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Nonlinear Optical Response of SDW Insulators

S. Akbar Jafari1,2 ? , Takami Tohyama1 and Sadamichi Maekawa1,3

arXiv:cond-mat/0507556v2 [cond-mat.str-el] 7 Feb 2006

1 2

Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan

(Received February 2, 2008)

Department of Physics, Isfahan University of Technology, Isfahan 84156, Iran

3

We calculate the third order nonlinear optical response in the Hubbard model within the spin density wave (SDW) mean ?eld ansatz in which the gap is due to onsite Coulomb repulsion. We obtain closed-form analytical results in one dimension (1D) and two dimension (2D), which show that nonlinear optical response in SDW insulators in 2D is stronger than both 3D and 1D. We also calculate the two photon absorption (TPA) arising from the stress tensor term. We show that in the SDW, the contribution from stress tensor term to the lowenergy peak corresponding to two photon absorption becomes identically zero if we consider the gauge invariant current properly. KEYWORDS: Third harmonic generation, Two photon absorption, Gauge invariance, SDW insulator

1. Introduction The realization of ultra fast networking through all-optical switching in modern optical technology requires advanced optical materials with large third-order nonlinear optical susceptibility χ(3) (Ref. 1). Quasi one dimensional (1D) π -conjugated polymers o?er χ(3) values of 10?12 to 10?7 e.s.u. (electronic system of units). The quasi 1D Mott insulators such as Sr2 CuO3 , o?er χ(3) values in the range 10?8 to 10?5 e.s.u.2, 3 The canonical model for strongly correlated materials is the standard Hubbard model. In the spin density wave (SDW) mean ?eld approximation, this Hamiltonian reduces to a simple quadratic form that can be diagonalized exactly. Such an SDW Hamiltonian is indeed relevant to several groups of organic materials.4, 5 The SDW approximation assumes long range order. On the other hand, since such a model is essentially non-interacting, there are no vertex corrections in the current loops and the calculation of χ(3) is greatly simpli?ed. This simpli?cation allows us to calculate various nonlinear optical processes, including those due to the stress tensor. Although in one dimensional organic materials, where SDW states have been observed, people have already studied the optical conductivity (linear response), nonlinear optical response in this approximation which admits closed form expressions in 1D and 2D has not

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been addressed yet. In this contribution we examine the nonlinear optical response of SDW insulators in one, two and three dimensions. Contrary to commonly accepted intuition that lower dimensions is equivalent to larger optical responses, we ?nd that among SDW insulators, optical response in 2D larger that both one and three dimensions. Among 1D, 2D and 3D systems with large on-site Coulomb interaction, the 1D system has the largest optical nonlinearity because of the decoupling of spin and charge degrees of freedom.9, 10 In contrast to this, among SDW-ordered systems, the largest third order optical response appears in 2D. Another purpose of this contribution is to clarify the importance of gauge-invariant treatment in a simple model. From symmetry point of view, some of optically allowed peaks (such as the case of two photon absorption) may become zero, provided there is charge conjugation symmetry.6 But, in SDW case, the mean ?eld factorization of the Hubbard model that leads to SDW Hamiltonian, breaks this symmetry. However, when dealing with the contribution arising from stress tensor terms, we ?nd that the gauge symmetry gives identically zero contribution to mid-gap peak in TPA. This result implies that the mid-gap peak in TPA is solely due to the four-current correlations. This observation is a symmetry property, and as we will explicitly show, is independent of dimension. This paper is organized as follows: In section 2 we review the SDW mean ?eld treatment of the Hubbard model. In section 3 we discuss the choice of gauge and method of calculation. In section 4 we discuss the four-current contribution to the third harmonic generation (THG) spectrum in 1D, 2D and 3D. In section 5 we consider the e?ects of stress tensor terms that come through quadratic couplings of gauge ?eld to the electron system and also through the dependence of gauge invariant current to the stress tensor. Finally in section 6, we summarize the results. 2. Model Hamiltonian The SDW Hamiltonian is obtained from the Hubbard Hamiltonian in the mean ?eld approximation, where the gap is driven by Coulomb repulsion. The Hubbard Hamiltonian is written as: H=

ks

?k c? ks cks + U

j

(nj ↑ ? n/2)(nj ↓ ? n/2),

(1)

where c? ks creates an electron with momentum k and spin s =↑, ↓. The dimension of the lattice

can be arbitrary, but in this derivation let us focus on 1D case. We are interested in half-?lled case (n = 1). Hereafter we will ?x the scale of energy by setting 2t0 = 1, where t0 is the hoping amplitude. In this paper we also use the system of units in which = c = e = a = 1, where a is the lattice constant. The hopping part of this Hamiltonian is characterized by the

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dispersion ?k = ? cos k. The SDW mean ?eld approximation amounts to requiring njs = n + smeiQj , 2 (2)

where n is the average particle density, and s = ± for ↑ and ↓, respectively, and Q = π . Hamiltonian11 H SDW = H0 ? U m eiQj (nj ↑ ? nj ↓ ),

Ignoring ?uctuations, and dropping additive constant of energy, we obtain the quadratic SDW

(3)

j

where H0 is the tight-binding band part. This can be written in a more compact form as H SDW =

k ?c ?v ? ? where χ? ks = (cks , ck +Qs ) = (cks , cks ) and s

χ? ks Hks χks ,

(4)

