Generalized susceptibility of quasi-one dimensional system with periodic potential: model for the organic superconductor (TMTSF) 2 ClO 4
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Generalized susceptibility of quasi-one dimensional system with periodic potential:model for the organic superconductor (TMTSF) ClO Yasumasa Hasegawa and Keita Kishigi Department of Material Science, Graduate School of Material Science, University of Hyogo, Hyogo 678-1297, Japan Faculty of Education, Kumamoto University, Kurokami 2-40-1, Kumamoto, 860-8555, Japan (Dated: November 1, 2018)The nesting vector and the magnetic susceptibility of the quasi-one-dimensional system havingimperfectly nested Fermi surface are studied analytically and numerically. The magnetic suscep-tibility has the plateau-like maximum in “ sweptback ” region in the momentum space, which issurrounded by Q = (2 k F , π ) + q i ( k F is the Fermi wave number, i = 1 , ,
4, and q , q and q aregiven in this paper). The best nesting vector, at which the susceptibility χ ( Q ) has the absolutemaximum at T = 0, is obtained near but not at the inflection point, Q = (2 k F , π ) + q . The effectof the periodic potential V on the susceptibility is studied, which is important for the successivetransitions of the field-induced spin density wave in (TMTSF) ClO . We obtain that the sweptbackregion (surrounded by q , q and q when V >
0) becomes small as V increases and it shrinksto q for V ≥ t ′ b , where t ′ b gives the degree of imperfect nesting of the Fermi surface, i.e. thesecond harmonics of the warping in the Fermi surface. The occurrence of the sign reversal of theHall coefficient in the field-induced spin density wave states is discussed to be possible only when V < t ′ b − t , where t is the amplitude of the fourth harmonics of the warping in the Fermi surface.This gives the novel limitation for the magnitude of V . PACS numbers: 75.30.Fv, 78.30.Jw, 71.10.Pm
I. INTRODUCTION
Various interesting properties, such as field-inducedspin density wave (FISDW), quantum Hall effect andsuperconductivity, have been observed in the quasi-one-dimensional organic conductors, (TMTSF) X, where Xis PF , ClO etc. The successive transitions betweendifferent FISDW phases occur as the magnetic field isincreased. The FISDW has been understood as a con-sequences of the reduction of the dimensionality due tothe magnetic field and the quantization of the nestingvector . The FISDW phases are charac-terized by the integer N , by which the wave number ofFISDW is given as Q x = 2 k F + N G , where k F is theFermi wave number, G = beB/ ~ , b is the lattice constant(we take b = 1 in this paper), e is the electron charge, B is the magnetic field and ~ = h/ π ( h is the Planck −1−0.500.51 k y / π k F −k F x k x(R) k x(L) k x(L’) Q =(2k F , π ) FIG. 1: Fermi surface for V = 0. −1−0.500.51 k y / π F −k F Q =(2k F , π )k x(R−) k x(L−) k x(L+) k x(R+) FIG. 2: Fermi surface for V = 0. constant). We take ~ = 1 hereafter in this paper. TheHall conductivity is quantized as σ xy = 2 N e /h with thequantum number N of the nesting vector . Thequantization of the x component of the nesting vector, Q x , can be seen as the sharp peaks in the susceptibilityfor the non-interacting system, χ ( Q ), at Q x = 2 k F + N G in the magnetic field.The peaks of χ ( Q ) in the magnetic field can be un-derstood to some extent by the peaks of χ ( Q ) in theabsence of the magnetic field. If the nesting of the Fermisurface is perfect, χ ( Q ) in the absence of the magneticfield diverges at the nesting vector as temperature be-comes zero. In that case the successive transitions ofFISDW does not happen. If the nesting of the Fermisurface is not perfect, the best nesting vector at B = 0,which gives the maximum of χ ( Q ), is located in thereciprocal space at Q = Q + q , (1)where Q = (2 k F , π ) , (2)and q = . If q x >
0, the quantum number N of FISDWis positive. If q x < N is possible in some region of themagnetic field .Although (TMTSF) PF is well understood by thequasi-one-dimensional model, (TMTSF) ClO is a littlemore complicated. Below T AO ≈
24 K the anion ClO ,which has no inversion symmetry, orders alternativelyin y direction, resulting the periodic potential V in theelectron system. Actually, the magnetic field and tem-perature phase diagram in (TMTSF) PF isdifferent from that in (TMTSF) ClO . The originof the different phase diagrams in (TMTSF) PF and(TMTSF) ClO is caused by the periodic potential, V .The magnitude of V is first estimated to be the order of T AO = 24 K , i.e. V ≪ t b . The suppression of the N = 0 FISDW state and even- N FISDW states hasbeen shown by the perturbation in V . On the other hand,the magnitude of V has been estimated to be V = 0 . t b from the angle dependence of the magnetoresistance byYoshino et al. . By treating V not in perturbation, a lotof interesting features, such as existence of several nestingvectors and the phase diagram of the FISDWstates , has been obtained. Recently, Yoshino et al. has estimated the value to be V = 0 . t a ( V = 0 . t b with their estimation t a = 12 t b ). Lebed et al. haveestimated the value as V = 0 . t b . The novel estimationof V is given in this paper from the existence of the signreversal of the Hall effect.In this paper we study the nesting vector and the sus-ceptibility in the quasi-one dimensional system with im-perfectly nested Fermi surface in the absence of the mag-netic field. The analytic expression of the susceptibilityand the nearly flat region in the reciprocal space are givenanalytically for the first time in the simple model with V = 0. The effect of V on the nesting vector and thesusceptibility are studied in detail numerically. II. MODEL
