We study low-energy excitations in antiferromagnetic Heisenberg chains with a staggered field which splits the spectrum into a longitudinal and a transverse branch. Bound states are found to exist inside the field induced gap in both branches. They originate from the edge effects and are inherent to spin-chain materials. The sine-Gordon scaling h(2/3)(s)[log(h(s)](1/6) (h(s), the staggered field) provides an accurate description for the gap and midgap energies in the transverse branch for S=1/2 and the midgap energies in both branches for S=3/2 over a wide range of magnetic field; however, it can fit other low-energy excitations only at much lower field. Moreover, the integer-spin S=1 chain displays scaling behavior that does not fit this scaling law. These results reveal intriguing features of magnetic excitations in spin-chain materials that deserve further investigation.
We study the low-energy excitations of the spin-
21 antiferromagnetic Heisenberg chain and
N-leg
(N=2,3,4) ladders in a staggered magnetic field
hs. We show that
hs induces gap and midgap states in all the cases and we will examine their field scaling behavior. A modified boundary scheme is devised to extract accurate bulk excitation behavior. The gap values converge rapidly as
N increases, leading to a field scaling exponent $\ensuremath{\gamma}=\frac{1}{2}$ for both the longitudinal and transverse gaps of the square lattice $(N\ensuremath{\rightarrow}\ensuremath{\infty})$. The midgap states induced by the boundary edge effects share the bulk gap scaling exponents but their overall scaling behavior in the large-
N limit needs further investigation.
We study the spin-
21 antiferromagnetic Heisenberg chain in both uniform and (perpendicular) staggered magnetic fields using the density-matrix renormalization-group method. This model has been shown earlier to describe the physics of the copper benzoate materials in magnetic field. In the present work, we extend the study to more general case for a systematic investigation of the field-induced gap and related properties of the spin-
21 antiferromagnetic Heisenberg chain. In particular, we explore the high magnetic field regime where interesting behaviors in the field-induced gap, magnetization, and spin correlation functions are found. Careful examination of the low energy properties and magnetization reveals interesting competing effects of the staggered and uniform fields. The incommensurate behavior in the spin correlation functions is demonstrated and discussed in detail. The present work reproduces earlier results in good agreement with experimental data on copper benzoate and predicts new interesting field-induced features at very high magnetic field.
Field-induced excitation gaps in quantum spin chains are an interesting phenomenon related to confinements of topological excitations. In this paper I present a type of this phenomenon. I show that an effective magnetic field with a fourfold screw symmetry induces the excitation gap accompanied by dimer orders. The dimer order parameter and the excitation gap exhibit characteristic power-law dependence on the fourfold screw-symmetric field. Moreover, the field-induced dimer order and the field-induced N\'eel order coexist when the external uniform magnetic field, the fourfold screw-symmetric field, and the twofold staggered field are applied. This situation is in close connection with a compound $[\mathrm{Cu}(\mathrm{pym}){({\mathrm{H}}_{2}\mathrm{O})}_{4}]{\mathrm{SiF}}_{6}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{H}}_{2}\mathrm{O}$ [J. Liu et al., Phys. Rev. Lett. 122, 057207 (2019)]. In this paper I discuss a mechanism of field-induced dimer orders by using a density-matrix renormalization group method, a perturbation theory, and quantum field theories.
Linear Heisenberg antiferromagnets (HAFs) are chains of spin-
S sites with isotropic exchange
J 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 when $S\ensuremath{\ge}1$. The ground state of open chains of
N spins has
SG=0 or
S, respectively, for even or odd
N. Density-matrix renormalization-group calculations with different algorithms for even and odd
N are presented up to
N=500 for the energy and spin densities $\ensuremath{\rho}(r,N)$ of edge states in HAFs with
S=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,...,N. The SDWs are in phase when
N is odd, are out of phase when
N is even, and have finite excitation energy $\mathrm{\ensuremath{\Gamma}}(N)$ that decreases exponentially with
N for integer
S and faster than
1/N for half integer
S. The spin densities and excitation energy are quantitatively modeled for integer
S 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=1 and 49.0 for
S=2. The BI-SDWs of
S=3/2 chains are not localized and are qualitatively different for even and odd
N. Exchange between the ends for odd
N 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=1 with even
N, 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=1 chains with odd
N are fully consistent with the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$;
S=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=3/2 chains with even
N and is modified for exchange between ends for odd
N.
The response of an open spin chain with isotropic antiferromagnetic interactions to a uniform magnetic field is studied by classical Monte Carlo simulations. It is observed how the induced on-site magnetization is nonuniform, due to the occurrence of edge staggered terms which decay exponentially over a distance equal to the zero-field correlation length of the infinite chain. The total magnetic moment associated to each staggered term is found to be about half of the original single-spin magnitude and to decrease as the inverse of temperature (i.e., to behave as a Curie-like moment). The numerical results are compared to recent NMR findings in spinless-doped
Y2BaNiO5; the remarkable agreement found shows that, for temperatures above the Haldane gap, the classical approach gives a correct picture of the boundary effects observed in the Heisenberg
S=1 chain.