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Variety of order-by-disorder phases in the asymmetric J1−J2 zigzag ladder: From the delta chain to the J1−J2 chain

Tomoki Yamaguchi, Stefan-Ludwig Drechsler, Yukinori Ohta, and Satoshi Nishimoto
Phys. Rev. B 101, 104407 – Published 9 March 2020
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  • INTRODUCTION
  • MODEL AND METHOD
  • PREVIOUS STUDIES FOR LIMITING CASES
  • RESULTS
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We study an asymmetric J1−J2 zigzag ladder consisting of two different spin-12 antiferromagnetic (AFM; J2, γJ2>0) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; J1<0) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of γ and J2/|J1|. It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki–type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger–liquid behavior in the AFM Heisenberg legs.

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    • Received 12 November 2019
    • Revised 19 February 2020
    • Accepted 20 February 2020

    DOI:https://doi.org/10.1103/PhysRevB.101.104407

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    FerrimagnetismFrustrated magnetismMagnetic orderMagnetic phase transitions
    1. Physical Systems
    1-dimensional systems
    1. Techniques
    Density matrix renormalization groupHeisenberg model
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Tomoki Yamaguchi1, Stefan-Ludwig Drechsler2, Yukinori Ohta1, and Satoshi Nishimoto2,3

    • 1Department of Physics, Chiba University, Chiba 263-8522, Japan
    • 2Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany
    • 3Department of Physics, Technical University Dresden, 01069 Dresden, Germany

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    Issue

    Vol. 101, Iss. 10 — 1 March 2020

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    Images

    • Figure 1
      Figure 1

      (a) Lattice structure of the asymmetric J1−J2 zigzag ladder. The indices A and B denote apical and basal chains, respectively. The lattice spacing a is set as a distance between neighboring sites along the chains. The AFM interaction in the apical chain is controlled by γ. (b) Lattice structure of the so-called delta chain (or sawtooth chain) which is realized in the limit of γ=0. (c) Schematic representation of the ferrimagnetic state with global spin-rotation symmetry breaking.

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    • Figure 2
      Figure 2

      (a) Classical ground-state phase diagram of the asymmetric J1−J2 zigzag ladder. The phases are characterized by propagation vector q=qmin minimizing Jq [Eq. (3)]: FM (qmin=0); commensurate (qmin=π); incommensurate (0<qmin<π). (b) Ground-state phase diagram of the asymmetric J1−J2 zigzag chain [Eq. (1)] determined by DMRG calculations. Inset: enlarged view around the PF phase.

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    • Figure 3
      Figure 3

      Finite-size scaling analysis of the total spin per site for the delta chain (γ=0), where (a) OBC and (b) PBC are applied. The dotted lines are guide for eyes.

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    • Figure 4
      Figure 4

      Spin-spin correlation function 〈Sα,izSβ,jz〉 for the delta chain (γ=0) as a function of distance |i−j| at (a) J2=0.6, (b) 1, and (c) 2. The total Sz sector is set to be Stotz=L/4. The legends denote as A-A: (α,β)=(A,A), B-B: (α,β)=(B,B), and A-B: (α,β)=(A,B). (d) Averaged values of 〈Sα,iz〉 on the apical and basal sites as a function of J2/|J1|.

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    • Figure 5
      Figure 5

      Energy difference between the lowest Stot=0 state and ferrimagnetic ground state for the delta chain, as a stabilization gap of the ferrimagnetic state. (a) Finite-size scaling and (b) the extrapolated values Δ/|J1| as a function of J2/|J1|. Inset: Semilog plot of Δ/|J1| as a function of J2/|J1|. The dotted line is a fit for the large J2/|J1| region: Δ/|J1|=0.037exp(−1.3J2/|J1|).