Hkσ = (

?k + ?k + Q ?k ? ?k + Q z )+ σ ? U mσσ x . 2 2

Here k runs over the half BZ and σ ’s are Pauli spin matrices. If the perfect nesting property ?k+Q = ??k holds, we have a much simpler Hamiltonian

SDW = ?k σ z ? s ? σ x , Hks

(5)

where the k?independent gap parameter ? = U m is determined by U . U? = with is uk = √ 2 ?i vk = √ 2 εk = We also need to note the relations: ?k , εk s? ? , ?vk uk ? u? k vk = 2uk vk = εk

2 2 2 ? u2 vk k = |uk | ? |vk | =

The unitary transformation ψks = U ? χks , such that U Hk U ? is diagonal, is given by uk

? ?vk

vk u? k

,

|uk |2 + |vk |2 = 1, ?k , εk ?k , εk

(6)

1+ 1?

(7)

? 2 + ?2 k.

(8)

giving U ?σx U = ? s ? z ?k x σ ? σ , εk εk ?k s? x U ?σz U = + σz ? σ , εk εk

(9)

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that imply: H SDW =

ks ? ψks εk σ z ψks .

(10)

In 1D the density of states for this model is given by (Appendix B) |ω | ρ(ω ) = √ ω 2 ? ?2 1

2 w1

√ where w1 = 1 + ?2 . One can also consider other quadratic models with gap and coherence factors, such as the U = 0 limit of the ionic Hubbard model12

H b = ? t0

? a? ? b ? ±1 + b ? ±1 a? + ? ?

? ω2

,

(11)

? A a? ? a? +

?

?B b ? ? b? .

(12)

where ’b’ stands for band insulator. This Hamiltonian describes simple tight-binding insulator in which the gap is due to di?erence ?A = ??B ≡ ? in site energy, not the Coulomb correlation in their coherence factors. Such a di?erence will a?ect the ?rst order responses dramatically, (in SDW ? = U m). This model will di?er from SDW insulator in the ?k ? ? replacements

but it is easy to see that third order optical response of this model is identical to SDW model. 3. Choice of gauge The coupling of electromagnetic ?eld to matter can be described in two gauges. One is a gauge in which vector potential is zero, but the scalar potential is non-zero and given by A0 = ?E.r. In this gauge, the electric ?eld of radiation couples to the dipole moment of electrons, and hence one needs the matrix elements of the position operator r to calculate the response of the matter to electromagnetic perturbation. Working in this gauge is suitable for molecules and small clusters. Because of the r operator, the calculations in this gauge are sensitive to boundary conditions. Moreover, for periodic boundary conditions, one has problem in choosing the origin of the r coordinate. Therefore this gauge is ill-de?ned in thermodynamic limit, and sensitive to boundary conditions.15 On the other hand, we have an alternative choice of working in a gauge in which scalar potential is zero, while the vector potential A is non zero. In this gauge the coupling between the external ?eld and electrons at the ?rst order is via the current operator: ?.A. In second order the gauge ?eld couples to electrons via the stress tensor operator as A.τ.A, etc. Working in this gauge is actually equivalent to Peierls substitution, and hence we call it the Peierls gauge. Without taking into account the e?ect of nonlinear couplings, and nonlinear dependence of the gauge-invariant current on the vector potential, the response function at n’th order is given by: χ

(n) (n)

ne2 δn1 ? χjj (?; ω1 , . . . , ωn ) , (?; ω1 , . . . , ωn ) = ? 2I + ?0 i?ω1 . . . ωn ?0 mω1

(13)

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where n?current correlation function is given by

χjj (?; ω1 , . . . , ωn ) = dt1 . . . dtn

(n)

1 n!

i

n

1 V

dr1 . . . drn

(14)

drdt ei?t?ik.r Tc ?(r, t)?(r1 , t1 ) . . . ?(rn , tn ) .

Here Tc is the time-ordering operator along the Keldysh path13 and ?(r, t) is particle current operator. branches. A unitary transformation to the ”physical” representation will give Gαβγ... where α, β, γ ∈ {a, r } Here a stands for ”advanced”, while r means ”retarded”. The optical experiments measure the fully retarded components which are given by nested commutators. On the other hand, the sum of nested commutators is generated by Garr... , etc., where only one of the indices is equal to a and the rest are equal to r index.8 As will be shown in appendix A, due to the commutators and appropriate θ functions, this component of Keldysh Green’s function is very special, in the sense that we do not really need to get into Keldysh machinery in order to calculate the nonlinear optical response. As is shown in appendix A, the fully retarded Garr...r component can always be calculated within the framework of equilibrium quantum ?eld theory. We ?rst calculate the time ordered expectation values and then analytically continue the result to ensure the correct behavior of the poles. If we need other components of Keldysh Green’s functions (the ?uctuation functions) that involve anti-commutators, and are related to noise spectroscopy, Keldysh formulation becomes inevitable. Therefore, in the case of optical response, we can forget the two time branches, and denote the response function at say, third order by ???? , keeping in mind that this is an ordinary time ordered expectation value and should be analytically continued according to prescription of appendix A. Another important point is that the four-current scheme for calculation of the optical response is reliable far from zero frequency. Therefore, we do not have to worry about the so called zero frequency divergence (ZFD) in our calculations.14 To remove the unphysical ZFD one has to calculate a few more correlation functions, but as far as the behavior near resonance region is concerned, the ???? is su?cient. To appreciate this point, let us look at the ?rst order response in two gauges (? is the dipole operator and is proportional to r; ? is ˙ ): current operator, and proportional to r

