.. _P3: ==================== Physics Background ==================== .. contents:: :local: :depth: 2 .. _P3Sec01: Non-equilibrium quantum many-particle physics ============================================= Simulating the time evolution of strongly driven many-body quantum systems is challenging because behavior distinct from the equilibrium properties can emerge on vastly different timescales. .. figure:: _static/Timescales001v01.png :width: 800px :alt: Time scales for different light-induced phenomena :align: center Time scales for different light-induced phenomena in lattice systems. Such simulations are relevant in a broad variety of contexts: - By driving condensed matter with tailored light one can **engineer novel quantum phases**. Light-induced superconductivity or Floquet states are among the tantalizing examples. See D. N. Basov, R. D. Averitt, and D. Hsieh, *Towards properties on demand in quantum materials*, Nature Materials **16**, 1077 (2017). - **Analog quantum simulation platforms** allow exploration of genuine non-equilibrium phenomena, such as dynamics at the boundary between integrable and ergodic behavior. - **Dissipative driven quantum systems** are relevant for quantum transport and nanotechnology, including quantum computing architectures. - **Time-resolved pump-probe spectroscopy** reveals the interplay of quasiparticles and collective excitations on microscopic timescales. See Claudio Giannetti *et al.*, *Ultrafast optical spectroscopy of strongly correlated materials...*, Advances in Physics **65**, 58 (2016). .. figure:: _static/exciton_pes.gif :width: 800px :alt: Example tr-ARPES spectrum :align: center Example: Simulated time- and angular-resolved photoemission spectrum (tr-ARPES) of an excitonic insulator. The movie (GIF) illustrates how tr-ARPES reveals electronic structure out of equilibrium, including filling, broadening, internal relaxation, gap closing, and thermalization. Time-dependent GW simulations performed using the ``NESSi`` library. See Denis Golež, Philipp Werner, and Martin Eckstein, *Photo-induced gap closure in an excitonic insulator*, Phys. Rev. B **94**, 035121 (2016). :ref:`Back to top ` .. _P3Sec02: Keldysh Formalism and Nonequilibrium Green's Functions ====================================================== Non-equilibrium Green's functions (NEGF) ---------------------------------------- .. _NEGF_def: Field-theoretical approaches based on **Green's functions** provide a versatile framework for deriving systematic approximations to quantum many-particle systems. Green's functions measure spectra and occupations of quasiparticles and therefore directly give spectroscopic quantities like tr-ARPES. This complements *exact* many-body methods whose Hilbert-space scaling is exponential. The **NEGF** approach, pioneered by Keldysh, Kadanoff and Baym, extends equilibrium many-body tools (diagrams, functional integrals) to nonequilibrium phenomena. For basic introductions, see: - A. Kamenev, *Field Theory of Non-equilibrium Systems*, CUP (2011). - G. Stefanucci and R. van Leeuwen, *Nonequilibrium Many-Body Theory of Quantum Systems*, CUP (2013). - H. Haug and A.-P. Jauho, *Quantum Kinetics in Transport and Optics of Semiconductors*, Springer (2008). The Keldysh formalism underlies the quantum Boltzmann equation, fluctuation–dissipation relations, and numerical two-time Green’s function methods. This is where ``NESSi`` enters. Kadanoff–Baym (KB) equations ---------------------------- .. _KB_def: Very schematically, a non-equilibrium propagator :math:`G_{ij}(t,t')` describes a **two-time correlation** between excitations. Its equation of motion is .. math:: :label: dyson_schematic i \partial_t G(t,t') - H_{mf}(t) G(t,t') - \int_{\text{previous time}} d\bar t\, \Sigma(t,\bar t) G(\bar t, t') = \delta(t,t'). Here :math:`H_{mf}` is an effective one-body Hamiltonian including mean-fields, and :math:`\Sigma(t,t')` is the **self-energy**, describing interaction effects and coupling to environments. Equation :eq:`dyson_schematic` is non-Markovian. .. figure:: _static/kb_small.gif :align: center Causal propagation of a two-time NEGF: computing :math:`G(t,t')` at the red point requires knowledge of all previous Green’s functions and self-energies (yellow region). ``NESSi`` implements these equations using the formulation successful in nonequilibrium DMFT (see Aoki *et al.