Towards a topological quantum chemistry description of correlated systems: The case of the Hubbard diamond chain

The recently introduced topological quantum chemistry (TQC) framework has provided a description of universal topological properties of all possible band insulators in all space groups based on crystalline unitary symmetries and time reversal. While this formalism filled the gap between the mathematical classification and the practical diagnosis of topological materials, an obvious limitation is that it only applies to weakly interacting systems, which can be described within band theory. It is an open question to which extent this formalism can be generalized to correlated systems that can exhibit symmetry-protected topological phases which are not adiabatically connected to any band insulator. In this work, we address the many facets of this question by considering the specific example of an extended version of a Hubbard diamond chain. This model features a Mott insulator, a trivial insulating phase, and an obstructed atomic limit phase. Here we first discuss the nature of the Mott insulator and determine the phase diagram and topology of the interacting model with infinite density matrix renormalization group calculations, variational Monte Carlo simulations, and with many-body topological invariants. We then proceed by considering a generalization of the TQC formalism to Green's functions combined with the concept of a topological Hamiltonian to identify the topological nature of the phases. Here we use cluster perturbation theory to calculate the Green's functions. The results are benchmarked with the above-determined phase diagram, and we discuss the applicability and limitations of the approach and its possible extensions in the diagnosis of topological phases in materials, in contrast to the use of many-body topological invariants.

Figure 1

(a) Atomic configuration of the extended diamond chain. The unit cell is marked with the gray background, WPs 1$a$ and 1$b$ are denoted with crosses, and atomic sites at WPs 2$i$ and 2$m$ are denoted with solid black circles. The enumeration of orbitals adopted to write the Hamiltonian is also shown. Intracell hoppings $t_1$ and $t_2$ are indicated in black and blue lines, respectively, while intercell hopping $t_3$ is indicated in red. Black lines and the circular arrow in the left diamond denote reflection planes and twofold rotation with respect to the $z$-axis, respectively. (b)–(d) Dominant coupling parameters for three different limiting cases: (b) $t_2\gg t_1,t_3$, (c) $t_3\gg t_1,t_2$, and (d) $t_1\gg t_2,t_3$.

Figure 2

Phase diagram of the noninteracting diamond chain model at half-filling. The blue represents the AI phase, red the OAL, and gray the metallic phase.

Figure 3

Band structure of the HDC for $U=0$, where the lowest occupied band has been omitted as it is disconnected from the rest and it contains the irreps $\{{\mathrm{\Gamma }}_1^{+},X_1^{+}\}$ in the studied range of parameters [50]. (a) AI phase ($t_2/t_1=1.2$, $t_3/t_1=0.2$). (b) Transition point between the AI and OAL phases ($t_2/t_1=1.2$, $t_3/t_1=0.264$). (c) OAL phase ($t_2/t_1=1.2$, $t_3/t_1=0.3$). (d) Metallic phase ($t_2/t_1=0.8$, $t_3/t_1=0.2$).

Figure 4

Phase diagram of the interacting diamond chain for $U/t_1=0.4$ and 1 at half-filling determined from the calculation of the correlation length $\xi$ in iDMRG (color map), as well as data points obtained by VMC at $t_2/t_1=0.8$ and 0.5 for various $t_3/t_1$ indicating whether the system is in a metallic (crosses) or an insulating (circles) phase. The phase boundaries of the noninteracting phase diagram are given by the white, dashed lines.

Figure 5

(a) Labeling of the paths considered in the CPT calculations over the iDMRG/VMC phase diagram obtained with $U/t_1=1.0$. Parts (b), (c), and (d) show the evolution of the charge gap $\mathrm{\Delta }/t_1$ along the different paths as a function of the hopping parameter for different values of the $u=U/t_1$. A cluster containing a single unit cell has been used in CPT calculations.

Figure 6

CPT spectral functions $A(\omega ,k)$ calculated for $U/t_1=1$ at the marked circles in Fig. 5. The Fermi level is shown with dashed lines. Each spectral function is normalized to satisfy $\int d\omega {\sum }_{k}A(\omega ,k)=1$, where the integral runs over the whole frequency domain. (a) AI ($t_2/t_1=1.2$, $t_3/t_1=0.1$), (b) OAL ($t_2/t_1=1.2$, $t_3/t_1=0.7$), (c) MI ($t_2/t_1=0.5$, $t_3/t_1=0.4$), and (d) metallic phase ($t_2/t_1=0.8$, $t_3/t_1=0.8$).

Figure 7

Spectrum and irreps of the topological Hamiltonian in the correlated insulating phases for $U/t_1=1$ in (a) the AI phase ($t_2/t_1=1.2$, $t_3/t_1=0.1$) and (b) the OAL phase ($t_2/t_1=1.2$, $t_3/t_1=0.7$). The lowest occupied band has been omitted [50].

Figure 8

Sum of the absolute value of the real part of self-energy's eigenvalues ${\sigma }_i(\omega ,k)$ in the Mott SPT phase. The drastic increase close to $\omega =0$ is identified as a divergence of the self-energy.

Figure 9

iDMRG calculations for the HDC model showing the correlation length $\xi$ for Hubbard interaction strengths (a) $U/t_1=0.4$, (b) $U/t_1=1.0$, (c) $U/t_1=2.0$, and (d) $U/t_1=4.0$. The maximal bond dimension is set to $\chi =128$.

Figure 10

Correlation length $\xi$ (solid line) and entanglement entropy $S$ (dashed line) plotted against $t_3/t_1$ for $U/t_1=1.0$ and $t_2/t_1=0.3$ for different values of $\chi$.

Figure 11

Localization parameter $z_L$ [Eq. (B5)] computed by variational Monte Carlo for $U/t_1=0.4$ and $t_2/t_1=0.8$. Left panel: $|z_L|$ for a system of $N=40$ diamonds (160 sites) and different values of $t_3/t_1$. We observe a metallic phase ($|z_L|\approx 0$) sandwiched between two insulating phases (finite $|z_L|$). Right panel: Finite-size scaling of $|z_L|$ at $t_3/t_1=0.1\text{(insulator)},0.8\text{(metal)},1.4\text{(insulator)}$.

Figure 12

Different clusters and supercells of diamondlike chains. Supercells are marked with black lines, clusters with blue lines. (a) Diamond chain, where a cluster contains a single diamond and a supercell contains a single cluster. (b) Diamond chain, with a cluster containing two diamonds and a supercell containing a single cluster. (c) Diamondlike chain, constructed by placing successively two different diamonds. Each cluster contains a single diamond, and the smallest supercell that can be chosen contains two clusters.

Figure 13

Illustration of the notation. (a) Example of a possible choice of the supercell (blue) in the HDC, where the unit cell is marked in gray. (b) Reciprocal structure, with the BZ in gray and sites of the superlattice denoted by blue dots.

Figure 14

Analysis of the single diamond in the AI phase with $t_2/t_1=1.5$. (a) Single-particle spectrum with $U=0$, where the bonding states whose combinations give rise to states transforming as ${\mathrm{\Gamma }}_1^{+}$ and antibonding states transforming as ${\mathrm{\Gamma }}_3^{-}$ and ${\mathrm{\Gamma }}_4^{-}$ are shown. Parts (b) and (c) show the traces of the single-particle spectral function and the Green's function computed with $U/t_1=1$, respectively.

Figure 15

Analysis of the diamond with $t_2/t_1=0.5$. Parts (a), (b), and (d) show the single-particle spectrum and traces of the spectral function and Green's function in the metal phase with $U=0$, while (c) and (e) contain the traces of the spectral function and the Green's function in the Mott phase with $U/t_1=1$.