Frustrated spin-1/2 model in two dimensions with a known ground state

1991 ◽  
Vol 44 (1) ◽  
pp. 443-445 ◽  
Author(s):  
Indrani Bose ◽  
Partha Mitra
Keyword(s):  
Ground State ◽  
Two Dimensions ◽  
Frustrated Spin

scholarly journals Boundary states for chiral symmetries in two dimensions

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Philip Boyle Smith ◽  
David Tong
Keyword(s):  
Boundary Conditions ◽  
Ground State ◽  
Zero Mode ◽  
Central Charge ◽  
Two Dimensions ◽  
Dirac Fermions ◽  
Chiral Symmetries ◽  
Boundary States ◽  
Ground State Degeneracy ◽  
Left And Right

Abstract We study boundary states for Dirac fermions in d = 1 + 1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (−1)F , which are characterised by the existence of an unpaired Majorana zero mode.

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.

Degenerate classical minima and instantons

2021 ◽  
pp. 942-959
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Ground State ◽  
Stochastic Dynamics ◽  
Minimum Energy ◽  
Two Dimensions ◽  
Quantum Theories ◽  
Four Dimensions ◽  
Quantum Tunnelling ◽  
Charge Conjugation Parity ◽  
Double Well Potential

Instantons play an important role in the following situation: quantum theories corresponding to classical actions that have non-continuously connected degenerate minima. The simplest examples are provided by one-dimensional quantum systems with symmetries and potentials with non-symmetric minima. Classically, the states of minimum energy correspond to a particle sitting at any of the minima of the potential. The position of the particle breaks (spontaneously) the symmetry of the system. By contrast, in quantum mechanics (QM), the modulus of the ground-state wave function is large near all the minima of the potential, as a consequence of barrier penetration effects. Two typical examples illustrate this phenomenon: the double-well potential, and the cosine potential, whose periodic structure is closer to field theory examples. In the context of stochastic dynamics, instantons are related to Arrhenius law. The proof of the existence of instantons relies on an inequality related to supersymmetric structures, and which generalizes to some field theory examples. Again, the presence of instantons again indicates that the classical minima are connected by quantum tunnelling, and that the symmetry between them is not spontaneously broken. Examples of such a situation are provided, in two dimensions, by the charge conjugation parity (CP) (N − 1) models and, in four dimensions, by SU(2) gauge theories.

scholarly journals Fermi-liquid ground state of interacting Dirac fermions in two dimensions

2019 ◽  
Vol 99 (12) ◽  
Author(s):  
Kazuhiro Seki ◽  
Yuichi Otsuka ◽  
Seiji Yunoki ◽  
Sandro Sorella
Keyword(s):  
Ground State ◽  
Fermi Liquid ◽  
Two Dimensions ◽  
Dirac Fermions

scholarly journals ROTATING FERMIONS IN TWO DIMENSIONS: A THOMAS–FERMI APPROACH

2001 ◽  
Vol 15 (19n20) ◽  
pp. 2799-2810
Author(s):  
SANKALPA GHOSH ◽  
M. V. N. MURTHY ◽  
SUBHASIS SINHA
Keyword(s):  
Ground State ◽  
Analytical Approach ◽  
State Energy ◽  
Critical Size ◽  
Energy Functional ◽  
Two Dimensions ◽  
Fermionic System ◽  
Thomas Fermi ◽  
Fermionic Systems ◽  
Analytical Expressions

Properties of confined mesoscopic systems have been extensively studied numerically over recent years. We discuss an analytical approach to the study of finite rotating fermionic systems in two dimension. We first construct the energy functional for a finite fermionic system within the Thomas–Fermi approximation in two dimensions. We show that for specific interactions the problem may be exactly solved. We derive analytical expressions for the density, the critical size as well as the ground state energy of such systems in a given angular momentum sector.

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