Statics and dynamics of weakly coupled antiferromagnetic spin-$\frac{1}{2}$ ladders in a magnetic field

We investigate weakly coupled spin-$1/2$ ladders in a magnetic field. The work is motivated by recent experiments on the compound ${(}_{\mathrm{C}}{\mathrm{H}}_{12}{\mathrm{N}}_{)}{\mathrm{CuBr}}_4$ (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 $t$-$J$ 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.

Figure 1

Single ladder structure: $J_{\perp }$ ($J_{\parallel }$) is the coupling along the rungs (legs) represented by thick (thin) dashed lines and ${\mathbf{S}}_{l,k}$ are the spin operators acting on the site $l$ of the leg $k=1,2$.

Figure 2

(a) Energy of the triplets $|t^{+}\rangle$, $|t^0\rangle$, $|t^{-}\rangle$ (solid lines) and singlet $|s\rangle$ (dashed line) vs the applied magnetic field in the absence of an interrung coupling ($J_{\parallel }=0$). The dotted lines represent the limits of the triplet excitation band when $J_{\parallel }\ne 0$. (b) Phase diagram of weakly coupled spin ladders: Crossovers (dotted lines) and the phase transition (solid line) that only exists in the presence of an interladder coupling are sketched.

Figure 3

Dependence of the magnetization per rung $m^z$ on the magnetic field $h^z$ at zero temperature for the single ladder with the BPCB couplings (see Sec. 6a) (dashed red line), the spin chain mapping (dotted blue line) rescaled to fit with the single ladder critical fields (dashed-dotted blue line), and for the weakly coupled ladders treated by the mean-field approximation (solid black line). The insets emphasize the different behavior of the magnetization curves for the single (dashed red line) and coupled (solid black line) ladders close to the critical fields, which are indistinguishable in the main part of the figure. The dotted lines in the insets correspond to the linear and square-root-like critical behavior.

Figure 4

Rung state density vs the applied magnetic field $h^z$ at zero temperature for the single ladder with the BPCB couplings (zero-temperature DMRG calculations averaging on the central sites of the ladder). The dashed-dotted (black) lines correspond to the singlet density $\langle {\rho }_s\rangle$. The triplet densities are represented by the solid (red) lines for $\langle {\rho }_{+}\rangle$, the dashed (blue) lines for $\langle {\rho }_0\rangle$, and the dotted (green) lines for $\langle {\rho }_{-}\rangle$.

Figure 5

Temperature dependence of the magnetization per rung, $m^z(T)$, for (a) the ladder with the BPCB couplings (39) and (b) the spin chain mapping at different applied magnetic fields $h^z=9\hspace{0.5em}\mathrm{T}$ (solid blue lines), $h^z=10\hspace{0.5em}\mathrm{T}$ (dashed-dotted green lines), and $h^z=11\hspace{0.5em}\mathrm{T}$ (dashed red lines). The results were obtained using $t$-DMRG. The stars at $T=0\hspace{0.5em}\mathrm{K}$ are the ground-state magnetization per rung determined by zero-temperature DMRG. The triangles (squares) mark the low- (high-) energy extrema. (c) Crossover temperature $T_{\mathrm{LL}}$ of the LL to the quantum critical regime vs the applied magnetic field [blue dots for the extremum in $m^z(T)|{}_{h^z}$ criterium and red crosses for the maximum in $c(T)|{}_{h^z}$ criterium].

Figure 6

Specific heat per rung $c$ vs the temperature $T$ for different applied magnetic fields in the gapless regime. (a) Full ladder $t$-DMRG calculations with the BPCB couplings (39) for $h^z=9$ T (solid blue line), $h^z=10$ T (dashed-dotted green line), and $h^z=11$ T (dashed red line). Spin chain mapping at $h^z=10$ T solved by DMRG (dashed black line) and by BA (dotted black line). Note that the two lines are hardly distinguishable. The triangles (squares) mark the low- (high-) energy maxima of the specific heat vs temperature. The vertical dashed line marks the temperature $T=0.4$ K below which the DMRG results are extrapolated (see Appendix 2). (b) Comparison between measurements on the compound BPCB from Ref. 29 (dots) and the $t$-DMRG calculations[67] (solid lines) at (b1) $h^z=$9 T, (b2) $h^z=10$ T, and (b3) $h^z=11$ T.

