Quantum monotone metric was introduced by Petz,and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always one based on completely positive, trace preserving (CPTP) maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators,and did not have some ideal properties such as monotonicity and convexity with respect to the positive operators. In this paper, we introduce another extension of quantum monotone metrics which have monotonicity under completely positive, trace non-increasing (CPTNI) maps and additive noise. We prove that our extended monotone metrics can be characterized only by static operator monotone functions from few assumptions without assuming continuities of metrics. We show that our monotone metrics have some natural properties such as additivity of direct sum, convexity and monotonicity with respect to positive operators.
{"title":"Quantum monotone metrics induced from trace non-increasing maps and additive noise","authors":"Koichi Yamagata","doi":"10.1063/1.5129058","DOIUrl":"https://doi.org/10.1063/1.5129058","url":null,"abstract":"Quantum monotone metric was introduced by Petz,and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always one based on completely positive, trace preserving (CPTP) maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators,and did not have some ideal properties such as monotonicity and convexity with respect to the positive operators. In this paper, we introduce another extension of quantum monotone metrics which have monotonicity under completely positive, trace non-increasing (CPTNI) maps and additive noise. We prove that our extended monotone metrics can be characterized only by static operator monotone functions from few assumptions without assuming continuities of metrics. We show that our monotone metrics have some natural properties such as additivity of direct sum, convexity and monotonicity with respect to positive operators.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84617640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-04DOI: 10.1142/S0217732321501406
I. Gomez, E. S. Santos, O. Abla
In this work we explore a generalization of the Dirac and Klein-Gordon (KG) oscillators, provided with a deformed linear momentum inspired in nonextensive statistics, that gives place to the Morse potential in relativistic contexts by first principles. In the (1+1)-dimensional case the relativistic oscillators are mapped into the quantum Morse potential. Using the Pekeris approximation, in the (3+1)-dimensional case we study the thermodynamics of the S-waves states (l=0) of the H2, LiH, HCl and CO molecules (in the non-relativistic limit) and of a relativistic electron, where Schottky anomalies (due to the finiteness of the Morse spectrum) and spin contributions to the heat capacity are reported. By revisiting a generalized Pekeris approximation, we provide a mapping from (3+1)-dimensional Dirac and KG equations with a spherical potential to an associated one-dimensional Schr"odinger-like equation, and we obtain the family of potentials for which this mapping corresponds to a Schr"odinger equation with non-minimal coupling.
{"title":"Morse potential in relativistic contexts from generalized momentum operator: Schottky anomalies, Pekeris approximation and mapping","authors":"I. Gomez, E. S. Santos, O. Abla","doi":"10.1142/S0217732321501406","DOIUrl":"https://doi.org/10.1142/S0217732321501406","url":null,"abstract":"In this work we explore a generalization of the Dirac and Klein-Gordon (KG) oscillators, provided with a deformed linear momentum inspired in nonextensive statistics, that gives place to the Morse potential in relativistic contexts by first principles. In the (1+1)-dimensional case the relativistic oscillators are mapped into the quantum Morse potential. Using the Pekeris approximation, in the (3+1)-dimensional case we study the thermodynamics of the S-waves states (l=0) of the H2, LiH, HCl and CO molecules (in the non-relativistic limit) and of a relativistic electron, where Schottky anomalies (due to the finiteness of the Morse spectrum) and spin contributions to the heat capacity are reported. By revisiting a generalized Pekeris approximation, we provide a mapping from (3+1)-dimensional Dirac and KG equations with a spherical potential to an associated one-dimensional Schr\"odinger-like equation, and we obtain the family of potentials for which this mapping corresponds to a Schr\"odinger equation with non-minimal coupling.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"115 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88795477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-04DOI: 10.21468/SCIPOSTPHYS.10.1.006
G. Niccoli, Hao Pei, V. Terras
We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.
{"title":"Correlation functions by separation of variables: the XXX spin chain","authors":"G. Niccoli, Hao Pei, V. Terras","doi":"10.21468/SCIPOSTPHYS.10.1.006","DOIUrl":"https://doi.org/10.21468/SCIPOSTPHYS.10.1.006","url":null,"abstract":"We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84036861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we are interested in understanding the impact of surface defects on a condensate of electron pairs in a quantum wire. Based on previous results we establish a simple mathematical model in order to account for such surface effects. For a system of non-interacting pairs, we will prove the destruction of the condensate in the bulk. Finally, taking repulsive interactions between the pairs into account, we will show that the condensate is recovered for pair densities larger than a critical one given the number of the surface defects is not too large.
