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arXiv:cond-mat/9507129 · Europhysics Letters (EPL)

Phase diagram of the spin-one Heisenberg system with dimerization and frustration

S Pati, R. Chitra, Diptiman Sen, H. R. Krishnamurthy, S. Ramasesha

Published Mar 20, 1996• 32 citations
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Abstract

We use the density matrix renormalization group method to study the ground-state "phase" diagram and some low-energy properties of an antiferromagnetic spin-1 chain with a next-nearest-neighbour exchange J 2 and an alternation δ of the nearest-neighbour exchanges. We find a line running from a gapless point at ( J 2 ,δ) = (0,0.25 ± 0.01) up to a "gapless" point at (0.73 ± 0.005,0) such that the open-chain ground state is fourfold degenerate below the line and is unique above it. A disorder line 2 J 2 + δ = 1 runs from δ = 1 to about δ = 0.136. To the left of this line, the peak in the structure factor S ( q ) is at π, while to the right of the line, it is at less than π.

DOI
10.1209/epl/i1996-00403-3

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•May 29, 2013•0 citations
We use the density matrix renormalization group method to study the ground-state "phase" diagram and some low-energy properties of an antiferromagnetic spin-1 chain with a next-nearest-neighbour exchange J 2 and an alternation δ of the nearest-neighbour exchanges. We find a line running from a gapless point at ( J 2 ,δ) = (0,0.25 ± 0.01) up to a "gapless" point at (0.73 ± 0.005,0) such that the open-chain ground state is fourfold degenerate below the line and is unique above it. A disorder line 2 J 2 + δ = 1 runs from δ = 1 to about δ = 0.136. To the left of this line, the peak in the structure factor S ( q ) is at π, while to the right of the line, it is at less than π.
The ground-state `phase' diagrams and some low-energy properties of isotropic antiferromagnetic spin- and spin-1 chains with a next-nearest-neighbour exchange and an alternation of the nearest-neighbour exchanges have been compared for the first time using the density matrix renormalization group method. In the spin- chain, the system is gapless for and , and is gapped everywhere else in the plane. At , for small , the gap increases as , where is a disorder line. To the left of this line, the structure factor S(q) peaks at (the Néel `phase'), while to the right, decreases from to (the spiral `phase') as increases. We also discover a ` phase' for large values of both and . In the spin-1 case, we find a line running from a gapless point at up to a `gapless' point at such that the open-chain ground state is fourfold degenerate below the line and is unique above it. There is a disorder line in this case also and it has the same equation as in the spin- case, but the line ends at about . Similarly to the spin- case, to the left of this line, the peak in the structure factor is at (the Néel `phase'), while to the right of the line, it is at less than (the spiral `phase'). For , the system corresponds to a spin ladder and the system is gapped for all values of the interchain coupling for both spin- and spin-1 ladders.
Using the density-matrix renormalization-group technique, we study the ground-state phase diagram and other low-energy properties of an isotropic antiferromagnetic spin-1/2 chain with both dimerization and frustration, i.e., an alternation \ensuremath{\delta} of the nearest-neighbor exchanges and a next-nearest-neighbor exchange J2{\mathit{J}}_{2}J2​. For \ensuremath{\delta}=0, the system is gapless for J2{\mathit{J}}_{2}J2​J2c{\mathit{J}}_{2\mathit{c}}J2c​ and has a gap for J2{\mathit{J}}_{2}J2​>J2c{\mathit{J}}_{2\mathit{c}}J2c​ where J2c{\mathit{J}}_{2\mathit{c}}J2c​ is about 0.241. For J2{\mathit{J}}_{2}J2​=J2c{\mathit{J}}_{2\mathit{c}}J2c​, the gap above the ground state grows as \ensuremath{\delta} to the power 0.667\ifmmode\pm\else\textpm\fi{}0.001. In the J2{\mathit{J}}_{2}J2​-\ensuremath{\delta} plane, there is a disorder line 2J2{\mathit{J}}_{2}J2​+\ensuremath{\delta}=1. To the left of this line, the peak in the static structure factor S(q) is at qmax{\mathit{q}}_{\mathrm{max}}qmax​=\ensuremath{\pi} (N\'eel phase), while to the right of the line, qmax{\mathit{q}}_{\mathrm{max}}qmax​ decreases from \ensuremath{\pi} to \ensuremath{\pi}/2 as J2{\mathit{J}}_{2}J2​ is increased to large values (spiral phase). For \ensuremath{\delta}=1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.
The spin-1/2 chain with isotropic Heisenberg exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed J1/J2 between gapless phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor S(q) that diverges at wave vector qm in QLRO(q(m)) phases with periodicity 2π/q(m) but remains finite in gapped phases. S(q(m)) is evaluated using ED and density matrix renormalization group (DMRG) calculations. Level crossing and the magnitude of S(q(m)) are independent and complementary probes of quantum phases, based respectively on excited and ground states. Both indicate a gapless QLRO(π/2) phase between -1.2 < J1/|J2| < 0.45. Numerical results and field theory agree well for quantum critical points at small frustration J2 but disagree in the sector of weak exchange J1 between Heisenberg antiferromagnetic chains on sublattices of odd and even-numbered sites.
The frustrated ferromagnetic spin-1∕21∕21∕2 Heisenberg chain is studied by means of a low-energy field theory as well as the density-matrix renormalization group and exact diagonalization methods. First, we study the ground-state phase diagram in a magnetic field and find an ``even-odd'' (EO) phase characterized by bound pairs of magnons in the region of two weakly coupled antiferromagnetic chains. A jump in the magnetization curves signals a first-order transition at the boundary of the EO phase, but otherwise the curves are smooth. Second, we discuss thermodynamic properties at zero field, where we confirm a double-peak structure in the specific heat for moderate frustrating next-nearest-neighbor interactions.
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We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.