We neglect the small dispersion in k z direction andstudy the tight binding model in the square lattice withanisotropic transfer integral elements t a ≫ t b . We takethe lattice constant to be 1. In the real system thecrystal is triclinic and we have to consider the multiple-transverse-transfer integrals but most of the essentialfeatures are obtained by studying the simple model inthe square lattice . The energy dispersion can be lin-earized with respect to k x and we take account of the higher harmonic terms for k y as ǫ ( k ) = v F ( | k x | − k F ) + t ⊥ ( k y ) , (3)where t ⊥ ( k y ) = − t b cos k y − t ′ b cos(2 k y ) − t cos(3 k y ) − t cos(4 k y ) . (4)and we study the case t b , t ′ b , t and t to be positive.The terms proportional to t and t are thought to beessential to understand the negative N phase ofFISDW in some region of the magnetic field. The Fermisurface consists of two “Fermi lines” near k x ≈ ± k F , asshown in Fig. 1. The Fermi surface is almost nested, i.e.when we translate the left part of the Fermi line withthe vector Q ≈ Q , it overlaps with the right part of theFermi line, but the overlap is not perfect due to the t ′ b and t terms.The Brillouin zone is divided into halves in the k y di-rection by the periodic potential. The Hamiltonian iswritten as a 2 × V as H = (cid:18) ǫ ( k ) VV ǫ ( k + Q A ) (cid:19) , (5)where Q A = (0 , π ). The energy E ( k ) is given by E ( k ) = 12 (cid:18) ǫ ( k ) + ǫ ( k + Q A ) ± q ( ǫ ( k ) − ǫ ( k + Q A )) + 4 V (cid:19) , (6)and the Fermi surface consists of four lines as shown inFig. 2.It is known that the susceptibility χ ( Q ) has max-imum near Q ≈ Q if V . . t b when t ′ b = 0 . t b (i.e. V . t ′ b ), while the absolute maximum of χ ( Q ) islocated near Q ≈ (2 k F ± V /v F , π/
2) if V & . t b .The peak of χ ( Q ) near Q ≈ Q is caused by thenesting between the outer Fermi surface and the in-ner Fermi surface ( k ( R +) x and k ( L − ) x ), i.e., the red andblue arrows in Fig. 2, while the peaks of χ ( Q ) near Q ≈ (2 k F ± V /v F , π/
2) are caused by the nesting theouter Fermi surfaces ( k ( R +) x and k ( L +) x ) or the inner Fermisurfaces ( k ( R − ) x and k ( L − ) x ) . The maximum valueof χ ( Q ) near Q ≈ Q depends weakly on V if V . . t b ,and it decreases as V increases if V & . t b . Senguptaand Dupuis and Zanchi and Bjelis obtained the sim-ilar results.In this paper we examine in detail the nesting prop-erties of the quasi-one dimensional systems without andwith the periodic potential ( V . . t b ). Thus we focuson the nesting condition for only Q ≈ Q . III. NESTING OF THE FERMI SURFACE FOR V = 0 In this section we study the nesting properties of thequasi-one dimensional system described by Eq. (3). The −1 0 1 v F q x /t b −1−0.500.51 K y / π V/t b =0q y / π =0q q FIG. 3: q x vs K y (Eq. 11) for q y = 0 and q y = π . −2 −1 0 1−1−0.500.51 K y / π −2 −1 0 1 v F q x /t b q y / π =0 q y / π =0.03 q y / π =0.12 q y / π =0.21 −2 −1 0 1 −2 −1 0 1 V/t b =0 FIG. 4: q x vs K y (Eq. (11)) for some values of q y . There aretwo minimums ( q min ( ± ) x ( q y )) and one maximum ( q maxx ( q y )) of q x as a function of K y for each 0 < | q y | < q y , while only oneminimum and one maximum of q x for | q y | > q y as shown bydotted vertical lines. Fermi surface consists of two curves (see Fig. 1). Theright and left part of the Fermi surface are given as afunction of k y , k ( R ) x ( k y ) = k F − v F t ⊥ ( k y ) , (7) k ( L ) x ( k y ) = − k F + 1 v F t ⊥ ( k y ) . (8)We translate the left part of the Fermi surface with thenesting vector, Q = Q + q . The translated curve isgiven by k ( L ′ ) x ( k y ) = k F + q x + 1 v F t ⊥ ( k y + q y + π ) . (9) −4 −2 0 2 4 6 v F q x / t b −1.0−0.50.00.51.0 q y / π q (=q )q q q xmin(+) q xmin(−) q xmax t b ’/t b =0.1 t =t =0V=0 −4 −2 0 2 4 6 v F q x / t b −1.0−0.50.00.51.0 q y / π q (=q )q q t b ’/t b =0.1 q xmin(+) q xmin(−) q xmax t /t b =0.02t /t b =0.002V=0 FIG. 5: q min ( − ) x ( q y ) (dashed lines in q x < q x ), q min (+) x ( q y )(thick blue lines in q x < q x < q x and dashed lines in q x >q x ) and q maxx ( q y ) (thick green lines in q x < q x < q x anddashed green lines in q x > q x ). We take t ′ b /t b = 0 . t = t = 0 (upper figure) and t /t b = 0 . t /t b = 0 .
