Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Research

Nodeless wave functions, spiky potentials, and the description of a quantum system in a quantum environment

Jakub Kocák, Axel Schild

Published Aug 27, 2021• 4 citations
Publisher

Abstract

A quantum system $\mathcal{Q}$ is characterized by a single potential $v$ and its eigenstates. While $v$ is usually postulated for a given physical problem, it represents the interaction with an implicit environment $\mathcal{E}$. We use the exact factorization to show how $v$ emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem $\mathcal{S}=\mathcal{Q}\ensuremath{\cup}\mathcal{E}$ corresponds to a different potential ${v}_{j}$ and state ${\ensuremath{\chi}}_{j}$ of $\mathcal{Q}$. Such a state ${\ensuremath{\chi}}_{j}$ typically has no nodes and is the ground state of ${v}_{j}$, even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting $\mathcal{Q}$ is much smaller than for exciting $\mathcal{E}$, the potentials ${v}_{j}$ are similar in shape and differ only by sharp spikes. An excitation of $\mathcal{S}$ can then be viewed as an excitation of $\mathcal{Q}$ with its environment being unaffected, and $\mathcal{Q}$ is approximately described by a single spikeless potential $v$ and its eigenstates. Second, ${\ensuremath{\chi}}_{j}$ can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of ${\ensuremath{\chi}}_{j}$ being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems.

DOI
10.1103/physrevresearch.3.033194

Related papers

Topology and Excitations in Low-Dimensional Quantum Matter

Ruben Verresen

The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.

•Jan 1, 2019•0 citations

Supersymmetry in Quantum Mechanics

Nelson Enrique Bolivar, Gabriel Abellán

The problem that usually arises in the study of a quantum system is the integration of the eigenvalue equation of the Hamiltonian, namely the search for eigenstates and the corresponding eigenvalues of energy. The eigenstates, in the representation of the position, are described by eigenfunctions with the same position variable. The relations work modulo a normalization constant. The number of nodes of a function allows to determine its parity: an odd number of nodes characterizes as an antisymmetric function, while a even number of nodes is a symmetric one. The state of a system can be characterized as bosonic or fermionic depending on whether the wave function is even or odd, respectively. The two operators are nilpotent, when applied more than once to an eigenstate, it generates a null ket, regardless of the starting state.

Apple Academic Press eBooks•May 23, 2018•8 citations

Quantal density functional theory of excited states: Application to an exactly solvable model

Marlina Slamet, Viraht Sahni

Abstract The quantal density functional theory (Q‐DFT) of excited states is the description of the physics of the mapping from any bound nondegenerate excited state of Schrödinger theory to that of the s‐system of noninteracting Fermions with equivalent density ρk(r), energy Ek, and ionization potential Ik. The s‐system may either be in an excited state with the same configuration as in Schrödinger theory or in a ground state with a consequently different configuration. The Q‐DFT description of the s‐system is in terms of a conservative field ℱk(r), whose electron‐interaction ℰee(r) and correlation‐kinetic 𝒵 (r) components are separately representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and correlation‐kinetic effects, respectively. The sources of these fields are expectations of Hermitian operators taken with respect to the system wavefunction. The local electron‐interaction potential vee(r) of the s‐system, representative of all the many‐body correlations, is the work done to move an electron in the force of the field ℱk(r). The electron interaction Eee and correlation‐kinetic Tc components of the total energy Ek may be expressed in integral virial form in terms of their respective fields. The difference between the s‐system in its ground or excited state representation is due entirely to correlation‐kinetic effects. The highest occupied eigenvalue of the s‐system differential equation in either case is minus the ionization potential Ik. In this work we demonstrate the transformation of an excited state of Schrödinger theory, as represented by the first excited singlet state of the exactly solvable Hooke's atom, to that of noninteracting Fermions in their ground state with equivalent excited state density, energy, and ionization potential. To further prove the Fermions are in a ground state, we solve the corresponding singlet s‐system differential equation numerically for the vee(r) determined, and obtain the excited state density from the zero node orbitals generated. The resulting total energy is also the same. In addition, the single eigenvalue determined corresponds to minus the ionization potential of the excited state. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

International Journal of Quantum Chemistry•Jan 1, 2001•31 citations

Nodeless wave function quantum theory

Geoffrey Hunter

Abstract The nodes where the wave function changes sign in conventional quantum theory can alternatively be regarded as places where the wave function descends to zero but rises again without changing its sign. This behavior is accomplished by adding delta function barriers to the potential energy wherever a node occurs in the wave function. This model is supported by the recognition that the wave functions of a subsystem, which is not truly isolated, are really marginal amplitudes which implicitly average out interactions of the system with its environment. The finite effect of these interactions is to replace the delta function barriers by barriers of finite width and height.

