It is conjectured that the Haldane phase of the
S=1 antiferromagnetic Heisenberg chain and the
S=1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain is stable against any strength of randomness, because of imposed breakdown of translational symmetry. This conjecture is confirmed by the density matrix renormalization group calculation of the string order parameter and the energy gap distribution.
We use a tensor network renormalization group method to study random
S=2 antiferromagnetic Heisenberg chains with alternating bond strength distributions. In the absence of randomness, an imposed dimerization with bond alternation induces two quantum critical points between the
S=2 Haldane phase, a partially dimerized phase, and a fully dimerized phase, depending on the strength of dimerization. These three phases, called ($\ensuremath{\sigma},4\ensuremath{-}\ensuremath{\sigma}$) = (2,2), (3,1), and (4,0) phases, are valence-bond solid (VBS) states characterized by $\ensuremath{\sigma}$ valence bonds (effective spin-1/2 singlet pairs) forming across even links and $4\ensuremath{-}\ensuremath{\sigma}$ valence bonds on odd links. Here we study the effects of bond randomness on the ground states of the dimerized spin chain, calculating disorder-averaged twist order parameters and spin correlations. We classify the types of random VBS phases depending on the strength of bond randomness
R and dimerization
D using the twist order parameter, which has a negative/positive sign for a VBS phase with odd/even $\ensuremath{\sigma}$. Our results demonstrate the existence of a multicritical point in the intermediate disorder regime with finite dimerization, where (2,2), (3,1), and (4,0) phases meet. This multicritical point is at the junction of three phase boundaries in the $R\text{\ensuremath{-}}D$ plane: the (2,2)-(3,1) and (3,1)-(4,0) boundaries that extend to zero randomness, and the (2,2)-(4,0) phase boundary that connects another multicritical point in the undimerized limit. The undimerized multicritical point separates a gapless Haldane phase and an infinite-randomness critical line with the diverging dynamic critical exponent in the large-
R limit at
D=0. Furthermore, we identify the (3,1)-(4,0) phase boundary as an infinite-randomness critical line even at small
R and find the signature of infinite randomness at the (2,2)-(3,1) phase boundary only in the vicinity of the multicritical point.
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.
We present a new perturbative real space renormalization group (RG) to study random quantum spin chains and other one-dimensional disordered quantum systems. The method overcomes problems of the original approach which fails for quantum random chains with spins larger than S=1/2. Since it works even for weak disorder, we are able to obtain the zero temperature phase diagram of the random antiferromagnetic Heisenberg spin-1 chain as a function of disorder. We find a random singlet phase for strong disorder. As the disorder decreases, the system shows a crossover from a Griffiths to a disordered Haldane phase.
By using Lanczos exact diagonalization and quantum Monte Carlo combined with stochastic analytic continuation, we study the dynamical properties of the S = 1 antiferromagnetic Heisenberg chain with different strengths of bond disorder. In the weak disorder region, we find weakly coupled bonds which can induce additional low-energy excitation below the one-magnon mode. As the disorder increases, the average Haldane gap closes at δ Δ ∼ 0.5 with more and more low-energy excitations coming out. After the critical disorder strength δ c ∼ 1, the system reaches a random-singlet phase with prominent sharp peak at ω = 0 and broad continuum at ω > 0 of the dynamic spin structure factor. In addition, we analyze the distribution of random spin domains and numerically find three kinds of domains hosting effective spin-1/2 quanta or spin-1 sites in between. These “spins” can form the weakly coupled long-range singlets due to quantum fluctuation which contribute to the sharp peak at ω = 0.