊥uN d2r1 d2r2
approximation by (see, e.g., App. E2 in Ref. [27])
×(cid:104)∂ ϕ˜ (r )∂ ϕ˜ (r )(cid:105)∂ ϕ˜ (r )∂ ϕ˜ (r ), (3.3)
x ±N 1 x ±N 2 x 0 1 x 0 2
mi = α1 (cid:18)πu8giK1 i(cid:19)2−1K1i . (2.25) winhteegrera(cid:126)tri1o/n2 o=verR(cid:126)a±re12l(cid:126)rati=ve(cid:0)cxo1o/r2d,iunNatτe1/2(cid:126)r(cid:1),.exWpleoitpinergfotrhme
factthatalltheϕ˜ fieldscorrelationsdecayexponentially
i
The marginal Gaussian terms of Hint (which couple the for gapped modes; this yields
b,b
variousbathmodes)arelessrelevantthanthemassterms
and can be treated perturbatively, leading to renormal- δSGauss (cid:39) 1 (cid:18) g2 (cid:19)2 logmN1α
ization of the values of mi. The low-energy dynamics is int,2 (2N +1)2 uN π3
therefore dominated by the gapless symmetric (0) mode, (cid:90) (cid:90)
andisdescribedbyH withcorrectionsresultingfromits × dx dτu (∂ ϕ˜ (x,τ))2. (3.4)
0 N x 0
coupling to the gapped bath modes. In the next section
we analyze these corrections systematically. Hereandintherestofthepaperm istheeffectivemass
i
of the i’th bath mode, accounting for its renormalization
by the marginal coupling terms between the bath modes
III. EFFECTIVE THEORY included in Hint [Eq. (2.16)].
b,b
OthercorrectionstotheLLparameterscomefromthe
Now, we are in a position to inspect the effect of the second part of Hint, namely from the operators Oˆa, Oˆb
b,0 i i
gapped modes on the symmetric 0-mode, and derive an and Oˆd [Eqs. (2.19), (2.20) and (2.22)]. Details of the
i
effective Hamiltonian H0eff in terms of the fields ϕ˜0,θ˜0. derivation are given in App B. Let us start with Oˆia,
ThebareHamiltonianH0 [Eq. (2.16)]describesagapless which results from coupling to the magnetic field:
mode, but couplings to the gapped modes may alter its
behavior. To investigate it, we apply perturbation the- δSa,2 (cid:39)(cid:18) BN (cid:19)2(cid:90) d2r (cid:90) d2r (cid:68)eiφ˜±N(r1)e−iφ˜±N(r2)(cid:69)
oryonHint uptosecondorderinallcouplingconstants, ±N παu 1 2
b,0 N
and write the corresponding correction terms in the ac-
tion. The resulting corrections δS are then obtained ×ei2N2+1φ˜0(r1)e−i2N2+1φ˜0(r2)e−2ikFN(x1−x2)+h.c
0
using a mean-field approximation for the bath operators (cid:18)B α(cid:19)2 1 8
described by Hb with renormalized parameters ui, Ki (cid:39)− uNN (2N +1)2 (mNα)12 π5 (3.5)
and m .
