Exact edge singularities and dynamical correlations in spin-1/2 chains

0.25 MB·English

by  Rodrigo G. Pereira

Exact edge singularities and dynamical correlations in spin-1/2 chains
Rodrigo G. Pereira,1 Steven R. White,2 and Ian Affleck1
1Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1
2Department of Physics and Astronomy, University of California, Irvine CA 92697, USA
(Dated: February 1, 2008)
Exactformulasforthesingularitiesofthedynamicalstructurefactor,Szz(q,ω),oftheS =1/2xxz
spinchainatallqandanyanisotropyandmagneticfieldinthecriticalregimearederived,expressing
theexponentsintermsofthephaseshiftswhichareknownexactlyfrom theBetheansatzsolution.
Wealso study thelong time asymptotics of the self-correlation function h0|Sz(t)Sz(0)|0i. Utilizing
j j
8 these results to supplement very accurate time-dependent Density Matrix Renormalization Group
0 (DMRG) for short to moderate times, we calculate Szz(q,ω)to very high precision.
0
2 PACSnumbers: 75.10.Pq,71.10.Pm

n
a
J The “xxz” S =1/2 spin chain, with Hamiltonian In this Letter we combine the methods of Ref. [13]
9 withtheBetheansatztoinvestigatethesingularityexpo-
1 L nents ofSzz(q,ω)for the xxzmodel forfinite interaction
H =J [SxSx +SySy +∆SzSz hSz], (1)
j j+1 j j+1 j j+1− j strength∆andgeneralmomentumq. Inaddition,wede-
] j=1
X terminetheexponentsofthelong-timeasymptoticsofthe
l
e is one of the most studied models of strongly correlated spinself-correlationfunction, which is not dominatedby
-
r systems. ItisequivalentbyaJordan-Wignertransforma- lowenergyexcitations. We checkourpredictionsagainst
t
s tionto a modelofinteractingspinless fermions,with the high accuracy numerical results calculated by DMRG.
.
t corresponding Fermi momentum k =π(1/2+ 0Sz 0 ) In the non-interacting,∆=0 case,only excited states
ma [1]. The model with ∆=1 describFes Heisenbergha|ntji|feir- with a single particle-hole pair contribute to Szz(q,ω).
romagnets. Theregime0<∆<1isalsoofexperimental Allthespectralweightisconfinedbetweenthelowerand
-
d interest;forexample,themodelwith∆=1/2canbe re- upper thresholds ωL,U(q) of the two-particle continuum.
n alized in S = 1/2 spin ladders near the critical field [2]. The choicesof momenta correspondingto the thresholds
o In optical lattices, it should be even possible to tune the depend on both kF and q. For zero field, kF = π/2,
c
[ anisotropy ∆ and explore the entire critical regime [3]. ωL(q) for any q > 0 is defined by the excitation with a
While some aspects of the model have been solved for hole at k1 = π/2 q and a particle right at the Fermi
2 exactly by Bethe ansatz [4], it has been very difficult to surface (or a hole−at the Fermi surface and a particle at
v
0 obtaincorrelationfunctions that way. Field theory (FT) k2 = π/2 + q), while ωU is defined by the symmetric
6 methods give the low energy behavior at wave-vectors excitation with a hole at k1 = π/2 q/2 and a particle
−
9 near0and2kF [1]. Fromtheexperimentalviewpoint[5], at k2 = π/2+q/2. For finite field and q < 2kF π ,
| − |
0 a relevant quantity is the dynamical structure factor ω (q) are defined by excitations with either a hole at
L,U
9. kF and a particle at kF +q or a hole at kF q and a
0 L +∞ particleatk . Forh=0andq > 2k π ,th−ereiseven
Szz(q,ω)= e−iqj dteiωt 0Sz(t)Sz(0)0 . (2) F 6 | F− |
07 j=1 Z−∞ h | j 0 | i a third “threshold” between ωL and ωU where Szz(q,ω)
X has a step discontinuity (see [7]).
