Disentanglement Approach to Quantum Spin Ground States: Field Theory and Stochastic Simulation

Abstract

We develop an analytical and numerical framework based on the disentanglement approach to study the ground states of many-body quantum spins systems. In this approach, observables are expressed as functional integrals over scalar fields, where the relevant measure is the Wiener measure. We identify the leading contribution to these integrals, given by the saddle point field configuration. Analytically, this can be used to develop an exact field-theoretical expansion of the functional integrals, performed by means of appropriate Feynman rules. The expansion can be truncated to the desired order to obtain approximate analytical results for ground state expectation values. Numerically, the saddle point configuration can be used to compute physical observables by means of an exact importance sampling scheme. We illustrate our methods by considering the quantum Ising model in 1, 2 and 3 spatial dimensions. Our analytical and numerical results are applicable to a broad class of many-body quantum spin systems, bridging concepts from quantum lattice models, continuum field theory, and classical stochastic processes.

Time-evolving Weiss fields in the stochastic approach to quantum spins

“…In essence, the Weiss field tracks the mean-field dynamics of the quantum spin system, which facilitates more efficient sampling. Similar conclusions have been drawn in imaginary time using saddle-point techniques [19]. We demonstrate these improvements by presenting results for the quantum Ising model in both one and two dimensions, with up to 121 spins.…”

“…A key feature of the representation (2) is that it is invariant under shifts of the Hubbard-Stratonovich fields ϕ(t) → ϕ(t) + ∆ϕ(t), since the fields correspond to dummy integration variables in the path integral. This leaves the time-evolution operator unchanged, as has been recently used to develop an importance sampling approach in imaginary time [19,30]. In this work, we show that a judicious choice of ∆ϕ(t) can significantly improve numerical simulations of real-time dynamics over a broad range of parameters.…”

“…More generally, we may choose the parameter m b j (t) to be time-dependent, in accordance with the dynamics of the neighboring spins. The shift of the fields ϕ also induces a transformation of the probability measure via the noise action (3) [19,30]:…”

“…where ... trajectories performed in imaginary time [19]. The vector n j (t) corresponds to the position of a spin on the Bloch sphere, expressed in terms of projective coordinates [18]:…”

“…Recently, an exact mapping between quantum spin dynamics and classical stochastic differential equations (SDEs) has emerged, based upon the Hubbard-Stratonovich decoupling of the exchange interactions [13][14][15][16][17][18][19]. This stochastic approach allows for the numerical evaluation of time-dependent quantum observables, in addition to analytical insights obtained from the classical stochastic formulae [16][17][18][19]. A notable feature is that it treats one-dimensional and higher-dimensional problems on a similar footing.…”