Magnetic properties of the second Mott lobe in pairing Hamiltonians

We explore the Mott insulating state of single-band bosonic pairing Hamiltonians using analytical approaches and large-scale density matrix renormalization group calculations. We focus on the second Mott lobe which exhibits a magnetic quantum phase transition in the Ising universality class. We use this feature to discuss the behavior of a range of physical observables within the framework of the one-dimensional quantum Ising model and the strongly anisotropic Heisenberg model. This includes the properties of local expectation values and correlation functions both at and away from criticality. Depending on the microscopic interactions it is possible to achieve either antiferromagnetic or ferromagnetic exchange interactions and we highlight the possibility of observing the $E_8$ mass spectrum for the critical Ising model in a longitudinal magnetic field.

Figure 1

Depiction of the second Mott lobe of the Hamiltonian (1) with ${\rho }_T=2$ showing either a pair of atoms or a molecule on each site. These may be regarded as an effective spin-down ($\Downarrow$) and spin-up ($\Uparrow$), respectively. Second-order virtual hopping processes lead to $J_{zz}$ interactions and an effective Ising model. The XY exchange, $J_{xy}$, occurs at third order and involves interchanging a pair of atoms and a molecule. The Feshbach term (2) converts a pair of atoms ($\Downarrow$) into a molecule ($\Uparrow$) and therefore acts like a transverse field.

Figure 2

Phase diagram of the 1D Hamiltonian (3) with ${\rho }_T=2$ obtained by DMRG with up to $L=256$ sites. We restrict the local Hilbert space to a maximum of one molecule and two atoms per site with $r_m=1$ and $r_a=2$, and we set ${\epsilon }_a={\epsilon }_m=0$, $U_{aa}=0$, and $U_{am}=1$, corresponding to $h=0$. We set $t_a=2t_m=t$ and vary $t$ to obtain different values of $J_{zz}=17t^2/4$ corresponding to the AFM exchange. This cross section shows a ${\mathbb{Z}}_2$ ordered AFM MI, a ${\mathbb{Z}}_2$ disordered AFM MI, and an $\mathrm{AC}+\mathrm{MC}$ SF. The solid squares are obtained from the vanishing of the first excitation gap, ${\Delta }_1\equiv E_1-E_0$, corresponding to an Ising quantum phase transition within the MI; the solid line is a spline fit which is extrapolated down to $\Gamma =0$. The dashed line corresponds to the strong coupling result, ${\Gamma }_c=J_{zz}/2$, which follows from the Hamiltonian (4) with $h=0$. The dotted line is obtained by ED of Hamiltonian (4) including the subleading XY exchange terms, $J_{xy}=-0.46J_{zz}^{3/2}$, which arise at higher order in the strong coupling expansion. The diamonds are obtained from the vanishing of the two-particle gap, $E_{2g}\equiv {\mu }_{+}-{\mu }_{-}$, where ${\mu }_{\pm }=\pm [E_0(L,N_{\mathrm{T}}\pm 2)-E_0(L,N_{\mathrm{T}})]$, and up to $L=128$, indicating the onset of an $\mathrm{AC}+\mathrm{MC}$ superfluid.

Figure 3

DMRG results for the excitation gaps, ${\Delta }_1\equiv E_1-E_0$ (circles) and ${\Delta }_2\equiv E_2-E_0$ (squares), for the bosonic Hamiltonian (3) with open boundaries and up to $L=256$ sites. We pass through the Ising quantum phase transition shown in Fig. 2 with $t=0.05$ and $J_{zz}\approx 0.01$. The transition occurs at ${\Gamma }_c\approx 4.6\times {10}^{-3}$, which is slightly below the strong coupling result, ${\Gamma }_c=J_{zz}/2\approx 5.3\times {10}^{-3}$. This reflects the curvature of the Ising transition shown in Fig. 2. The solid lines indicate the linear gap, $\Delta =|\Gamma -{\Gamma }_c|$, corresponding to the Ising critical exponent, $\nu =1$.

Figure 4

(a) DMRG results for the entanglement entropy, $S_L\left(l\right)$, at the Ising transition within the MI for a subsystem of length $l$ in a periodic chain with $L=64$. We use the same parameters as in Fig. 2 and set $t=0.17$ corresponding to $J_{zz}\approx 0.12$ and ${\Gamma }_c\approx 0.028$. The solid line is a fit to Eq. (11). The extracted value of the central charge $c=0.50\left(2\right)$ is consistent with a magnetic transition in the Ising universality class. (b) The entanglement entropy difference $\Delta S\left(L\right)$ for $t=0.17$ shows a peak at the Ising transition. The peak height corresponds to $c=1/2$ as indicated by the dashed lines.