{"title":"О влиянии поверхностных дефектов на конденсат электронных пар в квантовом проводе","authors":"Иоахим Кернер, Joachim Kerner","doi":"10.4213/tmf9720","DOIUrl":"https://doi.org/10.4213/tmf9720","url":null,"abstract":"In this paper we are interested in understanding the impact of surface defects on a condensate of electron pairs in a quantum wire. Based on previous results we establish a simple mathematical model in order to account for such surface effects. For a system of non-interacting pairs, we will prove the destruction of the condensate in the bulk. Finally, taking repulsive interactions between the pairs into account, we will show that the condensate is recovered for pair densities larger than a critical one given the number of the surface defects is not too large.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"55 1","pages":"300-310"},"PeriodicalIF":0.0,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80433258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Superconductivity in the presence of a step magnetic field has been recently the focus of many works. This contribution examines the behavior of a two-dimensional superconducting domain, when superconductivity is lost in the whole domain except near the intersection points of the discontinuity edge and the boundary. The problem involves its own effective energy. We provide local estimates of the minimizers in neighbourhoods of the intersection points. Consequently, we introduce new critical fields marking the loss of superconductivity in the vicinity of these points. The study is modelled by the Ginzburg--Landau theory, and large Ginzburg--Landau parameters are considered.
{"title":"Magnetic steps on the threshold of the normal state","authors":"W. Assaad","doi":"10.1063/5.0012725","DOIUrl":"https://doi.org/10.1063/5.0012725","url":null,"abstract":"Superconductivity in the presence of a step magnetic field has been recently the focus of many works. This contribution examines the behavior of a two-dimensional superconducting domain, when superconductivity is lost in the whole domain except near the intersection points of the discontinuity edge and the boundary. The problem involves its own effective energy. We provide local estimates of the minimizers in neighbourhoods of the intersection points. Consequently, we introduce new critical fields marking the loss of superconductivity in the vicinity of these points. The study is modelled by the Ginzburg--Landau theory, and large Ginzburg--Landau parameters are considered.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85251889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The notion of spectral localizer is extended to pairings with semifinite spectral triples. By a spectral flow argument, any semifinite index pairing is shown to be equal to the signature of the spectral localizer. As an application, a formula for the weak invariants of topological insulators is derived. This provides a new approach to their numerical evaluation.
{"title":"The spectral localizer for semifinite spectral triples","authors":"H. Schulz-Baldes, T. Stoiber","doi":"10.1090/proc/15230","DOIUrl":"https://doi.org/10.1090/proc/15230","url":null,"abstract":"The notion of spectral localizer is extended to pairings with semifinite spectral triples. By a spectral flow argument, any semifinite index pairing is shown to be equal to the signature of the spectral localizer. As an application, a formula for the weak invariants of topological insulators is derived. This provides a new approach to their numerical evaluation.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85224542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-18DOI: 10.1016/j.aop.2020.168157
Andres D. Bermudez Manjarres, M. Nowakowski, D. Batic
{"title":"Classical dynamics from a unitary representation of the Galilei group","authors":"Andres D. Bermudez Manjarres, M. Nowakowski, D. Batic","doi":"10.1016/j.aop.2020.168157","DOIUrl":"https://doi.org/10.1016/j.aop.2020.168157","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88301397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
With using the algebraic approach Lie symmetries of Schrodinger equations with matrix potentials are classified. Thirty three inequivalent equations of such type together with the related symmetry groups are specified, the admissible equivalence relations are clearly indicated. In particular the Boyer results concerning kinematical invariance groups for arbitrary potentials (C. P. Boyer, Helv. Phys. Acta, {bf 47}, 450--605 (1974)) are clarified and corrected.
