Nonequilibrium quantum impurity problems via matrix-product states in the temporal domain

Describing a quantum impurity coupled to one or more noninteracting fermionic reservoirs is a paradigmatic problem in quantum many-body physics. While historically the focus has been on the equilibrium properties of the impurity-reservoir system, recent experiments with mesoscopic and cold-atomic systems enabled studies of highly nonequilibrium impurity models, which require novel theoretical techniques. We propose an approach to analyze impurity dynamics based on the matrix-product state (MPS) representation of the Feynman-Vernon influence functional (IF). The efficiency of such a MPS representation rests on the moderate value of the temporal entanglement (TE) entropy of the IF, viewed as a fictitious “wave function” in the time domain. We obtain explicit expressions of this wave function for a family of one-dimensional reservoirs, and analyze the scaling of TE with the evolution time for different reservoir's initial states. While for initial states with short-range correlations we find temporal area-law scaling, Fermi-sea-type initial states yield logarithmic scaling with time, closely related to the real-space entanglement scaling in critical $1d$ systems. Furthermore, we describe an efficient algorithm for converting the explicit form of general reservoirs' IF to MPS form. Once the IF is encoded by a MPS, arbitrary temporal correlation functions of the interacting impurity can be efficiently computed, irrespective of the form of impurity interactions and bath geometry. The approach introduced here can be applied to a number of experimental setups, including highly nonequilibrium transport via quantum dots and real-time formation of impurity-reservoir correlations, as well as in nonequilibrium dynamical mean-field theory.

Figure 1

(a) Time evolution of an interacting impurity coupled to noninteracting fermionic reservoirs. (b) The dynamical influence of a reservoir on the impurity can be encoded in a tensor acting on the impurity trajectories on the Keldysh contour—the influence matrix (IM)—which can be evaluated analytically. This multitime tensor can be viewed as a many-body state in the temporal domain, and its “entanglement” can be analyzed. (c) When this temporal entanglement is low, the IM “wave function” can be represented as MPS, which allows to efficiently compute the real-time dynamics of the impurity.

Figure 2

(a) The impurity is coupled to two independent reservoirs to its left and right. The two tensors arising from the summation over all trajectories of the reservoirs (i.e., from the contractions of the two tensor networks) are the two IMs. Gray gates acting from bottom to top represent nearest-neighbor interactions $e^{-i\delta t{H}_{j,j+1}}$ (foreground) and $e^{+i\delta t{H}_{j,j+1}}$ (background). Green gates encode the coupling between impurity and environment; in this work they are chosen to be equivalent to gray gates. Blue gates represent impurity interactions $e^{-i\delta t{H}_{\mathcal{S}}}$ (foreground) and $e^{+i\delta t{H}_{\mathcal{S}}}$ (background). The labels on the external legs of the right IM define the impurity's trajectory (analogously for the left IM). (b) Floquet phase diagram of the trotterized $XY$ model $(\phi =0)$. Phase boundaries (diagonals) are associated with the quasienergy gap closing, and the labels refer to the types of edge modes in the corresponding phase.

Figure 3

By inserting resolutions of identity in terms of fermionic coherent states into Eq. (6), the IM can be written as Grassmann path integral. The leg at site $j$ and time $\tau$ in the tensor network is associated with Grassmann variables ${\overline{\xi }}_{j,\tau },{\xi }_{j,\tau }$.

Figure 4

Upper panels: Real and imaginary part of the response function $g_{\tau ,{\tau }^{\prime }}$ for the kicked Ising model with parameters $\mathcal{J}=\phi =0.3$ and nonequilibrium initial state ${\rho }_{\mathcal{E}}={\prod }_j{\rho }_j,{\rho }_j\propto e^{-\beta {\sigma }_j^z},\beta =10$. Lower panel: Response function $g_{\mathrm{\Delta }\tau +{\tau }^{\prime },{\tau }^{\prime }}$ for fixed $\mathrm{\Delta }\tau =18,$ corresponding to the direction indicated by the arrows in the upper panels. Inset: scaling of temporal entanglement entropy.

Figure 5

Spectrum ${\varphi }_k$ of the Floquet Hamiltonian and function $|{\mathcal{C}}_k|$ for the kicked Ising model in the (a) gapless and (b) gapped phase.

Figure 6

Response functions and temporal entanglement entropy scaling for the kicked Ising model with parameters $\mathcal{J}=\phi =0.3$ for (a) the infinite temperature initial state and (b) the critical initial state. Dashed and dotted lines serve as guide to the eye to identify power laws and logarithmic/saturation behavior, respectively.

Figure 7

“Temporal bulk” response functions and temporal entanglement entropy scaling for the trotterized $XY$ model with parameters ${\mathcal{J}}_x={\mathcal{J}}_y=0.3,T=800,$ for (a) the infinite temperature initial state and (b) the critical initial state. For the infinite temperature initial state in panel (a), all four functions $g_{\left[\mathrm{\Delta }\tau \right]}^x$ are purely real. For better distinguishability of the curves, we have multiplied the functions $g^{\alpha ,\beta ,\gamma ,\delta }$ with factors of 1.0,0.5,0.25,0.125, respectively. Dashed and dotted lines serve as guide to the eye to identify power laws and logarithmic/saturation behavior, respectively.

Figure 8

By applying the unitary two-site single-body rotations from one cycle of the algorithm (Steps 1 and 2), the original correlation matrix $\mathrm{\Lambda }$ can locally be brought into diagonal form, where it describes the vacuum state. Each horizontal line corresponds to one site of the chain defined in Eq. (51). Blue gates symbolize Givens rotations, orange gates represent Bogoliubov rotations.

Figure 9

Illustration of the MPS construction: After the single body-rotations are converted to quantum gates, they are applied to the vacuum many body state in reverse order and conjugated, which provides a MPS representation of the IM. From this illustration one can deduce the bound on the local bond dimension ${\chi }_{(j,j+1)}\le 2^{n_j-1}$.

Figure 10

Scaling of the typical bond dimension upper bound $\chi \left(T\right)=2^{n_{\mathrm{av}}-1}$, associated with the average subsystem block size $n_{\mathrm{av}}={\sum }_{j=1}^{4T-1}{n}_j/(4T-1)$ needed to diagonalize the correlation matrix $\mathrm{\Lambda }$ of the IM of the kicked Ising model over $T$ Floquet periods. The IM is calculated analytically in the thermodynamic limit from Eq. (35) for the same noncritical (infinite temperature) and critical initial states and for the same parameters as in Fig. 6. Dotted lines are power-law fits (notice the horizontal logarithmic scale).

Figure 11

Diagrammatic representation of Eq. (D9). Here, ${\widehat{O}}_1={\widehat{O}}_2=c_1-c_1^{†}.$ The nonunitary gate $F_1$ results from ${\mathcal{F}}_{0,1}$ with the source fields set to zero.

Figure 12

Response functions for the trotterized $XY$ model for the infinite temperature initial state, plotted from the same dataset as Fig. 7 (data not rescaled here).

Figure 13

Response functions for the trotterized $XY$ model for the critical initial state, plotted from the same dataset as Fig. 7 (data not rescaled here).

Figure 14

Typical bond dimension associated with the average subsystem block size needed to fully diagonalize the correlation matrix $\mathrm{\Lambda }$ of the IM of the kicked Ising model over $T$ Floquet periods [corresponding to the data presented Fig. 7]. For $T\gtrsim L$, the results show consistent strong finite-size effects: for $T\gg L=400$, long-range temporal correlations appear in the IM due to reflections at the edge of the reservoir [82].