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arXiv:1508.03118 · Physical review. B./Physical review. B

Efficient density matrix renormalization group algorithm to study Y junctions with integer and half-integer spin

Manoranjan Kumar, Aslam Parvej, Simil Thomas, S. Ramasesha, Z. G. Soos

Published Feb 3, 2016• 8 citations
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Abstract

An efficient density matrix renormalization group (DMRG) algorithm is presented and applied to Y junctions, systems with three arms of $n$ sites that meet at a central site. The accuracy is comparable to DMRG of chains. As in chains, new sites are always bonded to the most recently added sites and the superblock Hamiltonian contains only new or once renormalized operators. Junctions of up to $N=3n+1\ensuremath{\approx}500$ sites are studied with antiferromagnetic (AF) Heisenberg exchange $J$ between nearest-neighbor spins $S$ or electron transfer $t$ between nearest neighbors in half-filled Hubbard models. Exchange or electron transfer is exclusively between sites in two sublattices with ${N}_{A}\ensuremath{\ne}{N}_{B}$. The ground state (GS) and spin densities ${\ensuremath{\rho}}_{r}=\ensuremath{\langle}{S}_{r}^{z}\ensuremath{\rangle}$ at site $r$ are quite different for junctions with $S=1/2$, 1, 3/2, and 2. The GS has finite total spin ${S}_{G}=2S(S)$ for even (odd) $N$ and for ${M}_{G}={S}_{G}$ in the ${S}_{G}$ spin manifold, ${\ensuremath{\rho}}_{r}>0(<0)$ at sites of the larger (smaller) sublattice. $S=1/2$ junctions have delocalized states and decreasing spin densities with increasing $N$. $S=1$ junctions have four localized ${S}_{z}=1/2$ states at the end of each arm and centered on the junction, consistent with localized states in $S=1$ chains with finite Haldane gap. The GS of $S=3/2$ or 2 junctions of up to 500 spins is a spin density wave with increased amplitude at the ends of arms or near the junction. Quantum fluctuations completely suppress AF order in $S=1/2$ or 1 junctions, as well as in half-filled Hubbard junctions, but reduce rather than suppress AF order in $S=3/2$ or 2 junctions.

