Cluster Luttinger liquids and emergent supersymmetric conformal critical points in the one-dimensional soft-shoulder Hubbard model

We investigate the quantum phases of hard-core bosonic atoms in an extended Hubbard model where particles interact via soft-shoulder potentials in one dimension. Using a combination of field-theoretical methods and strong-coupling perturbation theory, we demonstrate that the low-energy phase can be a conformal cluster Luttinger liquid (CLL) phase with central charge $c=1$, where the microscopic degrees of freedom correspond to mesoscopic ensembles of particles. Using numerical density-matrix renormalization-group methods, we demonstrate that the CLL phase [first predicted in M. Mattioli et al., Phys. Rev. Lett. 111, 165302 (2013)] is separated from a conventional Tomonaga-Luttinger liquid by an exotic critical point with central charge $c=\frac{3}{2}$. The latter is expression of an emergent conformal supersymmetry, which is not present in the original Hamiltonian. We discuss the observability of the CLL phase in realistic experimental settings with weakly dressed Rydberg atoms confined to optical lattices. Using quantum Monte Carlo simulations, we show that the typical features of CLLs are stable up to comparatively high temperatures. Using exact diagonalizations and quantum trajectory methods, we provide a protocol for adiabatic state preparation as well as quantitative estimates on the effects of particle losses.

Figure 1

Sketch of the degenerate ground-state configuration for the Hamiltonian in Eq. (1) in the classical regime $\left(t=0\right)$ at density $n=\frac{2}{5}$ [panel (a)] and $n=\frac{3}{7}$ [panel (b)]. Insets: the structure factor $S\left(q\right)=\frac{1}{L}{\sum }_{\ell ,j}{e}^{i(\ell -j)q}\left[\langle n_{\ell }{n}_j\rangle -n^2\right]$ versus the lattice quasimomentum $q$ for various system sizes $L$ in the classical regime $\left(t=0\right)$ computed using Monte Carlo simulations. $S\left(q\right)$ is peaked at the $q$-vector characteristic of the formation of ground-state cluster structures (see main text) and does not shift even for very small numbers of particles.

Figure 2

(a) Energy of the ground state (GS) as a function of $V/t$ in the perturbative limit $t\ll V$ (here $t=1$). Symbols denote the exact GS energies obtained via exact diagonalization of Eq. (1), while lines denote perturbative estimates for different cluster configurations and system sizes. (b), (c) Scaling of the $E_d=(E_0-E)/\left(VL\right)$ as a function of $1/V^2$ for $n=\frac{2}{5}$ (b) and $n=\frac{3}{7}$. (c) Same sizes as in panel (a). This function, which is 0 in the classical limit, describes how the first-order correction is linear in $1/V^2$, in agreement with the perturbation theory. (d) Absolute value of the energy difference $\epsilon$ between the exact and the perturbative estimates. The energies are quite accurate down to low values of $V/t\lesssim 10$, where the strong-coupling expansion around the cluster manifold first starts to break down. The thin, black line is a guide for the eye indicating a power-law decay $t^4/V^3$. The latter demonstrates that higher-order terms are irrelevant at large couplings, but play a role around $V/t\simeq 10$. The small kink around $V/t\simeq 2$ is due to the fact that $\epsilon$ changes sign in this region.

Figure 3

Configurations allowed for the field $\phi \left(x\right)$. (a) In the Luttinger liquid scenario, $\phi \left(x\right)$ can describe all possible particle configurations and takes integer values at the position of each particle (red circles). The integrated particle density (thin orange line) follows the profile of $\phi$. (b) In the cluster Luttinger liquid scenario, ${\phi }_{cl}$ is constrained by the cluster structure, which assumes that a finite number of particles are packed into small clusters (blue circles). The integrated particle density (thick orange line) does not follow the behavior of ${\phi }_{cl}$, but jumps whenever a two-particle cluster is encountered.

