Topological phases of a dimerized Fermi–Hubbard model for semiconductor nano-lattices

www.nature.com/npjqi ARTICLE OPEN Topological phases of a dimerized Fermi–Hubbard model for semiconductor nano-lattices Nguyen H. Le1 ✉, Andrew J. Fisher2, Neil J. Curson 3 and Eran Ginossar1 Motivated by recent advances in fabricating artificial lattices in semiconductors and their promise for quantum simulation of topological materials, we study the one-dimensional dimerized Fermi–Hubbard model. We show how the topological phases at half-filling can be characterized by a reduced Zak phase defined based on the reduced density matrix of each spin subsystem. Signatures of bulk–boundary correspondence are observed in the triplon excitation of the bulk and the edge states of uncoupled spins at the boundaries. At quarter-filling, we show that owing to the presence of the Hubbard interaction the system can undergo a transition to the topological ground state of the non-interacting Su–Schrieffer–Heeger model with the application of a moderatestrength external magnetic field. We propose a robust experimental realization with a chain of dopant atoms in silicon or gatedefined quantum dots in GaAs where the transition can be probed by measuring the tunneling current through the many-body state of the chain. 1234567890():,; npj Quantum Information (2020)6:24 ; https://doi.org/10.1038/s41534-020-0253-9 INTRODUCTION Topological phases of matter are among the most exciting developments of modern condensed matter physics,1–4 owing to their rich phenomenology and wide-ranging potential applications from metrology5 to quantum computation.6 Many experimental platforms have been used to realize these exotic phases of matter such as cold atoms,7 photonic lattices,8,9 and engineered solid-state systems including graphene nanoribbons,10,11 arrays of carbon monoxide molecules,12,13 and chlorine monolayers14 on a copper surface. The band theory of topological insulators (TIs)15 based on the independentelectron approximation is well developed and has had many successes. However, in many of the possible experimental platforms for quantum simulation of TIs using electrons in solids, such as dopant atoms and gate-defined quantum dots in semiconductors,16,17 the electron–electron interaction is much stronger than the hopping amplitude of the electrons18,19 and therefore the independent-electron approximation is poor. Topological phases of strongly correlated models form a topic of ongoing active research with intense theoretical20 and experimental effort, including recent implementations in cold atoms21 and two-dimensional materials.22 There have been various proposals for the equivalent of the single-particle Berry phase (or Zak phase in one dimension) for the characterization of interacting topological phases, from the magnetic-fluxinduced Berry phase23,24 to Green’s functions25 and entanglement.26–28 Here we discuss one of the simplest one-dimensional (1D) models of strongly correlated TIs, the Su–Schrieffer–Heeger–Hubbard (SSHH) model, whose topological properties in various contexts have been investigated using the entanglement entropy,26,27 the entanglement spectrum,28 correlation functions,29 quench dynamics,30 and Berry phase.31 The SSHH model describes electrons hopping on a 1D superlattice with staggered hopping amplitudes but uniform local interaction. In this model, there exists a charge excitation gap at half- filling due to the on-site interaction (the Mott gap) and another gap at quarter-filling due to dimerization. This opens the possibility of realizing these fillings in experiments, for example by measuring transport while varying the chemical potential and looking for vanishing conductance when the chemical potential lies in the gaps.18 For this reason, we focus on these two fillings. We introduce the concept of the reduced many-body Zak phase based on the reduced density matrix of a subsystem and show that this phase, rather than the normal many-body Zak phase of the full system, should be used for classifying the topological phases at half-filling. This phase jumps from 0 to π as the hopping amplitude difference between the even and odd sites changes sign. At half-filling, the usual bulk–edge correspondence is manifested in the topological phase transition: the closing and reopening of the eigenenergy gap at the transition point accompanies the appearance of uncorrelated edge states. This is evident in the triplon-excitation spectrum of the dimer chain. In contrast, at quarter-filling the edges remain correlated to the bulk for both signs of the hopping amplitude difference, because of the presence of a long-range antiferromagnetic (AFM) order. There is also no gap in the eigenenergy spectrum due to the presence of gapless spin excitations. So the quarter-filled state does not show the characteristics of a TI. However, we show that applying an external magnetic field leads to a transition to the topological ground state of the non-interacting Su–Schrieffer–Heeger (SSH) model. Importantly, the strong on-site interaction significantly reduces the critical field strength required for the transition. Thus our analysis paves the way for the observation of electronic 1D topological insulator states in nanofabricated semiconductor devices. We propose a device architecture for observing this transition in a 1D chain of dopant atoms or quantum dots. The transition can be probed by measuring the tunneling current through the edges of the chain, which we estimate using a manybody formulation for the conductance of coupled quantum dots.32–34 1 Advanced Technology Institute and Department of Physics, University of Surrey, Guildford GU2 7XH, UK. 2Department of Physics and Astronomy and London Centre for Nanotechnology, University College London, Gower Street, London WC1E 6BT, UK. 3Department of Electronic and Electrical Engineering and London Centre for Nanotechnology, University College London, Gower Street, London WC1E 6BT, UK. ✉email: Published in partnership with The University of New South Wales N.H. Le et al. 2 RESULTS The SSHH model The SSHH Hamiltonian is H ¼ H0 þ V; where j (1) h i  1 þ ð1Þj Δt cyjþ1;σ cj;σ þ h:c:; X H0 ¼ Bloch form X X ψk ¼ eijkd=2 uk ðjÞ ¼ eijkd=2 eiθj ðkÞ cyj j+i; j;σ¼"# and V¼U X nj;" nj;# : 1234567890():,; j H0 is the well-known non-interacting SSH model35 of a particle hopping along a chain with staggered hopping amplitudes, t± = 1 ± Δt, as shown in Fig. 1a, and V is the on-site interaction. Here cyj;σ denotes the creation operator for the particle at site j and spin σ. All energies in this paper are scaled by the mean value of the two hopping amplitudes. We first describe briefly the topological phases of the SSH model given by H0.36,37 For 1D periodic systems of independent particles, the Berry phase picked up during an adiabatic process when the particle moves across the Bloch states in the Brillouin zone, first discussed by Zak,38 is Z π=d (2) dk huk j∂k juk i; ϕ¼i π=d where uk is the periodic part of the Bloch wavefunction, k the crystal mome (...truncated)