Temperature dependence of the NMR spin-lattice relaxation rate for spin-$\frac{1}{2}$ chains

We use recent developments in the framework of a time-dependent matrix product state method to compute the nuclear magnetic resonance relaxation rate $1/T_1$ for spin-1/2 chains under magnetic field and for different Hamiltonians (XXX, XXZ, isotropically dimerized). We compute numerically the temperature dependence of the $1/T_1$. We consider both gapped and gapless phases, and also the proximity of quantum critical points. At temperatures much lower than the typical exchange energy scale, our results are in excellent agreement with analytical results, such as the ones derived from the Tomonaga-Luttinger liquid (TLL) theory and bosonization, which are valid in this regime. We also cover the regime for which the temperature $T$ is comparable to the exchange coupling. In this case analytical theories are not appropriate, but this regime is relevant for various new compounds with exchange couplings in the range of tens of Kelvin. For the gapped phases, either the fully polarized phase for spin chains or the low-magnetic-field phase for the dimerized systems, we find an exponential decrease in $\mathrm{\Delta }/\left(k_{B}T\right)$ of the relaxation time and can compute the gap $\mathrm{\Delta }$. Close to the quantum critical point our results are in good agreement with the scaling behavior based on the existence of free excitations.

Figure 1

Schematic phase diagram of the XXZ model [Eq. (1)] at zero temperature as a function of the magnetic field $h$ and of the anisotropy parameter $\mathrm{\Delta }$. “Ferro” stands for the phase in which the spins are ferromagnetically aligned along the $z$ direction. XY denotes a massless phase with dominant in-plane antiferromagnetic correlations which is a TLL [3]. “Néel” denotes an Ising antiferromagnetically ordered phase along $z$. The behavior of the boundary as a function of $h$ around the point $\mathrm{\Delta }=1$ reflects the Berezinskii-Kosterlitz-Thouless (BKT) behavior of the gap at the transition. The figure is adapted from Fig 1.5 in Ref. [28].

Figure 2

Pictorial representation of the dimerized chain: a spin 1/2 is located on each black square, the strength of the coupling (between nearest-neighbor spins only) is alternated with values $J_s=J+\delta J$ and $J_w=J-\delta J$.

Figure 3

Sketch of the energy spectrum of excitations for the dimerized chain under the application of a magnetic field $h$ in the limit of large dimerization $\left(J\approx \delta J\right)$. In this limit, the nature of excitations can be approximated by considering the state of two spins 1/2 on a strong bond. At $h=0$ there is an energy gap of the order of $J_s$ between the singlet and the three triplet states. The magnetic field $h$ splits the triplets and brings down the excitation energy of the state $\left|t^{+}\right.〉$. Due to the presence of the weak bonds the triplets can be delocalized and, thus, have a dispersion in energy of the order of $J_w$ (the boundaries of which are represented by the dotted lines). At sufficiently high magnetic field, the energy of the lowest triplon band is close to the energy of the singlet state. This leads in the extended system to a quantum critical phase for $h>h_{c1}$ with gapless excitations. This phase exists up to the point $h_{c2}$ for which the triplon band is totally filled and a fully polarized phase arises. Picture adapted from Fig. 2 in Ref. [12].

Figure 4

Representation of the scheme used for the computation of dynamical correlations at finite temperature. The initial state at finite temperature is prepared via imaginary time evolution (a). A copy is created. The operator $\widehat{A}$ is applied on this copy (b) and then a double real-time evolution (c) is performed. At each time step the second operator is measured by sandwiching it through the two resulting states (d). This gives us the desired correlation. The figure is adapted from Fig. 2 in Ref. [23] with $\hslash =1$.

Figure 5

Phase diagram for the Heisenberg model $\left(\mathrm{\Delta }=1\right)$ as a function of the magnetic field $h$. For low magnetic field, the ground state is a TLL and the phase is gapless. In contrast, above the critical magnetic field $h_c=2J$ a ferromagnetic phase, which exhibits a gap in its excitation spectrum, arises.

