[Paper Review] Efficient quantum cluster algorithms for frustrated transverse field Ising antiferromagnets and Ising gauge theories

Sounak Biswas, Kedar Damle|arXiv (Cornell University)|Dec 13, 2018

Theoretical and Computational Physics4 citations

TL;DR

This paper presents efficient quantum cluster algorithms within the Stochastic Series Expansion (SSE) framework for frustrated transverse field Ising antiferromagnets on pyrochlore and planar pyrochlore lattices, and for dual Ising gauge theories. By introducing a microcanonical update based on premarked motifs and plaquette percolation, the method achieves significantly reduced autocorrelation times and enables the first direct evidence for a power-law ordered intermediate-temperature phase in the fully frustrated square lattice transverse field Ising model.

ABSTRACT

Working within the Stochastic Series Expansion (SSE) framework, we construct efficient quantum cluster algorithms for transverse field Ising antiferromagnets on the pyrochlore lattice and the planar pyrochlore lattice, for the fully frustrated square lattice Ising model in a transverse field (dual to the 2+1 dimensional odd Ising gauge theory), and for a transverse field Ising model with multi-spin interactions on the square lattice, which is dual to a 2+1 dimensional even Ising gauge theory (and reduces to the two dimensional quantum loop model in a certain limit). Our cluster algorithms use a microcanonical update procedure that generalizes and exploits the notion of "pre-marked motifs" introduced earlier in the context of a quantum cluster algorithm for triangular lattice transverse field Ising antiferromagnets. We demonstrate that the resulting algorithms are significantly more efficient than the standard link percolation based quantum cluster approach. We also introduce a new canonical update scheme that leads to a further improvement in measurement of some observables arising from its ability to make one-dimensional clusters in the "imaginary time" direction. Finally, we demonstrate that refinements in the choice of premarking strategies can lead to additional improvements in the efficiency of the microcanonical updates. As a first example of the physics that can be studied using these algorithmic developments, we obtain evidence for a power-law ordered intermediate-temperature phase associated with the two-step melting of long-range order in the fully frustrated square lattice transverse field Ising model.

Motivation & Objective

• To develop efficient quantum cluster algorithms for frustrated transverse field Ising models on geometrically frustrated lattices where standard link percolation fails due to freezing.
• To overcome the limitations of conventional cluster algorithms in systems with no conserved charge and high ground state degeneracy.
• To enable unbiased Monte Carlo simulations of T ≥ 0 quantum statistical mechanics for these models, particularly for systems dual to 2+1D Ising gauge theories.
• To demonstrate the existence of a power-law ordered intermediate phase in the fully frustrated square lattice transverse field Ising model.

Proposed method

• The method employs a microcanonical update procedure based on plaquette decomposition of the Hamiltonian within the SSE framework, generalizing the concept of 'premarked motifs' from triangular lattice antiferromagnets.
• Clusters are formed via a plaquette percolation process where operators on spatial plaquettes are assigned to space-time clusters based on premarked motifs, ensuring consistent cluster assignment.
• A new canonical update scheme is introduced that generates one-dimensional clusters in the imaginary time direction, improving measurement efficiency for observables like σˣ.
• The algorithm uses a Swendsen-Wang type update where each cluster is flipped independently with probability 1/2, preserving the configuration weight.
• The premarking strategy is refined to further enhance the efficiency of microcanonical updates by reducing cluster freezing and improving ergodicity.
• The method is applied to the fully frustrated square lattice transverse field Ising model (dual to odd Ising gauge theory) and the multi-spin interaction model (dual to even Ising gauge theory).

Experimental results

Research questions

• RQ1 Does a power-law ordered intermediate phase exist in the fully frustrated square lattice transverse field Ising model, as predicted by Landau-Ginzburg theory?
• RQ2 Can a microcanonical cluster update based on premarked motifs and plaquette percolation outperform standard link percolation in frustrated quantum spin systems?
• RQ3 How does the inclusion of a canonical update scheme affect the autocorrelation times of off-diagonal observables like σˣ in SSE simulations?
• RQ4 To what extent can refined premarking strategies improve the ergodicity and efficiency of quantum cluster algorithms in frustrated systems?
• RQ5 What is the nature of the finite-temperature phase transition sequence in the fully frustrated transverse field Ising model, particularly regarding the melting of columnar order?

Key findings

• The plaquette percolation-based cluster algorithm reduces autocorrelation times for σˣ by a factor of two or more compared to the standard link percolation method, even after accounting for the extra cost of canonical updates.
• Histograms of the phase θ of the complex order parameter ψ show eight distinct peaks at low temperatures (indicating columnar order) and flatten at intermediate temperatures, signaling emergent U(1) symmetry and a power-law ordered phase.
• Power-law fits of the susceptibility χψ/L² to the form kL⁻ᵠ yield critical exponents η in the range (1/16, 1/4), consistent with the expected universality class for a two-step melting scenario.
• The combined use of plaquette percolation and canonical updates leads to a significant improvement in the measurement of off-diagonal operators, resolving issues with slow decorrelation.
• The algorithm successfully reveals a power-law ordered intermediate phase in the fully frustrated square lattice transverse field Ising model, providing strong evidence for a two-step melting of long-range order.
• Refinements in premarking strategies further enhance algorithmic efficiency by reducing cluster freezing and improving cluster size distribution.