Extracting Conserved Operators from Projected Entangled Pair States Yields 4-Site-Plaquette Local Hamiltonians

Understanding the fundamental properties of complex quantum states remains a central challenge in physics, and researchers now present a new method for identifying the conserved operators, including the Hamiltonian, for a given quantum state. Wen-Tao Xu, Miguel Frías Pérez, and Mingru Yang demonstrate a technique to extract geometrically-local conserved operators from infinite projected pair states, a powerful way to represent quantum systems in two dimensions. This achievement moves beyond standard approaches by accurately reconstructing both simple and more complex parent Hamiltonians, even those with inherent quantum frustration, and importantly, achieves improved locality in the resulting models. The team’s work reveals a Hamiltonian with a short-range resonance valence bond (RVB) state as its ground state and identifies a Hamiltonian that supports deformed toric code states as excited eigenstates, potentially offering new insights into the nature of many-body scars.

Extracting Hamiltonians From Tensor Network States

Scientists have developed a method to determine the conserved operators, including Hamiltonians, for which a given tensor network state serves as an approximate eigenstate. This work centers on evaluating static structure factors of multi-site operators by differentiating a generating function, allowing the extraction of geometrically-local conserved operators. This achievement provides a powerful new tool for understanding the underlying physics of complex quantum systems. Applying this technique to the critical short-range RVB state on a square lattice, researchers discovered a 4-site-plaquette local Hamiltonian that approximately has the RVB state as its ground state.

Detailed analysis revealed consistent values and ranges for key parameters defining the Hamiltonian, and the energy expectation value of the RVB state closely matched the calculated ground state energy, confirming the accuracy of the extracted Hamiltonian. Furthermore, the team found a Hamiltonian that possesses deformed Ising or toric code wavefunctions as excited eigenstates. The lowest eigenvalue of the 4-site structure factor was found to be zero for all turning parameters, indicating that these wavefunctions lie in the middle of the Hamiltonian’s spectrum, suggesting they may represent quantum many-body scars. These results demonstrate the feasibility of learning Hamiltonians from tensor network states, even with approximations in the contraction schemes and the states themselves.

Local Hamiltonians from Tensor Network States

This work presents a new method to determine geometrically-local conserved operators, including Hamiltonians, given a tensor network state. Researchers successfully demonstrate the ability to extract these Hamiltonians by solving for the kernel of static structure factor matrices, which are evaluated through differentiation of generating functions. This represents a significant advance in the field of quantum many-body physics, offering a new approach to understanding complex quantum systems. Applying this approach to the critical short-range RVB state, the team obtained an approximate 4-site-plaquette local Hamiltonian and discovered a Hamiltonian possessing deformed Ising wavefunctions, or equivalently, deformed toric code states, as excited eigenstates.

This finding suggests potential connections to the emerging field of many-body scars, where certain excited states remain stable and do not thermalize. The results confirm that learning Hamiltonians from these complex tensor network states is feasible, even with the inherent approximations in both the contraction schemes and the states themselves. The authors acknowledge limitations in the precision of evaluating static structure factors for critical states and suggest future research directions to address this, including reducing computational complexity through parameterization of local terms and employing Monte Carlo methods for improved evaluation. Further investigation into the scaling of errors and tightening the learning complexity bound at zero temperature are also proposed, alongside exploring connections between tensor network norms and classical partition functions.

Extracting Conserved Operators from Projected Entangled Pair States

Scientists have developed a method to identify conserved operators, including Hamiltonians, for which a given state, represented as a Projected Entangled Pair State (PEPS), serves as an approximate solution. This work centers on evaluating static structure factors of multi-site operators by differentiating a generating function, allowing the extraction of geometrically-local conserved operators. This achievement provides a powerful new tool for understanding the underlying physics of complex quantum systems. The method involves analyzing the static structure factor, a measure of how different parts of the quantum state are correlated.

By differentiating a generating function, researchers can extract information about these correlations and identify the conserved operators that govern the system’s behavior. The team successfully applied this method to various states, demonstrating its ability to identify parent Hamiltonians. This work is important because it provides a rigorous framework for analyzing PEPS, especially in the context of complex quantum phases. It addresses a key challenge in the field: how to extract meaningful information from PEPS when the standard methods fail. The proposed methods can be used to identify conserved operators, characterize topological phases, and improve the accuracy of numerical simulations, correcting for ambiguities introduced by non-injectivity to obtain more reliable results. In summary, this paper presents a sophisticated set of techniques for analyzing PEPS, focusing on the challenges posed by non-injective states. It provides a rigorous framework for calculating static structure factors, finding conserved operators, and characterizing topological phases, representing a valuable contribution to the field of quantum many-body physics.