:
v This is the Fourier transform of the density correlation For ∆ = 0, Szz(q,ω) exhibits a tail associated with
Xi function in the fermionic model. For ∆ = 1 and h = 0, multiple p6 article-hole excitations [14]. However, the
the exact two-spinon contribution to Szz(q,ω) was ob- thresholds of the two-particle continuum are expected
r
a tained from the Bethe ansatz [6], partially agreeingwith to remain as special points at which power-law singular-
the Mu¨ller conjecture [7]. More recently a number of ities develop [13]. In order to describe the interaction
new methods have emerged which now make this prob- of the high energy particle and/or hole with the Fermi
lemmuchmoreaccessible. Theseincludetime-dependent surface modes, we integrate out all Fourier modes of the
DMRG [8, 9, 10], calculation of form factors from Bethe fermion field ψ(x) except those near k and near the
F
±
ansatz[11,12]andnewfieldtheoryapproacheswhichgo momentum of the hole, k1, or particle, k2, writing
beyond the Luttinger model [13, 14]. The results point
to a very nontrivial line shape at zero temperature for ψ(x) eikFxψR+e−ikFxψL+eik1xd1+eik2xd2. (3)
∼
Szz(q,ω) of the xxz model [14] and of one-dimensional
modelsingeneral[13]. Intheweakcouplinglimit∆ 1 Linearizing the dispersion relationabout k we obtain
F
≪ ±
and for small q, the singularities at the thresholds of the relativistic fermion fields which we bosonize in the usual
two-particle continuum have been explained by analogy way [1]. We also expand the dispersion of the d1,2 parti-
with the x-ray edge singularity in metals [13]. cles around k = k1,2 up to quadratic terms. This yields
2
the effective Hamiltonian density dα = −Bdλραimp(λ) +∞dλραimp(λ), (8)
= d† ε iu ∂ ∂x2 d imp Z−∞ 2 −ZB 2
H αX=1,2 α(cid:18) α− α x− 2mα(cid:19) α where B is the Fermi boundary and ραimp(λ) is the solu-
v tion to the integral equation
+2 (∂xϕL)2+(∂xϕR)2 +V12d†1d1d†2d2
+√h21πK α=1,2(καR∂xϕRi+καL∂xϕL)d†αdα. (4) ραimp(λ)−Z−+BB d2λπ′ ραimp(λ′)dΘ(λdλ−λ′) = Φα2π(λ), (9)
X
ThisHamiltoniandescribesaLuttingerliquidcoupledto where Θ(λ) = ilog[sinh(iζ + λ)/sinh(iζ λ)], with
−
one or two mobile impurities [15, 16]. In the derivation ∆= cosζ, is the two-particle scattering phase [4], and
of Eq. (4) from Eq. (1), we drop terms of the form Φ1,2(−λ) = dΘ(λ λ1,2)/dλ. The spectrum of Eq. (6)
∓ −
(d†d )2 because we only consider processes involving a describes a shifted c = 1 conformal field theory (CFT).
α α
single d1 and/or a single d2 particle. Here ϕR,L are the Thescalingdimensionsofthe variousoperatorscanthen
right and left components of the rescaled bosonic field. be expressed in terms of K, nαimp and dαimp. In the ef-
The long wavelength fluctuation part of Sz is given by fective model (4), the shift is introduced by the unitary
j
transformation of Eq. (5), which changes the boundary
Sz K/2π(∂ ϕ +∂ ϕ ). The spin velocity v and
j ∼ x R x L conditions of the bosonic fields. The equivalence of the
LuttingerparameterKareknownexactlyfromtheBethe
p two approaches allows us to identify
ansatz [4]. For zero field, v = (π/2)√1 ∆2/arccos∆
fianrsdt Kord=er [i2n−∆2, atrhcecocso(u∆p)l/inπg)]−co1n(swtaentsse−tdeJsc=rib1in)g. tThoe γRα,L/π=nαimp±2Kdαimp. (10)
scattering between the d particles and the bosons are The phase shifts can be determined analytically for zero
καR,L =2∆[1−cos(kF∓kα)]. Thedirectd1-d2interaction magneticfield. Inthis case,B →∞andwehavedαimp =
V12 is also of order ∆. The exact values of κR,L play a 0. Moreover,by integrating Eq. (9) over λ we find
crucial role in the singularities and will be determined
below. n1,2 = Θ(λ )/[π Θ(λ )]= (1 K). (11)
imp ∓ →∞ − →∞ ± −
We may eliminate the interactionbetweenthe dparti-
Once the exact phase shifts are known, the exponent
cles and the bosonic modes by a unitary transformation
for the (lower or upper) threshold determined by a sin-
dx gle high energy particle can be calculated straightfor-
U =exp i (γαϕ γαϕ )d†d , (5)
( √2πK R R− L L α α) wardly. For example, for a lower threshold defined by
α Z
X a deep hole, ω (q) = ǫ(k q), the correlation func-
L F
HwiatmhipltaornaimanetHe˜rs=γRαU,L†H=Uκ,αRϕ,LR/,L(va∓reufαre)e. uInptthoeirrreesluelvtainngt tdi1onprhodp†1aψgRa(tto,rx)aψnR†ddc1o(r0−r,e0la)iticoan−ns obfeefxapctoonreiznetdialisntoof aϕRfr,eLe.