Figure 5

Local expectation values deep within the Mott phase obtained by DMRG for the bosonic Hamiltonian (3) with periodic boundaries and restriction parameters $r_m=1$ and $r_a=2$. In order to demonstrate the broader validity of our results, while suppressing higher-order terms in the strong coupling expansion, we choose parameters different from those used in Fig. 2. We set ${\epsilon }_a=0$ and ${\epsilon }_m=U_{aa}=3.8$, corresponding to $h=0$, and we take $U_{am}=4$ and $t=0.005$. (a) Staggered magnetization $|\langle S_i^z\rangle |$. The theoretical result for the transverse field Ising model is given by Eq. (15) and is indicated by the solid line. Inset: The application of a small staggered field $h_{\mathrm{st}}$ yields the same results as in panel (a) in the limit $h_{\mathrm{st}}\to 0$. (b) Transverse magnetization $\langle S_i^x\rangle$. The solid line shows the Ising behavior given by Eq. (16). The dashed line corresponds to $\langle S_i^x\rangle =-1/\pi$, which holds at criticality.

Figure 6

(a) DMRG results for the local magnetization $\langle S_i^z\rangle$ for the parameters used in Fig. 2 with $h=0$ and $\Gamma =0.04$. We set $t=0.025,0.075,0.125,...,0.325$ from top to bottom with $L=32$ and periodic boundaries. The plateaus correspond to the development of a uniform magnetization while the staggered magnetization remains zero. (b) Uniform magnetization ${\mathcal{M}}_U={\sum }_i\langle S_i^z\rangle /L$. The nonzero value is attributed to higher-order terms in the strong coupling expansion. These may modify the leading-order magnetic field given in Eq. (8). The dotted line shows the approximate location of the MI to $\mathrm{AC}+\mathrm{MC}$ transition obtained from the gap data in Fig. 2. (c) We set $\Gamma =0.02$ and use the same values of the remaining parameters as in panel (a). The oscillations correspond to the onset of the ${\mathbb{Z}}_2$ ordered phase in Fig. 2. (d) Antiferromagnetic oscillations in the ${\mathbb{Z}}_2$ ordered phase with $t=0.125$ (circles), $t=0.155$ (squares), $t=0.175$ (diamonds), and $t=0.195$ (triangles). (e) Uniform magnetization ${\mathcal{M}}_U$ (circles) and staggered magnetization ${\mathcal{M}}_S={\sum }_i{(-1)}^i\langle S_i^z\rangle /L$ (squares) for $\Gamma =0.02$.

Figure 7

Correlation functions within the second Mott lobe of the bosonic Hamiltonian (3) obtained by DMRG with $L=512$ and open boundaries. We use the same parameters as in Fig. 3. The data are consistent with an Ising quantum phase transition between a magnetic state with long range order and a disordered state with a finite correlation length. The solid lines correspond to the critical two-point functions in Eqs. (17), (18) and (19). A quantitative demonstration of the Ising critical exponents is shown in Fig. 8 using periodic boundaries and finite-size scaling.

Figure 8

DMRG results for the rescaled bosonic correlation functions $L^{1/4}\left|{\langle S_i^z{S}_{i+n}^z\rangle }_L\right|$ (circles), $L^2{[\langle S_i^x{S}_{i+n}^x\rangle -\langle S_i^x\rangle \langle S_{i+n}^x\rangle ]}_L$ (squares), and $L^{9/4}\left|{\langle S_i^y{S}_{i+n}^y\rangle }_L\right|$ (triangles) with periodic boundaries at criticality for the parameter set of Fig. 7 with $t=0.05$. The data collapse over the entire system length with the Ising critical exponents. The nonuniversal prefactors differ slightly from the theoretical predictions of the lattice Ising model (4) as indicated by the solid lines. This is due to the presence of small additional XY contributions to the Ising description. By descending deeper into the Mott lobe one obtains a complete quantitative agreement including the nonuniversal prefactors as shown in Fig. 9.

Figure 9

DMRG results for the rescaled bosonic correlation functions $L^{1/4}\left|{\langle S_i^z{S}_{i+n}^z\rangle }_L\right|$ (circles), $L^2{[\langle S_i^x{S}_{i+n}^x\rangle -\langle S_i^x\rangle \langle S_{i+n}^x\rangle ]}_L$ (squares), and $L^{9/4}\left|{\langle S_i^y{S}_{i+n}^y\rangle }_L\right|$ (triangles) with periodic boundaries at criticality for the parameter set of Fig. 5 with $t=0.005$. The data show clear scaling collapse over the entire system length. The theoretical results for the lattice Ising model (4) in a finite-size geometry are indicated by solid lines. The data are in excellent agreement with both the universal Ising critical exponents and the nonuniversal amplitudes in Eqs. (17), (18), and (19).

