Discrete symmetry breaking transitions between paired superfluids

We explore the zero-temperature phase diagram of bosons interacting via Feshbach resonant pairing interactions in one dimension. Using DMRG (density matrix renormalization group) and field theory techniques we characterize the phases and quantum phase transitions in this low-dimensional setting. We provide a broad range of evidence in support of an Ising quantum phase transition separating distinct paired superfluids, including results for the energy gaps, correlation functions, and entanglement entropy. In particular, we show that the Ising correlation length, order parameter and critical properties are directly accessible from a ratio of the atomic and molecular two-point functions. We further demonstrate that both the zero-momentum occupation numbers and the visibility are in accordance with the absence of a purely atomic superfluid phase. We comment on the connection to recent studies of boson pairing in a generalized classical $\mathit{XY}$ model.

Figure 1

(a) Phase diagram of the 1D Hamiltonian (1) with total density ${\rho }_{\mathrm{T}}=N_{\mathrm{T}}/L=2$, showing a Mott insulator (MI), a molecular condensate (MC), and a coupled atomic plus molecular condensate (AC $+$ MC). We use DMRG with up to $L=128$ sites and open boundary conditions. We choose parameters ${\epsilon }_a=0$, $U_{aa}/2=U_{mm}/2=U_{am}=g=U$, $t_a=1$, $t_m=1/2$, for comparison with Ref. [37]. The squares and circles indicate the vanishing of the one-particle and two-particle gaps $E_{1g}$ and $E_{2g}$, respectively, as $L\to \infty$. The stars and crosses indicate where the molecular and atomic correlation exponents ${\nu }_m$ and ${\nu }_a$ reach $1/4$ in the MC and AC $+$ MC phases, respectively. These values correspond to a molecular KT transition and an atomic KT transition, respectively. The remaining panels (b), (c), and (d) show the variation of the extrapolated gaps $E_{1g}$ and $E_{2g}$ with $L\to \infty$ on passing through the phase boundaries at the corresponding points in panel (a). In particular, we provide detailed evidence for an Ising quantum phase transition occurring between the MC and AC $+$ MC phases.

Figure 2

(a) Local expectation values obtained by DMRG for the 1D Hamiltonian (1) with open boundaries and up to $L=128$ sites. We use the same parameters as in Fig. 1 and set $U=0.7$. We plot the local density of atoms $\langle a_i^{†}{a}_i\rangle$ (circles), molecules $\langle m_i^{†}{m}_i\rangle$ (squares), and the expectation value of the Feshbach conversion term $\langle m_i^{†}{a}_i{a}_i\rangle$ (triangles), evaluated at the system midpoint, $i=L/2$. All of these quantities are generically nonzero on both sides of the MC to AC $+$ MC transition as indicated by the dashed line. In the limit of large positive (negative) detuning ${\epsilon }_m$ we have mainly atoms (molecules) with a density determined by the canonical ensemble constraint ${\rho }_{\mathrm{T}}=2$. (b) Corresponding finite-size data and linear extrapolation as a function of $1/L$ for ${\epsilon }_m=-3$ (open) and ${\epsilon }_m=-5$ (filled).

Figure 3

Correlation functions in the ${\mathbb{Z}}_2$ disordered MC phase obtained by DMRG on the 1D Hamiltonian (1) with $L=512$ and open boundaries. Here and throughout the paper we consider sites displaced around the system midpoint in order to minimize boundary effects. We use the same parameters as in Fig. 1 and set $U=0.7$ (open), $U=0.5$ (filled), ${\epsilon }_m=-4$ (circles), and ${\epsilon }_m=-6$ (triangles). (a) Atomic Green's function $\langle a_i^{†}{a}_j\rangle$ showing exponential decay. (b) Molecular Green's function $\langle m_i^{†}{m}_j\rangle$ showing power-law behavior. (c) Bilinears of atoms $\langle a_i^{†}{a}_i^{†}{a}_j{a}_j\rangle$ showing power-law behavior with the same exponent as the molecular Green's function in panel (b); see Fig. 4. This establishes the MC phase as a pairing phase of atoms without power-law atomic condensation.

