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arXiv:1306.1671 · Physical Review B

Symmetric and asymmetric excitations of a strong-leg quantum spin ladder

D. Schmidiger, S. Mühlbauer, A. Zheludev, Pierre Bouillot, Thierry Giamarchi, Corinna Kollath, Georg Ehlers, Alexei M. Tsvelik

Published Sep 9, 2013• 51 citations
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Abstract

The zero-field excitation spectrum of the strong-leg spin ladder (C${}_{7}$H${}_{10}$N)${}_{2}$CuBr${}_{4}$ is studied with a neutron time-of-flight technique. The spectrum is decomposed into its symmetric and asymmetric parts with respect to the rung momentum and compared with theoretical results obtained by the density matrix renormalization group method. Additionally, the calculated dynamical correlations are shown for a wide range of rung and leg coupling ratios in order to point out the evolution of arising excitations, as, e.g., of the two-magnon bound state from the strong to the weak coupling limit.

DOI
10.1103/physrevb.88.094411

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The zero-field excitation spectrum of the strong-leg spin ladder (C7{}_{7}7​H10{}_{10}10​N)2{}_{2}2​CuBr4{}_{4}4​ is studied with a neutron time-of-flight technique. The spectrum is decomposed into its symmetric and asymmetric parts with respect to the rung momentum and compared with theoretical results obtained by the density matrix renormalization group method. Additionally, the calculated dynamical correlations are shown for a wide range of rung and leg coupling ratios in order to point out the evolution of arising excitations, as, e.g., of the two-magnon bound state from the strong to the weak coupling limit.
The strong-leg S=1/2 Heisenberg spin ladder system (C(7)H(10)N)(2)CuBr(4) is investigated using density matrix renormalization group calculations, inelastic neutron scattering, and bulk magnetothermodynamic measurements. Measurements showed qualitative differences compared to the strong-rung case. A long-lived two-triplon bound state is confirmed to persist across most of the Brillouin zone in a zero field. In applied fields, in the Tomonaga-Luttinger spin-liquid phase, elementary excitations are attractive, rather than repulsive. In the presence of weak interladder interactions, the strong-leg system is considerably more prone to three-dimensional ordering.
We consider asymmetric spin-12\frac{1}{2}21​ two-leg ladders with nonequal antiferromagnetic (AF) couplings ${J}_{\ensuremath{\parallel}}$ and $\ensuremath{\kappa}{J}_{\ensuremath{\parallel}}$ along legs $(\ensuremath{\kappa}\ensuremath{\le}1)$ and ferromagnetic rung coupling, ${J}_{\ensuremath{\perp}}$. This model is characterized by a gap $\ensuremath{\Delta}$ in the spectrum of spin excitations. We show that in the large ${J}_{\ensuremath{\perp}}$ limit this gap is equivalent to the Haldane gap for the AF spin-1 chain, irrespective of the asymmetry of the ladder. The behavior of the gap at small rung coupling falls in two different universality classes. The first class, which is best understood from the case of the conventional symmetric ladder at $\ensuremath{\kappa}=1$, admits a linear scaling for the spin gap $\ensuremath{\Delta}\ensuremath{\sim}{J}_{\ensuremath{\perp}}$. The second class appears for a strong asymmetry of the coupling along legs, $\ensuremath{\kappa}{J}_{\ensuremath{\parallel}}⪡{J}_{\ensuremath{\perp}}⪡{J}_{\ensuremath{\parallel}}$ and is characterized by two energy scales: the exponentially small spin gap $\ensuremath{\Delta}\ensuremath{\sim}{J}_{\ensuremath{\perp}}\text{ }\text{exp}(\ensuremath{-}{J}_{\ensuremath{\parallel}}/{J}_{\ensuremath{\perp}})$, and the bandwidth of the low-lying excitations induced by a Suhl-Nakamura indirect exchange $\ensuremath{\sim}{J}_{\ensuremath{\perp}}^{2}/{J}_{\ensuremath{\parallel}}$. We report numerical results obtained by exact diagonalization, density-matrix renormalization group and quantum Monte Carlo simulations for the spin gap and various spin correlation functions. Our data indicate that the behavior of the string order parameter, characterizing the hidden AF order in Haldane phase, is different in the limiting cases of weak and strong asymmetries. On the basis of the numerical data, we propose a low-energy theory of effective spin-1 variables, pertaining to large blocks on a decimated lattice.
