Twisted superfluid phase in the extended one-dimensional Bose-Hubbard model

In one-dimensional systems a twisted superfluid phase is found which is induced by a spontaneous breaking of the time-reversal symmetry. Using the density-matrix renormalization group allows us to show that the excitation energy gap closes exponentially causing a quasidegenerate ground state. The two degenerate ground states are connected by the time-reversal symmetry which manifests itself in an alternating complex phase of the long-range correlation function. The quantum phase transition to the twisted superfluid is driven by pair tunneling processes in an extended Bose-Hubbard model. The phase boundaries of several other phases are discussed including a supersolid, a pair superfluid, and a pair supersolid phase as well as a highly unconventional Mott insulator with a degenerate ground state and a staggered pair correlation function.

Figure 1

DMRG phase diagram of the extended Hubbard model for a filling of $N=4$ particles per lattice site. The depicted phases are the superfluid (SF), twisted superfluid (TSF), Mott insulator (MI), supersolid (SS), pair superfluid (PSF), pair supersolid (PSS), and pair-wave Mott insulator (PW). The TSF and the PW phase are characterized by a degenerate ground state (shaded areas, where the filled markers represent an excitation gap smaller than ${10}^{-5}U$). The boundaries between MI and SF/SS are defined by a critical exponent ${\nu }_1=1/4$ of the long-range correlation function $g_1\left(j\right)$ (the open squares are obtained for $8\le j\le 16$). The pair superfluid phase corresponds to ${\nu }_1<1/4$ and a critical exponent ${\nu }_p>1/4$ of the pair correlation function (open circles). The red arrows symbolize the decay of the correlation function $g_1\left(j\right)=\langle {\widehat{a}}_0^{†}{\widehat{a}}_j\rangle$ at a distance $j=0,...,4$ for different exponents ${\nu }_1$, where the vectors visualize amplitude and argument within the complex plane. The inset shows that the twisted superfluid phase extends substantially in parameter space for $N=6$ particles per site. Black (gray) markers correspond to 450 (256) states of the density matrix kept in each DMRG iteration.

Figure 2

Degeneracy of the twisted superfluid state and twisting of the correlation function. (a) Energy gap of the ground state as a function of the system length $L$ within the Mott insulator phase ($N=4, J=0.6U, P=0.16U$ to $0.22U$) and in the twisted superfluid ($P=0.24U$) on a double-logarithmic scale. (b) For the latter the gap closes exponentially with the length of the lattice. (c) In the twisted superfluid phase the complex argument ${\theta }_{0,j}$ of the correlation function $g_1\left(j\right)$ is alternating for the ground state ${\psi }_0^{\text{tr}}$. The red arrows visualize the values of the function $g_1\left(j\right)$ in the complex plane. (d) As a consequence, the nearest-neighbor correlation function shows an alternating twisting $|{\theta }_{j-1,j}|\approx \mathrm{const}$ in the bulk of the lattice with marginal boundary effects. The correlation functions are obtained for $P=0.28U$ using the finite-size DMRG algorithm (three full sweeps, 450 density-matrix states).

Figure 3

Long-range correlation functions across the transition from the Mott insulator ($N=4, J=0.6\mathrm{U}, P=0.20U$ and $0.22U$) to the twisted superfluid ($P=0.24U$). (a) The long-range correlation function $g_1\left(j\right)$ on a double-logarithmic and (b) a logarithmic scale. The red lines depict power-law fits $|g_1\left(j\right)|\propto {\left(\frac{a}{j}\right)}^{{\nu }_1}$ with ${\nu }_1=0.5$ ($P=0.2U$), ${\nu }_1=0.43$ ($P=0.22U$), and ${\nu }_1=0.08$ ($P=0.24U$). The black lines correspond to the real ground state ${\psi }_0$ and the blue lines to the complex ground state ${\psi }_0^{\text{tr}}$. (c), (d) Double-logarithmic and logarithmic plots of the long-range pair correlation function $g_p\left(j\right)$.