The mapping problem in nonequilibrium dynamical mean-field theory

Transcription

1 The mapping problem in nonequilibrium dynamical mean-field theory Masterarbeit im Fach Physik vorgelegt der Mathematisch-Naturwissenschaflichen Fakultät der Universität Augsburg von Christian Gramsch Januar 213 angefertigt am Lehrstuhl für Theoretische Physik III Zentrum für Elektronische Korrelationen und Magnetismus Institut für Physik der Universität Augsburg

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3 Contents 1 Nonequilibrium quantum mechanics Contour-ordered Green functions Equation of motion Analytical properties The Hubbard model Nonequilibrium dynamical mean-field theory Effective single-site action Analytical properties of the Weiss field Self-consistency condition for a semi-elliptical density of states The mapping problem Role of the bath geometry Existence of a valid mapping Existence for a semi-elliptical density of states Existence for an arbitrary density of states A numerically feasible mapping Memory of the initial state (the first bath) Approach to the steady state (the second bath) Construction for finite temperature Decomposition of the second Weiss field Continuous Cholesky decomposition Numerical results Low-rank Cholesky approximation Comparison to eigenvalue reduction Shifting Cholesky approximation Summary and outlook 54 A Appendix 56 A.1 Generating Functionals A.2 Lehmann representation of G(t, t ) A.3 Introduction of the baths

4 Introduction A solid consists of a large amount of electrons and nuclei ( 1 23 ) that interact through the Coulomb force. Its bonding is caused by the delocalized valence electrons whereas the nuclei and core electrons form the mostly immobile lattice ions. The properties of the solid are therefore largely determined by the behaviour of the valence electrons. A first approach to model them theoretically are band structure calculations [1]. Here one treats the valence electrons as independent particles, e.g. by neglecting their interaction completely or by taking it into account through a static mean field, that move through a periodic potential which is generated by the lattice ions. This single-particle picture allows, for example, to understand the good conductivity of metals by means of a partially filled energy band. However, it breaks down in materials with narrow energy bands, e.g. d-bands in transition metals, where the repulsive Coulomb interaction between the electrons leads to strong correlations. The simplest model that takes electronic correlations into account is the Hubbard model [2 4]. In this model the electron-electron interaction is purely local, i.e. restricted to electrons that occupy an orbital at the same lattice site. As one of its most prominent features it allows for Mott insulating states [5], i.e. interaction induced insulators. However, theoretical investigations are limited by the fact that the Hubbard model cannot be solved except in special one-dimensional cases [6]. This lead to the development of a variety of approximations to access its ground state properties [7]. A fascinating experimental realization of bosonic and fermionic Hubbard-type models has recently been achieved with ultracold atoms in optical lattices [8]. These experiments already reproduce metallic and Mott insulating phases [9,1] and also transport phenomena are becoming accessible [11]. Beyond equilibrium, the initial preparation of the system and the time dependence of external influences becomes of great importance. In a recent numerical study [12], for example, the Hubbard model was driven out of equilibrium by applying a time-dependent (sinusoidal) force on the electrons. Effectively, this inverted their repulsive interaction to become attractive and raised the question about the possibility of ac-induced superconductivity. Other interesting questions are closely related to the ergodic hypothesis, an important basis of statistical mechanics. Let us assume that we disconnect a system from its surrounding heat bath and suddenly change (at least) one of its parameters, performing a so called quantum quench. Will the system approach a new steady state? And if so, will the state be thermal 1? Analytically, a nonequilibrium situation is most easily treated in case of integrable models, but their infinite amount of constants of motion can hinder thermalization [14] (this was also observed experimentally [15]); sta- 1 Interestingly, the very first numerical experiment by Fermi, Pasta and Ulam focused on the question of thermalization in a classical solid [13]. 4

5 Contents tistical predictions for the long-time limit then require for generalized Gibbs ensembles (GGEs) [14,16]. It is therefore necessary to study more complicated, non-integrable systems in which thermalization is expected [17]. A natural candidate is the Hubbard model which is non-integrable in dimension two and higher due to its interaction term. A numerical study [18, 19] of the Hubbard model yielded indeed fast thermalization at an intermediate interaction strength. It separated two regimes where prethermal states were found on intermediate timescales. It was argued recently that these prethermal plateaus are caused by perturbed constants of motion and as such may be described by appropriate perturbed GGEs [2]. The theoretical treatment of the Hubbard model far from equilibrium is a complicated time-dependent many-body problem, but one can try to generalize techniques that were developed for equilibrium. The Gutzwiller wavefunction [4, 21], for example, was introduced to approximate ground-state properties of the Hubbard model and has recently been extended to a time-dependent nonequilibrium formalism [22, 23] by connecting it to the Dirac-Frenkel [24] instead of the Ritz variational principle. Dynamical mean-field theory (DMFT) [25,26] is a more powerful approach that was also generalized to nonequilibrium [27, 28]. Within DMFT, the lattice problem is represented by an effective local action which is determined self-consistently. This description becomes exact in the limit of infinite lattice dimension. The effective action describes an impurity problem, where a single interacting site is connected to a non-interacting bath. In equilibrium its associated Green function can not only be calculated by quantum Monte Carlo techniques but also by a mapping of this action onto an appropriate single-impurity Anderson model (SIAM). While continuous-time quantum Monte Carlo (CTQMC) is nowadays a key technique to work with DMFT in nonequilibrium [29], only an initial investigation has been performed regarding the mapping onto a SIAM Hamiltonian [3]. A further investigation of this mapping problem is topic of this work. The goal is to provide Hamiltonian-based impurity solvers with input data, e.g. a Krylov-space solver [31]. The previous analysis [3] featured the construction of a star-structured SIAM for a given DMFT action at zero temperature and found as a key result that at least two baths are required for a valid mapping. The first bath describes the correlations within the initial state and vanishes over time, while the second is developing to describe the final steady state. The transient behavior is influenced by both baths. In this thesis we continue the analysis. In particular, we attempt to transfer part of the information into a dynamical bath geometry with the goal to find a single-bath description. Such a description would require fewer bath sites and thus allow for an increased accuracy in a numerical simulation. However, it does not only turn out that the requirement of two baths is generic, but also that the star structure allows for the simplest analytical treatment of the mapping problem. Regarding the construction of the second bath a crucial question on the mathematical preconditions was left open in Ref. [3]. We tackle this problem by a general investigation regarding the existence of a valid mapping and find that the suggested procedure is indeed well-defined for an important class of situations. This further allows us to extend the construction of the second bath to finite temperatures. We then discuss the explicit construction of the second bath which requires an 5

6 Contents approximate Cholesky-like matrix decomposition. We discuss the advantages and disadvantages of different approaches and apply them to existing data. The developed code can be used in future numerical efforts. Outline In chapter 1 we briefly discuss the analytical tools for a treatment of nonequilibrium quantum mechanics. In particular, we shortly review contour-ordered Green functions and derive equations of motion for use with the effective DMFT action. We introduce the Hubbard model and derive a corresponding local action using the cavity method. In the limit of infinite dimensions the DMFT impurity action is recovered. In chapter 2 we study the role of the bath geometry in the mapping problem and find the equations to be simplest in case of a star structure. We further analyze the mathematical existence of a valid mapping. This is a necessary precondition for the following construction of a two bath solution, which we also develop for finite temperatures. Chapter 3 is devoted to the calculation of a matrix square root of the second Weiss field. While this can in principle be done by a Cholesky decomposition, numerical calculations require for approximate decompositions, which we develop and discuss. 6

7 1 Nonequilibrium quantum mechanics We first discuss the question how a nonequilibrium situation can be translated into the mathematical language of quantum mechanics. To be able to force a system out of equilibrium we consider its Hamiltonian H(t) to be time-dependent, so that we can change its parameters (e.g. the interaction, the hopping, external fields,...) dynamically. At the initial time t = t min we assume a configuration that can be described by an appropriate density matrix ρ(t min ). It can, for example, be given by the grand canonical ensemble ρ(t min ) = 1 Z exp ( β(h(t min) µn)), Z = tr(exp ( β(h(t min ) µn))), (1.1) describing a thermal state with temperature T = 1 k B β (k B is the Boltzmann constant) and chemical potential µ (N is the particle number operator). The assumption includes the possibility of a pure state, which is often chosen to be the ground state of the initial Hamiltonian H(t min ), so that ρ(t min ) = Ψ(t min ) Ψ(t min ). If we work within the Schrödinger picture, the time-evolution of ρ(t) follows through Von Neumann s equation ( 1 here and throughout this work) dρ(t) dt = i [H(t), ρ(t)]. (1.2) In the Heisenberg picture, on the other hand, ˆρ ρ(t min ) is a time-independent operator and the observables carry the time dependence dô(t) dt ] = i [Ĥ(t), Ô(t) + Ô t. (1.3) In this thesis we use the convention that operators with a hat are to be interpreted in the Heisenberg sense, i.e. Ô(t) = U (t, t min )O(t)U(t, t min ). (1.4) The propagator U(t, t ) is formally constructed by Picard iteration of the Schrödinger equation and takes the form (assuming t > t ) { ( t )} U(t, t ) = T exp i H(t)dt. (1.5) t In the last expression we introduced the time-ordering operator T. We understand it to work on every term of the series expansion. 7

8 1 Nonequilibrium quantum mechanics Of course, it is impossible to measure the exact state of a many-body system during an experiment (even if it were possible, the vast amount of information would be useless). Depending on the experiment only certain observables are accessible (e.g. momentum distribution, double occupancy, compressibility,...) and their expectation values can be expressed in our formal language as ) O(t) = tr(oρ(t)) = tr (Ô(t)ˆρ Ô(t). (1.6) Note that the presented formalism only allows for the contact with a heat bath for t t min. The time-evolution that takes place for t > t min is unitary, as such reversible, and therefore restricted to a closed environment. In practice, even the simplest observables, namely one-body observables, can only be calculated in rare cases. The appropriate tool to describe them is known from quantum many-particle theory and given by one-particle Green functions. They are defined as { } G ij (t, t ) i T ĉ i (t)ĉ j (t ), (1.7) where ĉ i (t) (ĉ j (t )) annihilates (creates) a particle in a state i (j) at time t (t ) and the set of single-particle states { i } (e.g. Wannier states) has to form a complete basis of the one-particle { Hilbert } space. In this thesis we only consider anticommuting fermionic operators, i.e. c i, c j = δ ij and {c i, c j } =. For interacting quantum systems in equilibrium one important approach to gain information about those correlation functions is given by perturbation theory. It allows for an expansion in the interaction and leads to Green functions of the non-interacting system of arbitrary high order. With the help of Wick s theorem [32] it is possible to reduce these to single-particle Green functions of the non-interacting system. This series expansion is usually performed by means of Feynman diagrams. 1.1 Contour-ordered Green functions The appropriate tool to extend perturbation theory to nonequilibrium is given by contourordered Green functions [33]. In this formalism two times t, t are not understood as ordinary real quantities. Instead we think of them to be lying on the contour C (cf. Fig. 1.1) which runs from t min to t max on the real axes and then back to t min. Finally, it runs to t = t min iβ into the complex plane. We call the time t to be bigger than t in sense of the contour (denoted by t > C t ) if it occurs later on the contour (this does not imply t > t ). Let now A, B be time-independent operators in the Schrödinger picture. To be able to position those operators in a contour-ordered product we add an explicit time argument (the operators themselves, however, remain constant) so that T C {A(t)B(t )} = { AB if t >C t, (±)BA if t < C t. (1.8) 8

