Spin dynamics of the generalized quantum spin compass chain Pontus Laurell1 2Gonzalo Alvarez2and Elbio Dagotto1 3 1Department of Physics and Astronomy University of Tennessee Knoxville Tennessee 37996 USA

2025-05-03 0 0 1.97MB 15 页 10玖币

侵权投诉

Spin dynamics of the generalized quantum spin compass chain

Pontus Laurell,1, 2, ∗Gonzalo Alvarez,2and Elbio Dagotto1, 3

1Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA

2Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

3Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

(Dated: March 28, 2023)

We calculate the dynamical spin structure factor of the generalized spin-1/2 compass spin chain using the

density matrix renormalization group. The model, also known as the twisted Kitaev spin chain, was recently

proposed to be relevant for the description of the spin chain compound CoNb2O6. It features bond-dependent

interactions and interpolates between an Ising chain and a one-dimensional variant of Kitaev’s honeycomb spin

model. The structure factor, in turn, is found to interpolate from gapped and non-dispersive in the Ising limit

to gapless with non-trivial continua in the Kitaev limit. In particular, the component of the structure factor

perpendicular to the Ising directions changes abruptly at the Kitaev point into a dispersionless continuum due

to the emergence of an extensive groundstate degeneracy. We show this continuum is consistent with analytical

Jordan-Wigner results. We also discuss implications for future inelastic scattering experiments and applications

to materials, particularly CoNb2O6.

I. INTRODUCTION

Orbital physics in Mott insulators can lead to a wide range

of important phenomena [1–5] including dimensionality re-

duction [5], orbital-selective Mott phases [6–8], and, in the

presence of spin-orbit coupling, bond-dependent magnetic ex-

change interactions [1,3,9]. The latter feature dramatically

in compass models [3], with Ising interactions along speciﬁc

spin-space directions depending on the spatial bond direction.

A famous example is Kitaev’s honeycomb spin model [10],

which realizes a quantum spin liquid ground state. Its possible

material realizations have been the subject of intense research

recently [11–13].

Another intriguing example is the 1D quantum compass

model (QCM) with alternating Sx

iSx

i+1and Sy

i+1Sy

i+2interac-

tions for diﬀerent bonds along the chain direction [14,15],

which provides an exactly solvable model presenting a quan-

tum multicritical point [16,17] in extended models. The

QCM can be viewed as arising from orbital order in systems

of weakly interacting zigzag chains [18], or simply as a 1D

version of Kitaev’s honeycomb model: a Kitaev spin chain.

Chain and ladder versions of the Kitaev honeycomb model

and its extensions (including e.g. Heisenberg and oﬀ-diagonal

Gamma interactions) have been studied theoretically [19–42],

mostly for their tractability and potential realizations in engi-

neered chains [43]. It was also proposed that charge order in

K-intercalated RuCl3may lead to eﬀective Kitaev-Heisenberg

chains [23,25], but a diﬀerent charge order was found in ex-

periments [44].

Given the above information, zigzag chains appear to be

the most promising way towards such 1D Kitaev-like models

in materials. Due to the variability of bond angles and lat-

tice distortions, it is natural to consider a generalized compass

∗plaurell@utk.edu

model (GCM) [18,45],

H=−K

L/2−1

i=0τˆn1

2iτˆn1

2i+1+τˆn2

2i+1τˆn2

2i+2,(1)

where τˆnj

i=ˆnj·~

τiis the projection of the pseudospin Pauli

operator vector on site ionto the bond-dependent Ising di-

rection ˆnj. Using a coordinate system where the two axes

ˆn1and ˆn2lie in a plane, we allow the angle 2θbetween ˆn1

and ˆn2to vary continuously. At θ=0, π/2 the Ising chain is

recovered, while θ=π/4 yields the QCM [46], which was

solved in the seventies as a special case of the alternating XY

model [47]. The interpolation between Ising and Kitaev spin

chains motivated Morris et al. [45] to introduce “twisted Ki-

taev spin chain” as an alternate name for the GCM away from

these limits. They also proposed the Hamiltonian (1) as a de-

scription of long-distance properties in the zigzag chain ma-

terial CoNb2O6[45], which is commonly considered the best

known realization of the ferromagnetic (FM) transverse-ﬁeld

spin-1/2 Ising chain due to its observed ﬁeld-induced criti-

cality [48–52]. The description as a pure FM Ising chain is,

however, insuﬃcient to explain the zero-ﬁeld behavior, the de-

scription of which motivates considering bond-dependent in-

teractions [45,53].

