Field-induced inhomogeneous ground states of antiferromagnetic ANNNI chains. P.N. Timonin. Southern Federal University, , Rostov-on-Don, Russia

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1 Field-iduced ihoogeeous groud states of atiferroagetic ANNNI chais. P.N. Tioi Souther Federal Uiversity, , Rostov-o-Do, Russia Fiite-size effects are studied i groud states of atiferroagetic (AF) ANNNI chais i a field. It is show that field ca iduce a variety of ihoogeeous states i fiite chais. They are coposed of two shifted AF states with the i at their juctio ad are highly degeerate with respect to the i positio. The phase diagra field exchage ratio for fiite chais is preseted.. Itroductio Fiite-size effects i the atiferroagetic (AF) groud states of short-rage Isig chais i a field have soe defiite peculiarity: the possible perturbatios of AF spi order caused by free boudaries caot be cofied to soe viciity of chai eds but spread throughout the whole chai. This is the cosequece of the AF groud states degeeracy i ifiite chais (or rigs) ad, hece, of the ifiite correlatio legth. The origi of degeeracy lies i the traslatioal ivariace of ifiite systes ad rigs, geerally there ca be several groud states shifted alog the chai. Cosider the siplest case of fiite chai with earest-eighbor (NN) AF exchage i a field H. If it has odd uber of spis N free boudaries lift the degeeracy aig the state with ed spis up (alog the field) the uique groud state at 0 < H < J, J is AF exchage. This apparetly chages spi order throughout the whole chai. Yet ore proouced effect appears i case of eve uber of spis: while at 0 < H < J there is still usual two-fold degeeracy, at J < H < J N/ degeerate groud states appear. Its costructio cosists i divisio of chai i two segets with odd ubers of spis ad edowig each seget with the spi structure of odd chai, i.e. ed spis up i both segets, see Fig. a. Thus each such state has a i coposed of two earby spis up with the usual AF order beyod the i. It ca be readily verified that it has lower eergy tha perfect AF order at J < H < J. Such i states were first foud i acroscopic odel of agetic layers with AF exchage []. I the eseble of earesteighbor AF chais the averagig over these states gives the followig liearly-odulated AF order for average Isig spis [], see Fig. b, s N N, 0,,..., N. () Here uber of spis N is eve. We ay ote that esebles of such AF chais ca be realized i dilute crystals with (quasi) d agetic structure, i solutios of agetic polyers ad i agetic liquid crystals. Itesity of agetic eutro diffractio o these objects i such bow-tie phase bears oly traces of the uderlyig AF order exhibitig a broad pea aroud []. Siilar field-iduced ihoogeeous groud states ay exist i ore coplex Isig odels o fiite chais ad chai-lie structures (stripes, tubes) with iteractios spreadig over several uit cells. I such odels it ay ot be easy to fid groud states

2 via eueratio of possible variats ad coparig their eergies. Here the regular calculatios withi trasfer-atrix (TM) foralis ca be used at low teperatures. I Sectio we cosider siple AF Isig chai with NN exchage to fid its groud state withi this stadard approach. I Sectio 3 we describe TM foralis for the Isig chai with NN ad ext-earest-eighbor (NNN) AF exchage (ANNNI odel) ad apply it i Sectios 4, 5 to reveal the appearace field-iduced ihoogeeous groud states i this odel. (b) Fig.. (a) Ki groud states of NN AF chai with N = 8 at J < H < J, dotted lies ar i positio; (b) resultig average agetizatio profile for N = 3 fro Eq. ().. Groud states of NN AF Isig chai. Let us see how TM foralis wors for a siple (NN) AF chai ad how Eq. () eerges i it. The Hailtoia of this odel defies the TM Uss, exp K ss h s s / which is used to fid average local spis as L N N J s s H s 0 0, K=J/T, h=h/t, () s Tr VU U / Tr VU, L N, L (3) z Here z is Pauli atrix, Vs, s vsvs, vs exp h s / V for free boudaries ad I - idetity atrix for a rig.

3 At low T i, H J we have 0 U x, exph K So at H < J 0 for T 0 ad eigevalues of U are Thus defies the correlatio legth /, =. Usig spectral decopositio of U [3] we fid / l / which diverges at 0 L L, / U E E U I I x z At sufficietly low T becoes uch larger tha L ( L ) ad here we have / L L L L L U E E L L L I x L x z I L L / Thus for N eve (L odd) ad free boudaries L h h h TrVU e e e L N 4e while K h h h z Tr VU U e N e So at 0 < H < J for N eve we have approxiately at T i H, J, T. N Kh N 4e justifyig the groud state expressio () for at J < H < J. At 0 < H < J free eergy of eve AF chais is (apart fro irrelevat costat) 3