Figure 7

Low-temperature dependence of the specific heat per rung vs the temperature, $c(T)$, for (a) $h^z=9$ T, (b) $h^z=10$ T, and (c) $h^z=11$ T. The $t$-DMRG calculations are shown as red thick lines for the ladder with the BPCB couplings (39) (black thick lines for the spin chain mapping). The two curves are hardly distinguishable. Their low-temperature polynomial extrapolation is plotted in thin lines below $T=0.4$ K (represented by a vertical dashed line). The linear low-temperature behavior of the LL is represented by dashed lines (red for the ladder, black for the spin chain mapping). The dashed yellow lines correspond to the BA computation for the spin chain mapping.

Figure 8

Magnetic-field dependence of the NMR relaxation rate, $T_1^{-1}(h^z)$, at $T=250$ mK. The solid red line is the bosonization determination using the ladder LL parameters for the BPCB couplings (the dashed blue line uses the LL parameters of the spin chain mapping). The black circles are the measurements from Ref. 30.

Figure 9

Magnetic-field dependence of the transition temperature between the gapless regime and the 3D-ordered phase, $T_c(h^z)$, is plotted as a solid red line for the ladder LL parameters of BPCB (dashed blue line for the LL parameters of the spin chain mapping). The NMR measurements from Ref. 30 are represented by black circles and the ND measurements from Ref. 31 are represented by green dots.

Figure 10

Magnetic-field dependence of the staggered magnetization per spin, $m_a^x(h^z)$, in the 3D-ordered phase determined for the ground state. (a) Its computation with the analytical bosonization technique for the LL parameters of the compound BPCB and $J^{\prime }=27\hspace{0.5em}\mathrm{mK}$ is represented by the dashed-dotted red line (dashed blue line for the LL parameters of the spin chain mapping). The DMRG result for $J^{\prime }=20\hspace{0.5em}\mathrm{mK}$($=$$J_{\mathrm{MF}}^{\prime })$ is represented by black dots (as a comparison, the bosonization result for $J^{\prime }=20\hspace{0.5em}\mathrm{mK}$ is plotted as a solid red line). Note that these two data are almost indistinguishable. (b) Comparison between NMR measurements (black circles) done at $T=40\hspace{0.5em}\mathrm{mK}$ from Ref. 30 and scaled on the theoretical results for $J^{\prime }=27\hspace{0.5em}\mathrm{mK}$, ND measurements on an absolute scale from Ref. 31 at $T=54\hspace{0.5em}\mathrm{mK}$ ($T=75\hspace{0.5em}\mathrm{mK}$) [red crosses (black dots)], and the computation for $J^{\prime }=27\hspace{0.5em}\mathrm{mK}$ [blue line]. Recent ND measurements as a function of temperature suggest that the data of Ref. 31 was taken at temperatures $~$10 mK higher than the nominal indicated temperature.

Figure 11

Momentum-energy-dependent correlation functions at $h^z=0$. (a) Symmetric part $S_0^{\alpha \alpha }(q,\omega )$ with $\alpha =x,y,z$. The dashed (black) line marks the $(q,\omega )$ position of the two-triplet bound state [Eq. (61b) in Ref. 14]. The dashed-dotted (white) lines correspond to the boundaries of the two-triplet continuum. (b) Antisymmetric part $S_{\pi }^{\alpha \alpha }(q,\omega )$. The dashed (black) line corresponds to the predicted dispersion relation of a single-triplet excitation [Eq. (8) in Ref. 13].

