scholarly journals PEPS as unique ground states of local Hamiltonians

2008 ◽  
Vol 8 (6&7) ◽  
pp. 650-663
Author(s):  
D. Perez-Garcia ◽  
F. Verstraete ◽  
J.I. Cirac ◽  
M.M. Wolf
Keyword(s):  
Ground State ◽  
Energy Gap ◽  
Hexagonal Lattice ◽  
Ground States ◽  
Local Region ◽  
The State ◽  
Lattice Geometry ◽  
Quantum Models ◽  
Local Parent ◽  
Aklt Model

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.

Zur Berechnung der chemischen Verschiebung unter besonderer Berücksichtigung des paramagnetischen Terms / Attempts to the Calculation of the Chemical Shift with Especial Consideration of the Paramagnetic Term

1977 ◽  
Vol 32 (12) ◽  
pp. 1541-1543
Author(s):  
H. Sterk ◽  
J. J. Suschnigg
Keyword(s):  
Chemical Shift ◽  
Excited States ◽  
Ground State ◽  
Energy Gap

Abstract Attempts to the Calculation of the Chemical Shift with Especial Consideration of the Paramagnetic Term The calculation of the paramagnetic term according to the Pople formalism of the chemical shift is expanded. The hitherto constant value of the energy gap between the ground state and the excited states is replaced by the value of the lowest lying excitation. This leads to a remarkably better differentiation of the paramagnetic terms of different compounds. The influence is shown on ethane, ethene and ethine.

scholarly journals LOCALIZATION OF THE NUMBER OF PHOTONS OF GROUND STATES IN NONRELATIVISTIC QED

2003 ◽  
Vol 15 (03) ◽  
pp. 271-312 ◽  
Author(s):  
FUMIO HIROSHIMA
Keyword(s):  
Radiation Field ◽  
Ground State ◽  
Fock Space ◽  
Adjoint Operator ◽  
Ground States ◽  
Electron System ◽  
Number Operator ◽  
Position Variable ◽  
Boson Fock Space ◽  
Quantized Radiation Field

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2 (ℝ3) ⊗ ℱ ≅ L2 (ℝ3; ℱ), where ℱ is the Boson Fock space over L2 (ℝ3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to [Formula: see text], where N denotes the number operator of ℱ. Moreover, it is shown that for almost every electron position variable x ∈ ℝ3 and for arbitrary k ≥ 0, ‖(1 ⊗ Nk/2) ψg (x)‖ℱ ≤ Dk e-δ|x|m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular [Formula: see text] for 0 < β < δ/2 is obtained.

scholarly journals A high-spin ground-state donor-acceptor conjugated polymer

2019 ◽  
Vol 5 (5) ◽  
pp. eaav2336 ◽  
Author(s):  
A. E. London ◽  
H. Chen ◽  
M. A. Sabuj ◽  
J. Tropp ◽  
M. Saghayezhian ◽  
...  
Keyword(s):  
Organic Semiconductors ◽  
High Spin ◽  
Ground State ◽  
Conjugated Polymer ◽  
Organic Materials ◽  
Energy Gap ◽  
Light Element ◽  
Paramagnetic Resonance ◽  
Practical Applications ◽  
Polymer Semiconductor

Interest in high-spin organic materials is driven by opportunities to enable far-reaching fundamental science and develop technologies that integrate light element spin, magnetic, and quantum functionalities. Although extensively studied, the intrinsic instability of these materials complicates synthesis and precludes an understanding of how fundamental properties associated with the nature of the chemical bond and electron pairing in organic materials systems manifest in practical applications. Here, we demonstrate a conjugated polymer semiconductor, based on alternating cyclopentadithiophene and thiadiazoloquinoxaline units, that is a ground-state triplet in its neutral form. Electron paramagnetic resonance and magnetic susceptibility measurements are consistent with a high-to-low spin energy gap of 9.30 × 10−3 kcal mol−1. The strongly correlated electronic structure, very narrow bandgap, intramolecular ferromagnetic coupling, high electrical conductivity, solution processability, and robust stability open access to a broad variety of technologically relevant applications once thought of as beyond the current scope of organic semiconductors.

