Quantum phase transitions in bosonic heteronuclear pairing Hamiltonians

We explore the phase diagram of two-component bosons with Feshbach resonant pairing interactions in an optical lattice. It has been shown in previous work to exhibit a rich variety of phases and phase transitions, including a paradigmatic Ising quantum phase transition within the second Mott lobe. We discuss the evolution of the phase diagram with system parameters and relate this to the predictions of Landau theory. We extend our exact diagonalization studies of the one-dimensional bosonic Hamiltonian and confirm additional Ising critical exponents for the longitudinal and transverse magnetic susceptibilities within the second Mott lobe. The numerical results for the ground-state energy and transverse magnetization are in good agreement with exact solutions of the Ising model in the thermodynamic limit. We also provide details of the low-energy spectrum, as well as density fluctuations and superfluid fractions in the grand canonical ensemble.

Figure 1

Zero-hopping phase diagram showing the minimum energy eigenstates in the basis $|n_{\downarrow },n_{\uparrow };n_m\rangle$. The Feshbach coupling mixes $|1,1;0\rangle$ and $|0,0;1\rangle$ to yield $|\pm \rangle$ with energies $E_{\pm }={\mathrm{\epsilon ̃}}_m-h/2\pm \sqrt{(h/2){}^2+g^2}$, where $h\equiv {\epsilon }_m-{\epsilon }_{\downarrow }-{\epsilon }_{\uparrow }-V$. The other states have $E(n_{\downarrow },n_{\uparrow };n_m)={\sum }_{\alpha }{\mathrm{\epsilon ̃}}_{\alpha }{n}_{\alpha }+{\mathit{Vn}}_{\uparrow }{n}_{\downarrow }+{\mathit{Un}}_m(n_{\uparrow }+n_{\downarrow })$. The total density ${\rho }_{\mathrm{T}}\equiv n_{\downarrow }+n_{\uparrow }+2n_m$ is pinned to integer values and increases with ${\mu }_{\mathrm{T}}$. Increasing (decreasing) ${\mu }_{\mathrm{D}}$ favors up (down) atoms. The topology changes with the system parameters and depends on the signs of the energy gaps, ${\Delta }_{\pm }$, defined in the text. We set ${\epsilon }_{\downarrow }={\epsilon }_{\uparrow }=g=1$, $U=0$ and (a) ${\epsilon }_m=2$, $V=0$, (b) ${\epsilon }_m=3.5$, $V=1$, (c) ${\epsilon }_m=3$, $V=1.2$, and (d) ${\epsilon }_m=2.5$ $V=1$. The panels shown in Fig. 2 correspond to vertical slices through diagram (a). Here and in subsequent figures, we take ${\epsilon }_{\uparrow }={\epsilon }_{\downarrow }=1$ as the unit of energy.

Figure 2

Evolution of the mean-field phase diagram with ${\mu }_{\mathrm{D}}$, corresponding to vertical scans through Fig. 1a. We set ${\epsilon }_{\uparrow }={\epsilon }_{\downarrow }=1$, ${\epsilon }_m=2$, $g=1$, $U=V=0$, $t_{\uparrow }=t_{\downarrow }=t$, and $t_m=t/2$. We indicate the one-component up, down, and molecular superfluids by ${\mathrm{SF}}_{\uparrow }$, ${\mathrm{SF}}_{\downarrow }$, and ${\mathrm{SF}}_m$, while MI denotes a Mott insulator. Phase 3SF has all three components superfluid. We denote first-order (continuous) transitions by double (single) lines. Junctions between phases are indicated by a dot, and the termination of first-order lines by a cross.

Figure 3

Mean-field phase diagram in the presence of interactions, $U$ and $V$. We set ${\epsilon }_{\uparrow }={\epsilon }_{\downarrow }=1$, ${\epsilon }_m=2$, $g=1$, $t_{\uparrow }=t_{\downarrow }=t$, $t_m=t/2$, and ${\mu }_{\mathrm{D}}=0.45$ as in Fig. 2c, and we consider the effects of $U$ and $V$ in turn. (a) $U=1$, $V=0$, showing the increase in extent of the $|-\rangle$ Mott lobe. (b) $U=0$, $V=1$, showing the appearance of asymmetry. (c) $U=1$, $V=1$, showing the combined effect of both interactions. The characteristic phases and phase transitions mirror those seen in the $U=V=0$ limit shown in Fig. 2.

Figure 4

Topologies of the reduced Landau theory (8) with $\alpha ,\beta \equiv 1,2$ and a term $({\varphi }_1^6+{\varphi }_2^6)/6$ added for stability. We set ${\Lambda }_{11}={\Lambda }_{22}=1$ and consider the evolution with ${\Lambda }_{12}$. We denote first-order (continuous) transitions by double (single) lines. Junctions between phases are indicated by a dot, and the termination of first-order lines by a cross. (a) ${\Lambda }_{12}^2<{\Lambda }_{11}{\Lambda }_{22}$: tetracritical point for ${\Lambda }_{12}=0.8$. (b) ${\Lambda }_{12}>\sqrt{{\Lambda }_{11}{\Lambda }_{22}}$: first-order transition between one-component condensates for ${\Lambda }_{12}=1.5$. (c) ${\Lambda }_{12}<-\sqrt{{\Lambda }_{11}{\Lambda }_{22}}$: first-order transition with condensation of both order parameters for ${\Lambda }_{12}=-1.5$.

