Volker Betz, Fumio Hiroshima, Jozsef Lorinczi, Robert A. Minlos, Herbert Spohn
We study properties of the ground state of the Nelson model through functional integration. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.
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The Nelson Hamiltonian is unitarily equivalent to a Hamiltonian defined through a closed, semibounded quadratic form, the unitary transformation being explicitly known and due to Gross. In this paper we study mapping properties of the Gross-transform in order to characterize regularity properties of vectors in the form domain of the Nelson Hamiltonian. Since the operator domain is a subset of the form domain, our results apply to vectors in the domain of the Hamiltonian was well. - This work is a continuation of our previous work on the Fröhlich Hamiltonian.
Let us consider the renormalized Nelson model at a fixed total momentum $P$: $H_{\mathrm{ren}}(P)$; The Hamiltonian $H_{\mathrm{ren}}(P)$ is defined through an infinite energy renormalization. We prove that $e^{-βH_{\mathrm{ren}}(P)}$ is positivity improving for all $P\in \mathbb{R}^3$ and $β>0$ in the Fock representation.
We construct Hamiltonians for systems of nonrelativistic particles linearly coupled to massive scalar bosons using abstract boundary conditions. The construction yields an explicit characterisation of the domain of self-adjointness in terms of boundary conditions that relate sectors with different numbers of bosons. We treat both models in which the Hamiltonian may be defined as a form perturbation of the free operator, such as Fröhlich's polaron, and renormalisable models, such as the massive Nelson model.
The Nelson model describes the interaction of nonrelativistic quantum particles with a relativistic quantum field of scalar bosons. Nelson rigorously demonstrated in 1964 the existence of a well-defined self-adjoint Nelson Hamiltonian by renormalisation. Recently, a Fock space description of the renormalised Nelson Hamiltonian and its domain were found based on a novel approach to defining Hamiltonians in quantum field theories. This novel approach involves an interior boundary condition (IBC) on the state vectors, which relates the value of the wave function on the boundary of the configurati…
We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in the non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, t…
In the theory of point interactions, one is given a formal expression for a quantum mechanical Hamiltonian. The interaction terms of the Hamiltonian are singular: they can not be rigorously defined as a perturbation (in the operator or form sense) of an unperturbed free operator. A similar situation occurs in Quantum Field Theory, where it is known as the ultraviolet problem. Recently, it was shown that some of the tools used in the context of point interactions can be adapted to solve the problem of directly defining a Hamiltonian for the Nelson model. This model provides a well studied examp…
Abstract interior-boundary conditions (IBC's) allow for the direct description of the domain and the action of Hamiltonians for a certain class of ultraviolet-divergent models in Quantum Field Theory. The method was recently applied to models where nonrelativistic scalar particles are linearly coupled to a quantised field, the best known of which is the Nelson model. Since this approach avoids the use of ultraviolet-cutoffs, there is no need for a renormalisation procedure. Here, we extend the IBC method to pseudorelativistic scalar particles that interact with a real bosonic field. We constru…
We prove that the resolvent of the renormalised Nelson Hamiltonian at fixed total momentum P improves positivity in the (momentum) Fock-representation, for every P. Our argument is based on an explcit representation of the renormalised operator and its domain using interior boundary conditions.
We prove that Nelson's massless scalar field model is infrared divergent in three dimensions. In particular, the Nelson Hamiltonian and the Hamiltonian obtained from Euclidean quantization are not unitarily equivalent. In contrast, for dimensions higher than three the Nelson Hamiltonian has a unique ground state in Fock space and the two Hamiltonians are unitarily equivalent. We also show that the Euclidean Hamiltonian has no spectral gap.
Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions
本文研究了二维相对论Nelson模型中纤维哈密顿量的费曼-卡茨公式，通过能量重整化定义哈密顿量，并推导了新的费曼-卡茨公式，提供了另一种全译不变哈密顿量的推导方法。
Ultraviolet (UV) renormalization of the Nelson model in quantum field theory is considered. A relationship between a ultraviolet renormalization term and the ground state energy of the Hamiltonian with total momentum zero is studied by functional integrations.
