Anderson orthogonality in the dynamics after a local quantum quench

We present a systematic study of the role of Anderson orthogonality for the dynamics after a quantum quench in quantum impurity models, using the numerical renormalization group. As shown by Anderson in 1967, the scattering phase shifts of the single-particle wave functions constituting the Fermi sea have to adjust in response to the sudden change in the local parameters of the Hamiltonian, causing the initial and final ground states to be orthogonal. This so-called Anderson orthogonality catastrophe also influences dynamical properties, such as spectral functions. Their low-frequency behavior shows nontrivial power laws, with exponents that can be understood using a generalization of simple arguments introduced by Hopfield and others for the x-ray edge singularity problem. The goal of this work is to formulate these generalized rules as well as to numerically illustrate them for quantum quenches in impurity models involving local interactions. As a simple yet instructive example, we use the interacting resonant level model as testing ground for our generalized Hopfield rule. We then analyze a model exhibiting population switching between two dot levels as a function of gate voltage, probed by a local Coulomb interaction with an additional lead serving as charge sensor. We confirm a recent prediction that charge sensing can induce a quantum phase transition for this system, causing the population switch to become abrupt. We elucidate the role of Anderson orthogonality for this effect by explicitly calculating the relevant orthogonality exponents.

Figure 1

(a) Cartoon of the Hamiltonian (2) for the LCM. (b)–(g) Cartoons of the occupation of the dot and a half-filled lead, for $U>0$, for several states discussed in the text. (b) and (c) give two equivalent depictions of the ground state $|G_0\rangle$ of ${\widehat{H}}_0$. (c) depicts the fact that $|g_0\rangle$ can be written as a superposition of the form ${|0\rangle }_{\mathrm{c}}{|Q\rangle }_{\mathrm{rest}}+{|1\rangle }_{\mathrm{c}}{|Q-1\rangle }_{\mathrm{rest}}$, indicating complementary occupations of the first site and the rest of a half-filled Wilson chain (defined in Sec. 2f below). Here, ${|0\rangle }_{\mathrm{c}}$ (which obeys $\widehat{c}{|0\rangle }_{\mathrm{c}}=0$) and ${|1\rangle }_{\mathrm{c}}={\widehat{c}}^{†}{|0\rangle }_{\mathrm{c}}$ describe the first site of the Wilson chain being empty or filled, respectively; the charge in the rest of the Wilson chain is correspondingly distributed in such a way that both components of the superposition have the same total charge, $Q$. (d) depicts the ground state $|G_1\rangle$ of ${\widehat{H}}_1$, indicating that charge on the dot pushes charge in the lead away from the dot site. (e) shows the effect of applying ${\widehat{d}}^{†}$ to $|G_0\rangle$, the latter depicted according to (b). Similarly, (f) and (g) show the effect of applying ${\widehat{c}}^{†}{\widehat{d}}^{†}$ or $\widehat{c}{\widehat{d}}^{†}$ to $|G_0\rangle$, the latter depicted according to (c). The displaced charge flowing inwards from infinity toward the dot as each of the states (e)–(g) evolves to the final ground state $|G_1\rangle$ of (d) is ${\Delta }_d<0$, ${\Delta }_d-1<0$ or ${\Delta }_d+1>0$, respectively. Comparison of (f) and (g) with (e) shows average charge differences of $+1$ and $-1$, respectively, in accord with the Hopfield-type argument summarized by Eq. (15).