Figure 5

Magnified portion of the lower left region of Fig. 2 showing the underlying tetracritical points as ${\mu }_{\mathrm{D}}$ is varied. The lower and middle tetracritical points bifurcate at ${\mu }_{\mathrm{D}}\approx 0.203$ and ${\mu }_{\mathrm{D}}\approx 0.445$, respectively. The former corresponds to the reduced Landau theory criterion ${\Lambda }_{\uparrow m}=-\sqrt{{\Lambda }_{\uparrow \uparrow }{\Lambda }_{\mathit{mm}}}$ obtained by saddle point elimination of ${\varphi }_{\downarrow }$ in the perturbation expansion around the vacuum state, $|0,0;0\rangle$. The latter corresponds to ${\Lambda }_{\downarrow m}=-\sqrt{{\Lambda }_{\downarrow \downarrow }{\Lambda }_{\mathit{mm}}}$ after eliminating ${\varphi }_{\uparrow }$ in the perturbation expansion around the second Mott lobe, $|-\rangle$.

Figure 6

Magnified portion of the second lobe tip in Fig. 2 showing the retreat of the first-order transitions and the emergence of a tetracritical point at ${\mu }_{\mathrm{D}}\approx 0.715$. This is given by the reduced Landau theory criterion, ${\Lambda }_{\uparrow \downarrow }=-\sqrt{{\Lambda }_{\uparrow \uparrow }{\Lambda }_{\downarrow \downarrow }}$, obtained after saddle point elimination of ${\varphi }_m$ in perturbation theory around the second Mott lobe, $|-\rangle$.

Figure 7

Schematic representation of the second Mott lobe with ${\rho }_{\mathrm{T}}=2$ showing the presence of a pair of atoms $\Uparrow$ or a molecule $\Uparrow$ at each site. $\mathit{XY}$ exchange involves interchanging a molecule and a pair of atoms and is suppressed relative to the Ising interaction for small hoppings.

Figure 8

(a) Density fluctuations and (b) superfluid fraction for the 1D bosonic model (1) obtained by exact diagonalization for $N=6$ sites. We set ${\epsilon }_{\downarrow }={\epsilon }_{\uparrow }=g=1$, ${\epsilon }_m=2$, $U=1$, $V=0$, $t=t_{\downarrow }=t_{\uparrow }=2t_m$, and ${\mu }_{\mathrm{D}}=0.45$. The extent of the Mott lobes is in good agreement with the mean-field predictions in Fig. 3a. The system sizes are insufficient to resolve the quantum phase transitions between the distinct superfluids shown in Figs. 2 and 3.

Figure 9

(a) Pseudostaggered magnetization, $m$, versus $\Gamma =2g$, obtained from exact diagonalization of the 1D bosonic model (1) for different system sizes, $N$, at density ${\rho }_{\mathrm{T}}=2$. We take ${\epsilon }_{\downarrow }={\epsilon }_{\uparrow }=1$, ${\epsilon }_m=2$ (corresponding to $h=0$), $U=1$, and $V=0$ and set $t=t_{\downarrow }=t_{\uparrow }=2t_m=0.01$ corresponding to $J=17t^2/4$. (b) Finite-size rescaling of ${\mathit{mN}}^{\beta }$ versus $\Gamma =2g$ showing the presence of the Ising quantum phase transition at $\Gamma ={\Gamma }_{\mathrm{c}}\approx 2.08\times {10}^{-4}$ and the critical exponent $\beta =1/8$ of the 2D classical Ising model. The critical coupling ${\Gamma }_{\mathrm{c}}\approx 0.49J$ is close to the small hopping Ising result, ${\Gamma }_c=J/2$. (c) Scaling collapse as a function of $(\Gamma -{\Gamma }_{\mathrm{c}})N^{1/\nu }$ corresponding to $\nu =1$.

Figure 10

(a) Dimensionless energy gap $\Delta /J$ versus $\Gamma =2g$ obtained by exact diagonalization of the 1D bosonic Hamiltonian (1) for different system sizes, $N$. We set ${\epsilon }_{\downarrow }={\epsilon }_{\uparrow }=1$, ${\epsilon }_m=2$, $U=1$, $V=0$, and $t=t_{\downarrow }=t_{\uparrow }=2t_m=0.01$. (b) Scaled energy gap $(\Delta /J)N^{\nu }$ versus $\Gamma =2g$ showing the presence of the Ising quantum phase transition at the numerically extracted value, ${\Gamma }_{\mathrm{c}}\approx 2.08\times {10}^{-4}$, and the correlation length exponent $\nu =1$. (c) Scaling collapse as a function of $(\Gamma -{\Gamma }_{\mathrm{c}})N^{1/\nu }$ corresponding to $\nu =1$. (d) The numerically extracted critical coupling, ${\Gamma }_{\mathrm{c}}\approx 2.08\times {10}^{-4}$, obtained from the crossing point in Fig. 10b is consistent with the gap closing in the thermodynamic limit, $N\to \infty$. We set $\Gamma$ equal to multiples of $J/2$ (or the numerically extracted bosonic coupling, ${\Gamma }_{\mathrm{c}}$) for the Ising (bosonic) model.

