Lindblad-driven discretized leads for nonequilibrium steady-state transport in quantum impurity models: Recovering the continuum limit

The description of interacting quantum impurity models in steady-state nonequilibrium is an open challenge for computational many-particle methods: the numerical requirement of using a finite number of lead levels and the physical requirement of describing a truly open quantum system are seemingly incompatible. One possibility to bridge this gap is the use of Lindblad-driven discretized leads (LDDL): one couples auxiliary continuous reservoirs to the discretized lead levels and represents these additional reservoirs by Lindblad terms in the Liouville equation. For quadratic models governed by Lindbladian dynamics, we present an elementary approach for obtaining correlation functions analytically. In a second part, we use this approach to explicitly discuss the conditions under which the continuum limit of the LDDL approach recovers the correct representation of thermal reservoirs. As an analytically solvable example, the nonequilibrium resonant level model is studied in greater detail. Lastly, we present ideas towards a numerical evaluation of the suggested Lindblad equation for interacting impurities based on matrix product states. In particular, we present a reformulation of the Lindblad equation, which has the useful property that the leads can be mapped onto a chain where both the Hamiltonian dynamics and the Lindblad driving are local at the same time. Moreover, we discuss the possibility to combine the Lindblad approach with a logarithmic discretization needed for the exploration of exponentially small energy scales.

Figure 1

Schematic depiction of the model for two physical leads $\alpha =\{L,R\}$. Each lead level $q$ couples to the impurity level $i$ with coupling strength $v_{iq}$. The reservoir $R=R_L+R_R$ consists of one Lindblad reservoir ${\mathcal{R}}_q$ for each lead level $q$, whose Lindblad driving rate is chosen such that it tends to drive that level's occupancy towards $f_{\alpha }\left({\varepsilon }_q\right)$ (though a small deviation from the latter will be induced by the level-dot coupling, see Sec. 3e for details). The value of $f_{\alpha }\left({\varepsilon }_q\right)$ is symbolized by the degree of filling of the corresponding open circle. The occupation numbers for the left and right leads differ for a system in nonequilibrium.

Figure 2

For a single lead with a continuum hybridization function as defined in Eq. (46), we plot the corresponding Lindblad result based on Eq. (42) and the definition (31) for different values of $\gamma$, which was chosen to be independent of $q$. A linear discretization is used with $M=2D/\delta$ lead levels for a level spacing of $\delta =0.1$. We need ${\displaystyle v_k=v=\sqrt{{\mathrm{\Gamma }}_0\delta /\pi }}$ to ensure the correct continuum limit ${\mathrm{\Delta }}_{\text{CL}}\left(\omega \right)$. The black curve represents the continuum limit (46a) and (46b), respectively. All energies are given in units of $D$.

Figure 3

The current through the local level of the RLM [Eq. (55)] with linearly discretized leads for several values of the level spacing $\delta$. The two panels show the same data, but in the left panel as function of $\gamma /\delta$ and in the right panel as function of $\gamma /{\mathrm{\Gamma }}_0$. This illustrates that the decrease in the current for small values of $\gamma$ is a discretization effect while the decrease for large $\gamma$ corresponds to an overdriving of the system. The correct physics can only be obtained if $\delta \lesssim \gamma \ll {\mathrm{\Gamma }}_0$.

Figure 4

The current [Eq. (55)] for several different level spacings $\delta$ using linearly discretized leads and ${\gamma }_q=\delta$ as a function (a) of voltage $V$, (b) temperature $T$, and (c) the level energy ${\varepsilon }_d$. For sufficiently small $\delta$, the exact black curve, calculated from the continuum limit of Eq. (56), is reproduced with a deviation of less than one percent. In (a), one can clearly see discretization artifacts for $\delta ={10}^{-2}$ and $\delta ={10}^{-3}$, which vanish where $\delta /V$ gets small enough. Analogously, also in (b), it is apparent that for larger temperatures $T$, larger level spacings $\delta$ can be used, while for small $T$ small level spacings are needed.

Figure 5

Relative error of the current for the parameters of Fig. 4, at ${\varepsilon }_d=0$. A linear fit, obtained from the data shown in the inset, yields an offset smaller than ${10}^{-4}$, showing that the LDDL scheme becomes exact in the continuum limit.

