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by Kang-Soo 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>cond-mat>arXiv:1301.2199.
##[.no-sec-N] 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 many-body 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
- 2022-07-06: Update (images, bold, italic and so on.)
- 2014-12-17: To the docuK.
- 2014-06-23: 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 many-body 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 many-body physics, incorporating quantum correlations beyond mean-field, 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, single-particle 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 mean-field theory, retaining just one orbital, gives the well-known Gross-Pitaevski$\check{i}$ equation. Without the aid of contemporary computers, it seemed to be inevitable until most recently, particularly in out-of-equilibrium 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 Gross-Pitaevski$\check{i}$ equation approach. With the increased interest in many-body physics, however, there arose the necessity to go beyond the all-too-simplified mean field approach of the Gross-Pitaevski$\check{i}$ equation. The accuracy of predictions on many-body 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 many-body 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 time-dependent 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 many-body evolution. The Dirac-Frenkel 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 many-body state $\langle\Phi|$ with respect to a given set of variational parameters, whereas McLachlan's principle requires that the error of many-body evolution must be minimized.
#####/ %\col{Sometimes you use $\Psi$, sometimes $\Phi$ for the many-body 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 Dirac-Frenkel's principle, the authors in have proposed a method they called MCTDHB (Multi-Configurational Time-Dependent Hartree method for Bosons).
This approach has, for example, provided tools for the description of the fragmentation of bosonic many-body states . We will describe below in detail that, besides its many beneficial properties, the MCTDHB method is incomplete in certain situations. Specifically, when the single-particle 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 many-body state. Although MCTDHB provides an important tool to describe the many-body 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 Dirac-Frenkel variational principle for many-body 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 Dirac-Frenkel's principle and the TDVP, and adopting alternatively McLachlan's principle for truncated many-body evolution, we improve on the previous multi-configurational 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 many-body 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
Schematic potential we used. It is an infinite potential with width $L$ and a finite bump with width $a$ and height $h(t)$ on center of the infinite potential. The ratio $a/L$ is chosen to be 0.1 in the following implementations.
Natural occupation number of the ground state on the first symmetric orbital according to the height $h$ of the bump on the center of the infinite potential. Here we set $L=10 \mu m$, $g=0.01$, $N=100$. As $h$ increases, particles get fragmented and gradually condensed from the symmetric orbital into the antisymmetric orbital. The details are discussed in the script.
Ground state without any bump. Therefore $h=0$, $\hbar = m = 1$ unit, and $N=100,$ $L=10 \mu m$, $g=0.01$. Red dot indicates real value and blue squre indicates imaginary value of complex number. Green x in orbital graphs indicates $|phi (x)|^2$, i.e. square norm of the complex orbital at that position. The graph shows that the bosonic particles in the ground state are nearly condensated on the first symmetric orbital in (a). Since there is unitary transformation freedom or arbitrariness, the state can be represented in several different ways with different orbital bases. Consequently (b) is unitarilly transformed one from (a) and therefore these two states are exactly the same many-body state.
Ground state with a bump of height $h=1000$. Although (b) representation is more physically understandable, (a) represents the natural orbitals of eigenvalues of single particle density matrix. Here we used $\hbar = m = 1$ unit, and $N=100, L=10 \mu m, g=0.01$.
Ground state with a bump of height $h=7000$. Although (b) representation is more physically understandable, (a) represents the natural orbitals of eigenvalues of single particle density matrix. Here we used $\hbar = m = 1$ unit, and $N=100$, $L=10 \mu m$, $g=0.01$.
Ground state with a bump of height $h=11000$. Here we used 1hbar = m = 1$ unit, and $N=100, L=10 \mu m, g=0.01$.
(a) the instantaneous error Eq. of $M=1$ simulation. (b,c) particle occupation along the time in different y-scale.
(Graph will be put.) The error divided by double derivative of the particle conservation.
\label{error vs dt2}
- Fermionic Algebra and Fock Space.pdf, The University of Texas at Austin
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- P. Bader and U. R. Fischer, Phys. Rev. Lett. 103, 060402 (2009). %Fragmented Many-Body Ground States for Scalar Bosons in a Single Trap
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- A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 106, 240401 (2011). %Swift Loss of Coherence of Soliton Trains in Attractive Bose-Einstein Condensates
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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 real-valued function $f$.
- $\vec{m}_k^l$ indicates the short-handed 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 many-body theory with complete self-consistency 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 Bose-Einstein Condensate by a Time-Dependent Barrier
- kipid's blog - Method of Lagrange multipliers
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