9 1.1 Contour-ordered Green functions t min C 2 C 1 t t t max C 3 t min iβ Figure 1.1: The L-shaped integration contour C consists out of three parts. The integration runs along C 1 from t min to t max on the real axis and then goes back along C 2. It then follows C 3 into the complex plane to t min iβ. In sense of the contour the time t C 2 is therefore bigger than t C 1 in this example. In our case A, B will be fermionic creation/annihilation operators, so that we obtain a negative sign when changing their order. The generalization for time-dependent (even in the Schrödinger picture) operators and operators in the Heisenberg picture is obvious. We follow Eckstein s [3] notation and define the contour-ordered Green function as G(1, 1 ) G ij (t, t ) = i c i (t)c j (t ) S i { }) tr (T C exp (S) c i (t)c j Z (t ), S Z S tr(t C {exp (S)}). (1.9) Note that in the expression i c i (t)c j (t ) S the contour ordering is hidden in the special definition of the expectation value. One therefore has to distinguish it carefully from the expectation value without subscript as it was defined in (1.6). For the action S we consider not only the term which arises from our nonequilibrium problem (S 1 ), but also an additional term S 2 that will arise for the effective DMFT action of the Hubbard model S = S 1 + S 2 = i H(t)dt i Λ ij (t 1, t 2 )c i (t 1)c j (t 2 )dt 1 dt 2. (1.1) C ij C C Equation (1.9) is to be understood in the sense that the matrix-exponential is expanded in powers of S, i.e. the contour ordering does also affect the time integrals. We note that this definition leaves G(1, 1 ) independent of t max which only has to fulfill t max max(t, t ). Of course, it is practical to choose it as small as possible. If Λ = holds true, the density matrix and the unitary time propagator are reproduced 1. In case of t > C t, for example, G(1, 1 ) = i tr Z S = i Z S tr { }) (T C exp (S)c i (t)c j (t ) ( exp ( βh(t min ))U (t, t min )c i U(t, t )c j U(t, t min ) (1.11) ) { } = i T C ĉ α (t)ĉ α (t ). 1 A detailed explanation how the contour ordering constructs density matrix and propagators at just the right points can be found in Haug and Jauho [33]. 9

10 1 Nonequilibrium quantum mechanics We emphasize again that operators with a hat are to be understood in the Heisenberg sense. Calculus of contour functions Contour functions require a slightly changed calculus since the variables are not ordinary real numbers. Depending on the position on the contour an infinitesimal change dt of the contour variable t can thus be positive (C 1 ), negative (C 2 ) or imaginary (C 3 ). Let now f(t), t C be a function on the contour. We can split it into the three parts: f 1 (t) for t C 1, f 2 (t) for t C 2 and f 3 (t) for t C 3. The derivative of f(t) can then be expressed in terms of real variables f(t) t s f 1 (s) s=t t C 1, s [t min, t max ], = s f 2 (s) s=t t C 2, s [t min, t max ], i τ f 1 (t min iτ) t = t min iτ C 3, τ [, β]. (1.12) The Heaviside step function Θ(t t ) can be naturally generalized to contour variables t, t C and the contour variant of the Dirac delta follows immediately by derivation Θ C (t, t ) = { 1 for t C t else δ C (t, t ) = t Θ C (t, t ). (1.13) Components of contour functions Similar to the relations above, it is convenient to introduce a variety of specialized Green functions whose arguments lie on the real axis (suppressing the spatial index): Matsubara component: G M (τ τ ) G 3,3 (t min iτ, t min iτ ), Mixed components: Lesser component: G < (t, t ) G 1,2 (t, t ), G (τ, t ) G 3,2 (t min iτ, t ) = G 3,1 (t min iτ, t ), G (t, τ ) G 2,3 (t, t min iτ ) = G 1,3 (t, t min iτ ), Greater component: G > (t, t ) G 2,1 (t, t ). (1.14) In common literature [33], one also finds retarded, advanced, time-ordered, anti timeordered and Keldysh Green functions. However, they are not necessary for our investigations Equation of motion Equipped with the mathematical tool of contour calculus, the equation of motion for G(1, 1 ) is in principle straightforward to derive. Still, the details are somewhat tedious. Its role in this work, however, is important and so we present it here. We first note that T C {exp (S)} = T C {exp (S 1 ) exp (S 2 )} holds true and therefore allows for an expansion 1

11 1.1 Contour-ordered Green functions of the second factor independently of the other. Doing so we find i t G(1, 1 ) = i [H(t), c i (t)]c j (t ) S (1.15) ( i) m + dt 1...dt 2m Λ n1,n m! 2 (t 1, t 2 )...Λ n2m 1,n 2m (t 2m 1, t 2m ) m= {n i } C { } ( t T C ) ĉ n 1 (t 1 )ĉ n2 (t 2 )...ĉ n 2m 1 (t 2m 1 )ĉ n2m (t 2m )ĉ i (t)ĉ j (t ). Here we used (1.11) in every expanded term leading to operators in Heisenberg representation. The derivation of ĉ i (t) then yielded a commutator with Ĥ(t) in each summand. Adding all of these together and going back to Schrödinger representation we got the first term in (1.16). Thus only the derivative of the contour ordering itself remains. It is readily calculated. We assume t 1,...,t 2m to be fixed and notice that ĉ i (t) anticommutes with ĉ 2k (t 2k ), k {1,..., m}. Starting from t = t min we increase t until it reaches its maximal (in the sense of the contour) value t = t min iβ. { Every } crossing of a value t = t 2k 1, k {1,...,m} results in a delta jump of height ĉ i, ĉ j = δ ij times the contour ordering of the remaining operators. This leads to ( t T C ) {... } = m δ C (t, t 2k 1 )δ in2k 1 T C {ĉ n 1 (t 1 )ĉ n2 (t 2 )...ĉ n 2k 3 (t 2k 3 )ĉ n2k 2 (t 2k 2 ) k=1 } ĉ n 2k+1 (t 2k+1 )ĉ n2k+2 (t 2k+2 )...ĉ n 2m 1 (t 2m 1 )ĉ n2m (t 2m )ĉ 2k (t 2k )ĉ j (t ) +δ C (t, t )δ ij T C {ĉ n1 (t 1 )ĉ n2 (t 2 )...ĉ n 2m 1 (t 2m 1 )ĉ n2m (t 2m ) }. (1.16) Plugging this result into (1.16) we notice that we can always substitute t 1 t 2k 1,t 2 t 2k if 2k 1 1. It is then possible to perform the integration over t 1, which is trivial because of the delta function. This yields i t G(1, 1 ) = δ C (t, t )δ ij + i [H(t), c i (t)] c j (t ) S + dt 2 Λ i,n2 (t, t 2 ) n 2 ( i) m dt 3...dt 2m Λ n3,n (m 1)! 4 (t 3, t 4 )...Λ n2m 1,n 2m (t 2m 1, t 2m ) {n i }\{n 1,n 2 } C } T C {ĉ n 3 (t 3 )ĉ n4 (t 4 )...ĉ n 2m 1 (t 2m 1 )ĉ n2m (t 2m )ĉ i (t)ĉ j (t ). m=1 The final result is obtained by shifting the index to match the definition of the exponential function. This allows to sum up all the terms. Using again (1.11) to go back to Schrödinger representation we find in agreement with Eckstein [3] i t G(1, 1 ) = δ(1, 1 ) + i [H(t), c i (t)]c j (t ) S + dt 2 Λ i,n2 (t, t 2 )G n2,j(t 2, t ), n 2 boundary condition: G ij (t min, t ) = G ij (t min iβ, t ) with t min C 1. (1.17) C C 11

12 1 Nonequilibrium quantum mechanics The boundary condition is a direct consequence of the cyclic property of the trace. Note that the annihilation operator c i (t) in the following equation is in the Schrödinger picture, i.e. its time dependence only has consequences inside the contour-ordering operator. Outside we have c i (t) = c i () for every t C and we can thus change the time argument at will, i.e. G ij (, t ) = i { }) tr (c i ()T C exp (S) c j Z (t ) S = i { } ) tr (T C exp (S)c j Z (t ) c i (t min iβ) = G ij (t min iβ, t ). S (1.18) For completeness, we also state the conjugate equation of motion and its corresponding boundary condition. It can be derived using the same scheme. [ i t G(1, 1 ) = δ(1, 1 ) + i c i (t) c j (t ), H(t ) ] S + n2 dt 2 G i,n2 (t, t 2 )Λ n2,j(t 2, t ), C boundary condition: G ij (t, t min ) = G ij (t, t min iβ) with t min C 1. (1.19) For Λ = both equations reduce to the simpler equations of motion of ordinary contour-ordered Green functions Analytical properties Based on Eckstein s [3] discussion on the analytical properties of contour-ordered Green functions, we will give a brief overview on this topic. Since we consider only thermal initial states, their analytical properties are very similar to those of equilibrium Green functions 2. For our purposes a discussion assuming Λ(t, t ) = is sufficient and we start with the Matsubara component of G ij (t, t ). It can be Fourier transformed in the usual way (ω n = nπ β are the Matsubara frequencies with β = 1 k B T as the inverse temperature) G ij ( iτ, iτ ) = G M ij (τ τ ) = i e H(t min)(τ τ )ĉ i eh(t min)(τ τ )ĉ j (1.2) = i β e iωn(τ τ ) gij M β (iω n), with gij M (iω n) = i dτe iωnτ G M ij (τ). n The Fourier components gij M(iω n) can be analytically continued into the upper or lower complex plane. In equilibrium one ensures by the requirement gij M (z) z 1 that this z continuation is equal to the Laplace transform of the retarded (for Im(z) > ) or advanced (for Im(z) < ) Green function [32]. We can use the same requirement for nonequilibrium. We find the known result g M ij (z) = 1 Z m,n (exp ( βe n ) + exp ( βe m )) n ĉ i m m ĉ j n z (E m E n ). (1.21) 2 For equilibrium a discussion can be found in common quantum many-body literature (e.g. [32,34]) 12

13 1.1 Contour-ordered Green functions It is obtained by expanding G M (τ) using the eigenbasis n with E n as the corresponding eigenenergy of the initial Hamiltonian H(t min ) and then performing the integration over τ. Replacing iω n z yields (1.21). The function g M (z) has a branch cut at the real axis which is purely imaginary and known as the spectral function C ij (ω), i.e. C ij (ω) i[gij M (w + i) gm ij (w i)] = 1 2 Im{gM ij (w + i)} (1.22) = 2π (exp ( βe n ) + exp ( βe m )) n ĉ i Z m m ĉ j n δ(w (E m E n )). mn The generalization for the mixed components is straightforward. There the Fourier transform is restricted to one of the variables and the Fourier components are thus timedependent G (τ, t ) = 1 G β n (iω n, t )e iωnτ, G (iω n, t ) = G (t, τ ) = 1 G (t, iω n )e iωnτ, G (t, iω n ) = β n β β dτg (τ, t )e iωnτ, (1.23) dτ G (t, τ )e iωnτ. (1.24) Both can be analytically continued in the lower and upper complex plane but we restrict the discussion to G ij (z, t ). Again, we require G ij (z, t ) z 1 and obtain using the same z scheme as in (1.21) Gij (z, t ) = i Z m,n (exp ( βe n ) + exp ( βe m )) n ĉ i m m ĉ j(t ) n. (1.25) z (E m E n ) A time-dependent generalization to the spectral function is given by its branch cut at the real axis. Because of the different definition of the Fourier components, we define it without a factor i Cij (ω, t ) Gij (w + i, t ) Gij (w i, t ) (1.26) = 2π (exp ( βe n ) + exp ( βe m )) n ĉ i Z m m ĉ j(t ) n δ(w (E m E n )). mn For a time-independent Hamiltonian this generalization is trivial. The additional time dependence just yields a phase factor Cij (ω, t ) = exp (iωt )C ij (ω) C ij (ω, ) = C ij (ω). (1.27) We note that the corresponding relations for G ij (t, z) and C ij (t, ω) can be followed from G ij (t, z) = (Gji(z, t)) C ij (t, ω) = (C ji (ω, t)). (1.28) 13

