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Physical Review A

Direct construction of path integrals in the lattice-space multiband dynamics of electrons in a solid

F. A. Buot

Published Apr 1, 1986• 18 citations
Publisher

Abstract

It is suggested that complex problems in ultrasubmicrometer electronics research may greatly benefit from use of the path-integral technique. The use of the Weyl-Wigner formalism of the quantum dynamics of electrons in solids provides a rigorous and straightforward derivation of the path integral in solid-state physics, both from the single-particle and from the many-body field-theoretical description of electron dynamics, without the need to postulate a priori the isomorphism between quantum operators and c-numbers of the base field. A rigorous construction of the path integral in many-body solid-state band theory necessitates a two-stage Weyl correspondence between quantum operators and c-numbers of the base field, namely, the Weyl correspondence of the base field of ``lattice-space'' particle-dynamical variables and that of the continuum many-body field-dynamical variables.

DOI
10.1103/physreva.33.2544

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It is suggested that complex problems in ultrasubmicrometer electronics research may greatly benefit from use of the path-integral technique. The use of the Weyl-Wigner formalism of the quantum dynamics of electrons in solids provides a rigorous and straightforward derivation of the path integral in solid-state physics, both from the single-particle and from the many-body field-theoretical description of electron dynamics, without the need to postulate a priori the isomorphism between quantum operators and c-numbers of the base field. A rigorous construction of the path integral in many-body solid-state band theory necessitates a two-stage Weyl correspondence between quantum operators and c-numbers of the base field, namely, the Weyl correspondence of the base field of ``lattice-space'' particle-dynamical variables and that of the continuum many-body field-dynamical variables.
This paper uses Schwinger's representation of quantum systems of a finite number of degrees of freedom to formulate discrete path integral representations of the dynamics. The representation starts with an observable with a finite number of distinct eigenvalues. Using that observable Schwinger constructs complementary pairs of finite dimensional unitary operators that are finite dimensional analogs of the irreducible Weyl algebra. These can be decomposed into products of irreducible sub-algebras that reduce to q-bit gates for the case of 2N2^N2N degrees of freedom. In the limit of a large number of discrete degrees of freedom these representations can be used to model quantum systems with continuous degrees of freedom. This limit recovers the continuum Weyl algebra. Path integrals can be formulated in the discrete case. They have the advantage that the number of paths for a finite number of time slices is finite. In this work the path integral is interpreted as the expectation value of a potential functional with respect to a complex probability distribution on the space of paths. An application to scattering from a short-range potential is given. Multi-resolution wavelet bases are used exactly represent a local ϕ4(x)\phi^4(x)ϕ4(x) quantum field theory as a theory with an infinite number of discrete local modes. The discrete path integral is illustrated by computing time evolution when this theory is truncated to two coupled modes.
Here, we clarify the physical aspects between the discrete Weyl-Wigner (W-W) formalism, well developed in condensed matter physics, and the so-called 'precise Weyl-Wigner calculus for lattice models' recently appearing in the literature. We point out that the use of compact continuous momentum space for a discrete lattice model is unphysically founded. It has an incommensurate phase space, highly unphysical, lacks the finite fields aspects, as exemplified by the Born-von Karman boundary condition of compactified Bravais lattice of solid-state physics, and leads to several ambiguities. This new W-W formalism simply lacks bijective Fourier transformation, which is well-known to support the uncertainty principle of canonical conjugate dynamical variables of quantum physics. Moreover, this new W-W formalism for lattice models failed to handle the quantum physics of qubits, representing two discrete lattice sites.
We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd{\mathbb{Z}_d} \times {\mathbb{Z}_d}Zd​×Zd​ (ddd an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3d=3d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.
The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q̇−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.