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arXiv:0910.1574 · Physical Review B

Flavor-twisted boundary condition for simulations of quantum many-body systems

Wei‐Guo Yin, Wei Ku

Published Nov 2, 2009• 3 citations
arXivHTMLPDF

Abstract

We present an approximative simulation method for quantum many-body systems based on coarse graining the space of the momentum transferred between interacting particles, which leads to effective Hamiltonians of reduced size with the flavor-twisted boundary condition. A rapid, accurate, and fast convergent computation of the ground-state energy is demonstrated on the spin-$\frac{1}{2}$ quantum antiferromagnet of any dimension by employing only two sites. The method is expected to be useful for future simulations and quick estimates on other strongly correlated systems.

DOI
10.1103/physrevb.80.180402

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We present an approximative simulation method for quantum many-body systems based on coarse graining the space of the momentum transferred between interacting particles, which leads to effective Hamiltonians of reduced size with the flavor-twisted boundary condition. A rapid, accurate, and fast convergent computation of the ground-state energy is demonstrated on the spin-12\frac{1}{2}21​ quantum antiferromagnet of any dimension by employing only two sites. The method is expected to be useful for future simulations and quick estimates on other strongly correlated systems.
An algorithm for simulation of quantum many-body dynamics having su(2)(2)(2) spectrum-generating algebra is developed. The algorithm is based on the idea of dynamical coarse-graining. The original unitary dynamics of the target observables---the elements of the spectrum-generating algebra---is simulated by a surrogate open-system dynamics, which can be interpreted as weak measurement of the target observables, performed on the evolving system. The open-system state can be represented by a mixture of pure states, localized in the phase space. The localization reduces the scaling of the computational resources with the Hilbert-space dimension nnn by factor ${n}^{3/2}(\mathrm{ln} n){}^{\ensuremath{-}1}$ compared to conventional sparse-matrix methods. The guidelines for the choice of parameters for the simulation are presented and the scaling of the computational resources with the Hilbert-space dimension of the system is estimated. The algorithm is applied to the simulation of the dynamics of systems of $2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ and $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ cold atoms in a double-well trap, described by the two-site Bose-Hubbard model.
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The symmetry and the locality are the two major sources of various general theorems in quantum many-body systems. We demonstrate that, in gapped phases of a U(1) symmetric Hamiltonian with finite-range interactions, the bulk properties such as the expectation value of local operators, the ground-state energy and the excitation gap, and the static and low-frequency dynamical responses in general do not depend on the U(1) phase of the twisted boundary condition in the limit of the large system size. Specifically, their dependence on the twisted angle is exponentially suppressed with the linear dimension of the system. Our argument is solely based on the exponential decay of various types of equal-time correlation functions and does not assume any details of the Hamiltonian, meaning that the statement applies quite generally regardless of the dimensionality or the interaction strength of the system.
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