i (cid:90) (cid:90) (cid:20) 1 (cid:21)
To first order, there are only two terms (the first term × dτ dxu −3(∂ ϕ˜ (x,τ))2+ (∂ ϕ˜ (x,τ))2 ,
in Hint) that couple the symmetric mode to the N and N x 0 u2 τ 0
b,0 N
−N gapped modes. They give rise to the following cor-
and for i=±1···±(N −1) we get
rection
(cid:32) (cid:33)2
Jzα 1 B˜ (cid:90) (cid:90) (cid:68) (cid:69)
δSGauss =− ⊥ (3.1) δSa,2 (cid:39) i d2r d2r eiφ˜±i(r1)e−iφ˜±i(r2)
int,1 π2 (2N +1) ±i παu 1 2
i
(cid:90) (cid:90)
(cid:68) (cid:69)
× dτ dx∂xϕ˜0((cid:104)∂xϕ˜N(cid:105)+(cid:104)∂xϕ˜−N(cid:105)), × eiφ˜±i+1(r1)e−iφ˜±i+1(r2) (3.6)
where the expectation values (evaluated with respect to ×ei2N2+1φ˜0(r1)e−i2N2+1φ˜0(r2)e−2ikFi(x1−x2)+h.c
HbN) are proportional to the ”magnetic field” acting on (cid:18)B α(cid:19)2 1 28log2kFi
theN, −N modes[seeEq. (2.24)]. Theresultingcorrec- (cid:39)− i (m α) mi
tion to H can be absorbed in the definition of B˜ (the ui (2N +1)2 i π5
last term0of H0 Eq. (2.16)). Its main effect is to pr0ovide ×(cid:90) dτ(cid:90) dxu (cid:20)3(∂ ϕ˜ (x,τ))2− 1 (∂ ϕ˜ (x,τ))2(cid:21)
an additional shift of the effective Fermi wave vector k0 i x 0 u2 τ 0
F i
by
where
δkB0 ∼(cid:18)J⊥uzα(cid:19)Ku (BN +B−N) . (3.2) 2kFi ≡2(kF −kBi )= απ − B˜uiK (3.7)
We next turn to the second order corrections, which [see Eq. (2.24)]. Here we have used a mean-field approx-
renormalize the Luttinger parameters of the symmetric imation where operators associated with different modes
6
i (cid:54)= 0 are decoupled, and replaced by their expectation where
values. Also, ki isapproximatedbyk tokeepthecoef-
F F g N K
ficients up to the second order in the coupling constants. δ = 2 (3.15)
Corrections from Oˆb have the same form as Oˆa apart 1 u 2N +1 π
i i
form a pre-factor: δ =4πK 1 (cid:88)N δB(cid:18)3ui − u0(cid:19)
2 2N +1 i u u
(cid:18) g (cid:19)2 1 25 i=1 0 i
δS±b,N2 (cid:39)− uN2 (2N +1)2 (mNα)−72 π3 (3.8) δ =−4πK 1 (cid:88)N δg2(cid:18)ui + u0(cid:19)
×(cid:90) dτ(cid:90) dxu (cid:20)(∂ ϕ˜ (x,τ))2+ 1 (∂ ϕ˜ (x,τ))2(cid:21), 3 2N +1 i=1 i u0 ui
i x 0 u2 τ 0 N (cid:18) (cid:19)
N δ =−4πK 1 (cid:88)δg3 ui + u0
4 2N +1 i u u
and for modes i=±1···±(N −1) we get 0 i
i=1
(cid:18)g (cid:19)2 1 2 and
δSb,2 (cid:39)− 2 (m α)−3 (3.9)
±(cid:90)i (cid:90) ui (cid:20)(2N +1)2 i 1 3π3 (cid:21) δB ≡(cid:16)BuNNα(cid:17)2(mNα)12 π25, i=N (3.16)
× dτ dxui (∂xϕ˜0(x,τ))2+ u2i (∂τϕ˜0(x,τ))2 . i (cid:16)Buiiα(cid:17)2(miα)π265 logmπiα i(cid:54)=N

Fthineaolplye,rawteoresvaOˆlud:ate the correction δS±d,i2 resulting from δig2 ≡(cid:16)(cid:16)ugN2(cid:17)(cid:17)222(mNα)−27 π23, i=N (3.17)
i  ug2i (miα)−3 6π13 i(cid:54)=N
δS±d,N2 (cid:39)−(cid:18)ugN3 (cid:19)2 (2N1+1)2 (mNα)−2 π44 (3.10) δg3 ≡(cid:16)ugN3 (cid:17)2(mNα)−2 4π44, i=N (3.18)
i (cid:16) (cid:17)2
×(cid:90) dτ(cid:90) dxu (cid:20)(∂ ϕ˜ (x,τ))2+ 1 (∂ ϕ˜ (x,τ))2(cid:21),  ug3i (miα)−3 π83c i(cid:54)=N.
N x 0 u2 τ 0
N
ThefullexpressionsaregiveninApp. B,Eqs. (B12)and
and (for i(cid:54)=N) (B13). The sign of δ depends on the relative strength of
the coupling constants.