DOI
10.1103/physrevb.93.075107

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An efficient density matrix renormalization group (DMRG) algorithm is presented and applied to Y junctions, systems with three arms of nnn sites that meet at a central site. The accuracy is comparable to DMRG of chains. As in chains, new sites are always bonded to the most recently added sites and the superblock Hamiltonian contains only new or once renormalized operators. Junctions of up to N=3n+1\ensuremath≈500N=3n+1\ensuremath{\approx}500N=3n+1\ensuremath≈500 sites are studied with antiferromagnetic (AF) Heisenberg exchange JJJ between nearest-neighbor spins SSS or electron transfer ttt between nearest neighbors in half-filled Hubbard models. Exchange or electron transfer is exclusively between sites in two sublattices with NA\ensuremath≠NB{N}_{A}\ensuremath{\ne}{N}_{B}NA​\ensuremath=NB​. The ground state (GS) and spin densities \ensuremathρr=\ensuremath⟨Srz\ensuremath⟩{\ensuremath{\rho}}_{r}=\ensuremath{\langle}{S}_{r}^{z}\ensuremath{\rangle}\ensuremathρr​=\ensuremath⟨Srz​\ensuremath⟩ at site rrr are quite different for junctions with S=1/2S=1/2S=1/2, 1, 3/2, and 2. The GS has finite total spin SG=2S(S){S}_{G}=2S(S)SG​=2S(S) for even (odd) NNN and for MG=SG{M}_{G}={S}_{G}MG​=SG​ in the SG{S}_{G}SG​ spin manifold, \ensuremathρr>0(<0){\ensuremath{\rho}}_{r}>0(<0)\ensuremathρr​>0(<0) at sites of the larger (smaller) sublattice. S=1/2S=1/2S=1/2 junctions have delocalized states and decreasing spin densities with increasing NNN. S=1S=1S=1 junctions have four localized Sz=1/2{S}_{z}=1/2Sz​=1/2 states at the end of each arm and centered on the junction, consistent with localized states in S=1S=1S=1 chains with finite Haldane gap. The GS of S=3/2S=3/2S=3/2 or 2 junctions of up to 500 spins is a spin density wave with increased amplitude at the ends of arms or near the junction. Quantum fluctuations completely suppress AF order in S=1/2S=1/2S=1/2 or 1 junctions, as well as in half-filled Hubbard junctions, but reduce rather than suppress AF order in S=3/2S=3/2S=3/2 or 2 junctions.
A density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity Z=3Z=3Z=3 and antiferromagnetic exchange between nearest-neighbor spins s=1/2s=1/2s=1/2 or 1 sites in successive generations ggg. The algorithm is accurate for s=1s=1s=1 sites. The ground states are magnetic with spin S(g)=2gsS(g)={2}^{g}sS(g)=2gs, staggered magnetization that persists for large g>20g>20g>20, and short-range spin correlation functions that decrease exponentially. A finite energy gap to S>S(g)S>S(g)S>S(g) leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for sss === 1/2 and 1 are interpreted in terms of an analytical three-site model.
The half-filled Hubbard model on the Bethe lattice with coordination number z=3z=3z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy EEE, average local magnetization $ $, its fluctuations $ - ^2$ and various spin correlation functions $ - $ are determined as a function of the Coulomb interaction strength U/tU/tU/t. The calculated local magnetic moments $ $ increase monotonically with increasing Coulomb repulsion U/tU/tU/t forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large U/tU/tU/t, $ $ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins (∣∣≃0.35| |\simeq 0.35∣∣≃0.35 for U/t→+∞U/t\to +\inftyU/t→+∞). $ - ^2$ shows a maximum for U/t=2.4U/t=2.4U/t=2.4--2.9 which results from the interplay between the usual increase of $ $ with increasing U/tU/tU/t and the formation of important permanent moments $ $ at large U/tU/tU/t. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of U/tU/tU/t. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
The half-filled Hubbard model on the Bethe lattice with coordination number z=3z=3z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy EEE, average local magnetization $$, its fluctuations $ - < \hat S_z>^2$ and various spin correlation functions $ - < S_z(i)> < S_z(j)>$ are determined as a function of the Coulomb interaction strength $U/t$. The calculated local magnetic moments $$ increase monotonically with increasing Coulomb repulsion U/tU/tU/t forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large U/tU/tU/t, $$ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins ($|< S_z(i)>|\simeq 0.35$ for $U/t\to +\infty$). $ - < S_z(i)>^2$ shows a maximum for $U/t=2.4$--2.9 which results from the interplay between the usual increase of $$ with increasing U/tU/tU/t and the formation of important permanent moments $$ at large U/tU/tU/t. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of U/tU/tU/t. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.
Linear Heisenberg antiferromagnets (HAFs) are chains of spin-SSS sites with isotropic exchange JJJ between neighbors. Open and periodic boundary conditions return the same ground-state energy per site in the thermodynamic limit, but not the same spin SG{S}_{G}SG​ when $S\ensuremath{\ge}1$. The ground state of open chains of NNN spins has SG=0{S}_{G}=0SG​=0 or SSS, respectively, for even or odd NNN. Density-matrix renormalization-group calculations with different algorithms for even and odd NNN are presented up to N=500N=500N=500 for the energy and spin densities $\ensuremath{\rho}(r,N)$ of edge states in HAFs with S=1S=1S=1, 3/2, and 2. The edge states are boundary-induced spin density waves (BI-SDWs) with $\ensuremath{\rho}(r,N)\ensuremath{\propto}{(\ensuremath{-}1)}^{r\ensuremath{-}1}$ for r=1,2,...,Nr=1,2,...,Nr=1,2,...,N. The SDWs are in phase when NNN is odd, are out of phase when NNN is even, and have finite excitation energy $\mathrm{\ensuremath{\Gamma}}(N)$ that decreases exponentially with NNN for integer SSS and faster than 1/N1/N1/N for half integer SSS. The spin densities and excitation energy are quantitatively modeled for integer SSS chains longer than $5\ensuremath{\xi}$ spins by two parameters, the correlation length $\ensuremath{\xi}$ and the SDW amplitude, with $\ensuremath{\xi}=6.048$ for S=1S=1S=1 and 49.0 for S=2S=2S=2. The BI-SDWs of S=3/2S=3/2S=3/2 chains are not localized and are qualitatively different for even and odd NNN. Exchange between the ends for odd NNN is mediated by a delocalized effective spin in the middle that increases $|\mathrm{\ensuremath{\Gamma}}(N)|$ and weakens the size dependence. The nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$) has been applied to the HAFs, primarily to S=1S=1S=1 with even NNN, to discuss spin densities and exchange between localized states at the ends as $\mathrm{\ensuremath{\Gamma}}(N)\ensuremath{\propto}{(\ensuremath{-}1)}^{N}exp(\ensuremath{-}N/\ensuremath{\xi})$. S=1S=1S=1 chains with odd NNN are fully consistent with the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$; S=2S=2S=2 chains have two gaps $\mathrm{\ensuremath{\Gamma}}(N)$ with the same $\ensuremath{\xi}$ as predicted whose ratio is 3.45 rather than 3; the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is more approximate for S=3/2S=3/2S=3/2 chains with even NNN and is modified for exchange between ends for odd NNN.
We consider a generalized Hubbard model with on-site interaction U, nearest-neighbor repulsion V, and nearest-neighbor hopping for spin \ensuremath{\sigma}, which depends on the sum of particles ${\mathrm{m}}_{\mathrm{\ensuremath{\sigma}}\ensuremath{-}|\mathrm{A}\mathrm{m}}$ with opposite spin in the two sites involved. The hopping matrix elements are denoted by tAA{\mathrm{t}}_{\mathrm{AA}}tAA​,tAB{\mathrm{t}}_{\mathrm{AB}}tAB​,tBB{\mathrm{t}}_{\mathrm{BB}}tBB​ for ${\mathrm{m}}_{\mathrm{\ensuremath{\sigma}}\ensuremath{-}|\mathrm{A}\mathrm{m}}$=0,1,2, respectively. For 0tAB{\mathrm{t}}_{\mathrm{AB}}tAB​tAA{\mathrm{t}}_{\mathrm{AA}}tAA​=tBB{\mathrm{t}}_{\mathrm{BB}}tBB​, we have determined the regions of parameters for which the ground-state (GS) of the one-dimensional system is a charge-density wave (CDW), a spin-density wave (SDW), or a metal (M), using Hartree-Fock, exact diagonalization of finite chains and quantum Monte Carlo. The results agree qualitatively with the exactly solvable limit of tAB{\mathrm{t}}_{\mathrm{AB}}tAB​=0. For 0tAB{\mathrm{t}}_{\mathrm{AB}}tAB​tAA{\mathrm{t}}_{\mathrm{AA}}tAA​=tBB{\mathrm{t}}_{\mathrm{BB}}tBB​, the GS is a M for sufficiently low values of U and V. In contrast, when tAA{\mathrm{t}}_{\mathrm{AA}}tAA​+tBB{\mathrm{t}}_{\mathrm{BB}}tBB​-2tAB{2\mathrm{t}}_{\mathrm{AB}}2tAB​=0, our results suggest that the GS is either a CDW or a SDW, with the boundary between them lying near the line U=2V.