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by KangSoo Lee and Uwe R. Fischer
I apologize for not finishing this paper. I just open this in the internet. But this document is flawed as it is originally written in LaTeX and not translated to the docuK format perfectly yet. This paper is published in Int. J. Mod. Phys. B 84, 1550021 (2014) and arXiv>condmat>arXiv:1301.2199.
##[.nosecN] Abstract
We introduce a scheme to describe the evolution of an interacting system of bosons, for which the field operator expansion is truncated after a finite number of modes, in a rigorously controlled manner. Using McLachlan's principle of least error , we find a set of equations for the manybody state. As a particular benefit, and in distinction to previously proposed approaches , our approach allows for the dynamical increase of the number of orbitals during the temporal evolution, because we can rigorously monitor the error made by increasing the truncation dimension. The additional orbitals, determined by the condition of least error of the truncated evolution relative to the exact one, are obtained from an initial trial state by a method we call steepest constrained descent.
## PH
 20220706: Update (images, bold, italic and so on.)
 20141217: To the docuK.
 20140623: First posting?
For fermions, the commutator is replaced by the anticommutator \{\ \ , \ \ \} ,
:
\begin{align*}
\{a^{\,}_i, a^\dagger_j\} &\equiv a^{\,}_i a^\dagger_j + a^\dagger_j a^{\,}_i = \delta_{i j}, \\
\{ a^\dagger_i, a^\dagger_j \} &= \{ a^{\,}_i, a^{\,}_j \} = 0.
\end{align*}
Therefore, exchanging disjoint (i.e. i \ne j ) operators in a product of creation or annihilation operators will reverse the sign in fermion systems, but not in boson systems Wiki :: Creation and annihilation operators :: Creation and annihilation operators in quantum field theories
and Wiki :: Second quantization
and Wiki :: Quantum field theory. But I am not currently understanding the reason why the composite of even fermions satisfy bosonic commutation relation
\begin{align*}
[ a^{\,}_i, a^\dagger_j ] &= a^{\,}_i a^\dagger_j  a^\dagger_j a^{\,}_i = \delta_{i j}, \\
[ a^\dagger_i, a^\dagger_j ] &= [ a^{\,}_i, a^{\,}_j ] = 0.
\end{align*}
I thought the composition of any particles in a set state $\alpha$, the creation and annihilation operators can be expressed like
\begin{align*}
z^\dagger_\alpha = \Big( a^\dagger_{1_\alpha} a^\dagger_{2_\alpha} b^\dagger_{3_\alpha} \cdots c^\dagger_{N_\alpha} \Big) \\
z^{\,}_\alpha = \Big( a^{\,}_{1_\alpha} a^{\,}_{2_\alpha} b^{\,}_{3_\alpha} \cdots c^{\,}_{N_\alpha} \Big) ,
\end{align*}
where $a^\dagger$ means creation of $a$ particle, $b^\dagger$ means creation of $b$ particle, and so on.
Since these $a$, $b$, $\cdots$ $c$ particles are fermions, I thought that the commutation relation of composite particle $z$ breaks down.
\begin{align*}
[ z^{\,}_\alpha, z^\dagger_\beta ] &= z^{\,}_\alpha z^\dagger_\beta  z^\dagger_\beta z^{\,}_\alpha = \delta_{\alpha \beta}, \\
[ z^\dagger_\alpha, z^\dagger_\beta ] &= [ z^{\,}_\alpha, z^{\,}_\beta ] = 0.
\end{align*}
\begin{align*}
[ z^{\,}_\alpha, z^\dagger_\beta ] &= \Big( a^{\,}_{1_\alpha} a^{\,}_{2_\alpha} b^{\,}_{3_\alpha} \cdots c^{\,}_{N_\alpha} \Big) \Big( a^\dagger_{1_\beta} a^\dagger_{2_\beta} b^\dagger_{3_\beta} \cdots c^\dagger_{N_\beta} \Big)  \Big( a^\dagger_{1_\beta} a^\dagger_{2_\beta} b^\dagger_{3_\beta} \cdots c^\dagger_{N_\beta} \Big) \Big( a^{\,}_{1_\alpha} a^{\,}_{2_\alpha} b^{\,}_{3_\alpha} \cdots c^{\,}_{N_\alpha} \Big) = \delta_{\alpha \beta}, \\
[ z^\dagger_\alpha, z^\dagger_\beta ] &= [ z^{\,}_\alpha, z^{\,}_\beta ] = 0.
