Mobile impurity approach to the optical conductivity in the Hubbard chain

0.9 MB·

by  Thomas Veness

Mobile impurity approach to the optical conductivity in the Hubbard chain
Thomas Veness and Fabian H. L. Essler
The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK
WeconsidertheopticalconductivityintheonedimensionalHubbardmodelinthemetallicphase
close to half filling. In this regime most of the spectral weight is located at frequencies above an
energy scale E that tends towards the optical gap in the Mott insulating phase for vanishing
opt
doping. Using the Bethe Ansatz we relate E to thresholds of particular kinds of excitations in
opt
theHubbardmodel. Wethenemployamobileimpuritymodelstoanalyzetheopticalconductivity
for frequencies slightly above these thresholds. This entails generalizing mobile impurity models to
excited states that are not highest weight with regards to the SU(2) symmetries of the Hubbard
chain, and that occur at a maximum of the impurity dispersion.
I. INTRODUCTION
6
Electron-electron interactions play a crucial rˆole in determining the physical response to external probes of various
1
quasi-one-dimensional materials e.g. organic semiconductors1. In order to successfully describe the mechanisms and
0
excitations responsible for distinct physical phenomena, it is imperative to have a microscopic model capturing the
2
essence of the physics involved; providing a framework within which realistic physical systems may be interpreted.
pr Theone-dimensionalHubbardmodel2 offersanexcellenttheoreticallaboratoryinwhichacomprehensivemicroscopic
A understandingoftheoriginofvariousbehaviourscanbedeveloped. TheHamiltonianfortheHubbardmodelisgiven
by

9
1 (cid:88) (cid:88)
H =−t c†i+1,σci,σ+c†i,σci+1,σ+U ni, ni,
] i,σ i ↑ ↓ (1)
l (cid:88) (cid:88)
e µ (n +n ) B (n n ).
- − i,↑ i,↓ − i,↑− i,↓
r i i
t
s
t. Here, cj,σ annihilates a fermion with spin σ =↑,↓ at site j, nj,σ = c†j,σcj,σ is the number operator, t is the hopping
a parameter, µ is the chemical potential, B is the magnetic field, and U 0 is the strength of the on-site repulsion.
≥
m The low-energy degrees of freedom in the metallic phase of the Hubbard chain are described2–4 by a (perturbed)
spin-charge separated Luttinger liquid5–8, with Hamiltonian
-
d
n (cid:90) (cid:34) (cid:18) (cid:19)2 (cid:18) (cid:19)2(cid:35)
o H = (cid:88) vα dx 1 ∂Φα +K ∂Θα +irrelevant operators. (2)
α
16π K ∂x ∂x
c α
α=c,s
[

Theparameters K , v canbecalculatedfortheHubbardmodelbysolvingasystemoflinearintegralequations(see
2 α α
v Appendix A). The Bose fields Φα(x) and dual fields Θα(x) obey the commutation relation
1
5 [Φα(x),Θβ(y)]=4πiδαβsgn(x y). (3)
−
2
1 The spectrum of low-lying excitations relative to the ground state for a large but finite system of length L in zero
0 magnetic field is given by2–4
.
