scholarly journals Typical Ground States for Large Sets of Interactions

2020 ◽  
Vol 181 (5) ◽  
pp. 1906-1914
Author(s):  
Aernout van Enter ◽  
Jacek Miȩkisz
Keyword(s):  
Recent Result ◽  
Ground State ◽  
Ground States ◽  
Weak Mixing ◽  
Singular Spectrum ◽  
Large Sets ◽  
Generic Measures

AbstractWe discuss what ground states for generic interactions look like. We note that a recent result, due to Morris, implies that the behaviour of ground-state measures for generic interactions is similar to that of generic measures. In particular, it follows from his observation that they have singular spectrum and that they are weak mixing, but not mixing.

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 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.

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.

Molecular Orbital Calculations on Penta Coordinated Ferric Dithiocarbamates Exhibiting Intermediate and Low Spin Ground States

1979 ◽  
Vol 34 (12) ◽  
pp. 1500-1506
Author(s):  
P. Ganguli ◽  
K. M. Hasselbach
Keyword(s):  
Energy Level ◽  
Molecular Orbital ◽  
Ground State ◽  
Ground States ◽  
Orbital Energy ◽  
Iron Nucleus ◽  
Molecular Orbital Calculations ◽  
Valence Shell ◽  
Molecular Orbital Energy ◽  
Orbital Energy Level

Abstract SCCEHMO calculations show that the ground state in [Fe(dtc)2X], X = Cl, Br, I and NCS, is 4A2: (x2 - y2)2(xz)1(yz)1(z2)1(xy)0 and for X = NO is 2A1: (xz)2(yz)2(x2-y2)2(z2)1(xy)0. The calculated quadrupole splittings of iron and iodine included the valence shell, overlap charge, and the ligand and lattice contributions to the EFG tensor at the nuclei. In addition, the elec-tron densities at the iron nucleus are compared with the measured isomer shifts. The spin densi-ties may be interpreted in terms of a π-delocalization. The results are discussed on the basis of the molecular orbital energy level schemes.

scholarly journals The large-Nc limit of borelized spectral sum rules and the slope of radial Regge trajectories

2018 ◽  
Vol 33 (18n19) ◽  
pp. 1850115 ◽  
Author(s):  
S. S. Afonin ◽  
T. D. Solomko
Keyword(s):  
Ground State ◽  
Sum Rules ◽  
Ground States ◽  
Phenomenological Method ◽  
Scalar Mesons ◽  
Regge Trajectories ◽  
Spectral Sum Rules ◽  
The Given ◽  
Vacuum Condensates

We put forward a new phenomenological method for calculating the slope of radial trajectories from values of ground states and vacuum condensates. The method is based on a large-[Formula: see text] extension of borelized spectral sum rules. The approach is applied to the light nonstrange vector, axial and scalar mesons. The extracted values of slopes proved to be approximately universal and are in the interval [Formula: see text] GeV2. As a by-product, the given method leads to prediction of the second radial trajectory with ground state mass lying near 0.6 GeV.

POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR -LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS

2019 ◽  
Vol 109 (2) ◽  
pp. 193-216 ◽  
Author(s):  
J. C. DE ALBUQUERQUE ◽  
JOÃO MARCOS DO Ó ◽  
EDCARLOS D. SILVA
Keyword(s):  
Large Class ◽  
Ground State ◽  
Variational Approach ◽  
Ground States ◽  
Nehari Manifold ◽  
Coupled Systems ◽  
Coupling Term ◽  
Ground State Solutions ◽  
Image Position ◽  
The Nehari Manifold

We study the existence of positive ground state solutions for the following class of $(p,q)$-Laplacian coupled systems $$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$ where $1<p\leq q<N$. Here the coefficient $\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition $|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$, where $\unicode[STIX]{x1D6FF}\in (0,1)$ and $\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$. Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.

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