interactionterms[15]. Asinthex-rayedgeproblem,γRα,L After Fourier transforming, we find that near the lower
mayberelatedtothephaseshiftsattheFermipointsdue edge Szz(q,ω) [ω ω (q)]−µ with exponent [17]
L
to the creation of the high energy d particle. ∼ −
α
Fortunately, we have access to the high energy spec- µ=1 (1 n1 )2/2K 2K(1/2 d1 )2. (12)
− − imp − − imp
trum of the xxz model by means of the Bethe Ansatz.
For h 0, we use Eq. (11) and obtain
Following the formalism of Ref. [16], we calculate the fi-
→
nite size spectrum from the Bethe ansatz equations with
µ=1 K, (h 0) (13)
an impurity term corresponding to removing (adding) − →
a particle with dressed momentum k1 = k(λ1) (k2 = independentofthemomentumofthehole. Thisformfor
k(λ2)), where λ1,2 are the corresponding rapidities. The the lower edge exponent had been conjectured long ago
term of O(1) yields ε = ǫ(k ), the dressed energy of
α α by Mu¨ller et al. [7]. It agrees (up to logarithmic correc-
the particle. For zero field, we have the explicit formula tions) with the exponent of the two-spinon contribution
ǫ(k) = vcosk. The excitation spectrum for a single to Szz(q,ω) for the Heisenberg point (K =1/2) [6].
−
impurity to O(1/L) reads
The general result of Eq. (12) is consistent with the
∆E = 2πv 1 ∆N nα 2+K D dα 2 weak coupling expression for µ [13]. To first order in ∆,
L 4K − imp − imp Eq. (12) reduces to
(cid:20)
+n+(cid:0)+n−], (cid:1) (cid:0) (cid:1) (6) κ1 2∆(1 cosq)
µ R − . (14)
withaconventionalnotationfor∆N,D andn± [4]. The ≈ π(v u1) ≈ π[sinkF sin(kF q)]
− − −
phase shifts nα and dα are given by
imp imp
For k = π/2, we expand for q k and get µ
F F
+B m∆q/π,6wherem=(cosk )−1(c.f.≪[13]). Fork =π/≈2,
nα = dλρα (λ), (7) F F
imp imp we obtain µ 2∆/π, which is 1 K to O(∆). Note the
Z−B ≈ −
3
cancellationof the q dependence of κ1R and v−u1 in the The term oscillating at 2W comes from a hole at k =
lattercase. Momentum-independentexponentshavealso 0 and a particle at k = π. For ∆ = 0, we have
been derived for the Calogero-Sutherlandmodel [18]. η2 = 1. The exponent η2 is connected with the sin-
We now consider a threshold defined by high-energy gularity at the upper threshold of Szz(q,ω) by G(t)
∼
particle and hole at k1,2 =π/2∓q/2. The relevant cor- dωeiωt dqSzz(q,ω) for q ≈ π and ω ≈ ωU(π) = 2v.
