Universal thermodynamics of the one-dimensional attractive Hubbard model

The one-dimensional (1D) Hubbard model, describing electrons on a lattice with an on-site repulsive interaction, provides a paradigm for the physics of quantum many-body phenomena. Here, by solving the thermodynamic Bethe ansatz equations, we study the universal thermodynamics, quantum criticality, and magnetism of the 1D attractive Hubbard model. We show that the compressibility and the susceptibility of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state obey simple additivity rules at low temperatures, indicating an existence of two free quantum fluids. The magnetic properties, such as magnetization and susceptibility, reveal three physical regions: quantum fluids at low temperatures, a non-Fermi liquid at high temperatures, and the quantum fluid to non-Fermi liquid crossover in between. The lattice interaction is seen to significantly influence the nature of the FFLO-like state in 1D. Furthermore, we show that the dimensionless Wilson ratio provides an ideal parameter to map out the various phase boundaries and to characterize the two free fluids of the FLLO-like state. The quantum scaling functions for the thermal and magnetic properties yield the same dynamic critical exponent $z=2$ and correlation critical exponent $\nu =1/2$ in the quantum critical region whenever a phase transition occurs. Our results provide a rigorous understanding of quantum criticality and free fluids of many-body systems on a 1D lattice.

Figure 1

A schematic configuration of the $k\text{−}\mathrm{\Lambda }$ strings of length $1,2,\text{and}3$. The $k\text{−}\mathrm{\Lambda }$ bound states are formed by the charge momenta and spin rapidities displayed within the dashed boundaries. In each $k\text{−}\mathrm{\Lambda }$ bound state, $sink$'s share a real part with the spin rapidities. A length-$mk\text{−}\mathrm{\Lambda }$ string contains $m$ rapidities in $\mathrm{\Lambda }$ space and $2m$ quasimomenta in $k$ space. In contrast to the two component Fermi gas, many electrons on a 1D lattice are allowed to form a bound state of multiparticles.

Figure 2

Ground-state phase diagram of the 1D attractive Hubbard model with $\left|u\right|=1$ in the $\mu$-B plane. In the phase diagram, the critical fields are ${\mu }_c=2\left|u\right|-2\sqrt{1+u^2}<0$, $B_{c1}=2\left|u\right|-2+2{\int }_{-\infty }^{\infty }d\omega \frac{J_1\left(\omega \right)exp(-|u\left|\omega \right)}{wcosh\left(u\omega \right)}$, and $B_{c2}=2+2\left|u\right|$. The different phases are denoted by (I) vacuum, (II) fully polarized state, (III) half-filling state, (IV) partially polarized state, and (V) fully paired state. The phase boundaries are defined by Eqs. (25, 26, 27, 28, 29, 30, 31, 32, 33). For comparison, the low-temperature phase diagram is given in Fig. 9.

Figure 3

A comparison between the analytic results (42) and (43) and the numerical results obtained from the TBA equations (17, 18, 19). We set up natural units in the plots. The upper and the lower panels respectively show the density and susceptibility vs magnetic field across phases V, IV, II, and III at a fixed chemical potential $\mu =-0.08$, temperature $T={10}^{-4}$ and interaction strength $u=-10$. The sudden changes in the density and susceptibility show subtle scaling behavior near phase transitions.

Figure 4

The lattice interacting parameters for the length-$1k-\mathrm{\Lambda }$ strings as a function of the interaction strength $u$. The parameter ${\alpha }_1$ strongly affects the band dispersion of bound pairs. The parameter ${\beta }_1$ presents a lattice contribution to the free energy of the pairs.

Figure 5

Contour plot entropy vs magnetic field $B$ for the 1D attractive Hubbard model. The numerical calculation is performed by solving the TBA equations (17, 18, 19) with a fixed chemical potential $\mu =-0.828$ and interaction $u=-1$. The crossover temperatures (white dashed lines) fanning out from the critical points separate different TLL phases from the quantum critical regimes. The linear temperature-dependent entropy breaks down when the temperature is greater than these crossover temperatures. Here, ${\mathrm{TLL}}_{\mathrm{u}}$ and ${\mathrm{TLL}}_{\mathrm{b}}$, respectively, stand for the TLLs of unpaired fermions and bound pairs. ${\mathrm{TLL}}_{\mathrm{m}}$ stands for the two-component TLL of the FFLO-like state.

Figure 6

Scaling laws for thermodynamic quantities vs chemical potential at different temperatures. The intersection points in (a), (b), (c), and (d) give the critical points for phase transitions (I-II), (II-III), (II-IV), and (I-V), respectively.

Figure 7

Additivity rules: (a) compressibility $\kappa$ and (b) spin susceptibility $\chi$ vs magnetic field $B$ for the attractive Hubbard model with $u=-1$ and $\mu =-0.8282$. The red dashed lines show the result obtained from the additivity rules (92) and (93). At low temperatures, all compressibility and susceptibility curves collapse into the zero temperature ones obeying the additivity rules. In the vicinity of the critical points, such free fluids nature beaks down.

Figure 8

Numerical results for the magnetization vs logarithm of the temperature for different magnetic fields. Here we have set a fixed chemical potential $\mu =-0.14$ and interaction strength $u=-7$. For magnetic field $B>B_c=12.11065$ (phase IV), three regions are clearly displayed: The free fluids region at low temperatures, non-Fermi liquid region at higher temperatures, and a crossover in between. For magnetic field $B<B_c$ (phase V), the magnetization displays the gapped nature of a non-Fermi liquid phase.

Figure 9

Finite temperature phase diagrams obtained from contour plots of the Wilson ratios. The plots in (a) and (b) are determined from the susceptibility Wilson ratio $R_{\mathrm{W}}^{\chi }$ and from the compressibility Wilson ratio $R_{\mathrm{W}}^{\kappa }$, respectively. Here, $u=-1$ and $T=0.001$. The red balls and green balls represent up spin and down spin, respectively. Both diagrams agree well with the zero temperature phase diagram Fig. 2, despite the fact that the plot in (a) cannot distinguish phase I and V due to the vanishing susceptibility in these two phases.