$z=2$ Quantum Critical Dynamics in a Spin Ladder

By means of inelastic neutron scattering we investigate finite temperature dynamics in the quantum spin ladder compound $({\mathrm{C}}_5{\mathrm{H}}_{12}\mathrm{N}{)}_2{\mathrm{CuBr}}_4$ (BPCB) near the magnetic field induced quantum critical point with dynamical exponent $z=2$. We observe universal finite-temperature scaling of the transverse local dynamic structure factor in spectacular quantitative agreement with long-standing theoretical predictions. At the same time, already at rather low temperatures, we observe strong nonuniversal longitudinal fluctuations. To separate the two, we make use of an intrinsic leg-exchange symmetry of the spin ladder. Complementary measurements of specific heat also reveal striking scaling behavior near the quantum critical point.

Figure 1

Schematic of the crystal structure of BPCB showing the ladders formed by the magnetic ${\mathrm{Cu}}^{2+}$ cations and linking ${\mathrm{Br}}^{-}$ anions. Hydrogen atoms of the organic ligands are not shown. The vectors ${\mathbf{d}}_{1,2}$ define the rung orientation of the two inequivalent ladders in each unit cell.

Figure 2

(a) Representative constant-field measurements of the magnetic contribution to specific heat in BPCB for a magnetic field along the $b$ axis. (b) Symbols: measured magnetic specific heat for $0.17\le T\le 0.5$ plotted in scaled variables with the scaling exponents $a=1/2$ and $z\nu =1$. The solid line is the exact scaling function for free fermions in one dimension plotted with no adjustable parameters. Inset: False color plot of the empirical measure of the quality of the data collapse vs the critical exponents as described in the text. Optimal scaling is found for $a=0.57(10)$ and $b=1.01(10)$.

Figure 3

Inelastic neutron scattering spectra as measured in the spin ladder compound BPCB near the critical field $H_c=6.66(6)\mathrm{T}$. In the plotted range of energy transfer, the lower two of the three Zeeman-split triplet branches are visible. (a)–(d) Low temperature ($T=0.35\mathrm{K}$) evolution of the spectrum across the critical field. (b), (e), (f) Temperature evolution of the spectrum at $H=6.5\mathrm{T}$. (g) Antisymmetric rung structure factor $s^{-}(\mathbf{Q})$ in the covered range of reciprocal space. (h) Low energy part of the $H=6.75\mathrm{T}$ spectrum. (i), (j) Antisymmetric and symmetric structure factor contributions determined for the same data set as described in the text.

Figure 4

Scaling plot of the antisymmetric transverse local dynamic structure factor component ${\mathcal{S}}_{-}^{\perp }(\omega )$ near the critical field at $H=6.5\mathrm{T}$. For the predicted scaling exponent $b=1/2$ all three data sets collapse onto a single continuous line. The inset shows the quality of the data collapse for different values of the critical exponent. Best scaling is found for $b=0.57(10)$. The solid line corresponds to the calculated scaling function.