\n For w-legged antiferromagnetic spin-1/2 Heisenberg\nladders, a long-range spin pairing order can be identified which\nenables the separation of the space spanned by finite-range\n(covalent) valence-bond configurations into w+1 subspaces.\nSince every subspace has an equivalent counter subspace connected by\ntranslational symmetry, twofold degeneracy, breaking translational\nsymmetry is found except for the subspace where the ground state of\nw = even belongs to.\nIn terms of energy ordering, (non)degeneracy and the discontinuities\nintroduced in the long-range spin pairing order by topological spin\ndefects, the differences between even and odd ladders are explained\nin a general and systematic way. \n\n
We consider a resonating-valence-bond state as a trial wave function for the ground state of the antiferromagnetic Heisenberg model on the square lattice. The energy and the staggered spin-spin correlation functions are calculated on a small lattice and are compared with the results of the exact diagonalization of Oitmaa and Betts. The trial energy is only about 4.7% different from the exact ground-state energy on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The importance of the singlet pairs between the next-nearest antiferromagnetic sites is inferred. An argument against the previous interpretation of the numerical data as being evidence of long-range order is presented.
The quantum phases of one-dimensional spin
s=1/2 chains are discussed for models with two parameters, frustrating exchange
g=J2>0 between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at
g=0 has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin
S=SA+SB, have singlet GS for any
g, $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at
1/g=0 to linear Heisenberg antiferromagnets on sublattices
A and
B of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap
Em to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at
gc=1/4ln2 from a GS with perfect LRO to a decoupled phase with
SA=SB=0 for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.
The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.
The large-N limit of a nearest-neighbor SU(N) antiferromagnet on a bipartite lattice exhibits in dimensions d\ensuremath{\ge}2 a zero-temperature phase transition between a N\'eel ordered state and a resonating-valence-bond state. Here it is shown in d=1,2 that topological effects produce spin-Peierls or valence-bond-solid order in the non-N\'eel phase with a ground-state degeneracy which varies periodically with ``spin'' for fixed N with periodicity given by the coordination number of the lattice. Thus a non-N\'eel phase of the spin-(1/2 Heisenberg model on a square lattice would be a spin-Peierls state with a fourfold degeneracy due to broken lattice rotational symmetry.
To make the transition from the quasi-long-range order in a chain of antiferromagnetically coupled S = 1/2 spins to the true long-range order that occurs in a plane, one can assemble chains to make ladders of increasing width. Surprisingly, this crossover between one and two dimensions is not at all smooth. Ladders with an even number of legs have purely short-range magnetic order and a finite energy gap to all magnetic excitations. Predictions of this ground state have now been verified experimentally. Holes doped into these ladders are predicted to pair and possibly superconduct.
We consider the exchanged Hamiltonian HST=−J∑〈rr′〉(2Sr⋅Sr′−12)(2Tr⋅Tr′−12), describing two isotropic spin-1/2 Heisenberg antiferromagnets coupled by a quartic term on equivalent bonds. The model is relevant for systems with orbital degeneracy and strong electron-vibron coupling in the large Hubbard repulsion limit. To investigate the ground state properties we use a Green’s Function Monte Carlo, calculating energy gaps and correlation functions, the latter through the forward walking technique. In one dimension we find that the ground state is a “crystal” of valence bond dimers. In two dimensions, the spin gap appears to remain finite in the thermodynamic limit, and, consistently, the staggered magnetization—signal of Néel long range order—seems to vanish. From the analysis of dimer–dimer correlation functions, however, we find no sign of a valance bond crystal. A spin liquid appears as a plausible scenario compatible with our findings.