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arXiv:1802.07827 · Physical review. B./Physical review. B

Method to identify parent Hamiltonians for trial states

Martin Greiter, Vera Schnells, Ronny Thomale

Published Aug 22, 2018• 46 citations
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Abstract

We describe a general method to identify exact, local parent Hamiltonians for trial states such as quantum Hall or spin-liquid states, which we have used extensively during the past decade. It can be used to identify exact parent Hamiltonians, either directly or via the construction of simpler annihilation operators from which a parent Hamiltonian respecting all the required symmetries can be constructed. Most remarkably, however, the method provides approximate parent Hamiltonians whenever an exact solution is not available within the space of presumed interaction terms.

DOI
10.1103/physrevb.98.081113

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quant-pharXiv:2608.30272•Sep 1, 2026•0 citations
We describe a general method to identify exact, local parent Hamiltonians for trial states such as quantum Hall or spin-liquid states, which we have used extensively during the past decade. It can be used to identify exact parent Hamiltonians, either directly or via the construction of simpler annihilation operators from which a parent Hamiltonian respecting all the required symmetries can be constructed. Most remarkably, however, the method provides approximate parent Hamiltonians whenever an exact solution is not available within the space of presumed interaction terms.
We construct a parent Hamiltonian for the family of non-Abelian chiral spin liquids proposed recently by two of us [Phys. Rev. Lett. 102, 207203 (2009)] which includes the Abelian chiral spin liquid proposed by Kalmeyer and Laughlin as the special case s=12s=\frac{1}{2}s=21​. As we use a circular disk geometry with an open boundary, both the annihilation operators we identify and the Hamiltonians we construct from these are exact only in the thermodynamic limit.
Variational wave functions have been a successful tool to investigate the properties of quantum spin liquids. Finding their parent Hamiltonians is of primary interest for the experimental realization of these strongly correlated phases, and for gathering additional insights on their stability. In this work, we systematically reconstruct approximate spin-chain parent Hamiltonians for Jastrow-Gutzwiller wave functions, which share several features with quantum spin liquid wave functions in two dimensions. Firstly, we determine the different phases encoded in the parameter space through their correlation functions and entanglement properties. Secondly, we apply a recently proposed entanglement-guided method to reconstruct parent Hamiltonians to these states, which constrains the search to operators describing relativistic low-energy field theories - as expected for deconfined phases of gauge theories relevant to quantum spin liquids. The quality of the results is discussed using different quantities and comparing to exactly known parent Hamiltonians at specific points in parameter space. Our findings provide guiding principles for experimental Hamiltonian engineering of this class of states.
We study the inverse problem of constructing an appropriate Hamiltonian from a physically reasonable set of orthogonal wave functions for a quantum spin system. Usually, we are given a local Hamiltonian and our goal is to characterize the relevant wave functions and energies (the spectrum) of the system. Here, we take the opposite approach; starting from a reasonable collection of orthogonal wave functions, we try to characterize the associated parent Hamiltonians, to see how the wave functions and the energy values affect the structure of the parent Hamiltonian. Specifically, we obtain (quasi) local Hamiltonians by a complete set of (multilayer) product states and a local mapping of the energy values to the wave functions. On the other hand, a complete set of tree wave functions (having a tree structure) results to nonlocal Hamiltonians and operators which flip simultaneously all the spins in a single branch of the tree graph. We observe that even for a given set of basis states, the energy spectrum can significantly change the nature of interactions in the Hamiltonian. These effects can be exploited in a quantum engineering problem optimizing an objective functional of the Hamiltonian.
We introduce a method for the search of parent Hamiltonians of input wave functions based on the structure of their reduced density matrix. The two key elements of our recipe are an ansatz on the relation between the reduced density matrix and parent Hamiltonian that is exact at the field theory level, and a minimization procedure on the space of relative entropies, which is particularly convenient due to its convexity. As examples, we show how our method correctly reconstructs the parent Hamiltonian correspondent to several nontrivial ground state wave functions, including conformal and symmetry-protected-topological phases, and quantum critical points of two-dimensional antiferromagnets described by strongly coupled field theories. Our results show the entanglement structure of ground state wave functions considerably simplifies the search for parent Hamiltonians.
Purified Gibbs states provide a bridge between finite-temperature physics, dissipative dynamics, and ground-state methods. In this work, we study the exact finite sum-of-squares (SoS) construction of their parent Hamiltonians and the associated Lindbladian based on modular annihilators. Given a finite set of Hermitian generators, the corresponding modular annihilators yield a frustration-free SoS representation without continuous time integrals or an explicit decomposition into Bohr-frequency sectors. The purified Gibbs state remains a common zero mode while the freedom to choose and combine the generators can be used to optimize the spectral properties of the parent Hamiltonian. For free-fermion Hamiltonians, modular transformations act linearly on Majorana operators, leading to an analytically solvable family of parent Hamiltonians parameterized by a real symmetric coefficient matrix SSS. For the scalar-functional subclass S=f(h)S=f(h)S=f(h), we show that, at fixed operator norm, the choice Sopt∝1/cosh⁡(2βh)S_{\mathrm{opt}}\propto 1/\sqrt{\cosh(2βh)}Sopt​∝1/cosh(2βh)​ has mixing time upper bound 2log⁡(2N/ε)2\log(2N/ε)2log(2N/ε) for any βββ, which exhibits rapid mixing and is irrelevant to the inverse temperature βββ. For interacting systems, where the modularly dressed generators are not available in closed form, we introduce a Krylov--Lanczos approximation scheme and bound the resulting ground-state error in terms of the modular-approximation error and the parent-Hamiltonian gap. Numerical results illustrate the free-fermion spectral advantage and show how the accuracy of the interacting construction depends on temperature, interaction strength, and Krylov dimension.