β β 0 β

In general a m?point Keldysh Green’s function has a tensor structure due to two time

χ?? (ω ; ω1 ) =

0

dτ eiωτ dτ eiωτ

dτ1 eiω1 τ1 ?(τ )?(τ1 ) dτ1 eiω1 τ1 ?(τ )?(τ1 )

(15) (16)

β 0

χ?? (ω ; ω1 ) =

0

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where a factor of 2πδ(ω + ωσ ), with ωσ = ω1 does also multiply the right hand side. We integrate by parts with respect to both time variables τ and τ1 to obtain:

β β

χ?? (ω ; ω1 ) =

0

dτ eiωτ

0

?(τ )?(τ1 ) d

eiω1 τ1 iω1

= ?

1 iω1

β 0

dτ eiωτ [ ?(τ )?(β ) ? ?(τ )?(0) ]

β 1 1 dτ1 eiω1 τ1 [ ?(β )?(τ1 ) ? ?(0)?(τ1 ) ] iω1 iω 0 1 1 χ?? (ω ; ω1 ) + iω1 iω

(17)

where we have used the fact that both ω1 and ω are bosonic Matsubara frequency, eiω1 β = 1. Hence we see that the relation χ?? (ω ; ω1 ) =

1 1 iω1 iω χ?? (ω ; ω1 ),

often used to relate the response

in two gauges is not quite correct. Inclusion of the boundary terms (?rst two lines of the above equation) compensates the frequency denominators appearing in ?? scheme. Similar consideration applies to higher order correlations such as ???? . If we keep nonlinear coupling between the gauge ?eld and electrons that comes through the stress tensor operator τ , and also dependence of gauge invariant current to powers of gauge ?eld A, a simple perturbation theory gives expressions of the type ??τ that again appear in fully retarded combination of nested commutators. The simplicity of SDW and our band model allows us to consider these terms as well. 4. Four-current response The particle current operator is given by ? = it0

? ? c? ?+1 c? ? c? c?+1

(18)

The momentum space representation of this operator for the band and SWD insulators is given by: ?=

ks z χ? ks γk σ χks ,

γk = 2t0 sin k

(19)

which in terms of new fermions is ?SDW =

ks ? ψks

γk ?k z γk ? x σ ? σ ψks εk εk

(20)

The coe?cient of σ z in the above expression describes intra-band transitions, while the coef?cient of σ x causes inter-band transitions. Let us de?ne the corresponding coe?cients by gεk = sin k cos k, hεk = s? sin k, (21)

The THG susceptibility corresponding to photon frequency ν (or iν in imaginary time) is given by

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A few comments are in order: The trace operator arises from closing the current loop, and enormously simpli?es the calculations. This trace is taken with respect to the indices of Pauli matrix. But since Pauli matrices along with unit matrix have a closed algebra, and that only the unit operator survives the trace, the sums of 16 terms in the case of ?rst order response, and of 256 terms in the case of third order response are considerably simpli?ed. We need one more physical considerations to further simplify the calculations. At half ?lled situation of interest to us, the intra-band terms at zero temperature do not contribute to optical absorption. In calculating general expectation value of say ABCD , where A, B, C, D can be any operators contributing to the coupling of light with matter, we are interested in dominant processes in which terms containing g (intra-band matrix element) in the rightmost and leftmost operators A and D do not contribute. Similarly in B and C operators the h term should be dropped. To calculate the time ordered product within the framework of equilibrium quantum ?eld theory, we use the Matsubara technique. The summation over fermion loop frequency iωn can be done with standard contour integration techniques. Also, if we ignore the momentum from the incident light, the momentum k running in the current loop must be the same for all fermion propagators. After contracting various fermion spinors to get the appropriate Greens’ functions we obtain

1 ???? = ν4 1

k

ν 4 ε3 k

×

(22)

The analytic continuation ν → ν + i0+ is implicitly understood.

2 2 2 4 8(4ε2 k + ν )? cos(k ) sin(k ) . (2εk ? 3ν )(2εk + 3ν )(2εk ? 2ν )(2εk + 2ν )(2εk ? ν )(2εk + ν )

It is also important to note that, as far as the behavior of response functions near the

resonance is concerned (that is away from ν = 0), four-current response functions, ???? give the same qualitative features as that of four-dipole response functions, ???? , where ? is the dipole moment operator. To appreciate this point, let us go back to the expression of the current operator for the SDW insulator. The inter-band matrix element for the current operator is ?k = s? sin k/εk , that via the equation of motion for the dipole moment operator ?, would imply that the dipole matrix elements are: ?k = s? sin k/2ε2 k. Similar procedure that lead to equation (23) gives:

???? =

k

(23)

1 × 2 ε7 k

(24)

Now we can see that up to a numerical factor of the order of unity, the ”near resonance” behavior of both expression is the same. Because after analytic continuation, the imaginary

