Population imbalance in the extended Fermi-Hubbard model

We study the interplay between population imbalance in a two-component fermionic system and nearest-neighbor interaction using the matrix product states method. Our analysis reveals a parameter regime for the existence of the Fulde-Ferrell-Larkin-Ovchinnikov phase. Furthermore, we find distinct evidence for the presence of hidden order in the system. We present an effective model to understand the emergent oscillations in the string correlations due to the imbalance and show how they can become an efficient tool to investigate systems with imbalance.

Figure 1

Schematic representation of the different phases for the EHM being modified by the presence of imbalance. Here $U$ and $V$ are the on-site and nearest neighbor interaction energies, respectively. The diagram shows the emergence of the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) phase in the region occupied by singlet and triplet superfluid phases in the balanced case. Shown are also the phase separated (PS), charge-density wave (CDW), spin-density wave (SDW), and bond-order wave (BOW) phases. The appearance of a finite string correlation is discussed later in this paper. The figure is an adaptation of the corresponding phase diagram in the balanced case in Ref. [18].

Figure 2

(a) The density-density structure factor, $S_{\mathrm{CDW}}\left(k\right)$ for different polarizations at $U=-2.0,V=2.0$. (b) The monotonic decrease of the peak value of $S_{\mathrm{CDW}}$ at $k=\pm \pi$ shown as a function of the polarization P. (c) The spin-spin structure factor $S_{\mathrm{SDW}}\left(k\right)$ for different polarizations at $U=5.0,V=1.0$. Black circles enclose the positions of the second feature in the presence of imbalance. The color coding for the different polarizations are the same as in (a). Inset shows the momentum values corresponding to the location of the second (bump) as a function of the polarization $P$ along with the function $f\left(P\right)=\pi (1-P)(N_{\uparrow }+N_{\downarrow })/L$. (d) The bond order wave order parameter $O_{\mathrm{BOW}}$ for different polarizations at $U=4.0,V=2.1$. The red diamond denotes the value of $O_{\mathrm{BOW}}$ when degeneracy is lifted by using 99 sites in the simulations.

Figure 3

(a) and (b) Plot of the pair momentum distribution, ${\mathrm{S}}_{\mathrm{pair}}\left(\mathrm{k}\right)$ as a function of momentum $k$ for different values of $P$ for $U=0.5,V=-0.5$ and $U=-1.5,V=-0.25$, respectively. The dotted lines indicate the values of the FFLO wave vector, $q$ (see text) corresponding to the different values of $P$. Single particle density matrix for spin-up [(c), (e)] and -down [(d), (f)] particles (black curves) with different values of $P$ for the same set of values of $U,V$ as in (a) and (b). The red lines indicate a power law fit to these curves with the exponent mentioned in the respective plots.

Figure 4

(a) The absolute value of the string correlation with different values of $V$ for $U=5$ in the balanced case, showing the emergence of string order. (b) Schematic diagram to show how string correlation corresponds to the positions of the holes in the $\downarrow$ component. In the balanced case, the string correlation $S_{\mathrm{string}}$ has a constant value of $-1$ shown in the top part of the figure. The bottom part shows the presence of imbalance $\left(N_{\downarrow }=N_{\uparrow }/2\right)$, resulting in the appearance of holes as denoted by blue dashed boxes. The string correlator has nodes corresponding to the positions of the holes as indicated by the black dotted lines. (c) An effective model of a series of box potentials as seen by a single $\downarrow$ component/hole for strong interactions. (d) and (e) Plot of string correlation function for two values of polarization with $U=5,V=1$ and $U=5,V=5$, respectively. (f) The corresponding fast Fourier transforms done on the data to extract the frequencies. (g) Primary peak positions of the Fourier transforms of the string correlations as a function of spin imbalance along with the function $\pi \times (N_{\uparrow }-N_{\downarrow })/L$.

Figure 5

The plot of the energy spectrum from the exact diagonalization code for a single particle in a series of box potentials. It clearly shows two bands with a gap between them.

Figure 6

Fast Fourier transform (FFT) of the absolute value of the ground state and the highest state in the lower band (which is the hole wave function) showing that they have the same frequency component.

Figure 7

Plot of the wave function corresponding to the ground state of Eq. (B1) for particles, ${\psi }_p^G$, and the highest state for the particles, ${\psi }_p^H$, which is also equal to the state of the first hole created, ${\psi }_h$. The absolute value of the hole wave function, $|{\psi }_h|$, is also plotted. State is also plotted.