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Dichotomy between Attractive and Repulsive Tomonaga-Luttinger Liquids in Spin Ladders

We present two methods to determine whether the interactions in a Tomonaga-Luttinger liquid (TLL) state of a spin-$1/2$ Heisenberg antiferromagnetic ladder are attractive or repulsive. The first method combines two bulk measurements, of magnetization and specific heat, to deduce the TLL parameter that distinguishes between the attraction and repulsion. The second one is based on a local-probe, NMR measurements of the spin-lattice relaxation. For the strong-leg spin ladder compound $\mathrm{(C_7H_{10}N)_2CuBr_4}$ we find that the isothermal magnetic field dependence of the relaxation rate, $T_1^{-1}(H)$, displays a concave curve between the two critical fields that bound the TLL regime. This is in sharp contrast to the convex curve previously reported for a strong-rung ladder $\mathrm{(C_5H_{12}N)_2CuBr_4}$. Within the TLL description, we show that the concavity directly reflects the attractive interactions, while the convexity reflects the repulsive ones.