What would originate such interactions in CoNb2O6? Their

Co2+ions are surrounded by oxygen octahedral cages and

form zigzag chains along the caxis; see Fig. 1. Hund’s cou-

pling favors a high-spin d7conﬁguration (t5

2ge2

g), which may

be viewed as a S=3/2, L=1 state. Spin-orbit coupling then

splits the energy levels further, resulting in a pseudospin-1/2

ground state Kramers doublet, just as in proposals for Kitaev

physics in honeycomb cobaltate systems [54,55]. Although

CoNb2O6is not a honeycomb system, its symmetry permits

identiﬁcation of two alternating Ising directions [45]. Distor-

tion of the octahedra splits the energy levels further, but the

ground state Kramers doublet remains [56]. We note that the

GCM, Eq. (1), is general and not restricted to materials such

as CoNb2O6. It may also emerge in d9, high-spin d4, and low-

spin d7conﬁgurations, where the egorbital degree of freedom

arXiv:2210.00357v2 [cond-mat.str-el] 27 Mar 2023

FIG. 1. Zigzag chain in CoNb2O6featuring a two-site unit cell with

lattice constant calong the chain direction. The lattice symmetry

allows for diﬀerent interactions between spins along the two dis-

tinct bonds. Similar bond-dependent interactions may emerge also

in other zigzag chain systems with speciﬁc electron conﬁgurations.

replaces the Kramers doublet degree of freedom [18]. Further

potential applications include Co zigzag chains on surfaces

[57] and quantum simulation in optical lattices [58,59].

Since Eq. (1) and variations of the model are exactly solv-

able using Jordan-Wigner fermions [60,61], many proper-

ties have been studied. These include ground state properties

[14,15,18,62–67] , thermodynamic properties [18,68,69],

and aspects of quantum quench dynamics [70–72]. Numerical

results were also reported in Refs. [17,64,73] using Lanc-

zos exact diagonalization and Ref. [69] using matrix product

state methods. However, to the best of our knowledge, the

full dynamical spin structure factor S(k, ω) has not yet been

studied except in the Ising limit, although time-dependent re-

sults for the spin dynamics of the QCM were obtained ana-

lytically for spin components in the plane spanned by ˆn1and

ˆn2⊥ˆn1[74] and for spin components transverse to the same

plane [75][76]. The goal of the present paper is thus to study

the frequency-dependent dynamics at zero temperature and as

function of the angle θ.

Using the density matrix renormalization group (DMRG)

[77,78] we obtain all components of S(k, ω) as a function of

the angle θ. The spectra interpolate from gapped and non-

dispersive at the Ising points towards a gapless continuum as

the Kitaev point is approached, with gapped and dispersive be-

havior in-between. There are abrupt qualitative changes in the

spectra at the Kitaev point, related to an underlying macro-

scopic degeneracy. In particular, the transverse component

Syy(k, ω) becomes gapless and dispersionless in the Kitaev

limit. These spectral features are understood via the Jordan-

Wigner ground state solution. Our S(k, ω) results can help the

design and interpretation of future experiments employing, for

example, inelastic neutron scattering (INS) or resonant inelas-

tic xray scattering (RIXS) techniques.

The paper is organized as follows. Sec. II introduces global

coordinate systems for Eq. (1) to interrelate the conventions

of Refs. [18,45]. We review relevant Jordan-Wigner results

in Sec. III and describe the numerical methods in Sec. IV.

We present our results in Sec. V, discuss their consequences

and summarize the conclusions in Sec. VI. A derivation of the

dispersionless continuum at the Kitaev point is provided in

Appendix A.

