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Diagonal quantum Ising interaction and classical conditional energies — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Analyticity of Holomorphic Functions; Liouville and Morera
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Banach Alaoglu Goldstine and Krein Milman
- Banach Algebras Spectrum and Holomorphic Functional Calculus
- Banach Valued Integration and the Radon Nikodym Property
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Bounded quantum lattice pressure on cubes and van Hove sets
- Bounded Variation and the Riemann–Stieltjes Integral
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Norms and Realification
- Complex Power Series and Analytic Functions
- Complex Subspace Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Continuous Functional Calculus for Self Adjoint and Normal Operators
- Contour Integration
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Diagonal quantum Ising interaction and classical conditional energies
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces Adjoint Operators and Annihilators
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Ensembles And Limits
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Finite-range lattice interaction and conditional energy
- Foundations of the Real Numbers for Analysis
- Function Space Topologies and the Exponential Law
- Fundamental Trigonometric Identities
- Gelfand Theory and Commutative C Star Algebras
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group C Star Algebras and the Fell Unitary Dual
- Group Homomorphisms and the Isomorphism Theorems
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Local spin matrix algebras and isometric inclusions
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal density states and spectral moments
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Physical-time KMS strip equilibrium condition
- pi: the Equivalent Characterizations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Properties of the Integral and the Working FTC
- Pure and Mixed States, Ensembles, and Schmidt Decomposition
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Splitting Fields
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Fundamental Theorems of Calculus
- The Galois Correspondence
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Uniformly bounded finite-range quantum spin interaction
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
The counterexample separates the unique Gibbs density of each fixed finite Ising Hamiltonian from the multiple KMS states of one specified infinite two-dimensional diagonal Ising model. It is an exact conditional mathematical model and gives no empirical claim.
3 · Logical flowchart
4 · Definitions, postulates, theorems and proofs
None yet.
5 · Examples, experiments and counterexamples
Finite Gibbs uniqueness does not imply infinite KMS uniqueness
Statement refuted
For every bounded finite-range quantum spin model, uniqueness of the finite-volume Gibbs state for each finite region implies uniqueness of the infinite-volume KMS state.
Facts & Assumptions
Given: The two-dimensional diagonal nearest-neighbor quantum Ising model with , zero field, , , Countable Choice, the finite-matrix C*-identity, and its finite- and infinite-volume equilibrium suppliers.
Diagonal quantum Ising interaction and classical conditional energies defines the finite-volume Hamiltonians as finite self-adjoint matrices on the tensor-product spin spaces and the infinite quasi-local model; Local spin matrix algebras and their factor inclusions defines the finite spin Hilbert spaces, local matrix algebras, product bases, and ordinary finite trace.
Countable Choice and the finite-matrix C*-identity are inherited as exact hypotheses from the infinite-volume theorem (The Axiom of Countable Choice (), Finite matrix C-star algebra and density-state duality).
Finite matrix spectral and state calculus supplies the finite Hermitian spectral calculus and trace cyclicity required by the finite KMS characterization; this current batch-36 result is used conditionally.
Finite KMS condition exactly determines the Gibbs density states that under [A1]–[A2], on every nonzero finite-dimensional Hilbert space and for every finite self-adjoint Hamiltonian, the normalized Gibbs trace is the unique -KMS state.
Two-dimensional diagonal quantum Ising KMS coexistence proves that for the stipulated infinite diagonal Ising model, under Countable Choice and the finite-matrix C*-identity, there are at least two distinct full-algebra -KMS states.
Counterexample
Given: [A1], [F1]–[F3], and the parameter conditions above.
Fix any finite . Its Hilbert space is nonzero and finite-dimensional, and by [F1]. The finite characterization [F2] gives exactly one finite-volume -KMS state, namely . This holds for every finite , regardless of finite-volume energy degeneracies.
The same coefficients and satisfy the hypotheses of [F3], so the infinite quasi-local system has at least two distinct -KMS states. Together, the finite-volume conclusion and this infinite-volume conclusion provide a counterexample to the refuted implication: every finite-volume Gibbs state is unique, while the infinite-volume KMS state is not unique. The distinction is between a fixed finite matrix Hamiltonian and the thermodynamic-limit KMS problem.
Remarks
This is an exact mathematical counterexample for the stated diagonal interaction. It does not say that finite-volume Gibbs states are ill-defined, nor does it identify a real material or report an observation.
Sources
- Pieter Naaijkens, Quantum Spin Systems on Infinite Lattices, extended and corrected lecture notes, arXiv:1311.2717v2
- David Ruelle, Topics in Quantum Statistical Mechanics and Operator Algebras, complete lecture notes, arXiv:math-ph/0107009v1
- Sascha Friedli and Yvan Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, complete author PDF, revised 2017
- Junior Seminar in Mathematical Physics, Proofs from the Book: Ising, DLR, Dobrushin and Peierls sections, complete seminar notes