Using a general-order many-body Green's-function method for molecules, we illustrate numerically three pathological behaviors of the Feynman-Dyson diagrammatic perturbation expansion of one-particle many-body Green's functions as electron propagators. First, the perturbation expansion of the frequency-dependent self-energy is nonconvergent at the exact self-energy in wide domains of frequency. Second, the Dyson equation with an odd-order self-energy has a qualitatively wrong shape and, as a result, most of their satellite roots are complex and nonphysical. Third, the Dyson equation with an even-order self-energy has an exponentially increasing number of roots as the perturbation order is raised, which quickly exceeds the correct number of roots. Infinite partial summation of diagrams by vertex or edge modification exacerbates these problems. Not only does the nonconvergence render higher-order perturbation theories useless for satellite roots, but it also calls into question the validity of their combined use with the ans"{a}tze requiring the knowledge of all poles and residues. Such ans"{a}tze include the Galitskii-Migdal formula, self-consistent Green's-function methods, Luttinger-Ward functional, and some models of the algebraic diagrammatic construction.
{"title":"Failures of the Feynman-Dyson diagrammatic perturbation expansion of propagators","authors":"So Hirata, Ireneusz Grabowski, Rodney J. Bartlett","doi":"arxiv-2312.03157","DOIUrl":"https://doi.org/arxiv-2312.03157","url":null,"abstract":"Using a general-order many-body Green's-function method for molecules, we\u0000illustrate numerically three pathological behaviors of the Feynman-Dyson\u0000diagrammatic perturbation expansion of one-particle many-body Green's functions\u0000as electron propagators. First, the perturbation expansion of the\u0000frequency-dependent self-energy is nonconvergent at the exact self-energy in\u0000wide domains of frequency. Second, the Dyson equation with an odd-order\u0000self-energy has a qualitatively wrong shape and, as a result, most of their\u0000satellite roots are complex and nonphysical. Third, the Dyson equation with an\u0000even-order self-energy has an exponentially increasing number of roots as the\u0000perturbation order is raised, which quickly exceeds the correct number of\u0000roots. Infinite partial summation of diagrams by vertex or edge modification\u0000exacerbates these problems. Not only does the nonconvergence render\u0000higher-order perturbation theories useless for satellite roots, but it also\u0000calls into question the validity of their combined use with the ans\"{a}tze\u0000requiring the knowledge of all poles and residues. Such ans\"{a}tze include the\u0000Galitskii-Migdal formula, self-consistent Green's-function methods,\u0000Luttinger-Ward functional, and some models of the algebraic diagrammatic\u0000construction.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545964","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}
Giovanni Scala, Anindita Bera, Gniewomir Sarbicki, Dariusz Chruściński
A family of linear positive maps in the algebra of $3 times 3$ complex matrices proposed recently in Bera et al. arXiv:2212.03807 is further analyzed. It provides a generalization of a seminal Choi nondecomposable extremal map in $M_3$. We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.
进一步分析了Bera et al. arXiv:2212.03807最近提出的$3 × 3$复矩阵代数中的一类线性正映射。它提供了在$M_3$中具有开创性的Choi不可分解极值映射的推广。我们研究了广义Choi映射何时是最优的，即不能表示为正和完全正映射的和。这种性质比极值性弱，然而，事实证明，它在探测量子纠缠方面起着关键作用。
{"title":"Optimality of generalized Choi maps in $M_3$","authors":"Giovanni Scala, Anindita Bera, Gniewomir Sarbicki, Dariusz Chruściński","doi":"arxiv-2312.02814","DOIUrl":"https://doi.org/arxiv-2312.02814","url":null,"abstract":"A family of linear positive maps in the algebra of $3 times 3$ complex\u0000matrices proposed recently in Bera et al. arXiv:2212.03807 is further analyzed.\u0000It provides a generalization of a seminal Choi nondecomposable extremal map in\u0000$M_3$. We investigate when generalized Choi maps are optimal, i.e. cannot be\u0000represented as a sum of positive and completely positive maps. This property is\u0000weaker than extremality, however, it turns out that it plays a key role in\u0000detecting quantum entanglement.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"111 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525275","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}
A computational method is proposed to calculate bound and resonant states by solving the Klein-Gordon and Dirac equations for real and complex energies, respectively. The method is an extension of a non-relativistic one, where the potential is represented in a Coulomb-Sturmian basis. This basis facilitates the exact analytic evaluation of the Coulomb Green's operator in terms of a continued fraction. In the extension to relativistic problems, we cast the Klein-Gordon and Dirac equations into an effective Schr"odinger form. Then the solution method is basically an analytic continuation of non-relativistic quantities like the angular momentum, charge, energy and potential into the effective relativistic counterparts.
