Modified density matrix renormalization group algorithm for the zigzag spin- 12 chain with frustrated antiferromagnetic exchange: Comparison with field theory at large J2/J1
Manoranjan Kumar, Z. G. Soos, Diptiman Sen, S. Ramasesha
A modified density matrix renormalization group (DMRG) algorithm is applied to the zigzag spin-$\frac{1}{2}$ chain with frustrated antiferromagnetic exchange ${J}_{1}$ and ${J}_{2}$ between first and second neighbors. The modified algorithm yields accurate results up to ${J}_{2}/{J}_{1}\ensuremath{\approx}4$ for the magnetic gap $\ensuremath{\Delta}$ to the lowest triplet state, the amplitude $B$ of the bond order wave phase, the wavelength $\ensuremath{\lambda}$ of the spiral phase, and the spin correlation length $\ensuremath{\xi}$. The ${J}_{2}/{J}_{1}$ dependences of $\ensuremath{\Delta}$, $B$, $\ensuremath{\lambda}$, and $\ensuremath{\xi}$ provide multiple comparisons to field theories of the zigzag chain. The twist angle of the spiral phase and the spin structure factor yield additional comparisons between DMRG and field theory. Attention is given to the numerical accuracy required to obtain exponentially small gaps or exponentially long correlations near a quantum phase transition.
A modified density matrix renormalization group (DMRG) algorithm is applied to the zigzag spin-21 chain with frustrated antiferromagnetic exchange J1 and J2 between first and second neighbors. The modified algorithm yields accurate results up to J2/J1\ensuremath≈4 for the magnetic gap \ensuremathΔ to the lowest triplet state, the amplitude B of the bond order wave phase, the wavelength \ensuremathλ of the spiral phase, and the spin correlation length \ensuremathξ. The J2/J1 dependences of \ensuremathΔ, B, \ensuremathλ, and \ensuremathξ provide multiple comparisons to field theories of the zigzag chain. The twist angle of the spiral phase and the spin structure factor yield additional comparisons between DMRG and field theory. Attention is given to the numerical accuracy required to obtain exponentially small gaps or exponentially long correlations near a quantum phase transition.
A frustrated spin-1/2 XXZ chain model comprising a ferromagnetic nearest-neighbor coupling with the bond alternation, ${J}_{1}(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta})<0$, and an antiferromagnetic second-neighbor exchange coupling J2>0 is studied at zero and weak magnetic fields by means of density-matrix renormalization-group calculations of order parameters, correlation functions, and the entanglement entropy, as well as an Abelian bosonization analysis. At zero magnetic field, the bond alternation $\ensuremath{\delta}>0$ suppresses the gapless phase characterized by a vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate spin spiral, whereas this phase occupies a large region in the space of J1/J2 and the easy-plane exchange anisotropy for $\ensuremath{\delta}=0$ [S. Furukawa et al., Phys. Rev. Lett. 105, 257205 (2010)]. Then, four gapped phases are found to appear as the exchange anisotropy varies from the SU(2)-symmetric case to the U(1)-symmetric case: the Haldane dimer (D+) phase with the same sign of the x,y- and z-component dimer order parameters, two VC dimer (VCD+/VCD${}_{\ensuremath{-}}$) phases with the sign of the z-component dimer order parameter being unaltered/reversed, and the even-parity dimer (D${}_{\ensuremath{-}}$) phase. At small magnetic fields, a field-induced ring-exchange interaction, which is proportional to a staggered scalar chirality and a magnetic flux penetrating the associated triangle, drives a transition from the D${}_{\ensuremath{-}}$ phase into a VC-Neel-dimer (VCND) phase, but not from the D+ phase. This VCND phase is stable up to the large magnetic field at which the Zeeman term closes the spin gap. A possible relevance to Rb2Cu2Mo3O12 is discussed.
