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arXiv:1503.03871 · Physical Review Letters

Heat Conductivity of the Heisenberg Spin- 1/2 Ladder: From Weak to Strong Breaking of Integrability

Robin Steinigeweg, J. Herbrych, X. Zotos, Wolfram Brenig

Published Jan 5, 2016• 44 citations
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Abstract

We investigate the heat conductivity $\ensuremath{\kappa}$ of the Heisenberg spin-$1/2$ ladder at finite temperature covering the entire range of interchain coupling ${J}_{\ensuremath{\perp}}$, by using several numerical methods and perturbation theory within the framework of linear response. We unveil that a perturbative prediction $\ensuremath{\kappa}\ensuremath{\propto}{J}_{\ensuremath{\perp}}^{\ensuremath{-}2}$, based on simple golden-rule arguments and valid in the strict limit ${J}_{\ensuremath{\perp}}\ensuremath{\rightarrow}0$, applies to a remarkably wide range of ${J}_{\ensuremath{\perp}}$, qualitatively and quantitatively. In the large ${J}_{\ensuremath{\perp}}$ limit, we show power-law scaling of opposite nature, namely, $\ensuremath{\kappa}\ensuremath{\propto}{J}_{\ensuremath{\perp}}^{2}$. Moreover, we demonstrate the weak and strong coupling regimes to be connected by a broad minimum, slightly below the isotropic point at ${J}_{\ensuremath{\perp}}={J}_{\ensuremath{\parallel}}$. Reducing temperature $T$, starting from $T=\ensuremath{\infty}$, this minimum scales as $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ down to $T$ on the order of the exchange coupling constant. These results provide for a comprehensive picture of $\ensuremath{\kappa}({J}_{\ensuremath{\perp}},T)$ of spin ladders.