Figure 6

${(1/T_1)}_{+-}$ as defined in Eq. (9), multiplied by $J/\hslash$ to have a dimensionless quantity, as a function of $k_{B}T/J$ for an XXX model under magnetic field. Dots with error bars are MPS results; solid lines are the analytical predictions and/or fits. $a=0.71\left(2\right)$ (fit parameter).

Figure 7

Magnetization per site as a function of $k_{B}T/J$ for the Heisenberg model at the critical field $h=2J$. Blue stars are MPS results (error bars are too small to be seen), red circles are the analytical predictions.

Figure 8

Logarithm of $J{(1/T_1)}_{zz}/\hslash$ (unitless quantity) from Eq. (10) as a function of $J/k_{B}T$. We considered the Heisenberg and the XX model, with $h=5J$. Blue dots with error bars are MPS results, red dots with error bars are analytical results, and solid lines are the fits according to the expected behaviors. $a=0.5\left(1\right)$ and $c=1.0\left(1\right)$ are the fit parameters.

Figure 9

${(1/T_1)}_{+-}$ as defined in Eq. (9), multiplied by $J/\hslash$ to have a dimensionless quantity, as a function of $k_{B}T/J$ for the XXZ model at different anisotropies. Dots with error bars are MPS results; solid lines are analytical predictions.

Figure 10

${(1/T_1)}_{-+}$ as defined in Eq. (9), multiplied by $J_w/\hslash$ to obtain a dimensionless quantity, plotted as a function of $k_{B}T/J_w$ for the isotropically dimerized spin-1/2 chain at $h=0$. Dots with error bars are MPS results; solid lines are combination of a fit with the analytical prediction. The fit parameter is $a\approx 4.7\left(3\right)$. Inset: logarithmic representation of the same quantities when positive.

Figure 11

${(1/T_1)}_{-+}$ as defined in Eq. (8), multiplied by $J_w/\hslash$ to obtain a dimensionless quantity, as a function of $k_{B}T/J_w$, for the isotropically dimerized spin-1/2 chain at $h\approx 3.02J_w$. Dots with error bars are MPS results; solid lines are fits or combinations of fit and analytics. The fit parameters are $a\approx 0.62\left(1\right),b\approx 0.63\left(2\right),c\approx 1.42\left(2\right),d\approx -0.84\left(1\right),g\approx 0.87\left(1\right)$, and $l\approx -0.25\left(1\right)$.

Figure 12

Integral over time from $-t_{\text{max}}$ to $+t_{\text{max}}$ of the on-site $S^{-+}$ correlations as a function of $\hslash /\left(t_{\text{max}}{J}_w\right)$ at $h\approx 3.02J_w\gtrsim h_{c1}$, at the temperature $k_{B}T\approx 0.081J_w$ for the dimerized model. The extrapolation is shown as a solid (red) line. The extrapolated point is reported with its error bar. Inset: the correlations as a function of $t{J}_w/\hslash$.

Figure 13

Integral over time from $-t_{\text{max}}$ to $+t_{\text{max}}$ of the on-site $S^{-+}$ correlations as a function of $\hslash /\left(t_{\text{max}}{J}_w\right)$ at $h=0$, at the temperature $k_{B}T\approx 0.081J_w$ for the dimerized model. The extrapolation is shown as a solid (red) line. The extrapolated point is reported with its error bar. Inset: the correlations as a function of $t{J}_w/\hslash$.

Figure 14

${(1/T_1)}_{zz}$ as defined in Eq. (B2) (multiplied by $J/\hslash )$ as a function of $k_{B}T/J$ for the XX model under a magnetic field $h=0$ and $h=5J$. The analytical results correspond to infinite system size and $t_{\text{max}}=20\hslash /J$. The numerical results are obtained with $t_{\text{max}}=20\hslash /J$ and $L=100$. The agreement found is excellent.