1
0 2πv (cid:34)(∆N )2 (cid:18) D (cid:19)2 (cid:35) 2πv (cid:34)(cid:0)∆N ∆Nc(cid:1)2 D2 (cid:35)
6 ∆E = Lc 8Kc +2Kc Dc+ 2s +Nc++Nc− + Ls s−2 2 + 2s +Ns++Ns− , (4)
1 c
:
v where ∆N , D and N are integers or half-odd integers subject to the “selection rules”
i α α α±
X
∆N +∆N ∆N
c s c
ar Nα± ∈N0, ∆Nα ∈Z, Dc = 2 mod1, Ds = 2 mod1. (5)
Atlowenergies,correlationfunctionscanbecalculatedfrom(2)andgenericallyexhibitsingularitiesatthethresholds
oftheallowedcollectivespinandchargedegreesoffreedom,withpower-lawexponentsgivenintermsofthequantities
∆N , D and N . However, when working at a finite energy scale RG irrelevant terms have a non-zero coupling
α α α±
and may (and in fact generically do) significantly alter the predictions of the unperturbed Luttinger-liquid9–35. Over
the last decade or so a fairly general method for taking into account the effects of certain irrelevant operators in
2
the vicinities of kinematic thresholds has been developed, which is reviewed in Refs. 36 and 38. The case of spin-
charge separated Luttinger liquids has very recently been revisited35 in order to make it explicitly compatible with
exactly known properties of the Hubbard model. The essence of this approach is that, when considering a response
function,therearethresholdsinthe(k,ω)-planethatcorrespondtoparticularexcitations. Inintegrablemodels,these
excitations hold privileged positions: they are stable (i.e. have infinite lifetimes) and can be identified in terms of the
exact solution. If the kinematics near the threshold are described by a case in which small number of high-energy
excitations carry most of the energy (in the precise sense of up to corrections of (L 1)) then the problem becomes
−
O
analogous to that of the X-ray edge singularity problem for a mobile impurity39.
In this work we employ mobile impurity methods to study the optical conductivity
σ (ω)= ImχJ(ω), χJ(ω)= ie2(cid:90) ∞dt eiωt L(cid:88)/2−1 GS [J (t),J (0)]GS , (6)
1 l 0
− ω − (cid:104) | | (cid:105)
0
l= L/2
−
where J is the density of the current operator
j
(cid:88)(cid:104) (cid:105)
Jj =−it c†j,σcj+1,σ−c†j+1,σcj,σ . (7)
σ
In the Mott insulating phase of the Hubbard model the optical conductivity has been previously determined40–44:
σ (ω)vanishesinsidetheopticalgap2∆,where∆istheMottgap. Atfrequenciesω >2∆thereisasuddenpower-law
1
onset σ (ω) √ω 2∆. Away from half-filling, the system is a metal and therefore has a finite conductivity for all
1
∼ −
ω, specifically acquiring a Drude peak45,46 at ω = 0. The low-frequency behaviour has been previously studied47–49
in the framework of Luttinger liquid theory, predicting ω3 behaviour for 0<ω t. Close to half-filling one expects
(cid:28)
most of the spectral weight in σ (ω) to be located above an energy scale E that tends to 2∆ as we approach
1 opt
half-filling. The scale E has been previously correctly identified in Ref. 50. In the same work it was conjectured
opt
that the optical conductivity increases in a power-law fashion above E
opt
σ (ω) (ω E )ζ Θ(ω E ) . (8)
1 opt opt
∼ − −
As we will see in the following, the mobile impurity approach leads to different results.
Theoutlineofthispaperisasfollows. InSec.II,weconsiderthespectralrepresentationoftheopticalconductivity
andidentifythequantumnumbersofthestatescontributingnon-zerospectralweight. InSec.IIIwereviewtheBethe
Ansatz description of the ground state and construct the excited states considered in Sec. II, specifically identifying
the thresholds of these continua. In Sec. IV we calculate the threshold/edge behaviour for the associated excitations
via the mobile impurity approach, fixing the coupling constants using the Bethe Ansatz to determine the finite-size
corrections to the energy in the presence of the high-energy excitation.
II. SPECTRAL REPRESENTATION OF THE CURRENT-CURRENT CORRELATOR
In considering the optical conductivity as defined in (6), the basic quantity of interest is
(cid:88)
GS Jj+(cid:96)(t)Jj(0)GS = GS Jj+(cid:96) n nJj GS e−i(En−EGS)t, (9)
(cid:104) | | (cid:105) (cid:104) | | (cid:105)(cid:104) | | (cid:105)
n
where n constitute a complete set of energy eigenstates. To understand threshold behaviours, we wish to identify
{| (cid:105)}
the states contributing to this sum. A crucial insight to this end are global continuous symmetries and their relation
to the energy eigenstates provided by the exact Bethe Ansatz solution51–55. In the case of zero magnetic field and
chemical potential, the Hubbard model possesses two independent SU(2) symmetries2,56,57:
L L L
1(cid:88) (cid:88) (cid:88)
Sz = 2 (c†i, ci, −c†i, ci, ), S+ = c†i, ci, , S− = c†i, ci, ,
↑ ↑ ↓ ↓ ↑ ↓ ↓ ↑
i=1 i=1 i=1
(10)
L L L
1(cid:88) (cid:88) (cid:88)
ηz = 2 (c†i, ci, +c†i, ci, −1), η+ = (−1)ic†i, c†i, , η− = (−1)ici, ci, .