relation function is the propagator of the transformed DuetothediscontinuityoftheexponentatωU,η2 jumps
R R
d†2d1. For simplicity, here we focus on the zero field from η2 =1 to η2 =2 for any nonzero ∆. This behavior
case, in which ε2 = ε1 = vsin(q/2), u2 = u1 and should be observed for t 1/(m1V122) 1/∆2. As a re-
m2 = m1 = [vsin(q−/2)]−1. Particle-hole symmetry sult,theasymptoticsofG≫(t)isgoverned∼bytheexponent
−
then implies that γ1 = γ2 and d†d is invariant η < 3/2 for 0 < ∆ < 1. For ∆ < 0, we must add to Eq.
R,L R,L 2 1
under the unitary transformation of Eq. (5). In the (15) the contribution from the bound state.
noninteracting case, there is a square root singularity We can also study Szz(q,ω) with time-dependent
at the upper threshold due to the divergence of the DMRG (tDMRG) [8, 9]. The tDMRG methods di-
jointdensity ofstates: Szz(q,ω) m1/[ωU(q) ω]for rectlyproduceSzz(x,t)anditsspatialFouriertransform
ω ω (q)=2vsin(q/2)[7]. For∝∆=0,weneed−totreat Szz(q,t) for short to moderate times. This information
U
≈ p6
the direct interaction V12 between the particle and the nicelycomplementstheasymptoticinformationavailable
hole, which is not modified by U. This problem is anal- analytically. The DMRG calculation begins with the
ogous to the effect of Wannier excitons on the optical standard finite system calculation of the ground state
absorption rate of semiconductors [19, 20]. This sim- φ(t=0)onafinite lattice oftypicallengthL=200-400,
ple two-body problem can be solved exactly for a delta where a few hundred states are kept for a truncation er-
function interaction. The result is that the upper edge ror less than 10−10. One of the sites at the center of the
exponent changes discontinuously for ∆=0: the square lattice is selected as the origin, and the operator S0z is
6
root divergence turns into a universal (for any q and ∆) applied to the ground state to obtain a state ψ(t = 0).
square root cusp, Szz(q,ω) ω (q) ω. This behav- Subsequently,thetimeevolutionoperatorforatimestep
U
∝ −
ior contradicts the Mu¨ller ansatz [7], but is consistent τ, exp(i(H E0)τ) where E0 is the groundstate energy,
p −
with the analytic two-spinonresultfor ∆=1 [6]. Unlike isappliedviaafourthorderTrotterdecomposition[10]to
the originalexcitonproblem,aboundstateonlyappears evolvebothφ(t) andψ(t). At eachDMRGstepcentered
for V12 < 0 (∆ < 0) [21], because the particle and hole on site j we obtaina data point for the Green’s function
haveanegativeeffectivemass. For∆=0,theupperedge G(t,j) by evaluating φ(t)Sz ψ(t) . As the time evolu-
6 h | j| i
cuspshouldintersectahigh-frequencytaildominatedby tion progresses, the truncation error accumulates. The
four-spinon excitations as proposed in [22]. This picture integrated truncation error provides a useful estimate of
must be modified for h = 0, since then γ1 =γ2 and the error, and so longer times require smaller trunca-
6 R,L 6 R,L
oneneedstoincludethebosonicexponentials. Theupper tion errorsat eachstep, attained by increasing the num-
edge singularity then becomes ∆- and q-dependent. The ber of states kept m. The truncation error grows with
general finite field case, including the middle singularity time for fixed m, and is largest near the center where
[7] for q > 2k π , will be discussed elsewhere. the spin operator was applied. We specify the desired
F
| − |
We canapply the Hamiltonian ofEq. (4) to study the truncation error at each step and choose m to achieve
self-correlation function G(t) 0Sz(t)Sz(0)0 . Even it, within a specified range. Typically for later times we
in the noninteracting case, the≡lohng| tjime ajsym|pitotics is have m 1000. Finite size effects are small for times
≈
a high energy property,since it is dominatedby a saddle less than (L/2)/v. We are able to obtain very accurate
point contribution with a hole at the bottom and a par- results for G(t,j), with errors between 10−4 and 10−5,
ticle atthe top ofthe band [23]. In this case,k1 =0 and for times up to Jt∼ 30-60.
k2 = π and d1im,2p vanish by symmetry (γRα = γLα). Here ForJt>10−20,wefind the behaviorof Szz(q,t)and
we restrict to zero field, but the method can be easily G(t)iswellapproximatedbyasymptoticexpressions,de-
generalized. For h=0 and ∆ 0, G(t) takes the form termined by the singular features of Szz(q,ω) and G(ω).
