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arXiv:0804.2639 · Physical Review Letters

Controlling Luttinger Liquid Physics in Spin Ladders under a Magnetic Field

M. Klanjšek, H. Mayaffre, C. Berthier, M. Horvatić, BRUNETTO CHIARI, O. Piovesana, Pierre Bouillot, Corinna Kollath, E. Orignac, R. Citro, Thierry Giamarchi

Published Sep 26, 2008• 217 citations
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Abstract

We present a 14N nuclear magnetic resonance study of a single crystal of CuBr4(C5H12N)2 (BPCB) consisting of weakly coupled spin-1/2 Heisenberg antiferromagnetic ladders. Treating ladders in the gapless phase as Luttinger liquids, we are able to fully account for (i) the magnetic field dependence of the nuclear spin-lattice relaxation rate T1(-1) at 250 mK and for (ii) the phase transition to a 3D ordered phase occurring below 110 mK due to weak interladder exchange coupling. BPCB is thus an excellent model system where the possibility to control Luttinger liquid parameters in a continuous manner is demonstrated and the Luttinger liquid model tested in detail over the whole fermion band.

DOI
10.1103/physrevlett.101.137207

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We present a 14N nuclear magnetic resonance study of a single crystal of CuBr4(C5H12N)2 (BPCB) consisting of weakly coupled spin-1/2 Heisenberg antiferromagnetic ladders. Treating ladders in the gapless phase as Luttinger liquids, we are able to fully account for (i) the magnetic field dependence of the nuclear spin-lattice relaxation rate T1(-1) at 250 mK and for (ii) the phase transition to a 3D ordered phase occurring below 110 mK due to weak interladder exchange coupling. BPCB is thus an excellent model system where the possibility to control Luttinger liquid parameters in a continuous manner is demonstrated and the Luttinger liquid model tested in detail over the whole fermion band.
Abstract We present a 14N nuclear magnetic resonance study of a single crystal of CuBr4(C5H12N)2 consisting of weakly coupled spin‐1/2 Heisenberg antiferromagnetic ladders. When placed in a magnetic field, such a ladder is theoretically predicted to exhibit a quantum critical Luttinger liquid (LL) behaviour in the gapless phase, i.e. between the two critical fields. Treating ladders as LLs and interaction between them as a perturbation, we are indeed able to fully account for (i) the spin dynamics accessed by measuring the spin–lattice relaxation rate T1−1T_1^{ - 1}T1−1​ , and for (ii) the phase transition to a 3D ordered phase occurring below 110 mK due to the weak interladder coupling.
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 investigate weakly coupled spin-1/2 ladders in a magnetic field. The work is motivated by recent experiments on the compound (C5H12N)2CuBr4\mathrm{(C_5H_{12}N)_2CuBr_4}(C5​H12​N)2​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 (NMR) relaxation rate of single ladders. For coupled ladders, treating the weak interladder coupling within a mean field 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 directly describe 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 (LL) or onto a t-J 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 present NMR measurements of a strong-leg spin-1/2 Heisenberg antiferromagnetic ladder compound (C7H10N)2CuBr4 under magnetic fields up to 15 T in the temperature range from 1.2 K down to 50 mK. From the splitting of NMR lines, we determine the phase boundary and the order parameter of the low-temperature (three-dimensional) long-range-ordered phase. In the Tomonaga-Luttinger regime above the ordered phase, NMR relaxation reflects characteristic power-law decay of spin correlation functions as 1/T1∝T(1/2K-1), which allows us to determine the interaction parameter K as a function of field. We find that field-dependent K varies within the 1<K<2 range, which signifies attractive interaction between the spinless fermions in the Tomonaga-Luttinger liquid.
We present an extensive NMR study of the spin-1/2 antiferromagnetic Heisenberg ladder Cu2(C5H12N2)2Cl4\mathrm{Cu}{}_{2}({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}\mathrm{Cl}{}_{4}Cu2​(C5​H12​N2​)2​Cl4​ in a magnetic field range 4.5--16.7 T. By measuring the proton NMR relaxation rate 1/T11/{T}_{1}1/T1​ and varying the magnetic field around the critical field ${H}_{c1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Delta}/g{\ensuremath{\mu}}_{B}\ensuremath{\approx}7.5\mathrm{T}$, we have studied the transition from a gapped spin liquid ground state to a gapless magnetic regime which can be described as a Luttinger liquid. We identify an intermediate regime $T\ensuremath{\ge}|{H\ensuremath{-}H}_{c1}|$, where the spin dynamics is (possibly) only controlled by the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 critical point Hc1{H}_{c1}Hc1​.
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