Finite correlation length scaling with infinite projected entangled-pair states (1803.08445v1)

Abstract: We show how to accurately study 2D quantum critical phenomena using infinite projected entangled-pair states (iPEPS). We identify the presence of a finite correlation length in the optimal iPEPS approximation to Lorentz-invariant critical states which we use to perform a finite correlation-length scaling (FCLS) analysis to determine critical exponents. This is analogous to the one-dimensional (1D) finite entanglement scaling with infinite matrix product states. We provide arguments why this approach is also valid in 2D by identifying a class of states that despite obeying the area law of entanglement seems hard to describe with iPEPS. We apply these ideas to interacting spinless fermions on a honeycomb lattice and obtain critical exponents which are in agreement with Quantum Monte Carlo results. Furthermore, we introduce a new scheme to locate the critical point without the need of computing higher order moments of the order parameter. Finally, we also show how to obtain an improved estimate of the order parameter in gapless systems, with the 2D Heisenberg model as an example.

Finite Correlation Length Scaling with Infinite Projected Entangled-Pair States

The paper "Finite correlation length scaling with infinite projected entangled-pair states" explores the utility of infinite projected entangled-pair states (iPEPS) in studying quantum critical phenomena in two-dimensional systems. The authors present a comprehensive argument supporting the notion that iPEPS — a tensor network variational ansatz — can accurately approximate Lorentz-invariant critical points despite inherently introducing a finite correlation length due to the finite bond dimension $D$. This paper parallels established methods that use infinite matrix product states in one-dimensional systems, extending the applicability of tensor network approaches in higher dimensions.

Central Premises

The paper identifies key aspects that enable iPEPS to describe 2D quantum critical phenomena:

1. Finite Correlation Length Scaling (FCLS): The approach leverages the intrinsic finite correlation length induced by the finite bond dimension in critical states to perform scaling analyses similar to finite-size scaling. The authors argue convincingly that this method is theoretically valid for 2D systems under specific conditions.
2. Novel Critical Point Location Scheme: They introduce an efficient method to locate critical points without needing higher-order moments of the order parameter, simplifying computational procedures in tensor networks.
3. Improved Order Parameter Estimates in Gapless Systems: The research demonstrates enhanced accuracy in estimating order parameters using correlation length scaling, further validating the practical applications of iPEPS in strongly correlated lattice systems.

Numerical Results

Through simulations of interacting spinless fermions on a honeycomb lattice, the authors achieve critical exponents consistent with those obtained via Quantum Monte Carlo (QMC) methods. The critical exponent estimates for the model, part of the chiral Ising Gross-Neveu universality class, are consistent with existing literature, verifying the robustness and accuracy of their approach.

Theoretical Implications

The authors articulate a compelling geometry-based argument explaining why finite $D$-dimension iPEPS cannot fully capture Lorentz invariant critical states. They propose that the system is encoded on a transformed imaginary time landscape embodying quasi-one-dimensional channels. These channels introduce finite correlation time, thus imposing an infrared cutoff analogous to a gap, affecting the low-energy phenology observed numerically.

Impact and Future Perspectives

This contribution is significant in advancing computational techniques in quantum many-body physics, especially for models suffering from the infamous sign problem in QMC methodologies. iPEPS circumvents some limitations by providing an effective means to paper critical dynamics in 2D systems, indicating potential in exploring new angles in topological quantum phases or exotic emergent phenomena not yet fully understood.

Future research may focus on quantifying the effects of the induced geometry on the operator content of corresponding field theories and investigating the continuum limits regulated by iPEPS bond dimension. As tensor network methods continue to evolve, they might unlock alternative paradigms in quantum simulation and critical phenomenon exploration in hitherto inaccessible systems.