Strongly correlated states of trapped ultracold fermions in deformed Landau levels

We analyze the strongly correlated regime of a two-component trapped ultracold fermionic gas in a synthetic non-Abelian $U\left(2\right)$ gauge potential, that consists of both a magnetic field and a homogeneous spin-orbit coupling. This gauge potential deforms the Landau levels (LLs) with respect to the Abelian case and exchanges their ordering as a function of the spin-orbit coupling. In view of experimental realizations, we show that a harmonic potential combined with a Zeeman term gives rise to an angular momentum term, which can be used to test the stability of the correlated states obtained through interactions. We derive the Haldane pseudopotentials (HPs) describing the interspecies contact interaction within a lowest LL approximation. Unlike ordinary fractional quantum Hall systems and ultracold bosons with short-range interactions in the same gauge potential, the HPs for sufficiently strong non-Abelian fields show an unconventional nonmonotonic behavior in the relative angular momentum. Exploiting this property, we study the occurrence of new incompressible ground states as a function of the total angular momentum. In the first deformed Landau level (DLL) we obtain Laughlin and Jain states. Instead, in the second DLL three classes of stabilized states appear: Laughlin states, a series of intermediate strongly correlated states, and finally vortices of the integer quantum Hall state. Remarkably, in the intermediate regime, the nonmonotonic HPs of the second DLL induce two-particle correlations which are reminiscent of paired states such as the Haffnian state. Via exact diagonalization in the disk geometry, we compute experimentally relevant observables such as density profiles and correlations, and we study the entanglement spectra as a further tool to characterize the obtained strongly correlated states.

Figure 1

The energy levels ${\varepsilon }_1,{\varepsilon }_2,{\varepsilon }_3$ (blue) and ${\varepsilon }_0^{\prime }$ (red) are shown as a function of $q^2/B$ for $\Delta /B=0.02$ and $m=0$ (the energy levels ${\varepsilon }_1^{\prime }$ and ${\varepsilon }_2^{\prime }$ are not shown). The crossing points $q_1^2$ and $q_2^2$ are slightly displaced from their values $q^2/B=3,5$ at $\Delta =0$. The numerical analysis in Secs. 4 and 5 will be performed at the values $q^2/B=0.5$ and $q^2/B=3.8$ which are close to the maxima of the gaps between the ground state energy and the first excited DLL.

Figure 2

The coefficients $V_m^{\left(n\right)}=v·{\tilde{v}}_m^{\left(n\right)}/2^{2n}$, related to the HPs by $W_m^{\left(n\right)}=V_m^{\left(n\right)}{sin}^2\left({\phi }_n\right)/2$, are plotted for $v=1$ and $n=1,2,3,4$. The integer numerators ${\tilde{v}}_m^{\left(n\right)}$ are specified in the chart. Only fermionic odd relative angular momenta $m$ are considered. $V_m^{\left(n\right)}$ increases with $m$, and then drops to zero for $m\ge 2n$.

Figure 3

Radial density profile for the up (dashed blue line) and down (dotted red line) components of the Laughlin state with 10 atoms at $L_{\mathrm{tot}}=135$ in a system with $q^2=B/2$. The total radial density is plotted as a continuous black line. Inset: Gap $\mathcal{D}$, in units of $v$, for the Laughlin state at filling $1/3$ as a function of $N$.

Figure 4

The low-energy part of the yrast spectrum (blue) and the related compressibility gap (red) are plotted as a function of the total angular momentum $L_{\mathrm{tot}}$ for $N=9$ atoms. The energies are expressed in units of the interaction constant $v$ ($\Delta =0$). On the horizontal axis we reported the angular momenta of the main incompressible states. $L=108$ corresponds to the Laughlin state; $L_{\mathrm{tot}}=99,92,87,84$ correspond respectively to the presence of $1,2,3$, and 4 quasiparticles. In the insets the density profiles of the states at $L_{\mathrm{tot}}=108,99$, and 92 are plotted for $q^2=B/2$. The Laughlin state presents an almost flat plateau, whereas the states at $L_{\mathrm{tot}}=99$ and $L_{\mathrm{tot}}=92$ show the typical profile of an odd or even number of quasiparticles in the center of the system.

Figure 5

Full yrast spectrum (blue) and related compressibility gap (red) for $N=10$ fermions. In the insets the total density profile of the IQH state, of the candidate Jain states at filling $3/7$ and $2/5$, and of the Laughlin state are plotted for $q^2=B/2$ and $\Delta =0$. On the horizontal axis the momenta of the states stabilized by the trapping potential are reported.