Figure 2

Numerical results for the LCM of Eq. (2), for the type 2 quench of Eq. (8), whose initial, final, and postquench initial states $|G_{\mathrm{i}}\rangle$, $|G_{\mathrm{f}}\rangle$, and $|{\psi }_{\mathrm{i}}\rangle$ are depicted in Figs. 1b, 1c, 1d and 1e–1(g), respectively. (a) Comparison of the decay exponent ${\Delta }_d$ obtained from Eq. (35) (crosses) with the displaced charge ${\Delta }_d^{\mathrm{ch}}$ from Eq. (36) (pluses) for a number of different values of $U$. The two values agree very well (they differ by less than $0.1\%$), also with the analytic prediction Eq. (37) (solid line). As expected, ${\Delta }_d\to -1/2$ for $U\to \infty$. (b) Comparison of two ways of determining the AO exponents $\eta$ that govern the low-energy asymptotic behaviour $\mathcal{A}\sim {\omega }^{-1+2\eta }$ of the spectral functions of Eqs. (39), related to Figs. 1e–1(g): exponents obtained by fitting a power law to the corresponding spectra [shown in (c)] are shown as crosses (marked “spec” for “spectra”); the corresponding exponents expected from Eq. (39), using the results of (a) for ${\Delta }_d$, are shown as dots (marked “exp” for “expected”). We find a maximal deviation of less than $1\%$. Here and in all similar figures below, the dashed lines are only guides to the eye. (c) Asymptotic low-frequency dependence of the spectra Eqs. (39), for $U=1$, on a double logarithmic plot, allowing the corresponding exponents $\eta$ to be extracted.

Figure 3

(a) Cartoon of the Hamiltonian (40) for the IRLM. (b) The renormalized level width ${\Gamma }_{\mathrm{ren}}$, calculated via the dot's charge susceptibility, $1/\pi {\chi }_c$,[30] (dots) or via Eq. (41) (solid line), shown as a function of $U$ for ${\varepsilon }_d=0$ and several values of $\Gamma$. As $U$ increases from 0, ${\Gamma }_{\mathrm{ren}}/\Gamma$ begins to differ significantly from its initial value, namely, 1, only once $U$ becomes comparable to the bandwidth, reaching its maximal value ${(\Gamma /D)}^{-1/2}$ for $U\gg D$.

Figure 4

(a)–(c) The equilibrium spectral functions ${\mathcal{A}}_d^{\mathrm{eq}}\left(\omega \right)$, ${\mathcal{A}}_{dc}^{\mathrm{eq}}\left(\omega \right)$, and ${\mathcal{A}}_{d{c}^{†}}^{\mathrm{eq}}\left(\omega \right)$ for the IRLM, showing a crossover from trivial power laws, ${\omega }^{-1+2{\eta }^{\mathrm{eq}}}$, for $\omega <{\omega }^{*}$, to AO power laws, ${\omega }^{-1+2{\eta }^0}$, for ${\omega }^{*}<\omega <D$, with the crossover frequency ${\omega }^{*}$ given by ${\Gamma }_{\mathrm{ren}}$. (d) Comparison of the exponents ${\eta }^{\mathrm{eq}}$ (triangles) and ${\eta }^0$ or ${\eta }_d^{\prime }$ (crosses) extracted from the spectra shown in (a)–(c), with the values expected from Eqs. (43) for ${\eta }^{\mathrm{eq}}$ (solid lines), and from Eqs. (44) for ${\eta }^0$ or from Eq. (45) for ${\eta }_d^{\prime }$ (dashed lines with dots), for several values of $U$. In (a), arrows indicate the scale ${\overline{\omega }}^{*}$ that separates the regimes ${\omega }^{*}<\omega <{\overline{\omega }}^{*}$ and ${\overline{\omega }}^{*}<\omega <D$, where ${\mathcal{A}}_d^{\mathrm{eq}}$ scales according to Eqs. (45) or (44a), respectively.

Figure 5

(a) Cartoon of the quench which occurs when an electron-hole pair is created at time $t=0$, see Eq. (47). (The cartoon depicts the situation relevant for exciton creation by absorption of a photon, which excited an electron from a valence-band to a conduction band level of a semiconducting quantum dot.) (b) The exponent ${\Delta }_h$ [from Eq. (11)] and the displaced charge ${\Delta }_h^{\mathrm{ch}}$ [from Eq. (12)], for the quench of Eq. (46), as a function of the quench range $W$. (c) Corresponding values of the AO exponents ${\eta }_{hd}$, ${\eta }_{hdc}$, and ${\eta }_{hd{c}^{†}}$, extracted from the asymptotic behavior ${\omega }^{-1+2\eta }$ of spectral functions (crosses), or as expected from Eqs. (50) (dots). Typically, relative errors are less than $1\%$.