{"title":"Calculation of Relativistic Single-Particle States","authors":"D. Wingard, B. Kónya, Z. Papp","doi":"arxiv-2312.02500","DOIUrl":"https://doi.org/arxiv-2312.02500","url":null,"abstract":"A computational method is proposed to calculate bound and resonant states by\u0000solving the Klein-Gordon and Dirac equations for real and complex energies,\u0000respectively. The method is an extension of a non-relativistic one, where the\u0000potential is represented in a Coulomb-Sturmian basis. This basis facilitates\u0000the exact analytic evaluation of the Coulomb Green's operator in terms of a\u0000continued fraction. In the extension to relativistic problems, we cast the\u0000Klein-Gordon and Dirac equations into an effective Schr\"odinger form. Then the\u0000solution method is basically an analytic continuation of non-relativistic\u0000quantities like the angular momentum, charge, energy and potential into the\u0000effective relativistic counterparts.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"7 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525276","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}
Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case it corresponds to having a time-dependent Hamiltonian. However, the quantum simulation of these systems often needs to appeal to rather complicated procedures involving the Dyson series, considerations of time-ordering, requirement of time steps to be discrete and/or requiring multiple measurements and postselection. These procedures are generally much more complicated than the quantum simulation of time-independent Hamiltonians. Here we propose an alternative formalism that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e., quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrodingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients.
{"title":"Quantum simulation for time-dependent Hamiltonians -- with applications to non-autonomous ordinary and partial differential equations","authors":"Yu Cao, Shi Jin, Nana Liu","doi":"arxiv-2312.02817","DOIUrl":"https://doi.org/arxiv-2312.02817","url":null,"abstract":"Non-autonomous dynamical systems appear in a very wide range of interesting\u0000applications, both in classical and quantum dynamics, where in the latter case\u0000it corresponds to having a time-dependent Hamiltonian. However, the quantum\u0000simulation of these systems often needs to appeal to rather complicated\u0000procedures involving the Dyson series, considerations of time-ordering,\u0000requirement of time steps to be discrete and/or requiring multiple measurements\u0000and postselection. These procedures are generally much more complicated than\u0000the quantum simulation of time-independent Hamiltonians. Here we propose an\u0000alternative formalism that turns any non-autonomous unitary dynamical system\u0000into an autonomous unitary system, i.e., quantum system with a time-independent\u0000Hamiltonian, in one higher dimension, while keeping time continuous. This makes\u0000the simulation with time-dependent Hamiltonians not much more difficult than\u0000that of time-independent Hamiltonians, and can also be framed in terms of an\u0000analogue quantum system evolving continuously in time. We show how our new\u0000quantum protocol for time-dependent Hamiltonians can be performed in a\u0000resource-efficient way and without measurements, and can be made possible on\u0000either continuous-variable, qubit or hybrid systems. Combined with a technique\u0000called Schrodingerisation, this dilation technique can be applied to the\u0000quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain\u0000nonlinear PDEs, with time-dependent coefficients.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"48 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525264","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}
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, Josip Žubrinić
In this article, we study the dynamic variant and the equilibrium variant of the model of Rydberg atoms on a square ladder. In the dynamic case, we obtain the jamming limit for all values of $b ge 1$, where $b$ represents the so-called blockade range of a Rydberg atom. In the equilibrium case, we derive the complexity function for all values of $b ge 1$. By comparing these results, we highlight significant differences in the behavior of the two models as $b$ approaches infinity.
