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Symmetry restoration in mean-field approaches

Sheikh, Javid, J. Dobaczewski, Peter A. Ring, Luis Miguel Robledo, Yannouleas, Constantine

Published Nov 1, 2021• 18 citations
OpenAlex

Abstract

The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrodinger equation for the exact many-body wave function is mapped onto a non-linear density-dependent one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. In this way, additional correlations are subsumed in the system. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the energy density functional theories are highlighted in the present work.

Related papers

Symmetry restoration in mean-field approaches

J A Sheikh, J. Dobaczewski, P Ring +2 more

Abstract The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrödinger equation for the exact many-body wave function is mapped onto a non-linear one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. In this way, additional correlations are subsumed in the system. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional (EDF) theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the EDF theories are highlighted in the present work.

arXiv:1901.06992•Journal of Physics G Nuclear and Particle Physics•Sep 20, 2021•128 citations

Symmetry projection methods in strongly correlated systems

Ruiheng Song

Quantum chemistry aims to understand and predict molecular structure and properties by solving the time-independent Schr\"odinger equation within fixed nuclei (Born-Oppenheimer) approximation. The Hamiltonian may have continuous symmetries (e.g., spin) or discrete symmetries (for example, point group). Mean-field wave functions may break some or all of these symmetries. By taking the symmetry-adapted pieces of these broken symmetry states, one can restore the physical character of the wave function while also treating relevant strong correlations. On the other hand, mean-field methods lack dynamic correlation arising from short-range interactions. Still, single-reference methods such as configuration interaction or coupled cluster theory can handle these correlations efficiently. Ideally, we would like to combine these approaches: performing configuration interaction or coupled cluster theory on a broken symmetry reference while restoring the desired symmetries. This work introduces a new approach to integrating symmetry projection and coupled cluster theory. By borrowing techniques from previous research, the new method can restore continuous symmetries like $U$(1) and $SU$(2) and discrete symmetries like point group and time reversal. This method achieves accurate results in challenging systems and has the same computational scaling as traditional coupled cluster theory. Jordan-Wigner transformation is a potent tool in spin models, but its application is challenging due to the inherent strings. The same techniques we use to combine symmetry projection and coupled cluster theory can be used to combine Jordan-Wigner transformation and coupled cluster. The strings are the ' symmetry projection ' operator when constructing the Hartree-Fock and coupled cluster wave functions for a Jordan-Wigner transformed Hamiltonian. Leveraging the method of symmetry projection, we obtain excellent results for both 1-D and 2-D spin models.

•Sep 30, 2024•0 citations

Mean fields and auxiliary systems

Richard M. Martin, Lucia Reining, David M. Ceperley

It is a characteristic of wisdom not to do desperate things. Henry David Thoreau Summary The essence of a mean-field method is to replace an interacting many-body problem with a set of independent-particle problems having an effective potential. It can be chosen either as an approximation for effects of interactions in an average sense or as an auxiliary system that can reproduce selected properties of an interacting system. The effective potential can have an explicit dependence on an order parameter for a system with a broken symmetry such as a ferro- or antiferromagnet. Mean-field techniques are vital parts of many-body theory: the starting points for practical many-body calculations and often the basis for interpreting the results. This chapter provides a summary of the Hartree–Fock approximation, the Weiss mean field, and density functional theory that have significant roles in the methods described in this book. Mean-field methods denote approaches in which the interacting many-body system is treated as a set of non-interacting particles in a self-consistent field that takes into account some effects of interactions in some way. In the literature such methods are often called “one-electron”; however, in this book we use “non-interacting” or “independent-particle” to refer to mean-field concepts and approaches. We reserve the terms “one-electron” or “one-body” to denote quantities that involve quantum mechanical operators acting independently on each body in a many-body system. Mean-field approaches are relevant for the study of interacting, correlated electrons because they lead to approximate formulations that can be solved more easily than more sophisticated approaches; when judiciously chosen, mean-field solutions can yield useful, physically meaningful results, and they can provide the basis and conceptual structure for investigating the effects of correlation. The particles that are the “bodies” in a many-body theory can be the original particles with their bare masses and interactions, or, most often, they may be the solutions of a set of mean-field equations chosen to facilitate the solution of the many-body problem. A large part of many-body theory in condensed matter involves the choice of the most appropriate independent particles. Hence, it is essential to define clearly the particles that are created and annihilated by the operators and in which the many-body theory is formulated.

