Quantum critical behavior and thermodynamics of the repulsive one-dimensional Hubbard model in a magnetic field

Even though the Hubbard model is one of the most fundamental models of highly correlated electrons, analytical and numerical data describing its thermodynamics at nonzero magnetization are relatively scarce. We present a detailed investigation of the thermodynamic properties for the one-dimensional repulsive Hubbard model in the presence of an arbitrary magnetic field for all values of the filling fraction and temperatures as low as $T\sim 0.005t.$ Our analysis is based on the system of integral equations derived in the quantum transfer matrix framework. We determine the critical exponents of the quantum phase transitions and also provide analytical derivations for some of the universal functions characterizing the thermodynamics in the vicinities of the quantum critical points. Extensive numerical data for the specific heat, susceptibility, compressibility, and entropy are reported. The experimentally relevant double occupancy presents an interesting doubly nonmonotonic temperature dependence at intermediate values of the interaction strength and also at large repulsion and magnetic fields close to the critical value. The susceptibility in zero magnetic field has a logarithmic singularity at low temperatures for all filling factors similar to the behavior of the same quantity in the spin-1/2 isotropic Heisenberg model. We determine the density profiles for a harmonically trapped system and show that while the total density profile seems to depend mainly on the value of chemical potential at the center of the trap the distribution of phases in the inhomogeneous system changes dramatically as we increase the magnetic field.

Figure 1

Ground state phase diagram in $\mu \text{−}H$ coordinates for $u=1$. The value of ${\mu }_{-}\left(0\right)$ is given by Eq. (13) and $H_0\left(u\right)=2{\left(1+u^2\right)}^{1/2}-2u$. In terms of the densities and magnetization the five phases are characterized by I: $n=m=0$, II: $n_{\uparrow }>0,n_{\downarrow }=0$, III: $n_{\uparrow }=1,n_{\downarrow }=0$, IV: $0<n<1,m\ge 0$, and V: $n=1,m\ge 0$.

Figure 2

Density dependence of the charge and spin velocities ($a_0$ is the lattice spacing) for $H=0$ (upper panels), $H=0.3t/{\mu }_B$ (middle panels), and $H=0.5t/{\mu }_B$ (bottom panels).

Figure 3

Magnetic field dependence of the charge and spin velocities for two fixed values of the density $n=0.5$ (upper panels) and $n=0.8$ (lower panels). For large values of the magnetic field the charge velocities are $v_c\left(n\right)=2sin\left(\pi n\right).$

Figure 4

Left panel: The spin velocity at half filling as a function of the magnetic field for different values of $U$. In the vicinity of $H_0\left(U\right)$ the velocity behaves like $v_s\sim {[H_0\left(U\right)-H]}^{1/2}$. Right panels: The logarithmic behavior of the spin velocity at small values of $H$.

Figure 5

(a) Chemical potential and temperature dependence of the grand-canonical specific heat for $U=4t$ and zero magnetic field. The boundaries of the critical regions identified with the maxima of $c_V^{\left(g\right)}$ are highlighted by dashed white lines. The two QCPs are ${\mu }_c^{(I\to IV)}=-4t$ and ${\mu }_c^{(IV\to V)}\equiv {\mu }_{-}\left(0\right)=-0.6433t.$ (b) Chemical potential and temperature dependence of the Wilson ratio $R_W^{\kappa }$. Note the anomalous enhancement in the quantum critical regions.

Figure 6

Plots of the scaled densities $(n-n_r)T^{-1/2}$ as functions of the chemical potential at different temperatures in the vicinities of the QCPs: ${\mu }_c^{(I\to IV)}=-4t$ (a) and ${\mu }_c^{(IV\to V)}=-0.6433t$ (c). For the first transition $n_r=0$ and for the second $n_r=1$. The dashed vertical lines pass through the QCPs. Plotting the scaled densities as functions of $(\mu -{\mu }_c)/T$ all curves collapse to the universal function ${\mathcal{P}}_H^{\prime }\left(x\right)$ (b) for the first transition and (d) for the second transition. In panels (b) and (d) the black diamonds represent the analytical predictions for the universal function ${\mathcal{P}}_H^{\prime }\left(x\right)$ obtained from the derivative of Eq. (36) (b) and Eq. (37) (d) with $a=0.462$ and $b=-1$.

