Dynamical response and dimensional crossover for spatially anisotropic antiferromagnets

Theoretically challenging, the understanding of the dynamical response in quantum antiferromagnets is of great interest, in particular for both inelastic neutron scattering (INS) and nuclear magnetic resonance (NMR) experiments. In such a context, we theoretically address this question for quasi-one-dimensional quantum magnets, e.g., weakly coupled spin chains for which many compounds are available in nature. In this class of systems, the dimensional crossover between a three-dimensional ordered regime at low temperature towards one-dimensional physics at higher temperature is a nontrivial issue, notably difficult concerning dynamical properties. Here we present a comprehensive theoretical study based on both analytical calculations (bosonization + random phase and self-consistent harmonic approximations) and numerical simulations (quantum Monte Carlo + stochastic analytic continuation) which allows us to describe the full temperature crossover for the NMR relaxation rate $1/T_1$, from one-dimensional Tomonaga-Luttinger liquid physics to the three-dimensional ordered regime, as a function of interchain couplings. The dynamical structure factor, directly probing the INS intensity, is also computed in the different regimes.

Figure 1

Different temperature regimes and crossovers for the transverse component of the NMR relaxation rate $1/T_1^{\perp }$, as defined in Eq. (2.14) for an anisotropic three-dimensional antiferromagnet made of weakly coupled chains with an ordering temperature $T_c$. The coupling strengths are $J$ along the chain direction and $J_{\perp }$ in the transverse direction, see Eq. (2.1). (a) Deep in the ordered phase, the NMR relaxation rate increases linearly $\propto T$ from the absolute zero temperature due to spin-waves contributions. (b) Right below the critical temperature $T_c$, the NMR relaxation rate goes through a strong algebraic suppression $\propto T^{\alpha }$ ($\alpha \simeq 4-5$) due to its “$\left[{\mathbf{q}}_{\mathrm{AF}}\right]$-component suppression.” The change of behavior from (b) to (a) sets a first crossover temperature. (c) When approaching the transition from above the critical temperature, the NMR relaxation rate diverges with critical exponents $\nu , \eta$, and $z_t$ characterizing the universality class of the transition, i.e., $\propto |T-T_c{|}^{-\nu (z_t-1-\eta )}$. The divergence associated to the transition is observed up to approximately $\simeq 3T_c$. (d) For $J_{\perp }/J\ll 1$ we can expect a crossover towards one-dimensional physics with a diverging NMR relaxation rate $\propto T^{1/2K-1}$ where $K$ is the Tomonaga-Luttinger liquid parameter. (e) At high temperature, larger than $\sim J/10$, the $1/T_1^{\perp }$ behavior is nonuniversal. Note that if $3T_c\gtrsim J/10$, the region (d) of the diagram is squashed, and no universal TLL physics is present in the system, at least regarding the NMR relaxation rate.

Figure 2

(a) Three-dimensional tetragonal lattice with the spatial $\mathbf{a}$ direction nonequivalent to $\mathbf{b}$ and $\mathbf{c}$. The $\mathrm{spin}-1/2$ degrees of freedom live on the vertices. The Brillouin zone and irreducible Brillouin zone (red region) are shown in panels (b) with wave vectors $\mathbf{q}=(q_a,q_b,q_c)$. The vertices $Z=(\pi ,0,0), R=(\pi ,0,\pi ), A\equiv {\mathbf{q}}_{\mathrm{AF}}=(\pi ,\pi ,\pi ), \mathrm{\Gamma }=(0,0,0), X=(0,0,\pi )$, and $M=(0,\pi ,\pi )$ are high-symmetry points of the Brillouin zone.

Figure 3

The lower panels show the transverse inelastic neutron scattering intensity $I^{\perp }$ for weakly coupled spin chains in three dimensions with $J_{\perp }/J=0.001$ and an Ising anisotropy $\mathrm{\Delta }=0.5$ along the spatial $\mathbf{a}$ direction. The $\mathbf{q}$ points follow the high symmetry lines of the BZ of Fig. 2. The temperature of the system is $T=0.1J$, such that we are in the universal one-dimensional regime with $3T_c\lesssim T\lesssim 0.1J$, making the BZ lines $\mathrm{\Gamma }Z, XR$, and $MA$ equivalent and corresponding all to the single chain spectrum. The white dot symbols show the first moment of the spectrum and the plus symbols the position of the maximum of intensity at a given $\mathbf{q}$ point. We also show the two sine branches of the des Cloizeaux-Pearson dispersion relations in Eq. (3.4) where the prefactor of the lower one corresponds to the TLL velocity $u\simeq 1.299J$ of a single chain with $\mathrm{\Delta }=0.5$. Note that the critical temperature for this system is $T_c/J\simeq 0.007$. The upper panels correspond to the static structure factor. The data are from quantum Monte Carlo simulations on the largest available system of size $N=96\times 8\times 8=6144$ spins.