Cambridge University Press eBooks•May 31, 2016•0 citations

Combining symmetry breaking and restoration with configuration interaction: A highly accurate many-body scheme applied to the pairing Hamiltonian

J. Ripoche, Denis Lacroix A, D. Gambacurta +2 more

Background: Ab initio many-body methods have been developed over the past ten years to address mid-mass nuclei. In their best current level of implementation, their accuracy is of the order of a few percent error on the ground-state correlation energy. Recently implemented variants of these methods are operating a breakthrough in the description of medium-mass open-shell nuclei at a polynomial computational cost while putting state-of-the-art models of internucleon interactions to the test.Purpose: As progress in the design of internucleon interactions is made, and as questions one wishes to answer are refined in connection with increasingly available experimental data, further efforts must be made to tailor many-body methods that can reach an even higher precision for an even larger number of observable quantum states or nuclei. The objective of the present work is to contribute to such a quest by designing and testing a new many-body scheme.Methods: We formulate a truncated configuration-interaction method that consists of diagonalizing the Hamiltonian in a highly truncated subspace of the total $N$-body Hilbert space. The reduced Hilbert space is generated via the particle-number projected BCS state along with projected seniority-zero two- and four-quasiparticle excitations. Furthermore, the extent by which the underlying BCS state breaks $\text{U}(1)$ symmetry is optimized in the presence of the projected two- and four-quasiparticle excitations. This constitutes an extension of the so-called restricted variation after projection method in use within the frame of multireference energy density functional calculations. The quality of the newly designed method is tested against exact solutions of the so-called attractive pairing Hamiltonian problem.Results: By construction, the method reproduces exact results for $N=2$ and $N=4$. For $N=(8,\phantom{\rule{0.16em}{0ex}}16,\phantom{\rule{0.16em}{0ex}}20)$, the error in the ground-state correlation energy is less than (0.006%, 0.1%, 0.15%) across the entire range of internucleon coupling defining the pairing Hamiltonian and driving the normal-to-superfluid quantum phase transition. The presently proposed method offers the advantage of automatic access to the low-lying spectroscopy, which it does with high accuracy.Conclusions: The numerical cost of the newly designed variational method is polynomial (${N}^{6}$) in system size. This method achieves unprecedented accuracy for the ground-state correlation energy, effective pairing gap, and one-body entropy as well as for the excitation energy of low-lying states of the attractive pairing Hamiltonian. This constitutes a sufficiently strong motivation to envision its application to realistic nuclear Hamiltonians in view of providing a complementary, accurate, and versatile ab initio description of mid-mass open-shell nuclei in the future.

arXiv:1610.04063•Physical Review C•Jan 24, 2017•21 citations

BRST quantization for the restoration of broken symmetries: a pedagogical example

Pranav Sharma, R. J. Furnstahl

We present a pedagogical showcase of BRST quantization for restoring symmetries in nuclear many-body systems as a reformulation and potential alternative to conventional projection methods. The formalism is illustrated using translational invariance for a simple system of two interacting masses in one dimension, but with an eye toward generalizing to more particles, higher dimensions, and other symmetries. We explain the considerations underlying particular choices within the BRST construction, and develop both Hamiltonian and path integral formulations to provide guidance for the variety of many-body and effective field theory contexts where gauge fixing for symmetry restoration might be useful. For the demonstration system we show how to diagonalize within the extended BRST phase space, how variation after projection is recovered for product reference states, how the corresponding gauge-fixed functional integral is constructed, and how collective zero modes are isolated and controlled. Throughout, we keep in mind extensions of BRST symmetry to various approximation schemes as a guide for consistent symmetry restoration.

nucl-thCross-listedarXiv:2609.10172•Sep 10, 2026•0 citations

Projected Bogoliubov Many-Body Perturbation Theory : Overcoming formal and technical challenges

Julien Ripoche

Given elementary inter-nucleon interactions,the resolution of the A-body Schrödinger equation gives access to the properties of the nuclear eigenstates. Many-body perturbation theory (MBPT) based on a Hartree-Fock mean field allows one to treat the dynamical correlations at play in doubly-closed-shell nuclei. Beyond shell closures,breaking U(1) symmetry associated with the conservation of neutron and proton numbers further permits to capture static correlations at the meanfield level and to formulate the well-behaved Bogoliubov many-body perturbation theory (BMBPT).Nevertheless, BMBPT results show contaminations associated with the broken symmetry that is only emergent on finite quantum system such as theatomic nucleus. Thus, the restoration of U(1) symmetry beyond the mean field is necessary for acorrect description and gives rise to the projected BMBPT method (PBMBPT). The goal is to implement PBMBPT in order to perform ab initio calculations of singly-open-shell mid-mass nuclei.The present work provides systematic solutions to the formal and technical problems arising during the implementation of PBMBPT: an automatic and safe method to generate PBMBPT diagrams and associated expressions, a framework to calculate other observables than the energy, an extension of the normal ordering approximation to symmetry breaking methods as well as an automatic and safe tool for spherical reduction of tensor networks.