↑ ↑ ↓ ↓ ↓ ↑ ↑ ↓
i=1 i=1 i=1
The Sα generate the well known spin rotational SU(2) symmetry, while the ηα are known as η-pairing generators.
TheBetheAnsatzprovidesuswiththelowest weight states51,whichwedenoteby LWS;m . Heremisamulti-index
| (cid:105)
3
whichlabelsalldistinctregularBetheAnsatzstatesinthesenseofRef. 51. Thestatesarelowest-weightwithrespect
to the two SU(2) algebras in the sense that
η LWS;m =0=S+ LWS;m . (11)
−
| (cid:105) | (cid:105)
Eachstate LWS;m isdefinedonasystemoflengthLandhasawell-definednumberofelectronsN andz-component
| (cid:105) (cid:8) (cid:9)
of spin S . A complete basis of states is given by (η+)k(S )l LWS;m k =0,...,L N; l=0,...,2Sz . For the
z −
| (cid:105)| −
repulsive Hubbard model below half-filling, the ground state in zero magnetic field and finite chemical potential is a
spin singlet and a lowest-weight η-pairing state i.e.
S GS =S+ GS =η GS =0. (12)
− −
| (cid:105) | (cid:105) | (cid:105)
Using the algebra defined in (10) it is readily verified that [η ,[η ,J ]]=0 and therefore for integer m 0
− − j
≥
LWS;n(η )m+1J GS = LWS;n(η )m[η ,J ]GS =δ LWS;n[η ,J ]GS . (13)
− j − − j m,0 − j
(cid:104) | | (cid:105) (cid:104) | | (cid:105) (cid:104) | | (cid:105)
This shows that the only states that may have a non-zero overlap with J GS are lowest weight states LWS;m or
j
| (cid:105) | (cid:105)
η-pairing descendant states of the form η+ LWS;m , which implies the expansion
| (cid:105)
J GS =(cid:88)(cid:0)a LWS;m +b η+ LWS;m (cid:1), (14)
j m m
| (cid:105) | (cid:105) | (cid:105)
m
where a , b are complex coefficients. Substituting this into (9) provides further constraints on the subset of energy
m m
eigenstates that may make non-vanishing contributions to the correlator. The subset consists of
1. Lowest-weight states with N electrons with S2 =Sz =0;
GS
2. States of the form η+ LWS;m , with LWS;m having N 2 electrons and S2 =Sz =0.
GS
| (cid:105) | (cid:105) −
Using that [H,η+]= 2µη+ we can thus express the current-current correlator in the form
−
C ((cid:96),t)= GS J (t)J (0)GS
JJ j+(cid:96) j
(cid:104) | | (cid:105)
(cid:88)
= GS Jj+(cid:96) LWS;m LWS;mJj GS e−i(Em−EGS)t
(cid:104) | | (cid:105)(cid:104) | | (cid:105)
m
(cid:88) 1
+ 2ηz (cid:104)GS|Jj+(cid:96)η+|LWS;m(cid:105)(cid:104)LWS;m|η−Jj|GS(cid:105)e−i(Em−EGS−2µ)t. (15)
m m
The factor of (2ηz ) 1 arises from the normalization of the state η+ LWS,m . We note that µ<0 and hence 2µ is
m − | (cid:105) −
a positive energy shift. It is not obvious how to understand the second term in the framework of a mobile impurity
model. However, using the lowest-weight property η GS =0, we can rewrite (15) in the form
−
| (cid:105)
(cid:88)
CJJ((cid:96),t)= GS Jj+(cid:96) LWS;m LWS;mJj GS e−i(Em−EGS)t
(cid:104) | | (cid:105)(cid:104) | | (cid:105)
m
(cid:88) 1
+ 2ηz (cid:104)GS|[Jj+(cid:96),η+]|LWS;m(cid:105)(cid:104)LWS;m|[η−,Jj]|GS(cid:105)e−i(Em−EGS−2µ)t. (16)
m m
The main advantage of the representation (16) is that it only involves regular Bethe Ansatz states, which can be