002 (lowerfigure). In the sweptback region enclosed by q , q and q , χ ( Q ) has large values. The difference of the right part of the Fermi surface andthe translated left part of the Fermi surface is given by k ( L ′ ) x ( k y ) − k ( R ) x ( k y )= q x + 1 v F ( t ⊥ ( k y ) + t ⊥ ( k y + q y + π )) . (10)If t ′ b = t = 0, the nesting of the Fermi surface isperfect with q x = q y = 0, i.e. k ( L ′ ) x ( k y ) − k ( R ) x ( k y ) = 0for all values of k y . If t ′ b = 0 or t = 0, the nesting ofthe Fermi surface is not perfect. In this case the Fermisurface intersect with the translated one with the nestingvector Q + q , if q x and q y satisfy q x = − v F [ t ⊥ ( k y ) + t ⊥ ( k y + q y + π )]= 4 v F " t b sin( K y ) sin( q y t ′ b cos(2 K y ) cos( q y )+ t sin(3 K y ) sin( 3 q y t cos(4 K y ) cos(2 q y ) , (11)for some value of K y , where K y = k y + q y . (12)Eq. (11) is the condition for the nesting vector ( Q = Q + q ) to realize the intersection of the translated leftpart of the Fermi surface with the right part of the Fermisurface at k y . In Fig. 3 we plot q x vs K y for q y = 0. Wedefine two vectors, q and q , as q y = q y = 0 and q x and q x being the minimum and the maximum of q x asa function of K y at q y = 0, respectively. When t ≤ t ′ b / q x as a function of K y for q y = 0 is given at K y = 0and ± π , and the minimum of q x as a function of K y for q y = 0 is given at K y = ± π/
2, as shown in Fig. 3; q = (cid:18) v F ( − t ′ b + t ) , (cid:19) , (13) q = (cid:18) v F ( t ′ b + t ) , (cid:19) . (14)We define q = q for V = 0 and we will define q for V = 0 in section V.We plot q x vs. K y (Eq. (11)) for some values of q y in Fig. 4. As seen in Fig. 4, q x as a function of K y has two minimums at K y = ± π/ q min ( ± ) x ( q y )) and onemaximum at 0 ≤ K y ≤ π/ q maxx ( q y )), if 0 < | q y | < q y ( q will be given later). There are one minimum at K y = − π/ K y = π/ | q y | > q y . Weobtain q min (+) x ( q y ) and q min (+) x ( q y ) as q min (+) x ( q y ) = 4 v F − t ′ b cos q y + t b sin | q y | − t sin 3 | q y | t cos 2 q y ! , (15) q min ( − ) x ( q y ) = 4 v F − t ′ b cos q y − t b sin | q y | t sin 3 | q y | t cos 2 q y ! . (16)If t and t are finite, we have to solve the fourth-degreeequation to obtain the expression of q maxx ( q y ), but it iseasy to obtain q maxx ( q y ) numerically. We define q =( q x , q y ) by the equation q min (+) x ( q y ) = q maxx ( q y ) = q x . (17)If t = t = 0, the simple expressions of q maxx ( q y ) and q are obtained as q maxx ( q y ) = 4 v F ( t ′ b cos q y + t b sin q y t ′ b cos q y ) , (18) q x = 1 v F t ′ b r (cid:16) t ′ b t b (cid:17) + 1 , (19) and q y = ± − t ′ b t b r (cid:16) t ′ b t b (cid:17) + 1 . (20)Note that q maxx ( q y ) has the physical meaning only if | q y | ≤ q y , since the analytical form Eq. (18) obtainedin the case of t = t = 0 and the numerically obtainedvalues at | q y | > q y corresponds to the local maximumof q x as a function of sin( K y /