Starting from the N-particle Nelson Hamiltonian defined by imposing an ultraviolet cutoff, we perform ultraviolet renormalization by showing that in the zero cutoff limit a self-adjoint operator exists after a logarithmically divergent term is subtracted from the original Hamiltonian. We obtain this term as the diagonal part of a pair interaction appearing in the density of a Gibbs measure derived from the Feynman-Kac representation of the Hamiltonian. Also, we show existence of a weak coupling limit of the renormalized Hamiltonian and derive an effective Yukawa interaction potential between t…
In this note we study a class of specifications over $d$-dimensional Wiener measure which are invariant under uniform translation of the paths. This degeneracy is removed by restricting the measure to the $σ$-algebra generated by the increments of the coordinate process. We address the problem of existence and uniqueness of Gibbs measures and prove a central limit theorem for the rescaled increments. These results apply to the study of the ground state of the Nelson model of a quantum particle interacting with a scalar boson field.
We propose an abstract framework describing energy-renormalized Hamiltonians in terms of local algebras. Within the framework, we examine the positivity improvingness of the semigroup generated by the renormalized Hamiltonian. As examples, we discuss the renormalized Nelson Hamiltonian and the renormalized Nelson Hamiltonian at fixed total momentum. The characteristic features of our approach are as follows:(i) in contrast to the probabilistic approach in the Schrödinger representation, our method works well in the Fock representation; (ii) the method covers the massless case.
We give a formula in terms of a joint Gibbs measure on Brownian paths and the measure of a random-time Poisson process of the ground state expectations of fractional (in fact, any real) powers of the boson number operator in the Nelson model. We use this representation to obtain tight two-sided bounds. As applications, we discuss the polaron and translation invariant Nelson models.
We consider the Nelson model on some static space-times and investigate the problem of absence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the absence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass $m(x)$ tends to $0$ at spatial infinity. Using path space techniques, we show that if $m(x)\leq C |x|^{-μ}$ at infinity for some $C>0$ and $μ>1$ then the Nels…
On the Ergodicity of Renormalized Translation-Invariant Nelson-Type Semigroups
该研究提出基于泛函积分的简洁证明，论证紫外重整化的平移不变非相对论与半相对论Nelson哈密顿量生成的半群对任意总动量关于Fröhlich锥正性改进即遍历，简化了已有证明且半相对论结果为新发现。
We consider the Nelson model with variable coefficients and investigate the problem of existence of a ground state and the removal of the ultraviolet cutoff. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. A physical example is obtained by quantizing the Klein-Gordon equation on a static space-time coupled with a non-relativistic particle. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assum…
We prove that in the Euclidean representation of the three dimensional massless Nelson model the t = 0 projection of the interacting measure is absolutely continuous with respect to a Gaussian measure with suitably adjusted mean. We also determine the Hamiltonian in the Fock space over this Gaussian measure space.
Time-dependent scattering theory for a large class of translation invariant models, including the Nelson and Polaron models, restricted to the vacuum and one-particle sectors is studied. We formulate and prove asymptotic completeness for these models. The translation invariance imply that the Hamiltonians considered are fibered with respect to the total momentum. On the way to asymptotic completeness we determine the spectral structure of the fiber Hamiltonians, establish a Mourre estimate and derive a geometric asymptotic completeness statement as an intermediate step.
We mathematically study the infrared catastrophe for the Hamiltonian of Nelson's model when it has the external potential in a general class. For the model, we prove the pull-through formula on ground states in operator theory first. Based on this formula, we show both non-existence of any ground state and divergence of the total number of soft bosons.
