Symmetry-broken states in a system of interacting bosons on a two-leg ladder with a uniform Abelian gauge field

We study the quantum phases of bosons with repulsive contact interactions on a two-leg ladder in the presence of a uniform Abelian gauge field. The model realizes many interesting states, including Meissner phases, vortex fluids, vortex lattices, charge density waves, and the biased-ladder phase. Our work focuses on the subset of these states that breaks a discrete symmetry. We use density matrix renormalization group simulations to demonstrate the existence of three vortex-lattice states at different vortex densities and we characterize the phase transitions from these phases into neighboring states. Furthermore, we provide an intuitive explanation of the chiral-current reversal effect that is tied to some of these vortex lattices. We also study a charge-density-wave state that exists at 1/4 particle filling at large interaction strengths and flux values close to half a flux quantum. By changing the system parameters, this state can transition into a completely gapped vortex-lattice Mott-insulating state. We elucidate the stability of these phases against nearest-neighbor interactions on the rungs of the ladder relevant for experimental realizations with a synthetic lattice dimension. A charge-density-wave state at 1/3 particle filling can be stabilized for flux values close to half a flux quantum and for very strong on-site interactions in the presence of strong repulsion on the rungs. Finally, we analytically describe the emergence of these phases in the low-density regime, and, in particular, we obtain the boundaries of the biased-ladder phase, i.e., the phase that features a density imbalance between the legs. We make contact with recent quantum-gas experiments that realized related models and discuss signatures of these quantum states in experimentally accessible observables.

Figure 1

(a) Sketch of the two-leg ladder model and interaction and tunneling terms as defined in Eqs. (1), (3), and (19). Current patterns and on-site density in (b)–(d) the different vortex lattices with vortex density (b) ${\rho }_v=1/2$, (c) ${\rho }_v=1/3$, and (d) ${\rho }_v=1/4$, (e) the biased-ladder phase (BLP) phase, and (f) in the charge-density-wave (${\mathrm{CDW}}_{1/4}$) phase at particle density $\rho =1/4$. The arrows indicate the direction and by their length, the strength of local currents. The density is represented by the size of the circles and the background shading.

Figure 2

Sequence of quantum phases at $\rho =0.8$, $J_{\perp }=1.6J$, $U=2J$ as a function of flux $\varphi$ probed by several measures: (a) Chiral current $j_c$ and average rung current $\mathrm{avg}|j_R|$, (b) central charge $c$, and (c) estimates of vortex density ${\rho }_v$. The vortex density is computed from the Fourier transform of the rung current pattern on systems with open boundary conditions using Eq. (9) or from analyzing local density fluctuations and plotting the wave number $k_{\max }^n$ of the maximum in the corresponding Fourier transformation (see Sec. 4a). $k_{\max }^n$ denotes the position of the maximum in the Fourier transform of the local density fluctuations (see the text). The inset in (b) shows the block-entanglement entropy $S_{\mathrm{vN}}$ for (from top to bottom) $\varphi /\pi =0.8$ (V-SF), 0.74 (${\mathrm{VL}}_{1/3}$-SF), 0.62 (M-SF), and 0.96 (${\mathrm{VL}}_{1/2}$-SF).

Figure 3

Momentum distribution for $\rho =0.8$, $J_{\perp }=1.6J$, $U=2J$ for (a) the leg gauge [see Eq. (3)] and (b) the rung gauge [see Eq. (1)]. Note that (a) and (b) are related by a linear shift by $\varphi /2$ due to the exact gauge transformation of Eq. (17).

Figure 4

Phase diagram for $\rho =0.5$ and $\varphi =0.9\pi$ in the $U/J$ versus $J_{\perp }$ plane. Symbols denote estimated points of the vortex-fluid-to-vortex lattice ($\circ$), the ${\mathrm{VL}}_{1/2}$-MI-to-M-MI ($\mathrm{\Delta }$), and the MI-SF ($\times$) phase transitions (see Sec. 4c). Straight lines and shadings are guides to the eye.

Figure 5

Quantum phases at $\rho =0.5$, $\varphi =0.9\pi$: Cut through the phase diagram Fig. 4 at $U/J=4$. (a) Chiral current $j_c/J$ and average rung current $\mathrm{avg}|j_R|$; (b) vortex density ${\rho }_v$, all versus $J_{\perp }/J$. Dashed lines denote the positions of the phase transitions.

Figure 6

Scaling of the peak of the momentum distribution $n_1\left(k_{\max }\right)$ as a function of $U/J$ close to the BKT transition from the ${\mathrm{VL}}_{1/2}$-SF to the ${\mathrm{VL}}_{1/2}$-MI phase ($J_{\perp }=1.6J$, $\rho =0.5$, $\varphi =0.9\pi$). For the exponent, we choose $\alpha =1/4$ corresponding to the expected scaling of the single-particle correlation functions. The crossing point of all curves marks the BKT-transition point. The inset shows $k_{\max }$ over a larger range of $U/J$ including the V-SF to ${\mathrm{VL}}_{1/2}$-SF transition where $k_{\max }=\pi /2$.