constructed by standard methods. As we concern ourselves only with the threshold behaviours of the optical con-
ductivity, we need only focus on the lower edges of the various excitation continua. As a consequence of kinematic
constraints and matrix-element effects, processes with a small number of excitations above the ground state give the
dominant contributions to response functions. Defining
=[η ,J ]=2it( 1)j(c c +c c ), (17)
j − j j, j+1, j+1, j,
O − ↓ ↑ ↓ ↑
we can recast (16) in the form
(cid:88)
CJJ((cid:96),t)= GS Jj LWS;m 2e−i(Em−EGS)t+iPm(cid:96)
|(cid:104) | | (cid:105)|
m
(cid:88) 1
+ 2ηz |(cid:104)GS|Oj†|LWS;m(cid:105)|2e−i(Em−EGS−2µ)t+i(Pm−π)(cid:96) ≡CJ(1J)((cid:96),t)+CJ(2J)((cid:96),t). (18)
m m
4
Here the additional contribution to the momentum arises because acting with η+ shifts the momentum by π. If the
ground state contains N fermions, the contribution C(2)((cid:96),t) is proportional to 1/(L N +2), and can therefore be
JJ −
dropped in the thermodynamic limit away from half filling. However, as we are interested in densities close to one
fermion per site it is useful to retain it in view of potential comparisons to numerical results for finite-size systems.
The optical conductivity can then be written as
(cid:34) (cid:35)
σ (ω)= e2 (cid:88)2 (cid:88)(cid:90) ∞ dt eiωt C(a)((cid:96),t) (cid:8)ω ω (cid:88)2 σ(a)(ω). (19)
1 2ω JJ − →− } ≡ 1
a=1 (cid:96) −∞ a=1
III. BETHE ANSATZ FOR THE HUBBARD MODEL
To gain further insight into the representation (18) we now construct the ground state and low-lying excitations
above it. We first calculate the energy of such excitations in the thermodynamic limit. This will allow us to identify,
on kinematic grounds, which states within the manifold identified earlier are important with respect to the threshold
behavioursweaimtodescribe. WethereforerecapitulatesomeresultsfromRef. 2toallowaself-containeddiscussion.
Forlarge systemsizes, the eigenstates ofthe repulsive Hubbard modelcan beexpressed interms of solutions of the
Takahashi equations, expressed in terms of so-called counting functions. In the case of N electrons, M of which are
spin-down, these are defined by
zc(kj)=kj + 1 (cid:88)∞ (cid:88)Mn θ(cid:18)sinkj −Λnα(cid:19)+ 1 (cid:88)∞ (cid:88)Mn(cid:48) θ(cid:18)sinkj −Λ(cid:48)nα(cid:19), j =1,...,N 2M(cid:48), (20)
L nu L nu −
n=1α=1 n=1α=1
z (Λn)= 1 N(cid:88)−2M(cid:48)θ(cid:18)Λnα−sinkj(cid:19) 1 (cid:88)∞ (cid:88)MmΘ (cid:18)Λnα−Λmβ (cid:19), α=1,...,M , (21)
n α L nu − L nm u n
j=1 m=1β=1
zn(cid:48)(Λ(cid:48)nα)=−L1 N(cid:88)−2M(cid:48)θ(cid:18)Λ(cid:48)nα−nusinkj(cid:19)− L1 (cid:88)∞ (cid:88)Mm(cid:48) Θnm(cid:18)Λ(cid:48)nα−uΛ(cid:48)mβ (cid:19)
j=1 m=1β=1
n
+2Re[arcsin(Λ +niu)], α=1,...,M , (22)
(cid:48)α n(cid:48)
where θ(x)=2arctan(x), u=U/4t,
 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
θ x +2θ x + +2θ x +θ x , n=m
Θnm(x)=2θ(cid:0)|nx−(cid:1)m+| 2θ(cid:0)x(cid:1)+|n−m+|+22θ(cid:16) ·x·· (cid:17)+θ(cid:0)n+xm(cid:1)−,2 n+m n(cid:54)=m, (23)