{"title":"Rydberg atoms on a ladder","authors":"Tomislav Došlić, Mate Puljiz, Stjepan Šebek, Josip Žubrinić","doi":"arxiv-2312.02747","DOIUrl":"https://doi.org/arxiv-2312.02747","url":null,"abstract":"In this article, we study the dynamic variant and the equilibrium variant of\u0000the model of Rydberg atoms on a square ladder. In the dynamic case, we obtain\u0000the jamming limit for all values of $b ge 1$, where $b$ represents the\u0000so-called blockade range of a Rydberg atom. In the equilibrium case, we derive\u0000the complexity function for all values of $b ge 1$. By comparing these\u0000results, we highlight significant differences in the behavior of the two models\u0000as $b$ approaches infinity.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"38 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138521804","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 study the upper tail large deviation for the one-dimensional frog model. In this model, sleeping and active frogs are assigned to vertices on $mathbb Z$. While sleeping frogs do not move, the active ones move as independent simple random walks and activate any sleeping frogs. The main object of interest in this model is the asymptotic behavior of the first passage time ${rm T}(0,n)$, which is the time needed to activate the frog at the vertex $n$, assuming there is only one active frog at $0$ at the beginning. While the law of large numbers and central limit theorems have been well established, the intricacies of large deviations remain elusive. Using renewal theory, B'erard and Ram'irez have pointed out a slowdown phenomenon where the probability that the first passage time ${rm T}(0,n)$ is significantly larger than its expectation decays sub-exponentially and lies between $exp(-n^{1/2+o(1)})$ and $exp(-n^{1/3+o(1)})$. In this article, using a novel covering process approach, we confirm that $1/2$ is the correct exponent, i.e., the rate of upper large deviations is given by $n^{1/2}$. Moreover, we obtain an explicit rate function that is characterized by properties of Brownian motion and is strictly concave.
{"title":"Upper tail large deviation for the one-dimensional frog model","authors":"Van Hao Can, Naoki Kubota, Shuta Nakajima","doi":"arxiv-2312.02745","DOIUrl":"https://doi.org/arxiv-2312.02745","url":null,"abstract":"In this paper, we study the upper tail large deviation for the\u0000one-dimensional frog model. In this model, sleeping and active frogs are\u0000assigned to vertices on $mathbb Z$. While sleeping frogs do not move, the\u0000active ones move as independent simple random walks and activate any sleeping\u0000frogs. The main object of interest in this model is the asymptotic behavior of\u0000the first passage time ${rm T}(0,n)$, which is the time needed to activate the\u0000frog at the vertex $n$, assuming there is only one active frog at $0$ at the\u0000beginning. While the law of large numbers and central limit theorems have been\u0000well established, the intricacies of large deviations remain elusive. Using\u0000renewal theory, B'erard and Ram'irez have pointed out a slowdown phenomenon\u0000where the probability that the first passage time ${rm T}(0,n)$ is\u0000significantly larger than its expectation decays sub-exponentially and lies\u0000between $exp(-n^{1/2+o(1)})$ and $exp(-n^{1/3+o(1)})$. In this article, using\u0000a novel covering process approach, we confirm that $1/2$ is the correct\u0000exponent, i.e., the rate of upper large deviations is given by $n^{1/2}$.\u0000Moreover, we obtain an explicit rate function that is characterized by\u0000properties of Brownian motion and is strictly concave.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"94 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138521805","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 study Brownian paths perturbed by semibounded pair potentials and prove upper bounds on the mean square displacement. As a technical tool we derive infinite dimensional versions of key inequalities that were first used in [Sellke; arXiv:2212.14023] in order to study the effective mass of the Fr"ohlich polaron.
