Combining phase-space and time-dependent reduced density matrix approach to describe the dynamics of interacting fermions

Abstract

The possibility to apply phase-space methods to many-body interacting systems might provide accurate descriptions of correlations with a reduced numerical cost. For instance, the so-called stochastic mean-field phase-space approach, where the complex dynamics of interacting fermions is replaced by a statistical average of mean-field like trajectories is able to grasp some correlations beyond the mean-field. We explore the possibility to use alternative equations of motion in the phase-space approach. Guided by the BBGKY hierarchy, equations of motion that already incorporate part of the correlations beyond mean-field are employed along each trajectory. The method is called hybrid phase-space because it mixes phase-space techniques and the time-dependent reduced density matrix approach. The novel approach is applied to the one-dimensional Fermi–Hubbard model. We show that the predictive power is improved compared to the original stochastic mean-field method. In particular, in the weak-coupling regime, the results of the HPS theory can hardly be distinguished from the exact solution even for long time.

Appendices

Equation of motion used for the Fermi–Hubbard model

The EOMs in the Fermi–Hubbard model with sharp boundary conditions (see the Hamiltonian (11)) can conveniently be written in the basis set of site orbitals with spin associated with the fermionic operators $({\hat{a}}^\dagger _{i\sigma }, {\hat{a}}_{i\sigma })$. We denote by $N= N^\uparrow + N^\downarrow$ the number of particles where $N^\uparrow$ (resp. $N^\downarrow$) is the number of particles with spin up (resp. down). For $N_s$ sites, the size of the Hilbert N-body space is given by $\genfrac(){0.0pt}0{N_s}{N^\uparrow } \times \genfrac(){0.0pt}0{N_s}{N^\downarrow }$. Some symmetries can eventually be used to reduce the numerical complexity of the problem:

• The number of particles $N = \sum _i \left( n_{i \uparrow } + n_{i \downarrow } \right)$ is conserved, i.e.$\left[ N,H \right] =0$,

• The projection on the z-axis of the total spin $S_{z} = \frac{1}{2} \sum _{i} \left( n_{i \uparrow } - n_{i \downarrow } \right)$ is conserved, i.e. $\left[ S_{z}, H \right] = 0$.

• As a consequence of the two symmetries above, the number of $+1/2$ particles and $-1/2$ particles are both conserved.

These symmetries imply that the Hamiltonian matrix will be block diagonal where a given block corresponds to a given value of the N and $S_z$ projections. In particular, if the system has a given particle number and $S_z$ at initial time, its time-evolution only requires the corresponding part of the Hamiltonian in this sub-block, reducing significantly the numerical effort for the exact solution.

The symmetries of the initial state that are preserved in time automatically imply some symmetries on the matrix elements of the one-, two-, $\ldots$ density matrices. Denoting the spin up (resp. spin down) with a $+$ (resp. −), and considering that the initial state corresponds to the $S_z=0$ (symmetry spin up/spin down) case, we have schematically:

$$\begin{aligned} R^{++}= & {} R^{--} ,\nonumber \\ R^{+-}= & {} R^{-+} = 0 ,\nonumber \\ D^{+-+-}= & {} D^{-+-+} ,\nonumber \\ D^{++++}= & {} D^{----} = D^{+-+-} + D^{+--+} , \nonumber \\ D^{+--+}= & {} D^{-++-} , \end{aligned}$$

(A1)

where R and D denote respectively the one and two-body density matrices (note that here the labels associated to site number are implicit). We can see that one only needs to propagate $R^{++}$ or $R^{--}$, and a careful analysis shows that $D^{+-+-}$ is the only component of the two-body density matrix that will affect the dynamics when propagating both the one-body and two-body degrees of freedom in the BBGKY hierarchy. Note that the quantity $\mathcal {D}^{(n)}_{12}$ introduced in this article follows the same symmetry properties as $D_{12}$.

We give below the different EOMs that are used in the present work (with the convention $\hbar =1$). Omitting the spin indices on R for clarity since no confusion can be made, and considering that the latin subscripts $i,j,\dots$ denote the $i\mathrm{th},j\mathrm{th},\dots ,$ sites starting from the left of the 1D lattice, one can write the EOMs for the TDHF, SMF and HPS theories:

• Mean-field EOM  Assuming spin symmetry at initial time and using the notations $R_{ij} = R^{++}_{ij} =R^{--}_{ij}$, the TDHF evolution is given by:

  $$\begin{aligned} i \dot{R}_{ij}= & {} -J \left( R_{i+1j} (1-\delta _{iN_s}) + R_{i-1j} (1-\delta _{i1})\right. \nonumber \\&\left. - R_{ij+1} (1-\delta _{jN_s})- R_{ij-1} (1-\delta _{j1}) \right) \nonumber \\&+ U R_{ij} \left( R_{ii} - R_{jj}\right) . \end{aligned}$$

