Table Of ContentInteger-spin Heisenberg Chains in a Staggered Magnetic Field.
A Nonlinear σ-Model Approach.
E. Ercolessi(1), G. Morandi(1), P. Pieri(2), and M. Roncaglia(1)
(1) Dipartimento di Fisica, Universit`a di Bologna, INFN and INFM, V.le Berti Pichat 6/2, I-40127, Bologna, Italy
(2) Dipartimento di Matematica e Fisica and INFM, Universit`a di Camerino, I-62032 Camerino, Italy
0
We present here a nonlinear sigma-model (NLσM) study of a spin-1 antiferromagnetic Heisenberg
0
0 chain in an external commensurate staggered magnetic field. We find, already at the mean-field
2 level, excellent agreement with recent and very accurate Density Matrix Renormalization Group
(DMRG) studies, and that up to the highest values of the field for which a comparison is possible,
n
for the staggered magnetization and the transverse spin gap. Qualitative but not quantitative
a
J agreement is found between the NLσM predictions for the longitudinal spin gap and the DMRG
results. The origin of the discrepancies is traced and discussed. Our results allow for extensions
8
to higher-spin chains that have not yet been studied numerically, and the predictions for a spin-2
2
chain are presented and discussed. Comparison is also made with previous theoretical approaches
] that led instead to predictions in disagreement with theDMRGresults.
l
e
PACS numbers: 75.10.Jm, 75.0.Cr, 75.40.Cx
-
r
t
s
.
t
a One and quasi-one-dimensional magnets (spin chains field appears to be even more interesting. Of course
m
and ladders) have become the object of intense analyt- static, staggered fields cannot be manufactured from
- ical, numerical and experimental studies since Haldane the outside. However, recently a class of quasi-one-
d
n putforward[1]hisbynowfamous“conjecture”according dimensional compounds has been investigated that can
o to which half-odd-integer spin chains should be critical be described as spin-1 chains acted upon by an effective
c with algebraically decaying correlations, while integer- internal staggered field. Such materials have Y BaNiO
2 5
[
spin chains should be gapped with a disordered ground as the reference compound and have the general for-
1 state [2,3]. Haldane’s conjecture has received strong ex- mula R BaNiO where R (= Y) is one of the magnetic
2 5
6
v perimental support from neutron scattering experiments rare-earth ions. The reference compound is found to
6
onquasi-one-dimensionalspin-1materialssuchasNENP be highly one-dimensional, hence a good realization of
1
and Y BaNiO [4]. It is by now well established that a Haldane-gap system (remember that the Ni2+ ions
4 2 5
1 pure one-dimensional Haldane systems (i.e. integer-spin have spin S = 1) [9]. The magnetic R3+ ions are po-
0 chains) have a disordered ground state with a gap to a sitioned between neighboring Ni chains and weakly cou-
0 degenerate triplet (for spin-1 systems) of magnon exci- pled with them. Moreover, they order antiferromagneti-
0
tations. There is also general consensus on the value of: callybelow acertainN´eeltemperatute T (typically: 16
t/ ∆ = 0.41048(2)J (with J the exchange constant) for K< T < 80 K [9]). This has the effectNof imposing an
a 0 N
m the gapin spin-1chains,that hasbeen obtainedby both eff∼ective∼staggered (and commensurate) field on the Ni
DensityMatrixRenormalizationGroup(DMRG)[5]and chains. The intensity of the field can be indirectly con-
-
d finite-size exact diagonalization[6] methods. trolledbyvaryingthe temperaturebelowTN. Thestag-
n The effects on Haldane systems of external magnetic gered field lifts partially the degeneracy of the magnon
o fields have also been the object of intense studies, again triplet, leading to different spin gaps in the longitudinal
c
both experimental, numerical and analytical. A uni- (i.e. parallel to the field) and transverse channels. Neu-
:
v form field induces a Zeeman splitting of the degener- tron scattering experiments on samples of Nd BaNiO
2 5
i
X ate magnon triplet, with the Haldane (gapped) phase andPr2BaNiO5 [10]showanincreaseoftheHaldanegap
remaining stable,with zero magnetization,up to a lower asafunctionofthestaggeredfield. Experimentalresults
r
a criticalfieldH =∆ ,whenoneofthemagnonbranches have also been obtained for the staggeredmagnetization
c1 0
becomes degenerate with the ground state and a Bose [11].
