Topological Phases of the Spin-1/2 Ferromagnetic-Antiferromagnetic Alternating Heisenberg Chain with Frustrated Next-Nearest-Neighbour Interaction

Transcription

1 Topological Phases of the Spin-1/2 Ferromagnetic-Antiferromagnetic Alternating Heisenberg Chain with Frustrated Next-Nearest-Neighbour Interaction Kazuo Hida (Saitama University) Ken ichi Takano (Toyota Technological Institute) Hidenori Suzuki (Nihon University) J. Phys. Soc. Jpn. 82 (213) 6473

2 1 Introduction Spin 1/2 ferro( )-antiferro( ) alternating Heisenberg chain H = L ( S 2l 1 S 2l + S 2l S 2l+1 ) l=1 Ground State: continuously connected to Haldane state as L ( H JŜlŜl+1 Ŝ l = S 2l 1 + S 2l : building block, J = 4 l=1 ) S=1 spin Spin-1/2 edge spin KH (1992) valence bond

3 Add ferromagnetic next-nearest neightbour interaction frustration ( < ) H = L ( S 2l 1 S 2l + S 2l S 2l+1 ) + 2L S l S l+2. l=1 l=1 or : Haldane GS, : Ferromagnetic GS Material with this structure Cu II Cl(O-mi) 2 (µ-cl) 2 M. Kato et al. Eur. J. Inorg. Chem. 211, 495 (211) (Saitama Univ. Dept. Chem.) Ferromagnetic GS

4 [Ground State Phase Diagram] - What is known- S=1AFHC S=1FHC 1 2 =1 Ferro :special point (Dmitriev et al.) m=2 Haldane m=3 m=4 m=5 m=6 Non magnetic Ferromagnetic Phase (F) 2. Haldane phase (H) edge spin 1/2 = 2 : exact singlet dimers on bonds (D. V. Dmitriev et al. Eur. Phys. J. B14 (2) 91.) 3. Ferromagnetic-nonmagnetic phase boundary = Nonmagnetic exact solutions (D. V. Dmitriev et al. Phys. Rev. B56 (1997) 5985.) / = m/2 (special point) AF short range correlation / m/2 : long range spiral correlation k = 2π L

5 S=1AFHC S=1FHC 1 2 =1 [Ground State Phase Diagram] - What we find- F :special point (Dmitriev et al.) m=2 m=3 m=4 m=5 m=6 H (I 1/2 ) I 5/2 I 2 I 3/2 I 1 1. Infinite series of intermediate spin gap phases I Se edge spin S e = 1/2, 1, 3/2, 2,.. 2. Exact ground state at the special points on the ferromagneticnonmagnetic phase boundary ( / = m/2) edge spin S e = (m 1)/2 Representative points of I Se phases

6 2 Intermediate Spin Gap Phase Scaled gap L E (PBC : Numerical diagonalization) 3 L E 2 =1 = =1 F Nonmag 1 L=6 L=14 L=8 L=16 L=1 L= Weak size-dependence around 1.3 L E increases with size on both sides Phase transition between two different spin-gap phases Haldane phase Intermediate spin gap phase

7 Open chain (DMRG) 2 S z tot = 2, quasi-ground state = 1.5 = 1.4 =1 L=72 M l Quasi-degenerate GS in nonmagnetic phases Haldane phase : Edge spins 1/2 (Kennedy triplet) Intermediate phase : Quasidegeneracy remains Local magnetization in the intermediate phase 1 <S z 2l> <S z 2l 1> M l l k=1 Sz k Edge spin-1 on both ends l

8 T 1 S 2 Add extra spins on both ends to compensate the edge spins S 2L 2 S 2L T 2 T 1 S 2 S 2L 2 S 2L S 1 S 3 J S L 2L 1 ~ T 1 ~ S 1 S 3 S 2L 1 T 1 ~ T 2 3 (L+1) E 2 =1 = 1.5 L=24 L=36 L=48 L=6 L=72 L=84 L=96 Scaled gap 4 (L+2) E 2 =1 = 1.5 L=22 L=34 L=46 L=58 L=7 L=82 L= Haldane phase : Finite gap Interm.phase : quasi-degeneracy edge spins not compensated Haldane phase : quasi-degeneracy edge spins not compensated. Intermediate Phase : Finite gap in original open chains Haldane phase : edge spin S e = 1/2 Intermediate phase : edge spin S e = 1 I 1 phase