\end{align*}
And I thought any more than double creation of the same state $\alpha$ of $z$ kills any manybody Fock state.
z^\dagger_\alpha z^\dagger_\alpha  \text{any} \rangle = 0 ,
since the individual fermions are created twice on the same state.
So treating a composite particle, which is composed of even fundamental fermion particles, as a boson seems to be wrong for me.
The extension to a true manybody physics, incorporating quantum correlations beyond meanfield, requires, however, vast computational resources when both the number of particles and the interaction between those particles increases. Therefore, a simplification of the problem by truncating the field operator expansion to a finite number of modes (or, as an equivalent term, singleparticle orbitals) has been commonly utilized to obtain results relevant to the prediction of experiments in trapped bosonic quantum gases.
The most extreme truncation, the semiclassical form of meanfield theory, retaining just one orbital, gives the wellknown GrossPitaevski$\check{i}$ equation. Without the aid of contemporary computers, it seemed to be inevitable until most recently, particularly in outofequilibrium situations far away from the ground state, to reduce the complexity of the problem at hand as much as possible, and hence to use the GrossPitaevski$\check{i}$ equation approach. With the increased interest in manybody physics, however, there arose the necessity to go beyond the alltoosimplified mean field approach of the GrossPitaevski$\check{i}$ equation. The accuracy of predictions on manybody correlations and the corresponding response functions will obviously increase with a less severe degree of truncation, though the solutions will not be exact still.
#####/ Necessity of truncation of field operator expansion to render problems computable
To derive the equations of manybody evolution, various variational approaches can be employed. Historically the first was the variational ansatz of Dirac and Frenkel , followed by McLachlan's variational principle and the timedependent variational principle (TDVP), which is a principle of stationary action . Therefore, there are various, not necessarily equivalent, choices of variational principle for finding the equation of motion of the truncated manybody evolution. The DiracFrenkel principle imposes $\langle\delta\Phi \hat{H}i\partial_t \Phi\rangle = 0$ ($\hbar \equiv 1$), where $\langle\delta \Phi$ denotes any possible variations of the manybody state $\langle\Phi$ with respect to a given set of variational parameters, whereas McLachlan's principle requires that the error of manybody evolution must be minimized.
#####/ %\col{Sometimes you use $\Psi$, sometimes $\Phi$ for the manybody state; is there supposed to be a difference, for example one is truncated the other exact? If not, we should use one symbol, e.g. $\Phi$ and its variation.}
On the other hand, the TDVP, as stated, requires stationarity of a given action. The three principles thus support quite different doctrines.
Applying either the TDVP or the DiracFrenkel's principle, the authors in have proposed a method they called MCTDHB (MultiConfigurational TimeDependent Hartree method for Bosons).
This approach has, for example, provided tools for the description of the fragmentation of bosonic manybody states . We will describe below in detail that, besides its many beneficial properties, the MCTDHB method is incomplete in certain situations. Specifically, when the singleparticle density matrix (SPDM) becomes singular, i.e. noninvertible, the method fails. As a consequence, MCTDHB does not provide a way to propagate, for example, a \emph{pure} single condensate into a fragmented condensate manybody state. Although MCTDHB provides an important tool to describe the manybody physics of interacting bosons, the method therefore lacks the possibility to directly connect the phenomena of condensation and fragmentation.
#####/ %will show: ambiguity of [previously employed DiracFrenkel variational principle for manybody evolution
%MacLachlan principle offers possibility to constrain error accumulated during evolution
%demonstrate that during time evolution SPDM (Single Particle Density Matrix) potentially becomes singular (noninvertible),
Here, critically examining DiracFrenkel's principle and the TDVP, and adopting alternatively McLachlan's principle for truncated manybody evolution, we improve on the previous multiconfigurational Hartree methods, and solve the singularity problem of a noninvertible SPDM. In the process, we will also validate the resulting equations of MCTDHB in a different manner, however additionally offering a straightforward handling of the exceptional evolution points related to the singularity of the SPDM.
## Variational Principles
Let us now discuss the possible variational principles in more detail. We are aiming at finding an approximate solution of the manybody Schr$\ddot{o}$dinger equation when the state $\Phi\rangle$ is restricted (or truncated). McLachlan's principle , which was presented in 1963 as a new version of Frenkel's principle, requires the minimization of the error or remainder of this approximate solution from the exact evolution.
