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Figure 6. Cold-atom QS of the topological Schwinger model: (a) Bose-Fermi mixture trapped in a 1D optical lattice formed by a red- detuned very deep lattice for the bosons (red circles trapped in two internal states at the maxima), and a blue-detuned shallower lattice for the fermions (blue circles trapped in two internal states at the minima). (b) Dominant scattering processes, including on-site boson-boson and fermion-fermion Hubbard interactions, as well as boson-fermion spin-changing collisions allowed by the common overlap of neighboring Wannier functions for the fermions, weighted by the probability to find a boson on the intermediate link. (c) Inhibition of the bosonic and fermionic bare tunnelings, either due to the very deep bosonic lattice, or to the application of a lattice tilting for the fermions. The spin-changing collisions can be assisted against the lattice tilting introducing a periodic modulation of the bosonic energy levels.

Figure 6 Cold-atom QS of the topological Schwinger model: (a) Bose-Fermi mixture trapped in a 1D optical lattice formed by a red- detuned very deep lattice for the bosons (red circles trapped in two internal states at the maxima), and a blue-detuned shallower lattice for the fermions (blue circles trapped in two internal states at the minima). (b) Dominant scattering processes, including on-site boson-boson and fermion-fermion Hubbard interactions, as well as boson-fermion spin-changing collisions allowed by the common overlap of neighboring Wannier functions for the fermions, weighted by the probability to find a boson on the intermediate link. (c) Inhibition of the bosonic and fermionic bare tunnelings, either due to the very deep bosonic lattice, or to the application of a lattice tilting for the fermions. The spin-changing collisions can be assisted against the lattice tilting introducing a periodic modulation of the bosonic energy levels.

Related Figures (7)

Figure 1. Phase diagram of the topological Schwinger model: The bosonization with the vacuum 6 operator (6) allows us to predict three distinct phases: a symmetry-protected topological (SPT) phase, corresponding to a correlated topological insulator, separated from a confined phase (C) through a continuous quantum phase transi- tion. This confined phase is itself separated from a symmetry-broken fermion condensate (FC) by another continuous phase transition.
Figure 1. Phase diagram of the topological Schwinger model: The bosonization with the vacuum 6 operator (6) allows us to predict three distinct phases: a symmetry-protected topological (SPT) phase, corresponding to a correlated topological insulator, separated from a confined phase (C) through a continuous quantum phase transi- tion. This confined phase is itself separated from a symmetry-broken fermion condensate (FC) by another continuous phase transition.
Figure 2. DMRG phase diagram for Zy topological Schwinger models: (a) Z3 model, including the central charges of the critical lines. (b) Zs and Z7 models, showing that the extension of the SPT phase grows as N is increased.
Figure 2. DMRG phase diagram for Zy topological Schwinger models: (a) Z3 model, including the central charges of the critical lines. (b) Zs and Z7 models, showing that the extension of the SPT phase grows as N is increased.
Figure 3. Block entanglement entropy of the Z3 model: (a) Scal- ing of the entanglement entropy of a subsystem of size / on the crit- ical point A = —0.162 and ga = 0.6. Through a logarithmic fit (15), it is possible to extract the central charge c = 0.506. (b) Same as (a), but for g =A=0.
Figure 3. Block entanglement entropy of the Z3 model: (a) Scal- ing of the entanglement entropy of a subsystem of size / on the crit- ical point A = —0.162 and ga = 0.6. Through a logarithmic fit (15), it is possible to extract the central charge c = 0.506. (b) Same as (a), but for g =A=0.
Figure 4. _ Basis of gauge-invariant Hilbert space for the Z5 model: taking into account Gauss’s law, we implement all possibile configurations of matter/antimatter fields on even/odd sites.
Figure 4. _ Basis of gauge-invariant Hilbert space for the Z5 model: taking into account Gauss’s law, we implement all possibile configurations of matter/antimatter fields on even/odd sites.
Table I. Z5 topological Schwinger model: critical values of A (re- lated to the two transitions FC-C and SPT-C) obtained for different values of g. The numerical error is equal to 10-3. Table II. Z7 topological Schwinger model: critical values of A (re- lated to the two transitions FC-C and SPT-C) obtained for different values of g. The numerical error is equal to 10-3. numerical curves for different system sizes N; collapse onto the same universal function A (x) in the shaded region. (d) Scaling quantity N ve of the electric field order parameter calculated for ga = 0.6 as a function of the dimerization for various system sizes Ny = [24,28,32, 36, 40] (from blue (24) to purple (40)). The crossing point of all curves allows us to determine the critical point separating the FC and C phases A~ =~ —0.193 and A® ~ 2.105 in accordance with the symmetry around A = 1.
Table I. Z5 topological Schwinger model: critical values of A (re- lated to the two transitions FC-C and SPT-C) obtained for different values of g. The numerical error is equal to 10-3. Table II. Z7 topological Schwinger model: critical values of A (re- lated to the two transitions FC-C and SPT-C) obtained for different values of g. The numerical error is equal to 10-3. numerical curves for different system sizes N; collapse onto the same universal function A (x) in the shaded region. (d) Scaling quantity N ve of the electric field order parameter calculated for ga = 0.6 as a function of the dimerization for various system sizes Ny = [24,28,32, 36, 40] (from blue (24) to purple (40)). The crossing point of all curves allows us to determine the critical point separating the FC and C phases A~ =~ —0.193 and A® ~ 2.105 in accordance with the symmetry around A = 1.
Figure 5. Observables and scaling relations for the Zs; topological Schwinger model: (a) Behavior of the topological correlator for A = 0.5 as a function of the gauge coupling constant g with Ns = [40,50, 60, 70, 80] sites (from blue (40) to purple (80)). (b) Scaling quantity NV ° O_a of the topological correlator calculated for A = 0.5 as a function of the gauge coupling for various system sizes N, = [24, 28,32, 36,40] (from blue (24) to purple (40)). The crossing point of all curves allows us to determine the critical point separating the SPT and C phases g,a = 1.786. (c) Universal scaling of the topological correlator with the critical exponents of the 2D Ising universality class B = 1/8 and v = 1: all the numerical curves for different system sizes N; collapse onto the same universal function A (x) in the shaded region. (d) Scaling quantity Nv L
Figure 5. Observables and scaling relations for the Zs; topological Schwinger model: (a) Behavior of the topological correlator for A = 0.5 as a function of the gauge coupling constant g with Ns = [40,50, 60, 70, 80] sites (from blue (40) to purple (80)). (b) Scaling quantity NV ° O_a of the topological correlator calculated for A = 0.5 as a function of the gauge coupling for various system sizes N, = [24, 28,32, 36,40] (from blue (24) to purple (40)). The crossing point of all curves allows us to determine the critical point separating the SPT and C phases g,a = 1.786. (c) Universal scaling of the topological correlator with the critical exponents of the 2D Ising universality class B = 1/8 and v = 1: all the numerical curves for different system sizes N; collapse onto the same universal function A (x) in the shaded region. (d) Scaling quantity Nv L
Table III. Different slopes for small g of the two critical lines.
Table III. Different slopes for small g of the two critical lines.
Related topics:
Condensed Matter PhysicsQuantum PhysicsLattice gauge theoryStrongly-correlated electron systemsTopological phases of quantum matterTopological InsulatorsQuantum SimulationFermions and Bosons
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