14 1 Nonequilibrium quantum mechanics 1.2 The Hubbard model During their investigations on electronic correlations in transition metals, Hubbard [2], Kanamori [3] and Gutzwiller [4] independently introduced the Hubbard model in As its most important feature it includes the competition between the kinetic energy of the valence electrons and their interaction. This interaction is assumed to be screened so strongly that it only affects electrons at the same lattice site. It poses nonetheless a very complicated many-body problem and can describe a variety of interesting correlation induced phenomena (the Mott metal-insulator transition [5,7] being the most prominent one). In a general nonequilibrium situation its Hamiltonian takes the form H(t) = ijσ t ij (t)c iσ c jσ + U(t) i ( n i ) ( n i ). (1.29) The operator c iσ (c jσ) creates (annihilates) an electron with spin σ in an Wannier state at site i (j). The local density of spin type σ {, } can be measured using n iσ = c iσ c iσ. The non-interacting part of the Hamiltonian is defined by the hopping matrix t ij (t). It corresponds to the movement of the electrons in the periodic potential that is generated by the lattice ions. It can be diagonalized at every time t by switching into the Bloch basis. Two electrons at the same lattice site interact through the Coulomb repulsion U(t). A less explicit ingredient is the Pauli exclusion principle which enters naturally within second quantization through the anticommutation rules of the creation (annihilation) operators } } {c iσ, c = δ jσ ij δ σσ, {c iσ, c jσ } = {c iσ, c jσ =. (1.3) The Hubbard model cannot be solved analytically 3. However, it can be mapped onto a single-site problem using dynamical mean-field theory (DMFT), which we discuss in the following section. 1.3 Nonequilibrium dynamical mean-field theory An important basis for DMFT was built in 1989 by Metzner and Vollhardt [25], who investigated the Hubbard model in the limit of infinite space dimensions. They realized that only the so called quantum scaling conserves the competition between kinetic and interaction energy while simplifying diagrammatic calculations drastically. Georges and Kotliar [26] then showed that the introduction of an auxiliary impurity problem allows for the computation of the self-energy in this limit using a self-consistency condition. They also suggested to use this procedure, which is nowadays known as dynamical meanfield theory or DMFT, as an approximation for finite dimensional systems. Diagrammatic perturbation theory is analogous for contour- and time-ordered Green functions and it was realized by Schmidt and Monien [27] in 22 that a self-consistency condition therefore 3 In special one-dimensional cases a solution is possible, see e.g. [6,35]. 14

15 1.3 Nonequilibrium dynamical mean-field theory also exists in nonequilibrium. An appropriate formulation for real times was developed by Freericks, Turkowski and Zlatić [28] in 26. Since then, nonequilibrium DMFT has proven to be a powerful tool to study time-dependent phenomena in correlated electron systems (see e.g. [12, 18, 19, 28, 3, 36]). In the following we derive its dynamical mean field (the so called Weiss field) for the Hubbard model using the cavity method (Sec ) and briefly discuss the selfconsistency cycle (Sec ) Effective single-site action To give a self-contained treatment of the mapping problem we show a possible derivation of the DMFT action for the Hubbard model. The idea is known as the cavity method. For equilibrium DMFT it is typically formulated using the mathematical tool of Grassmann variables (e.g. Ref. [37]), which we have not introduced here. In the following we will adapt it to the language of contour-ordered products. We start from the grand-canonical partition function which is given by { ( )}) Z = tr(exp ( β(h() µn))) = tr (T C exp i (H(t) µn(t))dt. (1.31) C Note that the integration along the real time axis cancels itself because H(t) and N(t) are identical on the upper and lower branch of the contour C. We define the action S as S = i dt(h(t) µn(t)) = S + S + S (). (1.32) C The contributions are given by (each operator is in the Schrödinger picture and the time argument is only required to guarantee the correct ordering when expanding (1.31)) ( S = i dt µ ( n σ (t) + U(t) n (t) 1 ) ( n (t) C 2 2) ) 1, (1.33) σ ( ) S = i dt t i (t)c iσ (t)c σ(t) + h.c., S () = i C C dt ( iσ µ n iσ (t) + t ij (t)c iσ (t)c jσ(t) (1.34) i,σ ij,σ + U(t) i ( n i (t) 1 ) ( n i (t) 1 2 2) ). The partition function can thus be rewritten as Z = tr(t C {exp (S)}) = tr ( T C { exp ( S + S + S ())}). (1.35) 15

16 1 Nonequilibrium quantum mechanics The idea of the cavity method is to pick out one single site and trace out the remaining lattice. This results in a purely local action which, in the limit d, takes a simple form. We first note that the Fock space of our system is given by the tensor product F = F rest F {n iσ } = {n iσ, i 1} {n σ }, (1.36) with F as the Fock space of the isolated site and F rest as the Fock space of the remaining sites. {n iσ } represents a state in the occupation-number basis. We can thus define the partial trace tr rest for an operator O which runs over all sites except site tr rest (O) = {n iσ, i 1} O {n iσ, i 1}, (1.37) {n iσ,i 1} where O is allowed to be an operator on the full Fock space F. Tracing out the remaining lattice will then yield an operator on F. In our case we find by expanding the trace in equation (1.35) Z = {n σ } T C exp (S ) {n iσ, i 1} exp ( S + S ()) {n iσ, i 1} {n σ} {n σ } {n iσ,i 1} = { ( ( {n σ } T C exp (S )tr ))} rest exp S + S () {n σ }. (1.38) {n σ } To calculate the trace over F rest we define (in accordance with (1.9)) O(t) S () 1 ( { ( tr ) rest TC exp S () O(t) }) ( { (, Z Z S () = tr )}) rest TC exp S (), S () (1.39) and note that it is allowed to perform the time-ordering on each subspace independently from the other. This is because the creation/annihilation operators that work on F rest anticommute with those working on F. It is thus possible to calculate the trace over F rest separately as long as one keeps track of the correct sign. Expansion in terms of S yields T C { trrest ( exp ( S + S () ))} = Z S () n= 1 n! ( S)n S (). (1.4) By definition of S only terms with an equal number of c iσ, c j,σ with i, j are non-zero and thus only terms which are of an even power of S contribute. Having in mind that the time ordering for operators working on F still has to be performed, we treat the operators c σ, c σ as (anticommuting) constants when evaluating the expectation value. For easier notation we will suppress the spin index for creation/annihilation operators working on F rest in the following; it can be reinserted using c in c inσ n, c j m c j mσ m. For 16

17 1.3 Nonequilibrium dynamical mean-field theory the operators at site we drop the spatial index, i.e. c σ c σ. We define the n-particle contour ordered Green function as G () i 1...i n,j 1,...,j n (t 1,...,t n) ( i) n c i1 (t 1 )...c in (t n )c j 1 (t 1)...c j n (t n) S (). (1.41) The contour ordering is again hidden in the special definition of the expectation value. To evaluate (1.4) one has to do some combinatorics to get the right amount of contributing terms which are generated by ( S) (2n). From the definition of S we can see that only product terms where n operators c i are multiplied with n operators c j are not equal to zero. Each term can be constructed by choosing n creation operators c j out of the 2n possibilities. This fixes the annihilation operators one has to choose exactly. So we get (2n)! n-particle Green functions that contribute. When we order those operators n!n! to match the definition of an n-particle Green function, we also reorder the operators at site, so that the sequence of the time variables is the same. For instance, a term in lowest order takes the form c j (t)c σ(t)c σ (t )c i (t ) = c i (t )c j (t)c σ (t )c σ (t). (1.42) Terms of higher order can be thought to be a product of n such terms and one readily realizes that the total sign is not affected by the reordering. The definition of the Green function consumes a factor ( i) n so the same factor is left (from ( S) 2n we got a ( i) 2n ). Multiplying it all together we find 1 Z n! ( S)n S () = n= n= = dt 1... dt n ( i) nt i1(t 1 )...t jn(t n= C C n!n! i 1,...,j n 1 (2n)! ( S)2n S () (1.43) n ) G () i 1,...,j n (t 1,...,t n )c σ 1 (t 1 )...c σ n (t n ). In the appendix A.1 it is shown that a reformulation of this expression in terms of connected Green functions (denoted by G (),c ) is possible. It leads to ( ) Z = exp i dt 1... dt nλ σ1...σ (t n 1,...,t n)c σ 1 (t 1 )...c σ n (t n), (1.44) C C n=1 where we defined the new quantity (reintroducing the spin index) Λ σ1...σ n (t 1,...,t n ) ( i)n 1 n!n! t i1 (t 1 )...t jn(t n )G(),c (i 1 σ 1 ),...,(j nσ n) (t 1,...,t n ). (1.45) i 1,...,j n In analogy to (1.4) we define tr, i.e. the trace over F, and insert our last result into equation (1.38). This yields the effective partition function for site Z eff Z Z S () = tr (T C {exp (S eff )}), (1.46) 17

18 1 Nonequilibrium quantum mechanics and the associated effective action S eff = i dt 1... Λ σ1...σ n (t 1,..., t n )c σ 1 (t 1 )...c σ n (t n ) + S. (1.47) n=1 C dt n C σ 1...σ n We emphasize that this result is valid independent of the system s dimension d. The cavity method therefore allows to describe the correlations at one single site by an effective local action. In general, this action involves interactions (non-local in time) of an unlimited amount of particles. As a further difficulty we note that there is no obvious way to determine the quantities Λ σ1...σ n (t 1,...,t n) as they depend on the (unknown) connected Green functions of the original system. We note the interpretation of Z as a functional of Λ encodes the full dynamics of the system. As an example we derive the one-particle Green function by functional derivative δ ln(z[λ]) δλ σσ (t, t ) = Z S δ ln(z () eff [Λ]) Z S () δλ σσ (t, t ) = i c σ (t)c σ (t ) Seff = G σ σ(t, t). (1.48) The limit of infinite lattice dimension (d ) The expression (1.47) is quite complicated. The big advantage becomes visible when the limit d is taken. Applying the quantum scaling [25] t ij 1 i j it can be shown d by counting powers of d that only first order terms (i.e. one-particle Green functions) contribute to the effective action. We follow the argumentation of Georges et al. [38] to show that for the special case of next-neighbour hopping. We take a look at the contributions of n-th order. The amount of next-neighbours at site is proportional to the dimension d. The hopping to sites which are farther away is equal to zero and so the summation yields a total factor d 2n. The quantum scaling defines t i 1 d for next-neighbours and thus the product of 2n such quantities contributes a factor d n. The Green function only contains connected diagrams. Because all the sites it connects are next-neighbours of the shortest path between them is of length 2 (using the Manhattan metric, i.e. R = i R i ). To connect 2n sites we need at least 2n 1 such paths and so the biggest terms are of order d 2(2n 1) = d 2n 1 if all sites are different. Using those results on (1.45) leads to Λ σ1...σ n (t 1,...,t n) i 1,...,j n }{{} d 2n t i1 (t 1 )...t jn(t }{{ n) } d 2n (),c G(i 1 σ 1 ),...,(j nσ n )(t 1,...,t n) } {{ } d 2(2n 1) 1 dn 1. (1.49) If only 2n m sites are different, the order reduces to 2(2n m 1). However, this constraint also reduces the factor given by the summation to d 2n m. In total, we always get the relation Λ σ1...σ n dn 1, which proves that only first order terms contribute. We further have Λ σσ (t, t) = δ σσ Λ σ (t, t ) since the Hubbard Hamiltonian does not involve 18