Figure 4

Transverse component of the NMR relaxation rate $1/T_1^{\perp }$ defined in Eq. (2.13) for weakly coupled chains in $3\mathrm{D}$ with an Ising anisotropy $\mathrm{\Delta }=0.5$ along the spatial direction $\mathbf{a}$, for various transverse couplings between the chains $J_{\perp }/J=0.1$ (blue), 0.01 (green), and 0.001 (red). The temperature axis has been rescaled by the critical temperature $T_c$ of each model, respectively, $T_c/J\simeq 0.224$, 0.04, and 0.007. The bold straight lines correspond to the purely one-dimensional TLL prediction (2.21), the small diamonds to the three-dimensional mean-field (RPA) calculations (3.2), and the circles to quantum Monte Carlo simulations. In the latter case, the largest system available is considered for each $J_{\perp }/J$ value: $N=96\times 12\times 12, 96\times 8\times 8$, and $96\times 8\times 8$, for $J_{\perp }/J=0.1$, 0.01, and 0.001, respectively. It shows that if any, the universal one-dimensional regime of the NMR relaxation rate is visible for $T\gtrsim 3T_c$.

Figure 5

(a) Rescaled transverse dynamical spin structure factor at the AF wave vector in the limit $\omega \to 0$ for different system sizes $N=L^3/8^2$ made of weakly coupled chains with $J_{\perp }/J=0.1$ and an Ising anisotropy $\mathrm{\Delta }=0.5$ along the spatial direction $\mathbf{a}$. The transverse dynamical spin structure factor has been rescaled according to the scaling law (3.8). The anomalous critical exponent takes the value of the $3\mathrm{D}$ XY universality class $\eta =0.0381$ [84], and $z_t=1.62$ [82] was considered. The crossing point for the different sizes using these exponents is close to the expected critical temperature $T_c/J\simeq 0.224$ for this system. (b) Transverse component of the NMR relaxation of Eq. (3.2) from RPA calculations versus $|T-T_c|$ with $T>T_c$. A divergence with the mean-field exponents $\nu =0.5$ and $\eta =0$ as well as $z_t=2$ is observed as $T\to T_c$ according to Eq. (3.7).

Figure 6

Transverse component of the NMR relaxation rate $1/T_1^{\perp }$ Eq. (3.10) where the ${\mathbf{q}}_{\mathrm{AF}}$ component has been removed (see discussions in main text) for weakly coupled chains in $3\mathrm{D}$ with an Ising anisotropy $\mathrm{\Delta }=0.5$ along the spatial direction $\mathbf{a}$. It has been computed numerically using QMC+SAC on different system sizes $N=L^3/8^2$ with an interchain coupling $J_{\perp }/J=0.1$, leading to a critical temperature $T_c/J\simeq 0.224$ (vertical dotted line). In panel (a) and its inset (b), the stochastic analytic continuation has been performed independently on all $\mathbf{q}$ components of $S_{\mathbf{q}}\left({\omega }_0\right)$ and summed thereafter to obtain the NMR relaxation rate. In panels (c) and (d), the sum over $\mathbf{q}$ of the imaginary time QMC data is performed before doing the analytic continuation. The inset (b) shows the relative weight of the $\mathbf{q}\ne {\mathbf{q}}_{\mathrm{AF}}$ components in the $1/T_1$ as a function of the temperature with $W_{{\mathbf{q}}_{\mathrm{AF}}}$ defined in Eq. (3.9). The inset (d) is the same as (c) in log-log scale where the power-law compatible with $\propto T^4$ can be observed. At lower temperature, the spin-waves contribution of the $1/T_1\propto T$ is plotted with the prefactor computed by SCHA in Eq. (3.16) (with no free parameter).

Figure 7

Lower panels: transverse inelastic neutron scattering intensity $I^{\perp }$ for weakly coupled spin chains in three dimensions with $J_{\perp }/J=0.1$ and an Ising anisotropy $\mathrm{\Delta }=0.5$ along the spatial $\mathbf{a}$ direction. The $\mathbf{q}$ points follow the high symmetry lines of the BZ of Fig. 2, focusing on regions where the spectral weight is the more significant. The temperature of the system is $T=0.1J$, below the critical temperature $T_c/J\simeq 0.224$, explaining why all the spectral weight is located at the AF wave vector. The white dot symbols show the first moment of the spectrum. The straight white line is the spin-waves dispersion relation ${\omega }_{\mathrm{sw}}\left(\mathbf{q}\right)$ with a gapless mode at the AF wave vector. The plus symbols correspond to the maximum of intensity in the spectrum at a given $\mathbf{q}$ point. The dotted white lines around the AF wave vector show the linear dispersion relation around ${\mathbf{q}}_{\mathrm{AF}}$ of the SW spectrum but with corrected (hydrodynamic) velocities, compared to the bare spin-waves ones, see text. For visibility, the color intensity has been saturated to 0.0005. The upper panels correspond to the static structure factor whose value at $A\equiv {\mathbf{q}}_{\mathrm{AF}}$ is the modulus square of the order parameter which clearly develops for $T<T_c$ (a careful finite-size scaling analysis would need to be performed in order to obtain the order parameter value in the thermodynamic limit $N\to \infty$). The data are from quantum Monte Carlo simulations on the largest available system of size $N=96\times 12\times 12=13824$ spins.