2 4 ··· 2n 2 2n
−
and
(cid:88)∞ (cid:88)∞
M = n(M +M ), M = nM . (24)
n n(cid:48) (cid:48) n(cid:48)
n=1 n=1
Takahashi’s equations are
2πI 2πJn 2πJ n
z (k )= j, z (Λn)= α, z (Λn)= (cid:48)α. (25)
c j L n α L n(cid:48) (cid:48)α L
Here the sets I , Jn , J n consist of integers or half-odd integers depending on the particular state under
{ j} { α} { (cid:48)α}
consideration, obeying the “selection rules”
I (cid:40)Z+ 12 if (cid:80)m(Mm+Mm(cid:48) ) odd , L <I L,
j (cid:80) j
∈ Z if m(Mm+Mm(cid:48) ) even − 2 ≤ 2
(cid:40)
Jαn ∈ ZZ+ 12 iiff NN −−MMnn eovdedn, |Jαn|≤ 12(N −2M(cid:48)−m(cid:88)∞=1tnmMm−1), (26)
(cid:40) (cid:32) (cid:33)
J(cid:48)nα ∈ ZZ+ 21 iiff LL−−NN ++MMnn(cid:48)(cid:48) eovdedn, |Jα(cid:48)n|≤ 12 L−N +2M(cid:48)−m(cid:88)∞=1tnmMm(cid:48) −1 ,
5
where t = 2min(m,n) δ . The energy and momentum, measured in units of t, of an eigenstate characterised
nm mn
by the set of roots k ,Λn−,Λm are given by
{ j α (cid:48)β }
N(cid:88)−2M(cid:48) (cid:88)∞ (cid:88)Mn(cid:48) (cid:113)
E = (2cosk +µ+2u+B)+2BM +4 Re 1 (Λn+niu)2+Lu, (27)
− j − (cid:48)β
j=1 n=1β=1
 
N(cid:88)−2M(cid:48) (cid:88)∞ (cid:88)Mn(cid:48) (cid:0) (cid:0) n (cid:1) (cid:1)
P = kj − 2Rearcsin Λ(cid:48)β +niu −(n+1)π mod2π. (28)
j=1 n=1β=1
The monotonicity of the counting functions ensures that specifying a set of integers/half-odd integers in accordance
with the “selection rules” uniquely determines a solution of the Takahashi equations.
A. Ground state
We consider the case where L is even, the total number of electrons N is even and the number of down spins
GS
M is odd. The ground state is then obtained by choosing the set I ,Jn,J m to be2
GS { j α (cid:48)β }
N 1
GS
I = +j, j =1,...,N , (29)
j GS
− 2 − 2
M 1
J1 = GS +α, α=1,...,M . (30)
α − 2 − 2 GS
This configuration is shown for the example L = 16, N = 2M = 10 in Fig. 1. We denote the ground state, in
GS GS
the previously established notation, by
GS = LWS; I , J1 . (31)
| (cid:105) | { j} { α}(cid:105)
−NG2S−1 NG2S−1
Ij ∈Z+ 12
L 9 7 5 3 1 1 3 5 7 9 L
−2 −2 −2 −2 −2 −2 2 2 2 2 2 2
−MG2S−1 MG2S−1
J1 Z
α ∈
2 1 0 1 2
− −
Figure 1: Configuration of the integers for the ground state, explicit numbers given are for L=16, N =10,
GS
M =5
GS
1. Thermodynamic limit
On taking the thermodynamic limit at fixed density n and magnetisation m the roots become dense and we
GS GS
can describe the ground state in terms of root densities ρ , ρ , which satisfy linear integral equations2
c,0 s,0
1 (cid:90) A
ρ (k)= +cosk dΛa (sink Λ)ρ (Λ), (32)
c,0 1 s,0
2π −
A
−
(cid:90) Q (cid:90) A
ρ (Λ)= dka (Λ sink)ρ (k) dΛ a (Λ Λ)ρ (Λ). (33)
s,0 1 c,0 (cid:48) 2 (cid:48) s,0 (cid:48)
− − −
Q A
− −
6
Here a (x)= 2nu 1 and the integration boundaries Q and A are determined by
n 2π (nu)2+x2
(cid:90) Q (cid:90) A 1
dkρ (k)=n , dΛρ (Λ)= (n 2m ). (34)
c,0 GS s,0 GS GS
2 −
Q A
− −
The energy density of the system is given to o(1) by2
(cid:90) Q dk
e = ε (k)+u, (35)
GS c
2π
Q
−
where
(cid:90) A