19 1.3 Nonequilibrium dynamical mean-field theory spin flips and so we find for the effective action in the limit d S eff = i dt dt Λ σ (t, t )c σ (t)c σ(t ) + S. (1.5) C C σ The action (1.5) describes an impurity that is coupled to the so called Weiss field Λ σ (t, t). This field summarizes the influence of all other lattice sites. The question how to obtain it for a given Hubbard Hamiltonian is a different problem. In Sec we briefly discuss how the Weiss field can be self-consistently related to the local Green function. This technique is known as the DMFT self-consistency cycle and allows to iterate towards the correct solution in a numerical treatment. Connection to equilibrium To relate our results to equilibrium, we take a look at the Matsubara component of G σ (t, t ). Initially the system is equilibrium and so G M σ (τ τ ) should be describable by the equilibrium DMFT action. This is indeed possible. Since the time-arguments are elements of C 3, i.e. the imaginary part of the contour, the contour ordering reduces to the imaginary-time ordering of the (equilibrium) Matsubara Green function. All real-time integrations cancel itself and we are allowed to replace the effective action by Seff C M = i dt dt Λ σ (t, t )c σ (t)c σ(t ) + S (1.51) 3 C 3 σ β β dτ dτ λ M σ (τ τ )c σ (τ)c σ(τ ) + S. σ In the last step we defined λ M σ (τ τ ) = iλ M σ (τ τ ), which is equal to the Matsubara Green function as it is defined in equilibrium (see e.g. [32]) Analytical properties of the Weiss field For the later investigations the analytical properties of the Weiss field Λ σ (t, t ) are of importance. They follow from the underlying connected nonequilibrium Green functions G (),c σ (t, t ) Λ σ (t, t ) = t i (t)g (),c ijσ (t, t )t j (t ). (1.52) ij By applying the linked cluster theorem backwards these can be replaced by G () ijσ (t, t ), i.e. usual nonequilibrium Green functions. This allows us to use the results obtained in section The Fourier transform of the Matsubara component follows as Λ M σ (τ τ ) = i e iωn(τ τ ) λ M σ β (iω n), n λ M σ (iω n) ij t i (t min ) g M,() ijσ (iω n ) t j (t min ). (1.53) 19

20 1 Nonequilibrium quantum mechanics Convergence of the Fourier component λ M σ (iω n) in the limit d is ensured because of the quantum scaling [25] t i 1 d n. Its analytical continuation follows from g M,() (z) ijσ λ M σ (z) = ij t i (t min ) g M,() ijσ (z) t j (t min ) (1.54) and inherits both λ M σ (z) z 1 and the branch cut at the real axis. The branch cut can z be seen as an analog of the spectral function (cf. Sec ). We define A σ (ω) i[λ M σ (w + i) λ M σ (w i)] = ij t i (t min ) C M,() ijσ (ω) t j (t min ) (1.55) For later reference we state the relation Λ M σ (τ τ ) = i 2π dωa σ (ω)(f(ω) Θ(τ τ ))exp ( ω(τ τ )), (1.56) which is easily proven by plugging in (1.55), using (1.22) for C M,() ijσ (ω) and identifying the Lehmann representation (cf. appendix (A.2)). The mixed component of Λ σ (t, t ) is related to the underlying Green functions in a similar manner. Using the same convention for the Fourier transform as in (1.23) we find for the analytical continuation of the Fourier components Λσ (z, t ) = ij,() t i (t min ) Gijσ (z, t ) t j (t ). (1.57) During the later analysis of the mapping problem, its branch cut at the real axis will be of special importance. It is closely related to the corresponding functions C ijσ (ω, t ) Aσ (ω, t ) Λ σ (w + i, t ) Λσ (w i, t ) = ij,() t i (t min )Cijσ (ω, t )t j (t ), with A σ (t, ω) = (Aσ (ω, t)) because of (1.28). (1.58) We note the relations i Λ σ (τ, t ) = 2π Λ σ (t, τ ) = i 2π dωaσ (ω, t )(f(ω) 1)exp ( ωτ) (1.59) dωa σ (t, ω)f(ω)exp(ωτ ). (1.6) for later reference. They are proven in the same way as (1.56). The quantity A has a trivial time-dependency for H () (t) = H () (t min ) in analogy to (1.27) (ω, t ) Aσ (ω, t ) = exp (iωt )A σ (ω) A σ (ω, ) = A σ(ω). (1.61) 2

21 1.3 Nonequilibrium dynamical mean-field theory Self-consistency condition for a semi-elliptical density of states In general, it is impossible to calculate the Weiss field Λ σ (t, t ) from the expression given by the cavity method. It is, however, possible to relate the Weiss field self-consistently to the Green function that results from the corresponding effective action. This was discussed in detail by Eckstein [3] and we only want to sketch the idea. It is based on the fact that the impurity Green function and the associated self-energy are connected by two different Dyson equations, one that belongs to the effective impurity problem and one that belongs to the original lattice problem. Let us assume that we were able to calculate the impurity Green function from a given Λ σ (t, t ). Then the self-energy follows from the impurity Dyson equation and can be inserted into the lattice Dyson equation. The local lattice Green function can be calculated and, since it has to coincide with the impurity Green function for the exact solution, be used to get an update on the Weiss field. This closes the self-consistency. There is one special situation where all those steps can be performed analytically. It allows to give an explicit relation that connects the Weiss field and the resulting impurity Green function. It happens when hopping matrix and lattice structure of the underlying Hamiltonian (1.29) lead to a semi-elliptical density of states 4V 2 ǫ ρ(ǫ) = 2, (1.62) 2πV 2 where V is the quarter-bandwidth. Such a form is, for example, obtained for nextneighbour hopping on the Bethe lattice [39]. For a time-independent kinetic term of such a structure it was show by Eckstein et al. [4] that the Weiss field Λ σ (t, t ) is related to its associated Green function by Λ σ (t, t ) = V 2 G σ (t, t ). (1.63) Despite the apparent simplicity of this relation it is usually not possible to find a selfconsistent solution analytically. An interesting exception is the Falicov-Kimball model that was solved analytically by Eckstein and Kollar [36] in case of an interaction quench. It is possible to allow for a time-dependent hopping if the time dependence just enters as a global scaling factor. The Hamiltonian then takes the form H(t) = f(t) t ij c iσ c jσ + U(t) ( n i + 1 ) ( n i + 1 ) f(t) 2 2 H(t). (1.64) ijσ i In the Hamiltonian H(t) only the interaction Ũ(t) is time-dependent. If we assume f(t) >, we can transform the Schrödinger equation to a different time argument, i.e. id Ψ(t) = f(t) H(t)dt = H(t)dF(t), with F(t) = t t min f(t )dt. (1.65) Because of f(t) >, F(t) is bijective and so we can interpret t as a function of F. The 21

22 1 Nonequilibrium quantum mechanics Schrödinger equation can thus be formulated as id Ψ(F) = H(F)dF, (1.66) i.e. F is our new time variable. Since H(F) has a time-independent hopping, which we assume to correspond to a semi-elliptic density of states with bandwidth 4V, we can use equation (1.63) Λ σ (F, F ) = V 2 G σ (F, F ). (1.67) The Weiss field enters the effective action S eff as S loc = dfdf Λ σ (F, F )c σ (F)c σ(f ) C C σ = dtdt f(t)λ σ (t, t )f(t )c σ(t)c σ (t ). (1.68) C C σ We can therefore identify Λ σ (F(t), F(t )) = f(t)λ σ (t, t )f(t ) and find the result Λ σ (t, t ) = V 2 f(t)g σ (F(t), F(t ))f(t ) = V 2 f(t)g σ (t, t )f(t ). (1.69) 22

23 2 The mapping problem In section we have derived an effective local action for the Hubbard Hamiltonian in the limit of infinite lattice dimension, where it reduces to a single-site impurity problem. However, because of the interaction it remains a complicated task to obtain the associated impurity Green function. A successful approach for equilibrium problems maps the action onto the Hamiltonian of an appropriate single-impurity Anderson model (SIAM) [26]. This allows to use Hamiltonian-based solvers, e.g. exact diagonalization, for the calculation of the local Green function. In this chapter we discuss the corresponding nonequilibrium mapping problem. We recall that the single-site action takes the form ( S 1 = i dt µ ) ( n σ (t) + H imp (t), H imp (t) = U(t) n (t) 1 ) ( n (t) 1 ), C 2 2 σ S 2 = i Λ σ (t 1, t 2 )c σ (t 1)c σ (t 2 )dt 1 dt 2, (2.1) σ C C and remind us that the time dependence of, for example, n σ (t) only leads to a correct placement within a contour integral (cf. section 1.1). The operator itself is in the Schrödinger picture. Only operators with a hat are in the Heisenberg picture and have a real time-dependence. For simplicity we will assume t min = in the following. In analogy to the cavity description of the Hubbard model we split the SIAM Hamiltonian into three parts. An impurity Himp SIAM that is coupled to a surrounding bath H bath by the hybridization H hyb, i.e. H SIAM = H SIAM imp + H bath + H hyb. (2.2) It will be very useful to investigate the bath subsystem independently from the impurity. We therefore introduce two sets, M SIAM = {, 1, 2,... } and M bath = M SIAM \{}, so that the impurity is located at site. The operator a pσ (a pσ ) annihilates (creates) an electron with spin σ at bath site p for p M bath, at the impurity for p =. The contributions to the SIAM Hamiltonian are then given by ( Himp SIAM = U(t) n 1 ) ( n 1 ), with n iσ = a iσ 2 2 a iσ, H bath = Vpp σ (t)a pσa p σ, H hyb = ( ) Vp(t)a σ σa pσ + h.c.. (2.3) p,p M bath σ p M bath σ For a star geometry, i.e. V σ pp (t) = δ pp V σ pp(t) for p M bath (cf. Fig. 2.1), Eckstein [3] 23