Figure 7

Fidelity susceptibility close to the ${\mathrm{VL}}_{1/2}$-MI to M-MI transition for $\rho =0.5$, $\varphi =0.9\pi$, and $U/J=8$, showing the Ising character of the transition.

Figure 8

Stability of the vortex lattice at ${\rho }_v=1/3$: (a) Chiral current and (b) vortex density from Eq. (9) versus $U/J$ ($J_{\perp }=1.6J$, $\varphi /\pi =0.75$, $\rho =0.8$, $L=120$).

Figure 9

Current pattern and density modulations for the hard-core boson limit of Fig. 8 ($J_{\perp }=1.6J$, $\varphi /\pi =0.75$, $\rho =0.8$, $L=120$). The length and width of the arrows are proportional to the local currents and the radii of the green dots to $\langle n_{\ell ,r}-0.75\rangle$ so as to highlight the density modulations.

Figure 10

Vortex lattice with ${\rho }_v=1/4$ (${\mathrm{VL}}_{1/4}$-SF): (a) Chiral current $j_c$, (b) vortex density ${\rho }_v$, and (c) average rung current $\mathrm{avg}|j_R|$ versus flux $\varphi$ for $\rho =0.8$, $J_{\perp }=1.6J$, $U=J$.

Figure 11

Stability of vortex phases against nearest-neighbor interactions on the rungs: Phase diagram for $\rho =0.5$ for $\varphi =0.9\pi$, $J_{\perp }=1.6J$ as a function of $U/J$ and $V/J$. Symbols denote estimated points of the V-VL ($\circ$), VL-M ($\mathrm{\Delta }$), and the MI-SF ($\times$) phase transitions (see the discussion in Sec. 4g). Straight lines and shadings are guides to the eye.

Figure 12

Two-leg ladder lattices: The small circles indicate lattice sites. Solid lines connecting lattice sites indicate bonds along which hopping is allowed. (a) Sketch of uniform two-leg ladder with flux $\varphi$ per elementary plaquette $\square$. (b) Sketch of the ladder lattice with explicitly enlarged unit cell of $q\square$ plaquettes. Along the rungs indicated by dashed lines hopping is blocked. (c) Uniform two-leg ladder with $q$ times larger lattice constant in the direction of the legs, $q\varphi$ flux per plaquette, and $q$ times fewer links along the legs. In particular, if $\varphi \in (0,\pi )$ and $q\varphi \in (-\pi ,0)$, mod $\left(2\pi \right)$, the chiral current circulates counterclockwise around the system in (a) and clockwise in (b) and (c).

Figure 13

Chiral currents for noninteracting bosons for the cases $\delta =1$, $\delta =0$, and flux $q\varphi$ of Figs. 12–12, respectively, for a rung-dimerized ladder $q=2$. We also include results for $\delta =0.5$.

Figure 14

Chiral currents for noninteracting bosons for the cases $\delta =1$, $\delta =0$, and flux $q\varphi$ of Figs. 12–12, respectively, for a rung-trimerized ladder $q=3$. We also include results for $\delta =0.5$.

Figure 15

Temperature dependence of the chiral current $j_c/J$ within the weak-coupling approximation for various values of the flux $\varphi /\pi$.

Figure 16

Comparison of the behavior of the chiral current in vortex-lattice states obtained for weakly interacting bosons in the high-density limit at small temperature by the transfer matrix approach (continuous curve) to Eq. (20) (dotted curves).

Figure 17

Phase diagram for $\rho =0.25$ and hard-core bosons ($U/J\to \infty$) as a function of $\varphi /\pi$ versus $J_{\perp }/J$. Symbols represent estimated points of the Meissner-to-vortex ($\mathrm{\Delta }$) and the Meissner-to-${\mathrm{CDW}}_{1/4}$ ($\times$) phase transitions (see the discussion in Sec. 6). Straight lines and shadings are a guide to the eye.

Figure 18

${\mathrm{CDW}}_{1/4}$ phase: Emergence of insulating phases at quarter filling $\rho =0.25$ as a function of $U/J$ for $\varphi =0.98\pi$ and $J_{\perp }=3.2J$. (a) Average rung current $\mathrm{avg}|j_R|$ and ${\mathrm{CDW}}_{1/4}$ order parameter $S^n(k=\pi )$, i.e., the value of the static density structure factor $S^n(k=\pi )$, for several system sizes $L=80$ ($+$ symbols), $L=60$ ($\mathrm{\Delta }$), and $L=40$ ($\nabla$). (b) Scaling of the fidelity susceptibility ${\chi }_{FS}/L$ for $L=40$, 60, and 80 rungs. The inset shows examples for the entanglement entropy $S_{\mathrm{vN}}\left(r\right)$ for (from top to bottom) the ${\mathrm{VL}}_{1/2}$-SF [$U=J$, with a fit of Eq. (8) for $c=1$ to the data indicated by the black solid line], ${\mathrm{CDW}}_{1/4}$ ($U=58J$), and ${\mathrm{VL}}_{1/2}$-MI ($U=20J$). Dashed lines indicate the estimated locus of phase transitions.

