The spin-1/2 chain with isotropic Heisenberg exchange $J_1$, $J_2 > 0$ between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed $J_1/J_2$ between gapless phases with nondegenerate
ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor $S(q)$ that diverges at wave vector $q_m$ in QLRO($q_m$) phases with periodicity $2pi/q_m$ but remains finite in gapped phases. $S(q_m)$ is evaluated using ED and density matrix renormalization group (DMRG) calculations. Level crossing and the magnitude of $S(q_m)$ are independent and complementary probes of quantum phases, based respectively on excited and ground states. Both indicate a gapless QLRO($pi/2$) phase between $-1.2 < J_1/|J_2| < 0.45$. Numerical results and field theory agree well for quantum critical points at small frustration $J_2$ but disagree in the sector of weak exchange $J_1$ between Heisenberg antiferromagnetic chains on sublattices of odd and even-numbered sites.
Exact diagonalization of finite spin-1/2 chains with periodic boundary conditions is applied to the ground state (gs) of chains with ferromagnetic (F) exchange $J_1 < 0$between first neighbors, antiferromagnetic (AF) exchange $J_2 = alpha J_1 > 0$bet
ween second neighbors, and axial anisotropy $0 le Delta le 1$. In zero field, the gs is in the $S_z = 0$ sector for the relevant parameters and is doubly degenerate at multiple points $gamma_m = (alpha_m, Delta_m)$ in the $alpha$, $Delta$ plane. Degeneracy under inversion at sites or spin parity or both leads, respectively, to a bond order wave (BOW), to staggered magnetization or to vector chiral (VC) order. Exact results up to $N = 28$ spins directly yield order parameters and spin correlation functions whose weak N dependencies allow inferences about infinite chains. The high-spin gs at $J_2 = 0$ changes discontinuously at $gamma_1 = (-1/4, 1)$ to a singlet in the isotropic ($Delta = 1$) chain. The transition from high to low spin $S(alpha, Delta)$ is continuous for $ Delta < Delta_B = 0.95 pm 0.01$ on the degeneracy line $alpha_1(Delta)$. The gs has staggered magnetization between $Delta_A = 0.72$ and $Delta_B$, and a BOW for $Delta < Delta_A$. When both inversion and spin parity are reversed at $gamma_m$, the correlation functions $C(p)$ for spins separated by $p$ sites are identical. $C(p)$ minima are shifted by $pi/2$ from the minima of VC order parameters at separation $p$, consistent with right and left-handed helices along the z axis and spins in the xy plane. Degenerate gs of finite chains are related to quantum phase diagrams of extended $alpha$, $Delta$ chains, with good agreement for order parameters along the line $alpha_1(Delta)$.
An efficient density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity $Z = 3$ and antiferromagnetic exchange between nearest neighbor spins $s= 1/2$ or 1 sites in successive generations $g$. The algor
ithm is accurate for $s = 1$ sites. The ground states are magnetic with spin $S(g) = 2^g s$, staggered magnetization that persists for large $g > 20$ and short-range spin correlation functions that decrease exponentially. A finite energy gap to $S > S(g)$ leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for $s$ = 1/2 and 1 are interpreted in terms of an analytical three-site model.
The bond order wave (BOW) phase of the extended Hubbard model (EHM) in one dimension (1D) is characterized at intermediate correlation $U = 4t$ by exact treatment of $N$-site systems. Linear coupling to lattice (Peierls) phonons and molecular (Holste
in) vibrations are treated in the adiabatic approximation. The molar magnetic susceptibility $chi_M(T)$ is obtained directly up to $N = 10$. The goal is to find the consequences of a doubly degenerate ground state (gs) and finite magnetic gap $E_m$ in a regular array. Degenerate gs with broken inversion symmetry are constructed for finite $N$ for a range of $V$ near the charge density wave (CDW) boundary at $V approx 2.18t$ where $E_m approx 0.5t$ is large. The electronic amplitude $B(V)$ of the BOW in the regular array is shown to mimic a tight-binding band with small effective dimerization $delta_{eff}$. Electronic spin and charge solitons are elementary excitations of the BOW phase and also resemble topological solitons with small $delta_{eff}$. Strong infrared intensity of coupled molecular vibrations in dimerized 1D systems is shown to extend to the regular BOW phase, while its temperature dependence is related to spin solitons. The Peierls instability to dimerization has novel aspects for degenerate gs and substantial $E_m$ that suppresses thermal excitations. Finite $E_m$ implies exponentially small $chi_M(T)$ at low temperature followed by an almost linear increase with $T$. The EHM with $U = 4t$ is representative of intermediate correlations in quasi-1D systems such as conjugated polymers or organic ion-radical and charge-transfer salts. The vibronic and thermal properties of correlated models with BOW phases are needed to identify possible physical realizations.