24 2 The mapping problem V 3 (t) V 4 (t) U(t) V 5 (t)... V 2 (t) V 1 (t) Mapping Λ(t 2, t 1 ) U(t) t 1 G(t, t ) t 2 Figure 2.1: The left graphic displays a SIAM with a star geometry. The bath sites (empty circles) are not coupled with each other but only to the impurity site (filled circle). The hopping from site i to the impurity is described by V i (t), the reverse process by its complex conjugate Vi (t). The right graphic shows the DMFT action that is generated by the Weiss field Λ(t 2, t 1 ). The continuous lines represent the contour-time integrations. The Hubbard interaction U(t) is in both cases restricted to the impurity site. For the correct hybridization both yield the same Green function G(t, t ). constructed a mapping that required for two baths to cover the dynamics of an arbitrary DMFT action. The first describes the correlations within the initial state and fades for t, while the second develops to describe the steady state at t =. In a numerical simulation it is crucial to reduce the complexity of the system as much as possible and so a single-bath solution, i.e. fewer sites, would be very desirable. One degree of freedom that Eckstein [3] did not use, is the freedom of choice for the bath geometry, which he fixed to have a star structure right from the beginning. So the question arises if a different or even dynamical geometry does allow for a single-bath solution. To tackle it we will discuss the mapping problem in full generality by extending Eckstein s [3] analysis (Sec. 2.1). However, it will not only turn out that the star structure is generic, but also that it is the canonical choice because of its simplicity. For the following discussion we require the SIAM system to have the same temperature T = 1 k B at time t =. To be consistent with the DMFT action we further measure any β observable O using the grand-canonical ensemble. Its expectation value thus takes the form (N is the particle number operator and the trace runs over the whole Fock space) ( ) Ô(t) SIAM = tr ˆρ SIAM Ô(t), with ˆρ SIAM = exp ( β(h() µn)) tr(exp ( β(h() µn))). (2.4) For easier notation we suppress the spin index. It can be reinserted using the replacement rule p (p, σ), V pp δ σ,σ Vpp σ. We note that a matrix M pp can be understood as an operator on a Hilbert space H 1, defined by the property that the indexes p, p are 24

25 2.1 Role of the bath geometry restricted to M bath. The matrix is then an element of the vector space L(H 1, H 1 ) of all linear functionals on H 1 and will be denoted by M. The matrix product is readily defined (MO) pp = p M bath M p p O pp. (2.5) The bath Hamiltonian can be diagonalized at every time t but for us only its diagonal form at t = will be of importance. It can be found by diagonalizing V (), which is a hermitian matrix. We call the corresponding unitary transformation O, so that V () = ODO, f p O p pa p H bath () = d p f p f p, (2.6) p M bath p M bath where d p are the elements of the diagonal matrix D and equal to the single-particle eigenenergies of H bath. In our discussion we will denote the contour-ordered correlation } function between two arbitrary SIAM sites by F pp (t, t ) i T C {â p (t)â p (t ) SIAM with p, p M SIAM, and the DMFT impurity Green function by G(t, t ) i c(t)c (t ) S. 2.1 Role of the bath geometry A necessary relation that the correct set of mapping parameters has to fulfill can be found using the theory of differential equations. If two functions satisfy the same boundary conditions and the same equation of motion, then they are equal everywhere. The differential equation and its corresponding boundary condition for the DMFT action were derived in section and the same equations can be used for the SIAM if we set Λ =. The initial state is thermal with temperature T = 1 k B. Therefore both impurity Green functions, i.e. β G(t, t ) for the DMFT action and F (t, t ) for the SIAM Hamiltonian, satisfy the same boundary condition, namely H(, t ) = H( iβ, t ), H F, G. The equations of motion are given by (i t + µ)g(t, t ) = δ C (t, t ) + i [H imp (t), c(t)] c (t ) S + dt 2 Λ(t, t 2 )G(t 2, t ), (2.7) {[ĤSIAM C ] (i t + µ)f (t, t ) = δ C (t, t ) + i T C imp (t), â (t) â (t ) } SIAM + V p (t)f p (t, t ). p M bath First, we analyze the higher order terms on the right hand side. Using their corresponding self-energies Σ(t, t ), they can be written as follows [H imp (t), c(t)]c (t ) S = dt 1 Σ DMFT (t, t 1 )G(t 1, t ), {[ĤSIAM C ] } T C imp (t), â (t) â (t ) SIAM = dt 1 Σ SIAM (t, t 1 )F (t 1, t ). (2.8) C 25

26 2 The mapping problem It is known from perturbation theory that the self-energy can be expressed as the sum over all skeleton diagrams when we use for each particle line the interacting propagator (i.e. G(t, t ), F (t, t )). The DMFT action and the SIAM Hamiltonian describe the same interaction U(t) at a single impurity site and thus the meaning of each diagram is the same in both cases. The higher order terms on the right hand are therefore equivalent, i.e. they describe the same functional of the corresponding impurity Green function. It remains to show that also the remaining term becomes equivalent for the right choice of the hybridization V p (t), i.e. dt 2 Λ(t, t 2 )G(t 2, t ) V p (t)f p (t, t ). (2.9) C p M bath To continue, we need to decouple the impurity Green function of the SIAM from the other (non-interacting) sites, i.e. we need to express F p (t, t ) in terms of F (t, t ) itself. This is possible by taking a look at the equation of motion of F pp (t, t ) for p M bath, p M SIAM (i t + µ)f pp (t, t ) = p M bath V p p (t)f pp (t, t ) + δ p V p (t)f (t, t ). (2.1) The impurity Green function enters only as an inhomogeneity into this set of differential equations. The differential operator L pp (t) (with p, p M bath ) of the corresponding homogeneous equation is given by L pp (t) = (i t + µ)δ p p V p p (t). (2.11) A fundamental solution to this operator (i.e. a mathematical Green function) fulfills the relation L p p (t)g M pp (t, t ) = ((i t + µ)δ p p V p p (t)) G M pp (t, t ) = δ C (t, t )δ p,p. (2.12) p M bath p M bath Without the inhomogeneity this equation would structurally be equivalent to the timedependent Schrödinger equation. A solution for the homogeneous equation is therefore found by Picard iteration and fulfills the usual group properties that are known for a propagator (however, it is not norm conserving in H 1 for imaginary times) { [ t ]} L p p (t)u pp (t, t ) =, where U(t, t ) = T exp i (V (t 1 ) µ) dt 1. (2.13) p M t bath We recall that V (t) is the hopping matrix and as such constant for imaginary arguments, i.e. V ( iτ) = V (). As a consequence the propagator fulfills the property U( iτ, iτ ) = U( i(τ τ )). An arbitrary fundamental solution of the differential operator L pp (t) is readily constructed, e.g. G F (t, t ) = Θ C (t, t )U(t, t ) L(t)G F (t, t ) = δ C (t, t ). (2.14) 26

27 2.1 Role of the bath geometry It is important to think about the properties that the (physical) solution to (2.1) has to satisfy. From its definition as the expectation value of the propagation amplitude the boundary conditions F pp (, t ) = F pp ( iβ, t ) follow and must be fulfilled for every possible value of t. For p = this can be achieved by choosing the fundamental solution to be the physical Green function of the decoupled bath with a thermal initial state at temperature T = 1, i.e. (p, k B β p M bath ) } g pp (t, t ) i T C {â p (t)â p (t ) bath = i [ U(t, ) (f[v ()] Θ C (t, t )) U (t, ) ] pp. (2.15) In this expression f[v ()] refers to the matrix Fermi distribution, i.e. f[v ()] = f[od()o ] = Of[D()]O, where f[d()] pp = δ pp f(d p ), (2.16) where f(ǫ) is the Fermi function. The relation g(, t ) = g( iβ, t ) holds by definition and we so can integrate (2.1) F p (t, t ) = dt 1 [g(t, t 1 )V (t 1 )] p G(t 1, t ), (2.17) C where F p (t, t ) inherits antiperiodic boundary conditions from g(t, t ). It is known from the theory of differential equations that only one solution exists which fulfills this antiperiodicity and such F p (t, t ) is indeed equal to the physical Green function of the interacting SIAM. Connection between the SIAM and the Weiss field Based on this result, it is possible to derive a necessary relation that connects bath geometry (which enters through g pp (t, t )) and hybridization to the Weiss field Λ(t, t ). Using the expression (2.17) for F p (t, t ), it is found by comparison of the equations of motion (2.8) Λ(t, t ) = V p (t)g pp (t, t )Vp (t ). (2.18) p,p M bath A slightly simpler result was given by Eckstein [3], who discussed a star geometry and thus obtained a diagonal Green function g pp (t, t ) δ pp. Our result reduces to his expression if we choose the same geometry. Using the unitary matrix O, which diagonalizes V () (cf. (2.6)), we can also obtain a diagonal formulation for the general case. We first define a new quantity v p (t) e i(dp µ)t V p (t)u pp (t, )O p p. (2.19) p,p M bath It is important to note that this definition ensures v p ( iτ) = v p () for every τ. That results from V p ( iτ) = V p () holding true by definition. Thus the propagator U( iτ, ) 27

28 2 The mapping problem can be written as U( iτ, ) = exp [ iτ ] i (V (t 1 ) µ)dt 1 = Oexp [ (D µ)τ] O, (2.2) where D is the diagonal matrix containing the one-particle eigenenergies (cf. (2.6)). Inserting this into (2.19) cancels the unitary transformation O on the right and one finds v p ( iτ) = e (dp µ)τ V p ()O pp exp [ (D µ)τ] p p p,p M bath = V p ()O pp = v p (). (2.21) p M bath It is therefore possible to use v p (t) as the hopping parameter for a new geometry. But first we notice that our definition leads to the following diagonal form for Λ σ (t, t ) (for completeness we reintroduce the spin index in the following discussion), i.e. Λ σ (t, t ) = p M bath v pσ (t)h pσ (t, t )v pσ(t ). (2.22) h pσ (t, t ) is the Green function of the time-independent diagonal bath (V σ (t) D σ ()) describing a non-interacting equilibrium situation h pσ (t, t ) h(d pσ, t, t ) i(f(d pσ µ) Θ C (t, t ))exp ( i(d pσ µ)(t t )). (2.23) We would have found the exact same expression for Λ σ (t, t ) if we had started from the following bath and hybridization matrices ( ) d pσ a pσa pσ, v pσ (t)a σa pσ + h.c.. (2.24) H star bath = p M bath σ H star hyb = p M bath σ This turns out to be the star structure suggested by Eckstein [3]. So the easiest structure, namely the time-independent star geometry, already covers the whole range of DMFT actions accessible to the SIAM. The result implies further that any solution to the mapping problem is not unique. Starting from a star geometry we can always introduce a unitary transformation U σ pp (t, t ) which, by inverting equation (2.19), leads to a dynamical bath geometry resulting in the same correlation function for the impurity site. We emphasize that the transformation onto the star geometry was not possible on the level of the SIAM Hamiltonian. Although it is possible to diagonalize its bath part at every time t, this would lead to time-dependent creation/annihilation operators ˆf i (t)/ ˆf j (t ) and so the geometry would still be dynamical. The derivatives of those operators with respect to time would then show up in equation (2.1). Instead, we replaced the whole system by a different one which has the same local correlation function for the impurity site (and in general, only this site). It is obvious that the star structure with time-independent 28