(cid:18)g (cid:19)2 1 25c Apart from modifying the LL parameters of the sym-
δSd,2 (cid:39)− 3 (m α)−3 (3.11)
±i ui (2N +1)2 i π3 metric mode, coupling to the bath degrees of freedom
may induce interactions. Recalling that all correlation
(cid:90) (cid:90) (cid:20) 1 (cid:21)
× dx uidτ (∂xϕ˜0(x,τ))2+ u2 (∂τϕ˜0(x,τ))2 , ftuianlclyt,iointsisofenthoeugdhistoordeexreadmifineeldosnlϕ˜yi(cid:54)=lo0cdalectaeyrmesxpoofntehne-
i
form cos(βϕ˜ ). To do so, we recall the following prop-
0
where c is a real number of order unity.
erty of the expectation value:
Collecting the contributions from Eqs. (3.4)–(3.6)
and (3.8)–(3.11), we obtain a correction (δS02) to the (cid:88)Aij (cid:54)=0⇒(cid:104)(cid:89)ei(cid:80)jAijϕ˜i(rj)(cid:105)=0. (3.19)
bare Gaussian action (S , corresponding to H from Eq.
0 0 j i(cid:54)=0
(2.16)), describing the symmetric mode. The resulting
effective action can be cast in the form This imposes many restrictions on possible interaction
terms of the form
1
S(cid:90)0+δ(cid:90)S02 = 2π(cid:40)K0(cid:48) 1 (3(cid:41).12) (cid:42)(cid:89)(cid:16)Oˆia(cid:17)|nai|(cid:16)Oˆib(cid:17)|nbi|(cid:16)Oˆic(cid:17)|nci|(cid:16)Oˆid(cid:17)|ndi|(cid:43) (3.20)
dx u(cid:48)dτ (∂ ϕ˜ (x,τ))2+ (∂ ϕ˜ (x,τ))2 i
0 x 0 u(cid:48)2 τ 0
0
wherenf withf =a,b,c,dcountthenumberofoperators
i
with modified LL parameters K(cid:48) and u(cid:48). The general ofaspecifictype. Replacingallbathmodesoperatorsby
0 0
expressions for the LL parameters are complicated (see theirexpectationvalues,thisgeneratesatermoftheform
B12 and B13 in App. B). However, to leading order in cos(βϕ˜0) with β = 2N1+1(cid:80)i(cid:0)4nbi +2nai +4ndi(cid:1). After
the coupling constants, K(cid:48) can be cast in the compact imposing all the restrictions [Eq. (3.19)], we obtain
0
form
β =2×n, (3.21)
K(cid:48) (cid:39)K(2N +1)(1−δ), (3.13)
0 where n is an integer number and the full derivation is
(cid:88)
δ ≡ δν (3.14) summarized in App. C. The above analysis only states
ν that cos(βϕ˜0) exists; the specific terms with n = 1,2,
7
which are the most relevant, are generated, e.g., from
1
the following contributions
(cid:42) N (cid:43)
(cid:89) Oˆa ∝cos(cid:0)2ϕ˜ +2k0x(cid:1)
ν 0 F 0.8
ν=−N
(cid:42) N (cid:43)
(cid:89) Oˆd ∝cos(cid:0)4ϕ˜ +4k0x(cid:1). (3.22)
ν 0 F 0.6
ν=−N
Another property, that can be inferred from the analy-
sis of the restrictions Eq. (3.19), is that absolute anti- 0.4
symmetry of the DW configuration, namely, the case
of B = −B , leads to the cancellation of the cos2ϕ˜
i −i 0
terms. Inthefollowinganalysisweassumeamoregeneric
0.2
case, where some asymmetry is always present. 1 2 3 4 5 6 7
The derivation described above leads to an effective
action of the symmetric 0-mode, of a relatively simple
1
form:
Seff =S +δS2 (3.23)
0 0 0
+ 2g(cid:48) (cid:90) dx(cid:90) u(cid:48)dτcos(cid:0)2ϕ˜ +2k0x(cid:1) 0.9
(2πα)2 0 0 F
+ 2g(cid:48)(cid:48) (cid:90) dx(cid:90) u(cid:48)dτcos(cid:0)4ϕ˜ +4k0x(cid:1), 0.8
(2πα)2 0 0 F
where only one of two cosine terms will contribute de-
pending on the corresponding oscillatory factors. Since 0.7
k0 = π/2α−k0 where k0 ∝ B˜ [see Eq. (2.24)], this
F B B 0
depends crucially on the commensuration condition set
bytheeffectivemagneticfieldB˜0. Theexactexpressions 0.61 2 3 4 5 6 7
forthenewcouplingconstantsg(cid:48) andg(cid:48)(cid:48) arecomplicated,
howevertheirvaluesareoflesssignificancetothelowen-
ergy properties and are left unspecified.
