Feynman path integrals for discrete-variable systems: Walks on Hamiltonian graphs

TL;DR

The paper introduces a universal, parameter-free discrete-variable analogue of Feynman path integrals by reinterpreting matrix elements of functions of the Hamiltonian as sums over walks on the Hamiltonian graph via the off-diagonal series expansion. In the continuum limit $a\to0$ of a discretized one-dimensional system, this walk-sum formulation reproduces the Feynman path integrals for both partition functions and transition amplitudes, including the potential term. A concrete demonstration with the transverse-field Ising model shows how walks on the spin-configuration graph yield the corresponding amplitudes and thermodynamic sums, while highlighting connections to continuous-time quantum Monte Carlo methods. Overall, the work provides a graph-based, canonical discretization of quantum dynamics in discrete systems with potential broad impact on simulation and analysis of complex quantum spin models.

Abstract

We propose a natural, parameter-free, discrete-variable formulation of Feynman path integrals. We show that for discrete-variable quantum systems, Feynman path integrals take the form of walks on the graph whose weighted adjacency matrix is the Hamiltonian. By working out expressions for the partition function and transition amplitudes of discretized versions of continuous-variable quantum systems, and then taking the continuum limit, we explicitly recover Feynman's continuous-variable path integrals. We also discuss the implications of our result.

Figures (3)

• Figure 1: (a) Matrix elements as walks over the Hamiltonian graph. A matrix element $\langle z_{\textrm{out}}| f(H) |z_{\textrm{in}}\rangle$ is calculated by summing up divided-difference contributions from all walks leading from $|z_{\textrm{in}}\rangle$ to $|z_{\textrm{out}}\rangle$ along the edges of the Hamiltonian graph. Here, each node represents a basis state and each (directed) edge -- an off-diagonal Hamiltonian matrix element. (b) Walks on the Hamiltonian graph as the continuum limit is approached. In the continuum limit, walks become Feynman paths.
• Figure 2: The displacement of a many-step i.i.d. random walk may be approximated by a (continuous) normal variable. Left: Displacement as a function of number of steps for a thousand i.i.d. random walkers. The total number of steps for each walker is $n=1000$, where each step displaces the walker by one unit either up or down. Right: A histogram of the final displacement values of the random walkers depicted in the left panel. In the limit of infinitely-long walks, the distribution approaches a normal one with mean zero and variance $n$.
• Figure 3: A diagrammatic representation of walks contributing to a transition amplitude between $|z_{\textrm{in}}\rangle =|\cdots \uparrow \uparrow \uparrow \cdots \rangle$ and $|z_{\textrm{out}}\rangle =|\cdots \downarrow \downarrow \downarrow \cdots \rangle$ in the transverse-field Ising model. Since the in and out states are three spin-flips apart, the shortest walks contributing to the transition amplitude are the six possible paths on the hypercube of the basis states corresponding to the six different orderings in which the three spins may be flipped. Three of the six paths are shown in the figure as sequences of arrows. Each arrow, which points from one node to its neighbor, corresponds to a single spin flip, i.e., the action of a single permutation operator, namely a Pauli-$X$, on a basis state.