Mean field phase diagrams for a balanced-spin mixture at unity filling, ρ = 2. The calculations are performed for Us/Uv ≈ 4.64 (a,b) and at Us/Uv ≈ 0.33 (c,d). (a,c) Dependence of superfluid order parameter ψ. The solid lines are perturbative estimations for the transition. (b,d) Dependence of density and antiferromagnetic order parameters θD and θS. (e,f,g) Cuts along the phase diagrams for Us/Uv ≈ 4.64 at constant UL/U = 0.63 and UL/U = 1.41 (e,f) and for Us/Uv ≈ 0.33 at UL/U = 1.3 (g), indicated by thin horizontal lines in (a,c). The basis truncation nmax = mmax = 3 leads to a saturation of the superfluid order parameter at large tunneling.

Contexts in source publication

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... m =↑, ↓. The choice to work at fixed density is motivated by experiments with ultracold atoms, although a qualitatively similar phase diagram arises in a grand canonical ensemble [37,47]. In this section, we assume U 12 = U . The different order parameters are shown in Fig. 2. The competition of scalar and vectorial longrange interactions gives rise to two qualitatively different ...

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... Fig. 2(a,b)], we observe two distinct insulating phases (ψ = 0) at low tunneling rates zt/U : For large U L , a spin-degenerate charge density wave (CDW), with θ D > 0 and θ S = 0. For small U L , an antiferromagnetic Mott insulator (AFM), with θ S > 0 and θ D = 0. Remarkably, the system favors an AFM for arbitrarily small vectorial contributions ...

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... the regime U v > U s [ Fig. 2(c,d)], the system exhibits solely spin ordered phases (θ S > 0, and θ D = 0), as the vectorial long-range and the contact interactions dominate over the scalar long-range interaction. For small U L , we identify a first-order AFM-SF phase transition, signaled by a discontinuous jump of ψ and θ S [ Fig. 2(e)]. This is in contrast to the ...

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... the regime U v > U s [ Fig. 2(c,d)], the system exhibits solely spin ordered phases (θ S > 0, and θ D = 0), as the vectorial long-range and the contact interactions dominate over the scalar long-range interaction. For small U L , we identify a first-order AFM-SF phase transition, signaled by a discontinuous jump of ψ and θ S [ Fig. 2(e)]. This is in contrast to the second-order MI-SF transition in the absence of the long-range interactions (U L = 0). For larger U v /U the AFM phase extends towards higher tunneling strengths. AFM-SF transitions in the context of entanglement properties have recently been studied in three-component BH models with long-range interactions ...

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... the AFM phase extends towards higher tunneling strengths. AFM-SF transitions in the context of entanglement properties have recently been studied in three-component BH models with long-range interactions [38]. For larger U L , we observe second-order phase transitions from AFM to AF-SS and from CDW to SS phases, along lines of constant U L /U [ Fig. 2(f,g)]. The second-order phase transitions from AFM to AF-SS and CDW to SS are supported by perturbative estimations, cf. black lines in Fig. 2(a,c) [47,56]. ...

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... in three-component BH models with long-range interactions [38]. For larger U L , we observe second-order phase transitions from AFM to AF-SS and from CDW to SS phases, along lines of constant U L /U [ Fig. 2(f,g)]. The second-order phase transitions from AFM to AF-SS and CDW to SS are supported by perturbative estimations, cf. black lines in Fig. 2(a,c) [47,56]. ...

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... first discuss the results for U s /U v > 1, see Fig. 3(ac). For dominating intra-spin interactions [ Fig. 3(a)], U 12 < U , the phases are identical as those obtained for U 12 = U assuming a uniform mixture as discussed in the context of Fig. 2. We additionally find a region of instability in the SS phase for U 12 ≤ U . Our observations of phase instability are qualitatively different from the results for spinless systems [17,46], which predict stable supersolid phases at integer filling in two-dimensional systems. For In the case of dominating vectorial long-rage ...