condensation of magnons takes place [7,8]. At higher Recently, an extensive DMRG study of an S = 1
fields the system enters into a gapless phase and magne- Heisenberg chain in a commensurate staggered field has
tizes, with the magnetization reaching saturation at an appeared[12]. TheauthorsinRef.[12]haveobtainedac-
upper critical field H =4SJ. curateresults for the staggeredmagnetizationcurve,the
c2
In view of the underlying, short-range antiferromag- longitudinal and transverse gaps and the static correla-
netic (AFM) ordering that is present in the Haldane tion functions (both longitudinal and transverse). They
phase, the study of the effects of a staggered magnetic found however a strong disagreement in the high-field
1
regimebetweentheirresultsandprevioustheoreticalap- The euclidean action S is quadratic in the field n
E
proaches [13,14] whose starting point was a mapping of which can be integrated out exactly. To this end, we
the chainontoanonlinearsigma-model,andthis hasled introduce the notation x=(x,τ) and write
them to openly questionthe entire validity ofthe NLσM
approach, at least in the high-field regime. S = dxdx′K(x,x′)n(x) n(x′)
E
This strong criticism has partly motivated our study ·
Z
of the same model as in Ref. [12]. We will actually show
SH dxn(x)+i dxλ(x) , (4)
s
that an accurate treatment of the NLσM does indeed − ·
Z Z
lead to an excellent agreement between the analytical
where the kernel K is given by:
and the DMRG results both for the magnetization and
the transverse gap for all values of the staggered field. 1
Some discrepancies are instead present for the longitu- K(x,x′)=−2gc[c2∂x2+∂τ2+2igcλ(x)]δ(x−x′) . (5)
dinal gap between our results and the data of Ref. [12],
but our identification of the latter relies on an approxi- Thesquareinequation(4)canbecompletedbyperform-
mation that, as we shall argue in the concluding part of ing the linear shift n(x) n′(x)=n(x) n¯(x), where
→ −
this Letter, may be questionable in the presence of an
S
external (staggered) field. After presenting the deriva- n¯(x)= H dx′K−1(x,x′) , (6)
s
tion of our results, we will also discuss what are the rea- 2
Z
sons for the disagreement between previous theoretical
and K−1 is the inverse of the kernel (5). After the in-
approaches and the DMRG results.
tegration over the shifted field we obtain the effective
We startfrom the following Hamiltonian for a Heisen-
action for the field λ
berg chain coupled to a staggered magnetic field:
3
N S(λ)= TrlnK+i dxλ(x)
H = JSi Si+1+Hs ( 1)iSi (1) 2 Z
· · − 1
Xi=1(cid:2) (cid:3) − 4S2H2s dxdx′K−1(x,x′) (7)
where S2 = S(S +1) (we set ¯h = 1 from now on), the Z
i
staggeredmagneticfieldH includesBohrmagnetonand The partition function is now given by = [Dλ]e−S(λ)
s Z 2π
gyromagneticfactor,andJ >0istheexchangeconstant. and can be calculated by a saddle-point approximation
R
In this Letter we will consider the case of integer spin forthe integraloverλ. We lookforastaticandconstant
S. Under this assumption, the Haldane mapping of the saddle-point solution: the kernel K depends in this case
Heisenberg chain onto a (1+1) nonlinear σ-model can only on the relative coordinates K(x,x′) = K(x x′).
−
be readily obtained [14]. The topological term is absent The equation ∂S =0 reads then:
∂λ
since the spin S is integer, while the staggered magnetic
fieldcoupleslinearlytotheNLσMfieldn. Theeuclidean 3 S2H2 2
K−1(x=0)=1 s dxK−1(x) . (8)
NLσM Lagrangianis thus given by [14]: 2 − 4
(cid:20)Z (cid:21)
1 1 ∂n 2 ∂n 2 It is convenient to work now in Fourier space, which is
E = +c 2gSn Hs , (2) defined according to
L 2g "c (cid:18)∂τ(cid:19) (cid:18)∂x(cid:19) − · #
L β
where the NLσM field n is a three-component unit vec- n˜(q,n)= dx dτe−i(qx−Ωnτ)n(x,τ) (9)
tor pointing in the direction of the local staggered mag- Z0 Z0
netization. The bare coupling constant g and spin-wave
(the frequencies Ω = 2πn/β are Matsubara Bose fre-
n
velocity c are related to the parameters of the Heisen-
quencies), and similarly for the field λ and kernel K.
berg Hamiltonian by g = 2/S and c = 2JSa, with a
The saddle-point equation becomes:
the lattice spacing. The constraint n2 = 1 can be taken
into account in the path-integral expression for the par- 3 S2H2 2
Stitio=n fuLndcxtionβdbτy w(xri,tτin)g(LZ==Na[Dbnei]ngD2πtλheet−oStEal,lwenhgetrhe 2(βL)Xq,n K˜−1(q,n)=1− 4 s hK˜−1(q =0,n=0)i (10)
ofEthe c0hain)0andL R (cid:2) (cid:3) K˜−1(q,n)= 2gc . (11)
R R Ω2 +c2q2 iλ2gc
n −
(x,τ)= (x,τ) iλ(x,τ)[n(x,τ)2 1] . (3)
L LE − − We now make the assumption iλ2gc = c2/ξ2, with ξ2
−
real and positive. Under this assumption, that will be
The functional integration over the field λ(x,τ) imple-
ments the constraint n2 =1. consistentlyverifiedshortlybelow,thesumoverfrequen-
ciesinEq.(10)canbeperformedbystandardtechniques.
2
By taking the thermodynamic and zero-temperature 1.0
limit(L,β )wefinallyobtainthefollowingequation
→∞
for ξ: 0.8
3g S2g2H2
ln Λξ+ 1+(Λξ)2 =1 sξ4 , (12) 0.6
2π − c2 mS
h p i
wherewehaveintroducedthe ultravioletcutoffΛtoreg- 0.4
ularize the momentum integration. The staggered mag-
netization (in units of S) will be determined instead by 0.2
the equation for the linear shift, which now reads
SH gSξ2 0.0
m = n¯ = sK˜−1(q =0,n=0)= H . (13) 0.0 0.1 0.2 0.3 0.4 0.5
s s
h i 2 c H
S
In order to get rid of the ultraviolet cutoff Λ we observe FIG.1. Thestaggered-magnetization curveforS =1. Our
results (solid line) are compared with DMRG data (circles)
that from Eq. (11) it follows that, when H = 0, the
s
andwiththeNLσMresults(dottedline)ofRefs.[12]and[14],
Green functions for the field n are given by
respectively.
1
G (q,n)= n˜ (q,n)n˜ ( q, n) =δ K˜−1(q,n)
α,β α β α,β
h − − i 2
gc
=δ , (14)
α,βΩ2n+c2q2+c2/ξ2 where A = ∆043πsSin(hJ(Sπ)S1//32). It is obvious from Eq. (18)
that the magnetization saturates to its maximum value
with ξ = sinh(2π/3g)/Λ, as obtained from Eq. (12) for
of m = 1 only asymptotically for H . This is
Hs = 0. From the structure of the Green functions (14) s s → ∞
consistentwiththefactthatitisonlyinthatasymptotic
itfollowsthat the excitationsaregapped, withthe value
limit that a fully polarized N´eel state becomes an exact
Λc eigenstate ofthe originalHeisenbergHamiltonian (1), as
∆ =∆(H =0)= (15)
0 s
sinh(2π/3g) well as with the fact that our mean-field approachkeeps
trackoftheNLσMconstraint,albeitonlyontheaverage.
for the zero-field Haldane gap [1] ∆ . The Haldane gap
0 In Figure 1 we compare, for S = 1, our staggered-
is the only free parameter in our theory, that we will as-
magnetizationcurve with DMRG data [12]and with the
sumefrompreviousnumericalestimates[5,6]. Inthisway
NLσM treatment of Ref. [14]. The magnetization curve
we fix the cutoff. Eqs. (12) and (13) (with the addition
was obtained by solving numerically Eq. (12) (with Λ
of Eq. (15)) provide our staggered-magnetization curve.
given by Eq. (15)) and by inserting the resulting ξ into
Eq.(12)canbe solvedanalyticallyfor largeorsmallval-
Eq.(13). Asalreadyanticipated,theagreementwiththe
ues H , while in general it has to be solved numerically
s DMRG data is excellent over the whole range of fields.
(albeit with no effort).
We have also solved Eq. (12) for S = 2, by fix-
For small H we obtain
s ing again Λ via Eq. (15) with the value, specific to
gSc 4π/3 c2gS2 S = 2, ∆ = 0.0876J [16]. At present, the staggered-
m = H 1 H2+O(H4) . 0
s ∆2 s − tanh(2π/3g) ∆4 s s magnetizationcurveforS =2hasnotyetbeenobtained
0 (cid:18) (cid:19)
by numerical methods. Given the excellent agreement
(16)
between our curve and numerical data for S = 1, we
The zero-field staggered susceptibility is then given by expect that our magnetization curve (shown in Figure
2) will agree with future numerical investigations also
dm gSc 4JS
χ (0)= s = = (17) in the case S = 2. Indeed, on general grounds, Hal-
s dH ∆2 ∆2
s(cid:12)Hs=0 0 0 dane’s mapping is expected to work even better when
(cid:12)
For S = 1, by taking (cid:12)∆ = 0.41048J from Ref. [5], we the value of the spin S is increased. Note that the mag-
(cid:12) 0
netization increases faster with H (or, equivalently, the
obtain χ (0) = 23.74/J which agrees fairly well with s
s
staggeredsusceptibilitybecomes larger)whenthe spinS
Monte-Carlo result χ (0) = 21(1)/J [15] and DMRG
s
is increased. This is consistent with the fact that, when
calculation χ (0) = 18.5/J [12]. For S = 2, we take
s
S , the system approaches the classical one, which
∆0 = 0.0876J from Ref. [16] and get χs(0) = 1043/J, → ∞
is criticalat T =0 and H =0. Note also that the value
whichshouldbecheckedbynewnumericalinvestigations. s
ofthezero-fieldHaldanegaproughlysetsthescaleofthe
For large H the staggered magnetization is instead
s
staggered-magnetizationcurve.
given by
A 1A2
m =1 + +O(A3/H3/2) (18)
s − √H 2H s
s s
3
1.0 8
7
0.8
6
5
0.6
0
mS D - 4
0.4 D 3
2
0.2
1
0.0 0
0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.01 0.02 0.03 0.04 0.05
H
H S
S FIG. 3. Plots of the transverse (solid curve) and longitu-
FIG.2. The staggered-magnetization curvefor S=2. dinal (dashed curve) gap predicted by the NLσM approach,
compared with the corresponding DMRG data (circles and
Introducing an external space and time-dependent triangles, respectively).
source field J=J(x) and promoting the partition func-
tion to a generating functional [J] obtained by re- In the parallel channel, on the other hand, the depen-
Z Z
placing in the original path-integral the Euclidean La- dence of λ on J yields additional contributions to the
sp
grangian of Eq. (2) with: Green functions, which do not allow us to calculate the
longitudinalGreenfunctionsexplicitly. Toobtainanesti-
1 1 ∂n 2 ∂n 2 mateforthelongitudinalgapweresortedtothesocalled
[J]= +c n(x) J(x) , (19)
L 2g "c (cid:18)∂τ(cid:19) (cid:18)∂x(cid:19) #− · “single-mode approximation” [14], that is we assumed
the gapand susceptibilities to be connected by the same
the connected Green functions for the staggeredfield n: law both in the transverse and parallel channel. It can
be readily shown that for an isotropic system, for which
Gα,β(x,x′)= T [nα(x)nβ(x′)] nα(x) nβ(x′) (20)
c h τ i−h ih i the magnetization is directed along the external field,
canbeobtainedbydoublefunctionaldifferentiationwith χ⊥ = ∂∂mHsxxs = ms/Hs, while χk = ∂∂Hmszzs = ddmHss. From
respect to the source field as: Eq. (13) we thus find χ = Sgξ2/c = Sgc/(∆ )2. If we
⊥ ⊥
assumethesamerelationtoholdalsointheparallelchan-
δ2ln [J]
Gα,β(x,x′)= Z . (21) nel, ∆ can be calculated as a derived quantity from the
c δJα(x)δJβ(x′) k
(cid:12)(cid:12)J=SHs magnetization curve by using ∆k = (Sgc/χk)1/2. Our
The saddle-point analysis can be pe(cid:12)(cid:12)rformed in this case data for ∆k are also shown in Figure 3. The agreement
as well. The saddle-point equation reads now: withthe DMRGdataisdefinitely worseinthiscase,and
this may attributed to a shortcoming of the single-mode
3 1
K−1(x,x)=1 dydy′K−1(y,x)K−1(x,y′) approximationinthelongitudinalchannel. Thecomplete
2 − 4 structure of the Green functions along the lines outlined
Z
J(y) J(y′) . (22) above in Eqs. (19) and (21) is being presently under-
× ·
taken, and more extended and complete results will be
This equation will yield in general a space and time-
published elsewhere [17].
dependent saddle point λ = λ (x), which will reduce
sp sp In conclusion, we come to a brief comparison between
however to a constant for J=const.=SH when, as it
s our approachand that of Refs. [13] and [14].
can be checked easily, Eq. (22) reproduces our previous
Whatthetwoapproacheshaveincommonisthestart-
Eq. (8). What matters here is the fact that the sad-
ing point, namely the mapping of the original problem
dle point will depend quadratically on the source field.
onto a NLσM, slightly modified by the external stag-
From this it follows that the transverse Green functions
gered field. In the abovemntioned references, however,
G (x,x′)=G (x,x′)donotgetanyadditionalcontri-
xx yy the authors resolved to relaxing the NLσM constraint
butionfromthe dependence ofλ onJ(we assumehere
sp by replacing it with a polynomial self-interaction of the
Hs = zˆHs) and can be calculated by taking J = SHs n-field, keeping terms in the expansion of the interac-
from the beginning. The transverse Green functions are
tion up to eight order. This leads to a (generalized)
thus still given by the same expression valid for H =0,
s Ginzburg-Landau-typetheorywithuptosixfreeparam-
Eq. (14) where now ξ depends on H via Eq. (12). The
s eters which the authors took from previous numerical
transverse gap will be accordingly ∆ (H )= c , and
⊥ s ξ(Hs) and Renormalization-Group studies. Their results for
inFigure3wecomparedthetransversegapascalculated the staggered magnetization saturate at a finite value of
by this relation with the one obtained by DMRG data. the staggered field (see Figure 1), which indicates that
The agreement is again excellent.
4
softening the NLσM constraint is inappropriate at high [17] E.Ercolessi, G.Morandi, P.Pieri and M.Roncaglia, in
fields. Also,their results for the gapsdeviate badly from preparation.
the DMRG results. All this has been evidenced in Ref.
[12]andjustifiesthenegativecommentsoftheauthorsin
thatreference,thathowevershouldbeaddressedmoreto
the additional approximations adopted in Refs. [13] and
[14] than to the NLσM approach itself.
ACKNOWLEDGMENTS
TheauthorsaregratefultoProf.YuLuandDr.Jizhong
Lou for letting them have access to the files of their
DMRG data.
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5