Figure 12

Momentum-energy-dependent $\mathit{zz}$-correl-ation function (1) at $m^z=0.25$ ($h^z=3.153J_{\parallel }$), (2) at $m^z=0.5$ ($h^z=4.194J_{\parallel }$), and (3) at $m^z=0.75$ ($h^z=5.192J_{\parallel }$). (a) Symmetric part $S_0^{\mathit{zz}}(q,\omega )$ without Bragg peak at $q=0$. The dashed black lines correspond to the location of the slow divergence at the lower edge of the continuum predicted by the LL theory. The dashed white curve corresponds to the predicted two-triplet bound-state location. (b) Antisymmetric part $S_{\pi }^{\mathit{zz}}(q,\omega )$. The dashed black lines correspond to the position of the high-energy divergences or cusps predicted by the approximate mapping on the $t$-$J$ model. The vertical white dashed-dotted lines mark the momenta of the minimum energy of the high-energy continuum and the black cross is the energy of its lower edge (Ref. 20) at $q=\pi$.

Figure 13

Momentum-energy-dependent $+-$ correlation function (1) at $m^z=0.25$ ($h^z=3.153J_{\parallel }$), (2) at $m^z=0.5$ ($h^z=4.194J_{\parallel }$), and (3) at $m^z=0.75$ ($h^z=5.192J_{\parallel }$). (a) Symmetric part $S_0^{+-}(q,\omega )$. The vertical dashed-dotted white lines mark the momenta of the minimum energy of the high energy continuum and the horizontal ones the frequency of its lower edge (Ref. 20) at $q=0,2\pi$. The dashed white lines correspond to the position of the high energy divergences or cusps predicted by the approximate mapping on the $t$-$J$ model. The dotted white curve corresponds to the predicted two-triplet bound-state location. The high-energy excitations at $\omega =3h_z$ are hardly visible. (b) Antisymmetric part $S_{\pi }^{+-}(q,\omega )$. The dashed and dashed-dotted (dotted) white lines correspond to the location of the strong divergence (cusp) at the lower edge of the continuum predicted by the LL theory.

Figure 14

Momentum-energy-dependent $-+$ correlation function (1) at $m^z=0.25$ ($h^z=3.153J_{\parallel }$), (2) at $m^z=0.5$ ($h^z=4.194J_{\parallel }$), and (3) at $m^z=0.75$ ($h^z=5.192J_{\parallel }$). (a) Symmetric part $S_0^{-+}(q,\omega )$. The vertical dashed-dotted white lines correspond to the momenta at which the minimum energy of the high energy continuum occurs and the horizontal line to the frequency of its lower edge (Ref. 20) at $q=0,2\pi$. The dashed black curve corresponds to the predicted two-triplet bound-state location. (b) Antisymmetric part $S_{\pi }^{-+}(q,\omega )$. The dashed and dashed-dotted (dotted) white lines correspond to the location of the strong divergence (cusp) at the lower edge of the continuum predicted by the LL theory.

Figure 15

Fermionic picture for the effect of the magnetic field. Filling of (a) the singlet band $|s\rangle$ and (b) the triplet band $|t^{+}\rangle$ in the gapless phase for a given magnetization $m^z$.

Figure 16

Cuts at fixed momentum $q=\pi$ and magnetization $m^z=0.5$ of the low-energy spectrum (a) $S_0^{\mathit{zz}}(q=\pi ,\omega )$ and (b) $S_{\pi }^{-+}(q=\pi ,\omega )$. The (red) circles and the (black) squares are the numerical results for the ladder and its spin chain mapping, respectively. The dashed lines correspond to the LL predictions, and the solid lines are the latter convolved with the same Gaussian filter than the numerical data. The DMRG frequency numerical limitation is of the order of the peak broadening of width $\delta \omega \approx 0.1J_{\parallel }$ (see Appendix 3).