Figure 4

DMRG results for the two-point functions of (a) molecules and (b) atomic bilinears in the MC phase with open boundaries and up to $L=512$. We extract the molecular and bilinear exponents ${\nu }_m$ and ${\nu }_b$ by finite-size scaling collapse of the data for different system sizes. In (c), (d), and (e) we show the resulting evolution of ${\nu }_m$ (circles) and ${\nu }_b$ (stars) for vertical scans through Fig. 1 with fixed values of ${\epsilon }_m$. The vertical dashed lines correspond to the location of the MC to MI transition obtained from the gap data. The molecular exponent reaches the value of ${\nu }_m=1/4$ at the MC to MI transition. This corresponds to a molecular KT transition and is analogous to the fixed density transition at the tips of the Mott lobes in the single-band Bose-Hubbard model. The critical exponent ${\nu }_b$ associated with the power-law decay of the atomic bilinears $\langle a_i^{†}{a}_i^{†}{a}_j{a}_j\rangle$ (stars) coincides with ${\nu }_m$.

Figure 5

Correlation functions in the ${\mathbb{Z}}_2$ ordered AC $+$ MC phase obtained by DMRG on the 1D Hamiltonian (1) with $L=512$ and open boundaries. We use the same parameters as in Fig. 1 and set $U=0.7$ (open), $U=0.5$ (filled), ${\epsilon }_m=-2$ (circles), and ${\epsilon }_m=-3$ (squares). (a) Atomic Green's function $\langle a_i^{†}{a}_j\rangle$ showing power-law decay, in contrast to Fig. 3a. (b) Molecular Green's function $\langle m_i^{†}{m}_j\rangle$ showing power-law behavior; the exponent tracks the atomic exponent in panel (a) up to a factor of 4; see Fig. 6. (c) Bilinears of atoms $\langle a_i^{†}{a}_i^{†}{a}_j{a}_j\rangle$ showing power-law behavior with the same exponent as the molecular Green's function in panel (b). This establishes the AC $+$ MC phase as a pairing phase of atoms in the presence of atomic condensation.

Figure 6

DMRG results for the two-point functions of (a) atoms and (b) molecules in the AC $+$ MC phase with open boundaries and up to $L=512$. We extract the atomic and molecular exponents ${\nu }_a$ and ${\nu }_m$ by finite-size scaling collapse of the data for different system sizes. In (c), (d), and (e) we show the resulting evolution of $4{\nu }_a$ (circles) and ${\nu }_m$ (stars) for vertical scans through Fig. 1 with fixed values of ${\epsilon }_m$. The vertical dashed lines correspond to the location of the AC $+$ MC to MI transition obtained from the gap data. The data confirm the locking of the atomic and molecular exponents via the relation ${\nu }_m=4{\nu }_a$. The exponents reach the values of ${\nu }_a=1/4$ and ${\nu }_m=1$ at the MI boundary. This is consistent with an atomic KT transition. It is analogous to the behavior at the tips of the Mott lobes in the single-band Bose-Hubbard model.

Figure 7

DMRG results for the mixed correlation function $-\langle m_i^{†}{a}_j{a}_j\rangle$ with $L=512$ sites, open boundaries and $U=0.5$. The data correspond to ${\epsilon }_m=-2$ (circles), ${\epsilon }_m=-3$ (squares), ${\epsilon }_m=-4$ (up triangles), ${\epsilon }_m=-5$ (diamonds), ${\epsilon }_m=-6$ (down triangles), and show power-law behavior in both the AC and AC $+$ MC phases. As predicted by Eq. (19), the exponents agree with those of the molecular Green's function in panels (b) of Figs. 3 and 5.