Figure 6

(a) The AO exponent ${\Delta }_h$ [extracted according to Eq. (11)] as a function of $W/{\Gamma }_{\mathrm{ren}}$ for several values of $U$. For $W/{\Gamma }_{\mathrm{ren}}\gg 1$, ${\Delta }_h$ approaches its maximal value ${\Delta }_h^{\max }$. As $W$ is reduced below ${\Gamma }_{\mathrm{ren}}$, ${\Delta }_h$ drops below its maximal value and decreases with $W$ [linearly so for $W/{\Gamma }_{\mathrm{ren}}\ll 1$]. (b) The final and initial occupancies $n_d^{\mathrm{f}}$ (squares, upper curves) and $n_d^{\mathrm{i}}$ (circles, lower curves) as functions of $W/{\Gamma }_{\mathrm{ren}}$ for the same values of $U$ [same color code as in (a)]. (c) The maximal value ${\Delta }_h^{\max }$ of the AO exponent ${\Delta }_h$, extracted from the $W/{\Gamma }_{\mathrm{ren}}\gg 1$ regime of (a) (crosses, “calc”), or expected from Eq. (48) (solid line, “exp”); the relative deviations are well below $1\%$.

Figure 7

(a) Cartoon of the Hamiltonian (51) for the asymmetric SIAM. (b) The occupations $n_{\mathrm{L}}$ (solid lines) and $n_{\mathrm{R}}$ (dashed lines) of the left and right levels, respectively, as functions of ${\varepsilon }_d$, for several values of ${\Gamma }_{\mathrm{R}}$, at a fixed ratio of ${\Gamma }_{\mathrm{R}}/{\Gamma }_{\mathrm{L}}=20$. As ${\varepsilon }_d$ is lowered past the particle-hole symmetric point at ${\varepsilon }_d=-U/2$, population switching occurs, with $n_{\mathrm{R}}$ changing from near 1 to near 0, and vice versa for $n_{\mathrm{L}}$. Inset: zoom into the switching region around ${\varepsilon }_d=-U/2$, showing that the switch is continuous (as a function of ${\varepsilon }_d$) even though the switching region becomes narrower for decreasing ${\Gamma }_{\mathrm{R}}$. (c) Comparison of $b_0{W}_{\mathit{PS}}$ [from Eqs. (52) and (56), crosses], $T_{\mathrm{K}}$ [from Eq. (57), solid], and the inverse pseudospin susceptibility $1/{\chi }_s$ (pluses). All three quantities evidently decrease similarly with decreasing ${\Gamma }_R/U$.

Figure 8

(a) The pseudospin-flip spectral function ${\mathcal{A}}_Y^{\mathrm{eq}}\left(\omega \right)$ [cf. Eq. (58)] for the PS model without charge sensor, for several values of ${\Gamma }_{\mathrm{R}}/U$ with fixed ratio ${\Gamma }_{\mathrm{R}}/{\Gamma }_{\mathrm{L}}$, calculated at ${\varepsilon }_d=-U/2$: plotting ${\mathcal{A}}_Y^{\mathrm{eq}}\left(\omega \right)/{\mathcal{A}}_Y^{\mathrm{eq}}\left(T_{\mathrm{K}}\right)$ versus $\omega /T_{\mathrm{K}}$ yields a scaling collapse. The frequency dependence of the curves qualitatively changes at $T_{\mathrm{K}}$; for $\omega <T_{\mathrm{K}}$, we find Fermi-liquid behavior, $\sim$${\omega }^3$ (dotted line), while for $\omega >T_{\mathrm{K}}$, each curve shows a nontrivial AO power law, $\sim$${\omega }^{-1+2{\eta }_Y^0}$ [see Eq. (61)], exemplified by the dashed and dash-dotted lines for ${\Gamma }_{\mathrm{R}}/U=0.2$ and $0.05$, respectively. (b) Comparison of the values for ${\eta }_Y^0$ expected from Eq. (61b) (solid line), or extracted from the spectral function ${\mathcal{A}}_Y^{\mathrm{eq}}\left(\omega \right)$ (crosses) by fitting Eq. (61a) to it in the intermediate-frequency regime between $T_{\mathrm{K}}$ and the high-frequency maximum. For each curve in panel (a), two arrows of corresponding color above and below the curve, indicate the fitting range's upper and lower ends, respectively. None of the fitting ranges extend below ${10}^3{T}_{\mathrm{K}}$ (hence the accumulation of arrows there), since below this value the logarithmic corrections discussed in the text become significant, causing the curves to bend. The effect of these logarithmic corrections increases with increasing ${\Gamma }_{\mathrm{R}}/U$, since this reduces the width of the fitting range [see (a)], causing the relative error between crosses and solid line to increases from $1\%$ for ${\Gamma }_{\mathrm{R}}/U=0.05$ to $10\%$ for ${\Gamma }_{\mathrm{R}}/U=0.2$.