Figure 17

Momentum-energy-dependent $+-$ correlations $[S_{q_y}^{+-}(q,\omega )]$ at $m^z=0.25$ for different ladder couplings ($\gamma =J_{\parallel }/J_{\perp }$): (1) $\gamma \to \infty$, (2) $\gamma =2$, (3) $\gamma =1$, (4) $\gamma =0.5$, (5) $\gamma \to 0$. The symmetric (antisymmetric) correlations with $q_y=0$ ($q_y=\pi$) are presented in the figures labeled by a (b). In (1), (2a)–(4a) the vertical dashed lines represent the incommensurate momenta of the low-energy branches of the single spin chain at $q=\pm \pi m^z=\pm \pi /4$ [they also correspond to the predicted momenta of the lowest-energy excitations of the symmetric correlations (Ref. 20)]. The horizontal solid (dotted) horizontal lines in (2a)–(4a) correspond to the approximate energy $J_{\perp }$ ($3J_{\perp }$) of the single-triplet excitations of type (ii) (two-triplet excitations of type (iii)). The horizontal dashed lines in (2b)–(4b) correspond to the approximate energy $2J_{\perp }$ of the excitations of type (ii). The dashed-dotted lines in (1) correspond to the linear low-energy boundaries of the continuum of excitations given by the LL theory applied on single Heisenberg chains $\gamma \to \infty$. The dashed-dotted lines in (2b)–(5b) correspond to the linear low-energy boundaries of the continuum of excitations given by the LL theory on spin ladder with finite $\gamma$ (see Appendix 2).

Figure 18

Coupling structure of BPCB where the unpaired electron spins of the ${\mathrm{Cu}}^{2+}$ atoms in the first (second) type of ladders are pictured by red (blue) spheres. The $J_{\perp }$, $J_{\parallel }$, and $J^{\prime }$ coupling paths are represented in turquoise, pink, and green, respectively. $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are the three unit-cell vectors of the structure. Gray arrows are the rung vectors of the two types of ladders ${\mathbf{d}}_{1,2}$. (a) 3D structure. (b) Projection of the 3D structure onto the $\mathbf{bc}$ plane. (c) Projection of the 3D structure onto the $\mathbf{ac}$ plane. (d) Projection of the 3D structure onto the $\mathbf{ab}$ plane.

Figure 19

Theoretical momentum-energy-depen-dent INS cross section for BPCB with $\mathbf{Q}·{\mathbf{d}}_i=\pi$ ($i=1,2$) and $q=\mathbf{Q}·\mathbf{a}$ at $m^z=0.5$ in (a) a ladder system and (b) the spin chain mapping. The horizontal dashed lines correspond to the constant energy scans at $\omega =0.2$ and 0.4 meV shown in Fig. 21.

Figure 20

Theoretical momentum-energy-dependent INS cross section for BPCB with $\mathbf{Q}·{\mathbf{d}}_i=\pi$ ($i=1,2$) and $q=\mathbf{Q}·\mathbf{a}$ at (a) $m^z=0.25$, (b) $m^z=0.25,0.75$ in the spin chain mapping, and (c) $m^z=0.75$. The horizontal dashed lines correspond to the constant energy scans at $\omega =0.2$ and 0.4 meV plotted in Fig. 22.

Figure 21

INS intensity measured along ${\mathbf{a}}^{\star }$ of BPCB (Ref. 32) with the momentum $\pi$ in the rung direction ($\mathbf{Q}·{\mathbf{d}}_{\mathbf{i}}=\pi$) at $h^z=10.1\hspace{0.5em}\mathrm{T}$ ($m^z\approx 0.5$) and $T=250\hspace{0.5em}\mathrm{mK}$ after substraction of the zero-field background. In each panel, fixed energy scans (shown by white dashed lines in Fig. 19) are plotted: (a) $\omega =0.2\hspace{0.5em}\mathrm{meV}$, (b) $\omega =0.4\hspace{0.5em}\mathrm{meV}$. The circles correspond to the experimental data. The red (black) solid lines are the $m^z=0.5$ theoretical data for the ladder (the spin chain mapping) convolved with the instrumental resolution. The shaded bands indicate the error bar in the experimental determination of a single proportionality constant valid for all fields, energies, and wave vectors. The width of these areas combines the statistics of all our scans with uncertainties in the exact magnetization values at the chosen fields and in the convolution procedure.