The dynamics of the effective theory Eq. (3.23) is de-
FIG.2: (Coloronline.) Aschematicphasediagram. Herethe
termined by the relevance of cosine operators. Let us, magnetic field B is given in units of uN. To plot the phase
first, inspect the cos(cid:0)2ϕ˜0+kF0x(cid:1) term, whichis relevant boundaries, we Nset δ1 = δ3 = δ4 = α0 and fix K so that
(cid:16) (cid:17)
when 4πK 1 (cid:80)N δ˜B 3ui − u0 ∼ 1, where δ˜B = uN δB.
2N+1 i=1 i u0 ui i BNα i
K0(cid:48) <2 (3.24) TB¯h(cid:39)ep0l.ot(a)isforthecaseofB¯ (cid:39) αuπ and(b)forthecaseof
provided the corresponding oscillatory factor is vanish-
ing. The condition for that is
This inequality is satisfied only in the case where δ [Eq.
2k0 mod 4k (cid:28)2k , (3.25) 2
F F F (3.15)] is the larger correction to the parameter K(cid:48); in-
0
deed, this corresponds to the case of strong Zeeman field
leading to
B¯ obeyingEq. (3.26). Underthiscondition,asδistuned
B¯ ≡(cid:88)B (cid:39) uπ ∼Jxy (3.26) above δc2 one obtains a quantum phase transition of the
j α (cid:107) Kosterlitz-Thouless type28 from a LL to a gapped phase
j
wherethecos(2ϕ˜ )operatoracquiresafiniteexpectation
0
[Eq. (2.24)] where B¯ may be seen as a measure of DW value. Recalling the definition of the Sz spin component
in terms of Euler angles (see Eq. (2.2)) we can identify
asymmetry. Plugging Eq. (3.13) into the condition of
the order induced by the above mentioned cosine terms
Eq. (3.24) and writing it in terms of δ, we obtain the
as spin-density wave (SDW) polarized along the z-axis.
following relation
This result may be summarized in a schematic phase di-
2 agram presented in Fig. 2.
δ >δ =1− . (3.27)
c2 K(2N +1) In the same fashion we derive a condition for the
8
cos(cid:0)4ϕ˜ +4k0x(cid:1) term to be relevant: IV. SUMMARY AND CONCLUDING
0 F
REMARKS
1
δ >δ =1− . (3.28)
c4 2K(2N +1) In this paper we have studied a quasi-1D model
for quantum spin fluctuations in a domain wall (DW)
However this time the appropriate condition for cancela-
configuration generated by a non-uniform transverse
tion of the oscillations
field imposed on a ferromagnet with easy-plane XXZ-
B¯ anisotropy. We find that the low-energy dynamics is
u (cid:28)2kF (3.29) dominated by a single soft mode propagating along the
DW,reflectingtheglobalU(1)symmetryforrotatingthe
is easily satisfied for an almost symmetric DW, as well spinsintheXY-plane,whichcouplestoabathofgapped
as for a random Zeeman field of zero average, even if its spin-fluctuation modes. An effective theory describing
mean-square( 1 (cid:80) B2)isratherlarge. Inthesecases the quantum dynamics of this mode is derived, yielding
2N+1 j j
as well, the tuning of δ above δ induces a transition to in a large part of the parameters space a Luttinger liq-
c4
a SDW phase. The ordered spin structure in the two uid. The corresponding Luttinger parameter K(cid:48) grows
0
typesofSDWphasesaredistinct: inthecaseofastrong systematically with the width of the DW, parametrized
Zeeman field with highly asymmetric profile (B¯ (cid:39) uπ), by the number of legs of the spin-ladder in our model.
