Frustration Free Gapless Hamiltonians for Matrix Product States

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• 作者：C. Fernández-González (1) (2)
  N. Schuch (3) (4)
  M. M. Wolf (5)
  J. I. Cirac (6)
  D. Pérez-García (2) (7)
  
• 刊名：Communications in Mathematical Physics
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：333
• 期：1
• 页码：299-333
• 全文大小：3,709 KB
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• 作者单位：C. Fernández-González (1) (2)
  N. Schuch (3) (4)
  M. M. Wolf (5)
  J. I. Cirac (6)
  D. Pérez-García (2) (7)
  
  1. Departamento de Física de los Materiales, Universidad Nacional de Educación a Distancia (UNED), 28040, Madrid, Spain
  2. Departamento de Análisis Matemático and IMI, Universidad Complutense de Madrid, 28040, Madrid, Spain
  3. Institut für Quanteninformation, RWTH Aachen, 52056, Aachen, Germany
  4. Institute for Quantum Information, California Institute of Technology, MC 305-16, Pasadena, CA, 91125, USA
  5. Department of Mathematics, Technische Universit?t München, 85748, Garching, Germany
  6. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748, Garching, Germany
  7. Instiuto de Ciencias Matemticas, ICMAT (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, 28049, Madrid, Spain
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Quantum Physics
  Quantum Computing, Information and Physics
  Complexity
  Statistical Physics
  Relativity and Cosmology
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0916

文摘

For every matrix product state (MPS) one can always construct a so-called parent Hamiltonian. This is a local, frustration free, Hamiltonian which has the MPS as ground state and is gapped. Whenever that parent Hamiltonian has a degenerate ground state space (the so-called non-injective case), we construct another ‘uncle-Hamiltonian which is also local and frustration free, has the same ground state space, but is gapless, and its spectrum is \({\mathbb{R}^+}\) . The construction is obtained by linearly perturbing the matrices building up the state in a random direction, and then taking the limit where the perturbation goes to zero. For MPS where the parent Hamiltonian has a unique ground state (the so-called injective case) we also build such uncle Hamiltonian with the same properties in the thermodynamic limit.