2 2 2 4 (4ε2 k + ν )? cos(k ) sin(k ) , (2εk ? 3ν )(2εk + 3ν )(2εk ? 2ν )(2εk + 2ν )(2εk ? ν )(2εk + ν )

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?χTHG (ν )

50

(ν ) 10 × ?χTHG 1 ?χTHG ( ν ) 2 (ν ) 100 × ?χTHG 3

0

-50 0 0.2 0.4

ν

0.6

0.8

1

Fig. 1. Plot of imaginary part of χTHG (ν ) vs. ν in the SDW model in D = 1, 2, 3 dimensions. The D unit of energy is 2t0 = 1, and in all calculations we have assumed ? = 0.3. The gap is given by Eg = 2? = 0.6. We see that the response in D=2 is stronger than D=1,3. Values of D=1,3 are magni?ed by factors of 10 and 100, respectively, for eye assistance.

of the same order of magnitude as ε7 k in ???? expression. Note that four-current formula makes sense near the resonance conditions only. It gives the unphysical ν ?4 divergence in the static limit ν → 0. Fixing this problem as discussed in equation (17), requires calculation of a few more expectation values, but we are not interested in this limit here. Since we are dealing with a gapped situation in which the frequencies of interest are of the order of gap and zero frequency is avoided. The imaginary part of THG susceptibility can now be written as ?χTHG (ν ) = 27 fD 2 3ν 2 ν 1 ? 8fD (ν ) + fD 2 2 (25)

parts give delta functions peaking near ? εk . Therefore ν 4 ε3 k in the denominator of ???? is

fD (ν ) =

k

π ?2 cos2 k sin4 k δ(εk ? ν ), 24ε6 k

where D is the dimension of space. In 1D this integral can be evaluated as follows: f1 (ν ) = = dερ(ε)

2 ? ε2 )2 π ?2 (ε2 ? ?2 )(w1 δ(ε ? ν ) 12ε6 3/2

Using the Kramers-Kronig relation

π ?2 2 w1 ? ν2 24|ν |5 ?χ(ω ) =

ν 2 ? ?2 ?χ(ν ) , ν?ω

1/2

.

(26)

1 P π

dν

(27)

one can obtain a closed form expression for the real part of the above retarded susceptibility in terms of Elliptic functions of various kind. In 2D still we can obtain closed form results bye transforming to tight binding coordinates

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(appendix B) f2 (ν/2) = = ?2 6πνλ

w2 π/4

dε

? 0

dξ

kinds, respectively.17 The second term near the band edge behaves like 1/ ν/2 ? ?. The main THG peak is due to f2 (3ν/2) ? 1/ ν = 2?/3, with 2? being the excitation gap.

?2 E β 548β 2 ? 1258β ? 708 90πνλ ?2 λ β (60β 2 ? 226β + 177) . (28) + K 90πν 4 In this expression λ2 = ν 2 /4 ? ?2 . Here E and K are elliptic functions of second and ?rst ν/2 ? ? log(ν/2 ? ?)

1 ? β cos2 (2ξ )

δ(ν/2 ? ε)

which remains ?nite, while the ?rst term give a divergent contribution of the form f2 (ν/2) ? 3ν/2 ? ? which occurs at

shows ?χTHG (ν ) for D = 1, 2, 3 and the gap parameter ? = 0.3. The 3D result is magni?ed D 100 times (dotted line), and the 1D result (dashed line) is magni?ed 10 times. In the SDW this, notice that, one can always replace a k integration with a energy integration weighted by DOS, and so such an enhancement in 2D compared to 1D and 3D can be traced back to the nature of singularities of DOS (appendix B). This ?gure demonstrates that if we have a insulator in which gap is due to SDW type of order, the phase space e?ect (DOS) along with

In three dimensions f3 (ν ) and hence ?χTHG (ν ) can be calculated numerically. Figure 1

system, inverse square root divergence in 2D has no counterpart in 1D and 3D. To understand

SDW coherence factors (uk , vk ) generate larger nonlinear response in 2D compared to 1D and 3D. 5. Stress tensor terms The simplicity of SDW model, along with possibility of obtaining closed form results in one and two dimensions allows us to investigate the role of stress tensor terms in nonlinear optical processes. The ?rst place that stress tensor appears is via the coupling of external ?eld to matter at second order which is τ.A.τ . Then the gauge invariant current (particle current plus a correction from external ?eld) involves the stress tensor itself, and to ?rst order is given by:15 J m = ?m ? τ mn An , (29)

Therefore the third order response of gauge invariant current involves the response of both particle-current operator (?m above) and also the response of stress tensor operator (τ mn above). The third order response due to particle-current is ?τ ? + τ ?? . while the third order response due to τ operator is ??τ (31) (30)

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These expressions are of the type given in (A·10). The ?rst two corresponds to A, Q → ?, B → τ , and A → τ, B, Q → ?, respectively, while the third one corresponds to A, B → ?, Q → τ . To obtain the appropriate matrix elements, let us begin by writing the stress tensor in 1D

as τ = ? t0 or equivalently τ =

? z ks χks ?k σ χks , ? c? ns cn+1s + cn+1s cns , ns

(32)

that eventually becomes

? ψks

τ SDW =

ks

?2 s ? ?k x k z σ ? σ ψks . εk εk

(33)

The matrix elements we need are given in table I, where s is ±1 for ↑ and ↓ spins, ?inter ?intra τ inter SDW

Table I.

