Probabilistic solution of Pauli type equations

Angelis, Gian Fabrizio De; Jona-Lasinio, G.; Sirugue, M.

doi:10.1088/0305-4470/16/11/015

Cited by 39 publications

(46 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…the corresponding probability, is now meant to have support on the whole hypercube [0, 1] m . In other words, the function ω(ν) in the system (38) is the analytic continuation to the set [0,1] m of the function defined in Eq. (32).…”

Section: Analytical Probabilistic Approachmentioning

confidence: 99%

See 1 more Smart Citation

The exact ground state for a class of matrix Hamiltonian models: quantum phase transition and universality in the thermodynamic limit

Ostilli

Presilla

2006

J. Stat. Mech.

View full text Add to dashboard Cite

Abstract. By using a recently proposed probabilistic approach, we determine the exact ground state of a class of matrix Hamiltonian models characterized by the fact that in the thermodynamic limit the multiplicities of the potential values assumed by the system during its evolution are distributed according to a multinomial probability density. The class includes i) the uniformly fully connected models, namely a collection of states all connected with equal hopping coefficients and in the presence of a potential operator with arbitrary levels and degeneracies, and ii) the random potential systems, in which the hopping operator is generic and arbitrary potential levels are assigned randomly to the states with arbitrary probabilities. For this class of models we find a universal thermodynamic limit characterized only by the levels of the potential, rescaled by the ground-state energy of the system for zero potential, and by the corresponding degeneracies (probabilities). If the degeneracy (probability) of the lowest potential level tends to zero, the ground state of the system undergoes a quantum phase transition between a normal phase and a frozen phase with zero hopping energy. In the frozen phase the ground state condensates into the subspace spanned by the states of the system associated with the lowest potential level. Exact ground state for a class of matrix Hamiltonian models 2

show abstract

Section: Analytical Probabilistic Approachmentioning

confidence: 99%

“…In this Section, we specialize the discussion to the uniformly fully connected models defined by Eqs. (1)(2)(3). For these models, both the sets T and L have a single element, namely T = (M − 1)η/ǫ and λ = 1, whereas we may count, in general, M distinct values V in the set V .…”

Section: Uniformly Fully Connected Modelsmentioning

confidence: 99%

The exact ground state for a class of matrix Hamiltonian models: quantum phase transition and universality in the thermodynamic limit

Ostilli

Presilla

2006

J. Stat. Mech.

View full text Add to dashboard Cite

show abstract

“…Then, we assign to each site i and spin component σ a stochastic process N t iσ which is the sum of all the processes associated with the links incoming at that site and having the same spin component. A jump in the link process N t ijσ implies a jump in both the site processes N We now show that the representation (4-5) follows from the general formula to represent probabilistically the solution of an ODE system [8] and the expression of the matrix elements of H. The matrix elements n ′ |e −Ht |n obey the ODE system…”

mentioning

confidence: 80%

“…In particular, the real time or the imaginary time evolution operator of a Fermi system or a Berezin integral can be expressed in terms of an associated stochastic dynamics of a collection of Poisson processes. This approach depends on an older general formula to represent probabilistically the solution of a system of ordinary differential equations (ODE) in terms of Poisson processes [8]. In this paper, we present a simple derivation of a similar formula to study fermionic systems, in particular, the Hubbard model.…”

mentioning

confidence: 99%

An exact representation of the fermion dynamics in terms of Poisson processes and its connection with Monte Carlo algorithms

