Research ArticlePHYSICS

Many-body topological invariants from randomized measurements in synthetic quantum matter

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Science Advances  10 Apr 2020:
Vol. 6, no. 15, eaaz3666
DOI: 10.1126/sciadv.aaz3666
  • Fig. 1 Measuring the MBTI ZR for the extended bosonic SSH model.

    (A) Schematic illustration of the model Eq. 1, where the nearest-neighbor spin-exchange coefficients alternate between the bonds. (B) The partial reflection invariant ZR (Eq. 2) is defined as the expectation value of a partial reflection operator RI (visualized by the blue lines) for the many-body state |ψ〉. The dashed line between the intervals I1 and I2 indicates the reflection center. (C) In terms of the normalized invariant Z˜R, the full-phase diagram of the extended bosonic SSH model is revealed here for a system size of N = 48 spins and n = 6 reflected pairs of spins. We find three phases with different quantized values of Z˜R. (D) Protocol to measure Z˜R via statistical correlations between randomized measurements, implemented with local random unitaries applied symmetrically around the central bond. (E) The results of simulated experiments allow us to identify topological phase transitions. The solid lines are results from DMRG, whereas the dots with error bars represent estimations from simulated randomized measurements with NU = 512 unitaries and NM = 256 measurements per unitary.

  • Fig. 2 Probing the MBTI ZT with randomized measurements.

    (A) Graphical representation of the definition of the time-reversal invariant ZT (Eq. 5) involving partial transpose (red lines) and partial swap (blue lines) operations. (B) Experimental protocol to measure ZT with two experiments, which are correlated using randomized measurements. To account for the anti-unitarity of the time-reversal symmetry, the local random unitaries applied in I1 (red) in the two experiments are complex conjugate to each other. (C) Simulated measurements of Z˜T (dots with statistical error bars, with NU = 768, NM = 512), revealing the topological phase transitions in the extended bosonic SSH model as a function of J′ = J for two values of δ. Solid lines are calculated with the DMRG method, in a system with N = 48 sites, and n = 6 per interval I1 and I2. (D) Z˜T converges as a function of the partition size n to the quantized values ±1 for the case of δ = 0.25. Different colors represent different values of J′ = J. Inset: The divergence of the corresponding correlation length λ, extracted from an exponential fit on the first three values of n, can be used to detect the quantum critical point between the topological trivial (with Z˜T = 1) and nontrivial (with Z˜T = −1) phases.

  • Fig. 3 Monitoring the adiabatic preparation of an SPT state.

    (A) Starting from a trivial Néel state without reflection symmetry Z˜R(t), the ground state of HeSSH is adiabatically prepared. This is monitored by the evolution of Z˜R(t), which evolves to quantized values ±1 at late times. The dynamical buildup of long-range SPT order—for intermediate times up to a certain length scale—is indicated at intermediate times by the increasing magnitude of Z˜R(t) for decreasing number n of reflected pairs of spins. Here, we set JtF = 20. (B) The convergence of Z˜R(tF) to ±1 as a function of the total preparation time tF indicates that, for sufficiently long preparation times, the ground states in trivial and topological states are prepared with high fidelity. For the simulations, we use the time-evolving block decimation (TEBD) algorithm (as detailed in Materials and Methods) and set the parameters as δ = 0.25, Δ = 40J, and N = 48.

  • Fig. 4 Detecting the protecting symmetries for the SPT states.

    In the presence of the symmetry-breaking perturbation HB (Eq. 8), the topological phase in the modified Hamiltonian H = HeSSH + HB is (only) protected by the time-reversal symmetry. (A) This is detected by the partial time-reversal MBTI Z˜T—converging to the quantized values ±1 for increasing n—which still identifies the topological phase transition. (B) On the contrary, the partial reflection MBTI Z˜R—approaching 0 with increasing n—shows that the reflection symmetry is explicitly broken for a nonzero B in Eq. 8. We choose B = 0.1J, δ = 0.3, and N = 48.

Supplementary Materials

  • Supplementary Materials

    Many-body topological invariants from randomized measurements in synthetic quantum matter

    Andreas Elben, Jinlong Yu, Guanyu Zhu, Mohammad Hafezi, Frank Pollmann, Peter Zoller, Benoît Vermersch

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