?inter τ intra ?inter s?2 sin2 k cos2 k/ε3 k

?2 sin2 k cos2 k/ε3 k

Inter band and intra band matrix elements in the SDW model.

respectively. Since the matrix elements involve both sin and cos, in 1D the transitions at zone center and zone boundary are suppressed, and hence there is no divergence. But in 2D again SDW response will be divergent. Let us look at the two photon absorption (TPA) contribution arising from τ operator of stress tensor TPA corresponds to ω1 = ω2 = ν . If we denote the denominators of the ?rst, second and third lines of the expression (A·10) by ?1 , ?2 , ?3 , respectively, after decomposing to partial fractions we have ?1,2 = ?3 = 1 1 ? , 2εk (ν + iη ? εk ) 2εk (ν + iη ? 2εk ) 1 , 2εk (ν + iη ? 2εk ) (34) (35) carefully. In ordinary case of ???? , the TPA corresponds to ω1 = ω2 = ?ω3 = ν . In the case

where a (ν + iη → ?ν ? iη ) term also is present in the time ordered correlation function. Putting in the matrix elements, the TPA susceptibility becomes ?inter

2

τ intra [?1 + ?2 ? ?3 ]

(36)

+ ?inter ?intra τ inter [(?2 + ?3 ) + (?3 + ?1 ) ? (?1 + ?2 )] , where we have not simpli?ed the second line deliberately to emphasize the role of gauge invariant current. Obviously what we measure is the gauge invariant current. The minus sign in third terms of the each line in the above equation comes from the minus sign in equation (29), which is essential for the cancellation that takes place in the second line of the above equation, leaving a contribution proportional to ∝ ?3 .

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Now let us concentrate on the ?rst line of this equation. Here the matrix element have opposite signs for up and down spins (which ultimately comes from the coherence factors of the SDW state). Therefore their contributions identically cancel each other, and we are left only with a term coming from denominator of ?3 only, which peaks at the gap 2?. In 1D as we mentioned, the presence of sin and cos suppresses the transitions at band edge and zone center. So let us calculate this term in 2D in the SDW insulator:

(?x ?x τxx ) π ?2 4

SDW

=?

π ?2 4

kx ,ky

δ (ν/2 ? εk )

sin2 kx cos2 kx ε4 k

=?

dxdy sin2 x cos2 x δ (ν/2 ? ε) 2π 2 ε4

(37)

?2 =? 8π

dλdξ sin2 x cos2 x δ (ν/2 ? ε) J ε4 √ √ (23 ? 22β )E ( β ) 2 λ(12β ? 23)K ( β ) = ?4? ? 16 , 3πν 4 3πλν 4

where in the last line again delta function picks up the value of λ =

Therefore in the case of stress tensor terms too, the 2D SDW systems o?er larger response than 1D. We would also like to emphasize that, the requirement of gauge invariance cancels the singularity at ? = Eg /2, in τ ’s contribution and leaves us with a square root singularity in 2? = Eg in 2D SDW systems. In other words, in SDW insulator the peak at Eg /2 may solely be due to four-current correlations and the contribution from stress tensor to this peak is identically zero. 6. Summary and conclusions In conclusion, we have calculated the nonlinear optical responses in the SDW insulator in which the gap is due to on-site Coulomb repulsion. The linear response of SDW model has the characteristic inverse square root divergence in 1D which due to larger amount of nesting is further enhanced by a logarithmic factor in 2D and is entirely suppressed in 3D. The THG spectrum of SDW model has no divergence in 1D, but diverges as inverse square root in 2D and becomes ?nite again in 3D. Therefore the optical responses (linear and nonlinear) of SDW insulators is maximal in 2D as a function of dimensionality. The model calculations presented in this work suggests that nesting as a possible mechanism of nonlinearity enhancement which works best in 2D, contrary to common intuition that lower spatial dimensions are better for nonlinear optical materials. This mechanism does not work in 1D. It was found that in 1D such a enhancement can arise from the spin-charge separation.9, 10 The simplicity of the quadratic model treated in this investigation allowed us to exactly

1 ? λ2 /4 as in the appendix B. The logarithmic divergence of the ?rst term gets suppressed √ by λ ? ν ? 2? factor, but the second term still shows inverse square root divergence in 2D.