Figure 8

DMRG results for the connected density correlation functions in the ${\mathbb{Z}}_2$ disordered MC phase for the parameters used in Fig. 1. The values of ${\epsilon }_m$ are indicated in panel (b). We use open boundaries with $L=128$ and set $U=0.7$. (a) $|\langle n_{ia}{n}_{ja}\rangle -\langle n_{ia}\rangle \langle n_{ja}\rangle |$. (b) $|\langle n_{im}{n}_{jm}\rangle -\langle n_{im}\rangle \langle n_{jm}\rangle |$. (c) $|\langle n_{ia}{n}_{jm}\rangle -\langle n_{ia}\rangle \langle n_{jm}\rangle |$. The results are in agreement with the leading $1/x^2$ dependence predicted by Eq. (21).

Figure 9

DMRG results for the connected density correlation functions in the ${\mathbb{Z}}_2$ ordered AC $+$ MC phase for the parameters used in Fig. 1. The values of ${\epsilon }_m$ are indicated in panel (b). We use open boundaries with $L=128$ and set $U=0.7$. (a) $|\langle n_{ia}{n}_{ja}\rangle -\langle n_{ia}\rangle \langle n_{ja}\rangle |$. (b) $|\langle n_{im}{n}_{jm}\rangle -\langle n_{im}\rangle \langle n_{jm}\rangle |$. (c) $|\langle n_{ia}{n}_{jm}\rangle -\langle n_{ia}\rangle \langle n_{jm}\rangle |$. The results are in agreement with the leading $1/x^2$ dependence predicted by Eq. (21).

Figure 10

DMRG results for the dependence of the zero-momentum occupation numbers $n_a\left(0\right)$ and $n_m\left(0\right)$ on system size $L$ within the AC $+$ MC phase shown in Fig. 1. The results are consistent with algebraic correlations for both atoms and molecules with locked exponents ${\nu }_m=4{\nu }_a$. The presence of molecular superfluidity in the lower panels confirms the absence of an AC phase close to the MI boundary.

Figure 11

Atomic and molecular correlation functions within the AC $+$ MC phase obtained by DMRG with periodic boundary conditions. We set ${\epsilon }_m=-2$ and $U=1.5$. (a) Atomic correlation function $\langle a_i^{†}{a}_{i+r}\rangle$ as a function of the reduced separation $r/L$ for different system sizes. (b) Normalization factor $\mathcal{N}$ obtained from (a) using Eq. (26). (c) Correlation exponent $a$ obtained from (a) using Eq. (26). (d) Rescaling the data in (a) using the extracted exponent $a$ leads to data collapse. This confirms the applicability of the conformal result (26) within the AC $+$ MC phase. This corresponds to power-law atomic correlations for separations $r\gtrsim 3a_0$. The remaining panels show the corresponding results for molecules.

Figure 12

Finite-size scaling of the atomic and molecular visibilities within the AC $+$ MC phase. The circles correspond to ${\mathcal{V}}_a$ and the squares to ${\mathcal{V}}_m$ obtained by Fourier transformation of the correlation functions obtained by DMRG. The crosses and stars correspond to Fourier transformation of the conformal result (25) supplemented by exact DMRG results for the correlators at small separations $r⩽3a_0$. The solid line indicates the results of conformal extrapolation (described in the text and justified by the scaling collapse in Fig. 11) supplemented by the exact DMRG results for small separations $r⩽3a_0$. (a) With ${\epsilon }_m=-2$ and $U=1.5$ both ${\mathcal{V}}_a$ and ${\mathcal{V}}_m$ extrapolate to unity in the thermodynamic limit. (b) Close to the MI transition with ${\epsilon }_m=-2$ and $U=2.2$ both ${\mathcal{V}}_a$ and ${\mathcal{V}}_m$ approach unity as $L\to \infty$. This is in direct contrast to naive polynomial extrapolation (dashed) which erroneously suggests that the molecular visibility is less than unity.

Figure 13

(a) Atomic and molecular visibilities ${\mathcal{V}}_a$ (circles) and ${\mathcal{V}}_m$ (crosses) within the AC $+$ MC phase for ${\epsilon }_m=-2$ obtained by DMRG with up to $L=64$ and periodic boundaries. We use the conformal extrapolation procedure described in the text in order to obtain the asymptotic results as $L\to \infty$. Both ${\mathcal{V}}_a$ and ${\mathcal{V}}_m$ are unity right up to the MI boundary, indicating the presence of both atomic and molecular superfluidity. (b) Naive polynomial extrapolation erroneously suggests that the molecular visibility is less than unity in the AC $+$ MC phase.