Figure 11

(a) Staggered magnetic susceptibility obtained by exact diagonalization of the 1D bosonic Hamiltonian (1) revealing the Ising quantum phase transition at ${\Gamma }_{\mathrm{c}}\approx 2.08\times {10}^{-4}$ and the critical exponent $\gamma =7/4$. We set ${\epsilon }_{\downarrow }={\epsilon }_{\uparrow }=1$, ${\epsilon }_m=2$, $U=1$, $V=0$, and $t=t_{\downarrow }=t_{\uparrow }=2t_m=0.01$. (b) Scaling collapse showing $\nu =1$. (c) The transverse magnetic susceptibility, ${\chi }_{\perp }$, plays an analogous role to the specific heat capacity of the 2D classical Ising model. (d) The logarithmic divergence at the critical point yields $\alpha =0$ [43]. (e) Plot of $\ln (m_{\mathrm{st}})$ versus $\ln (h_{\mathrm{st}}/J)$ to extract the exponent $\delta =15$ for the bosonic model (1) using $m_{\mathrm{st}}~h_{\mathrm{st}}^{1/\delta }$ for $\Gamma ={\Gamma }_{\mathrm{c}}\approx 2.08\times {10}^{-4}$.

Figure 12

Ising quantum phase transition in the bosonic model (1) showing the emergence of the critical line, ${\Gamma }_c=J/2$, in the regime of small hoppings. We set ${\epsilon }_{\downarrow }={\epsilon }_{\uparrow }=1$, ${\epsilon }_m=2$, $U=1$, $V=0$, and $t=t_{\downarrow }=t_{\uparrow }=2t_m$ and vary the hopping, $t$, corresponding to different values of $J=17t^2/4$.

Figure 13

Comparison of (a) the shifted ground-state energy $E_{\mathrm{gs}}^{\prime }\equiv E_{\mathrm{gs}}-({\mathrm{\epsilon ̃}}_m-h/2-\mathit{Jz}/8)N$ and (inset) transverse magnetization $m_{\perp }={\sum }_i\langle S_i^x\rangle$/N of the bosonic model (1) and the exact thermodynamic results (16) and (17) for the Ising model. We set ${\epsilon }_{\downarrow }={\epsilon }_{\uparrow }=1$, ${\epsilon }_m=2$, $U=1$, $V=0$, and $t=t_{\downarrow }=t_{\uparrow }=2t_m=0.01$. (b) Finite-size corrections for $E_{\mathrm{gs}}^{\prime }$ of the bosonic model (1), with $\Gamma ={\Gamma }_{\mathrm{c}}\approx 2.08\times {10}^{-4}$. We use the finite-size scaling result, $E_{\mathrm{gs}}^{\prime }/\mathit{JN}=e_{\infty }^{\prime }/J-\pi c(v/J)/6N^2$. The intercept at $-0.3164$ is in good agreement with the exact Ising result (16) in the thermodynamic limit, $e_{\infty }^{\prime }/J=-1/\pi \approx -0.3183,$ when $\Gamma =J/2$. The slope, $-\pi c/12$, yields the central charge, $c\approx 0.52$, where we use the characteristic velocity $v=J/2$. (c) Comparison of the low-energy spectra obtained by exact diagonalization for $N=8$ sites in the bosonic model (1) with the same parameters as used in panel (a) with $\Gamma =2g=2J$ and the Ising model (12) with $J=17t^2/4$, $h=0$, and $\Gamma =2J$.

Figure 14

(a) Evolution of the Ising quantum phase transition with the longitudinal field, $h$, in the 1D bosonic model (1). Here ${\epsilon }_m=2+h$ and all other parameters are as in Fig. 9. (b) Quadratic correction to the ground-state energy for the bosonic Hamiltonian (1) due to a longitudinal magnetic field, $h$, in the vicinity of the $h=0$ Ising critical point. We set ${\epsilon }_{\downarrow }={\epsilon }_{\uparrow }=1$, ${\epsilon }_m=2$, $U=1$, $V=0$, $t=t_{\downarrow }=t_{\uparrow }=2t_m=0.01$, and $\Gamma ={\Gamma }_{\mathrm{c}}\approx 2.08\times {10}^{-4}$. The results are close to the theoretical value, $\delta E_{\mathrm{gs}}^{(2)}\approx -0.0706h^{2}N$, at $\Gamma ={\Gamma }_c$ [35].