et al. 1999

Self Cite

View full text Add to dashboard Cite

We present a simple derivation of a Feynman-Kac type formula to study fermionic systems. In this approach the real time or the imaginary time dynamics is expressed in terms of the evolution of a collection of Poisson processes. A computer implementation of this formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. From these an optimal algorithm can be chosen which coincides with the Green Function Monte Carlo method in the limit when the latter becomes exact.A crucial issue in quantum Monte Carlo methods is the choice of the most convenient stochastic process to be used in the simulation of the dynamics of the system. This aspect is particularly evident in the case of fermion systems [1][2][3][4] due to the anticommutativity of the variables involved which for long time have not lent themselves to direct numerical evaluation. In a recent paper [5] progress has been made in this direction by providing exact probabilistic expressions for quantities involving variables belonging to Grassmann or Clifford algebras. In particular, the real time or the imaginary time evolution operator of a Fermi system or a Berezin integral can be expressed in terms of an associated stochastic dynamics of a collection of Poisson processes. This approach depends on an older general formula to represent probabilistically the solution of a system of ordinary differential equations (ODE) in terms of Poisson processes [8]. In this paper, we present a simple derivation of a similar formula to study fermionic systems, in particular, the Hubbard model. However, the fermionic nature of the Hamiltonian plays no special role and similar representations can be written down for any system described by a finite Hamiltonian matrix. A computer implementation of this formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. For an optimal choice of these parameters we obtain an algorithm which coincides with the well known Green Function Monte Carlo method in the limit when the latter becomes exact [6]. In this way we provide a theoretical characterization of GFMC.Let us consider the Hubbard Hamiltonianwhere Λ ⊂ Z d is a finite d-dimensional lattice with cardinality |Λ|, {1, . . . , |Λ|} some total ordering of the lattice points, and c iσ the usual anticommuting destruction operators at site i and spin index σ. In this paper, we are interested in evaluating the matrix elements n ′ |e −Ht |n where n = (n 1↑ , n 1↓ , . . . , n |Λ|↑ , n |Λ|↓ ) are the occupation numbers taking the values 0 or 1 [7]. The total number of fermions per spin component is a conserved quantity, therefore we consider only configurations n and n ′ such thatIn the following we shall use the mod 2 addition n ⊕ n ′ = (n + n ′ ) mod 2. Let Γ = {(i, j), 1 ≤ i < j ≤ |Λ| : η ij = 0} and |Γ| its cardinality. For simplicity, we will assume that η ij = η if (i, j) ∈ Γ and γ i = γ. By introducingwhere 1 iσ = (0, . . . , 0, 1 iσ , 0, . . . , 0), andthe following representation holds...

show abstract

“…and concentrate on its probabilistic analysis, with an additional motivation coming from the series of papers due to other authors [10,11,12,13,14,15], where the Markov property has been attributed to analogous dynamical problems, see however Ref. [16].…”

mentioning

confidence: 99%

Comment on ‘‘Why quantum mechanics cannot be formulated as a Markov process’’

Garbaczewski

Olkiewicz

1996

Phys. Rev. A

View full text Add to dashboard Cite

In the paper with the above title, D. T. Gillespie [Phys. Rev. A 49, 1607, (1994] claims that the theory of Markov stochastic processes cannot provide an adequate mathematical framework for quantum mechanics. In conjunction with the specific quantum dynamics considered there, we give a general analysis of the associated dichotomic jump processes. If we assume that Gillespie's "measurement probabilities" are the transition probabilities of a stochastic process, then the process must have an invariant (time independent) probability measure. Alternatively, if we demand the probability measure of the process to follow the quantally implemented (via the Born statistical postulate) evolution, then we arrive at the jump process which can be interpreted as a Markov process if restricted to a suitable duration time. However, there is no corresponding Markov process consistent with the Z 2 event space assumption, if we require its existence for all times t ∈ R + .Before, [1], we have contested the general statement due to Gillespie [2] about the generic contradiction between the probabilistic concepts appropriate for quantum theory and those proper to the common-sense theory of stochastic (in particular, Markov) processes. Our argument was based on invoking the standard, configuration space, Schrödinger picture quantum dynamics which, if combined with the Born statistical interpretation postulate, allows for a consistent description in terms of Markov processes of diffusion type. In conformity with the rich theory developed so far, [3,4,5,6,7,8,9].

show abstract

Probabilistic solution of Pauli type equations

Cited by 39 publications

References 4 publications

The exact ground state for a class of matrix Hamiltonian models: quantum phase transition and universality in the thermodynamic limit

The exact ground state for a class of matrix Hamiltonian models: quantum phase transition and universality in the thermodynamic limit

An exact representation of the fermion dynamics in terms of Poisson processes and its connection with Monte Carlo algorithms

Comment on ‘‘Why quantum mechanics cannot be formulated as a Markov process’’

Contact Info

Product

Resources

About