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    • Figure 6
      Figure 6

      Dynamical spin structure factors for (a) apical and (b) basal chains at J2/|J1|=0.6. (c), (d) The same spectra at J2/|J1|=1. Finite broadening η is introduced: η=0.03|J1| in (a) and (c), η=0.02|J1| in (b), and η=0.05|J1| in (d). The dotted lines are approximate analytical expressions of the main dispersions (see text).

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    • Figure 7
      Figure 7

      (a) System-size dependence of total spin per site Stot/L for several γ values with J2/|J1|=0.6 fixed. The dashed line indicates the value of Stot/L for the full ferrimagnetic state. (b) The L→∞ extrapolated values of Stot/L as a function of γ.

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    • Figure 8
      Figure 8

      Spin-spin correlation function 〈SizSjz〉 of the asymmetric J1−J2 zigzag ladder as a function of |i−j| with fixed Stotz/L=14 and J2/|J1|=0.6 for (a) γ=0, (b) 0.04, and (c) 0.08. (d) Averaged values of 〈Siz〉 on the apical and basal sites as a function of γ for J2/|J1|=0.6. The circles and crosses denote iDMRG and DMRG results, respectively.

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    • Figure 9
      Figure 9

      (a) Finite-size scaling of energy difference between lowest Stot=0 state and ferrimagnetic ground state for the asymmetric J1−J2 zigzag ladder with fixed J2/|J1|=0.6. The solid and dotted lines show the fitting results with Δ(L)/|J1|=Δ/|J1|+A/L and Δ(L)/|J1|=Δ/|J1|+A/Lη, respectively. (b) Extrapolated values of Δ/|J1| as a function of γ. The width of error bar means the difference of Δ/|J1| obtained by the two fitting functions.

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    • Figure 10
      Figure 10

      (a) Static spin structure factor for J2/|J1|=0.6 as a function of γ. The lattice spacing a is set as shown in Fig. 1. (b) Enlarged figure of (a) for 0≤γ≤0.3. Inset: ground-state energy as a function of γ.

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    • Figure 11
      Figure 11

      (a) Schematic pictures of possible dimerization order. The states for δ=2 and 3 are characterized as VBS. A solid (dotted) ellipse denotes a spin-singlet (spin-triplet) dimer. (b) Averaged dimerization order parameters Odimer(δ) as a function of γ at J2/|J1|=1. Inset: each contribution to Odimer(2) from the apical and basal chains.

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    • Figure 12
      Figure 12

      (a) Schematic pictures of considered splitting of the system into two subsystems in the D2-VBS and D3-VBS state. A solid ellipse denotes a spin-singlet pair. The number of singlet pair crossing with each cut is shown in the green square. (b)–(e) Entanglement spectrum for the corresponding splitting as a function of γ at J2/|J1|=1.

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    • Figure 13
      Figure 13

      String order parameter as a function of γ at J2/|J1|=1 using iDMRG (circles) and DMRG (crosses) methods. The DMRG results are extrapolated values to the thermodynamic limit.

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    • Figure 14
      Figure 14

      (a) iDMRG results for spin-spin correlation function 〈Si·Sj〉 as a function of distance |i−j| with γ=0.2 fixed. (b) Comparison of the γ dependence of FM critical points J2,c/|J1| estimated by DMRG and spin-wave theory.

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    • Figure 15
      Figure 15

      Energies per site for lowest-lying spin-singlet and ferrimagnetic states at J1=−1, J2=0.8, and γ=0.0 as a function of inverse system size. The PBC is used.

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    • Figure 16
      Figure 16

      Spin-spin correlation function as a function of distance |i−j| with J2/|J1|=0.6.

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    • Figure 17
      Figure 17

      (a) Schematic pictures of possible dimerization order. The pictures I–III are related to the dimerization order parameter with δ=1–3, respectively. A solid (dotted) ellipse denotes a spin-singlet (spin-triplet) dimer. The symbols on bond correspond to those used in the following plots (b)–(d). (b) Spin-spin correlation function for the bonds marked with the symbols in I. (c), (d) Similar plots to (a) for II and III.

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