Figure 6

The occupation number of the local level for the RLM in the Lindblad approach as given by Eq. (58), as function of a symmetrically applied voltage. The discretization was chosen to be linear with different values for the level spacing $\delta$. The black curve represents the continuum limit of Eq. (60). If the level spacing $\delta$ is small enough compared to the voltage $V$, the exact result is recovered. For $\delta ={10}^{-2}$, one can clearly see discretization artifacts.

Figure 7

The occupation numbers $N_{q;\text{DL}}$ for the left and right channels of a symmetric RLM as defined in Eq. (57), choosing a linear discretization with level spacing ${\delta }_q=\delta$ and $q$-independent broadening ${\gamma }_q=\gamma$. In (a), we show $N_{q;\text{DL}}$ for several values of $\delta$ at a fixed ratio $\delta /\gamma =1$. If the level spacing is small enough, the exact current is reproduced, although $N_{q;\text{DL}}$ deviates from the Fermi distribution by a nonzero amount $\delta N_{q;\text{DL}}$, which for small enough $\delta$ is given by Eq. (62). In (b), $\gamma$ is kept fixed at a value that can resolve the physical relevant energy scales (here ${\mathrm{\Gamma }}_0$ and $V)$ while $\delta$ is varied. Reducing $\delta ,N_{q;\text{DL}}$ approaches the Fermi distribution, but as soon as $\delta$ becomes $\lesssim \gamma$ the accuracy of $J_{\text{DL}}$ (taking $J_{\text{CL}}$ as reference) does not improve.

Figure 8

Schematic depiction of the level-doubling construction scheme for two leads, $\alpha =L$ and $R$, assuming constant values of $v_{i\alpha k}$. (a) Original levels of the left and right leads, described by $c_{\alpha k}^{(†)}$ operators. (b) After level doubling, each lead $\alpha$ is represented by two sets of auxiliary levels, distinguished by $\eta =1$ and 2 and Lindblad-driven towards occupancy 0 and 1, respectively. These levels are described by ${\tilde{c}}_{\alpha k\eta }^{(†)}$ operators, whose coupling strengths ${\tilde{v}}_{i\alpha k\eta }$ (indicated by the width of the horizontal lines) depend on the Fermi function $f_{\alpha }\left({\varepsilon }_q\right)$ of that lead (depicted by smooth black curves). For $\eta =1$ (or 2), all those auxiliary levels decouple for which $f_{\alpha }\left({\varepsilon }_q\right)\approx 1$ (or 0), indicated by grey shading. (c) For a model involving just a single impurity level, only certain linear combinations of $L$ and $R$ auxiliary lead operators, the ${\tilde{b}}_{k\eta }^{(†)}$ operators of Eq. (71a), couple to the impurity; they are depicted here by double lines representing the couplings ${\tilde{v}}_{Lk\eta }$ and ${\tilde{v}}_{Rk\eta }$, with grey shading indicating vanishing couplings.

Figure 9

Sketch of the suggested discretization: the high-energy intervals are discretized logarithmically, while a window $[-E^{*},E^{*}]$ large enough compared to the dynamical window is discretized linearly.

Figure 10

(a) and (b) show the current through the local level of the RLM as given by Eq. (55) using a discretization, which is linear within the dynamical window $[-E^{*},E^{*}]$ and logarithmic outside as a function of $E^{*}$. In (a), the voltage is large compared to temperature and the correct value for the current can only be obtained if $E^{*}\gtrsim 1.2{\mu }_L=-1.2{\mu }_R$. In (b), temperature is larger than voltage and $E^{*}\gtrsim 4T$ is needed. In (c), the occupation of the local level in the RLM as given by Eq. (58) is shown as a function of the level position ${\varepsilon }_d$, again using the logarithmic-linear discretization. For ${\varepsilon }_d\gtrsim E^{*}$ we see deviations from the CL result, see Appendix pp1 for details. For all three panels, $\mathrm{\Lambda }=2$ and we used ${\gamma }_q=\gamma ={\delta }_{\text{lin}}$ for the linear and the logarithmic states. The number of lead levels is approximately given by $M_{\text{lin}}=2E^{*}/{\delta }_{\text{lin}}$ plus $M_{\text{log}}=-2\text{log}\left(E^{*}\right)/\text{log}\left(\mathrm{\Lambda }\right)$.