Two-sided bounds on the point-wise spatial decay of ground states in the renormalized Nelson model with confining potentials
本文研究重整化Nelson模型基态的逐点空间衰减，利用Feynman-Kac表示推导下界，并证明其与先前上界渐近匹配。
We provide explicit lower bounds for the ground-state energy of the renormalized Nelson model in terms of the coupling constant $α$ and the number of particles $N$, uniform in the meson mass and valid even in the massless case. In particular, for any number of particles $N$ and large enough $α$ we provide a bound of the form $-Cα^2 N^3\log^2(αN)$, where $C$ is an explicit positive numerical constant; and if $α$ is sufficiently small, we give one of the form $-Cα^2 N^3\log^2 N$ for $N \geq 2$, and $-Cα^2$ for $N = 1$. Whereas it is known that the renormalized Hamiltonian of the Nelson model is …
For the massless Nelson model we provide detailed information about the dependence of the normalized ground states $\checkψ_{P,σ}$ of the fiber single-electron Hamiltonians $H_{P,σ}$ on the total momentum $P$ and the infrared cut-off $σ$. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form \[ \|\partial_{P_i}\checkψ_{P,σ}\|,\ \| \partial_{P_i} \partial_{P_j}\checkψ_{P,σ} \|\leq \frac{c}{σ^{δ_{λ_0}}}, \] for some constant $c$ and a function of the maximal admissible coupling constant $λ_0\mapsto δ_{λ_0}$ s.t. $\li…
The translation invariant model in quantum field theory is considered by functional integrations. Ultraviolet renormalization of the translation invariant Nelson model with a fixed total momentum is proven by functional integrations. As a corollary it can be shown that the Nelson Hamiltonian with zero total momentum has a ground state for arbitrary values of coupling constants in two dimension. Furthermore the ultraviolet renormalization of the polaron model is also studied.
It is known that a linear hamiltonian system has too many invariant measures, thus the problem of convergence to Gibbs measure has no sense. We consider linear hamiltonian systems of arbitrary finite dimension and prove that, under the condition that one distinguished coordinate is subjected to dissipation and white noise, then, for almost any hamiltonians and almost any initial conditions, there exists the unique limiting distribution. Moreover, this distribution is Gibbsian with the temperature depending on the dissipation and of the variance of the white noise.
A remark on Gibbs measures with log-correlated Gaussian fields
本文利用变分公式证明了在 $d$ 维环面上具有对数相关高斯场和四次相互作用的聚焦Gibbs测度的不可正规化性，给出了发散速率，并探讨了三次相互作用及二维Zakharov系统的相关性质。
We study the limit of large volume equilibrium Gibbs measures for a rather general Hamiltonians. In particular we study Hamiltonians which arise in naturally in Nonlinear Elasticity and Hamiltonians (containing surface terms) which arises naturally in Fracture Mechanics. In both of these settings we show that an integral representation holds for the limit. Moreover, we also show a homogenization result for the Nonlinear Elasticity setting. This extends a recent result of R. Kotecký and S. Luckhaus \cite{LK-comm-math-phys}.
Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians
该研究证明在常温下，通过O(1)-局部哈密顿量的吉布斯态采样，量子计算机仍能实现超多项式加速。
We investigate, in a fairly general setting, the limit of large volume equilibrium Gibbs measures for elasticity type Hamiltonians with clamped boundary conditions. The existence of a quasiconvex free energy, forming the large deviations rate functional, is shown using a new interpolation lemma for partition functions. The local behaviour of the Gibbs measures can be parametrized by Young measures on the space of gradient Gibbs measures. In view of unboundedness of the state space, the crucial tool here is an exponential tightness estimate that holds for a vast class of potentials and the cons…
Mixing time of quantum Gibbs sampling for random sparse Hamiltonians
针对随机稀疏哈密顿量，在任意恒定温度下建立了量子吉布斯采样算法混合时间的polylog(n)上界，并分析了跳跃算子与谱性质的影响，使该方法与高效低能态制备算法相当。
Let $H_{P,σ}$ be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum $P$ and infrared cut-off $σ>0$. We establish detailed regularity properties of the corresponding $n$-particle ground state wave functions $f^n_{P,σ}$ as functions of $P$ and $σ$. In particular, we show that \[ |\partial_{P^j}f^{n}_{P,σ}(k_1,\ldots, k_n)|, \ \ |\partial_{P^j} \partial_{P^{j'}} f^{n}_{P,σ}(k_1,\ldots, k_n)| \leq \frac{1}{\sqrt{n!}} \frac{(cλ_0)^n}{σ^{δ_{λ_0}}} \prod_{i=1}^n\frac{ χ_{[σ,κ)}(k_i)}{|k_i|^{3/2}}, \] where $c$ is a numerical constant, $λ_0\mapsto δ_{λ_0}$ is a posit…
In the present paper and the companion paper [9] a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on a complex algebraic varieties X is introduced, by sampling "temperature deformed" determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kahler-Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point pro…
We consider general hamiltonian systems with quadratic interaction potential and $N<\infty$ degrees of freedom, only $m$ of which have contact with external world, that is subjected to damping and random stationary external forces. We show that, as $t\to\infty$, already for $m=1$, the unique limiting distribution exists for almost all interactions. Moreover, it is Gibbs if the external force is the white noise, but typically not Gibbs for gaussian processes with smooth trajectories. This conclusion survives also in the thermodynamic limit $N\to\infty$.
