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 revisit the spin-reduction transition in the random spin-
3/2 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-
1/2 moments at weaker disorder and the other by full spin-
3/2 moments at stronger disorder. Within the Damle--Huse classification, this transition corresponds to an emergent permutation-symmetric
P4 multicritical point on a
P2 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
Pn 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
P4 transition point and the reentrant behavior of the RS phase.
We show that the numerical strong disorder renormalization group algorithm of Hikihara et al. [Phys. Rev. B 60, 12116 (1999)] for the one-dimensional disordered Heisenberg model naturally describes a tree tensor network (TTN) with an irregular structure defined by the strength of the couplings. Employing the holographic interpretation of the TTN in Hilbert space, we compute expectation values, correlation functions, and the entanglement entropy using the geometrical properties of the TTN. We find that the disorder-averaged spin-spin correlation scales with the average path length through the tensor network while the entanglement entropy scales with the minimal surface connecting two regions. Furthermore, the entanglement entropy increases with both disorder and system size, resulting in an area-law violation. Our results demonstrate the usefulness of a self-assembling TTN approach to disordered systems and quantitatively validate the connection between holography and quantum many-body systems.
Using an adaptive strategy which enables the study of quenched disordered system via the density-matrix renormalization-group method, we compute the various ground-state spin-spin correlation measures of the spin-
21 antiferromagnetic Heisenberg chain with random coupling constants, namely, the mean values of the bulk and of the end-to-end correlations, the typical value of the bulk correlations, and the distribution of the bulk correlations. Our results are in agreement with the predictions of the strong-disorder renormalization-group method. We do not find any hint of logarithmic corrections either in the bulk average correlations, which were recently reported by Shu et al. [Phys. Rev. B 94, 174442 (2016)], or in the end-to-end average correlations. We report the existence of a logarithmic correction on the end-to-end correlations of the clean chain. Finally, we have determined that the distribution of the bulk correlations, when properly rescaled by an associated Lyapunov exponent, is a narrow and universal (disorder-independent) probability function.
We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study
S=1/2 spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that cannot be generated by the SDRG scheme. In the model with multispin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.
Tensor networks provide a powerful and elegant approach to quantum manybody simulation. The simplest example is the density matrix renormalisation group (DMRG), which is based on the variational update of a matrix product state (MPS). It has proved to be the most accurate approach for the numerical study of strongly correlated one dimensional systems. We use DMRG to study the one dimensional disordered Bose-Hubbard model at fillings N=L = 1=2, 1 and 2 and show that the whole phase diagram for each can be successfully obtained by analysing entanglement properties alone. We �nd that the average entanglement is insufficient to accurately locate all of the phases, however using the standard error on the mean we are able to construct a phase diagram that is consistent with previous studies. It has recently been shown that there is a connection between the geometry of tensor networks and the entanglement and correlation properties that it can encode, which is a generalisation of the so called area law for entanglement entropy. This suggests that whilst gapped quantum systems can be accurately modeled using an MPS, a tensor network with a holographic geometry is natural to capture the logarithmic entanglement scaling and power law decaying correlation functions of critical systems. We create an algorithm for the disordered Heisenberg Hamiltonian that self assembles a tensor network based on the disorder in the couplings. The geometry created is that of a disordered tree tensor network (TTN) that when averaged has the holographic properties characteristic of critical systems. We continue the analysis of holographic tensor network geometry by considering the average length of leaf-to-leaf paths in various tree graphs, which is related to two-point correlation functions in tensor networks. For regular, complete trees we analytically calculate the average path length and all statistical moments, and generalise it for any splitting number. We then turn to the Catalan trees, which is the set of unique binary trees with n vertices, as it has a similar geometry to the disordered TTNs. We calculate the average depth of a leaf and show that it is equal to the average path length. We compare these analytic results with the structures found in the TTN and randomly constructed trees to show that the renormalisation involved in the TTN algorithm is crucial in the selection of the tree structure.