*, *Rev. Mod. Phys.*, 2014; Eckstein 2010). :ref:`Back to top ` .. _P3Sec05: Mathematical formulation ======================== The Keldysh contour ------------------- Non-equilibrium many-body theory is naturally formulated on a **time contour**. - The **L-shaped contour** :math:`\mathcal{C}` represents a system that begins in a thermal state with density matrix :math:`\rho = e^{-\beta H(0_-)}/Z`. ``libcntr`` is tailored to this contour. - **Non-equilibrium steady states (NESS)** typically depend only on relative time, suggesting a contour pushed to :math:`t=-\infty`. .. figure:: _static/contour_discrete.png :width: 543px :align: center L-shaped Kadanoff–Baym contour with real-time and Matsubara branches, discretized for numerical evaluation. Contour-ordered Green’s functions --------------------------------- All relevant expectation values appear as contour-ordered objects: .. math:: T_{\mathcal{C}} \{ A(t_1) B(t_2) \} = \begin{cases} A(t_1) B(t_2) & t_1 \succ t_2 \\ \xi\, B(t_2) A(t_1) & t_2 \succ t_1 , \end{cases} where :math:`\xi=+1` for bosons and :math:`-1` for fermions. The basic two-time correlator is .. math:: :label: twotimegreens C_{A,B}(t,t') = -i \langle T_\mathcal{C} A(t) B(t') \rangle_\mathcal{S} = -i \frac{ \mathrm{tr}[T_\mathcal{C} e^\mathcal{S} A(t) B(t')] }{ \mathrm{tr}[T_\mathcal{C} e^\mathcal{S}] }. .. figure:: _static/unfolding_timeorder.png :align: center Unfolding contour-ordered expectation values into real-time correlators when :math:`\mathcal{S}` corresponds to Hamiltonian evolution. Important Keldysh components ---------------------------- - **Lesser**: :math:`C^<(t,t') = -i\xi \langle B(t') A(t) \rangle` - **Greater**: :math:`C^>(t,t') = -i \langle A(t) B(t') \rangle` - **Retarded**: :math:`C^R(t,t') = -i \theta(t-t') [A(t),B(t')]_\xi` - **Matsubara**: imaginary-time component In ``libcntr`` they are stored as :math:`\{C^<, C^R, C^\rceil, C^M\}` on the contour :math:`\mathcal{C}[h,N_t,h_\tau,N_\tau]`. :ref:`Back to top ` .. _P3Sec06: Numerical solution and accuracy =============================== Dyson equations are reduced to **Volterra integral equations** and solved using Gregory quadrature rules (Brunner & van Houven, *The numerical solution of Volterra equations*, North Holland, 1986). A rule of degree :math:`k` approximates .. math:: I = \int_0^{Nh} f(t)\, dt \approx \sum_{j=0}^{m(N,k)} w_j f(jh). The numerical error scales as :math:`\mathcal{O}(N^{-(k+2)})`. Order :math:`k=0` corresponds to trapezoidal integration with error :math:`\mathcal{O}(N^{-2})`. .. _P3Sec04: Previous use of NESSi ===================== Before public release (2019), ``NESSi`` was used extensively in nonequilibrium DMFT studies of correlated systems. Over 60 publications rely on it. Thermalization of a pump-excited Mott insulator ----------------------------------------------- .. figure:: _static/thermalization_small.png :width: 800px :align: center Dynamics of doublon density and relaxation times as function of :math:`U`. From Eckstein & Werner, *Phys. Rev. B* **84**, 035122 (2011). Doublon relaxation in photo-excited 1T-TaS₂ ------------------------------------------- .. figure:: _static/1TTaS2small.png :width: 800px :align: center Comparison between experiment and DMFT simulations for doublon dynamics. From Ligges *et al.*, *PRL* **120**, 166401 (2018). Benchmarking quantum simulators ------------------------------- .. figure:: _static/coldatom_small.png :width: 800px :align: center Doublon production in driven cold atom systems vs. nonequilibrium DMFT. From K. Sandholzer *et al.*, arXiv:1811.12826. :ref:`Back to top `