Temperature dependence of the NMR relaxation rate $1/T_1$ for quantum spin chains

We present results of numerical simulations performed on one-dimensional spin chains in order to extract the so-called relaxation rate $1/T_1$ accessible through NMR experiments. Building on numerical tensor network methods using the matrix product states formalism, we can follow the nontrivial crossover occurring in critical chains between the high-temperature diffusive classical regime and the low-temperature response described by the Tomonaga-Luttinger liquid (TLL) theory, for which analytical expressions are known. In order to compare analytics and numerics, we focus on a generic spin-$1/2$ $XXZ$ chain which is a paradigm of gapless TLL, as well as a more realistic spin-1 anisotropic chain, modeling the DTN material, which can be either in a trivial gapped phase or in a TLL regime induced by an external magnetic field. Thus, by monitoring the finite temperature crossover, we provide quantitative limits on the range of validity of TLL theory, that will be useful when interpreting experiments on quasi-one-dimensional materials.

Figure 1

(i) Upper panel: magnetization curve of the $XXZ$ Hamiltonian (2.1) as a function of the magnetic field for $\mathrm{\Delta }\in (-1,1]$. There is a gapless TLL phase below $h_c=J(\mathrm{\Delta }+1)/g{\mu }_B$ and a gapped one above when the system is fully polarized. (ii) Lower panel: magnetization curve of the 1D DTN Hamiltonian (2.2) as a function of the magnetic field. Two gapped phases (large-$D$ and polarized) are respectively located below $h_{c1}$ and above $h_{c2}$. The intermediate gapless phase can be described by a TLL theory.

Figure 2

We compare numerical results (circles) used to determine the $1/T_1$ with analytical results (straight lines) for the $XX$ model. For the transverse $(\perp )$ case, exact results are computed on a chain of size $L=64$ (OBC). As for the longitudinal $(\parallel )$ case, the chain size is $L=1000$ (OBC). Numerics on their side are performed on a chain of $L=64$ (OBC) sites. The left panel shows the real value of the dynamical correlations. For readability, we only display numerical results for the lowest temperature for the longitudinal correlations. Indeed, this is a priori the hardest to compute and thus the most subject to errors. The right panel shows the real part of the Fourier transform of the real time data. Although we only show data up to $t=40J^{-1}$ the Fourier transform of the exact $zz$ correlations was performed using data up to $t=1000J^{-1}$.

Figure 3

Transverse relaxation rate $1/T_1^{\perp }$ vs reduced temperature $T/J$ for the spin-$1/2$ $XXZ$ chain at various $\mathrm{\Delta }$ and $h=0$ obtained numerically using MPS techniques (circles, from top to bottom: $\mathrm{\Delta }=-0.8,-0.6,-0.5,-0.4,-0.2,0.0,0.2,0.4,0.5,0.6,0.8,1)$. Numerics are compared to TLL theory Eq. (2.12) at low temperature (thick lines) for $\left|\mathrm{\Delta }\right|<1$, and with Eq. (4.2) for the SU(2) Heisenberg point $\mathrm{\Delta }=1$. The thin lines between the circles are guides to the eyes.

Figure 4

Transverse and longitudinal relaxation rates $1/T_1^{\perp ,\parallel }$ plotted against reduced inverse temperature $\beta J$ for the spin-$1/2$ $XXZ$ chain in its gapped phase for the anisotropy $\mathrm{\Delta }=0.5$. The critical magnetic field is $h_c=3J/2g{\mu }_B$ and the value of the gap ${\mathrm{\Delta }}_g=g{\mu }_B\left(h-h_c\right)$. Numerical results are obtained using MPS techniques (circles and diamonds) and the exponentially decaying behavior is verified with the straight lines set with the expected gap value $1/T_1^{\perp ,\parallel }=c_{\perp ,\parallel }{e}^{-\beta {\mathrm{\Delta }}_g}$ and $c_{\perp ,\parallel }$ a nonuniversal free parameter.

Figure 5

Transverse relaxation rate $1/T_1^{\perp }$ plotted vs temperature $T$ for the spin-1 DTN chain obtained numerically using MPS techniques (circles). The low temperature behavior is compared to TLL prediction (straight lines). The magnetic field $h$ is given in T. The inset compares TLL prediction and numerical results for $T=0.4\mathrm{K}$ and covers the whole TLL phase from $h_{c1}$ to $h_{c2}$. The lower panel is a zoom on the low temperature asymptotic power-law regime.

Figure 6

Longitudinal $1/T_1^{\parallel }$ and transverse $1/T_1^{\perp }$ relaxation rates for the DTN spin-1 chain at $h=11.0\mathrm{T}$, corresponding to $m^z\simeq 0.85$. As $1/T_1^{\parallel }$ cannot be estimated for sure, we only provide a lower bound. The nonmonotonic behavior observed experimentally at high $T$ in Ref. [21] apparently comes from the large contribution of $1/T_1^{\parallel }$ at high temperature. Experimental data for DTN [21] at the same magnetization are shown for comparison, after a proper rescaling in order to match the low-$T$ regime. The 3D BEC transition temperature $T_N(m^z\simeq 0.85)\simeq 0.59$ K [36] is also shown.