Inelastic neutron scattering is used to measure the spin excitation spectrum of the Heisenberg S=1/2 ladder material (C7H10N)2CuBr4 in its entirety, both in the gapped spin liquid and the magnetic field-induced Tomonaga-Luttinger spin liquid regimes. A fundamental change of the spin dynamics is observed between these two regimes. Density matrix renormalization group calculations quantitatively reproduce and help understand the observed commensurate and incommensurate excitations. The results validate long-standing quantum field-theoretical predictions but also test the limits of that approach.
We investigate weakly coupled spin-1/21/21/2 ladders in a magnetic field. The work is motivated by recent experiments on the compound (CH12N)CuBr4{(}_{\mathrm{C}}{\mathrm{H}}_{12}{\mathrm{N}}_{)}{\mathrm{CuBr}}_{4}(C​H12​N)​CuBr4​ (BPCB). We use a combination of numerical and analytical methods, in particular, the density-matrix renormalization group (DMRG) technique, to explore the phase diagram and the excitation spectra of such a system. We give detailed results on the temperature dependence of the magnetization and the specific heat, and the magnetic-field dependence of the nuclear-magnetic-resonance relaxation rate of single ladders. For coupled ladders, treating the weak interladder coupling within a mean-field or quantum Monte Carlo approach, we compute the transition temperature of triplet condensation and its corresponding antiferromagnetic order parameter. Existing experimental measurements are discussed and compared to our theoretical results. Furthermore, we compute, using time-dependent DMRG, the dynamical correlations of a single spin ladder. Our results allow to describe directly the inelastic neutron scattering cross section up to high energies. We focus on the evolution of the spectra with the magnetic field and compare their behavior for different couplings. The characteristic features of the spectra are interpreted using different analytical approaches such as the mapping onto a spin chain, a Luttinger liquid or onto a ttt-JJJ model. For values of parameters for which such measurements exist, we compare our results to inelastic neutron scattering experiments on the compound BPCB and find excellent agreement. We make additional predictions for the high-energy part of the spectrum that are potentially testable in future experiments.
We study quantum phase transitions in the asymmetric variation of the three-leg Heisenberg tube for half-odd-integer spin, with a modulation of one of the rung exchange couplings ${J}_{\ensuremath{\perp}}^{\ensuremath{'}}$ while the other two are kept constant ${J}_{\ensuremath{\perp}}$. We focus on the strong rung-coupling regime ${J}_{\ensuremath{\perp}}\ensuremath{\gg}{J}_{\ensuremath{\parallel}}$, where ${J}_{\ensuremath{\parallel}}$ is the leg coupling, and analyze the effective spin-orbital model with a transverse crystal field in detail. Applying the Abelian bosonization to the effective model, we find that the system is in the dimer phase for the general half-odd-integer-spin cases without the rung modulation; the phase transition between the dimer and Tomonaga-Luttinger-liquid phases induced by the rung modulation is of the SU(2)-symmetric Berezinskii-Kosterlitz-Thouless type. Moreover, we perform a level spectroscopy analysis for the effective model for spin-1/2 using exact diagonalization, to determine the precise transition point $|{J}_{\ensuremath{\perp}}^{\ensuremath{'}}\ensuremath{-}{J}_{\ensuremath{\perp}}|/{J}_{\ensuremath{\parallel}}\ensuremath{\sim}0.283$ in the strong rung-coupling limit. The presence of the dimer phase in a small but finite region is also confirmed by a density-matrix renormalization group calculation on the original spin-tube model.
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