  (A2)

  Assuming that all particles are located on the left hand side of the lattice, the initial density is given by:

  $$\begin{aligned} R_{ij} (t_0)=\left\{ \begin{array}{ll} 1 &{}\quad \text {if}\ i=j \quad \text {and}\quad i \le N_s = N \\ \\ 0 &{}\quad \text {otherwise} \end{array}\right. . \end{aligned}$$

  (A3)

  In another test, the initial conditions were modified to simulate the collision of two groups of particles of equal sizes initially disposed on each extremities of the mesh:

  $$\begin{aligned} R_{ij} (t_0)=\left\{ \begin{array}{ll} 1 &{}\quad \text {if}\ i=j \quad \text {and}\quad i \not \in \left[ N^\uparrow /2,N_s-N^\uparrow /2 \right] \\ \\ 0 &{}\quad \text {otherwise} \end{array}\right. . \end{aligned}$$

  (A4)

• SMF EOM  In the original SMF phase-space approach, the EOM remains the TDHF one except that the initial density is fluctuating at initial time. We then have:

  $$\begin{aligned} i \dot{R}^{(n)}_{ij}= & {} -J \left( R_{i+1j}^{(n)} (1-\delta _{iN_s}) + R_{i-1j}^{(n)} (1-\delta _{i1}) \right. \nonumber \\&\left. - R_{ij+1}^{(n)} (1-\delta _{jN_s}) - R_{ij-1}^{(n)} (1-\delta _{j1}) \right) \nonumber \\&+ U R_{ij}^{(n)} \left( R_{ii}^{(n)} - R_{jj}^{(n)}\right) , \end{aligned}$$

  (A5)

  where the density at initial time:

  $$\begin{aligned} R_{ij}^{(n)} (t_0)= & {} \overline{R_{ij}^{(n)} (t_0)} + \delta R_{ij}^{(n)} (t_0), \nonumber \\ \overline{R_{ij}^{(n)} (t_0)}= & {} R_{ij} (t_0). \end{aligned}$$

  (A6)

  The properties of $\delta R_{ij}^{(n)} (t_0)$ are specified in Sect. 2.2. We would like to mention that we assume in the present SMF application as well as in the HPS presented below that spin up-spin down symmetry is respected along each path. Fluctuations that break the spin symmetry at initial time are allowed by the statistical properties of the one-body density $R_{ij}^{(n)}$ within SMF. For the SMF, this was tested and discussed in Ref. [16]. The conclusion is that allowing the breaking of spin symmetry at initial time increases the numerical effort while not increasing/decreasing the predicting power. For this reason, we consider here the case where the spin symmetry is respected event-by-event.

• The HPS EOM – In the HPS equation of motion, only $\mathcal {D}^{+-+-(n)}$ is coupled to $R^{(n)} = R^{++(n)} =R^{--(n)}$. For this reason, we use the compact notations $\mathcal {D}_{ijkl}^{(n)} = \mathcal {D}_{ijkl}^{+-+-(n)}$. The EOMs then read

  $$\begin{aligned} i \dot{R}^{(n)}_{ij}= & {} -J \Bigg (R_{i+1j}^{(n)} (1-\delta _{iN_s}) + R_{i-1j}^{(n)} (1-\delta _{i1}) \nonumber \\&- R_{ij+1}^{(n)} (1-\delta _{jN_s}) - R_{ij-1}^{(n)} (1-\delta _{j1}) \Bigg ) \nonumber \\&+ U \left( \mathcal {D}_{iiji}^{(n)} - \mathcal {D}_{ijjj}^{(n)} \right) , \nonumber \\ i \dot{\mathcal {D}}_{ijkl}^{(n)}= & {} - J \Bigg (\mathcal {D}_{i+1jkl}^{(n)} (1-\delta _{iN_s}) + \mathcal {D}_{i-1jkl}^{(n)} (1-\delta _{i1}) \nonumber \\&+ \mathcal {D}_{ij+1kl}^{(n)} (1-\delta _{jN_s}) + \mathcal {D}_{ij-1kl}^{(n)} (1-\delta _{j1}) \nonumber \\&- \mathcal {D}_{ijk+1l}^{(n)} (1-\delta _{kN_s}) \nonumber \\&- \mathcal {D}_{ijk-1l}^{(n)} (1-\delta _{k1}) - \mathcal {D}_{ijkl+1}^{(n)} (1-\delta _{l N_s}) \nonumber \\&- \mathcal {D}_{ijkl-1}^{(n)} (1-\delta _{l1}) \Bigg )\nonumber \\&+ U \left( R_{ii}^{(n)} + R_{jj}^{(n)} - R_{kk}^{(n)} - R_{ll}^{(n)} \right) \mathcal {D}_{ijkl}^{(n)} \nonumber \\&+ U \left( \delta _{ij} \mathcal {D}^{(n)}_{iikl} - R_{ij} \mathcal {D}^{(n)}_{jjkl} - R_{ji} \mathcal {D}^{(n)}_{iikl}\right) \nonumber \\&- U \left( \delta _{kl} \mathcal {D}^{(n)}_{ijkk} - R_{kl} \mathcal {D}^{(n)}_{ijkk} - R_{lk} \mathcal {D}^{(n)}_{ijll}\right) . \end{aligned}$$