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... note here two limitations of our simulations. First, the identification of different phases in the phase diagrams relies on numerical minimization [47] in a high-dimensional landscape. This can lead to spurious solutions, such as the scattered instability points in Fig. 3(a,b) and the irregular phase boundaries in Fig. 2 and 3. Second, there is a small region of fully PS SS for U 12 = U and U s /U v > 1, cf. Fig. 3(b). This is due to the relatively small Hilbert space (n max = m max = 3) used for the simulations. We expect that a larger Hilbert space would lead to degenerate SS and PS SS solutions. Additional information on the different properties of the ...

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... particular, under the assumption of unity filling the CDW phase will not be formed if U s < U v or equivalently 1/ξ 2 < tan(φ) 2 . For the values used in Fig. 2(a,b) of the main text with |ξ| = 0.464 and φ = π/4 the transition occurs at (U L /U ) c ≈ ...

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... start by studying the dependence of different order parameters on the ratio U 12 /U . This is shown in Fig. S1 for the same parameters as used in Fig. 2(e-g) of the main text. Here, we do not consider phase separation or instabilities. The evolution of different order parameters as a 0.00 0.25 0.50 0.75 function of zt/U is similar with a slight increase in the critical tunnelling for the transition to supersolid or superfluid phases with increasing U 12 /U . Such an increase is expected as ...

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... p i , where the summation is over all the eigenvalues p i of ρ red . By construction, the entropy S j,↑ = S j,↓ for jj{e, o} for uniform density order solutions and S e,↑ = S o,↓ , S o,↑ = S e,↓ for uniform spin order solutions. So, we accordingly show only S e,↑ , S e,↓ and S e,↑ , S o,↑ for phases with density and spin order respectively, see Fig. S2. We see that except for CDW phase in the U 12 /U ≥ 1 cases, the insulating phases are locally product states and hence have zero von Neumann entropy. For U 12 /U > 1, the ground state is (|2, 0) + |0, 2/ √ 2 on one site and |0, 0 on the other which is the maximally entangled state in a two-dimensional subspace and hence has ρ red = ln ...

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... phase boundary displayed in Fig. 2 of the main part is obtained by employing a perturbation theory analysis in the GCE starting from the effective Hamiltonian Eq. (S35) [56]. We rewrite Eq. ...

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... m =↑, ↓. The choice to work at fixed density is motivated by experiments with ultracold atoms, although a qualitatively similar phase diagram arises in a grand canonical ensemble [37,47]. In this section, we assume U 12 = U . The different order parameters are shown in Fig. 2. The competition of scalar and vectorial longrange interactions gives rise to two qualitatively different ...

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... Fig. 2(a,b)], we observe two distinct insulating phases (ψ = 0) at low tunneling rates zt/U : For large U L , a spin-degenerate charge density wave (CDW), with θ D > 0 and θ S = 0. For small U L , an antiferromagnetic Mott insulator (AFM), with θ S > 0 and θ D = 0. Remarkably, the system favors an AFM for arbitrarily small vectorial contributions ...

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... the regime U v > U s [ Fig. 2(c,d)], the system exhibits solely spin ordered phases (θ S > 0, and θ D = 0), as the vectorial long-range and the contact interactions dominate over the scalar long-range interaction. For small U L , we identify a first-order AFM-SF phase transition, signaled by a discontinuous jump of ψ and θ S [ Fig. 2(e)]. This is in contrast to the ...

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... the regime U v > U s [ Fig. 2(c,d)], the system exhibits solely spin ordered phases (θ S > 0, and θ D = 0), as the vectorial long-range and the contact interactions dominate over the scalar long-range interaction. For small U L , we identify a first-order AFM-SF phase transition, signaled by a discontinuous jump of ψ and θ S [ Fig. 2(e)]. This is in contrast to the second-order MI-SF transition in the absence of the long-range interactions (U L = 0). For larger U v /U the AFM phase extends towards higher tunneling strengths. AFM-SF transitions in the context of entanglement properties have recently been studied in three-component BH models with long-range interactions ...