ε (k)= 2cosk µ 2u B+ dΛa (sink Λ)ε (Λ), (36)
c 1 s
− − − − −
A
−
(cid:90) Q (cid:90) A
ε (Λ)=2B+ dk coska (Λ sink)ε (k) dΛ a (Λ Λ)ε (Λ). (37)
s 1 c (cid:48) 2 (cid:48) s (cid:48)
− − −
Q A
− −
The dressed energies ε (k) and ε (Λ) satisfy ε ( Q)=ε ( A)=0. The dressed momenta are given by2
c s c s
± ±
(cid:90) A (cid:18)sink Λ(cid:19)
p (k)=k+ dΛρ (Λ)θ − , (38)
c s,0
u
A
(cid:90) Q − (cid:18)Λ sink(cid:19) (cid:90) A (cid:18)Λ Λ (cid:19)
(cid:48)
ps(Λ)= dkρc,0(k)θ − dΛ(cid:48)ρs,0(Λ(cid:48))θ − . (39)
u − 2u
Q A
− −
B. Excitations contributing to C(1)((cid:96),t).
JJ
We now turn to excited states that contribute to the spectral representation (18) of C(1)((cid:96),t). These are lowest
JJ
weight states of the spin and η-pairing SU(2) algebras with quantum numbers N =N , M =M .
GS GS
1. “Particle-hole” excitation with N =N , M =M .
GS GS
Creating a particle-hole excitation in the charge degrees of freedom yields a state with the same charge and spin
quantum numbers as the ground state, but with a finite momentum and energy difference. The (half-odd) integers
for this type of excitation are given by
(cid:40) NGS+1 +j+Θ(cid:0) NGS+1 +j Ih(cid:1), j =1,...,N 1
I = − 2 − 2 − GS − , (40)
j Ip, j =N
GS
M +1
J = GS +α, α=1,...,M , (41)
α GS
− 2
where Θ(x)=1 for x 0 and 0 otherwise. The arrangement for these integers is shown in Fig. 2. This excitation is
≥
two-parametric and has an energy and momentum of the form
E =e L+ε (kp) ε (kh)+o(1),
GS c c
− (42)
P =p (kp) p (kh)+o(1),
c c
−
where the rapidities are determined by z (kh) = 2πIh, z (kp) = 2πIp. This forms a continuum of excitations above
c L c L
the ground state, shown in Fig. 3.
7
Ip −NG2S−1 Ih NG2S−1
Ij ∈Z+ 12
L 9 7 5 3 1 1 3 5 7 9 L
−2 −2 −2 −2 −2 −2 2 2 2 2 2 2
−MG2S−1 MG2S−1
J1 Z
α ∈
2 1 0 1 2
− −
Figure 2: Configuration of the integers for the particle-hole excitation above the ground state, explicit numbers
given are for L=16, N =10, M =5
GS GS
Particle-hole continuum for U =8, n=0.8
4
3.5
3
2.5
S
G
e
L 2
−
E 1.5
1
0.5
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
p/π
Figure 3: Particle-hole excitation continuum above the ground state
2. “k-Λ string” excitation
We start by considering excitations with N =N , M =M involving a single (“k-Λ string”) bound state. This
GS GS
excitation has been considered previously e.g. in Section 7.7.2 of Ref. 2. It involves having a single (half-odd) integer
in the sector corresponding to the set J 1 . The lowest-energy bound state which can be created comprises of two
{ (cid:48)α}
ks and one Λ forming a string pattern in the complex plane. The Takahashi equations describe the real centres of
these and other root patterns. The case we consider is realised by the integer configuration
N 2 1
GS
I = − +j, j =1,...,N 2, (43)
j GS
− 2 − 2 −
M 1 1
J1 = GS − +α, α=1,...,M 1, (44)
α − 2 − 2 GS −
1 p
J =J , β =1, (45)
(cid:48)β (cid:48)
whichisdisplayedinFig.4. Inthenotationsusedabove,wecandenotethisexcitedstateby LWS; I , J1 , J 1 .
