Simultaneous fermion and exciton condensations from a model Hamiltonian

Fermion-exciton condensation in which both fermion-pair (i.e., superconductivity) and exciton condensations occur simultaneously in a single coherent quantum state has recently been conjectured to exist. Here, we capture the fermion-exciton condensation through a model Hamiltonian that can recreate the physics of this new class of highly correlated condensation phenomena. We demonstrate that the Hamiltonian generates the large-eigenvalue signatures of fermion-pair and exciton condensations for a series of states with increasing particle numbers. The results confirm that the dual-condensate wave function arises from the entanglement of fermion-pair and exciton wave functions, which we previously predicted in the thermodynamic limit. This model Hamiltonian—generalizing well-known model Hamiltonians for either superconductivity or exciton condensation—can explore a wide variety of condensation behavior. It provides significant insights into the required forces for generating a fermion-exciton condensate, which will likely be invaluable for realizing such condensations in realistic materials with applications from superconductors to excitonic materials.

Figure 1

A figure of the condensate phase diagram in the phase space of the signatures of particle-particle condensation, ${\lambda }_D$, and exciton condensation, ${\lambda }_G$, is shown. See Ref. [38] for the original figure.

Figure 2

A pictorial representation of the model Hamiltonian we introduce in which there are two $N$-degenerate energy levels—with energies $-\frac{\epsilon }{2}$ and $\frac{\epsilon }{2}$—with double excitations and de-excitations, scattering in which one particle is de-excited while another is simultaneously excited, and a pairwise interaction term between sites $2j-1$ and $2j$ for $j\in \{1,2,\cdots ,N\}$ (yellow circles) is shown. Note that the Lipkin-like excitations must occur within a site ($p\leftrightarrow p+N$, blue arrow).

Figure 3

Configurations representing each of the five classes of nonzero basis states for the FEC Hamiltonian for $N,r=4,8$ are shown where each label $x,y,bool$ represents the number of particles excited to the upper $N$-degenerate energy level ($x$), the number of BCS-like pairs ($y$), and whether the configuration is consistent with the Lipkin model ($bool$), where the degeneracy of each class of states is given in parenthesis, and where green, yellow, and blue configurations represent that the corresponding states are consistent with only the Lipkin Hamiltonian, only the pairing-force Hamiltonian, or both Lipkin and PF Hamiltonians, respectively.

Figure 4

Plots of ${\lambda }_G$ vs ${\lambda }_D$ where parameters in the FEC Hamiltonian are systematically varied are shown for systems involving (a) $N=4$, (b) 6, (c) 8, and (d) 10 particles in $r=2N$ orbitals.

Figure 5

The probabilities corresponding to each of the five classes of basis states (see Fig. 3) consistent with the FEC Hamiltonian for $N,r=4,8$ are shown where green, yellow, and blue bars correspond to the lowest eigenstate of the Lipkin Hamiltonian, the pairing-force Hamiltonian, and FEC Hamiltonian, respectively.

Figure 6

The probabilities corresponding to each of the fourteen classes of basis states consistent with the FEC Hamiltonian for $N,r=8,16$ are shown where green, yellow, and blue bars correspond to the lowest eigenstate of the Lipkin Hamiltonian, the pairing-force Hamiltonian, and FEC Hamiltonian, respectively. Each label $x,y,bool,\zeta ,\text{and}\tau$ represents the number of particles excited to the upper $N$-degenerate energy level ($x$), the number of BCS-like pairs ($y$), whether the configuration is consistent with the Lipkin model ($bool$), the number of times BCS-like pairs are “stacked” into the same site ($\zeta$), and the number of times a diagonal configuration occur in which either $2j-1/2j+N$ or $2j-1+N/2j$ are simultaneously occupied where $2j-1$ and $2j$ are adjacent, paired orbitals ($\tau$). These values act as quantum numbers that define the degenerate classes of nonzero basis functions composing the ground state to the FEC Hamiltonian.

Figure 7

Configurations representing how the Lipkin-like double excitation term ($\lambda$) and scattering term ($\gamma$) in the FEC Hamiltonian relate the $|4,4,F,1,2\rangle$ basis state for $N,r=8,16$ to BCS-like basis states.

Figure 8

A plot of ${\lambda }_G$ vs ${\lambda }_D$ where parameters in the Plastino Hamiltonian are systematically varied for $N=4$ particles in $r=8$ orbitals is shown.