29 2.2 Existence of a valid mapping eigenenergies d p has nice properties. For an arbitrary bath the Green function g pp (t, t ) can only be calculated numerically. For the star structure, on the other hand, this Green function is diagonal and well-known. Its explicit form is given by equation (2.23). The star structure therefore seems to be the canonical choice for the construction of an exact mapping. 2.2 Existence of a valid mapping In the previous section we have seen that the canonical choice for the SIAM has a star geometry. Its hybridization has to fulfill (2.22) to yield a Green function that corresponds to the action (2.1). A priori it is not clear that such a hybridization exists and we will discuss that question in the following subsection. For zero temperature and particle-hole symmetry Eckstein [3] suggested a construction that involved the splitting of the Weiss field into two parts Λ(t, t ) = Λ (t, t ) + Λ + (t, t ), (2.25) where the first Weiss field Λ (t, t ) stores the fading knowledge about the initial state and only the second Weiss field Λ + (t, t ) remains in the long time limit. An important question was left open regarding the construction of the second bath. The approach relies on the assumption that the difference i(λ < (t, t ) Λ < (t, t )) of the lesser components always yields a positive semi-definite matrix (with t, t as its indexes). Because of the complex structure of Λ(t, t ) and Λ (t, t ) it is difficult to tackle this question directly. We will therefore take a different route and first discuss the existence of a formal solution to (2.22). This solution, though impractical for numerical purposes, can be used to proof the required assumption for an important class of situations. We will further lift the restriction of particle-hole symmetry and extend the formalism to finite temperatures. Our search concerns a representation of the Weiss field that has the following form (we drop again the spin index in the following) Λ(t, t ) = j=1 dǫρ j (ǫ)v j (ǫ, t)h(ǫ, t, t )v j (ǫ, t ), (2.26) with ρ j (ǫ) as the density of states of the j-th bath. It is shown in the appendix A.3 that this expression can always be reduced to (2.22) Existence for a semi-elliptical density of states The strongest result is possible if the DMFT action (2.1) is evaluated for a semi-elliptic density of states. If we set the quarter-bandwidth V = 1, the self-consistency equation is given by (1.69) Λ(t, t ) = f(t)g(t, t )f(t ), (2.27) where f(t) is a time-dependent scaling factor of the underlying hopping matrix. The factors f(t), f(t ) (both are real quantities) can always be absorbed by the hopping between 29

30 2 The mapping problem bath and impurity v p (t), v p (t ) and so Λ(t, t ) can be expressed in the form (2.26) if and only if G(t, t ) itself has a representation of this form. To prove the existence we start from the Lehmann representation for the Green function G(t, t ), which holds for any time-dependent Hamiltonian H(t) if the system is in equilibrium at t =. A proof can be found in the appendix A.2. {ĉ(t)ĉ G(t, t ) = i T C (t ) } (2.28) = i n ĉ (t ) m m ĉ(t) n Z n=1 m=1 (exp ( βe m ) + exp ( βe n )) (f(e n E m ) Θ C (t, t )). If G(t, t ) is the exact solution, the eigenstates n and eigenenergies E n belong to the Hamiltonian H(t) of the original Hubbard model (cf. equation (1.29)) in the limit d. If G(t, t ) is an approximation stemming from the DMFT self-consistency loop, H(t) is the SIAM Hamiltonian that was calculated during the last step. Z = tr (exp ( βh())) is the partition function. We define the new quantity W(m, n, t) exp ( βe m ) + exp ( βe n ) m ĉ(t) n exp ( i(e m E n )t). (2.29) This definition ensures that W(m, n, t) is a function on the real numbers (i.e. t [, t max ]) and thus fulfills the property W(m, n, iτ) = W(m, n, ). (2.3) It therefore might be used as a hopping parameter in the SIAM. We plug this definition into (2.29) and find G(t, t ) = 1 Z = 1 Z W(m, n, t)w (m, n, t ) [i(f(ǫ nm ) Θ C (t, t ))exp ( iǫ nm (t t ))] n=1 m=1 n=1 m=1 W(m, n, t)w (m, n, t )h(ǫ nm, t, t ). (2.31) Here we defined ǫ nm E n E m and identified the non-interacting Green function h(ǫ nm, t, t ). This result is already close the desired form (2.26). It remains to introduce a proper density of states. First, we rearrange the summands in a way that keeps the energy difference ǫ nm = E n E m fixed. This can be achieved by introducing a sequence ǫ k which runs through every possible value of ǫ nm exactly once, i.e. G(t, t ) = 1 Z k n=1 m=1 W(m, n, t)w (m, n, t )h(ǫ k, t, t )δ ǫk,ǫ nm. (2.32) To every value ǫ k there might exist more than one combination of values (m, n) (possibly 3

31 2.2 Existence of a valid mapping an infinite number) so that ǫ k = ǫ nm. In any case we can define an energy depended sequence (a j (ǫ)) j N that runs for a given energy ǫ through all possible combinations (m, n). If their number is finite, we define a j (ǫ) (, ) for j > j max (j max labels the last valid combination) and W(,, t). We further introduce the density of states ρ(ǫ) = 1 δ(ǫ ǫ k ), where L = L This allows to rewrite the equation (2.31) as G(t, t ) = 1 Z = j=1 j=1 k=1 k dǫ k δ(ǫ ǫ k ). (2.33) W(a j (ǫ k ), t)w (a j (ǫ k ), t )h ǫk (t, t ) (2.34) dǫρ(ǫ)v j (ǫ, t)h(ǫ, t, t )V j (ǫ, t ), with V j (ǫ) L Z W(a j(ǫ), t). The comparison of this result to equation (2.26) shows that a valid representation is found. It is, of course, of a pure formal nature and we will not be able to use it in a numerical calculation. However, it guarantees the existence of a valid representation not only for the exact Weiss field Λ(t, t ), but also for any intermediate Λ(t, t ) obtained during an iteration Existence for an arbitrary density of states For an arbitrary density of states (DOS) the situation is less obvious. It was shown by Eckstein [3] that Λ(t, t ) = f n+1 [G] n (t, t ) (2.35) n follows from the DMFT self-consistency equations for a large class of DOS functions ([G] n (t, t ) is the n-fold convolution of G(t, t ) with itself and f n+1 are real coefficients dependent on the DOS). It is hard to tell if that expression conserves the property (2.34) and an iteration might get stuck at a Λ(t, t ) that has no valid representation. However, we will see that the exact solution can always be represented. And so, if the starting point for G(t, t ) is close enough, an approximate realization for Λ(t, t ) might be enough to ensure convergence. For the exact solution a discussion similar to the previous case is possible. The form of the Weiss field was derived in section using the cavity method (equation (1.45)) Λ(t, t ) = kl t k (t)g () kl (t, t )t l (t ). (2.36) 31

32 2 The mapping problem In analogy to the case of a semi-elliptic density of states, we define W(m, n, j, t) exp ( βe m ) + exp ( βe n )t j (t) m ĉ j (t) n exp ( i(e m E n )t), and W(m, n, t) = W(m, n, j, t). (2.37) j The sequences a kl j (ǫ) are the same for every Green function G() kl (t, t ), i.e. a kl j (ǫ) = a j(ǫ) and the same holds for ρ kl (ǫ) = ρ(ǫ) (cf. the discussion beneath (2.32)). This is because the same basis (i.e. the eigenbases of H()) is used in every case and therefore the corresponding eigenenergies E n, E m are independent of the Green function. Thus we can formulate Λ(t, t ) = 1 Z = j=1 kl dǫρ(ǫ) W(a j (ǫ), k, t)w (a j (ǫ), l, t )h(ǫ, t, t ) j=1 dǫρ(ǫ)v j (ǫ, t)h(ǫ, t, t )V j (ǫ, t ), with V j (ǫ, t) L Z W(a j(ǫ), t). (2.38) We compare this result to equation (2.26) and notice that it is a valid representation. This verifies that the exact Weiss field can, in principle, always be mapped onto an appropriate SIAM Hamiltonian. 2.3 A numerically feasible mapping In the last section we have seen that a representation of the form (2.26) is ensured to exist for the Weiss field if Λ(t, t ) is the exact Weiss field. Λ(t, t ) is an approximate iterative solution, and a semi-elliptic density of states is used. Based on this result we will construct a numerically feasible mapping in this section. Our idea is to interpret (2.26) as a scalar product of two vectors Λ(t, t ) = n=1 dǫv n (ǫ, t)h(ǫ, t, t )v n(ǫ, t ) dǫ v (ǫ, t ) v(ǫ, t)h(ǫ, t, t ), (2.39) where we substituted ρ n (ǫ)v n (ǫ, t) v n (ǫ, t) (ρ(ǫ) is non-negative). The vector v(ǫ, t) is seen as the element of a Hilbert space H and v n (ǫ, t) as its n-th component. The scalar product v (ǫ, t ) v(ǫ, t) is of course independent of the basis and our idea is to search for a especially convenient representation of v(ǫ, t) where all its coefficients can be calculated. The number of non-zero coefficients in this basis is then equal to the number of baths. 32

33 2.3.1 Memory of the initial state (the first bath) 2.3 A numerically feasible mapping We emphasize again the important constraint v(ǫ, iτ) = v(ǫ, ), which every vector inherits from its components. On the imaginary branch C 3 the time dependence thus vanishes, i.e. Λ( iτ, iτ ) = dǫ v(ǫ, ) 2 (f(ǫ) Θ(τ τ ))exp ( ǫ(τ τ )). (2.4) Eckstein [3] therefore suggested to determine the bath coefficients at time t = using the Matsubara component Λ( iτ, iτ ) = Λ M (τ τ ) of the Weiss field. This can be done by using representation (1.56) of Λ M (τ τ ) Λ M (τ τ ) = i 2π dǫa(ǫ)(f(ǫ) Θ(τ τ ))exp ( ǫ(τ τ )), (2.41) where A(ǫ) = i(λ M (ǫ + i) λ M (ǫ i)) is an analog of the spectral function (cf. section for the definition). Similar to his result we find by identifying the integrands 1 2π A(ǫ) = v(ǫ, ) 2 = v n (ǫ, ) 2. (2.42) The result A(ǫ) is a direct analog to a similar relation for the spectral function. Equation (2.42) can be further simplified by choosing a simple basis for H where one of the basis vectors is parallel to v(ǫ, ). All other basis vectors we choose arbitrarily, but of course orthonormal, so that n=1 v(ǫ, ) = κ 1 (ǫ, ) e 1 (ǫ), v(ǫ, t) = κ n (ǫ, t) e n (ǫ). (2.43) We can assume κ 1 (ǫ, ) to be real and non-negative since a possible phase factor can be absorbed by e 1 (ǫ). Thus (2.42) becomes 1 κ 1 (ǫ, ) = A(ǫ). (2.44) 2π For the mixed component we use the same scheme. An integral representation of Λ (t, τ ) is given by (1.59) and we can identify (cf. section for the definition of A (t, ǫ)) n=1 A (t, ǫ) = exp ( iǫt) v(ǫ, ) v(ǫ, t) = exp ( iǫt) κ 1 (ǫ, )κ 1 (ǫ, t), κ 1 (ǫ, t) = exp (iǫt) A (t, ǫ) κ(ǫ, ) = exp (iǫt) 2π A (t, ǫ) A(ǫ). (2.45) It is ensured by (1.58) that the same result is found if we identify Λ (τ, t ) instead. The result is further in agreement with (2.44) because of A (, ǫ) = A(ǫ), cf. (1.61). 33