The time evolution of any state is dictated by Schr$\ddot{o}$dinger's equation, $i\partial_t \Phi\rangle = \hat{H} \Phi\rangle$. In other words, the evolution of state is determined by the Hamiltonian at any moment. But to make the state $\Phi\rangle$ manipulable, we are generally forced to restrict or confine the state $\Phi\rangle$ into some simple and computationally feasible forms. With the state $\Phi\rangle$ in restricted form, $[i\partial_t  \hat{H}] \Phi\rangle$ cannot be exactly zero in general. Therefore, McLachlan's principle aims at finding the approximate solution which minimizes the positive semidefinite error measure $\langle\Phi [i\partial_t  \hat{H}]^{\dagger}[i\partial_t  \hat{H}] \Phi\rangle$. The details of the corresponding procedure will be rephrased in section and Wiki :: Second quantization
and Wiki :: Quantum field theory. But I am not currently understanding the reason why the composite of even fermions satisfy bosonic commutation relation
 Fermionic Algebra and Fock Space.pdf, The University of Texas at Austin

Science 269, 198 (1995);
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 P. Bader and U. R. Fischer, Phys. Rev. Lett. 103, 060402 (2009). %Fragmented ManyBody Ground States for Scalar Bosons in a Single Trap
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 We illustrate this statement as follows. The reason why we can use $df = \frac{\partial f}{\partial z}dz + \frac{\partial f}{\partial z^*}dz^*$ is not because $dz$ and $dz^*$ are linearly independent differentials, but because $dz$ and $dz^*$ are differentials, that is infinitesimal quantities; $dz^*$ is in fact 100\% dependent on $dz$ since $dz^*$ is simply the complex conjugate of $dz$. Therefore we cannot simply argue that because $dz$ and $dz^*$ are independent variations, the partial derivatives $\frac{\partial f}{\partial z}$ and $\frac{\partial f}{\partial z^*}$ must be zero for the stationarity of $f$. There is a rather widespread misunderstanding of this mathematical fact found in the literature.
 When the variables and constraints are given in complex form, the method of Lagrange multipliers is expressed by $\frac{\partial f} {\partial z_k^*} = \sum_l \bigg[ \lambda_l \frac{\partial g_l} {\partial z_k^*} + \lambda_l^* \frac{\partial g_l^*} {\partial z_k^*} \bigg]$. Though $g_l (z_k, z_k^*)=c$ and $g_l^* (z_k, z_k^*)=c^*$ are the same constraint, we have to add the complex conjugate of it in the Lagrange equation. Be careful to note that $(\frac{\partial g_l}{\partial z_k^*})^*$ is not equal to $\frac{\partial g_l^*}{\partial z_k^*}$. The above equation gives the condition for the stationarity of the realvalued function $f$.
 $\vec{m}_k^l$ indicates the shorthanded notation of $\hat{a}_{l}^{\dagger} \hat{a}_{k}  \vec{m} \rangle = \sqrt{(m_{l}+1)m_{k}}  \vec{m}_k^l \rangle$ which means that, from the given configuration $\vec{m}$, one particle is removed from $k$orbital and one particle is added to $l$orbital and the state $ \vec{m}_k^l \rangle$ is properly normalized.
 A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. A. 73, 063626 (2006). %General variational manybody theory with complete selfconsistency for trapped bosonic systems.
 J. J. McKeown, D. Meegan, and D. Sprevak, An Introduction to Unconstrained Optimization, IOP Publishing, 1990.
 The steepest descent direction in terms of real variables is given by the gradient: $d\vec{x} =  \vec{\nabla} f$. On the other hand, the steepest descent direction in terms of complex variables is given by the differential with respect to the complex conjugate of the variables: $dz_k =  \frac{\partial f}{\partial z_k^*}$. Then the steepest constrained descent is given by $\frac{d z_k} {d \tau} = \Delta(\tau) \bigg[ \frac{\partial f} {\partial z_k^*}  \sum_l \Big[\lambda_l \frac{\partial g_l} {\partial z_k^*} + \lambda_l^* \frac{\partial g_l^*} {\partial z_k^*} \Big] \bigg]$ where $\Delta(\tau)$ can be any arbitrary positive definite function.
 A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 99, 030402 (2007). %Role of Excited States in the Splitting of a Trapped Interacting BoseEinstein Condensate by a TimeDependent Barrier
 kipid's blog  Method of Lagrange multipliers
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