II. COORDINATE SYSTEMS

For concreteness, we ﬁrst consider the application of Eq. (1)

to CoNb2O6. The crystal structure features zigzag chains

along the crystallographic caxis as shown in Fig. 1, in which

the two Ising directions are constrained by symmetry to be

related by a twofold rotation symmetry about b,Cb

2. Follow-

ing Morris et al. [45] we use a global xyz coordinate system

where two Ising directions ˆn1,ˆn2deﬁne the xz-plane. This is

done by choosing ˆxparallel with the baxis, and ˆzsuch that

it bisects the angle 2θ≈34◦between ˆn1and ˆn2and is at an

angle φ≈31◦to the caxis. The ﬁrst Ising axis can be taken

as ˆn1=(sin θ, 0,cos θ), with ˆn2ﬁxed by Cb

2symmetry.

Substituting the ˆnjinto Eq. (1), transforming to pseudospin-

1/2 operators Sa

i=τa

i/2 and deﬁning ˜

K=4Kone obtains

H1=−˜

ihcos2(θ)Sz

iSz

i+1+sin2(θ)Sx

iSx

i+1

+sin (2θ)

2(−1)iSx

iSz

i+1+Sz

iSx

i+1#,(2)

as in Ref. [45]. In the absence of magnetic ﬁelds there is a

twofold ground state degeneracy due to invariance under spin

rotations around ˆyby π. We call this the Ising-like coordinate

system because the Ising nature of the Hamiltonian is manifest

at θ=0, π/2. However, since the bond alternation is in the

symmetric oﬀ-diagonal (or Γ) terms, the Kitaev nature at π/4

is obscured:

Hθ=π/4

1=−˜

ihSz

iSz

i+1+Sx

iSx

i+1+(−1)iSx

iSz

i+1+Sz

iSx

i+1i.

(3)

The connection to Kitaev or compass physics becomes

clearer by canonically transforming to an alternate coordinate

system (x0y0z0) by a π/4 counterclockwise rotation around −ˆy.

In this Kitaev-like coordinate system the bond-alternation is

moved to the Ising terms,

H2=−˜

inh1−(−1)isin (2θ)iSx0

iSx0

i+1(4)

+h1+(−1)isin (2θ)iSz0

iSz0

i+1−cos (2θ)hSx0

iSz0

i+1+Sz0

iSx0

i+1io,

making the Kitaev nature manifest at θ=π/4. The drawback

is that the Ising nature at θ=0, π/2 is now obscured, where

the Hamiltonian takes the form of an X’Y’ model with a Γin-

teraction term. We will report our spin dynamics results in the

Ising-like coordinate system, both because of its established

connection to experimentally relevant systems and because

the rotation to the Kitaev-like coordinate system generically

induces oﬀ-diagonal Sx0z0/z0x0(k, ω) correlations, which can be

signiﬁcant.

Finally, to connect with prior Jordan-Wigner analyses of the

GCM it is convenient to apply a π/2 spin rotation about ˆx,

Sx→˜

Sx,Sy→˜

Sz,Sz→ −˜

Sy,(5)

to Eq. (2), yielding

H3=−˜

ihsin2(θ)˜

i˜

i+1+cos2(θ)˜

i˜

i+1

−sin (2θ)

2(−1)i˜

i˜

i+1+˜

i˜

i+1#.(6)

In the following we will use H3in the discussion of the

Jordan-Wigner solution, but present spin dynamics results in

the coordinate system of H1. This approach gives both a con-

crete connection to CoNb2O6and similar systems, and in-

creased numerical eﬃciency from working with real-valued

Hamiltonians.