利用代数方法对具有矩阵势的薛定谔方程的李对称性进行了分类。给出了33个此类不等价方程及其相关对称群，并明确指出了可容许的等价关系。特别是关于任意势的运动不变性群的Boyer结果(C. P. Boyer, Helv.)。理论物理。《学报》，{bf 47}， 450—605(1974))作了澄清和更正。
{"title":"Symmetries of the Schrödinger–Pauli equation for neutral particles","authors":"A. Nikitin","doi":"10.1063/5.0021725","DOIUrl":"https://doi.org/10.1063/5.0021725","url":null,"abstract":"With using the algebraic approach Lie symmetries of Schrodinger equations with matrix potentials are classified. Thirty three inequivalent equations of such type together with the related symmetry groups are specified, the admissible equivalence relations are clearly indicated. In particular the Boyer results concerning kinematical invariance groups for arbitrary potentials (C. P. Boyer, Helv. Phys. Acta, {bf 47}, 450--605 (1974)) are clarified and corrected.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84485725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine a completely positive and trace preserving evolution of finite dimensional open quantum system, coupled to large environment via periodically modulated interaction Hamiltonian. We derive a corresponding Markovian Master Equation under usual assumption of weak coupling using the projection operator techniques, in two opposite regimes of very small and very large modulation frequency. Special attention is granted to the case of uniformly (globally) modulated interaction, where some general results concerning the Floquet normal form of a solution and its asymptotic stability are also addressed.
{"title":"Markovian dynamics under weak periodic coupling","authors":"K. Szczygielski","doi":"10.1063/5.0014078","DOIUrl":"https://doi.org/10.1063/5.0014078","url":null,"abstract":"We examine a completely positive and trace preserving evolution of finite dimensional open quantum system, coupled to large environment via periodically modulated interaction Hamiltonian. We derive a corresponding Markovian Master Equation under usual assumption of weak coupling using the projection operator techniques, in two opposite regimes of very small and very large modulation frequency. Special attention is granted to the case of uniformly (globally) modulated interaction, where some general results concerning the Floquet normal form of a solution and its asymptotic stability are also addressed.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"126 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87830776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The term "particle-hole symmetry" is beset with conflicting meanings in contemporary physics. Conceived and written from a condensed-matter standpoint, the present paper aims to clarify and sharpen the terminology. In that vein, we propose to define the operation of "particle-hole conjugation" as the tautological algebra automorphism that simply swaps single-fermion creation and annihilation operators, and we construct its invariant lift to the Fock space. Particle-hole symmetries then arise for gapful or gapless free-fermion systems at half filling, as the concatenation of particle-hole conjugation with one or another involution that reverses the sign of the first-quantized Hamiltonian. We illustrate that construction principle with a series of examples including the Su-Schrieffer-Heeger model and the Kitaev-Majorana chain. For an enhanced perspective, we contrast particle-hole symmetries with the charge-conjugation symmetry of relativistic Dirac fermions. We go on to present two major applications in the realm of interacting electrons. For one, we argue that the celebrated Haldane phase of antiferromagnetic quantum spin chains is adiabatically connected to a free-fermion topological phase protected by a particle-hole symmetry. For another, we review the recent proposal by Son for a particle-hole conjugation symmetric effective field theory of the half-filled lowest Landau level, and we comment on the emerging microscopic picture of the composite fermion.
{"title":"Particle–hole symmetries in condensed matter","authors":"M. Zirnbauer","doi":"10.1063/5.0035358","DOIUrl":"https://doi.org/10.1063/5.0035358","url":null,"abstract":"The term \"particle-hole symmetry\" is beset with conflicting meanings in contemporary physics. Conceived and written from a condensed-matter standpoint, the present paper aims to clarify and sharpen the terminology. In that vein, we propose to define the operation of \"particle-hole conjugation\" as the tautological algebra automorphism that simply swaps single-fermion creation and annihilation operators, and we construct its invariant lift to the Fock space. Particle-hole symmetries then arise for gapful or gapless free-fermion systems at half filling, as the concatenation of particle-hole conjugation with one or another involution that reverses the sign of the first-quantized Hamiltonian. We illustrate that construction principle with a series of examples including the Su-Schrieffer-Heeger model and the Kitaev-Majorana chain. For an enhanced perspective, we contrast particle-hole symmetries with the charge-conjugation symmetry of relativistic Dirac fermions. We go on to present two major applications in the realm of interacting electrons. For one, we argue that the celebrated Haldane phase of antiferromagnetic quantum spin chains is adiabatically connected to a free-fermion topological phase protected by a particle-hole symmetry. For another, we review the recent proposal by Son for a particle-hole conjugation symmetric effective field theory of the half-filled lowest Landau level, and we comment on the emerging microscopic picture of the composite fermion.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86252378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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