34 2 The mapping problem It is interesting to check if this leads to a time-independent κ 1 (ǫ, t) if we consider a time-independent Hamiltonian H(t) = H(), i.e. if we reduce the problem to the mapping problem in equilibrium. Indeed, we find κ 1 (ǫ, t) = κ 1 (ǫ, ) because of A (t, ǫ) (1.61) 1 = exp ( iǫt) A(ǫ) κ 1 (ǫ, t) = 2π A(ǫ) = κ 1(ǫ, ). (2.46) It is possible to transform (2.39) to our new basis. This results in a new bath structure (represented by Λ κ n(t, t )) Λ(t, t ) = Λ κ n (t, t ) dǫ v(ǫ, t ) h(ǫ, t, t ) v(ǫ, t) = Λ κ n (t, t ), n=1 dǫ κ n (ǫ, t)h(ǫ, t, t )κ n (ǫ, t ). (2.47) The first Weiss field, Λ κ 1(t, t ), is defined by κ 1 (ǫ, t). It takes the form Λ κ 1 (t, t ) = i 2π dǫ A (t, ǫ)a A(ǫ) (ǫ, t ) (f(ǫ) Θ C (t, t )). (2.48) and is thus completely determined by the correlations of the initial state with states at later times t (these correlations are described by the functions Λ (t, τ ), Λ (τ, t)). The expression can be evaluated for a system that is in equilibrium. Its Hamiltonian is time-independent and by using (1.61), i.e. A (t, ǫ) = exp ( iǫt)a(ǫ), it is thus possible to cancel the denominator. The resulting integral is nothing else but a backward Fourier transform and yields Λ(t, t ) itself. The bath κ 1 (ǫ, t) therefore describes the exact Weiss field exactly. In nonequilibrium, on the other hand, H(t) will have an explicit time dependence. Eckstein [3] gave an argument showing that this precludes the difference Λ κ rest(t, t ) Λ κ n(t, t ) = Λ(t, t ) Λ κ 1(t, t ) (2.49) n=2 from being equal to zero in general. For a system out of equilibrium the correlations between the initial state and states at t = usually vanish and this requires Λ (t, τ) t ( κ 1 (ǫ, t) t ). For the final state to be non-trivial, Λ κ rest(t, t ) therefore has to be non-zero. At t, t =, on the other hand, we have Λ κ rest (t, t ) = and so we conclude in agreement with Eckstein [3] that the first bath, κ 1 (ǫ, t), describes the initial state at t = and the fading memory about its properties at later times. For comparison with Eckstein s [3] analysis we note the relation Λ (t, t ) = Λ κ 1(t, t ) and Λ + (t, t ) = Λ κ rest (t, t ). In the next subsection we will discuss the second Weiss field Λ κ rest (t, t ) which is defined it by means of a basis transformation. We will see that this ensures the necessary relation i(λ < (t, t ) Λ < (t, t )) (cf. the discussion beneath (2.25)). 34

35 2.3.2 Approach to the steady state (the second bath) 2.3 A numerically feasible mapping The second Weiss field, Λ κ rest, has the property Λκ rest (t, t ) = for t C 3 (or t C 3 ; C 3 runs from t min to t min iβ, cf. Fig 1.1) because κ n (ǫ, ) = holds for n 2. This allows for the formulation Λ κ rest (t, t ) = i n=2 ) dǫ (κ n (ǫ, t)e iǫt (f(ǫ) Θ C (t, t )) (κ n (ǫ, t )e iǫt ) ( ) = i k n (f Θ C (t, t )) k n n=2 where (k n ) (t,ǫ) κ n (ǫ, t)e iǫt, (f Θ C (t, t )) (ǫ,ǫ ) (f(ǫ) Θ C(t, t ))δ(ǫ ǫ ). (t,t ), (2.5) It is the last step that relies on Λ κ rest(t, t ) = for t C 3 or t C 3 (for t = iτ follows exp ( ǫτ) exp (ǫτ) ). We further interpreted the sum as a scalar product of two vectors whose components are given by k n. However, these components are themselves matrices. The Fermi function f is a diagonal matrix in this notation. In the previous discussion we have seen that the initial state and the fading memory about it are described by a single bath, κ 1 (ǫ, t). For large times only the second Weiss field Λ κ rest remains and hence describes a possible steady state in the long time limit. We will discuss in the following how it can be expressed through a single bath. In the subsequent analysis we will call the decomposition of any positive semi-definite, hermitian matrix M into the form M(t, t ) = (M) (t,t ) = (mm ) (t,t ) = dǫ m(t, ǫ)m (t, ǫ) (2.51) taking a square root of M. A square root m of M is not unique as can be seen by inserting ½ = UU between m and m (where U is any unitary transformation). T = At zero temperature the transformation onto a single bath turns out to be straightforward. The Fermi-function reduces to the Heaviside step function (f(ǫ) = Θ(ǫ)) and therefore we find two independent equations, one for the lesser and one for the greater component 1. iλ κ,< rest(t, t ) = 2. iλ κ,> rest(t, t ) = n=2 n=2 dǫ κ n (t, ǫ)e iǫ(t t ) κ n(t, ǫ) dǫ κ n (t, ǫ)e iǫ(t t ) κ n(t, ǫ) dǫv (t, ǫ)e iǫ(t t ) V (t, ǫ), dǫv (t, ǫ)e iǫ(t t ) V (t, ǫ). (2.52) A similar result was obtained by Eckstein [3], who investigated a particle-hole symmetric Hamiltonian. This requires V (t, ǫ) = V (t, ǫ) and thus solving one of the equations is 35

36 2 The mapping problem sufficient in this case. In addition to his result we realize that the decompositions (2.52) are always well defined. They require iλ κ,> rest(t, t ) and iλ κ,< rest(t, t ) to be positive semidefinite and hermitian, which they are both by construction. In chapter 3 we will discuss the Cholesky and the eigenvalue decomposition as possible candidates to calculate V (t, ǫ) in a numerical simulation. For any bijective function ǫ(s) (with the inverse ǫ 1 (ǫ(s)) = s) we note the property dǫv (t, ǫ)e iǫ(t t ) V (t, ǫ) = ǫ 1 ( ) ǫ 1 () dsṽ (t, s)e is(t t ) Ṽ (t, s) (2.53) with Ṽ (t, s) = ǫ (s)v (t, ǫ(s))e i(ǫ(s) s)t. The energy interval for the integration is thus arbitrary at zero temperature. T = There is another case where the Fermi-function takes a special form. At infinite temperature it reduces to a constant (f(ǫ) = 1 2 ). Λκ rest (t, t ) therefore takes the form Calculating a square root q ( ) ( ) Λ κ rest(t, t ) T= 1 = i 2 Θ C(t, t ) k n k n qq = n=2 (t,t ). (2.54) k n k n (2.55) n=2 immediately yields the desired form for the second Weiss field ( ( ) 1 Λ κ rest(t, t ) = i q 2 Θ C(t, t ) )q dǫ V (t, ǫ)(f(ǫ) Θ C (t, t ))e iǫ(t t ) V (t, ǫ). = (t,t ) with V (t, ǫ) q(t, ǫ)e iǫt = q (t,ǫ) e iǫt. (2.56) Numerically, the matrix q can be constructed by Cholesky or eigenvalue decomposition and the results of the following chapter 3 are therefore also applicable Construction for finite temperature It remains to discuss a finite temperature T,. We can eliminate the Fermi-function by calculating the difference i(λ κ,< rest(t, t ) Λ κ,> rest(t, t )). The result is guaranteed to yield a positive semi-definite matrix and can thus be decomposed ( ) (Γ) (t,t ) i(λκ,< rest(t, t ) Λ κ,> rest(t, t )) = k n k n = qq. (2.57) n=2 (t,t ) 36

37 2.3 A numerically feasible mapping However, this equation does not determine Λ κ rest (t, t ) completely and we cannot expect the first guess Λ κ rest (t, t )? = ( q(f Θ C (t, t ))q ) (t,t ) (2.58) to yield the correct result. For the following construction we will assume that q is invertible, i.e. n=2 k nk n strictly positive definite.1 This allows for the definition (q 1 k n )(q 1 k n ) n=2 c n c n = ½. (2.59) For the lesser component of Λ κ rest we notice the following property ( a is an arbitrary normalized vector in the corresponding vector space) in the sense that ( ) a c n fc n a = n=2 iq 1 Λ κ,< rest (q ) 1 = n=2 c n fc n ½, (2.6) n=2 a ( ) c n fc n a a ( c n c n) a = 1. (2.61) n=2 The inequality is a direct consequence of f(ǫ) 1 for every ǫ. The sum n=2 c nfc n is hermitian and thus an eigenstate decomposition can be performed. The resulting eigenvalue spectrum σ(ǫ) is by construction confined to σ(ǫ) [, 1]. Also, we can assume that σ(ǫ) decreases monotonic (an appropriate unitary transformation can always be inserted). We thus find a unitary transformation U so that (with (σ) ǫǫ = δ(ǫ ǫ )σ(ǫ)) iq 1 Λ κ,< rest (q ) 1 = UσU Λ κ,< rest = (qu)σ(qu). (2.62) We define q qu and realize, by using equation (2.57), that we found the following representation for Λ κ rest (t, t ) Λ κ rest(t, t ) = ( q(σ Θ C (t, t )) q ) = dǫ q(t, ǫ)(σ(ǫ) Θ (t,t ) C (t, t )) q (t, ǫ). (2.63) n=2 Continuous construction for σ (ǫ) < The last result is very close to the desired form where σ(ǫ) would be equal to the Fermi function. If the inequality dσ(ǫ) < holds strictly, then there exists an inverse function dǫ σ 1 (σ(ǫ)) = ǫ. The Fermi-function also has an inverse f 1 (f(ǫ)) = ǫ because of T,. 1 Mathematically, this assumption is quite subtle. We have Γ(, t) = Γ(t, ) = for any t by construction. However, a single point has zero measure and therefore dt 1 dt2 f (t 1 )Γ(t 1, t 2 )f(t 2 ) > for any function f can still hold. More intuitively one could argue that we require any discretization to yield a positive definite matrix Γ(t, t ) if we exclude the values t, t =. 37

38 2 The mapping problem We can therefore define ǫ f 1 (σ(ǫ)) ǫ = σ 1 (f( ǫ)) dǫ = and introduce an appropriate hopping f V (t, ǫ) ( ǫ) σ (σ 1 [f( ǫ)]) q(t, σ 1 (f( ǫ))e i ǫt. This leads to the desired form Λ κ rest (t, t ) = f ( ǫ) d ǫ, (2.64) σ (σ 1 [f( ǫ)]) dǫ V (t, ǫ)(f(ǫ) Θ C (t, t ))e iǫ(t t ) V (t, ǫ). (2.65) It is unclear if the relation σ (ǫ) < is reasonable and can be expected in practice. Therefore we also present a construction for the general case. Discrete construction for σ (ǫ) σ (ǫ) might not be non-zero everywhere. In fact there can even exist an infinite amount of intervals I n = [ǫ (n) 1, ǫ (n) 2 ] so that σ(ǫ) = const. for ǫ I n. For this case we derive an approximate discrete-time description, which is required for a numerical simulation in any case. We assume that there are N max time steps (i.e. t N max = t max ) and use our freedom to choose ǫ = 1 independently. The latter implies an integration interval ǫ [1, N max ]. A different choice, e.g. ǫ = 1 N max, is equivalent to rescaling this interval. In this discrete description equation (2.63) becomes N max Λ κ,< rest(t, t ) = n=1 q(t, n)σ n q (t, n). (2.66) We recall that the matrix q is defined as a square root of Γ (cf. equation (2.57)) and therefore has the same dimension, i.e. N max (assuming we used a Cholesky or eigenvalue decomposition to construct q). Consequently, we find an amount of N max eigenvalues σ n. For every n we can calculate We define the hopping between bath and impurity as and find the representation Λ κ rest (t, t ) = N max n=1 ǫ n = f 1 (σ n ) f(ǫ n ) = σ n. (2.67) V n (ǫ, t) e iǫnt q(t, n), (2.68) V n (ǫ n, t)(f(ǫ n ) Θ C (t, t ))e iǫn(t t ) V n (ǫ n, t ) ǫ. (2.69) 38