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... the AFM phase extends towards higher tunneling strengths. AFM-SF transitions in the context of entanglement properties have recently been studied in three-component BH models with long-range interactions [38]. For larger U L , we observe second-order phase transitions from AFM to AF-SS and from CDW to SS phases, along lines of constant U L /U [ Fig. 2(f,g)]. The second-order phase transitions from AFM to AF-SS and CDW to SS are supported by perturbative estimations, cf. black lines in Fig. 2(a,c) [47,56]. ...

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... in three-component BH models with long-range interactions [38]. For larger U L , we observe second-order phase transitions from AFM to AF-SS and from CDW to SS phases, along lines of constant U L /U [ Fig. 2(f,g)]. The second-order phase transitions from AFM to AF-SS and CDW to SS are supported by perturbative estimations, cf. black lines in Fig. 2(a,c) [47,56]. ...

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... first discuss the results for U s /U v > 1, see Fig. 3(ac). For dominating intra-spin interactions [ Fig. 3(a)], U 12 < U , the phases are identical as those obtained for U 12 = U assuming a uniform mixture as discussed in the context of Fig. 2. We additionally find a region of instability in the SS phase for U 12 ≤ U . Our observations of phase instability are qualitatively different from the results for spinless systems [17,46], which predict stable supersolid phases at integer filling in two-dimensional systems. For In the case of dominating vectorial long-rage ...

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... note here two limitations of our simulations. First, the identification of different phases in the phase diagrams relies on numerical minimization [47] in a high-dimensional landscape. This can lead to spurious solutions, such as the scattered instability points in Fig. 3(a,b) and the irregular phase boundaries in Fig. 2 and 3. Second, there is a small region of fully PS SS for U 12 = U and U s /U v > 1, cf. Fig. 3(b). This is due to the relatively small Hilbert space (n max = m max = 3) used for the simulations. We expect that a larger Hilbert space would lead to degenerate SS and PS SS solutions. Additional information on the different properties of the ...

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... particular, under the assumption of unity filling the CDW phase will not be formed if U s < U v or equivalently 1/ξ 2 < tan(φ) 2 . For the values used in Fig. 2(a,b) of the main text with |ξ| = 0.464 and φ = π/4 the transition occurs at (U L /U ) c ≈ ...

Context 22

... start by studying the dependence of different order parameters on the ratio U 12 /U . This is shown in Fig. S1 for the same parameters as used in Fig. 2(e-g) of the main text. Here, we do not consider phase separation or instabilities. The evolution of different order parameters as a 0.00 0.25 0.50 0.75 function of zt/U is similar with a slight increase in the critical tunnelling for the transition to supersolid or superfluid phases with increasing U 12 /U . Such an increase is expected as ...

Context 23

... p i , where the summation is over all the eigenvalues p i of ρ red . By construction, the entropy S j,↑ = S j,↓ for jj{e, o} for uniform density order solutions and S e,↑ = S o,↓ , S o,↑ = S e,↓ for uniform spin order solutions. So, we accordingly show only S e,↑ , S e,↓ and S e,↑ , S o,↑ for phases with density and spin order respectively, see Fig. S2. We see that except for CDW phase in the U 12 /U ≥ 1 cases, the insulating phases are locally product states and hence have zero von Neumann entropy. For U 12 /U > 1, the ground state is (|2, 0) + |0, 2/ √ 2 on one site and |0, 0 on the other which is the maximally entangled state in a two-dimensional subspace and hence has ρ red = ln ...

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... phase boundary displayed in Fig. 2 of the main part is obtained by employing a perturbation theory analysis in the GCE starting from the effective Hamiltonian Eq. (S35) [56]. We rewrite Eq. ...