The thermal equilibrium properties of physical systems can be described using Gibbs states. It is therefore of great interest to know when such states allow for an easy description. In particular, this is the case if correlations between distant regions are small. In this work, we consider 1D quantum spin systems with local, finite-range, translation-invariant interactions at any temperature. In this setting, we show that Gibbs states satisfy uniform exponential decay of correlations and, moreover, the mutual information between two regions decays exponentially with their distance, irrespectiv…
The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. The mai…
In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some prelimi…
An efficient and exact noncommutative quantum Gibbs sampler
本文针对任意非交换哈密顿量，构造了首个可高效实现且满足精确细致平衡的Lindbladian，用于制备量子Gibbs态，其演化时间正比于混合时间和逆温度β，并显著降低了格点哈密顿量的门复杂度，为经典MCMC采样提供了量子算法对应物。
We suggest a possible approach to proving the Mézard-Parisi formula for the free energy in the diluted spin glass models, such as diluted K-spin or random K-sat model at any positive temperature. In the main contribution of the paper, we show that a certain small modification of the Hamiltonian in any of these models forces all finite-RSB asymptotic Gibbs measures in the sense of the overlaps to satisfy the Mézard-Parisi ansatz for the distribution of spins. Unfortunately, what is still missing is a description of the general full-RSB asymptotic Gibbs measures. If one could show that the gener…
In the present paper the Ising model with competing binary ($J$) and binary ($J_1$) interactions with spin values $\pm 1$, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model considered is studied. We completely describe the set of all periodic Gibbs measures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the…
Efficient thermalization and universal quantum computing with quantum Gibbs samplers
本文证明了量子吉布斯采样器在高温下能高效热化至吉布斯态，并展示了其在低温下与多项式时间量子计算的等价性，为量子许多体系统的模拟提供了新方法。
Sharp universal death of entanglement threshold for Pauli Hamiltonians
本文确定了有界度 Pauli 哈密顿量高温可分离性的精确普适阈值，并证明阈值之上存在纠缠 Gibbs 态，阈值之下可经典高效采样逼近。
We analyze the problem of preparing quantum Gibbs states of lattice spin Hamiltonians with local and commuting terms on a quantum computer and in nature. Our central result is an equivalence between the behavior of correlations in the Gibbs state and the mixing time of the semigroup which drives the system to thermal equilibrium (the Gibbs sampler). We introduce a framework for analyzing the correlation and mixing characteristics of quantum Gibbs states and quantum Gibbs samplers, which is rooted in the theory of non-commutative Lp spaces. We consider two distinct classes of Gibbs samplers, on…
Code Swendsen-Wang Dynamics
提出一种新的马尔可夫链——Code Swendsen-Wang动力学，通过全局更新为任意码哈密顿量制备吉布斯态，解决了4D环面码的快速混合问题，并在一级相变处达到基本障碍。
Strong decay of correlations for Gibbs states in any dimension
本文证明任意维度下，若量子系统Gibbs态的每个边缘态具有短程局部有效哈密顿量，则满足强混合条件，导致远距离区域间相关性（如互信息）指数衰减，并给出对易哈密顿量情形下的充分条件。
Quantum Gibbs Sampling in Infinite Dimensions: Generation, Mixing Times and Circuit Implementation
本文提出一种严格且可实现的框架，用于无限维量子系统的吉布斯采样，解决了无限维中 dissipative 吉布斯采样面临的核心问题，通过构造 KMS 对称的量子马尔可夫半群，实现了高效且可靠的采样方法。
We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. We study periodic Gibbs measures of the model with period two. For $k=1$ we show that there is no any periodic Gibbs measure. In case $k\geq 2$ we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model have not any periodic Gibbs measure with period two. We construct several models which have at least two periodic Gibbs measures.
