The Composite Operator Method (COM)

Avella, Adolfo; Mancini, Ferdinando

doi:10.1007/978-3-642-21831-6_4

Cited by 17 publications

(59 citation statements)

References 110 publications

(235 reference statements)

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“…We have solved the Hamiltonian (2.4) by using the Green's function and the equations of motion formalisms within the COM framework 57,58 . One of the main ingredients of the method is the extremely sound observation that, in presence of strong electronic interactions, the focus should be moved from the bare electronic operators, in terms of which any perturbative calculation is doomed to fail, to new operators (composite operators).…”

Section: Operatorial Basis and Equations Of Motionmentioning

confidence: 99%

“…The COM framework is based on two main ideas: (i) use of propagators of relevant composite operators as building blocks for any subsequent approximate calculations; (ii) use of algebra constraints to fix the representation of the relevant propagators in order to properly preserve algebraic and symmetry properties; these constraints will also determine the unknown parameters appearing in the formulation due to the non-canonical algebra satisfied by the composite operators. In the last fifteen years, COM has been successfully applied to several models and materials: Hubbard [59][60][61][62][63] 57,58 , one has to choose a set of composite operators as operatorial basis and rewrite the electronic operators and the electronic Green's function in terms of this basis. Algebra constraints are relations among correlation functions dictated by the non-canonical operatorial algebra closed by the chosen operatorial basis 57,58 .…”

Section: Introductionmentioning

confidence: 99%

“…In the last fifteen years, COM has been successfully applied to several models and materials: Hubbard [59][60][61][62][63] 57,58 , one has to choose a set of composite operators as operatorial basis and rewrite the electronic operators and the electronic Green's function in terms of this basis. Algebra constraints are relations among correlation functions dictated by the non-canonical operatorial algebra closed by the chosen operatorial basis 57,58 . Other ways to obtain algebra constraints rely on the symmetries enjoined by the Hamiltonian under study, the Ward-Takahashi identities, the hydrodynamics, etc 57,58 .…”

Section: Introductionmentioning

confidence: 99%

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Emery vs. Hubbard model for cuprate superconductors: a composite operator method study

Avella

Mancini

et al. 2013

Eur. Phys. J. B

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Within the Composite Operator Method (COM), we report the solution of the Emery model (also known as p-d or three band model), which is relevant for the cuprate high-Tc superconductors. We also discuss the relevance of the often-neglected direct oxygen-oxygen hopping for a more accurate, sometimes unique, description of this class of materials. The benchmark of the solution is performed by comparing our results with the available quantum Monte Carlo ones. Both singleparticle and thermodynamic properties of the model are studied in detail. Our solution features a metal-insulator transition at half filling. The resulting metal-insulator phase diagram agrees qualitatively very well with the one obtained within Dynamical Mean-Field Theory. We discuss the type of transition (Mott-Hubbard (MH) or charge-transfer (CT)) for the microscopic (ab-initio) parameter range relevant for cuprates getting, as expected a CT type. The emerging single-particle scenario clearly suggests a very close relation between the relevant sub-bands of the three-(Emery) and the single-band (Hubbard) models, thus providing an independent and non-perturbative proof of the validity of the mapping between the two models for the model parameters optimal to describe cuprates. Such a result confirms the emergence of the Zhang-Rice scenario, which has been recently questioned. We also report the behavior of the specific heat and of the entropy as functions of the temperature on varying the model parameters as these quantities, more than any other, depend on and, consequently, reveal the most relevant energy scales of the system.