2) at | sin( K y / | >
1. Weplot q maxx ( q y ), q min (+) x ( q y ) and q i ( i = 1 ,
3, and 4) inFig. 5. There are large overlap between the Fermi lineand the translated one, if q is in the “ sweptback ” regionwith the apexes q and q enclosed by the thick lines inFig. 5. IV. SUSCEPTIBILITY IN THE Q1D SYSTEMWITH V = 0 The susceptibility χ ( Q ) = X k f ( E k + Q ) − f ( E k ) E k − E k + Q , (21)where f ( E k ) is the Fermi distribution function, is calcu-lated at T = 0 as χ ( Q ) = Z π − π dk y π Z k ( R ) x ( k y ) k ( L ) x ( k y ) dk x π ǫ ( k − Q ) − ǫ ( k )= 1 π Z π − π dk y π (cid:20)Z k ( L ) x ( k y ) dk x v F Q x + t ⊥ ( k y − Q y ) − t ⊥ ( k y )+ Z k ( R ) x ( k y )0 dk x v F ( − k x + Q x ) + t ⊥ ( k y − Q y ) − t ⊥ ( k y ) (cid:21) = 12 πv F Z π − π dk y π (cid:20) v F k F − t ⊥ ( k y ) v F Q x + t ⊥ ( k y − Q y ) − t ⊥ ( k y ) −
12 log (cid:12)(cid:12)(cid:12)(cid:12) v F ( Q x − k F ) + t ⊥ ( k y − Q y ) + t ⊥ ( k y ) v F Q x + t ⊥ ( k y − Q y ) − t ⊥ ( k y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) (22)The susceptibility is finite at T = 0 and has the singular-ity (kinks) as a function of Q . The singularity of χ ( Q )comes from the integration of the logarithmic term ineq. (22). For Q y = π (i.e. q y = 0) and t = t = 0, thesingular part of χ ( Q + Q ) is calculated as χ ,sing = 1 πv F Z π − π dk y π (cid:18) − (cid:19) log (cid:12)(cid:12)(cid:12)(cid:12) v F q x − t ′ b cos 2 k y k F v F (cid:12)(cid:12)(cid:12)(cid:12) = − πv F log (cid:12)(cid:12)(cid:12)(cid:12) q x v F + √ ( q x v F ) − (4 t ′ b ) k F v F (cid:12)(cid:12)(cid:12)(cid:12) if | q x v F | > t ′ b − πv F log (cid:12)(cid:12)(cid:12) t ′ b k F v F (cid:12)(cid:12)(cid:12) if | q x v F | < t ′ b . (23)It is obtained from Eq. (23), that χ ( q ) has a plateau asa function of q x when t = t = 0 and q y = 0. If t , t and q y are not zero, we have to integrate Eq. (22) numer-ically. In Fig. 6 we plot χ ( Q ) for several t and t and q y as a function of q x . It can be seen that if t = t = 0,nearly flat peak at q min (+) x ( q y ) < q x < q maxx first in-creases as q y increases, and have the absolute maximumbefore q y reaches q y (= 0 . π and v F q x /t b = 0 . t ′ b /t b = 0 .