4 F hk T h l L T e N e TrVU l N N It describes the fiite-size effects i the eseble of eve AF chais at low T. I particular, the etropy S hk N K e h N N l Kh N 4e experieces sharp growth fro rather low values at 0 < H < J h K S N l e K h / to greater oes at J < H < J S N l N / reflectig the appearace of N/ groud states. Also agetizatio N F M N H N 4e 0 Kh grows fro early zero to the costat value /N whe H becoes greater tha J.. AF ANNNI chai We cosider fiite Isig chai with first ad secod eighbor AF exchage havig the Hailtoia N N 3 N J s s J s s H s The groud states of this odel o the ifiite rig or chai are foud i Ref. [4] as J / J ad H > 0. There are four types of order: ordiary fuctio of exchage ratio AF with (up, dow) uit cell, AF with (up, up, dow, dow) cell, ferriagetic (up, up, dow) ad ferroagetic oe [4]. The TM of the odel coects the eighborig pairs of spis U exp K ss ss s s / h s s s s s,s ss / ad local agetizatio is l l L N Tr VU S U / Tr VU, L, l l L l, (4) 4

5 S s s s,s s,s s s s s / s,s (5) Square bracets i (4) deote the iteger part of a uber. The chai with free boudaries ad eve N has V v v, v exp K s s h s s s,s s s s (6) while for odd N V v v, v v exp s K ss h s,s s s s s (7) s Adoptig the followig covetio for the relatio betwee row (colu) ubers ad cofiguratios s, s,, 3, 4 we ca represet U at T i J, J, H as c c a 0 b 0 U a 0 ab c b ab a exp h K K, b exp h K K, c exp h K K (8) Apparetly, at low T cosidered a ax b, c, bc (9) so the approxiate equatio for eigevalues λ of U reads ab c a (0) The solutios to Eq. (0) have four differet types of behavior whe T 0 depedig o the relatios betwee a, b ad c. For the eigevalues with largest odules we have 4,, = 3 ab 3 b a h ab () 5

6 , a 0 h -, a /,, () 3 3 ax c, a b 3 h, /3 i /3 a e, 0,,. (3) a c 3 h, c. (4) Here h H / J. So i four regios of h α plae U has differet liitig fors at T 0, which we ca obtai leavig i the origial TM (8) oly those atrix eleets which give the eigevalues listed above, aely, a h 4, U a 0 ab ab 0 0 a h, U a , a a h U c h, U (5) (6) (7) (8) These differet liitig fors of U ae evidet that we have differet groud states i the correspodig regios of h α plae. Thus at h ad T 0 U trasfers ito the to ad so o, while is trasferred to the to etc. So here we have phase AF i the ifiite chai. At h 4 the sequece is or etc. ad it is the ordiary AF phase. 6

7 At 3 h spi structure has period 3 as atrix eleets of U geerate the sequece, that is the ferriagetic groud state. At h oly state survives so this is the regio of ferroagetic groud state. This phase diagra foud i Ref. [4] is show i Fig.. Fig.. Groud state phase diagra for the fiite AF ANNNI chai i a field. Solid lies are phase boudaries for ifiite chais or rigs foud i Ref. [4]. Dotted lies show the boudaries of ihoogeeous (odulated) phases of fiite chais. I the ferriagetic phase the fiite size effects are rather trivial cosistig i (partial) liftig the triple degeeracy of the groud states. Notrivial groud states i fiite chais ca appear i two AF phases of preset odel so further we cosider just these phases. 4. Groud states i AF phase. W ab U Here it is coveiet to use the oralized TM secod row ad secod colu with zeros, cf. (5), i which we oit the 0 0 b b 0 W 0 b,. (9) 3 ab At T 0 ε goes to zero ad for L we have fro the spectral decopositio 7

8 L L Z Tr VW Tr VG L Tr VG, G E E. Here Ê ad Ê are the idepotet copoets related to the eigevalues ad correspodigly. Apparetly, is iverse correlatio legth. I the lowest order i we have [3] 0 0 b G b b 0 0, G 0 0 b so for N eve ad K 0 K K K Z b L e e b b L e b L. Siilarly, we get Z Tr VG S G ltr VG S G l Tr VG S G L Here S differs fro that i Eq. (5) by the absece of secod row ad secod colu. For N eve (L odd) ad L 3, l, l, K 0 we have l l b L. (0) is zero as i ordiary doubly degeerate AF state but Thus for b ( h ) whe b 0 ( h 4, F regio i Fig. ) we have liearly odulated state. Returig to the site ubers i (0), cf. Eq. (4), we get N N 4, N 6, N 3 Fourier trasfor of this, N 3 i3 / e e i,0, cos / r, r 0,,..., N 5 N 4, depeds o N through values oly, so it has defiite therodyaic liit ad ca be cosidered as order paraeter for this ihoogeeous phase. Note that the sae has the bow-tie phase of NN AF chai () for r / N, cf. Ref. []. It defies the itesity 8