Figure 14

Entanglement entropy $S_L\left(l\right)$ obtained by DMRG with $L=64$ and periodic boundaries. We consider horizontal scans through Fig. 1 with $U=0.5$ and $U=0.7$. (a) Within the MC phase with ${\epsilon }_m=-3.5$ we find $c\approx 1$ corresponding to a gapless superfluid. (b) In the vicinity of the MC to AC $+$ MC quantum phase transition we find $c\approx 3/2$. This corresponds to the presence of additional gapless Ising degrees of freedom coexisting with superfluidity. (c) Within the AC $+$ MC phase with ${\epsilon }_m=-4$ we find $c\approx 1$ corresponding to an effective free boson. The panels on the right correspond to the same data as on the left, but are plotted against the conformal distance $\tilde{l}\equiv \ln \left[(L/\pi )\sin (\pi l/L)\right]$ in order to yield a linear plot with slope $c/3$. The offset between the different curves within each panel is due to the nonuniversal contribution in Eq. (27).

Figure 15

Entanglement entropy difference $\Delta S\left(L\right)$ on transiting across the phase boundaries shown in Fig. 1. We use periodic boundaries with up to $L=64$ and work away from the multicritical point. (a) The transition from MC to AC $+$ MC yields $c\approx 3/2$ corresponding to an Ising transition coexisting with a gapless superfluid. (b) The transition from MC to MI yields $c\approx 1$ and is consistent with a molecular KT transition. (c) The transition from AC $+$ MC to MI with ${\epsilon }_m=4$ yields $c\approx 1$ and is consistent with an atomic KT transition. (d) The transition from AC $+$ MC to MI with ${\epsilon }_m=-3.5$ appears to be compatible with the approach toward $c\approx 1$ with increasing $L$, although the finite-size effects are stronger than those in (c). (a) and (c) are adapted from Ref. [63].

Figure 16

DMRG results used to extract the Ising correlation length $\xi$ within the ${\mathbb{Z}}_2$ disordered MC phase with $L=256$ and open boundaries. (a) With ${\epsilon }_m=-3.85$ the molecular correlation function $\langle m_i^{†}{m}_j\rangle \approx 0.81{|i-j|}^{-0.18}$ decays as a power law. (b) Using the previous exponent ${\nu }_m\approx 0.18$ we extract the Ising correlation length $\xi \approx 43.6$ from the exponential decay of $\langle a_i^{†}{a}_j\rangle \sim {|i-j|}^{-{\nu }_m/4}{\mathcal{K}}_0\left(\right|i-j|/\xi )$. (c) Repeating the above procedure we plot ${\xi }^{-1}$ (circles) vs the departure of the molecular density ${\rho }_m$ from its value ${\rho }_m^c$ at the MC to AC $+$ MC transition. Close to the transition the results are in good agreement with the Ising relation ${\xi }^{-1}\sim {|{\rho }_m-{\rho }_m^c|}^{\nu }$ with $\nu =1$. The triangles correspond to extracting $\xi$ directly from the ratio $\mathcal{R}\left(\right|i-j\left|\right)\equiv {\langle a_i^{†}{a}_j\rangle }^4/\langle m_i^{†}{m}_j\rangle \sim [{\mathcal{K}}_0{\left(\right|i-j|/\xi )]}^4$.

Figure 17

(a) Atomic correlation functions within the AC $+$ MC phase obtained by DMRG with up to $L=512$ and open boundaries. We set $U=0.7$ and consider ${\epsilon }_m=-3.75$ (left) and ${\epsilon }_m=-3.79$ (right). A direct fit to Eq. (31) yields $\mathcal{A}{\langle \phi \rangle }^2$ for each value of ${\epsilon }_m$, where $\langle \phi \rangle$ is the Ising order parameter and $\mathcal{A}$ is a nonuniversal constant prefactor. Changing the fitting interval gives an estimate of the error bars. (b) Extrapolation of $\mathcal{A}{\langle \phi \rangle }^2$ to the thermodynamic limit using linear extrapolation of the largest three system sizes is indicated by the solid line. An estimate of the error bars in the thermodynamic limit is obtained by comparing to a quadratic fit of the data shown by the dashed line. These results are plotted as a function of the molecular density in Fig. 18 in order to confirm Ising behavior with $\beta =1/8$.