DOI
10.1103/physrevlett.116.017202

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We investigate the heat conductivity \ensuremathκ\ensuremath{\kappa}\ensuremathκ of the Heisenberg spin-1/21/21/2 ladder at finite temperature covering the entire range of interchain coupling J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​, by using several numerical methods and perturbation theory within the framework of linear response. We unveil that a perturbative prediction \ensuremathκ\ensuremath∝J\ensuremath⊥\ensuremath−2\ensuremath{\kappa}\ensuremath{\propto}{J}_{\ensuremath{\perp}}^{\ensuremath{-}2}\ensuremathκ\ensuremath∝J\ensuremath⊥\ensuremath−2​, based on simple golden-rule arguments and valid in the strict limit J\ensuremath⊥\ensuremath→0{J}_{\ensuremath{\perp}}\ensuremath{\rightarrow}0J\ensuremath⊥​\ensuremath→0, applies to a remarkably wide range of J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​, qualitatively and quantitatively. In the large J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ limit, we show power-law scaling of opposite nature, namely, \ensuremathκ\ensuremath∝J\ensuremath⊥2\ensuremath{\kappa}\ensuremath{\propto}{J}_{\ensuremath{\perp}}^{2}\ensuremathκ\ensuremath∝J\ensuremath⊥2​. Moreover, we demonstrate the weak and strong coupling regimes to be connected by a broad minimum, slightly below the isotropic point at J\ensuremath⊥=J\ensuremath∥{J}_{\ensuremath{\perp}}={J}_{\ensuremath{\parallel}}J\ensuremath⊥​=J\ensuremath∥​. Reducing temperature TTT, starting from T=\ensuremath∞T=\ensuremath{\infty}T=\ensuremath∞, this minimum scales as \ensuremathκ\ensuremath∝T\ensuremath−2\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}2}\ensuremathκ\ensuremath∝T\ensuremath−2 down to TTT on the order of the exchange coupling constant. These results provide for a comprehensive picture of \ensuremathκ(J\ensuremath⊥,T)\ensuremath{\kappa}({J}_{\ensuremath{\perp}},T)\ensuremathκ(J\ensuremath⊥​,T) of spin ladders.
The heat conductivity $\ensuremath{\kappa}(T)$ of integrable models, like the one-dimensional spin-1/21/21/2 nearest-neighbor Heisenberg model, is infinite even at finite temperatures as a consequence of the conservation laws associated with integrability. Small perturbations lead to finite but large transport coefficients which we calculate perturbatively using exact diagonalization and moment expansions. We show that there are two different classes of perturbations. While an interchain coupling of strength ${J}_{\ensuremath{\perp}}$ leads to $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}_{\ensuremath{\perp}}^{2}$ as expected from simple golden-rule arguments, we obtain a much larger $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}^{\ensuremath{'}4}$ for a weak next-nearest-neighbor interaction ${J}^{\ensuremath{'}}$. This can be explained by a new approximate conservation law of the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ Heisenberg chain.
We present a general theory for the intermediate-temperature (T) properties of Heisenberg antiferromagnets of spin-SSS ions on p-leg ladders, valid for 2Sp2Sp2Sp even or odd. Following an earlier proposal for 2Sp2Sp2Sp even [Damle and Sachdev, Phys. Rev. B 57, 8307 (1998)], we argue that an integrable, classical, continuum model of a fixed-length, three-vector applies over an intermediate-temperature range; this range becomes very wide for moderate and large values of 2Sp.2Sp.2Sp. The coupling constants of the effective model are known exactly in terms of the energy gap above the ground state $\ensuremath{\Delta}$ (for 2Sp2Sp2Sp even), or a crossover scale T0{T}_{0}T0​ (for 2Sp2Sp2Sp odd). Analytic and numeric results for dynamic and transport properties are obtained, including some exact results for the spin-wave damping. Numerous quantitative predictions for neutron scattering and NMR experiments are made. A general discussion on the nature of T>0T>0T>0 transport in integrable systems is also presented: an exact solution of a toy model proves that diffusion can exist in integrable systems, provided proper care is taken in approaching the thermodynamic limit.
Heisenberg antiferromagnetic spin ``ladders'' (two coupled spin chains) are low-dimensional magnetic systems which for S=1/2 interpolate between half-integer-spin chains, when the chains are decoupled, and effective integer-spin one-dimensional chains in the strong-coupling limit. The spin-1/2 ladder may be realized in nature by vanadyl pyrophosphate, (VO)2{)}_{2})2​P2{\mathrm{P}}_{2}P2​O7{\mathrm{O}}_{7}O7​. In this paper we apply strong-coupling perturbation theory, spin-wave theory, Lanczos techniques, and a Monte Carlo method to determine the ground-state energy and the low-lying excitation spectrum of the ladder. We find evidence of a nonzero spin gap for all interchain couplings ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$>0. A band of spin-triplet excitations above the gap is also analyzed. These excitations are unusual for an antiferromagnet, since their long-wavelength dispersion relation behaves as (k-k0{\mathit{k}}_{0}k0​)2{)}^{2})2 (in the strong-coupling limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\gg}J, where J is the in-chain antiferromagnetic coupling). Their band is folded, with a minimum energy at k0{\mathit{k}}_{0}k0​=\ensuremath{\pi}, and a maximum between k1{\mathit{k}}_{1}k1​=\ensuremath{\pi}/2 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=0) and 0 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=\ensuremath{\infty}). We also give numerical results for the dynamical structure factor S(q,\ensuremath{\omega}), which can be determined in neutron scattering experiments. Finally, possible experimental techniques for studying the excitation spectrum are discussed.
We employ a classical limit grounded in SU(4) coherent states to investigate the temperature-dependent dynamical spin structure factor of the S=1/2S=1/2S=1/2 ladder consisting of weakly coupled dimers. By comparing the outcomes of this classical approximation with density matrix renormalization group and exact diagonalization calculations in finite-size ladders, we demonstrate that the classical dynamics offers an accurate approximation across the entire temperature range when the interdimer coupling is weak and a good approximation in the high-temperature regime even when the interdimer coupling is strong. This agreement is achieved after appropriately rescaling the temperature axis and renormalizing expectation values to satisfy a quantum sum rule, following D. Dahlbom et al. [Phys. Rev. B 109, 014427 (2024)]. We anticipate the method will be particularly effective when applied to 2D and 3D lattices composed of weakly coupled dimers, situations that remain challenging for alternative numerical methods.
Understanding the physics of the integrable spin-1/2 XXZ chain has witnessed substantial progress, due to the development and application of sophisticated analytical and numerical techniques. In particular, infinite-temperature magnetization transport has turned out to range from ballistic, over superdiffusive, to diffusive behavior in different parameter regimes of the anisotropy. Since integrability is rather the exception than the rule, a crucial question is the change of transport under integrability-breaking perturbations. This question includes the stability of superdiffusion at the isotropic point and the change of diffusion constants in the easy-axis regime. In our work, we study this change of diffusion constants by a variety of methods and cover both, linear response theory in the closed system and the Lindblad equation in the open system, where we throughout focus on periodic boundary conditions. In the closed system, we compare results from the recursion method to calculations for finite systems and find evidence for a continuous change of diffusion constants over the full range of perturbation strengths. In the open system weakly coupled to baths, we find diffusion constants in quantitative agreement with the ones in the closed system in a range of nonweak perturbations, but disagreement in the limit of weak perturbations. Using a simple model in this limit, we point out the possibility of a diverging diffusion constant in such an open system.
We set up a framework in which controlled lower bounds to conductivities of strongly correlated quantum systems may be calculated perturbatively in situations where the conductivities are infinite in the unperturbed model. This framework is then applied to study the transport properties of almost integrable systems, in general, and heat and spin conductivities in the integrable XXZ Heisenberg spin chain perturbed by various integrability breaking couplings, in particular. Furthermore, we systematically develop a microscopic description of the contribution of spin chains to experimentally measured heat conductivity in spin chain compounds, with an emphasis on the effect of weak disorder.