Figure 9

AO for the PS model without charge sensing. The exponent ${\Delta }_{\mathit{AO}}$ [extracted from Eq. (1)] (solid lines with dots) is shown as function of quench size $W/U$ [see Eq. (62)], for several values of ${\Gamma }_{\mathrm{R}}/U$ with fixed ratio ${\Gamma }_{\mathrm{R}}/{\Gamma }_{\mathrm{L}}$, showing that AO becomes significant once $W$ increases past $T_{\mathrm{K}}/b_0$ (indicated by dashed vertical lines). The exponent ${\Delta }_{\mathit{AO}}$ increases linearly with $W$ for $W\ll T_{\mathrm{K}}/b_0$, and saturates to a maximal value of $\sqrt{2}$ [see Eq. (63)] for $W\gg T_{\mathrm{K}}/b_0$. The corresponding values of ${\Delta }_{\mathrm{ch}}$ [from Eq. (7)] are not shown but agree with ${\Delta }_{\mathit{AO}}$ with relative errors of a few percent.

Figure 10

Cartoon of the Hamiltonian (64), describing an asymmetric SIAM with an additional sensor lead coupled electrostatically to the left dot.

Figure 11

Population switching for the charge sensor model of Eq. (64). (a) $n_{\mathrm{R}}\left({\varepsilon }_d\right)$ (solid lines) and $n_{\mathrm{L}}\left({\varepsilon }_d\right)$ (dashed lines) for several values of $U_{\mathrm{S}}/U$, plotted versus $({\varepsilon }_d+U/2)/W_{\mathit{PS}}$ in a pseudologarithmic fashion (“pseudo” in that the $x$ axis is plotted logarithmic with positive and negative values to the left and right of the switching point, respectively, represented by the vertical solid line). The horizontal light solid lines indicate the values of $n_{\mathrm{R}}$ which define the widths $W_{\mathit{PS}}^{\mathrm{S}}$ of the PS regimes. The noisy behavior of the curves for $U_{\mathrm{S}}=5U$ at small values of ${\varepsilon }_d$ indicates that our analysis cannot resolve smaller values for ${\varepsilon }_d$ as we are reaching the limits of double precision numerical accuracy. (b) Inset: $T_{\mathrm{K}}^{\mathrm{S}}/T_{\mathrm{K}}$ as a function of $U_{\mathrm{S}}/U$, showing the rapid decrease of the Kondo temperature with increasing coupling. Main panel: $\ln T_{\mathrm{K}}^{\mathrm{S}}/\ln T_{\mathrm{K}}$ versus $(U_{\mathrm{S}}^{*}-U_{\mathrm{S}})/U$, plotted on a log-log scale (dashed line with dots), together with a linear fit using Eq. (66) (solid line).