ν 2 /4 ? ?2 , β =

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calculate the contributions of stress tensor in nonlinear optical response. We showed that (i) in TPA measurements in the SDW systems, the structure at the mid-gap (Eg /2) has essentially no contribution from the stress tensor term, and (ii) when these contributions are non-zero, have comparable e?ect to that of usually considered four-current terms. Stress term has even parity and in nonlinear processes can lead to dipole-forbidden transition. This hints to the importance of gauge invariant treatment of currents in nonlinear optics, which is usually neglected in the literature. 7. Acknowledgments This work was supported by Grant-in-Aid for Scienti?c Research, MEXT of Japan, CREST, and NAREGI. S.A.J. was supported by JSPS fellowship P04310. Appendix A: Analytic continuation In nonlinear response theory, we need to calculate fully retarded expectation value of nested commutators. For example in the case of nonlinear dielectric response of an electron gas to a fast moving ion, the nested commutator of density operators at di?erent times appears.16 In the general theory of nonlinear response, we might have the fully retarded combination of nested commutators of arbitrary operators. These operators are determined by nature of coupling (linear, quadratic, etc.) between the system and the external perturbation, and also the observable being studied. For example the second order coupling of the electromagnetic ?eld to matter via the stress tensor operator τ , leads to a fully retarded current response of the form [?, [?, τ ]] . The general framework to study this type of expectation values is the non-equilibrium quantum ?eld theory. However, within the standard formulation of quantum ?eld theory, with appropriate analytic continuation, one can obtain these kind of expectation values that correspond to Garr...r component in Keldysh Green’s function language. In optical measurements always this kind of expectation values appear. In noise spectroscopies the other components of Keldysh Green’s function appear that can not be treated as straightforward as Garr...r component, and use of Keldish formulation becomes necessary. For the problem in which external perturbation couples linearly to the system, through operator Aj , and quadratically through Bkl , where we are interested in variations of quantity Qi , one can write the retarded response at third order as:

φR ijkl (t; t1 , t2 ) = +θ (t ? t2 )θ (t2 ? t1 ) [Aj (t1 ), [Bkl (t2 ), Qi (t)]] + θ(t ? t1 )θ(t1 ? t2 ) [Bkl (t2 ), [Aj (t1 ), Qi (t)]] + (j → k → l → j ) + (j → k → l → j )2 . (A·1)

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On the other hand, the time-ordered product of these three operators is given by: φT ijkl (t; t1 , t2 ) = T Aj (t1 )Bkl (t2 )Qi (t) = + θ (t1 ? t2 )θ (t2 ? t) Aj (t1 )Bkl (t2 )Qi (t) + θ (t ? t2 )θ (t2 ? t1 ) Qi (t)Bkl (t2 )Aj (t1 ) + θ (t2 ? t1 )θ (t1 ? t) Bkl (t2 )Aj (t1 )Qi (t) + θ (t ? t1 )θ (t1 ? t2 ) Qi (t)Aj (t1 )Bkl (t2 ) + θ (t1 ? t)θ (t ? t2 ) Aj (t1 )Qi (t)Bkl (t2 ) + θ (t2 ? t)θ (t ? t1 ) Bkl (t2 )Qi (t)Aj (t1 ) + permutations. The signs are all positive, since operators A, B, Q are quadratic in fermion operators. Now let us expand the commutators in de?nition of retarded expectation value to obtain φR ijkl (t; t1 , t2 ) = + θ (t ? t2 )θ (t2 ? t1 ) Aj (t1 )Bkl (t2 )Qi (t) ? θ (t ? t2 )θ (t2 ? t1 ) Aj (t1 )Qi (t)Bkl (t2 ) ? θ (t ? t2 )θ (t2 ? t1 ) Bkl (t2 )Qi (t)Aj (t1 ) + θ (t ? t2 )θ (t2 ? t1 ) Qi (t)Bkl (t2 )Aj (t1 ) + θ (t ? t1 )θ (t1 ? t2 ) Bkl (t2 )Aj (t1 )Qi (t) ? θ (t ? t1 )θ (t1 ? t2 ) Bkl (t2 )Qi (t)Aj (t1 ) ? θ (t ? t1 )θ (t1 ? t2 ) Aj (t1 )Qi (t)Bkl (t2 ) + θ (t ? t1 )θ (t1 ? t2 ) Qi (t)Aj (t1 )Bkl (t2 ) . Now using the identity

θ(t ? t2 )θ(t2 ? t1 ) + θ(t ? t1 )θ(t1 ? t2 ) = θ(t ? t2 )θ(t ? t1 ), (A·3)

(A·2)

the sum of second and seventh terms simpli?es to ?θ (t ? t2 )θ (t ? t1 ) AQB , while the sum of third and sixth terms becomes ?θ (t ? t2 )θ (t ? t1 ) BQA , (A·5) (A·4)

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so that we obtain φR ijkl (t; t1 , t2 ) = + θ (t ? t2 )θ (t2 ? t1 ) Aj (t1 )Bkl (t2 )Qi (t) + θ (t ? t2 )θ (t2 ? t1 ) Qi (t)Bkl (t2 )Aj (t1 ) + θ (t ? t1 )θ (t1 ? t2 ) Bkl (t2 )Aj (t1 )Qi (t) + θ (t ? t1 )θ (t1 ? t2 ) Qi (t)Aj (t1 )Bkl (t2 ) ? θ (t ? t1 )θ (t ? t2 ) Aj (t1 )Qi (t)Bkl (t2 ) ? θ (t ? t1 )θ (t ? t2 ) Bkl (t2 )Qi (t)Aj (t1 ) . Now apart from the θ function that determines the analytical structure, the time ordered and retarded nested commutators have similar structures. Therefore one can obtain the expectation value of fully retarded nested commutator by appropriate analytic continuation of the corresponding time ordered one. So, let us obtain the spectral representations of the retarded and time ordered expectation values and compare them. Our conventions for Fourier transforms are

+∞

(A·6)

f (t) =

?∞

dω ?iωt ? ?(ω ) = e f (ω ) ? f 2π dω 2π dω2 2π

+∞ ?∞

dte+iωt f (t), (A·7)