International Journal of Quantum Chemistry•Jan 1, 1980•24 citations

Geometric potential of the exact electron factorization: Meaning, significance, and application

Jakub Kocák, Eli Kraisler, Axel Schild

The theoretical and computational description of materials properties is a task of utmost scientific and technological importance. A first-principles description of electron-electron interactions poses an immense challenge that is usually approached by converting the many-electron problem to an effective one-electron problem. There are different ways to obtain an exact one-electron theory for a many-electron system. An emergent method is the exact electron factorization (EEF) -- one of the branches of the Exact Factorization approach to many-body systems. In the EEF, the Schrödinger equation for one electron, in the environment of all other electrons, is formulated. The influence of the environment is reflected in the potential $v^{\rm H}$, which represents the energy of the environment, and in a potential $v^{\rm G}$, which has a geometrical meaning. In this paper, we focus on $v^{\rm G}$ and study its properties in detail. We investigate the geometric origin of $v^{\rm G}$ as a metric measuring the change of the environment, exemplify how translation and scaling of the state of the environment are reflected in $v^{\rm G}$, and explain its shape for homo- and heteronuclear diatomic model systems. Based on the close connection between the EEF and density functional theory, we also use $v^{\rm G}$ to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.

arXiv:2010.14885•Oct 28, 2020•2 citations

Geometric potential of the exact electron factorization: Meaning, significance, and application

Jakub Kocák, Eli Kraisler, Axel Schild

The theoretical and computational description of materials properties is a task of utmost scientific and technological importance. A first-principles description of electron-electron interactions poses an immense challenge that is usually approached by converting the many-electron problem to an effective one-electron problem. There are different ways to obtain an exact one-electron theory for a many-electron system. An emergent method is the exact electron factorization (EEF) -- one of the branches of the exact factorization approach to many-body systems. In the EEF, the Schr\"odinger equation for one electron, in the environment of all other electrons, is formulated. The influence of the environment is reflected in the potential ${v}^{\mathrm{H}}$, which represents the energy of the environment, and in a potential ${v}^{\mathrm{G}}$, which has a geometrical meaning. In this paper, we focus on ${v}^{\mathrm{G}}$ and study its properties in detail. We investigate the geometric origin of ${v}^{\mathrm{G}}$ as a metric measuring the change of the environment, exemplify how translation and scaling of the state of the environment are reflected in ${v}^{\mathrm{G}}$, and explain its shape for homo- and heteronuclear diatomic model systems. Based on the close connection between the EEF and density functional theory, we also use ${v}^{\mathrm{G}}$ to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.