{"title":"Mean square displacement of Brownian paths perturbed by bounded pair potentials","authors":"Volker Betz, Tobias Schmidt, Mark Sellke","doi":"arxiv-2312.02709","DOIUrl":"https://doi.org/arxiv-2312.02709","url":null,"abstract":"We study Brownian paths perturbed by semibounded pair potentials and prove\u0000upper bounds on the mean square displacement. As a technical tool we derive\u0000infinite dimensional versions of key inequalities that were first used in\u0000[Sellke; arXiv:2212.14023] in order to study the effective mass of the\u0000Fr\"ohlich polaron.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138521807","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}
Rafał Bistroń, Michał Eckstein, Shmuel Friedland, Tomasz Miller, Karol Życzkowski
The complex projective space $mathbb{P}(mathbb{C}^n)$ can be interpreted as the space of all quantum pure states of size $n$. A distance on this space, interesting from the perspective of quantum physics, can be induced from a classical distance defined on the $n$-point probability simplex by the `earth mover problem'. We show that this construction leads to a quantity satisfying the triangle inequality, which yields a true distance on complex projective space belonging to the family of quantum $2$-Wasserstein distances.
{"title":"A new class of distances on complex projective spaces","authors":"Rafał Bistroń, Michał Eckstein, Shmuel Friedland, Tomasz Miller, Karol Życzkowski","doi":"arxiv-2312.02583","DOIUrl":"https://doi.org/arxiv-2312.02583","url":null,"abstract":"The complex projective space $mathbb{P}(mathbb{C}^n)$ can be interpreted as\u0000the space of all quantum pure states of size $n$. A distance on this space,\u0000interesting from the perspective of quantum physics, can be induced from a\u0000classical distance defined on the $n$-point probability simplex by the `earth\u0000mover problem'. We show that this construction leads to a quantity satisfying\u0000the triangle inequality, which yields a true distance on complex projective\u0000space belonging to the family of quantum $2$-Wasserstein distances.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"8 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525265","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 provide formulas calculating the partition functions of two types of plane partitions using the crystal melting model method introduced by Okounkov, Reshetikhin and Vafa. As applications, we obtain a product formula for the partition function of the plane partitions with a limit shape boundary. A corollary of this formula is the demonstration of the equivalence between this partition function and the open-closed string amplitude of the double$-mathbb{P}^1$ model. We also derive a product formula for the partition function of symmetric plane partitions with a limit shape boundary.
{"title":"A remark on certain restricted plane partitions and crystal melting model","authors":"Chenglang Yang","doi":"arxiv-2312.02749","DOIUrl":"https://doi.org/arxiv-2312.02749","url":null,"abstract":"In this paper, we provide formulas calculating the partition functions of two\u0000types of plane partitions using the crystal melting model method introduced by\u0000Okounkov, Reshetikhin and Vafa. As applications, we obtain a product formula\u0000for the partition function of the plane partitions with a limit shape boundary.\u0000A corollary of this formula is the demonstration of the equivalence between\u0000this partition function and the open-closed string amplitude of the\u0000double$-mathbb{P}^1$ model. We also derive a product formula for the partition\u0000function of symmetric plane partitions with a limit shape boundary.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"49 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525271","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 investigate new integrable extensions of two-center Coulomb systems. We study the most general $n$-dimensional deformation of the two-center problem by adding arbitrary functions supporting second order commuting conserved quantities. The system is superintegrable for $n>4$ and, for certain choices of the arbitrary functions, reduces to known models previously discovered. Then, based on this extended system, we introduce an additional integrable generalisation involving Calogero interactions for $n=3$. In all examples, including the two-center problem, we explicitly present the complete list of Liouville integrals in terms of second-order integrals of motion.
{"title":"Integrable extensions of two-center Coulomb systems","authors":"Francisco Correa, Octavio Quintana","doi":"arxiv-2312.02013","DOIUrl":"https://doi.org/arxiv-2312.02013","url":null,"abstract":"In this paper, we investigate new integrable extensions of two-center Coulomb\u0000systems. We study the most general $n$-dimensional deformation of the\u0000two-center problem by adding arbitrary functions supporting second order\u0000commuting conserved quantities. The system is superintegrable for $n>4$ and,\u0000for certain choices of the arbitrary functions, reduces to known models\u0000previously discovered. Then, based on this extended system, we introduce an\u0000additional integrable generalisation involving Calogero interactions for $n=3$.\u0000In all examples, including the two-center problem, we explicitly present the\u0000complete list of Liouville integrals in terms of second-order integrals of\u0000motion.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"41 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138521726","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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