scholarly journals Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture

2013 ◽  
Vol 13 (5&6) ◽  
pp. 393-429
Author(s):  
Matthew Hastings
Keyword(s):  
Ground State ◽  
Coding Theory ◽  
Energy State ◽  
Ground States ◽  
Coarse Graining ◽  
Quantum Circuit ◽  
Technical Condition ◽  
Two Dimensions ◽  
Low Energy ◽  
Coarse Grained

We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have ``trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states\cite{bv}. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an ``interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a $1$-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the $1$-complex). While this includes the cases considered by Ref.~\onlinecite{bv}, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref.~\onlinecite{bv} but still can be mapped continuously to a $1$-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a $1$-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to $1$-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere\cite{fh}, and have useful properties in quantum coding theory.

scholarly journals Non-local to local transition for ground states of fractional Schrödinger equations on $$\mathbb {R}^N$$

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Bartosz Bieganowski ◽  
Simone Secchi
Keyword(s):  
Ground State ◽  
Ground States ◽  
Ground State Solution ◽  
State Solution ◽  
Local Problem ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Fractional Schrödinger Equations ◽  
Non Local ◽  
Local Transition

Abstract We consider the nonlinear fractional problem $$\begin{aligned} (-\Delta )^{s} u + V(x) u = f(x,u)&\quad \hbox {in } \mathbb {R}^N \end{aligned}$$ ( - Δ ) s u + V ( x ) u = f ( x , u ) in R N We show that ground state solutions converge (along a subsequence) in $$L^2_{\mathrm {loc}} (\mathbb {R}^N)$$ L loc 2 ( R N ) , under suitable conditions on f and V, to a ground state solution of the local problem as $$s \rightarrow 1^-$$ s → 1 - .

PHASE DIAGRAM OF THE FALICOV-KIMBALL MODEL WITH INFINITE-RANGE HOPPING

1992 ◽  
Vol 06 (13) ◽  
pp. 793-801 ◽  
Author(s):  
PAVOL FARKAŠOVSKÝ
Keyword(s):  
Phase Diagram ◽  
Ground State ◽  
Lattice Structure ◽  
Ground States ◽  
Chemical Potentials ◽  
Ground State Properties ◽  
Electrons And Ions ◽  
Infinite Range

We have studied the ground state properties of the Falicov-Kimball model with unconstrained hopping. It is shown that the model still behaves non-trivially, although it no longer depends on the actual lattice structure and dimensionality of the system. For arbitrary ion configurations with total number of ions Ni, we have been able to determine domains in the plane of the chemical potentials of electrons and ions where these ion configurations are ground states. The phase diagram of the model is discussed.

Vibronische Kopplung und dynamisch verzerrte Strukturen in Hexahalogenotelluraten(IV): Ergebnisse aus Tieftemperatur-Röntgenbeugungsuntersuchungen (300—160 K) und aus FTIR-spektroskopischen Experimenten (300—5 K) [1] / Vibronic Coupling and Dynamically Distorted Structures in Hexahalogenotellurates(IV): Low Temperatur X-Ray Diffraction (300—160 K) and FTIR-Spectroscopic (300—5 K) Results [1]

1987 ◽  
Vol 42 (10) ◽  
pp. 1273-1281 ◽  
Author(s):  
Walter Abriel ◽  
Ernst-Jürgen Zehnder
Keyword(s):  
Ground State ◽  
Low Temperatures ◽  
Potential Surface ◽  
Energy Gap ◽  
Vibronic Coupling ◽  
High Symmetry ◽  
X Ray Diffraction ◽  
X Ray ◽  
First Excited State ◽  
Theoretical Considerations