α
the period of SDW goes to infinity, which corresponds For an arbitrary finite width, a sufficiently strong mean-
to ferromagnetic (FM) order along the ladder. On the square value of the transverse Zeeman field can induce
other hand, in the case B¯ (cid:28) 2kFu the period of the a quantum phase transition of the Kosterlitz-Thouless
α
SDWisoforderthelatticeconstant. Qualitatively, both type28 into a SDW phase, where the spins are polarized
phasediagramsFig. 2(a)andFig. 2(b)arethesamebut along the local field direction and their fluctuations are
the range of the parameter space is considerably large in gapped. However, with increasing width of the DW, the
the case of FM order. Finally, in the case when non of SDW phase progressively narrows. In the limit of infi-
the cosine terms is relevant, the low-energy dynamics is nitewidth(i.e. asthesystembecomestwo-dimensional),
determined solely by the LL parameters K(cid:48), u(cid:48) and one one obtains a gapless LL dynamics where the effective
0 0
obtains a gapless LL dynamics. Luttinger parameter K(cid:48) → ∞, restoring the Heisenberg
0
To probe the SDW order explicitly, one may measure FMlimit. ThegaplessmodethenbecomestheGoldstone
the local magnetization m(x) as a response to a local mode of the spontaneously symmetry-broken state.
change δB in the magnetic field at point x along the Inprinciple, aphysicalrealizationofourmodelispos-
DW: sible in a magnetic compound with anisotropic FM ex-
(cid:12) changeinteractions, inwhichcasethetransitionintothe
m=− ∂F (cid:12)(cid:12) (3.30) SDW phase can be probed by measurement of magneti-
∂(δB)(cid:12) zation or magnetic susceptibility. In addition, the same
(cid:12)
δB=0
model applies for alternative realizations where the dis-
where F = −1 lnZ and Z is a partition function cal- tinct phases are manifested by electric conduction prop-
β erties. One prominent example is a low-dimensional
culated for the effective action Eq. (3.23). The exact
superconducting (SC) device (e.g. a Josephson array)
dependence of the effective action on δB is complicated,
where theplanar anglefield θ represents thelocal phase
but the leading order comes from the induced modifica- i
of the complex order parameter, and S the charge den-
tion of g(cid:48) which acquires a correction linear in δB. This z
sityoperatorsothattheZeemanfieldB (X)corresponds
yields z
to a gate voltage. In such systems, the transition to the
m(x)∝(cid:104)cos(cid:0)2ϕ˜ +2k0x(cid:1)(cid:105) (3.31) SDW phase can be interpreted as a SC-insulator tran-
0 F
sition. Finally, in graphene devices and particularly bi-
which vanishes in the disordered LL phase. A transition layers in the quantum Hall regime, the conduction prop-
toeitheroftheSDWphasesismarkedbytheemergence erties of polarized spin or isospin bulk phases are dom-
of a finite m(x), which exhibits a modulation pattern inated by the 1D dynamic of DW’s on the edges of the
alongtheDWonascaledictatedbythewavevectork0. sample or the boundaries between domains of opposite
F
In particular, for B¯ (cid:39) uπ, i.e. when cos(2ϕ˜ ) becomes polarization21. Due to the helical nature of these DW
α 0
relevant, the magnetization is constant in space, namely, modes, the effective spin gap that opens in the SDW
FM order is established with total magnetization phase has quite the opposite interpretation than in the
SC analogue: it exponentially suppresses backscatter-
(cid:90)
ing at low temperatures, leading to a nearly perfect 1D
M ∝ dx(cid:104)cos(2ϕ˜ )(cid:105). (3.32)
0 conductance19.
Acknowledgements – Useful discussions with H. A.