Quantum Markov networks are a generalization of quantum Markov chains to arbitrary graphs. They provide a powerful classification of correlations in quantum many-body systems---complementing the area law at finite temperature---and are therefore useful to understand the powers and limitations of certain classes of simulation algorithms. Here, we extend the characterization of quantum Markov networks and in particular prove the equivalence of positive quantum Markov networks and Gibbs states of Hamiltonians that are the sum of local commuting terms on graphs containing no triangles. For more ge…
Cellular Automata are discrete-time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata (PCA), are discrete time Markov chains on lattice with finite single-cell states whose distinguishing feature is the \emph{parallel} character of the updating rule. We study the ground states of the Hamiltonian and the low-temperature phase diagram of the related Gibbs measure naturally associated with a class of reversible PCA, called the \textit{cross PCA}. In such a mod…
Reversible cellular automata are seen as microscopic physical models, and their states of macroscopic equilibrium are described using invariant probability measures. We establish a connection between the invariance of Gibbs measures and the conservation of additive quantities in surjective cellular automata. Namely, we show that the simplex of shift-invariant Gibbs measures associated to a Hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the Hamiltonian. A special case is the (well-known) invariance of the uniform Bernoulli measure …
Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains
提出多项式深度对偶变换概念，利用其将量子哈密顿量对偶到经典哈密顿量，从而构建高效吉布斯采样器，以二维环面码为例证明其对偶于两个解耦经典伊辛自旋链。
In this paper, we consider an Ising model with three competing interactions on a triangular chandelier-lattice (TCL). We describe the existence, uniqueness, and non-uniqueness of translation-invariant Gibbs measures associated with the Ising model. We obtain an explicit formula for Gibbs measures with a memory of length 2 satisfying consistency conditions. It is rigorously proved that the model exhibits phase transitions only for given values of the coupling constants. As a consequence of our approach, the dichotomy between alternative solutions of Hamiltonian models on TCLs is solved. Finally…
We present an elementary approach to concentration of disordered Hamiltonians. Assuming differentiability of the limiting free energy $F$ with respect to the inverse temperature $β$, we show that the Hamiltonian concentrates around the energy level $F'(β)$ under the free energy and Gibbs average.
Local quantum channels giving rise to quasi-local Gibbs states
本文研究由局域Kraus算符构成的量子信道稳态，证明其在一般条件下为准局域哈密顿量的吉布斯态，并给出高阶项指数衰减条件及经典计算算法。
Quantum Gibbs states are locally Markovian
本研究证明任意有界相互作用度哈密顿量的量子Gibbs态在任意温度下具有局部Markov性，通过细致平衡Lindbladian获得准局部恢复映射，由此导出条件互信息指数衰减并给出制备Gibbs态的量子电路深度上界。
We extend the notion of Gibbsianness for mean-field systems to the set-up of general (possibly continuous) local state spaces. We investigate the Gibbs properties of systems arising from an initial mean-field Gibbs measure by application of given local transition kernels. This generalizes previous case-studies made for spins taking finitely many values to the first step in direction to a general theory, containing the following parts: (1) A formula for the limiting conditional probability distributions of the transformed system. It holds both in the Gibbs and non-Gibbs regime and invokes a min…
Invariant Gibbs measure for Anderson nonlinear wave equation
本文通过随机最优控制构造Anderson高斯测度与高斯自由场的正则耦合，定义重整化幂，并证明四次Gibbs测度在带重整化三次非线性的波动方程下不变。
Gibbs Measures with Multilinear Forms
本文研究了由广义U统计量定义的多线性吉布斯测度，探讨了渐近自由能的优化条件，证明了复制对称性的必要性，并给出了局部场、哈密顿量等统计量的弱极限，同时揭示了温度参数下的相变点。
Slow Mixing of Quantum Gibbs Samplers
该研究提出量子瓶颈引理，证明多类低温哈密顿量的量子吉布斯采样器混合时间存在指数下界，将经典慢混合分析工具推广至量子系统。
Classical Estimation of the Free Energy and Quantum Gibbs Sampling from the Markov Entropy Decomposition
本文提出基于有效相互作用衰减条件的量子吉布斯采样方法，通过马尔可夫熵分解实现自由能估计，并在量子计算机上高效实施。
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