Figure 7

(a) Chemical potential and temperature dependence of the grand-canonical specific heat for $U=4t$ and $H=0.1t/{\mu }_B$. The boundaries of the critical regions identified with the maxima of $c_V^{\left(g\right)}$ are highlighted by dashed white lines. The three QCPs are ${\mu }_c^{(I\to II)}=-2u-2-H=-4.1t,{\mu }_c^{(II\to IV)}=-3.429t$, and ${\mu }_c^{(IV\to V)}=-0.6453t$. (b) Chemical potential and temperature dependence of the Wilson ratio $R_W^{\kappa }$.

Figure 8

Plots of the scaled total densities $(n-n_r)T^{-1/2}$ and density of down spins $(n_{\downarrow }-n_{\downarrow r})T^{-1/2}$ as functions of the chemical potential at different temperatures in the vicinities of the QCPs ${\mu }_c^{(I\to II)}=-4.1t$ (a), ${\mu }_c^{(II\to IV)}=-3.429t$ (c), and ${\mu }_c^{(IV\to V)}=-0.6453t$ (e). The regular parts are $n_r\left(\text{phase}\text{I}\right)=0$, $n_{\downarrow ,r}\left(\text{phase}\text{II}\right)=0$, and $n_r\left(\text{phase}\text{V}\right)=1$. The dashed vertical lines pass through the QCPs. Plotting the scaled quantities as functions of $(\mu -{\mu }_c)/T$ all curves collapse to the universal functions ${\mathcal{P}}_H^{\prime }\left(x\right)$ [(b), (d), and (f)]. In panel (b) the black diamonds represent the analytical predictions for the universal function ${\mathcal{P}}_H^{\prime }\left(x\right)$ obtained from the derivative of Eq. (38). In panels (d) and (f) the analytical predictions for the universal functions are obtained from Eq. (37) with $a=3.1,b=0.4$ (d) and $a=0.476,b=-1$ (f).

Figure 9

(a) Magnetic field and temperature dependence of the grand-canonical specific heat for $U=4t$ at half filling ($\mu =0$). The boundaries of the critical region identified with the maxima of $c_V^{\left(g\right)}$ are highlighted by dashed white lines. The QCP is $H_c^{(V\to III)}\equiv H_0\left(U\right)=0.8284t/{\mu }_B.$ (b) Magnetic field and temperature dependence of the Wilson ratio $R_W^{\chi }$.

Figure 10

(a) Plot of the scaled magnetization $(m-m_r)T^{-1/2}$ versus magnetic field at different temperatures in the vicinity of the QCP $H_c^{(V\to III)}\equiv H_0\left(U\right)=0.8284t/{\mu }_B.$ The regular part of the magnetization is $m_r\left(\text{phase}\text{III}\right)=1/2.$ (b) Universal function ${\mathcal{P}}_{\mu }^{\prime }\left(x\right)$ obtained from the collapse of all the curves plotted as functions of $(H-H_c)/T$. The black diamonds represent the analytical predictions for the universal function ${\mathcal{P}}_{\mu }^{\prime }\left(x\right)$ obtained from the derivative of Eq. (37) for $a=0.8$ and $b=-2$.

Figure 11

Temperature dependence of the specific heat in zero magnetic field for various filling fractions and interaction strengths.

Figure 12

Temperature dependence of the specific heat in the presence of a magnetic field. The insets present a zoom of the data at low temperatures. The shaded regions contain the theoretical predictions $c_V^{\left(c\right)}=\gamma T$ with specific heat coefficients given in Tables 1 and 2.