Ontheotherhand,forB¯ (cid:39)0wheretheorderisdictated Fertig, G. Murthy and R. Santos are gratefully acknowl-
by the cos(4ϕ˜0) term, the magnetization oscillates on a edged. P. T. thanks the Bar-Ilan Institute for Nanotech-
scale of π ∼α, which is interpreted as AFM order. nologyandAdvancedMaterialsforfinancialsupportdur-
2k0
F
9
ingtheacademicyear2017. E.S.thankstheAspenCen- To see this, we inspect terms of the matrix DCDT:
terforPhysics(NSFGrantNo. 1066293)foritshospital-
ity. ThisworkwassupportedbytheUS-IsraelBinational
Science Foundation (BSF) grant 2012120, and the Israel
Science Foundation (ISF) grant 231/14. (cid:0)DCDT(cid:1) =(cid:88)D C (cid:0)DT(cid:1)
ab ai ij jb
i,j
(cid:88)(cid:8) (cid:9)
= δ +δ −δ {δ +δ }
a,i a,0 a−sg(a),i i,j−1 i−1,j
Appendix A: Properties of the transformation D i,j
(cid:8) (cid:9)
× δ +δ −δ
b,j b,0 b−sg(b),j
In this Appendix we derive properties of the transfor- (cid:0)DCDT(cid:1) =(cid:88) {δ +δ } =2N
00 i,j−1 i−1,j
mation D defined in Eq.(2.10). First, we show that it is (cid:124) (cid:123)(cid:122) (cid:125)
i,j
sum ovel all C elements
indeed a canonical transformation
(cid:0)DCDT(cid:1) =(cid:88){δ +1−δ }{δ +δ }
0,N 0,i 0,i i,j−1 i−1,j
(cid:104)ϕ˜ (x),∂ θ˜ (x(cid:48))(cid:105)=(cid:88)(cid:104)(cid:0)DT(cid:1)−1ϕ (x),∂ D θ (x(cid:48))(cid:105) i,j
i x(cid:48) k i,j j x(cid:48) k,l l ×(cid:8)δ −δ (cid:9) (A4)
j,l b,j b−sg(b),j
=(cid:88)(cid:0)DT(cid:1)−i,j1Dk,l[ϕj(x),∂x(cid:48)θl(x(cid:48))] =(cid:88)δi,j−1{δN,j −δN−1,j}+(cid:88)δi−1,j{δN,j −δN−1,j}
j,l i,j i,j
=(cid:88)(cid:0)DT(cid:1)−i,j1Dk,lδj,liπδ(x(cid:48)−x) (A1) =(cid:88){(cid:124)δN,i+1−(cid:123)(cid:122)δN−1,i+1}(cid:125)+(cid:88){(cid:124)δN,i−1−(cid:123)(cid:122)δN−1,i−1}(cid:125)=−1
j,l i 1−1=0 i 0−1
=iπδ(x(cid:48)−x)(cid:88)(cid:0)DT(cid:1)−1D (cid:0)DCDT(cid:1) =(cid:88){δ −δ }
i,l k,l 0,−N −N,i+1 −N+1,i+1
l i (cid:124) (cid:123)(cid:122) (cid:125)
=iπδ(x(cid:48)−x)(cid:88)(D)−1D =iπδ(x(cid:48)−x)δ (cid:88) 0−1
l,i k,l i,k + {δ −δ }=−1
−N,i−1 −N+1,i−1
l (cid:124) (cid:123)(cid:122) (cid:125)
i
1−1=0
For the treatment of the Hamiltonian (2.6) we have cho- (cid:0)DCDT(cid:1) =(cid:88)(cid:8)δ −δ (cid:9)
sen a specific from of the matrix D (2.10): 0,b(cid:54)=±N i (cid:124) b,i+1 (cid:123)b(cid:122)−sg(b),i+1(cid:125)
1−1=0
(cid:88)(cid:8) (cid:9)
... ... ... ... ... ...  + i (cid:124)δb,i−1−δ(cid:123)b(cid:122)−sg(b),i−1(cid:125)=0
... 1 −1 0 0 0 ... 1−1=0
 
... 0 1 −1 0 0 ...
 
D =... 1 1 1 1 1 ... (A2)
 
... 0 0 −1 1 0 ...
  andtheresultingcouplingtothesymmetricmodeisgiven
... 0 0 0 −1 1 ...
  by the first term in Hint [Eq. (2.16)].
