Finite-Temperature Topological Invariant for Interacting Systems

We generalize the ensemble geometric phase, recently introduced to classify the topology of density matrices, to finite-temperature states of interacting systems in one spatial dimension (1D). This includes cases where the gapped ground state has a fractional filling and is degenerate. At zero temperature the corresponding topological invariant agrees with the well-known invariant of Niu, Thouless, and Wu. We show that its value at finite temperatures is identical to that of the ground state below some critical temperature $T_c$ larger than the many-body gap. We illustrate our result with numerical simulations of the 1D extended superlattice Bose-Hubbard model at quarter filling. Here, a cyclic change of parameters in the ground state leads to a topological charge pump with fractional winding $\nu =1/2$. The particle transport is no longer quantized when the temperature becomes comparable to the many-body gap, yet the winding of the generalized ensemble geometric phase is.

Figure 1

(a) Phase diagram of Ex-SLBHM for $t_2=0.5t_1$, $V_1=2V_2=0.2U$. Blue and white areas indicate Mott-insulating (MI) and superfluid (SF) phases, respectively. The MI at average filling $\rho =1/4$ per lattice site is doubly degenerate corresponding to superpositions of two density waves indicated in (b) in blue and orange. (b) Cyclic adiabatic variations of $t_1-t_2$ and $\mathrm{\Delta }$ encircling the point $\mathrm{\Delta }=t_1-t_2=0$ lead to a fractionally quantized charge pump [42, 43].

Figure 2

(a) Integrated particle current in the Ext-SLBHM as function of time for a small system of $L=12$ unit cells and different temperatures obtained by exact diagonalization in the $\rho =1/4$ MI phase. Here, $t_{1/2}=5[1\pm \cos \lambda (t)]$, and $\mathrm{\Delta }=-60\sin \lambda (t)$, with angle $\lambda (t)=2\pi t/\mathcal{T}+3\pi /2$ and $V_1=40$, $V_2=20$, and $U$ is infinite. The inset shows the same for $L\to \infty$ obtained by LCRG (Light-Cone Renormalization Group, see Supplemental Material [45] for details) [53, 54, 55]. (b) Generalized EGP for the twofold degenerate system also obtained by LCRG. One notices that the winding remains strictly quantized even at temperatures where there is a substantial occupation of excited states.

Figure 3

Temperature scaling of polarization amplitude $|z(T)|$ for the extended SLBHM for the parameters of Fig. 2 and different system sizes (a) for small systems obtained with exact diagonalization, (b) for larger systems obtained with LCRG, which is an infinite size method. Here, $\lambda (t)=3\pi /4$ but we verified that the results hold for any value of $\lambda$. Also a finite length $L$ was cut out and used for the calculation of $|z(T)|$. The inset shows the behavior in the low temperature regime. As shown in [41] $|z(T\to 0)|\to 1$ for increasing system size.