| { j} { α} { (cid:48)β}(cid:105)
We can again take the thermodynamic limit and compare the energy of this excited state with that of the ground
state. Followingsimilarmanipulationstothecaseofthegroundstateenergy,the (1)correctionscanbecalculated2.
O
The energy is given by
E =Le +ε (Λp)+o(1), (46)
GS kΛ
8
−NG2S−3 NG2S−3
Ij ∈Z+ 12
L 7 5 3 1 1 3 5 7 L
−2 −2 −2 −2 −2 2 2 2 2 2
−MG2S−2 MG2S−2
J1 Z
α ∈
3 1 1 3
−2 −2 2 2
L N L N
− −2 J0p −2
J01α ∈Z
3 2 1 0 1 2 3
− − −
Figure 4: Configuration of the integers for the k-Λ string excited state
where
(cid:90) Q
(cid:112)
ε (Λ)=4Re 1 (Λ iu)2 2µ 4u+ dkcosk a (sink Λ)ε (k). (47)
kΛ 1 c
− − − − −
Q
−
The momentum is given by P =p (Λp), where
kΛ
(cid:90) Q (cid:18)Λ sink(cid:19)
(cid:48)
pkΛ(Λ(cid:48))= 2Rearcsin(Λ(cid:48)+iu)+ dkρc,0(k)θ − , (48)
− u
Q
−
and Λp is determined by z (Λp)= 2πJ(cid:48)p. This form can be readily interpreted physically as a particle-like excitation
1(cid:48) L
above the ground state. The k-Λ string dispersion describes the threshold of an excitation continuum obtained by
adding e.g. particle-hole excitations in the charge sector. The dispersion relation for this excitation and the particle-
holecontinuumisshowninFig.5. Theexistenceofsuchacontinuumatp=0isnecessarytounderstandtheproblem
within the mobile impurity approach to threshold singularities.
0.12 k-Λstringandparticle-holecontinuumforU=8,n=0.8
U =8
9.5
0.09 U =10 k-Λ+particle-holecontinuum
U =20 9 k-Λdispersion
0.06
n=0.8 8.5
0.03
µ2 GS 8
+ 0 Le 7.5
εkΛ 0.4 UU==180 E− 7
0.3
U =20
0.2 n=0.6 6.5
0.1
6
0
-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 5.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
p/π p/π
(a) DispersionrelationεkΛ(Λ(p))forthek-Λstringforvarious (b)k-Λandchargeparticle-holeexcitationcontinuum.
U andn. Eachcurvehasbeenshifteddownby−2µ.
Figure 5: k-Λ string dispersion for various U and n, and particle-hole excitation continuum above this for U =8,
n=0.8. For small momenta, the k-Λ string dispersion marks the lower edge of a continuum described by additional
excitations e.g. particle-hole excitations in the charge sector.
9
C. Excitations contributing to C(2)((cid:96),t).
JJ
We now turn to excited states that contribute to the spectral representation (18) of C(2)((cid:96),t). As we have re-
JJ
expressed CJ(2J)((cid:96),t) in terms of matrix elements of the operator Oj† defined in (17), we will focus on excited states
|LWS;m(cid:105) that have non-vanishing matrix elements (cid:104)GS|Oj†|LWS;m(cid:105) (cid:54)= 0 These are lowest weight states of the
spin and η-pairing SU(2) algebras and their quantum numbers are N = N 2, M = M 1. It is of course
GS GS
− −
straightforward to translate back to excitations contributing to the original spectral representation (15): all that is
required is to act with η on the states we discuss in the following.