Figure 10

Off-critical order parameter correlations $|\langle S_i^z{S}_{i+n}^z\rangle |$ obtained by DMRG on the 1D system (3) with $L=48$ and periodic boundaries. We use the same parameters as in Fig. 5. The solid lines correspond to the quantum Ising model (4) and are given by Eq. (24) for $\Gamma <{\Gamma }_c$ and Eq. (25) for $\Gamma >{\Gamma }_c$. The agreement confirms the presence of long-range order for $\Gamma <{\Gamma }_c$ and exponential decay for $\Gamma >{\Gamma }_c$.

Figure 11

Correlation function $C_{am}\left(n\right)/2$ at and away from criticality obtained by DMRG on the 1D system (3) with $L=48$ and periodic boundaries. We use the same parameters as in Fig. 5. (a) At criticality clear oscillations are present in conformity with the Ising description. The solid line corresponds to the theoretical prediction in Eq. (31). (b) Away from the critical point with $\Gamma =2{\Gamma }_c$, $C_{am}\left(n\right)$ shows a finite correlation length due to the Ising gap.

Figure 12

Ground-state energy density of the Hamiltonian (3) obtained by DMRG with $L=20$ (to aid comparison with ED results for the Ising model) and periodic boundaries. (a) We use the parameters in Fig. 5 and increase the local atomic Hilbert space restriction $r_a$, with $r_m=1$ held fixed. The lines are results for the energy density of the Ising Hamiltonian (4) obtained by ED ($L=20$) with $J_{zz}=5.02\times {10}^{-4}$ and $h=0$ for $r_a=2$ and with $J_{zz}=4.23\times {10}^{-4}$ and $h=7.89\times {10}^{-5}$ for $r_a=3$ and 4. The change from $r_a=2$ to $r_a=3$ is due to the presence of additional virtual states which modify the Ising model coefficients. The absence of any change beyond $r_a=3$ is consistent with second-order perturbation theory around the second Mott lobe. (b) Ground-state energy density for the same parameters as in panel (a) with the additional interaction $U_{mm}=4$. We increase the local Hilbert space from $r_a=3$ and $r_m=2$ to $r_a=4$ and $r_m=3$. The absence of any further change is consistent with the maximum occupancy for virtual states explored in second-order perturbation theory around the second Mott lobe. The solid line is the energy density of the Ising Hamiltonian (4) obtained by ED ($L=20$) with $J_{zz}=4.16\times {10}^{-4}$ and $h=7.27\times {10}^{-5}$ as given by Eqs. (32) and (33), respectively.

Figure 13

Entanglement entropy difference $\Delta S\left(L\right)\equiv S_L(L/2)-S_{L/2}(L/4)$ showing an Ising quantum phase transition within the second Mott lobe for restricted softcore bosons. We truncate the local Hilbert space to $r_a=3$ and $r_m=2$ which is the maximum occupancy explored at second order in perturbation theory. We use the same parameters as in Fig. 12b and set $t=0.005$. The peak height corresponds to a central charge $c\approx 1/2$.

Figure 14

ED results for the Hamiltonian (3) with hardcore molecules ($r_m=1$) and up to three atoms ($r_a=3$) per site. We set ${\epsilon }_a=0$, $U_{aa}=2$, $U_{am}=4$, and $t_a=2t_m$ and take $t_a=0.01$ corresponding to FM exchange with $J_{zz}\simeq -3.94\times {10}^{-4}$. For simplicity we set $h={\epsilon }_m-2+6t_a^2=0$ by taking ${\epsilon }_m=1.9994$. (a) The rescaled energy gap ${\Delta }_1\equiv E_1-E_0$ shows a clear intersection at ${\Gamma }_c\approx 1.969\times {10}^{-4}\approx 0.4997\left|J_{zz}\right|$ corresponding to a FM transition. (b) Scaling collapse with the Ising critical exponent, $\nu =1$. (c) The rescaled pseudomagnetization $m=\langle \left|{\sum }_i{S}_i^z\right|\rangle /L$ indicates a transition at the same value of ${\Gamma }_c$. (d) Scaling collapse with the Ising critical exponent $\beta =1/8$.

Figure 15

(a) Rescaled ground-state energy $E_0^{\prime }\equiv E_0-C$ of the bosonic Hamiltonian (3) with the FM parameters used in Fig. 14. The DMRG data are in good agreement with Eq. (35) for the thermodynamic limit of the Ising model. (b) The residual finite-size corrections, $E_0^{\prime }/L|J_{zz}|=e_{\infty }/\left|J_{zz}\right|-\pi c/12L^2+\cdots$, are consistent with the central charge $c\approx 0.499$ of an Ising transition to a FM state in the absence of a longitudinal magnetic field.