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Section: Operatorial Basis and Equations Of Motionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Emery vs. Hubbard model for cuprate superconductors: a composite operator method study

Avella

Mancini

et al. 2013

Eur. Phys. J. B

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“…Despite this curse, the machine learning community has developed a number of techniques with remarkable abilities to recognize, classify, and characterize complex sets of data. In light of this success, it is natural to ask whether such techniques could be applied to the arena of condensed-matter physics, particularly in cases where the microscopic Hamiltonian contains strong interactions, where numerical simulations are typically employed in the study of phases and phase transitions [2,3]. We demonstrate that modern machine learning architectures, such as fully-connected and convolutional neural networks [4], can provide a complementary approach to identifying phases and phase transitions in a variety of systems in condensed matter physics.…”

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confidence: 99%

Machine learning phases of matter

2017

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Neural networks can be used to identify phases and phase transitions in condensed matter systems via supervised machine learning. Readily programmable through modern software libraries, we show that a standard feed-forward neural network can be trained to detect multiple types of order parameter directly from raw state configurations sampled with Monte Carlo. In addition, they can detect highly nontrivial states such as Coulomb phases, and if modified to a convolutional neural network, topological phases with no conventional order parameter. We show that this classification occurs within the neural network without knowledge of the Hamiltonian or even the general locality of interactions. These results demonstrate the power of machine learning as a basic research tool in the field of condensed matter and statistical physics. arXiv:1605.01735v1 [cond-mat.str-el] 5 May 20162 Condensed matter physics is the study of the collective behavior of massively complex assemblies of electrons, nuclei, magnetic moments, atoms or qubits [1]. This complexity is reflected in the size of the classical or quantum state space, which grows exponentially with the number of particles. This exponential growth is reminiscent of the "curse of dimensionality" commonly encountered in machine learning. That is, a target function to be learned requires an amount of training data that grows exponentially in the dimension (e.g. the number of image features). Despite this curse, the machine learning community has developed a number of techniques with remarkable abilities to recognize, classify, and characterize complex sets of data. In light of this success, it is natural to ask whether such techniques could be applied to the arena of condensed-matter physics, particularly in cases where the microscopic Hamiltonian contains strong interactions, where numerical simulations are typically employed in the study of phases and phase transitions [2,3]. We demonstrate that modern machine learning architectures, such as fully-connected and convolutional neural networks [4], can provide a complementary approach to identifying phases and phase transitions in a variety of systems in condensed matter physics. The training of neural networks on data sets obtained by Monte Carlo sampling provides a particularly powerful and simple framework for the supervised learning of phases and phase boundaries in physical models, and can be easily built from readily-available tools such as Theano [5] or TensorFlow [6] libraries.Conventionally, the study of phases in condensed matter systems is performed with the help of tools that have been carefully designed to elucidate the underlying physical structures of various states. Among the most powerful are Monte Carlo simulations, which consist of two steps: a stochastic importance sampling over state space, and the evaluation of estimators for physical quantities calculated from these samples [3]. These estimators are constructed based on a variety of physical impetuses; e.g. the ready availability of an analogous experi...

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“…We report on a solution of the twodimensional Hubbard model in the framework of the Composite Operator Method (COM) [10][11][12] within a novel three-pole approximation [12]. The COM, which is based on the equations of motion and Green's function formalisms, is highly tunable and expressly devised for the characterization of strongly correlated electronic states and the exploration of novel emergent phases.…”

Section: Introductionmentioning

confidence: 99%

Single-particle properties of the Hubbard model in a novel three-pole approximation

Ciolo

Avella

2018

Physica B: Condensed Matter

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We study the 2D Hubbard model using the Composite Operator Method within a novel three-pole approximation. Motivated by the long-standing experimental puzzle of the single-particle properties of the underdoped cuprates, we include in the operatorial basis, together with the usual Hubbard operators, a field describing the electronic transitions dressed by the nearestneighbor spin fluctuations, which play a crucial role in the unconventional behavior of the Fermi surface and of the electronic dispersion. Then, we adopt this approximation to study the single-particle properties in the strong coupling regime and find an unexpected behavior of the van Hove singularity that can be seen as a precursor of a pseudogap regime.

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The Composite Operator Method (COM)

Abstract: The composite operator method (COM) is formulated, its internals illustrated in detail and some of its most successful applications reported

Cited by 17 publications

References 110 publications

Emery vs. Hubbard model for cuprate superconductors: a composite operator method study

Emery vs. Hubbard model for cuprate superconductors: a composite operator method study

Machine learning phases of matter

Single-particle properties of the Hubbard model in a novel three-pole approximation

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