χ(ω ; ω1 , ω2 ) =

dω1 iωt iω1 t1 iω2 t2 χ(t; t1 , t2 ). e e e 2π

Note that the operator B (t2 ) acting at time t2 couples the second power of the external perturbation to the system. Therefore, in principle it could involve two frequencies ω2 , ω3 . But since the external ?elds are supposed to act at the same time, we always have the combination ω2 + ω3 . Therefore here we have dropped the ω3 in our calculations. Using the representation

+∞

θ (t) = i

?∞

dω e?iωt , 2π ω + iη

(A·8)

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for the step function, after some algebra we get χT (ω ; ω1 , ω2 ) = 2πδ(ω + ωσ )

a,b a ab b0 A0 j Bkl Qi (ωσ + Eb0 ? iη )(ω1 ? E0a ? iη )

? ? ? ? +

a ab b0 Q0 i Bkl Aj (ωσ + E0a + iη )(ω1 ? Eb0 + iη ) 0a Aab Qb0 Bkl j i

(ωσ + Eb0 ? iη )(ω2 ? E0a ? iη )

(A·9)

a ab b0 Q0 i Aj Bkl (ωσ + E0a + iη )(ω2 ? Eb0 + iη ) a ab b0 A0 j Qi Bkl (ω2 ? Eb0 + iη )(ω1 ? E0a ? iη )

while

0a Qab Ab0 Bkl i j + , (ω2 ? E0a ? iη )(ω1 ? Eb0 + iη )

χR (ω ; ω1 , ω2 ) = 2πδ(ω + ωσ )

a,b a ab b0 A0 j Bkl Qi (ωσ + Eb0 + iη )(ω1 ? E0a + iη ) a ab b0 Q0 i Bkl Aj

? ? ?

(ωσ + E0a + iη )(ω1 ? Eb0 + iη )

0a Aab Qb0 Bkl j i

a ab b0 Q0 i Aj Bkl ? (ωσ + E0a + iη )(ω2 ? Eb0 + iη ) a ab b0 A0 j Qi Bkl + (ω2 ? Eb0 + iη )(ω1 ? E0a + iη )

(ωσ + Eb0 + iη )(ω2 ? E0a + iη )

(A·10)

+

where we have de?ned ωσ = ω2 + ω1 . We see the exact parallelism between the time ordered and fully retarded expectation values. The important di?erence is due to the nature of step functions that give rise to ωn + iη (n = 1, 2, σ ) structure in the retarded function. This is what was expected from causality imposed by appropriate θ functions in fully retarded one. However, as a consequence of having θ functions along with commutators, equal time contractions in diagrammatic perturbation theory do not contribute, and should be excluded. We can also write the above result in a more compact form if we note that the frequencies are associated with operators as (A, ω1 ), (B, ω2 ), (Q, ?ωσ ). If we ignore the iη factor, the above

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0a Qab Ab0 Bkl i j , (ω2 ? E0a + iη )(ω1 ? Eb0 + iη )

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result can be written as

χR (ω ; ω1 , ω2 ) = 2πδ (ω + ωσ )

a,b P a ab b0 A0 j Bkl Qi , (?ωσ + E0b )(ω1 ? E0a )

(A·11)

remember to use the appropriate iη factors to ensure ωn + iη structure. One can easily see that this prescription gives the correct spectral representation in case of two-current correlation with A = B = ?: χR (ω ; ω1 ) = 2πδ(ω + ω1 )

a

where P stands for all di?erent permutations of (A, ω1 ), (B, ω2 ), (Q, ?ωσ ). At the end we must

| 0|?|a |2 . ω1 ? (Ea ? E0 ) + i0+

We can derive a similar prescription for higher order correlation functions of fully retarded nature in a straightforward way: χR (ω ; ω1 , ω2 , ω3 ) = 2πδ(ω + ωσ )

a,b,c

P

A0a B ab C bc Qc0 . (?ωσ + E0c )(ω2 + ω1 ? E0b )(ω1 ? E0a )

(A·12)

Appendix B: Tight binding coordinates In this appendix, we denote the kx and ky coordinates in the reciprocal space by x, y for convenience. Since constant energy surfaces cos x +cos y appear very frequently in calculations related to 2D tight binding systems, it is useful to de?ne a natural orthogonal transformation, (x, y ) → (λ, ξ ), so that constant coordinate surfaces correspond to constant energy surfaces. The ?rst coordinate obviously must be λ = cos x + cos y. (B·1)

In order to guess an appropriate form for the second coordinate ξ , we require the constant ξ surfaces to be orthogonal to the constant λ surfaces, that is vξ ∝ ?λ = ? sin x e ?x ? sin y e ?y , where vξ is a ’velocity’ tangent to the constant ξ surface. This equation implies that dy dx = ? sin x, = ? sin y, dt dt (B·3) (B·2)

stant to be tan ξ :

the division of which gives dx dy = ? d ln (tan(x/2)) = d ln (tan(y/2)) . (B·4) sin x sin y Integrating the above equation gives tan(y/2) = const × tan(x/2), where we de?ne this contan ξ = tan(y/2) cot(x/2). Equations (B·1) and (B·5) imply that dxdy = Jdλdξ, J = 1 1 ? β cos2 (2ξ ) (B·5) , (B·6)

