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PEPS as unique ground states of local Hamiltonians
David Perez-Garcia, Frank Verstraete, J. Ignacio Cirac, Michael M. Wolf
TL;DR
The paper asks when PEPS admit local parent Hamiltonians with a unique ground state and how this relates to spectral gaps. It constructs parent Hamiltonians, identifies injectivity as the relevant uniqueness condition, and uses classical-to-quantum mappings to show that higher-dimensional PEPS can have unique critical ground states despite injectivity.
Problem
The paper addresses the scarcity of exactly solvable higher-dimensional spin models by studying when PEPS have unique ground states and whether injectivity ensures an energy gap.
Method
The paper constructs local parent Hamiltonians for PEPS and analyzes injectivity using boundary-to-bulk relations and a classical-to-quantum mapping.
Results
Injective PEPS are unique ground states of local Hamiltonians, while a 2D hexagonal-lattice example is a unique critical ground state, showing that injectivity need not imply a gap.
Takeaways & Limitations
In higher-dimensional PEPS, ground-state uniqueness and the existence of an energy gap are distinct properties.
Abstract
from arXiv · showhide
In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.
I. INTRODUCTION
The paper studies quasi-exactly solvable PEPS models and identifies injectivity as sufficient for unique parent-Hamiltonian ground states. It also shows that lattice geometry can permit critical unique ground states and that injectivity is not equivalent to a spectral gap beyond one dimension.
- The work investigates quasi-exactly solvable models based on PEPS, focusing on uniqueness of their ground states.
- Every PEPS satisfying injectivity is the unique ground state of a local Hamiltonian.
- For PEPS that are coherent versions of classical Gibbs states, injectivity depends solely on lattice geometry.
- A classical-to-quantum mapping yields a unique critical ground state of a local Hamiltonian on a 2D hexagonal lattice.
- In contrast to 1D MPS, injectivity does not imply an energy gap in higher-dimensional PEPS.
- The paper also proves uniqueness without injectivity on the 2D square lattice and provides a computable sufficient condition for an energy gap.
II. PRELIMINARIES
PEPS assign virtual entanglement to graph edges and local maps from virtual systems to physical sites. Contracting the resulting site tensors according to the graph produces the many-body quantum state.
- A PEPS is a quantum state constructed on a graph whose vertices represent physical sites and whose geometry typically follows a lattice.
- Each graph edge receives a maximally entangled state, creating virtual subsystems at the adjacent vertices.
- At each vertex, a local map sends the virtual systems associated with adjacent edges to the physical system.
- The PEPS is formed by contracting the site tensors according to the graph's edges.
- A translationally invariant PEPS has a representation with site-independent tensors, although the paper does not assume this symmetry generally.
III. PARENT HAMILTONIANS
On lattices, PEPS parent Hamiltonians arise because sufficiently large regions have physical Hilbert spaces larger than their boundary-supported state spaces. The resulting Hamiltonian is local and frustration free.
- Lattice geometry makes region volumes grow faster than boundary areas, enabling parent-Hamiltonian construction for PEPS.The boundary support grows at most as D^eR, while the physical Hilbert space grows with the region volume.
- Figure 1 illustrates grouping vertices into injective regions and placing interactions between connected neighboring regions.
- A sufficiently large region has a nontrivial kernel because its reduced-state support is bounded by D^eR, whereas its Hilbert space grows as d^|R|.Hamiltonian terms can be chosen to act on this kernel.
- The local terms are assigned across the lattice, producing a local frustration-free Hamiltonian for which the PEPS minimizes every interaction term.The construction gives the PEPS zero expectation value for each local interaction.
IV. INJECTIVITY
Injectivity means that the boundary-to-bulk map of a region is injective, so boundary data generate the full relevant tensor space. This property is generic under a dimension-counting condition and can be combined across regions.
- The map ΓR sends virtual boundary data to the physical bulk, and a region is injective when this map is injective.Equivalently, the tensors generated by the region span the full tensor space on its eR boundary indices.
- Injectivity is expected generically when the number of physical tensors satisfies d^|R| ≥ D^eR.The physical tensor count grows with volume, while the target tensor-space dimension grows with boundary size.
- Disjoint injective regions remain injective after union, allowing injectivity to lift from smaller regions to larger ones.
- When a region and its complement are injective, the boundary-generated space GR equals the support SR of the region’s reduced density operator.
- For three regions, the generated bulk space is contained in intersections of spaces generated by overlapping unions, with equality under specified injectivity and connectivity conditions.If all three regions are injective, three such intersection constraints characterize the generated space.