The frustrated ladder with alternate ferromagnetic exchange $\ensuremath{-}{J}_{F}$ and antiferromagnetic exchange JA to first neighbors and ferromagnetic exchange $\ensuremath{-}{J}_{L}$ to second neighbors is studied by exact diagonalization and density matrix renormalization group calculations in systems of 2N spins-21 with periodic boundary conditions. The ground state is a singlet (S=0) and the singlet-triplet gap ${\ensuremath{\varepsilon}}_{T}$ is finite for the exchanges considered. Spin-21 string correlation functions g1(N) and g2(N) are defined for an even number N of consecutive spins in systems with two spins per unit cell; the ladder has string order ${g}_{2}(\ensuremath{\infty})>0$ and ${g}_{1}(\ensuremath{\infty})=0$. The minimum N∗ of g2(N) is related to the range of ground-state spin correlations. Convergence to ${g}_{2}(\ensuremath{\infty})$ is from below, and g1(N) decreases exponentially for $N\ensuremath{\ge}{N}^{*}$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of JF,JL, and JA reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-21 systems. The nondegenerate singlet ground state of the ladder is a bond-order wave, a Kekul\'e VB diagram at ${J}_{L}={J}_{F}/2\ensuremath{\le}{J}_{A}$, that is reversed on interchanging $\ensuremath{-}{J}_{F}$ and JA. Inversion symmetry is spontaneously broken in the dimer phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model where the Kekul\'e diagrams are the doubly degenerate ground states at J2/J1=1/2.
Using the density-matrix renormalization-group technique, we study the ground-state phase diagram and other low-energy properties of an isotropic antiferromagnetic spin-1/2 chain with both dimerization and frustration, i.e., an alternation \ensuremath{\delta} of the nearest-neighbor exchanges and a next-nearest-neighbor exchange J2. For \ensuremath{\delta}=0, the system is gapless for J2J2c and has a gap for J2>J2c where J2c is about 0.241. For J2=J2c, the gap above the ground state grows as \ensuremath{\delta} to the power 0.667\ifmmode\pm\else\textpm\fi{}0.001. In the J2-\ensuremath{\delta} plane, there is a disorder line 2J2+\ensuremath{\delta}=1. To the left of this line, the peak in the static structure factor S(q) is at qmax=\ensuremath{\pi} (N\'eel phase), while to the right of the line, qmax decreases from \ensuremath{\pi} to \ensuremath{\pi}/2 as J2 is increased to large values (spiral phase). For \ensuremath{\delta}=1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.
Using the density matrix renormalization group and a bosonization approach, we study a spin-1/2 antiferromagnetic Heisenberg chain with near-neighbor coupling J1 and frustrating second-neighbor coupling J2, particularly in the limit J2\ensuremath{\gg}J1. This system exhibits both dimerization and incommensurate spiral spin correlations. We argue that this system is closely related to a doped, spin-gapped phase of the one-dimensional Kondo lattice. \textcopyright{} 1996 The American Physical Society.
The frustrated ladder with alternate ferromagnetic(F) exchange −JF and AF exchange JA to first neighbors and F exchange −JL to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of 2N spins-1/2 with periodic boundary conditions. The ground state is a singlet (S=0) and the singlet-triplet gap εT is finite for the exchanges considered. Spin-1/2 string correlation functions g1(N) and g2(N) are defined for an even number N of consecutive spins in systems with two spins per unit cell; the ladder has string order g2(∞)>0 and g1(∞)=0. The minimum N∗ of g2(N) is related to the range of ground-state spin correlations. Convergence to g2(∞) is from below, and g1(N) decreases exponentially for N≥N∗. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of JF, JL and JA reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the J1−J2 model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekulé VB diagrams at JL=JF/2≤JA, that is reversed on interchanging −JF and JA. Inversion symmetry is spontaneously broken in the dimer phase of the J1−J2 model where the Kekulé diagrams are the doubly degenerate ground states at J2/J1=1/2.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of N 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≥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.