•Sep 24, 2019•2 citations
The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrodinger equation for the exact many-body wave function is mapped onto a non-linear density-dependent one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. In this way, additional correlations are subsumed in the system. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the energy density functional theories are highlighted in the present work.
Abstract The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrödinger equation for the exact many-body wave function is mapped onto a non-linear one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. In this way, additional correlations are subsumed in the system. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional (EDF) theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the EDF theories are highlighted in the present work.
Quantum chemistry aims to understand and predict molecular structure and properties by solving the time-independent Schr\"odinger equation within fixed nuclei (Born-Oppenheimer) approximation. The Hamiltonian may have continuous symmetries (e.g., spin) or discrete symmetries (for example, point group). Mean-field wave functions may break some or all of these symmetries. By taking the symmetry-adapted pieces of these broken symmetry states, one can restore the physical character of the wave function while also treating relevant strong correlations. On the other hand, mean-field methods lack dynamic correlation arising from short-range interactions. Still, single-reference methods such as configuration interaction or coupled cluster theory can handle these correlations efficiently. Ideally, we would like to combine these approaches: performing configuration interaction or coupled cluster theory on a broken symmetry reference while restoring the desired symmetries. This work introduces a new approach to integrating symmetry projection and coupled cluster theory. By borrowing techniques from previous research, the new method can restore continuous symmetries like UUU(1) and SUSUSU(2) and discrete symmetries like point group and time reversal. This method achieves accurate results in challenging systems and has the same computational scaling as traditional coupled cluster theory. Jordan-Wigner transformation is a potent tool in spin models, but its application is challenging due to the inherent strings. The same techniques we use to combine symmetry projection and coupled cluster theory can be used to combine Jordan-Wigner transformation and coupled cluster. The strings are the ' symmetry projection ' operator when constructing the Hartree-Fock and coupled cluster wave functions for a Jordan-Wigner transformed Hamiltonian. Leveraging the method of symmetry projection, we obtain excellent results for both 1-D and 2-D spin models.
It is a characteristic of wisdom not to do desperate things. Henry David Thoreau Summary The essence of a mean-field method is to replace an interacting many-body problem with a set of independent-particle problems having an effective potential. It can be chosen either as an approximation for effects of interactions in an average sense or as an auxiliary system that can reproduce selected properties of an interacting system. The effective potential can have an explicit dependence on an order parameter for a system with a broken symmetry such as a ferro- or antiferromagnet. Mean-field techniques are vital parts of many-body theory: the starting points for practical many-body calculations and often the basis for interpreting the results. This chapter provides a summary of the Hartree–Fock approximation, the Weiss mean field, and density functional theory that have significant roles in the methods described in this book. Mean-field methods denote approaches in which the interacting many-body system is treated as a set of non-interacting particles in a self-consistent field that takes into account some effects of interactions in some way. In the literature such methods are often called “one-electron”; however, in this book we use “non-interacting” or “independent-particle” to refer to mean-field concepts and approaches. We reserve the terms “one-electron” or “one-body” to denote quantities that involve quantum mechanical operators acting independently on each body in a many-body system. Mean-field approaches are relevant for the study of interacting, correlated electrons because they lead to approximate formulations that can be solved more easily than more sophisticated approaches; when judiciously chosen, mean-field solutions can yield useful, physically meaningful results, and they can provide the basis and conceptual structure for investigating the effects of correlation. The particles that are the “bodies” in a many-body theory can be the original particles with their bare masses and interactions, or, most often, they may be the solutions of a set of mean-field equations chosen to facilitate the solution of the many-body problem. A large part of many-body theory in condensed matter involves the choice of the most appropriate independent particles. Hence, it is essential to define clearly the particles that are created and annihilated by the operators and in which the many-body theory is formulated.