Figure 13

Temperature dependence of the susceptibility in zero magnetic field for various filling fractions and interaction strengths.

Figure 14

Temperature dependence of the susceptibility in the presence of a magnetic field. The values at $T=0$ denoted by squares $(H=0)$, disks $(H=0.1t/{\mu }_B)$, triangles $(H=0.2t/{\mu }_B)$, and diamonds $(H=0.3t/{\mu }_B)$ are computed using Eqs. (17) and (20) and given in Tables 3 and 4. The insets show the behavior at high temperatures.

Figure 15

Logarithmic dependence of the susceptibility in zero magnetic field. The disks are numerical data and the lines represent best fits using Eq. (50).

Figure 16

Temperature dependence of the compressibility in zero magnetic field for various filling fractions and interaction strengths.

Figure 17

Temperature dependence of the compressibility in the presence of a magnetic field. The values at $T=0$ denoted by squares $(H=0)$, disks $(H=0.1t/{\mu }_B)$, triangles $(H=0.2t/{\mu }_B)$, and diamonds $(H=0.3t/{\mu }_B)$ are computed using Eqs. (17) and (21) and given in Tables 5 and 6. The insets show the behavior at high temperatures.

Figure 18

Temperature dependence of the specific heat, susceptibility, and compressibility at half filling in the presence of a magnetic field. The insets present a zoom of the specific heat data at low temperatures. The shaded regions are inaccessible numerically and contain the theoretical predictions $c_V^{\left(c\right)}\sim \gamma T$ for $H<H_0\left(U\right),c_V^{\left(c\right)}\sim a{T}^{1/2}$ for $H=H_0\left(U\right)t/{\mu }_B$, and $c_V^{\left(c\right)}\sim T^{3/2}{e}^{-\alpha /T}$ for $H=0.9t/{\mu }_B.$ The susceptibilities at $T=0$ denoted by squares $(H=0)$ and disks $(H=0.3t/{\mu }_B)$ are computed using Eq. (22).

Figure 19

Double occupancy as a function of the density for different values of temperature, magnetic fields, and $U=\{4t,8t\}.$ The insets contain the results for the density in the (1,2] interval.

Figure 20

Double occupancy at half filling as a function of the temperature for different values of $U$ and magnetic fields $H=\{0,0.15t/{\mu }_B,0.3t/{\mu }_B\}.$ The double occupancy presents two minima for intermediate values of $U$ (a) and for large values of $U$ and magnetic field close to the critical value (d) $[H_0\left(12\right)\sim 0.3245t/{\mu }_B]$.

Figure 21

(a), (b), (e), (f): Dependence of the entropy on the density for different temperatures and magnetic fields. (c), (d), (g), (h): Dependence of the ratio of the entropy of the Hubbard model to the entropy of lattice free fermions on the filling factor.

Figure 22

Phase diagram at zero temperature for $U=8$ and both positive and negative values of the chemical potential. The three horizontal lines pass through $H=\{0,0.25t/{\mu }_B,0.5t/{\mu }_B\}$ and the disks on each one correspond to values of the chemical potential for which the density is (from left to right) $n=\{0.8,1,1.5,2\}$.

Figure 23

Density profiles for the Hubbard model in the presence of a trapping potential at $T=0.025t$, $U=8t$, and $H=0.5t/{\mu }_B$ (top row); $H=0.25t/{\mu }_B$ (middle row); and $H=0$ (bottom row). The profiles are symmetric with respect to $x$. Here we present only the $x>0$ region. At this temperature the density profiles are effectively indistinguishable from the ones at $T=0$. The black, violet, and green lines represent the total density $n\left(x\right)=n_{\uparrow }\left(x\right)+n_{\downarrow }\left(x\right)$, the density of spin-up electrons $n_{\uparrow }\left(x\right)$, and the density of spin-down electrons $n_{\downarrow }\left(x\right)$. Each phase of the system is highlighted in the same color as in the phase diagram presented in Fig. 22.