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Physical review. B, Condensed matter

Quantum phase transition in the random antiferromagnetic spin-1 chain

A. Saguia, B. Boechat, M. A. Contínentino

Published Sep 1, 2000• 7 citations
Publisher

Abstract

We study the random antiferromagnetic spin-1 chain following the evolution of the bond probability distributions under a renormalization group transformation. We use a mapping of the spin-1 chain into an effective spin-1/2 chain with both ferromagnetic (odd bonds) and antiferromagnetic (even and odd bonds) interactions. We obtain a recursion relation for the coupling constants, solving exactly up to a four-spin cluster. Our improved perturbation treatment on these larger clusters shows that the random singlet phase in the spin-1 chain, differently from previous results, is obtained only when $100%$ of the odd bonds are strong ferromagnetic, i.e., larger than the even antiferromagnetic bonds. Otherwise the ground state is that of a dimerized spin-1/2 chain.

DOI
10.1103/physrevb.62.5541

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We study the random antiferromagnetic spin-1 chain following the evolution of the bond probability distributions under a renormalization group transformation. We use a mapping of the spin-1 chain into an effective spin-1/2 chain with both ferromagnetic (odd bonds) and antiferromagnetic (even and odd bonds) interactions. We obtain a recursion relation for the coupling constants, solving exactly up to a four-spin cluster. Our improved perturbation treatment on these larger clusters shows that the random singlet phase in the spin-1 chain, differently from previous results, is obtained only when 100100%100 of the odd bonds are strong ferromagnetic, i.e., larger than the even antiferromagnetic bonds. Otherwise the ground state is that of a dimerized spin-1/2 chain.
We study the random antiferromagnetic spin-1 chain by mapping to a random dimerized spin-1/2 chain with antiferromagnetic and ferromagnetic bonds and applying an asymptotically exact real space renormalization procedure. We find that the chain undergoes an impurity driven second order phase transition from the Haldane phase to the random singlet phase as the bond distribution broadens. In the Haldane phase and near the critical point, there is a Griffiths region in which the excitation gap is filled and the magnetic susceptibility diverges in a nonuniversal manner. The correlation length critical exponent is $\ensuremath{\nu}\ensuremath{\approx}2.3$.
The low energy properties of the spin-1/2 random Heisenberg chain with ferromagnetic and antiferromagnetic interactions are studied by means of the density matrix renormalization group (DMRG) and real space renormalization group (RSRG) method for finite chains. The results of the two methods are consistent with each other. The deviation of the gap distribution from that of the random singlet phase and the formation of the large-spin state is observed even for relatively small systems. For a small fraction of the ferromagnetic bond, the effect of the crossover to the random singlet phase on the low temperature susceptibility and specific heat is discussed. The crossover concentration of the ferromagnetic bond is estimated from the numerical data.
The S=1/2 ferromagnetic (F) - antiferromagnetic (AF) random alternating Heisenberg quantum spin chain is investigated by the quantum Monte Carlo method. The randomness is only present on the strong bonds taking values of ±2J randomly, and the weak bonds are uniformly antiferromagnetic with J. The model corresponds to the quasi one-dimensional compound, (CH3)2CHNH3Cu(Clx, Br1-x)3, whose ground state interpolates the Haldane state of the F-AF alternating spin chain at x=1 and the singlet-dimer state of the AF-AF alternating spin chain at x=0. The nonequilibrium relaxation function of the staggered magnetic susceptibility exhibits algebraic divergence in the intermediate region of 0.44 ≪ x ≪ 0.87, which explains the experimental observation of the magnetic phase transition. The results suggest that the antiferromagnetic order becomes critical by the randomness of the bond distributions even in the presence of the rich ferromagnetic bonds.
We formulate a real-space renormalization scheme that allows a study of the effects of bond randomness in the Heisenberg antiferromagnetic spin-1 chain. There are four types of bonds that appear during the renormalization flow. We implement the decimation procedure numerically. We give a detailed study of the probability distributions of all these bonds in the phases that occur when the strength of the disorder is varied. Approximate flow equations are obtained in the weak-disorder regime as well as in the strong-disorder case, where the physics is that of the random-singlet phase.
By using mainly numerical methods, we investigate the ground-state phase diagram (GSPD) of an S=1S=1S=1 ferromagnetic-antiferromagnetic bond-alternating chain with the XXZXXZXXZ and the on-site anisotropies. This system can be mapped onto an anisotropic spin-2 chain when the ferromagnetic interaction is much stronger than the antiferromagnetic interaction. Since there are many quantum parameters in this system, we numerically obtained the GSPD on the plane of the magnitude of the antiferromagnetic coupling versus its XXZXXZXXZ anisotropy, by use of the exact diagonalization, the level spectroscopy as well as the phenomenological renormalization group. The obtained GSPD consists of six phases. They are the $XY1,thelarge−1, the large-1,thelarge−D$ (LD), the intermediate-DDD (ID), the Haldane (H), the spin-1 singlet dimer (SD), and the Néel phases. Among them, the LD, the H, and the SD phases are the trivial phases, while the ID phase is the symmetry-protected topological phase. The former three are smoothly connected without any quantum phase transitions. It is also emphasized that the ID phase appears in a wider region compared with the case of the GSPD of the anisotropic spin-2 chain with the XXZXXZXXZ and the on-site anisotropies. We also compare the obtained GSPD with the result of the perturbation theory.
We study disordered antiferromagnetic spin-12\frac{1}{2}21​ chains with nearest- and further-neighbor interactions using the real-space renormalization-group method. We find that the system supports two different phases, depending on the ratio of the strength between nearest-neighbor and further-neighbor interactions as well the bond randomness strength. For weak further-neighbor coupling the system is in the familiar random singlet phase, while stronger further-neighbor coupling drives the system to a large spin phase similar to that found in the study of random antiferromagnetic-ferromagnetic spin chains. The appearance of the large spin phase in the absence of ferromagnetic coupling is due to the frustration introduced by further-neighboring couplings, and is unique to the disordered chains.