9 of eutro diffractio paraeter. I for wave vector expressed i uits of iverse cell The calculatios for l =0 ad l = L result i 0, for K 0 ad N N we have a profile show i Fig. 3 for N = 36. Evidetly, here we have (N - 4)/ degeerate groud states fored fro the ier N - 4 spis by the process siilar to that i siple NN AF chai. The couples of the boudary spis stay fixed at the above values as NNN AF exchage forbids two parallel spis at the edges. For N odd i all AF regio Fig.3. Magetizatio of liearly odulated state of ANNNI chai with N = 36 at h 4 (F regio i Fig.). 5. Groud states i AF phase. Accordig to (6) U i this phase ( h -) ca be represeted as U E ag, x E, G 0. x a for T 0, we ca evaluate the powers of As 0 decopositio. Thus at a U without resortig to spectral, D E, G U I ad 0 I x 0 () Fro this equatio it follows that chais with L eve do ot have ihoogeeous groud states as i such case we ca eglect the cotributios of order a i calculatios. However, for L odd (i. e. for N = 4M or N = 4M + ) we have 9

10 a Z L TrVE TrV L G L EGE () ad TrVE 0 for N = 4M while for N = 4M + TrVE ay go to zero as T 0, see (6, 7). Thus further we cosider L odd. I such a case l l l Z L TrV I a D ES I a D al GS al SG (3) For N = 4M we get fro these equatios l l L L l L 4e K h (4) Thus at h (F regio i Fig. ) we have l N N (5) Fig. 4 shows (5) for N = 64. Oe ca see that this is the result of averagig over N /4 M degeerate groud states which are fored fro AF state via flippig odd uber of successive spis i eve or odd sublattice. They are show i Fig. 5(b)-(e) for N = Fig.4. Magetizatio profile at h (F regio i Fig. ) for N = 64. Fourier trasfor of (5) is 0 N i/ i e cos / e, cos r r, r 0,,..., N, N M Its sigularities at /ad 3 / reflects the uderlyig early perfect period-4 order of the i states. 0

11 Accordig to (4) at h i, i doubly degeerate 0 whe T 0 which eas that chai is AF groud states. For N = 8 oe of these states is show i Fig. 5 (a), its couterpart has all spi reversed. Fig.5. Procedure of groud states foratio at h for N = 8. (a) The paret state. The groud states are obtaied via flippig i it oe spi (b), (c) or three spis (d), (e). Flipped spis are ared by the dots. For N = 4M + ad h i, we get fro (7,, 3) l l L e 4e l l K h K h l l L 4 e e K h K h Calculatig N = 4M + : for other regios of h h, F regio i Fig., plae where AF phase exists we fially get for l ll l L ; (6) M l, h, F3 regio i Fig.,,4r ; (7) L M r

12 ax,h, the rest of AF phase i Fig., l. (8) The profile (6) for h (F regio) is show i Fig. 6 for N = 65. We see that eve sublattice have AF order with up spis at the edges but the odd oe is fored via averagig over N / 4 M i states. I cotrast, at ax,h the AF order i odd sublattice stay itact up to a global reversal so spis i it have zero average value, see Eq. (8) Fig.6. Magetizatio profile at h (F regio) for N = 65. Fourier trasfor of (6) is N i / cos, r / N r / M, r 0,,..., N e 0, It also shows idepedece of its for o N ad sigularities at / ad 3 /., h (F3 regio) for N = Fig.7. Procedure of groud state foratio at 9. (a) Two paret states, below there are groud states obtaied fro the via flippig of oe spi (b) ad three spis (c). Flipped spis are ared by the dots.

13 The profile (7) at, h (F3 regio) is the result of the averagig over N / M groud states. The procedure of their foratio is show i Fig. 7 for N = 9. It aes the uber of the satisfied NN bods larger at the expese of the NNN oes. This provides the eergy gai as. 6. Discussio ad coclusios Above results are suarized o the phase diagra i Fig.. F phase exists i eve chais ad is quite siilar to that i ordiary NN AF chai. I F regio the chais with N = 4M ad N = 4M + has odulated phases. F3 phase of N = 4M + chai is forally periodic (of 0,0,,0 type, cf. Eq. (7)) but it is also the result of the averagig over ihoogeeous i states. Their uber is always of order N due to the degeeracy with respect to the i positio. I the above exaples the for of i states ca be readily guessed fro the results of TM calculatios but i ore coplex cases it ca be also calculated via addig local fields liftig their degeeracy. The echais of appearace of field-iduced ihoogeeous groud states i fiite AF chais lies i the degeeracy of their (ifiite) AF groud states i a field. I the above exaples the i states are fored via cojuctio of two shifted AF states which results i appearace of ozero agetic oet alog the field. The eergy gai fro this ay overcoe the eergy eeded for the i foratio at large eough H. Oe ay be iterested if this echais ca wor i higher diesios, o-isig systes or quatu odels havig degeerate groud states. Now there is o defiite aswer to this, yet it sees worthwhile to study the fiite-size effects i such systes. REFERENCES. C. Micheletti, R. B. Griffiths, J. M. Yeoas, Phys. Rev. B, 59, 639 (999).. P.N. Tioi, JETP, 5, 00 (0). 3. P. Laaster, Theory of Matrices, Acadeic Press: New Yor - Lodo (969), ch.. 4. P. Ruja, W. Sele, G.V. Uii, Z. Phys. B, 53, (983) 3