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arXiv:cond-mat/0509646 · Physical Review B

Disorder-induced phases in the S=1 antiferromagnetic Heisenberg chain

Péter Lajkó, Enrico Carlon, Heiko Rieger, Ferenc Iglói

Published Sep 19, 2005• 14 citations
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Abstract

We use extensive density matrix renormalization group (DMRG) calculations to explore the phase diagram of the random $S=1$ antiferromagnetic Heisenberg chain with a power-law distribution of the exchange couplings. We use open chains and monitor the lowest gaps, the end-to-end correlation function and the string order parameter. For this distribution at weak disorder, the system is in the gapless Haldane phase with a disorder dependent dynamical exponent, $z$, and $z=1$ signals the border between the nonsingular and singular regions of the local susceptibility. For strong enough disorder, which approximately corresponds to a uniform distribution, a transition into the random singlet phase is detected, at which the string order parameter as well as the average end-to-end correlation function are vanishing and at the same time the dynamical exponent is divergent. Singularities of physical quantities are found to be somewhat different in the random singlet phase and in the critical point.

DOI
10.1103/physrevb.72.094205

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We use extensive density matrix renormalization group (DMRG) calculations to explore the phase diagram of the random S=1S=1S=1 antiferromagnetic Heisenberg chain with a power-law distribution of the exchange couplings. We use open chains and monitor the lowest gaps, the end-to-end correlation function and the string order parameter. For this distribution at weak disorder, the system is in the gapless Haldane phase with a disorder dependent dynamical exponent, zzz, and z=1z=1z=1 signals the border between the nonsingular and singular regions of the local susceptibility. For strong enough disorder, which approximately corresponds to a uniform distribution, a transition into the random singlet phase is detected, at which the string order parameter as well as the average end-to-end correlation function are vanishing and at the same time the dynamical exponent is divergent. Singularities of physical quantities are found to be somewhat different in the random singlet phase and in the critical point.
Extensive density-matrix renormalization-group calculations for spin S=1/2S=1/2S=1/2 and S=3/2S=3/2S=3/2 disordered antiferromagnetic Heisenberg chains show a rather distinct behavior in the two cases. While at sufficiently strong disorder both systems are in a random singlet phase, we show that weak disorder is an irrelevant perturbation for the S=3/2S=3/2S=3/2 chain, contrary to what expected from a naive application of the Harris criterion. The observed irrelevance is attributed to the presence of a new correlation length due to enhanced end-to-end correlations. This phenomenon is expected to occur for all half-integer S>1/2S>1/2S>1/2 chains. A possible phase diagram of the chain for generic S is also discussed.
It is conjectured that the Haldane phase of the S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 antiferromagnetic Heisenberg chain and the S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=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=2S=2S=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=2S=2S=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 RRR and dimerization DDD 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-RRR limit at D=0D=0D=0. Furthermore, we identify the (3,1)-(4,0) phase boundary as an infinite-randomness critical line even at small RRR and find the signature of infinite randomness at the (2,2)-(3,1) phase boundary only in the vicinity of the multicritical point.
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We revisit the spin-reduction transition in the random spin-3/23/23/2 antiferromagnetic Heisenberg chain using the tensor-network strong-disorder renormalization group (tSDRG). The predicted transition separates two random-singlet (RS) phases: one formed by effective spin-1/21/21/2 moments at weaker disorder and the other by full spin-3/23/23/2 moments at stronger disorder. Within the Damle--Huse classification, this transition corresponds to an emergent permutation-symmetric P4P_4P4​ multicritical point on a P2P_2P2​ critical line associated with the RS phases. We first derive the critical exponents governing the disorder-averaged bulk and end-to-end correlation functions in the PnP_nPn​ universality classes within the Damle--Huse framework. Using these theoretical predictions, together with a diagnostic constructed from the mean and variance of the energy-gap distribution, we identify the P4P_4P4​ transition point and the reentrant behavior of the RS 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.