The phase diagram in temperature and magnetic field of the metal-organic, two-leg, spin-ladder compound (C5H12N)2CuBr4 is studied by measurements of the specific heat and the magnetocaloric effect. We demonstrate the presence of an extended spin Luttinger-liquid phase between two fieldinduced quantum critical points and over a broad range of temperature. Based on an ideal spinladder Hamiltonian, comprehensive numerical modelling of the ladder specific heat yields excellent quantitative agreement with the experimental data across the complete phase diagram. Quantum spin systems display a remarkable diversity of fascinating physical behavior. This is especially true for systems such as spin ladders, which have a gapped or a gapless ground state, respectively, for an even or an odd number of ladder legs [1]. For two-leg ladders, and in general for any even leg number, quantum phase transitions (QPTs) between gapped and gapless phases can be driven by an external magnetic field. While these QPTs are generic in quantum magnets [2], the nature of the gapless phase depends crucially on the dimensionality of the spin system. In two and higher dimensions, a quantum critical point (QCP) separates the low-field, quantum disordered (QD) phase, with gapped triplet excitations, from a gapless phase with long-range antiferromagnetic (AF) order, which can be well described as a Bose-Einstein Condensate (BEC) of magnons [2,3,4].By contrast, for one-dimensional (1D) systems such as ladders, both long-ranged magnetic order and BEC are precluded due to phase fluctuations. In addition, spin excitations are best viewed as interacting fermions, whereas a bosonic representation pertains in higher dimensions. The physics of the gapless phase in 1D is thus quite different. It is a (spin) Luttinger liquid (LL) [5], and is a key component of the rich phase diagram presented in Fig. 1 [3 , 6, 7, 8, 9]. In the LL, the spectrum is gapless with algebraically decaying spin correlations. Because there is no finite order parameter, the LL regime is reached from the high-temperature, classical regime through a crossover rather than a phase transition. Nevertheless, clear manifestations of LL behavior are expected not only in the correlation functions but also in thermodynamic quantities such as the magnetization and specific heat. and Bs (spin system fully polarized). The contour plot shows the magnetic specific heat as Cm(T, B)/T . Local maxima from the reduction of the triplet gap by the Zeeman effect are indicated by crosses. Circles denote the LL crossover based on measurements of the magnetocaloric effect [ Fig. 4], black lines are fits to extract the critical fields, and the dashed blue line indicates the onset of long-ranged order below 100 mK [21,22]. Inset: lattice structure of (Hpip)2CuBr4 in projection along the b-axis, with Cu atoms blue and Br white.However, materials in which to explore such effects are rather rare. Investigations of the spin excitations and arXiv:0808.2715v2 [cond-mat.str-el]