9 For larger : add L e S = 1/2 spins ferromagnetically T Le T 2 T 1 S 2 S 2L 2 S 2L =1 S JL 1 S 3 S ~ 2L 1 T 1 3 (L+L e ) E 2 ~ T 2 T ~ Le L e :3 2 1 =1 = 2 L=22 L=34 L=46 L=58 Nonmag 1 F (L+L e ) E =1 = L=7 L=82 L=94 L e :4 3 2 L=22 L=34 L=46 L=58 L=7 L=82 L=94

10 [Ground State Phase Diagram] =1 1 H (I 1/2 ) 2 F I 1 I Se phase : Edge spin S e in open chain Large large ferromagnetic clusters large edge spin I 5/2 I 2 I 3/2

11 3 Relation with exact solutions on the phase boundary / = m/2 (special point: D. V. Dmitriev et al. PRB56 (1997) 5985.) Exact solution with antiferromagnetic short range correlation Local magnetization 1 =1 :special point (Dmitriev et al.) m=2 H (I 1/2 ) 4 2 M l <Sl> z S e m=5 2 N=24 m=4 m=3 3/2 1 N=2 N=1 Stot =2S e 2 F m=3 m=4 m=5 m= I 5/2 I 2 I 3/2 I l M l l k=1 Sz k /2 /2 Special point : representative of I Se phase S e = (m 1)/2 m = 2, 3,..., Infinite series of I Se phases

12 4 Nature of Intermediate Phases [Equivalence of I Se and I Se +1 phases] S 2 S 4 S 2L 4 S 2L 2 S 2L edge spin S e S 1 S 2L 3 S 3 S 5 S 2L 1 S 4 S 2L 2 S 2L 2 S 2L 1 S 2L JL edge spin S S 1 S e +1 2 S 3 S 5 S 2L 1 Bulk part remains the same Phases with integer S e and half-odd-integer S e are different phases. I Se +1/2 phase is always present between I Se and I Se +1 phases All I Se phases can be distinguished in the present model. Successive phase transitions between TWO topologically distinct ground states with integer and half-odd-integer edge spin

13 [Valence bond picture of I Se phases] Expected valence bond structure of I 1 phase (open chain S e = 1) J ad -dependence of energy gap 2 E =1 = 1.5 = Two end spin added (S e = 1, L e = 2) 1 J ad To stabilize this VBS structure, add antiferromagnetic bonds between S 2l and S 2l+3 J Jad F J ad : Isolated dimers on J ad - bonds Gap does not close down to J ad =.

14 [Spin-spin correlation] H.5 [Valence bond structure] Superposition of valence bond states consistent with edge spins including long range singlet pairs (a) I 1 = 1.5, =1 <S 2l 7 S 2l > <S 2l S 2l+1 > <S 2l 5 S 2l > <S 2l S 2l+3 > <S 2l 3 S 2l > <S 2l S 2l+5 > <S 2l 1 S 2l > <S 2l S 2l+7 > (b) J F S 2l S 2l+1 ( -bond) is the largest correlation even in I 1 phase (absolute value decreases) S 2l S 2l+3 increases in I 1 phase (c) etc.

15 5 Summary 1. Ferromagnetic phase 2. Haldane phase 3. Infinite series of I Se phases with edge spin S e Strong ferromagnetic NNN interaction I Se phases with increasing S e. Large Large ferromagnetic cluster Large edge spin Bulk ground states are classified into TWO topologically distinct phases with integer and half-odd-integer edge spins. VBS-structure : superposition including long range singlet pairs [Ground State Phase Diagram] 1 2 =1 4. Exact solutions on the phase boundary (Dmitriev et al.) F :special point (Dmitriev et al.) m=2 m=3 m=4 m=5 m=6 H (I 1/2 ) I 5/2 I 2 I 3/2 = m 2 (special point) Representative of I (m 1)/2 phase I 1

16 5. Future Problems Characterizaton by entanglement spectrum. Spin-1 case : Spiral short range order near ferromagnetic phase (S. Sahooa et al. arxiv: