[Paper Review] PT Symmetry and the Sign Problem

Peter N. Meisinger, Michael C. Ogilvie+1 more|arXiv (Cornell University)|Sep 3, 2010

Fractal and DNA sequence analysis2 references4 citations

TL;DR

This paper establishes that generalized $\mathcal{PT}$ symmetry provides a unifying framework for understanding the sign problem in quantum and classical statistical models with non-zero chemical potential. It shows that the sign problem reduces to a real-weight problem precisely in $\mathcal{PT}$-symmetric systems, where unbroken $\mathcal{PT}$ symmetry enables exact solution via similarity transformations, while broken $\mathcal{PT}$ symmetry leads to oscillatory correlation functions and first-order transitions.

ABSTRACT

Generalized PT symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT-symmetric Hamiltonian. There is a corresponding class of PT-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. For both quantum and classical models, the class of models with generalized PT symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT symmetry is unbroken, the sign problem can always be solved in principle. Moreover, there are models with PT symmetry which can be simulated for all parameter values, including cases where PT symmetry is broken.

Motivation & Objective

• To identify the class of models where the complex weight problem (sign problem) can be reduced to a real-weight problem.
• To establish $\mathcal{PT}$ symmetry as a unifying framework for understanding oscillatory and modulated phases in quantum and classical systems.
• To clarify the connection between $\mathcal{PT}$ symmetry breaking and the emergence of spatially modulated phases.
• To demonstrate that $\mathcal{PT}$-symmetric models can be simulated without sign problems in principle, even in broken symmetry regions.
• To provide a theoretical basis for solving the sign problem using similarity transformations when $\mathcal{PT}$ symmetry is unbroken.

Proposed method

• Formulate the grand canonical partition function as a trace of an exponential of a non-Hermitian Hamiltonian $H_\beta = H - i\mu \int j^d \, d^{d-1}x$, which exhibits generalized $\mathcal{PT}$ symmetry.
• Use the $\mathcal{CT}$ symmetry (charge conjugation and time reversal) to define the $\mathcal{PT}$ symmetry of the Hamiltonian, ensuring real or complex-conjugate eigenvalue pairs.
• Apply the similarity transformation theorem of Mostafazadeh to map $\mathcal{PT}$-symmetric Hamiltonians to isospectral Hermitian ones when $\mathcal{PT}$ symmetry is unbroken.
• Analyze spatial two-point functions to classify behavior as exponential decay (unbroken $\mathcal{PT}$), oscillatory decay, or periodic (broken $\mathcal{PT}$).
• Use the theory of partition function zeros to locate phase boundaries, showing that zeros accumulate on the boundary between unbroken and broken $\mathcal{PT}$ regions in the thermodynamic limit.
• Demonstrate that real, non-symmetric transfer matrices in $\mathcal{PT}$-symmetric models (e.g., ANNNI model) allow exact simulation across all phases.

Experimental results

Research questions

• RQ1 Which classes of quantum and classical statistical models are susceptible to the sign problem, and how can $\mathcal{PT}$ symmetry classify them?
• RQ2 How does $\mathcal{PT}$ symmetry breaking manifest in the correlation functions of many-body systems?
• RQ3 Can the sign problem be solved in principle for $\mathcal{PT}$-symmetric systems, and under what conditions?
• RQ4 What is the role of the similarity transformation in eliminating the sign problem when $\mathcal{PT}$ symmetry is unbroken?
• RQ5 Are there $\mathcal{PT}$-symmetric models that can be simulated without sign problems across all parameter regimes, including broken symmetry phases?

Key findings

• The sign problem reduces to a real-weight problem precisely in $\mathcal{PT}$-symmetric systems, where complex weights are associated with $\mathcal{PT}$ symmetry breaking.
• In the unbroken $\mathcal{PT}$ phase, the sign problem can be solved in principle via a similarity transformation to a Hermitian Hamiltonian.
• Spatial two-point functions exhibit exponential decay in the unbroken $\mathcal{PT}$ phase, oscillatory decay in the broken phase, and periodic behavior in the strongly broken regime.
• The transition to a spatially modulated phase corresponds to $\mathcal{PT}$ symmetry breaking of the ground state and is generically a first-order transition.
• Partition function zeros in region III (broken $\mathcal{PT}$) lie asymptotically on the boundary $\text{Im}(f_0) = 0$ in the thermodynamic limit, indicating a first-order transition.
• Real, non-symmetric transfer matrices in $\mathcal{PT}$-symmetric models (e.g., ANNNI) allow exact simulation across all phases, suggesting a broad class of solvable models.