Figure 19

Equation of state $\rho =\rho \left(\mu \right)$ for $U/J\to \infty$, $V/J\to \infty$ and $J_{\perp }=J$, $\varphi =0.99\pi$ ($L=120$ rungs). At fillings $\rho =1/4$, $\rho =1/3$, and $\rho =1/2$ (indicated by dashed lines) extended plateaus correspond to the gapped ${\mathrm{CDW}}_{1/4}$, ${\mathrm{CDW}}_{1/3}$, and MI phase, respectively.

Figure 20

Two branches of the single-particle dispersion with doubly degenerate minima at $\pm k_0$ in the lowest band for parameters $J_{\perp }=J$ and $\varphi =0.8\pi$. $k_0,k_1$ and ${\tilde{k}}_0,{\tilde{k}}_1$ are the incoming and outgoing momenta in the two-body scattering problem described in Sec. 7a3.

Figure 21

Ground-state phase diagram of the bosonic ladder in the dilute limit for $V=0$ for (a) $\varphi =0.5\pi$ and (b) $\varphi =0.8\pi$. The dashed line does not represent a phase transition; rather it indicates the line where the intraspecies scattering length vanishes. Above the dashed line, in the V-SF phase, the intraspecies interaction enters into the super-Tonks regime. The Lifshitz point, beyond which the M-SF sets in, is indicated by a filled circle at $J_{\perp }=2Jsin\frac{\varphi }{2}tan\frac{\varphi }{2}$.

Figure 22

Biased-ladder phase: (a) Order parameter $\mathrm{\Delta }n$ (imbalance) for the BLP phase, the chiral current $j_c$, and the averaged rung current $\mathrm{avg}|j_R|$ for $\rho =0.8$, $U/J=2$, and $J_{\perp }/J=3$ (DMRG calculation, $L=120$ rungs). (b) Enlarged view of the BLP-to-${\mathrm{VL}}_{1/2}$ transition region. Again, in the ${\mathrm{VL}}_{1/2}$-SF phase, the chiral current reverses its sign.

Figure 23

Momentum distribution $n_1\left(k\right)$ for $U/J\to \infty$, $\rho =0.2$ as a function of (a) $J_{\perp }/J$ for $\varphi /\pi =0.8$ and (b) the flux $\varphi /\pi$ for $J_{\perp }=1.6J$. The dotted line indicates the value of $J_{\perp }$(flux) beyond which we observe an incommensurate behavior (icM-SF) inside the Meissner phase. The transition into the vortex-liquid state (V-SF) occurs at $(J_{\perp }^c,{\varphi }_c)$, indicated by the dot-dashed line.

Figure 24

(a) Chiral current $j_c$ and (b) average rung currents $\mathrm{avg}|j_R|$ for $U/J\to \infty$, $\rho =0.2$, and $L=80$ as a function of the flux $\varphi /\pi$ for $J_{\perp }=1.6J$. For small values of $\varphi <{\varphi }_c$ (dot-dashed line), we are in the Meissner phase. For ${\varphi }_{\mathrm{ic}}<\varphi <{\varphi }_c$, we observe two broad maxima in the momentum distribution $n_1\left(k\right)$ shown in Fig. 23.

Figure 25

Scaling of the average rung current $\mathrm{avg}|j_R|/J$ with the system size $L$ for $\varphi =0.75\pi$, $\rho =0.2$, and $J_{\perp }=1.6J$. A fit to the data yields $\alpha =0.39$.

Figure 26

(a) Average rung currents $\mathrm{avg}|j_R|$, (b) position $k_{\max }$ of the maximum of $n_1\left(k\right)$ and (c) fidelity susceptibility $F\left[j_R\right]\left(k\right)$, (d) central charge $c$, (e) excitation gaps for $L=80$, and (f) Luttinger-liquid parameter $K_{\rho }$, as a function of $J_{\perp }/J$ for $U/J\to \infty$, $\rho =0.2$, and $\varphi /\pi =0.8$. For small values of $J_{\perp }<J_{\perp }^c$ (dot-dashed line), we are in the Meissner phase. Upon increasing $J_{\perp }$, we observe two broad maxima in the momentum distribution $n_1\left(k\right)$ shown in Fig. 23. The transition into the vortex-liquid phase takes place at $J_{\perp }^c/J\sim 3.2$ (dot-dashed line).