1) as shown in the top figure in Fig. 6. If t >
0, the peaks for q y = 0 are suppressed as shown inthe middle figure in Fig. 6. If t >
0, the degeneracy of χ ( Q + Q ) at q and q is lifted and the absolute max-imum of χ ( Q + q ) is obtained at q for the sufficientlylarge values of t and t , as seen in the bottom figure inFig. 6.As seen in Fig. 6, χ ( Q + q ) has plateau-like maximumin the region q min (+) x ( q y ) < q x < q maxx ( q y ). The absolutemaximum of χ ( Q + q ) occurs at q close to q but not at q = q , as seen in Figs. 7 and 8, where we plot χ ( Q + q )as a function of q x or q y on the curves of q x = q min (+) x ( q y )and q x = q maxx ( q y ), respectively. The three-dimensionalplot of χ ( Q + q ) is shown in Fig. 9. When t = t = 0and q y = q y , eq. (11) becomes q x = 1 k F t ′ b (cid:0) −
16 sin ( K y − π ) (cid:1)r (cid:16) t ′ b t b (cid:17) + 1 . (24)Therefore, q x as a function of K y has a maximum at K y = π/ q x ∝ − K y − π/ when q y = q y . Withthe vector Q = Q + q the nesting of the Fermi surfaceis better than other q ’s, which will make the expectationof the large χ ( Q + q ). However, the region of q y where χ ( Q + q ) is mainly contributed, is larger at q x / q x and | q y | / q y than at q = q . This is the reason whythe absolute maximum of χ ( Q + q ) is not located atthe inflection point ( q = q ). V. NESTING OF THE FERMI SURFACE FOR V = 0 In this section we study the effects of periodic potential V on the nesting of the Fermi surface and the suscepti-bility. When V = 0, there are two pairs of the Fermilines in k x − k y plane (see Fig. 2), which are given by k x as a function of k y , i.e., k L ± x ( k y ) and k R ± x ( k y ) for the leftand the right parts of the Fermi lines, respectively. Thenesting vectors are characterized into four types accord-ing to the pairs of the left and right parts of the Fermilines, i.e. (+ , − ), ( − , +), (+ , +), and ( − , − ) as shown inFig. 2. The left and right parts of the Fermi lines aregiven by k ( L ± ) x ( k y ) = − k F − v F (cid:18) − t ⊥ ( k y ) − t ⊥ ( k y + π ) ± q [ t ⊥ ( k y ) − t ⊥ ( k y + π )] + 4 V (cid:19) , (25) −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t =0, t =0q y / π =0,0.02,0.04,...,0.4q y / π =0 q y / π =0.2q y / π =0.4V/t b =0q y / π =0.1 q y / π =0.3 q xmin(+) (q y ) q xmax (q y ) −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t /t b =0.02, t =0q y / π =0,0.02,0.04,...,0.4q y / π =0 q y / π =0.2q y / π =0.4V/t b =0q y / π =0.1 q y / π =0.3 −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t /t b =0.02, t /t b =0.002q y / π =0,0.02,0.04,...,0.4q y / π =0 q y / π =0.2q y / π =0.4V/t b =0q y / π =0.1 q y / π =0.3 FIG. 6: χ ( Q ) at T = 0 (eq. 22) as a function of q x . We take t ′ b /t b = 0 and, t = t = 0 (the upper figure), t /t b = 0 . t = 0 (middle figure), and t /t b = 0 . t /t b = 0 .
002 (lowerfigure). As obtained by Zanchi and Montambaux , t reducesthe peak height near q and t lifts the degeneracy at q and q . and k ( R ± ) x ( k y ) = k F + 1 v F (cid:18) − t ⊥ ( k y ) − t ⊥ ( k y + π ) ± q [ t ⊥ ( k y ) − t ⊥ ( k y + π )] + 4 V (cid:19) . (26)The condition for the Fermi surface intersect by thetranslation of the left part (Eq. (11) for V = 0) is written −0.5 0 0.5 1 q y / π χ ( Q ) q q t b ’/t b =0.1, t =t =0q x =q xmin(+) (q y )q x =q xmin(+) (q y )q FIG. 7: χ ( Q ) at T = 0 (eq. 22) as a function of q x for ( q x , q y )on the curves ( q min (+) x ( q y ) , q y ) (filled green diamonds) and( q maxx ( q y ) , q y ) (open blue circles) in Fig. 5. For q x > q x weuse Eq. (18), although curves ( q maxx , q y ) terminate at q = q .Note that the absolute maximum is not realized at ( q x , q y ).We take t ′ b /t b = 0 . t = t = 0. q y / π χ ( Q ) t b ’/t b =0.1, t =t =0q x =q xmin(+) (q y )q x =q xmin(+) (q y )q , q q FIG. 8: χ ( Q ) at T = 0 (eq. 22) as a function of q y for ( q x , q y )on the curves ( q min (+) x ( q y ) , q y ) (filled green diamonds) and( q maxx ( q y ) , q y ) (open blue circles) in Fig. 5. We take t ′ b /t b =0 . t = t = 0. as the four equations (++ , + − , − +, and −− ), q ( ±± ) x = 12 v F (cid:20) − t ⊥ ( k y ) − t ⊥ ( k y + π ) − t ⊥ ( k y + q y ) − t ⊥ ( k y + q y + π ) ± q [ t ⊥ ( k y ) − t ⊥ ( k y + π )] + 4 V ± q [ t ⊥ ( k y + q y ) − t ⊥ ( k y + q y + π )] + 4 V (cid:21) . (27) FIG. 9: 3D plot of χ ( Q ) at T = 0 (eq. 22) as a function of q x and q y . We take t ′ b /t b = 0 . t = t = 0. When q y = 0, we obtain Eq. (27) for (+ , − ) and ( − , +)to be the same as that for V = 0 (Eq. (11)), q (+ − ) x = q ( − +) x = 1 v F [ − t ⊥ ( k y ) − t ⊥ ( k y + π )] . (28)The condition for the intersect of (+ , +) is obtained as q (++) x = 1 v F [ − t ⊥ ( k y ) − t ⊥ ( k y + π )]+ 1 v F q ( t ⊥ ( k y ) − t ⊥ ( k y + π )) + 4 V , (29)and the condition for the intersect of ( −− ) is obtainedas q ( −− ) x = 1 v F [ − t ⊥ ( k y ) − t ⊥ ( k y + π )] − v F q ( t ⊥ ( k y ) − t ⊥ ( k y + π )) + 4 V . (30)We define q x , q x , q x , and q x as the maximum −4 −2 0 2 4 6 v F q x /t b −1−0.500.51 K y / π V/t b =0.1 q y / π =0 q q q q q x(+−) , q x(−+) q x(++) q x(−−) FIG. 10: q x vs. K y (Eq. 27) for q y = 0 and V /t b = 0 . of q ( −− ) x (at K y = ± π/ q (+ − ) x (at −4 −2 0 2 4 6 v F q x /t b −1−0.500.51 K y / π V/t b =0.1q y / π =0.1 FIG. 11: q x as a function of K y for q y /π = 0 . −4 −2 0 2 4 6 v F q x /t b −1−0.500.51 K y / π V/t b =0.1q y / π =0.3q x(++) q x(−−) q x(+−) q x(−+) q x(−+) q x(−+) q x(−+) q x(+−) q x(+−) q x(+−) FIG. 12: q x as a function of K y for q y /π = 0 . K y = ± π/ q (++) x (at K y = ± π/
2) andthe maximum of q (+ − ) x (at K y = 0 and π ) as a function of K y when q y = 0 ( q y = q y = q y = q y = 0), respectively(see Fig. 10), i.e., q = (cid:18) v F ( − t ′ b + 4 t − V ) , (cid:19) , (31) q = (cid:18) v F ( − t ′ b + 4 t ) , (cid:19) , (32) q = (cid:18) v F ( − t ′ b + 4 t + 2 V ) , (cid:19) , (33) q = (cid:18) v F (4 t ′ b + 4 t ) , (cid:19) . (34)When q y is given, the maximums and minimums of q (+ − ) x are obtained as a function of K y as shown by the filledgreen circles and the filled squares in Figs. 10, 11, 12and 13. We define q maxx ( q y ) and q min (+) x ( q y ) by themaximums (filled green circles) and minimums (filledsquares) of q (+ − ) x for each q y , respectively. We also de-fine q min ( − ) x ( q y ) by the value of q ( −− ) x at K y = ± π/ q min (++) x ( q y ) by the value of −4 −2 0 2 4 6 v F q x /t b −1−0.500.51 K y / π V/t b =0.5 q y / π =0q q FIG. 13: q x as a function of K y for q y /π = 0 . V /t b = 0 . −2 −1 0 1 2 v F q x /t b −0.4−0.3−0.2−0.100.10.20.30.4 q y / π (V=0)q xmin(+) q xmin(++) q xmax q xmin(−) q q q V/t b =0.1 t b ’/t b =0.1 t =t =0 q q FIG. 14: The same as Fig. 5 for
V /t b = 0 . q is given asthe minimum of q (+ − ) x as a function of K y for each q y . q isgiven as q (++) x at K y = π/ q y . q is given as themaximum of q (+ − ) x as a function of K y for each q y . q is givenas q ( −− ) x at K y = π/ q y . q (++) x at K y = ± π/ q maxx ( q y ), q min (+) x ( q y ), q min ( − ) x ( q y ), and q min (++) x ( q y ) in the plane of q x and q y for V /t b = 0 .
1, 0 . . .
5. As V becomes zero, q min (++) x ( q y ), q maxx ( q y ),and q min ( − ) x ( q y ) approach to q min (+) x ( q y ), q maxx ( q y ) and q min ( − ) x ( q y ) at V = 0, respectively (cf. Fig. 5). On theother hand q min (+) x ( q y ) has no partner at V = 0, since thefilled squares in Figs. 10, 11, 12 and 13 become not theminimum but just the crossing points due to the foldingin K y as V becomes zero. We define q as the crossingpoints of q min (++) x ( q y ) and q maxx ( q y ), which is the exten-sion of that in V = 0.We plot χ ( Q + q ) as a function of q x for several valuesof q y in Fig. 18 ( V /t b = 0 .
2) and Fig. 19 (
V /t b = 0 .