39 2.3 A numerically feasible mapping It is possible to reformulate this result as Λ κ rest(t, t ) = N max n=1 dǫρ n (ǫ)v n (ǫ, t)(f(ǫ) Θ C (t, t ))e iǫ(t t ) V n(ǫ, t ), where the corresponding densities of states ρ n (ǫ) are given by ρ n (ǫ) δ(ǫ ǫ n ). (2.7) Before, we interpreted each ρ n (ǫ) to correspond to a contributing bath. On the other hand, each pair (ρ n (ǫ), V n (ǫ)) just contributes a single bath site to the effective SIAM Hamiltonian. In total we need N max sites if we want to calculate the same number of time steps (numerically) exactly. The same amount of sites is needed for a discrete version of the previous method (where σ (ǫ) < is required) and therefore the set of all pairs (ρ n (ǫ), V n (ǫ)) can be seen to form a single bath. Remarks The finite temperature construction of the second bath involves not only the calculation of a matrix square root (cf. equation (2.57)) but also an eigenstate decomposition (cf. equation (2.62)). The former can be obtained using a Cholesky decomposition, which has the nice feature of leaving previous results untouched if the input matrix is updated by one line and column. The latter, however, does not have this feature. An increase of t max would therefore require for a new hybridization matrix. This very unpractical for a numerical simulation. It will have to be clarified in the future if it is possible to a work around this problem. 39

40 3 Decomposition of the second Weiss field The construction of the second bath relies on the calculation of square roots of matrices (cf. equations (2.52), (2.55), (2.57)), i.e. continuous decompositions of the form 1 A(t, t ) = tmax dǫv (t, ǫ)v (t, ǫ). (3.1) Such a decomposition requires A(t, t ) to be a hermitian and positive semi-definite matrix. A(t, t ) is of course only a placeholder, and we find that all matrices that we are interested in have the desired properties (cf. equation (2.5)) and note in addition A { iλ κ,< rest, iλ κ,> rest, i(λ κ,< rest Λ κ,> rest) } A(t, ) = A(, t ) = A(, ) =. (3.2) We will always assume that A(t, t ) is in fact strictly positive definite, i.e. tmax tmax dt 1 dt 2 f (t 1 )A(t 1, t 2 )f(t 2 ) > (3.3) for any integrable function f (this excludes distributions, e.g. the Dirac delta). We emphasize that A(t, t ) is not a contour function. t, t [, t max ] are ordinary real variables since we consider only lesser or greater components. We further remark that the decomposition (3.1) cannot be unique. Multiplication of V with an arbitrary unitary transformation U, i.e. V (t, ǫ) dǫ V (t, ǫ )U(ǫ, ǫ), yields another square root. Finding a square root of A is equivalent to finding the second bath for zero and infinite temperature (cf. section 2.3.2; for finite temperatures, the situation is more difficult). For a numerical simulation it would be very convenient to construct this bath stepwise, i.e. by starting from t max =, which is nothing else but the equilibrium problem, and then extend this representation to t max = t and so forth. This requires that all V (t, ǫ) with t t max stay untouched if we increase t max to t max + t. Otherwise we have to perform a new simulation for a different SIAM model in each time step. Furthermore, we want our discrete approximation to converge against an exact continuous description in the limit t dt. These requirements lead naturally to the Cholesky decomposition. 1 Assuming time and energy to be dimensionless quantities, we rescaled the integration interval of ǫ to match the maximal time t max and further absorbed the phase factor e iǫt into V (t, ǫ) for simplicity. 4

41 3.1 Continuous Cholesky decomposition 3.1 Continuous Cholesky decomposition For discrete matrices the Cholesky decomposition (e.g. Press et al. [41]) is known very well. If its prerequisites are fulfilled, it is commonly used as a replacement for the LRdecomposition because it is about twice as fast [41]. A differential equation for its continuous counterpart can be derived directly from equation (3.1), as was shown by Kollar [42]. We review this approach here. Differential equation for a continuous Cholesky decomposition In complete analogy to the discrete case, one has to require V (t, ǫ) to be lower triangular (i.e. V (t, ǫ) = for ǫ > t) and to fulfill V (t, t) > (for t > ; in particular, this requires the diagonal elements to be real). For t < t, this leads to the equation A(t, t ) = t dǫv (t, ǫ)v (t, ǫ). (3.4) Within the continuous Cholesky decomposition, we find an additional precondition (compared to the discrete case), namely A(, ) =. This can be seen by taking the limit t which causes the integral to vanish. However, for the matrices we are interested in it is always fulfilled (cf. equation (3.2)). A set of differential equations for V (t, ǫ) can now be derived as follows. We calculate the derivative of equation (3.4) with respect to t, which is given by The diagonal part is found to be A(t, t) t A(t, t ) t = V (t, t )V (t, t ) + dǫ V (t, ǫ) V (t, ǫ). (3.5) t t = V (t, t) 2 + t V (t, ǫ) dǫ [V (t, ǫ) + t By solving for V (t, t ) we find the differential equations V (t, t 1 ) = V (t, t ) V (t, t) = A(t, t) t [ A(t, t ) t dǫ V (t, ǫ) V (t, ǫ) t t t dǫ [ V (t, ǫ) V (t, ǫ) + t ] V (t, ǫ) V (t, ǫ). (3.6) t ], < t t, ] V (t, ǫ) V (t, ǫ) t. (3.7) We emphasize that V (t, t ) is only dependent on values at smaller times. An extension t max t max > t max therefore leaves the previous results V (t, ǫ) with t, ǫ < t max untouched. Note that equation (3.7) only ensures the diagonal elements V (t, t) and the elements in each column, i.e. V (t, ǫ) with ǫ fixed, to be differentiable functions. The elements within a line, i.e. V (t, ǫ) with t fixed, only need to be integrable. 41

42 3 Decomposition of the second Weiss field Discrete Cholesky decomposition The input function A(t, t ) is, in practice, only known numerically and so one has to rely on a discrete description. We discretize the integral using t n n t, ǫ = t = tmax N max A nn A(t n, t n ) ǫ = N max m=1 V (t n, t m )V (t m, t n ) N max m=1 V n,m V m,n with n, n 1. (3.8) The requirement n, n 1 removes the first row and column of A (both are equal to zero) and thus ensures that the resulting matrix is strictly positive definite (cf. equation (3.3)). The corresponding values V m are set to zero. On the first look one might be concerned about the division on the left hand side because A nn diverges for ǫ if we keep the values of t, t fixed. However, this divergence ensures that all the matrix elements V nm stay finite in this limit and do not decrease to zero. An interesting special case is given for constant n, n with the values n = n = 1 t n, t N max n, where only the first summand of the right hand side contributes since V nm is triangular. Because of A(, ) =, the left hand side yields the derivative = A(t, t) t = V11 2 = V (, )2, (3.9) t= which is in agreement with equation (3.7) at t =. The derivative is equal to zero because of the constraint A(, t ) = A(t, ) = (cf. equation (3.2)). The differential equations (3.7) can be recovered from the discrete description. The discrete Cholesky decomposition [41] is looked up as V nn = 1 V n n [ V nn = Ann A nn n 1 m=1 V nm V n m ], for 1 n < n, n V nm 2. (3.1) m=1 By solving these equations for A nn, A nn and plugging back the results for A n 1,n 1 and A n,n 1, we find in agreement with Kollar [42] V nn = 1 V n n [ A nn A nn 1 V nn = Ann A n 1n 1 n 1 m= n 1 m= V nm (V n m V n 1m ) ], for 1 n < n, (3.11) [ (Vnm V n 1m )V n 1m + V nm(v nm V n 1m )]. These equations reduce to (3.7) in the limit ǫ. A fast implementation of (3.1) can, for example, be found in LAPACK [43]. 42

43 3.1 Continuous Cholesky decomposition Numerical results In the top panels of Fig. 3.1 numerical data for the lesser component of the second Weiss field, i.e. A = iλ κ,< rest, is shown. It was calculated by Eckstein [3] using self-consistent perturbation theory 2. The physical setup is an interaction quench in the Hubbard model from a non-interacting system, U =, to an interacting system with U = 5. The hopping was chosen to lead to a semi-elliptical density of states with quarter bandwidth V = 1 and the time was measured in units of /V = 1. Further, a particle-hole and spin symmetric situation was considered. This requires V (ǫ, t) = V ( ǫ, t) for the hybridization and determines the greater component as Λ κ,> rest(t, t ) = Λ κ,< rest(t, t) (cf. equation (2.52)). For t, t we see that A(t, t ) approaches zero as expected. We note that the first Weiss field, Λ κ 1 (t, t ), (not depicted) vanishes for large t, t whereas Λ κ,< rest(t, t ) can be seen to take the form Λ κ,< rest(t t ). This indicates a relaxation towards a steady state within the approximation. The lower panels show the Cholesky decomposition that we obtained for this data. It has the characteristical triangular shape and features most of its structure for ǫ < 1 corresponding to the relaxation process. In particular, V (t, ǫ) as a function of ǫ with fixed t changes very fast for ǫ. A definite decision if the change is continuous or not is not possible within our numerical calculation. For large times V (ǫ < 1, t) approaches zero and the description of the Weiss field is taken over by V (ǫ > 1, t). The hybridization shows far less structure in this part and suggests that a decomposition of the form A(t, t ) t,t A(t t ) = i dǫv (ǫ t)v (ǫ t ) (3.12) could be a compatible description for the long time limit. In this respect we note that the Cholesky decomposition for the presented data is numerically not unique. It is an intrinsic property of the second Weiss field that A nn for n, n. In the input data the numerical error was of the order 1 4. This caused the matrix A nn to be indefinite (depending on the available accuracy, this phenomenon might occur in general). An inspection is possible by performing an eigenstate decomposition A nn = N max m=1 U nm a m U n m, (3.13) where a m are the eigenvalues and U mn the corresponding unitary transformation. a m is plotted in Fig It turns out that there are a few very large eigenvalues that determine the matrix A nn almost completely. The remaining eigenvalues are close to zero and, as a consequence of the numerical errors in the input data, some of them become negative. In the following sections we will discuss different approximate decompositions of a positive definite and hermitian, i.e. exact, second Weiss field. We will measure the error that is introduced by such approximations with respect to the matrix A pd in which all negative 2 Self-consistent perturbation theory involves skeleton diagrams for the self-energy up to second order. See e.g. Eckstein et al. [18]. 43

44 3 Decomposition of the second Weiss field t ε Re(-iΛ < (t,t )) t Re(V(t,ε)) t t ε Im(-iΛ < (t,t )) t Im(V(t,ε)) t Figure 3.1: The two panels in the top show 1 time steps with t =.4 for the Weiss field of the second bath Λ κ,< rest(t, t ). The data was calculated by Eckstein [3] using self-consistent perturbation theory and belongs to an interaction quench, U = 5, in the Hubbard model at zero temperature. The lower panels show the corresponding Cholesky decomposition that we obtained for this data. Re(V(t,ε)) V(t,ε 1 ) V(t,ε 2 ) V(t,ε 3 ) Im(V(t,ε)) t t Figure 3.2: Time evolution of the first three entries of the hybridization V (t, ǫ), i.e. ǫ 1 =.4, ǫ 2 =.8, ǫ 3 =.12. All of them drop to zero for large times. 44