Background: Ab initio many-body methods have been developed over the past ten years to address mid-mass nuclei. In their best current level of implementation, their accuracy is of the order of a few percent error on the ground-state correlation energy. Recently implemented variants of these methods are operating a breakthrough in the description of medium-mass open-shell nuclei at a polynomial computational cost while putting state-of-the-art models of internucleon interactions to the test.Purpose: As progress in the design of internucleon interactions is made, and as questions one wishes to answer are refined in connection with increasingly available experimental data, further efforts must be made to tailor many-body methods that can reach an even higher precision for an even larger number of observable quantum states or nuclei. The objective of the present work is to contribute to such a quest by designing and testing a new many-body scheme.Methods: We formulate a truncated configuration-interaction method that consists of diagonalizing the Hamiltonian in a highly truncated subspace of the total NNN-body Hilbert space. The reduced Hilbert space is generated via the particle-number projected BCS state along with projected seniority-zero two- and four-quasiparticle excitations. Furthermore, the extent by which the underlying BCS state breaks U(1)\text{U}(1)U(1) symmetry is optimized in the presence of the projected two- and four-quasiparticle excitations. This constitutes an extension of the so-called restricted variation after projection method in use within the frame of multireference energy density functional calculations. The quality of the newly designed method is tested against exact solutions of the so-called attractive pairing Hamiltonian problem.Results: By construction, the method reproduces exact results for N=2N=2N=2 and N=4N=4N=4. For N=(8,16,20)N=(8,\phantom{\rule{0.16em}{0ex}}16,\phantom{\rule{0.16em}{0ex}}20)N=(8,16,20), the error in the ground-state correlation energy is less than (0.006%, 0.1%, 0.15%) across the entire range of internucleon coupling defining the pairing Hamiltonian and driving the normal-to-superfluid quantum phase transition. The presently proposed method offers the advantage of automatic access to the low-lying spectroscopy, which it does with high accuracy.Conclusions: The numerical cost of the newly designed variational method is polynomial (N6{N}^{6}N6) in system size. This method achieves unprecedented accuracy for the ground-state correlation energy, effective pairing gap, and one-body entropy as well as for the excitation energy of low-lying states of the attractive pairing Hamiltonian. This constitutes a sufficiently strong motivation to envision its application to realistic nuclear Hamiltonians in view of providing a complementary, accurate, and versatile ab initio description of mid-mass open-shell nuclei in the future.
We present a pedagogical showcase of BRST quantization for restoring symmetries in nuclear many-body systems as a reformulation and potential alternative to conventional projection methods. The formalism is illustrated using translational invariance for a simple system of two interacting masses in one dimension, but with an eye toward generalizing to more particles, higher dimensions, and other symmetries. We explain the considerations underlying particular choices within the BRST construction, and develop both Hamiltonian and path integral formulations to provide guidance for the variety of many-body and effective field theory contexts where gauge fixing for symmetry restoration might be useful. For the demonstration system we show how to diagonalize within the extended BRST phase space, how variation after projection is recovered for product reference states, how the corresponding gauge-fixed functional integral is constructed, and how collective zero modes are isolated and controlled. Throughout, we keep in mind extensions of BRST symmetry to various approximation schemes as a guide for consistent symmetry restoration.
Given elementary inter-nucleon interactions,the resolution of the A-body Schrödinger equation gives access to the properties of the nuclear eigenstates. Many-body perturbation theory (MBPT) based on a Hartree-Fock mean field allows one to treat the dynamical correlations at play in doubly-closed-shell nuclei. Beyond shell closures,breaking U(1) symmetry associated with the conservation of neutron and proton numbers further permits to capture static correlations at the meanfield level and to formulate the well-behaved Bogoliubov many-body perturbation theory (BMBPT).Nevertheless, BMBPT results show contaminations associated with the broken symmetry that is only emergent on finite quantum system such as theatomic nucleus. Thus, the restoration of U(1) symmetry beyond the mean field is necessary for acorrect description and gives rise to the projected BMBPT method (PBMBPT). The goal is to implement PBMBPT in order to perform ab initio calculations of singly-open-shell mid-mass nuclei.The present work provides systematic solutions to the formal and technical problems arising during the implementation of PBMBPT: an automatic and safe method to generate PBMBPT diagrams and associated expressions, a framework to calculate other observables than the energy, an extension of the normal ordering approximation to symmetry breaking methods as well as an automatic and safe tool for spherical reduction of tensor networks.