4) forthe parameters t ′ b /t b = 0 . t and t . The contour plots of the χ ( Q + q ) in the k x − k y plane are shown in Fig. 20 ( t = t = 0) and Fig. 21 −2 −1 0 1 2 v F q x /t b −0.4−0.3−0.2−0.100.10.20.30.4 q y / π (V=0)q xmin(+) q xmin(++) q xmax q xmin(−) V/t b =0.3 t b ’/t b =0.1 t =t =0 FIG. 15: The same as Fig. 14 for
V /t b = 0 . −2 −1 0 1 2 v F q x /t b −0.4−0.3−0.2−0.100.10.20.30.4 q y / π (V=0)q xmin(+) q xmin(++) q xmax q xmin(−) V/t b =0.4 t b ’/t b =0.1 t =t =0 FIG. 16: The same as Fig. 14 for
V /t b = 0 . ( t /t b = 0 . t /t b = 0 . V /t b = 0, 0 . V /t ′ b = 2)and 0 . V /t ′ b = 4). When 0 < V < t ′ b , q x < q x < q x .In this case χ ( Q + q ) has a plateau-like maximum inthe “ sweptback ” region enclosed by q , q , and q , asshown in Figs. 14 and 15. This region shrinks to thepoint q when V /t ′ b = 4 as shown in Fig. 16. The absolutemaximum of χ ( Q + q ) occurs near q if t = t = 0.The effects of t and t on χ ( Q + q ) are the same asthese at V = 0; A finite t suppresses χ ( Q + q ) at q y = 0 and t lifts the degeneracy at q x ≤ q x ≤ q x .If V > t ′ b , we obtain q x < q x < q x and there areno region where χ ( Q ) has a plateau-like maximum asshown in Figs. 16, 17 and 19. In that case the effects of t and t are small. In Fig. 22, q and q which gives themaximum of χ ( Q + q ) (i.e., the best nesting vector) areshown for some values of V /t b in the case of t = t = 0.The best nesting vector moves to q as V /t b approachesto 0 . has been explained by the t and t terms . When V = 0, the terms with t /t b = 0 . t /t b = 0 .
002 make the absolute maximum of χ ( Q )in the zero magnetic field to be at q (best nesting vec- −2 −1 0 1 2 v F q x /t b −0.4−0.3−0.2−0.100.10.20.30.4 q x / π (V=0)q xmin(+) q xmin(++) q xmax q xmin(−) V/t b =0.5 t b ’/t b =0.1 t =t =0 FIG. 17: The same as Fig. 14 for
V /t b = 0 . tor), while the best nesting vector is located near q if t = t = 0, as shown in Fig. 6. The negative Hall con-stant is possible, since q x <
0. If
V /t ′ b > t and t are the same as above, the best nesting vector is q (see the lower figures in Fig. 18 and the middle figurein Fig. 21). As far as V < t ′ b − t , the negative Hallconstant is possible since q x <
0. If
V > t ′ b − t ,however, the best nesting vector q has the positive x component, as seen in the lower figures in Fig. 18 andFig. 19. Therefore, the negative Hall constant is diffi-cult to be stabilized when V > t ′ b − t . Recently, theauthors have numerically obtained the phase diagramfor the quantum Hall effect as a function of the magneticfield and periodic potential V . We have shown that thenegative Hall constant ( N = −
2) appears only in theregion 0 . . V /t b . . . . V /t ′ b .
2) for the pa-rameters t ′ b /t b = 0 . t /t b = 0 .
02 and t /t b = 0 .
002 (theupper figure of Fig. 12 in Ref. [32]). That result canbe understood by the fact that for
V > t ′ b − t thebest nesting vector has the positive x component. Theexistence of the negative Hall constant for V /t ′ b & . V that will make χ ( Q + q )at q ≈ q to be smaller. Experimentally, a negative Halleffect is observed when the system is cooled slowly (lessthan 0 . . It is expected that the magnitude of the peri-odic potential V becomes larger at the slower colling rate.Therefore, we can conclude from the existence of the neg-ative Hall effect in (TMTSF) ClO that V < t ′ b − t .The value of V estimated from the magnetic-field-angledependence of the conductivity is close to the bor-der of this condition. −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t =t =0q y / π =0,0.02,0.04,...,0.4q y / π =0 q y / π =0.4V/t b =0.2q y / π =0.1 q y / π =0.2q y / π =0.3 −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t /t b =0.02, t =0q y / π =0,0.02,0.04,...,0.4 q y / π =0.2q y / π =0.4V/t b =0.2 q y / π =0.1q y / π =0.3q y / π =0 −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t /t b =0.02, t /t b =0.002q y / π =0,0.02,0.04,...,0.4q y / π =0 q y / π =0.2q y / π =0.4V/t b =0.2q y / π =0.1 q y / π =0.3 FIG. 18: χ ( Q ) at T = 0 as a function of q x . The parametersare the same as in Fig. 6 but V /t b = 0 . −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t =t =0q y / π =0,0.02,0.04,...,0.4 q y / π =0.2q y / π =0.4V/t b =0.4 q y / π =0.1q y / π =0.3q y / π =0 −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t /t b =0.02, t =0q y / π =0,0.02,0.04,...,0.4 q y / π =0.2q y / π =0.4V/t b =0.4 q y / π =0.1q y / π =0.3q y / π =0 −2 −1 0 1 2 v F q x / t b χ ( Q ) t b ’/t b =0.1, t /t b =0.02, t /t b =0.002q y / π =0,0.02,0.04,...,0.4 q y / π =0.2q y / π =0.4V/t b =0.4 q y / π =0.1q y / π =0.3q y / π =0 FIG. 19: The same as Fig. 18 with