Figure 22

INS intensity measured along ${\mathbf{a}}^{\star }$ of BPCB (Ref. 32) with a $\pi$ momentum in the rungs ($\mathbf{Q}·{\mathbf{d}}_{\mathbf{i}}=\pi$) at $T=250\hspace{0.5em}\mathrm{mK}$ after substraction of the zero-field background. In each panel, cuts at fixed energy (shown by white dashed lines in Fig. 20) are plotted: (a) $\omega =0.4\hspace{0.5em}\mathrm{meV}$ and $m^z=0.24$, (b) $\omega =0.4\hspace{0.5em}\mathrm{meV}$ and $m^z=0.72$, (c) $\omega =0.2\hspace{0.5em}\mathrm{meV}$ and $m^z=0.24$, and (d) $\omega =0.2\hspace{0.5em}\mathrm{meV}$ and $m^z=0.72$. The circles correspond to the experimental data. The solid red (black) curves are the theoretical data for the ladder (the spin chain mapping) convolved with the instrumental resolution. The shaded bands indicate the error bar in the experimental determination of a single proportionality constant valid for all fields, energies, and wave vectors. The width of these areas combines the statistics of all our scans with uncertainties in the exact magnetization values at the chosen fields and in the convolution procedure.

Figure 23

LL parameters $u$, $K$, and the prefactors of the spin operators $A_x$, $B_x$, and $A_z$ vs the magnetization per rung $m^z$ computed for a spin ladder with the BPCB couplings (39) (red crosses) and for the spin chain mapping (blue stars). The strong coupling expansion of $u$ and $K$ up to the second order in $\gamma$ (discussed in Appendix 4) is plotted as black dashed lines.

Figure 24

Map of the momentum-frequency low-energy correlations of the LL model, where the white areas represent the continuum of excitations. In the striped areas no excitations are possible. (a) $S_0^{\mathit{zz}}(q,\omega )$: The dashed-dotted lines (blue) are the excitation peaks close to $q=0,2\pi$ and the dashed lines (red) are the continuum lower boundary with the edge exponent $1-K$ close to $q=2\pi m^z,2\pi (1-m^z)$. (b) $S_{\pi }^{+-}(q,\omega )$, (c) $S_{\pi }^{-+}(q,\omega )$: The continuum lower boundary close to $q=\pi ,\pi (1\pm 2m^z)$ is represented by solid lines (black) (edge exponent $1-1/4K$), dashed lines (red) (edge exponent $1-{\eta }_{-}=2-1/4K-K$), and dashed-dotted lines (blue) (edge exponent $1-{\eta }_{+}=-1/4K-K$).

Figure 25

(a) Different exponents that appear in the LL correlation functions, Eqs. (B4) and (B5), vs the magnetization $m^z$. The solid (dashed) lines are determined from the ladder (spin chain mapping) exponent $K$ in Fig. 23. The exponent $1-K$ of the $S_0^{\mathit{zz}}$ correlations is shown in (a1), and the exponent $1-1/4K$ of the $S_{\pi }^{\pm \mp }$ correlations at the $q=\pi$ branch is shown in (a2) (lower red curves). The exponents $1-{\eta }_{-}=2-1/4K-K$ (upper black curves) in (a2) and $1-{\eta }_{+}=-1/4K-K$ in (a3) correspond to both sides of the incommensurate branches of the $S_{\pi }^{\pm \mp }$ (see Fig. 24). (b) Shift of the ground-state energy per rung vs the magnetization $\Delta E_0(m^z)=E_0(0)-E_0(m^z)$.

Figure 26

QMC simulation results for the spin stiffness scaling (upper panel), and the order parameter scaling (lower panel). The vertical dashed line denotes the coinciding estimate of $T_c$ from the crossing curves in both panels.