Phase diagram and structure of the charge-density-wave state in a high magnetic field in quasi-one-dimensional materials: A mean-field approach

We develop the mean-field theory of a charge-density wave (CDW) state in a magnetic field and study the properties of this state below the transition temperature. We show that the CDW state with shifted wave vector in a high magnetic field (${\mathrm{CDW}}_x$ phase) has at least double harmonic modulation on most of the phase diagram. In the perfect nesting case, the single harmonic CDW state with shifted wave vector exists only in a very narrow region near the tricritical point where the fluctuations are very strong. We show that the transition from ${\mathrm{CDW}}_0$ to ${\mathrm{CDW}}_x$ state below the critical temperature is accompanied by a jump of the CDW order parameter and of the wave vector rather than by their continuous increase. This implies a first order transition between these CDW states and explains the strong hysteresis accompanying this transition in many experiments. We examine how the phase diagram changes in the case of imperfect nesting.

Figure 1

Transition lines $T_c\left(H\right)$ at different values of $t_b^{\prime }$ at three different values of the coupling constant ratio $\nu$. The values of $t_b^{\prime }$ are given in units of $T_{c0}$; for the tight-binding dispersion (2), the critical value of $t_b^{\prime }$ at which CDW disappears without the magnetic field is $t_b^{\prime *}={\Delta }_0∕2\approx 0.88T_{c0}$. The triangles show the tricritical points (normal-${\mathrm{CDW}}_0\text{−}{\mathrm{CDW}}_x$).

Figure 2

The values $H_{c1}\left(\nu \right)$ and $H_{c2}\left(\nu \right)$ of the magnetic field at two tricritical points, normal-${\mathrm{CDW}}_0\text{−}{\mathrm{CDW}}_c$ and normal-${\mathrm{CDW}}_c\text{−}{\mathrm{CDW}}_{2c}$, as functions of the coupling constant ratio $\nu =U_s∕U_c$ for two different values of $t_b^{\prime }$. The solid and dotted lines denote $H_{c2}\left(\nu \right)$ and $H_{c1}\left(\nu \right)$ for perfect nesting $(t_b^{\prime }=0)$, while the dashed and dashed-dotted lines denote $H_{c2}\left(\nu \right)$ and $H_{c1}\left(\nu \right)$ at a finite value of the antinesting term $(t_b^{\prime }=0.5T_{c0})$. The cosine phase exists in the regions between $H_{c1}\left(\nu \right)$ and $H_{c2}\left(\nu \right)$ lines for the same value of $t_b^{\prime }$.

Figure 3

Phase diagram in $H-T$ coordinates at $\nu =0.7$ at perfect nesting near the tricritical point. The ${\mathrm{CDW}}_c$ region disappears rapidly as temperature decreases below $T_c$.

Figure 4

The shifts $q_x$ of wave vectors of the phases ${\mathrm{CDW}}_{2c}$ (solid line) and ${\mathrm{CDW}}_c$ (dash line) along the ${\mathrm{CDW}}_c\text{−}{\mathrm{CDW}}_{2c}$ transition line (at $0.713>T∕T_{c0}>0.66$) or ${\mathrm{CDW}}_0\text{−}{\mathrm{CDW}}_{2c}$ transition line (at $T∕T_{c0}<0.66$) as a function of temperature. These shifts coincide only at the normal-${\mathrm{CDW}}_c\text{−}{\mathrm{CDW}}_{2c}$ tricritical point at $T∕T_{c0}=0.713$. The difference between the solid and dashed lines gives the jump of the CDW wave vector on the ${\mathrm{CDW}}_c\text{−}{\mathrm{CDW}}_{2c}$ transition line that makes this transition of first order and leads to hysteresis. The dashed-dotted line stands for the normal-CDW transition temperature, and the dashed-double-dotted line shows the temperature at the ${\mathrm{CDW}}_0\text{−}{\mathrm{CDW}}_c\text{−}{\mathrm{CDW}}_{2c}$ tricritical point.

Figure 5

The schematic picture of the density of electron states in the ${\mathrm{CDW}}_0$ (a), cosine (b), and double-cosine phase (c). The DoS for the cosine phase (b) has one energy gap at the Fermi level for each spin orientation, but the values of this gap is different for spin-up and spin-down electrons. The DOS for the double-cosine phase (c) has at least two energy gaps for each spin orientation with the largest gap being at the Fermi level. It is the same for spin-up and spin-down electrons, but corresponds to the CDW condensates with different wave vectors.

Figure 6

Phase diagram in $H-T$ coordinates at perfect nesting in the whole essential region of CDW at $\nu =0.7$. The solid lines have been calculated from Eqs. (28, 47, 48, 34, 43). The ${\mathrm{CDW}}_c$ region is so narrow that it can hardly be distinguished on the total phase diagram. The dotted line is a smooth link between the ${\mathrm{CDW}}_0\text{−}{\mathrm{CDW}}_{2c}$ boundary near $T_c$ and its value $H_c=2{\Delta }_0∕\pi \approx 1.12T_{c0}$ at $T=0$.