In 1D, there is a class of quantum phase called Tomonaga-Luttinger liquid (TLL). The particular feature of this liquid is that its basic property is determined solely by the acoustic wave or a bosonic low energy excitation of many-body type. The thermodynamic properties are characterized a velocity v of this acoustic wave and a Tomonaga-Luttinger parameter K, which represents the effective interactions between the bosons. This holds regardless of the details of the system, what the degrees of freedom are (e.g. for electrons we consider two component TLL, and for magnets the single-component TLL). For example, if K exceeds some critical value, there is a phase transition breaking the discrete symmetry of the system. In the TLL, all the correlation functions behave critical and their exponents are determined by K's.

Now since this TLL does not break the symmetry of the system, there should be a crossover from the high temperature disordered/para phase to the TLL? How could we detect it? In our paper [Maeda-Hotta-Oshikawa, PRL

[Hong, Kim, Hotta, Takano, Tremelling, Landee, Kang, Christensen, Schmidt, Lefmann, Uhrig, Broholm, PRL

[Ninios, Hong, Manabe, Hotta, Herringer, Turnbull, Landee, Takano, Chan, PRL

Unfortunately, there is not much examples of ideal 1D magnets in reality. One example is a spin ladder material DIMPY. Yasu Takano and collaborators worked with us to measure the TLL parameters in this material.

Part 1 :[Tao Hong et. al] the spin gap is evaluated from the specific heat measurement and from the neutron scattering experiment, and combined with DMRGm, we determined the model parameter J. Part2 [Ninios, et al.]; Based on the value of J determined experimentally, we calculated the TLL parameter K. Along with that, we found a universal relation between the so-called Wilson ratio R

Going back to our TLL, we also have a T-linear specific heat and a constant χ, meaning that R

We evaluated R

Along with that, we compared the magnetization as functions of temperature in the experiment and from QMC calculation(by Manabe, our master student), and derived a phase diagram in a magnetic field by making use of our previous theory.

We finally briefly introduce the work by Yohei Kohno in ISSP(phD thesis)
on the ideal spin-1/2 Heisenberg chain material, CuPZN
[Kono, Sakakibara, et.al. PRL **114** 037202 (2015)] .

Surprizingly, the magnetization of this material shows almost exact coincidence with the theoretical exact solution we calculated by using a quantum transfer matrix method (red circles versus square). The critical phenomena near the saturation field are also very clear. The work demonstrates the critical phenomena of the 1D quantum Heisenberg model for the first time in laboratories.

Surprizingly, the magnetization of this material shows almost exact coincidence with the theoretical exact solution we calculated by using a quantum transfer matrix method (red circles versus square). The critical phenomena near the saturation field are also very clear. The work demonstrates the critical phenomena of the 1D quantum Heisenberg model for the first time in laboratories.