Figure 12

(a) The pseudospin-flip spectral function ${\mathcal{A}}_Y^{\mathrm{eq}}\left(\omega \right)$ [cf. Eq. (58)] for the PS model with charge sensor, for several values of $U_{\mathrm{S}}/U$, calculated at ${\varepsilon }_d=-U/2$: plotting ${\mathcal{A}}_Y^{\mathrm{eq}}\left(\omega \right)/{\mathcal{A}}_Y^{\mathrm{eq}}\left(T_{\mathrm{K}}^{\mathrm{S}}\right)$ vs $\omega /T_{\mathrm{K}}^{\mathrm{S}}$ yields a scaling collapse. The general shape of the curves is similar to those shown in Fig. 8: for $\omega <T_{\mathrm{K}}^{\mathrm{S}}$ we find Fermi-liquid behavior, $\sim$${\omega }^3$ (dotted line), while for $\omega >T_{\mathrm{K}}^{\mathrm{S}}$, each curve shows a nontrivial AO power law, $\sim$${\omega }^{-1+2{\eta }_Y^{\mathrm{S}}}$ [cf. Eq. (61)], exemplified by the dashed and dash-dotted lines for $U_{\mathrm{S}}/U=0$ and 5, respectively. (b) Comparison of the values for ${\eta }_Y^{\mathrm{S}}$ expected from Eq. (67) (solid line), or extracted from power-law fits to the spectral function ${\mathcal{A}}_Y^{\mathrm{eq}}\left(\omega \right)$ in the intermediate-frequency regime between $T_{\mathrm{K}}$ and the high-frequency maximum (crosses). Arrows in panel (a) indicate the fitting ranges, as in Fig. 8a. The relative errors are below $5\%$, where the errors decrease with increasing $U_{\mathrm{S}}$ for similar reasons as in Fig. 8. The light horizontal line indicates ${\eta }_Y^{\mathrm{S}}=1$. (We were unable to obtain reliable data for $U_{\mathrm{S}}$ around $7U$, presumably because this is too close to $U_{\mathrm{S}}^{*}$.) (c) ${\mathcal{A}}_Y^{\mathrm{eq}}\left(\omega \right)/{\mathcal{A}}_Y^{\mathrm{eq}}\left(T_{\mathrm{K}}\right)$ vs $\omega /T_{\mathrm{K}}$ for $U_{\mathrm{S}}=8U$. The AO power-law behavior ${\omega }^{-1+2{\eta }_Y^S}$ extends down to the smallest frequencies accessible, illustrating that the crossover scale $T_{\mathrm{K}}^{\mathrm{S}}$ has become undetectable small.

Figure 13

AO for the PS model with charge sensing. The exponent ${\Delta }_{\mathit{AO}}$ [extracted from Eq. (1)] (solid lines with dots) is shown as function of quench size $W/U$ [see Eq. (62)], for several values of $U_{\mathrm{S}}/U$, with fixed values of ${\Gamma }_{\mathrm{R}}$ and ${\Gamma }_{\mathrm{L}}$. We see that $T_{\mathrm{K}}^{\mathrm{S}}/b_0$ (indicated by dashed vertical lines) is pushed to zero as $U_{\mathrm{S}}$ increases past $U_{\mathrm{S}}^{*}/U\simeq 6.78$. Already for $U_{\mathrm{S}}/U⩾6$, the curves are essentially indistinguishable, in that they do not deviate from their constant value for all $W/U$ values accessible to our analysis. For $W\gg T_{\mathrm{K}}^{\mathrm{S}}/b_0$, the exponent ${\Delta }_{\mathit{AO}}$ saturates to a maximal value given by Eq. (71). The corresponding values of ${\Delta }_{\mathrm{ch}}$ [from Eq. (7)] are not shown but agree with ${\Delta }_{\mathit{AO}}$ with relative errors of a few percent.

Figure 14

Schematic depiction of the equilibrium spectral function ${\mathcal{A}}_Y^{\mathrm{eq}}\left(\omega \right)$ for the cases that the local charge relaxation rate ${\omega }^{*}$ is (a) larger or (b) smaller than the lead level spacing $\delta E$.