†
1. “Particle-hole” excitation with N =N −2, M =M −1.
GS GS
The integer configuration for this type of excitation is given by
I =(cid:40)−NG2S +j+Θ(cid:0)−N2GS +j−Ih(cid:1), j =1,...,NGS −3, (49)
j Ip, j =N 2
GS
−
M
J = GS +α, α=1,...,M 1 . (50)
α GS
− 2 −
This is shown graphically in Fig. 6.
Ip −NG2S−2 Ih NG2S−4
Ij Z
∈
−L2 −4 −3 −2 −1 0 1 2 3 L2
−MG2S−2 MG2S−2
Jα ∈Z+ 12
3 1 1 3
−2 −2 2 2
Figure 6: Integer configuration for the particle-hole excitation, explicit numbers for L=16, N =10, M =5.
GS GS
In complete analogy to the previous case, the energy and momentum of this state are given by
E =Le +ε (kp) ε (kh)+o(1),
GS c c
−
P =p (kp) p (kh) 2k +o(1), (51)
c c F
− ±
where kp and kh are determined by z (kp)= 2πIp, z (kh)= 2πIh. The contributions 2k arise from the asymmetry
c L c L ± F
of the charge “Fermi sea”, leaving a choice of two parity-related states. The continuum of excitations given by (51)
is shown in Fig. 7, and consists of the union of two copies of the continuum depicted in Fig. 3 shifted by 2k
F
±
respectively. We note that in order to make closer contact with the spectral representation (18) we have shifted the
momentum by π.
2. “Two particle” excitation with N =N −2, M =M −1.
GS GS
A closely related type of excitation corresponds to the choice of (half-odd) integers

−NG2S−4 +j, j =1,...,NGS −4
Ij = Ip1, j =NGS 3, , (52)
Ip2, j =NGS −2
−
M
J = GS +α, α=1,...,M 1. (53)
α GS
− 2 −
10
“Particle hole” excitation continuum for U =8, n=0.8
4.5
4
3.5
3
S
G 2.5
e
L
− 2
E
1.5
1
0.5
0
-1 -0.5 0 0.5 1
(p π)/π
−
Figure 7: Continuum for particle-hole excitation with momentum shifted for clarity.
Ip1 −NG2S−6 NG2S−4 Ip2
Ij Z
∈
−L2 −2 −1 0 1 2 3 L2
−MG2S−2 MG2S−2
Jα ∈Z+ 12
3 1 1 3
−2 −2 2 2
Figure 8: Integer configuration for the particle-particle excitation, explicit numbers for L=16, N =10, M =5.
GS GS
Such a configuration is shown in Fig. 8 and can be thought of as involving two particles associated with Ip1 and Ip2
respectively. The energy and momentum of this excitation are
E =Le +ε (kp1)+ε (kp2)+o(1),
GS c c
P =p (kp1)+p (kp2) 2k +o(1), (54)
c c F
±
with zc(kpi) = 2πLIpi. The continua corresponding to (54) are shown in Fig. 9. We note that both possible choices
2k have been taken into account, and we have again shifted the total momentum by π in order to make closer
F
±
contact with the spectral representation (18) of our correlator.
3. “Two hole” excitation with N =N −2, M =M −1.
GS GS
Finally, we consider excitations characterised by the distribution of (half-odd) integers
(cid:18) (cid:19) (cid:18) (cid:19)
N N N
I = GS +j+Θ GS +j Ih1 +Θ GS +j Ih2 , j =1,...,N 2, (55)
j GS
− 2 − 2 − − 2 − −
M
GS
J = +α, α=1,...,M 1, (56)
α GS
− 2 −
which is displayed in Fig. 10. We see that these states can be viewed as involving two holes associated with Ih1 and
Ih2 respectively. The energy and momentum of this excitation are given by
E =Le ε (kh1) ε (kh2)+o(1),
GS c c
− −
P = p (kh1) p (kh2) 2k +o(1), (57)
c c F
− − ±