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with β = 1 ? λ2 /4. As a cross check for this formula, one can calculate the area of the BZ ( dxdy = 4π 2 ) in the new coordinate system. A straightforward numerical integration count the original BZ only once. reassures us that the above Jacobian is correct and the intervals 0 < ξ < 2π, ? 2 < λ < 2 Also the inverse transformation is given by cos x = cos y = λ J ?1 ? 1 + , 2 cos(2ξ ) λ J ?1 ? 1 ? , 2 cos(2ξ ) (B·7) (B·8)

where the Jacobian J is already de?ned in equation (B·6). The variable ξ can be interpreted

can actually see that tan ξ ? y/x and hence near to Γ point ξ can be identi?ed with the our coordinate system is a natural extension of polar coordinates (r, φ). Near the zone center

as an angle. In fact near the Γ point where x, y ≈ 0 and hence, tan x ? x, tan y ? y , we

polar coordinate φ. In this limit also we can write λ ≈ 2 ? (x2 + y 2 )/2 = 2 ? r 2 /2. Therefore the energy contours become circular, but near the Fermi energy λ = 0, the contours are rectangular. As an example of the application of this coordinate system, let us calculate the exact DOS for SDW systems in 2D. All we have to do is to switch to the new coordinate system so that

ρ2D (ν ) = dxdy δ (ε ? ν ) = 4π 2

2π 2

dξ

0 0

dλ

4π 2

δ (ε ? ν ) . 1 ? β cos2 (2ξ )

(B·9)

The λ integration can be performed by changing to the new variable ε = gives dλ = εdε/λ. Hence the ?nal result is ρ2D (ν ) = ν √ K π 2 ν 2 ? ?2 1? ν 2 ? ?2 4 ,

√

?2 + λ2 that

(B·10)

where K is the elliptic integral of ?rst kind,17 and the restriction 0 < λ < 2 translates to

2 = 4 + ?2 de?ning the upper band edge. This result in the limit of ? < ν < w2 , with w2

log(ν ) singularity at the Fermi surface. But for ? = 0, the nature of singularity near the lower √ band edge for 2D SDW system is log(ν ? ?)/ ν ? ?. In 1D, the logarithmic contribution is absent. Hence the lower band edge in 2D spin density wave systems o?ers more DOS than 1D nonlinearity in SDW systems. Calculation of DOS in 1D is much easier.18 We need to calculate the following: ρ(ω ) =

k

? → 0 reduces to the appropriate result for the tight binding bands.18 In this limit we have a

case. We see in the text that this simple observation has profound implications on third order

δ(εk ? ω ) =

d?ρ0 (?)δ(ε ? ω ).

(B·11)

√ The DOS corresponding to gapless situation ? = 0, is given by ρ0 (ω ) = 1/ 1 ? ω 2 . Now we √ use the relation ? = ε2 ? ?2 between energies ? and ε corresponding to ? = 0 and ? = 0

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situations, respectively. This change of variables gives: ρ(ω ) = where w1 = √ εdε |ω | δ(ε ? ω ) √ =√ 2 2 2 2 2 ε ?? ω ? ?2 w1 ? ε 1

2 w1

? ω2

,

(B·12)

Coulomb correlation.

1 + ?2 . For SDW insulator, the gap parameter ? = ?U m is determined by

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References

1) D. H. Auston, et al. App. Opt. 26 (1987) 211. 2) H. Kishida, et al. Nature, 405 (2000) 929 and references therein. 3) H. Kishida et al., Phys. Rev. Lett. 87 (2001) 177401. 4) G. Gr¨ uner, Density Waves in Solids, Perseus Publishing (2000) 5) D. Baeriswyl, D. K. Campbell, and S. Mazumdar in Conjugated Conducting Polymers, Ed. H. Kiess, Springer Verlag, Heidelberg (1992) 6) F. Guo, D. Guo, and S. Mazumdar, Phys. Rev. B 49 (1994) 10102. 7) L. V. Keldysh, Sov. Phys. JETP, 20 (1965) 1018. 8) K.-C.Chou, et al. Phys. Rep. 118 (1985) 1. 9) Y. Mizuno, K. Tsutsui, T. Tohyama and S. Maekawa, Phys. Rev. B 62 (2000) R4769. 10) T. Tohyama and S. Maekawa, Jour. Luminescence, 94-95 (2001) 659. 11) P. Fazekas, Lecture notes on Electron Correlation and Magnetism, World Scienti?c, 1999 12) M. Fabrizio, A.O. Gogolin, A.A. Nersesyan, Phys. Rev. Lett, 83 (1999) 2014. 13) W. Wu, Phys. Rev. Lett. 61 (1988) 1119. 14) M. Xu and X. Sun, J. Phys. Condens. matter 11 (1999) 9823, Phys. Lett. A 257 (1999) 215 and references thereinnnn 15) F. Gebhard, et al., Phil. Mag. B 75 (1997) 1. 16) A. Bergara, et al., Phys. Rev. B 56 (1997) 15654. 17) M. Abramowitz and A. Segun, Handbook of Mathematical Functions, Dover Publication Inc., New York (1965) 18) E. N. Economou, Green’s functions in Quantum Physics, Springer Verlag, 1979

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