V /t b = 0 . -0.4-0.200.20.4-2-1012 (a) V/t b = 0 00.2-0.20.4-0.4 q y / π (b) V/t b = 0.200.20.4 / π t = t = 0 -0.4-0.20-2-1012 -0.4-0.200.20.4-2-1012 q y / (c) V/t b = 0.4 00.2-0.2-0.4 q y / π v F q x /t b -1 2-2 10 FIG. 20: The contour plot of χ ( Q + q ). The filled circlesshow the location of the maximum (best nesting vector). Thediamonds, the open circles, the triangles, and the squaresare q , q , q , and q , respectively. We take t ′ b /t b = 0 . t = t = 0. -0.4-0.200.20.4-2-1012 (a) V/t b = 0 00.2-0.20.4-0.4 q y / π (b) V/t b = 0.200.20.4 π t /t b = 0.02, t /t b = 0.002 -0.4-0.20-2-1012 -0.4-0.200.20.4-2-1012 q y / π (c) V/t b = 0.4 00.2-0.2-0.4 q y / π v F q x /t b -1 2-2 10 FIG. 21: The same as Fig. 20 with t ′ b /t b = 0 . t /t b = 0 . t /t b = 0 . -1.0 -0.5 0.0 0.5 1.0-0.20.00.2 V F t =t =0V/t b =0.4 q y / π q x /t b q for χ (cid:1) max q q q q for V=0 FIG. 22: Open diamond, open triangle and open squaresare q , q and q for V = 0, respectively. Open circles andclosed circles are q and the locations of the maximum of χ ( Q ) (best nesting vector), respectively, for V /t b = 0, 0 . .
2, 0 . .
4. We take t ′ b /t b = 0 . t = t = 0. VI. SUMMARY AND DISCUSSIONS
We have studied the nesting vector and χ ( Q ) in thequasi-one dimensional systems having the imperfectlynested Fermi surface (the imperfectness is measured by t ′ b ). We have obtained the plateau-like maximum of χ ( Q ) when Q is in the sweptback region with the apexes q and q . The absolute maximum of χ ( Q ) is obtainednear but not at Q = Q + q if t = t = 0. Whenthe periodic potential V is finite but not as large as 4 t ′ b (which is thought to be the case in (TMTSF) ClO ), the“sweptback” region (with apexes q and q ) becomessmaller as V increases and shrinks to q when V = 4 t ′ b .The best nesting vector moves to Q ≈ Q + q . Theabsolute maximum of χ ( Q ) is located at Q = Q + q when V > t ′ b . The negative Hall coefficient observedin the field-induced spin density wave states in some re-gion of the magnetic field is shown to be possible onlywhen V < t ′ b − t , in which case the vectors q ’s givingthe plateau-like maximum of χ ( Q + q ) (“ sweptback ”region) can have the negative x component, ( q x < V should be smaller than2 t ′ b − t in (TMTSF) ClO , where the sign reversal ofthe Hall effect has been observed.Recently, a lot of interest is attracted by the quasi-one-dimensional conductor (Per) M (mnt) (where Per= perylene, mnt = maleonitriledithiolate and M = Auand Pt) . The charge density wave (CDW)state is realized in (Per) M (mnt) , and the successive transitions of the field-induced CDW has been observedin high magnetic field in contrast to the field-inducedSDW in (TMTSF) ClO . This material has a simi-lar band structure as (TMTSF) ClO , but the origin ofthe pairs of the quasi-one-dimensional Fermi surface in(Per) M (mnt) is different from that in (TMTSF) ClO .The origin of the four pairs of the quasi-one-dimensionalFermi surface in (Per) M (mnt) is the existence of fourperylene molecules in the unit cell in the perpendicularplane to the conduction axis , while the origin of thetwo pairs of the quasi-one-dimensional Fermi surface in(TMTSF) ClO is the periodic potential caused by theanion ordering. It will be interesting to study the sim-ilarity and the difference between two materials, sincethe spin susceptibility χ ( Q ) and the charge susceptibil-ity χ c ( Q ) for the non-interacting system have the same Q dependence caused by the nesting properties of theFermi surface, except for the effects of the Zeemen split-ting of the Fermi surface, which play important role onlyfor CDW. Acknowledgments
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