... ... ... ... ... ... b,0
ThefirstpartoftheHamiltonianH ischangedby
Gauss
the symmetric matrix DDT, as can be seen by exploring
a general element of this matrix
(cid:0)DDT(cid:1)a,b =(cid:88)(D)a,c(cid:0)DT(cid:1)c,b (A3) Appendix B: Corrections to the LL parameters of
c the symmetric mode
(cid:88)(cid:0) (cid:1) (cid:0) (cid:1)
= δ +δ −δ × δ +δ −δ
a,c a,0 a−sg(a),c b,c b,0 b−sg(b),c
c
(cid:40)(cid:80) In this Appendix we present details for the derivation
(δ +1−δ )(δ +1−δ ) ,a=b=0
= c 0,c 0,c 0,c 0,c of corrections to the LL parameters. All the corrections
(cid:80) (cid:0) (cid:1)
(δ +1−δ ) δ +0−δ ,a=0,∀b(cid:54)=0
c 0,c 0,c b,c b−sg(b),c come from the second order in perturbation theory. It
(cid:40)(cid:80) is natural to treat this problem in the functional path
1=2N +1 ,a=b=0
= c . integral language, namely, it is convenient to write the
(cid:80) (cid:80)
δ − δ =1−1=0 ,a=0,∀b(cid:54)=0
c b,c c b−sg(b),c problem in terms of imaginary-time action. The induced
corrections then result from re-exponentiation of a ter-
The Gaussian part of the interactions between modes, minatedperturbationseries(i.e.,acumulantexpansion).
HGauss,isalmostdecoupledfromthesymmetric0-mode. Symbolically, we can derive the perturbation theory in
int
10
the following way: where, following the integration over the relative coordi-
(cid:90) nate (cid:126)r, we obtain
(cid:89)
Z = Dϕ˜iDϕ˜0e−(S0[ϕ˜0]+Sb[{ϕ˜i(cid:54)=0}]+Sint[{ϕ˜i}])
(cid:90) i(cid:54)=0 F =(cid:90) d2rF (r)= m2N (cid:90) d2r(cid:0)K2(m r)−K2(m r)(cid:1)
= (cid:89)Dϕ˜iDϕ˜0e−S0[ϕ˜0]e−Sb[{ϕ˜i(cid:54)=0}] 2 1 N 0 N
(cid:26) (cid:27)
i(cid:54)=0 1 1 1 1
(cid:39)2π ln − (cid:39)πln (B7)
(cid:18) 1 (cid:19) 2 m α 2 m α
× 1−S [{ϕ˜ }]+ S2 [{ϕ˜ }]+... (B1) N N
int i 2 int i
(cid:39)(cid:90) Dϕ˜0e−S0[ϕ˜0](cid:18)1−(cid:104)Sint[{ϕ˜i}](cid:105)+ 12(cid:10)Si2nt[{ϕ˜i}](cid:11)(cid:19) lbeuatdiionngfrtoomEqth.e(O3.id3)c.orSriemctiliaornlsy,,ywieeldeivnagluate the contri-
(cid:90)
= Dϕ˜0e−S0[ϕ˜0]−(cid:104)Sint[{ϕ˜i}](cid:105)+12(cid:104)Si2nt[{ϕ˜i}](cid:105)c (cid:32) 2g (cid:33)2(cid:90) (cid:90) (cid:68) (cid:69)
=(cid:90) Dϕ˜0e−S0eff[ϕ˜0] δS±d,N2 = (2πα3)2 d2r1 d2r2 ei2φ˜±N(r1)e−i2φ˜±N(r2)
where (cid:10)S2 [{ϕ˜ }](cid:11) =(cid:10)S2 [{ϕ˜ }](cid:11)−(cid:104)S [{ϕ˜ }](cid:105)2, and ×e−2ikFN(x1−x2)ei2N4+1φ˜0(r1)e−i2N4+1φ˜0(r2)+h.c
allexpectianttionvialucesareiwntithreispecttointhtebaithaction (cid:32) 2g (cid:33)2 1 (cid:18) 4 (cid:19)2
=− 3 (m α)2 (B8)
Sb[{ϕ˜i(cid:54)=0}]. (2πα)2 u2 2N +1 N
To evaluate the expectation value (cid:10)S2 [{ϕ˜ }](cid:11), we N
int i (cid:90) (cid:26)(cid:16) (cid:16) (cid:17)(cid:17)2 (cid:16) (cid:16) (cid:17)(cid:17)2 (cid:27)
need expressions for various correlation functions of the dR(cid:126) ∂ ϕ˜ R(cid:126) F + ∂ ϕ˜ R(cid:126) F
ϕ˜ fields which are strongly fluctuating when their dual Rx 0 x Rτ 0 τ
i
fields θ˜ are ordered. These are given by29
i
(cid:68)ei2ϕ˜i(r1)e−i2ϕ˜i(r2)(cid:69)≈(m α)2(cid:0)K2(m r)+K2(m r)(cid:1) where
i 0 i 1 i
(B2) (cid:90)
(cid:104)∂xϕ˜i(r1)∂xϕ˜i(r2)(cid:105)c ≈ m22i (cid:8)K12(mir)−K02(mir)((cid:9)B3) FFxτ ==(cid:90) rrddrrddθθ(cid:2)(cid:2)KK1122((mmNNrr))++KK0022((mmNNrr))(cid:3)(cid:3)ee−−22iikkFFNNrrccoossθθ((rrscionsθθ))22.