III. JORDAN-WIGNER SOLUTION

We review here aspects of the exact solution of the model

in the Jordan-Wigner formalism [60,61], following mainly

Refs. [18,70]. Introducing the standard transformation

S+

i=˜

i+i˜

i=c†

iexp 





iπ

i−1

j=1

c†

jcj





,(7)

S−

i=˜

i−i˜

i=exp 



−iπ

i−1

j=1

c†

jcj





ci,(8)

i=c†

ici−1

2,(9)

where nci,c†

jo=δi,j, Eq. (6) is recast in terms of spinless

fermions,

H3=−K

i=1hc†

ici+1+H.c.i

L/2

i=1hc†

2ic†

2i+1e−i2θ+c†

2i+1c†

2i+2ei2θ+H.c.i,(10)

where Lis the length of the chain, L/2 is the number of unit

cells, and H.c.denotes Hermitian conjugate. We adopt a peri-

odic Fourier convention with

c2j−1=r2

e−ik jak,c2j=r2

e−ik jbk,(11)

and momenta given by

k=2nπ

L,n=−L

2−1,−L

2−3,...,L

2−1.(12)

Following the Fourier transform, Eq. (10) is rewritten in a

symmetrized Bogoliubov-de Gennes form,

H=1

Γ†

kh(k)Γk,Γ†

k=a†

k,a−k,b†

k,b−k,(13)

where

h(k)=





0 0 AkPk+Qk

0 0 −Pk−Qk−Ak

k−P?

k−Q?

k0 0

k+Q?

k−A?

k0 0







(14)

and

Ak=−K1+eik,(15)

Pk=Kcos (2θ)1−eik,Qk=iK sin (2θ)1+eik.(16)

Unitary diagonalization of Eq. (14) yields a spectrum sym-

metric around zero, with energies {±k,n},n=1,2 given by

k,1=qCk−pDk, k,2=qCk+pDk,(17)

where

Ck=Ak

2+Pk

2+Qk

2=4K2h1+cos (k)sin2(2θ)i(18)

and

Dk=A?

kPk+AkP?

k2−A?

kQk−AkQ?

k2

+P?

kQk+PkQ?

k2(19)

=16K4cos2 k

2!sin2(2θ)

h3+cos (4θ)+2 cos (k)sin2(2θ)i.(20)

k,1and k,2are called the acoustic and optical branches, re-

spectively, in analogy with phonon terminology. Positive en-

ergy states represent physical excitations, while negative en-

ergy states stem from the redundancy in the description and

are ﬁlled in the ground state, which has energy

E0=−1

kk,1+k,2.(21)

This function is plotted in black in Fig. 2(a).

Some important observations follow directly from the

eigenvalues (17). First of all, the energies are independent

of the sign of K. Second, since k,1≤k,2∀k, θ the exci-

tation gap is given by ∆(θ)=2 minkk,1(θ),which generi-

cally has extrema at k=0, π and is plotted in Fig. 2(b). We

note that the gap ∆(θ) is best understood as the physical en-

ergy gap in the thermodynamic limit, i.e. the gap between a

spontaneously Z2-symmetry-broken ground state and the ﬁrst

excited state above it. At ﬁnite system size, analysis of the gap

in the Jordan-Wigner formalism requires careful treatment of

boundary conditions and Bologiubov vacua [14], which is out-

side the scope of the current paper. In numerical calculations

on ﬁnite-size systems the physical gap may be identiﬁed via

∆2=E2−E0, where Enis the nth lowest eigenvalue and mul-

tiplicity is taken into account.

In the Ising limits at θ=0, π/2, the excitations are gapped,

doubly degenerate and nondispersive, with k,1=k,2=2|K|

(Dk=0, Ck=4K2). At the Kitaev point k,1(θ=π/4)=

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

SpindynamicsofthegeneralizedquantumspincompasschainPontusLaurell,1,2,GonzaloAlvarez,2andElbioDagotto1,31DepartmentofPhysicsandAstronomy,UniversityofTennessee,Knoxville,Tennessee37996,USA2ComputationalSciencesandEngineeringDivision,OakRidgeNationalLaboratory,OakRidge,Tennessee37831,USA3MaterialsScie...

展开>> 收起<<

Spin dynamics of the generalized quantum spin compass chain Pontus Laurell1 2Gonzalo Alvarez2and Elbio Dagotto1 3 1Department of Physics and Astronomy University of Tennessee Knoxville Tennessee 37996 USA.pdf

共15页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Spin dynamics of the generalized quantum spin compass chain Pontus Laurell1 2Gonzalo Alvarez2and Elbio Dagotto1 3 1Department of Physics and Astronomy University of Tennessee Knoxville Tennessee 37996 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: