a r X i v :c o n d -m a t /0403175v 1 [c o n d -m a t .s t r -e l ] 5 M a r 2004
EPJ manuscript No.
(will be inrted by the editor)
Quantitative expression of the spin gap via bosonization for a dimerized spin-1/2chain
E.Orignac
Laboratoire de Physique Th´e orique de l’´Ecole Normale Sup´e rieure CNRS–UMR854924,Rue Lhomond F-75231Paris Cedex 05
France
February 2,2008
Abstract.Using results on the mass gap in the sine-Gordon model combined with the exact amplitudes in the bosonized reprentation of the Heinberg spin-1/2chain and one-loop renormalization group,we derive a quantitative expression for the gap in a dimerized spin-1/2chain.T
his expression is shown to be in good agreement with recent numerical estimates when a marginally irrelevant perturbation is taken into account.
PACS.75.10.Pq spin chain models
Low dimensional antiferromagnets have been the sub-ject of inten scrutiny both theoretical and experimental for the last twenty years.The simplest model,the spin-1/2Heinberg antiferromagnet,is integrable[1]and can be mapped onto a continuum field theory[2,3,4]which al-lows the full determination of its zero temperature critical behavior.The prence of a marginally irrelevant operator in the continuum theory induces logarithmic corrections to the critical scaling[5].The corrections to scaling of the correlation functions[5,6,7],NMR relaxation rates [8,9],and susceptibilities[10,11]in this model have been inves-tigated in details.Further,when the Heinberg spin-1/2chain model is perturbed by a relevant operator such as an alternation of the exchange coupling,the marginal op-erator gives ri to a logarithmic correction to the power law dependence of the gap on the perturbation[5].Such logarithmic corrections to scaling in the gap in the con-text of two dimensional statistical mechanics of the four state Potts model who transfer matrix is related to the Hamiltonian of the alternating Heinberg chain[12,13,14,15].In [14]in particular,it was shown that the dependence of the gap ∆on the dimerization δ,was changed from the form ∆∼δ2/3[16,17]to
a the form ∆∼δ2/3/|ln δ|1/2.Such logarithmic behavior was confirmed by numerical calculations in [18,19,20].Alternatively,the dependence of the gap on the dimerization can be described by an effective power law form with an exponent that deviates from 2/3[21,22].For a dimerization not too small,it is found that the resulting effective exponent is clo to 2/3[21].Further,by considering a Heinberg chain with an additional next-nearest neighbor coupling finely tuned to cancel the marginal operator,a pure power law with ex-ponent 2/3can be obtained obtained for the gap [23].Recently,the logarithmic corrections were investigated in greater details using the DMRG[24].The data for the gap
could be fitted to the form:
∆=
α1/2gap
δ2/
3
2 E.Orignac:Quantitative expression of the spin gap via bosonization for a dimerized spin-1/2chain
continuum
Hamiltonian[4,14]:
H =
dx
K
(∂x φ)2
−2g 28φ,(3)
where [φ(x ),Π(x ′)]=iδ(x −x ′),u =π2πu .The latter conditions ensures SU (2)symmetry,and for g 2<0the operators (πΠ)2−(∂x φ)2and cos √
a
=(J ++J −)(na )+(−)x/a n (na ),
(4)J +r (x )=(J x r +iJ y r )(x )=
1
2(θ−rφ)(x )
η↑η↓,(5)
J z
r =
12
[rΠ−∂x φ],
(6)
n +(x )=(n x +in y )(x )=
λ
2θ(x )
η↑η↓,
(7)
n z (x )=λ2φσ(x ),(8)where a is a lattice spacing,η↑/↓reprent Majorana fermion operators that can be omitted in some cas (Ref.
[28]for a discussion of this point),and θis defined by θ(x )=π x
−∞dx ′Π(x ′).The constant λis a non-universal parameter that depends on the lattice model being con-sidered.Recently,this parameter has been determined in the ca of the isotropic Heinberg spin-1/2chain[25,26],
and it was found that:
windows资源管理器
λ=
π
a
S n ·S n +1=(uniform)+(−)n a [−(J ++J −)(na )·n ((n +1)a )+n (na )·(J ++J −)((n +1)a )](10)
The bosonized expression of the dimerization operator,is
thus obtained from the the short distance expansion of J R,L and n .Using Eqs.(5)and (7)with Glauber identi-ties,one finds the following expressions:n ±(x )(J ++J −)∓(x +a )=λ2φ(x )+...(11a)(J ++J −)±(x )n ∓(x +a )=−
λ
2φ(x )+...
(11b)The change of sign is a conquence of the application
of the Glauber identity taking into account the commu-tation relation [φ(x ),θ(x ′)]=iπY (x ′−x ),Y being the
Heaviside step function.Finally,n z (x +a )(J z ++J z
−)(x )
and n z (x )(J z ++J z大学gpa怎么算
−)(x +a )are respectively obtained from
Eqs.(6)and (8)the two following short distance expan-sion:
−
12∂x φ(x +a )sin √2πa cos √2π
√2φ(x +a )=−12φ(x )+...
(12b)which can be derived by normal ordering the product of
the two operators[29,30].A sketch of the derivation is given in the appendix.It is easily en that Eqs.(11)–(12)are compatible with spin rotational invariance.Combining the expressions (11)and (12)in (10),and using the value of λin Eq.(9)we finally obtain that:1
π2a
π
2φ(13)
Therefore,the continuum Hamiltonian describing the dimer-ized spin 1/2chain at low energy reads:
H = dx K
诚字组词(∂x φ)2
−2g 12φ−2g 2
8φ.(14)Note that in (14),the sign of g 1does not matter as it can
always been rendered positive by the shift φ→φ+π/√
2 1/4
δa.(15)As we noted before,g 2is a marginally irrelevant field which flows to 0if g 1=0.Let us assume for a moment that we can neglect completely the prence of this marginally irrelevant operator and take K =1,g 2=0in (14).Then,the Hamiltonian (14)becomes the sine-Gordon model.This model is integrable,and the expression of the gap can be found in [31],or in [32]Eq.(12).In the notations of [32],β2=K/4=1/4,and µ=3/π3(π/2)1/4δ(where
we have ud the fact that the velocity u =π√
Γ(2/3)
Γ(3/4)
π2
π
a M
<
∆
2
M ≃1.723δ2/3.
(17)
We note that the formula (16)has already been applied to calculate the gap of the dimerized spin 1/2chain in [33],but the value of λ,Eq.(9)was not known.The formula (17)is in reasonable agreement with the result quoted in [21]who reported that ∆/J =1.5δ0.65as two expression differ at most by 6%for 0.01≤δ≤0.1.Comparing our expression (17)to the one of Ref.[24],∆/J =1.94δ0.73,
E.Orignac:Quantitative expression of the spin gap via bosonization for a dimerized spin-1/2chain
3
0.010.1
1
0.0001
0.001
0.010.1
1
δ
∆/J
1.94 δ0.731.723 δ2/3α δ2/3
/(log(δ0/δ))1/2
Fig. 1.Comparison of the gap obtained in Ref.[24](solid line)with the
expressions
(17)
without
logarithmic
correc-
tions(dashed line)
and
the expression
装修方案
(29)including logarith-mic corrections with y 2(0)=−α=4.499and δ0=148(dash dotted line).Equation (17)gives a result nearly indis-tinguishable from the result of Ref.[24].
we find that they are in agreement within a 10%relative error when δ≥0.03as reprented on Fig.1.For lower values of δ,the two results deviate nsibly.As we shall e,this is the result of the logarithmic corrections.
In [32],the expression of the ground state energy was also given in dimensionless units in Eq.(14).Using this expression,we obtain for the ground state energy:
E 0
2J M 2
6
≃−0.2728δ4/3
(18)
This expression is compared to the one quoted in [24],
E 0/J =−0.39δ1.45on Fig.2.The two energy formulas are in better agreement than the gap formulas a
t low dimerization.Till now,we have totally neglected the pres-ence of the marginally irrelevant operator cos
√dl
1
8
y 21+
1
dl =
2−K
dl
=(2−2K )y 2+
y 21
1−y 2(0)l
(22)
Let us now assume[13]that we have turned on a very small y 1.Using the initial conditions with SU (2)symmetry,we can easily show that the RG equations reduce to:
dy 2
避孕套使用方法4
y 21,
(23)
dy 1
2+
3
2l
y 1(0)2/3
.(26)
For l >l 0the contribution of y 2to the renormalization of
y 1being negligible,y 1(l )=e 3/2(l −l 0)y 1(l 0).The scale l ∗at which y 1(l )∼1is thus given by:e
−l ∗
=e
−l 0
(1+|y 2(0)|l 0)2/3
(1+|y 2(0)|l 0)1/2
,(27)
4 E.Orignac:Quantitative expression of the spin gap via bosonization for a dimerized spin-1/2chain An approximate form of l0can be obtained by iterating
(26)leading to:
e−l∗≃|y1(0)|2/3
3|y2(0)|ln
y2(0)
J =
1.723δ2/3
3|y2(0)|ln
y2(0)纵横捭阖
J
=
0.2728δ4/3
3|y2(0)|ln
y2(0)
dφ
(φ(x)):∂x φ(x′)φ(x)−
酒店转让协议which is easily obtained by expanding V(φ)as a power
ries and applying Wick’s theorem[38].In the ca of the
massless free boson,we have:
φ(x′)φ(x)−φ2 =1a
,(32)
which leads to the expansion:
:∂xφ(x′)::V(φ(x)):=:∂xφ(x′)V(φ(x)):+
−1
dφ
(φ(x)):
(33)
Applying this formula in the ca of V(φ)=sin
√
E.Orignac:Quantitative expression of the spin gap via bosonization for a dimerized spin-1/2chain5
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