Enhancing violations of Leggett-Garg inequalities in nonequilibrium correlated many-body systems by interactions and decoherence

We identify different schemes to enhance the violation of Leggett-Garg inequalities in open many-body systems. Considering a nonequilibrium archetypical setup of quantum transport, we show that particle interactions control the direction and amplitude of maximal violation, and that in the strongly-interacting and strongly-driven regime bulk dephasing enhances the violation. Through an analytical study of a minimal model we unravel the basic ingredients to explain this decoherence-enhanced quantumness, illustrating that such an effect emerges in a wide variety of systems.

applied to different sites. The results are shown in Fig. S2. For maximal driving f = 1, the Leggett-Garg function has a very similar behavior to that evaluated at the central site ( Fig. 4(a) of the main text), but with a lower violation since excitations created closer to the boundaries propagate more slowly. This is more clearly seen for weak dephasing, due to the presence of the ferromagnetic domains. In fact, for zero dephasing, no violation is seen for measurement sites away from the centre. For intermediate driving, deep inside the chain, the Leggett-Garg function is essentially site-independent for low dephasing rates γ; differences emerge for larger γ, for which the violation of LGIs starts to be degraded. For sites closer to the boundary the violation enhancement compared to γ = 0 is weaker, but still notable. Thus the main results of violation enhancement by dephasing are not exclusive to the central spin. Now we consider the effect of non-local measurements. For this we take strings of operatorsQ =σ z lσ z l+1σ z l+2 · · ·σ z n of different length, located around the central site of the lattice. The results, shown in Fig. S3, are qualitatively very similar to those obtained for the central site (and different sites, as seen in Fig. S2). For maximal driving we again see an enhancement of the violation even for large values of γ. However, as the string ofσ z l operators includes sites closer to the boundaries, the maximal violation decreases; furthermore, for zero dephasing, no appreciable violation is seen for long-enough strings. For intermediate driving, the violation of the LGIs for several sites is also similar to that of a single site, with a very small but observable enhancement by dephasing. These results indicate that our observations remain qualitatively unaffected if non-local operators are considered instead of single-site operators.

S3 Analytic results on the minimal model
In the following we describe an analytic approach to obtain the Leggett-Garg functions of the simple minimal model, focused on large driving and weak dephasing. We first note that in the maximally-driven limit f = 1 and in the absence of dephasing, this model features an insulating NESS akin to that of the original model of the main text, of the form 2 with normalization constant This result shows that the population of a site exponentially decreases as it gets far from the rightmost site |K .

S3.1 Perturbative 1 − f solution to the minimal model
Now we move from the maximally-driven case, and consider the situation where a weak back-flow is introduced, while keeping γ = 0. We obtain the correction to the NESS to first order in µ = 1 − f 1, given bŷ For this we rewrite the Lindblad superoperator of Eq. (2) of the main text in the form whereL (0) (ρ) corresponds to the µ = 0 Lindblad superoperator, namelŷ andL (1) (ρ) to the remaining terms, where each dissipator D is given by To solve Eq. (S4) for the NESSρ ∞ , we use the anstaẑ Gathering terms of equal powers of µ, we obtain the equations that determine the different order corrections. For p = 0, this corresponds toL (0) ρ (0) = 0, which is the µ = 0 equation with the solution of Eq. (S1). For p > 0 we havê which indicates that to obtain the correction of O(p), that of O(p − 1) is required. Importantly, sinceρ (0) is a valid density matrix, its trace is one, so the p > 0 corrections are traceless. In the following calculation we restrict to p = 1, valid for a weak deviation from the maximally-driven system. Equation (S9) becomeŝ from which we can obtainρ (1) fromρ (0) . First we calculateL (1) (ρ (0) ), obtaininĝ

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(S11) To calculateL (0) (ρ (1) ) we consider the following general form forρ (1) , a m,n |K − m K − n|. (S12) Note that site |s has no coherences with other sites, as it is only incoherently coupled to other sites (|1 and |K ). We obtain (S13) After gathering the coefficients a m,n of all the different basis elements |K − m K − n| in Eqs. (S11) and (S13), and considering that a * m,n = a n,m so the density matrix is Hermitian, we obtain a linear system of equations whose solution gives the correction ρ (1) . Rewriting each correction coefficient as a m,n = r m,n + i j m,n and considering the Hermiticity of the density matrix (r m,n = r n,m and j m,n = − j n,m ), this is transformed into a linear system of 1 + K 2 equations with (K 2 + K + 2)/2 unknowns r m,n (including r s ≡ a s ) and (K 2 − K)/2 unknowns j m,n (the diagonal elements must be real). In addition, sinceρ (1) is traceless, the diagonal coefficients must also satisfy (S14) We solve this system of equations up to O(δ −2 ), for whichρ (1) has the (almost tridiagonal) form The lattice populations have four possible values, corresponding to boundary (r K−1 , r 1 , r 0 ) and bulk (r b ) values, given by (S16) The off-diagonal real values are (S17) The imaginary values correspond to the expectation value of the (homogeneous) NESS current, given by (S18) Finally, the population of the auxiliary state is r s = 1/2. We have verified numerically that these results indeed correspond to the correct solution to the linear system of equations, and thus give the correct form of the first-order correctionρ (1) .

S3.2 LGIs violation enhancement by driving
Here we discuss the Leggett-Garg functions of the NESS of the strongly-driven minimal model. We first consider the f = 1 limit. Applying the operatorQ =Î − 2|P P| toρ (0) we get The dominant term of the sum is of O((2δ ) −(K−p) ) (for n = 0), which is insignificant within the order O(δ −2 ) of our solution if site P is far from the right boundary. ThusQρ (0) ≈ρ (0) , and the time correlation is It is approximately constant, as expected from the insulating nature of the state. The corresponding Leggett-Garg function is indicating that the LGI is not violated 1 .
Now we consider the system slightly below maximal driving, namely with µ = 1 − f 1, whose NESS was discussed in Section S3.1. With the NESS of Eq. (S3) calculated up to first order in µ we can proceed to obtain the perturbative correction to the early-time time correlations C (1) (t) and Leggett-Garg function K (1) (t), so that Expanding the time correlations in Eq.
(3) of the main text up to second order in time, the perturbative correction to the f = 1 Leggett-Garg function is indicating that there is no linear correction in time for K . To calculate it, first we note that applyingQ toρ (1) giveŝ The correction to the Leggett-Garg function is then which is also positive and thus indicates that dephasing induces a violation of the LGIs, which increases with γ. This result is also in agreement with exact numerical calculations, as depicted in In Fig. S4(b). If both a weak backflow and dephasing are present, where both mechanisms enhance the LGI violations, the result is simply the sum of the independent contributions of Eqs. (S27) and (S29), namely This additive enhancement of the two mechanisms is verified by exact numerical simulations, which as shown in Fig. S4(c), present a very good agreement.