V. UNIQUENESS OF THE GROUND STATE
The paper proves uniqueness by regrouping an injective PEPS into an injective super-lattice and coupling neighboring regions. The resulting local frustration-free Hamiltonian has the PEPS as its unique ground state.
- An injective PEPS can be regrouped into injective super-lattice vertices formed from disjoint injective regions.Super-lattice edges preserve connectivity between the original regions.
- Nearest-neighbor interactions are chosen with kernels equal to the boundary-generated spaces GRα∪Rβ of neighboring super-lattice regions.
- Every injective PEPS on an arbitrary lattice has a local frustration-free Hamiltonian for which it is the unique ground state.
- The parent Hamiltonian is not unique, since positive modifications of interaction terms and alternative region sizes can preserve the construction.Smaller regions can sometimes reduce the number of sites involved in an interaction, as in the AKLT model.
- The uniqueness proof uses induction over connected collections of injective regions and the intersection identities established for such regions.
VI. QUANTUM STATES FROM CLASSICAL MODELS
The paper constructs PEPS from classical nearest-neighbor spin models so that the quantum states reproduce classical thermal correlations. It then studies injectivity of these states through the resulting product-form tensors.
- A classical nearest-neighbor Hamiltonian with d configurations per site defines an associated PEPS through local tensor data.
- The construction can enforce d = D and reproduce the correlations of the classical thermal state.The mapping replaces thermal fluctuations with quantum fluctuations.
- The PEPS construction uses an auxiliary maximally entangled state on each edge and local maps that convert joint physical-virtual systems into physical states.
- For the Ising example, the relevant local matrix is invertible for every β.
A. Injectivity and Criticality
For PEPS derived from classical spin models, injectivity is determined by lattice geometry and can coexist with a critical, gapless parent Hamiltonian on the 2D hexagonal lattice.
- Injectivity criterion: For classical-model PEPS, injectivity depends on lattice geometry rather than the interaction type.The PEPS lattice coincides with the classical interaction graph.
- Injectivity criterion: A region is injective when each site has at most one outgoing edge; otherwise, it is not injective.This criterion follows from comparing boundary edges with boundary sites in the contraction map.
- Lattice dependence: Hexagonal and tetrahedral geometries yield injectivity, whereas the ordinary square lattice does not without substructures or defects.Square-lattice injectivity can also arise through added substructures or missing bonds.
- Criticality: The hexagonal Ising PEPS is the unique ground state of its parent Hamiltonian while remaining critical and gapless.Critical Ising correlations decay by a power law, which implies gaplessness of the parent Hamiltonian.
B. Uniqueness for non-injective lattices
Uniqueness of the parent-Hamiltonian ground state does not require injectivity: for a square-lattice classical-model PEPS, range intersection properties can establish uniqueness directly.
- Non-injective uniqueness: Injectivity is sufficient but not necessary for proving uniqueness of a PEPS parent-Hamiltonian ground state.The square-lattice classical-model example lacks injectivity yet still has a unique ground state.
- Proof strategy: The non-injective square-lattice case is handled by proving the required intersection properties of the regional ranges.The proof uses the tensor representation and dimension considerations instead of injectivity.
- Parent Hamiltonian: The construction permits equivalent parent-Hamiltonian projectors, with the cross-shaped projector chosen because it reflects the central spin’s nearest neighbors.One projector has a square structure, while the other matches the local neighborhood more naturally.
VII. A COMPUTABLE SUFFICIENT CONDITION FOR AN ENERGY GAP
The paper gives a computable sufficient criterion for a uniform energy gap and applies classical Markov-dynamics mappings to establish gaps beyond the directly testable regime.
- Gap criterion: A local inequality of the form H^2 > ϵH provides a uniform energy gap for an injective PEPS parent Hamiltonian.The bound is independent of system size.
- Ising application: For the square-lattice Ising PEPS, the criterion proves a gap for β < 0.27, while analytic arguments establish a gap throughout β < βc = 1The broader result uses a Q-matrix associated with Glauber dynamics.
- Markov-dynamics mapping: The Q-matrix is built from local spin-flip dynamics whose operators act only on a site and its nearest neighbors.The associated transition rates are those of Metropolis dynamics.
- Markov-dynamics mapping: A size-independent lower bound on gap(Q), together with H_Q = −Q being bounded by the parent Hamiltonian, implies that the parent Hamiltonian is gapped.The comparison transfers the spectral-gap information from the classical dynamics to the quantum Hamiltonian.
VIII. APPENDIX: SITE-INDEPENDENT TENSORS
Every translationally invariant PEPS on an N × M square lattice admits a representation using site-independent tensors.
- Site-independent representation: Every translationally invariant PEPS on an N × M square lattice has a representation with site-independent tensors.The proof follows closely the one-dimensional case.