Figure 18

DMRG results for the Ising order parameter in the ${\mathbb{Z}}_2$ ordered AC $+$ MC phase with up to $L=512$ and $U=0.7$. (a) Variation of the Ising order parameter squared $\mathcal{A}{\langle \phi \rangle }^2$ where $\mathcal{A}$ is a constant prefactor, vs the deviation of the molecular density ${\rho }_m$ from its value ${\rho }_m^c$ at the MC to AC $+$ MC quantum phase transition. (b) Variation of ${\mathcal{A}}^4{\langle \phi \rangle }^8$ vs the molecular density difference. The results are in good agreement with the Ising magnetization relation $\langle \phi \rangle \sim |{\rho }_m-{\rho }_m^c{|}^{1/8}$ with $\beta =1/8$. The inset shows analogous results obtained from the plateau value of $\mathcal{R}\left(x\right)$ for $L=512$, as indicated in Fig. 19c. The error bars are estimated from the magnitude of $\left|\mathcal{R}\right(x=128,L=512)-\mathcal{R}(x=64,L=256\left)\right|$. (c) Finite-size scaling of the thermodynamic molecular density used in (a) and (b). (a) and (b) are adapted from Ref. [63].

Figure 19

Ratio $\mathcal{R}\left(\right|i-j\left|\right)\equiv {\langle a_i^{†}{a}_j\rangle }^4/\langle m_i^{†}{m}_j\rangle$ of the atomic and molecular correlation functions with $U=0.7$. (a) In the ${\mathbb{Z}}_2$ disordered MC phase with ${\epsilon }_m=-3.85$ the ratio $\mathcal{R}\left(\right|i-j\left|\right)\sim [{\mathcal{K}}_0{\left(\right|i-j|/\xi )]}^4$ exhibits exponential decay. (b) In the vicinity of the MC to AC $+$ MC quantum phase transition the ratio $\mathcal{R}\left(\right|i-j\left|\right)\sim 1/|i-j|$ decays with a universal power law corresponding to the Ising critical exponent $\eta =1/4$. The line is a fit to $\mathcal{R}=A_0{|i-j|}^{A_1}$ over the interval $3⩽|i-j|⩽48$ for $L=512$ with $A_0\approx 0.011$ and $A_1\approx -1.01$. (c) In the ${\mathbb{Z}}_2$ ordered AC $+$ MC phase with ${\epsilon }_m=-3.75$ the ratio $\mathcal{R}\left(\right|i-j\left|\right)\sim {\langle \phi \rangle }^8$ exhibits a plateau corresponding to a nonzero Ising order parameter.

Figure 20

DMRG results for a horizontal scan through Fig. 1 with $1/U=2.5$. We consider up to $L=128$ and extrapolate to the thermodynamic limit. (a) Evolution of the ground state energy $E_0$ with increasing local Hilbert space restriction $n_r=n_a=n_m$. (b) Evolution of the excitation gap $E_{1g}$ with increasing $n_r$, showing very little change beyond $n_r=5$.

Figure 21

DMRG results with $L=128$ and $U=0.5$ within the MC phase shown in Fig. 1. We show the evolution of the atomic and molecular correlation functions with increasing local Hilbert space restriction $n_r$. We set ${\epsilon }_m=-4$ (${\epsilon }_m=-6$) in the upper (lower) panels.

Figure 22

DMRG results with $L=128$ and $U=0.5$ within the AC $+$ MC phase shown in Fig. 1. We show the evolution of the atomic and molecular correlation functions with increasing local Hilbert space restriction $n_r$. We set ${\epsilon }_m=-2$(${\epsilon }_m=-3$) in the upper (lower) panels.