AbstractFrom theoretical considerations a dynamically distorted octahedron as a result of vibronic coupling between the ground state and the first excited state should exist for 14 electron AX6E systems like TeX62- . A high symmetry crystal field yielding at least a center of symmetry for the Te position stabilizes this fluctuating structure, otherwise statical distortion will be observed. From X-ray diffraction experiments on antifluorite type compounds A2TeX6 (A = Rb. Cs: X = Cl, Br) the averaged structure (m3̅m symmetry) of the anions was found even at very low temperatures. The thermal parameters are not significantly different from those of similar SnX62 compounds. Distortions therefore are very small and are evident from FTIR spectroscopic meas­urements only. Here very broad T1u-deformation vibration bands are observed down to tempera­tures <10 K without splitting: Astatically distorted species could not be frozen out. In contrast to XeF6 for TeX62- the energy gap between the threefold, fourfold or sixfold minima of the potential surface (according to the symmetry of one component of the T1u-vibration) is very small and shifted to temperatures lower than reached with the devices used for these experiments.

scholarly journals Accurate and Efficient Numerical Methods for Computing Ground States and Dynamics of Dipolar Bose-Einstein Condensates via the Nonuniform FFT

2016 ◽  
Vol 19 (5) ◽  
pp. 1141-1166 ◽  
Author(s):  
Weizhu Bao ◽  
Qinglin Tang ◽  
Yong Zhang
Keyword(s):  
Numerical Methods ◽  
Ground State ◽  
High Efficiency ◽  
Three Dimensional ◽  
Ground States ◽  
Dipole Interaction ◽  
Polar Coordinates ◽  
Pitaevskii Equation ◽  
Nonuniform Fft ◽  
Bose Einstein

AbstractWe propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.

EXACT GROUND STATES OF ONE-DIMENSIONAL VALENCE-BOND-SOLID HAMILTONIANS

1987 ◽  
Vol 01 (05n06) ◽  
pp. 231-237 ◽  
Author(s):  
P.L. Iske ◽  
W.J. Caspers
Keyword(s):  
Ground State ◽  
Valence Bond ◽  
Ground States ◽  
Open Chain ◽  
State Correlation ◽  
One Dimensional ◽  
Degenerate Ground State ◽  
Closed Chain ◽  
Ground State Correlation ◽  
Valence Bond Solid

The ground state(s) of a Hamiltonian, introduced by Affleck, Kennedy, Lieb and Tasaki, in connection with the Valence-Bond-Solid (VBS) states, are explicitly given for the spin-1 chains. The structure of these ground states is a rather simple one. For a closed chain we find a unique ground state; for the open chain we find a fourfold-degenerate ground state. The ground state correlation function for the ring is calculated.

scholarly journals Ground states and zero-temperature measures at the boundary of rotation sets

2017 ◽  
Vol 39 (1) ◽  
pp. 201-224
Author(s):  
TAMARA KUCHERENKO ◽  
CHRISTIAN WOLF
Keyword(s):  
Ground State ◽  
Line Segment ◽  
Invariant Measures ◽  
Ground States ◽  
Zero Temperature ◽  
Direction Vector ◽  
Compact Metric Space ◽  
Rotation Vectors ◽  
Continuous Potential ◽  
Rotation Set

We consider a continuous dynamical system $f:X\rightarrow X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{m}):X\rightarrow \mathbb{R}^{m}$. We study the set of ground states $GS(\unicode[STIX]{x1D6FC})$ of the potential $\unicode[STIX]{x1D6FC}\cdot \unicode[STIX]{x1D6F7}$ as a function of the direction vector $\unicode[STIX]{x1D6FC}\in S^{m-1}$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $\unicode[STIX]{x1D6F7}$. In particular, for each $\unicode[STIX]{x1D6FC}$ the set of rotation vectors of $GS(\unicode[STIX]{x1D6FC})$ forms a non-empty, compact and connected subset of a face $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ of the rotation set associated with $\unicode[STIX]{x1D6FC}$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any $m\in \mathbb{N}$ examples with an exposed boundary point (that is, $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face $F_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6F7})$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of $GS(\unicode[STIX]{x1D6FC})$ is a non-trivial line segment.

Sign in / Sign up

Export Citation Format

Share Document