Physical Review Research•Jan 17, 2023•10 citations
A quantum system Q\mathcal{Q}Q is characterized by a single potential vvv and its eigenstates. While vvv is usually postulated for a given physical problem, it represents the interaction with an implicit environment E\mathcal{E}E. We use the exact factorization to show how vvv emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem S=Q\ensuremath∪E\mathcal{S}=\mathcal{Q}\ensuremath{\cup}\mathcal{E}S=Q\ensuremath∪E corresponds to a different potential vj{v}_{j}vj​ and state \ensuremathχj{\ensuremath{\chi}}_{j}\ensuremathχj​ of Q\mathcal{Q}Q. Such a state \ensuremathχj{\ensuremath{\chi}}_{j}\ensuremathχj​ typically has no nodes and is the ground state of vj{v}_{j}vj​, even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting Q\mathcal{Q}Q is much smaller than for exciting E\mathcal{E}E, the potentials vj{v}_{j}vj​ are similar in shape and differ only by sharp spikes. An excitation of S\mathcal{S}S can then be viewed as an excitation of Q\mathcal{Q}Q with its environment being unaffected, and Q\mathcal{Q}Q is approximately described by a single spikeless potential vvv and its eigenstates. Second, \ensuremathχj{\ensuremath{\chi}}_{j}\ensuremathχj​ can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of \ensuremathχj{\ensuremath{\chi}}_{j}\ensuremathχj​ being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems.
The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.
The problem that usually arises in the study of a quantum system is the integration of the eigenvalue equation of the Hamiltonian, namely the search for eigenstates and the corresponding eigenvalues of energy. The eigenstates, in the representation of the position, are described by eigenfunctions with the same position variable. The relations work modulo a normalization constant. The number of nodes of a function allows to determine its parity: an odd number of nodes characterizes as an antisymmetric function, while a even number of nodes is a symmetric one. The state of a system can be characterized as bosonic or fermionic depending on whether the wave function is even or odd, respectively. The two operators are nilpotent, when applied more than once to an eigenstate, it generates a null ket, regardless of the starting state.
Abstract The quantal density functional theory (Q‐DFT) of excited states is the description of the physics of the mapping from any bound nondegenerate excited state of Schrödinger theory to that of the s‐system of noninteracting Fermions with equivalent density ρk(r), energy Ek, and ionization potential Ik. The s‐system may either be in an excited state with the same configuration as in Schrödinger theory or in a ground state with a consequently different configuration. The Q‐DFT description of the s‐system is in terms of a conservative field ℱk(r), whose electron‐interaction ℰee(r) and correlation‐kinetic 𝒵 (r) components are separately representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and correlation‐kinetic effects, respectively. The sources of these fields are expectations of Hermitian operators taken with respect to the system wavefunction. The local electron‐interaction potential vee(r) of the s‐system, representative of all the many‐body correlations, is the work done to move an electron in the force of the field ℱk(r). The electron interaction Eee and correlation‐kinetic Tc components of the total energy Ek may be expressed in integral virial form in terms of their respective fields. The difference between the s‐system in its ground or excited state representation is due entirely to correlation‐kinetic effects. The highest occupied eigenvalue of the s‐system differential equation in either case is minus the ionization potential Ik. In this work we demonstrate the transformation of an excited state of Schrödinger theory, as represented by the first excited singlet state of the exactly solvable Hooke's atom, to that of noninteracting Fermions in their ground state with equivalent excited state density, energy, and ionization potential. To further prove the Fermions are in a ground state, we solve the corresponding singlet s‐system differential equation numerically for the vee(r) determined, and obtain the excited state density from the zero node orbitals generated. The resulting total energy is also the same. In addition, the single eigenvalue determined corresponds to minus the ionization potential of the excited state. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
Abstract The nodes where the wave function changes sign in conventional quantum theory can alternatively be regarded as places where the wave function descends to zero but rises again without changing its sign. This behavior is accomplished by adding delta function barriers to the potential energy wherever a node occurs in the wave function. This model is supported by the recognition that the wave functions of a subsystem, which is not truly isolated, are really marginal amplitudes which implicitly average out interactions of the system with its environment. The finite effect of these interactions is to replace the delta function barriers by barriers of finite width and height.
The theoretical and computational description of materials properties is a task of utmost scientific and technological importance. A first-principles description of electron-electron interactions poses an immense challenge that is usually approached by converting the many-electron problem to an effective one-electron problem. There are different ways to obtain an exact one-electron theory for a many-electron system. An emergent method is the exact electron factorization (EEF) -- one of the branches of the Exact Factorization approach to many-body systems. In the EEF, the Schrödinger equation for one electron, in the environment of all other electrons, is formulated. The influence of the environment is reflected in the potential vHv^{\rm H}vH, which represents the energy of the environment, and in a potential vGv^{\rm G}vG, which has a geometrical meaning. In this paper, we focus on vGv^{\rm G}vG and study its properties in detail. We investigate the geometric origin of vGv^{\rm G}vG as a metric measuring the change of the environment, exemplify how translation and scaling of the state of the environment are reflected in vGv^{\rm G}vG, and explain its shape for homo- and heteronuclear diatomic model systems. Based on the close connection between the EEF and density functional theory, we also use vGv^{\rm G}vG to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.
The theoretical and computational description of materials properties is a task of utmost scientific and technological importance. A first-principles description of electron-electron interactions poses an immense challenge that is usually approached by converting the many-electron problem to an effective one-electron problem. There are different ways to obtain an exact one-electron theory for a many-electron system. An emergent method is the exact electron factorization (EEF) -- one of the branches of the exact factorization approach to many-body systems. In the EEF, the Schr\"odinger equation for one electron, in the environment of all other electrons, is formulated. The influence of the environment is reflected in the potential vH{v}^{\mathrm{H}}vH, which represents the energy of the environment, and in a potential vG{v}^{\mathrm{G}}vG, which has a geometrical meaning. In this paper, we focus on vG{v}^{\mathrm{G}}vG and study its properties in detail. We investigate the geometric origin of vG{v}^{\mathrm{G}}vG as a metric measuring the change of the environment, exemplify how translation and scaling of the state of the environment are reflected in vG{v}^{\mathrm{G}}vG, and explain its shape for homo- and heteronuclear diatomic model systems. Based on the close connection between the EEF and density functional theory, we also use vG{v}^{\mathrm{G}}vG to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.