Figure 4

Entanglement entropy scaling as a function of the block length for $n=\frac{4}{10}$. (a)–(c) Different interaction strengths $V/t=4.0,5.675,6.5$. Data for different sizes are indicated by symbols, while lines are linear fits. Deep in the TLL phase, the finite-size central charge quickly approaches 1. At the transition point, the value is $1.5$. Deep in the CLL phase, finite-size effects are strong, as evidenced by the bending of the entropy for different system sizes. (d) $L=50$, different interaction strengths. The black continuous and blue dotted-dashed lines are guides for the eye at $c=1$ and $\frac{3}{2}$, respectively.

Figure 5

Entanglement entropy scaling as a function of the block length for $n=\frac{3}{7}$. (a) Entropies at the transition point $V/t=5.5$. Data for different sizes are indicated by symbols, while the black line is a linear fit in good agreement with $c=\frac{3}{2}$. (b) $L=50$, different interaction strengths. The black continuous and blue dotted-dashed lines are guides for the eye at $c=1$ and $\frac{3}{2}$, respectively.

Figure 6

(a) Finite-size central charge as estimated from a linear interpolation according to the Cardy-Calabrese formula. Blocks of size $<9$ are neglected to avoid strong nonuniversal effects. Residuals are of order ${10}^{-4}$ in the worst cases, closer to the transition line and at large system sizes. The data evidence a phase transition around $V/t\simeq 5.63\pm 0.02$ with a $c=\frac{3}{2}$ central charge. Both liquid phases are compactified bosons, as expected. (b) Finite-size scaling for the critical point $V_L^c$ as a function of $1/L$. Errors in the estimates of $V_L^c$, obtained via local interpolations of $c_L$ in panel (a), are of the size of the symbols. In both panels, $n=\frac{4}{10}$.

Figure 7

(a): Finite-size scaling of the cluster gap deep into the different phases. The gap scales to 0 in both CLL and TLL phases. (b) Data close to the transition point. In both panels, $n=\frac{4}{10}$.

Figure 8

Finite-size scaling of the single-particle gap deep into the different phases at $n=\frac{4}{10}$. The gap scales to 0 in the TLL phase, but takes a finite expectation value in the CLL phase, in agreement with the field-theory predictions.

Figure 9

Finite-size scaling of the single-particle gap at $n=\frac{4}{10}$ in the vicinity of the transition point. Lines are best fits of the form $a_0+a_1(1/L^{a_2})$, with $a_0,a_1$, and $a_2$ constants; the caption indicates as a reference some of the extrapolated values.

Figure 10

Single-particle gap extrapolated to the thermodynamic limit as a function of the interaction strength at $n=\frac{4}{10}$. The red line is a linear fit in the vicinity of the transition point, while the dashed line is a linear fit improved with a logarithmic correction. Errors in the extrapolation are of order of the symbol size $\left(\simeq 0.01\right)$ except at $V/t=5.6,5.675$, where the errors estimated by least-square methods are $\simeq 0.015$.

Figure 11

Rescaled sound velocities $v_{\beta }^r=v^{\beta }/v_M$ (where $v_M$ is the maximum of the two sound velocities in the $L\to \infty$ limit) of the fermionic and bosonic models are extracted from the low-energy spectrum following Eqs. (31) and (32).

Figure 12

The structure factor $S\left(q\right)$ for $V/t=8,L=56$. For example, $\mu =-3.2t (-5t)$ for temperatures $T=1/t (8/t)$, such that $N\simeq 24$ is kept constant.

Figure 13

Comparison of the adiabatic state preparation at the final time ${\tau }_{\max }=200/t_1$ and starting from either $ABAB$ (dotted lines) or $AABB$ (dashed lines), with exact diagonalization results (continuous lines plus symbols). Different colors distinguish different Hamiltonian parameters $V/t_1$. The ratio $t_0/t_1=0.01$ is fixed. Black squares indicate the classical limit prediction.

Figure 14

Plot of $S\left(q\right)$ for dissipative dynamics. The loss rates $\gamma$ range over three order of magnitudes. Also, the case of purely Hamiltonian dynamics $\left(\gamma =0\right)$ is included. The initial state is $ABAB,{\tau }_{\max }=200/t_1$ and $V/t_1=6$. Notice how cluster features survive at $\gamma ={10}^{-3}{t}_1$.