  (A7)

  For an initial state that corresponds to a Slater determinant, we have the initial conditions:

  $$\begin{aligned} R_{ij}^{(n)} (t_0)= & {} \overline{R_{ij}^{(n)} (t_0)} + \delta R_{ij}^{(n)} (t_0), \\ \overline{R_{ij}^{(n)} (t_0)}= & {} R_{ij} (t_0) ,\\ \overline{\mathcal {D}_{ijkl}^{(n)} (t_0)}= & {} R_{ik} (t_0) R_{jl} (t_0). \end{aligned}$$

General remark on SMF and some properties

In Ref. [29], it has been shown that the SMF approach can be linked to a hierarchy of equations of the moments of the one-body density that resembles the BBGKY hierarchy. In the present section, we precise the link between the moments and the SMF approach. In SMF, one-body observables are treated as classical fluctuating objects that are given along each trajectory by:

$$\begin{aligned} A^{(n)}(t) = \sum _{ij} A_{ij} R^{(n)}_{ji}(t) \end{aligned}$$

(B.1)

where $R^{(n)}_{ji}(t)$ are the densities with initial fluctuations followed by TDHF evolution.

The SMF approach makes a mapping between quantum expectation values and classical statistical average. More precisely, let us consider a set of one-body operators, denoted by ${\hat{A}}$, ${\hat{B}}$, ${\hat{C}}$, ... The following mapping is made:

$$\begin{aligned} \displaystyle \langle {\hat{A}} \rangle&\longrightarrow \overline{A^{(n)}} = \sum _{ij } A_{ij} \overline{R^{(n)}_{ji}},\\ \displaystyle \langle \{ {\hat{A}}, \hat{B} \}_+ \rangle&\longrightarrow \overline{A^{(n)} B^{(n)}} = \sum _{ij kl} A_{ij} B_{kl} \overline{R^{(n)}_{ji} R^{(n)}_{lk}},\\ \displaystyle \langle \{ \hat{A}, \hat{B} , \hat{C} \}_+ \rangle&\longrightarrow \overline{A^{(n)} B^{(n)}C^{(n)}} \\&= \sum _{ij kl mn} A_{ij} B_{kl} C_{mn} \overline{R^{(n)}_{ji} R^{(n)}_{lk}R^{(n)}_{nm}}, \\&\cdots \end{aligned}$$

where we have used the notation:

$$\begin{aligned} \langle \{ \hat{A}, \hat{B} \}_+ \rangle\equiv & {} \frac{1}{2} \langle {\hat{A}} {\hat{B}} + {\hat{B}} {\hat{A}} \rangle \\ \langle \{ \hat{A}, {\hat{B}}, \hat{C}\}_+ \rangle\equiv & {} \frac{1}{6} \langle {\hat{A}} {\hat{B}} {\hat{C}} + \hat{A} {\hat{C}} {\hat{B}} + {\hat{B}} {\hat{A}} {\hat{C}}+ \hat{B} {\hat{C}} {\hat{A}} \\&+ \hat{C} {\hat{B}} {\hat{A}}+ {\hat{C}} {\hat{A}} \hat{B} \rangle \\&\cdots&\end{aligned}$$

The above quantum averages can be connected to the one-, two- and higher order many-body densities simply by setting ${\hat{A}} = \hat{N}_{ji}$, ${\hat{B}}= \hat{N}_{lk}$, ${\hat{C}} = \hat{N}_{nm}$ where we have introduced the notations ${\hat{N}}_{ij} = a^\dagger _j a_i$. A lengthy but straightforward calculation gives:

$$\begin{aligned} R_{ij}= & {} \langle N_{ij} \rangle , \end{aligned}$$

(B.2)