≥
By utilizing the leading and subleading terms for each
e−iWt e−i2Wt B3 B4 singularity,wehavebeenabletofitwithatypicalerrorin
G(t)∼B1 tη +B2 tη2 + tσ + t2 , (15) Szz(q,t) or G(t) for Jt 20-30 between 10−4 and 10−5.
∼
We can fit with the decay exponents determined analyt-
where W = ǫ(0) = v. The last two terms are the ically or as free parameters to check the analytic expres-
−
standard low-energy contributions, with σ = 2K. The sions. Table I shows the comparison between the expo-
amplitudes B3 and B4 are known [24]. The first term is nents for G(t) extracted independently from the DMRG
thecontributionfromtheholeatthebottomoftheband dataandtheFTpredictions. Inallcasestheagreementis
and the particle at k =π/2, with exponent very good. By smoothly transitioning from the tDMRG
F
data to the fit as t increases, we obtain accurate results
η =(1+K)/2+(1 n1 )2/2K =K+1/2. (16) for all times. A straightforward time Fourier transform
− imp
4
the particle-hole symmetric zero field case, we showed
TABLE I: Exponents for the spin self-correlation function
that the lower edge exponent is q-independent and the
G(t) for h = 0. The parameters W, η, η2 and σ were ob-
(“exciton-like”) upper edge has a universal square root
tained numerically by fitting the DMRG data according to
Eq. (15). These are compared with the corresponding FT singularity. The combination of analytic methods with
predictions (with v and K taken from the Betheansatz). the tDMRG overcomesthe finite t limitation on the res-
olutionofthetDMRGandcanbeusedtostudydynamics
∆ W v η 12 +K σ 2K η2 of other one-dimensional systems (integrable or not).
0 1 1 1.5 1.5 2 2 1 1
We thank L. Balents, J.-S. Caux, V. Cheianov, L.I.
0.125 1.078 1.078 1.451 1.426 1.954 1.852 1.761 2
Glazman, N. Kawakami, M. Pustilnik and J. Sirker for
0.25 1.153 1.154 1.366 1.361 1.811 1.723 2.034 2
discussions. We acknowledge the support of the CNPq
0.375 1.226 1.227 1.313 1.303 1.694 1.607 2.000 2
grant 200612/2004-2 (RGP), NSERC (RGP, IA), NSF
0.5 1.299 1.299 1.287 1.25 1.491 1.5 2.120 2 under grant DMR-0605444(SRW) and CIfAR (IA).
0.75 1.439 1.438 1.102 1.149 1.324 1.299 2.226 2
with a very long time window yields very accurate high
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ω
, [14] R.G. Pereira et al.,Phys.Rev.Lett. 96, 257202 (2006).
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π
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q 58, 3633 (1998).
(
S [17] After this paper was accepted we learned of the results
of Cheianov and Pustilnik (private communication and
arXiv:0710.3589). We have checked that their exponent
is exactly the same as ours for all q, ∆ and h6=0.
[18] M. Pustilnik, Phys.Rev. Lett. 97, 036404 (2006).
0
[19] G.D. Mahan, Many Particle Physics (Kluwer/Plenum,
0.8 1 1.2 1.4 1.6 1.8
ω New York,2000).
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FIG. 1: (Color online). DMRG results for Szz(q,ω) versus
[22] J.-S. Caux and R. Hagemans, J. Stat. Mech. P12013
ω for q = π/2, h = 0 and several values of anisotropy ∆.
(2006).
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(2003).
The width of thepeak is very small for small |∆|.
[25] M. Khodas et al.,cond-mat/0702505.