Figure 6

Angular momentum of the ground state for $N=10$ fermions as a function of the trapping parameter $\Delta /v$, keeping constant the ratio between $q^2$ and the magnetic field $\mathcal{B}$ at the value $q^2/\mathcal{B}=0.5$ in the first DLL (with $\mathcal{B}=1$). The horizontal lines define the incompressible ground states stabilized by the trapping potential. These cusp states include the two candidate Jain states identified in [72]. For higher densities several ground states alternate, consistent with eightfold rings with two fermions in the center [76].

Figure 7

The yrast spectrum (blue) and the incompressibility gap (red) of the system with $N=8$ atoms is plotted for $q_1^2<q^2<q_2^2$ as a function of the total angular momentum. The energies are expressed in terms of the interaction constant $v$. Inset: Detail of the lowest part of the spectrum presenting quasiparticle plateaus.

Figure 8

Angular momentum of the ground state for 8 fermions at $q^2=3.8\mathcal{B}$ as a function of the trapping parameter $\Delta /v$ for $\mathcal{B}=1$. The dotted lines are a guide for the eye to distinguish the three regimes discussed in the main text.

Figure 9

Spin-resolved profile densities and effective two-body correlation functions for the most relevant gapped states in the intermediate regime for $N=8$. Left: Spin-up (dashed blue), spin-down (dotted red), and total (black) density profiles. Right: Two-body effective correlation function with the reference atom in the position ${\vec{r}}_0=(6,0)$ (red dots). The states at $L_{\mathrm{tot}}=68,72,80,88$ are cusp states stabilized by the trap, whereas the incompressible state at $L_{\mathrm{tot}}=76$ is compatible with the Haffnian wave function (24) and it is shown for comparison. The states at $L_{\mathrm{tot}}=72,76,80$ show three distinct peaks corresponding to pairs of atoms whose presence is consistent with the pairing in the Haffnian wave function. The state at $L_{\mathrm{tot}}=68$ may be related to the WYQ state. The state at $L_{\mathrm{tot}}=88$ presents instead an almost flat correlation distribution. All the correlation functions are calculated for the single-particle wave functions ${\psi }_{0,m}$ instead of ${\chi }_{2,m}$ to make their structure more evident.

Figure 10

Effective two-body correlation function of the ground state for $N=10$ and $L_{\mathrm{tot}}=125$. The red dots represent the position of the reference fermion.

Figure 11

The density profiles (first column) and the spin-resolved 2-body density correlations for the spin-up (second column) and spin-down (third column) component of vortex states are showed for a system of 8 fermions. The considered states correspond to one, two, and three central vortices, in turn corresponding respectively to the cusp states with $L_{\mathrm{tot}}=36,44,52$. In the first column the blue lines represent the spin-up radial density, the red dashed lines are the spin-down densities, and the black ones correspond to the total densities. The two-body correlations are plotted in arbitrary units fixing the position of one atom in $(6,0)$ (red dots). The insertion of the first vortex does not cause a minimum in the densities and correlations in the center of the system. The second vortex has instead a skyrmionic behavior, with minima in the spin-up component only, whereas the third vortex present minima in both the up and down components.

Figure 12

Comparison of the orbital entanglement spectrum for the ground states at $L_{\mathrm{tot}}=135$ and $N=10$ for $q^2=0.5B$ (left) and $q^2=3.8B$ (right). The panel on the left shows the usual behavior of a Laughlin state at filling $1/3$. The corresponding state in the second DLL (right) appears to be radically different and, in particular, the decreasing behavior of the entanglement spectrum as a function of $L_A$ suggests an inversion of the chirality of the edge modes. The orbital entanglement spectrum is obtained by removing $m_A=12$ orbitals and $N_A=5$ atoms and it is plotted as a function of the residual angular momentum $L_A$.

Figure 13

Particle entanglement spectrum of the states with 10 atoms at $q^2=3.8$, with $N_A=5$ at $L_{\mathrm{tot}}=65$ (left) and $L_{\mathrm{tot}}=75$ (right). The states are characterized by the presence of 2 and 3 vortices, respectively. A gap separating the universal and nonuniversal parts of the spectrum is evident. We notice, in particular, that the universal part appears at $L_{A,\min }=N_A(N_A-1)/2+2N_A$ and $L_{A,\min }=N_A(N_A-1)/2+3N_A$, respectively, consistent with the wave function for multiple vortices.

Figure 14

Particle entanglement spectrum of the ground state with $N=8$ at $L_{\mathrm{tot}}=76$ and $q^2=3.8B$. Left panel: Spectrum with $N_A=4$. Right panel: Spectrum obtained in the conformal limit [88] under the same conditions.