$$\begin{aligned} D_{ik,jl}= & {} \langle \{ {\hat{N}}_{ij}, {\hat{N}}_{kl} \}_+ \rangle - \frac{1}{2} \left( \delta _{il} R_{kj} + \delta _{kj} R_{il} \right) , \nonumber \\ T_{jln;ikm}= & {} \langle \{ \hat{N}_{ji} ,\hat{N}_{lk}, \hat{N}_{nm} \}_+ \rangle - \frac{1}{2} \big ( \delta _{jk} D_{ln;im} \nonumber \\&+ \delta _{lm} D_{nj;ki} + \delta _{jm} D_{nl;ik} + \delta _{li} D_{jn;km}\nonumber \\&+ \delta _{ni} D_{jl;mk} + \delta _{nk} D_{lj,mi} \big ) \nonumber \\&- \frac{1}{6} \big ( \delta _{jk} \delta _{lm} R_{ni} + \delta _{li} \delta _{jm} R_{nk}\nonumber \\&+ \delta _{lm} \delta _{ni} R_{jk} + \delta _{nk} \delta _{li} R_{jm} \nonumber \\&+ \delta _{ni} \delta _{jk} R_{lm} + \delta _{jm} \delta _{nk} R_{li} \big ) \nonumber , \\&\cdots&\end{aligned}$$

(B.3)

xhere $R_1$, $D_{12}$ and $T_{123}$ denote the one-, two-, three-body density matrix respectively. We see in particular that the information content of the symmetric moments $\langle N_{ij} \rangle$, $\langle \{ {\hat{N}}_{ij}, {\hat{N}}_{kl} \}_+ \rangle$, $\langle \{ \hat{N}_{ji} ,\hat{N}_{lk}, \hat{N}_{nm} \}_+ \rangle$ , ... is equivalent to the information content of the one-, two-, three-body, ... density matrices.

These relationships on the quantum densities and quantum symmetric moments and the mapping between these moments and the density $R^{(n)}$ show that the equivalent of the two-, three- ... body densities can also be constructed in the SMF theory. Based on the above relationships, we introduce the matrices $D^{(n)}_{12}$, $T^{(n)}_{123}$,... that are defined from the quantity $R^{(n)}$ used in SMF using:

$$\begin{aligned} D^{(n)}_{ik,jl}= & {} R^{(n)}_{ij} R^{(n)}_{kl} - \frac{1}{2} \left( \delta _{il} R^{(n)}_{kj} + \delta _{kj} R^{(n)}_{il} \right) , \end{aligned}$$

(B.4)

$$\begin{aligned} T_{jln;ikm}^{(n)}= & {} + R_{ji}^{(n)} R_{lk}^{(n)} R_{nm}^{(n)} \nonumber \\&- \frac{1}{2} \big ( \delta _{jk} R^{(n)}_{il} R^{(n)}_{mn} + \delta _{lm} R^{(n)}_{kn} R^{(n)}_{ij}\nonumber \\&+ \delta _{jm} R^{(n)}_{in} R^{(n)}_{kl} + \delta _{li} R^{(n)}_{kj} R^{(n)}_{mn} \nonumber \\&+ \delta _{ni} R^{(n)}_{mj} R^{(n)}_{kl} + \delta _{nk} R^{(n)}_{ml} R^{(n)}_{ij} \big ) \nonumber \\&+ \frac{1}{3} \big ( \delta _{jk} \delta _{lm} R_{ni}^{(n)} + \delta _{li} \delta _{jm} R_{nk}^{(n)} + \delta _{lm} \delta _{ni} R_{jk}^{(n)} \nonumber \\&+ \delta _{nk} \delta _{li} R_{jm}^{(n)} + \delta _{ni} \delta _{jk} R_{lm}^{(n)} + \delta _{jm} \delta _{nk} R_{li}^{(n)} \big ), \nonumber \\&\cdots&\end{aligned}$$

(B.5)

1.1 Properties of the density matrices

The density matrices $D^{(n)}$ and $T^{(n)}$ defined in Eq. (B.4) and (B.5) do automatically fulfill some important properties. For instance, after a rather lengthy but straightforward calculation, it is possible to show that we have:³

$$\begin{aligned} \mathrm{Tr} R^{(n)}_1(t)= & {} N, \\ \mathrm{Tr}_2 D^{(n)}_{12}(t)= & {} (N-1) R^{(n)}_1(t), \\ \mathrm{Tr}_3 T^{(n)}_{123}(t)= & {} (N-2) D^{(n)}_{12}(t), \\&\cdots&\end{aligned}$$

These are important properties that holds for the exact evolution and are automatically fulfilled on an event-by-event basis and therefore also hold when averaging over events. Such requirements are known to be a critical issue when performing TDnRDM calculations [29]. In SMF, the statistical properties of the initial conditions are constructed to insure that the first and second moments of the quantum fluctuations match the one obtained through the statistical average. This automatically implies that we have the properties:

$$\begin{aligned} \overline{R^{(n)}_1 }(t=0)= & {} R_{1}(t=0) , \nonumber \\ \overline{D^{(n)}_{12}(t=0) }= & {} D_{12}(t=0) . \end{aligned}$$

(B.6)

However, the three-body average density does not a priori match the quantum three-body density, especially if a Gaussian approximation is made for the initial statistical ensemble (see for instance the discussion in [20])