(cid:68)eiϕ˜i(r1)e−iϕ˜i(r2)(cid:69)∝(miα)21 K0(mir), (B4) (B9)
where mi are the masses, r = |(cid:126)r1−(cid:126)r2| and Ki(x) are Performingintegrationover(cid:126)r weobtaintheresultofEq.
Besselfunctionsofthesecondkind. Thecorrelationfunc- (3.10).
tions B2 and B3 are obtained by fermionization of SG
Hamiltonian (which becomes exact for K =1), whereas
i (cid:32) (cid:33)2
B4 is calculated by mapping SG Hamiltonian to quan- δSd,2 = 2g3 (cid:90) d2r (cid:90) d2r (B10)
tum Ising chain for r (cid:29) 1 . All these correlation func- ±i (2πα)2 1 2
mi
tions exponentially suppress contributions from r > 1 . (cid:68) (cid:69)(cid:68) (cid:69)
mi ei2φ˜±i(r1)e−i2φ˜±i(r2) ei2φ˜±i+1(r1)e−i2φ˜±i+1(r2)
Hence, insidetheintegralsEqs. (3.3), (3.5), (3.6)wecan
make the approximation (cid:32) (cid:33)2
42 2g F
(cid:18) 1 (cid:19) (cid:18) 1 (cid:19) ei2N4+1φ˜0(r1)e−i2N4+1φ˜0(r2)+h.c(cid:39) 3 4
ϕ˜ ((cid:126)r )−ϕ˜ ((cid:126)r )=ϕ˜ R(cid:126) + (cid:126)r −ϕ˜ R(cid:126) − (cid:126)r (2N +1)2 (2πα)2 u2
0 1 0 2 0 2 0 2 N
(cid:39)∇ϕ˜0(cid:16)R(cid:126)(cid:17)·(cid:126)r (B5) ×(miα)4(cid:90) dτ(cid:90) dxui(cid:20)(∂xϕ˜0(x,τ))2+ u12 (∂τϕ˜0(x,τ))2(cid:21),
i
whichallowsustointegratedovertherelativecoordinate
(cid:126)r. Also whenever we have a product of two correlation where
functions with different velocities we approximate both
vmeolodceistiaesretaolmbeosetqeuqaul,alb.ecause velocities of consecutive F4 =(cid:90) ∞d2r(cid:0)K02(mir)+K12(mir)(cid:1)2(cid:12)(cid:12)rx/τ(cid:12)(cid:12)2
α
EmployingEqs. (B2)through(B5),wenextderivethe (cid:18) (cid:18) (cid:19)(cid:19)
1
varioussecondordercorrectionstoSeff. First,wederive (cid:39)πm−4 0.075+2×0.325+πln (B11)
0 i m α
corrections that come form Eq. (3.3): i
δSGauss =(cid:68)(cid:0)SGauss(cid:1)2(cid:69) (B6) leadingtoEq. (3.10). Alloftheabovecorrectionscanbe
int,2 int
c summarized as a set of equations of new LL parameters
= 1 (cid:18)J⊥zα(cid:19)2 1 (cid:90) d2r(cid:90) d2RF (r)(cid:16)∂ ϕ˜ (cid:16)R(cid:126)(cid:17)(cid:17)2 of the symmetric mode ϕ˜0:
(2N +1)2 π2 u2 x 0
N
1 (cid:18)Jzα(cid:19)2 F (cid:90) (cid:90)
= ⊥ dx dτu (∂ ϕ˜ (x,τ))2
(2N +1)2 π2 u2 N x 0
N