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Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods

Jutho Haegeman, Frank Verstraete

arXiv:1611.08519v1cond-mat.str-elcond-mat.stat-mechmath-phmath.MPquant-ph

TL;DR

Many-body physics uses transfer matrices and MPOs to study phases, partition functions, and operator eigenstructure, but their applications span both classic exact solutions and newer computational methods. This review organizes those topics through tensor networks, algebraic structures, and MPS-based eigenvector methods, while noting specific challenges in non-ergodic and excited-state settings.

  • Problem

    Transfer matrices and MPOs appear across many-body physics, where their eigenstructure is important for understanding classical and quantum systems.

    Source: §6 · p. 51

  • Method

    The paper reviews transfer-matrix and MPO applications, algebraic structures, and numerical MPS methods for finding MPO eigenvectors.

    Source: §5.3 · p. 37, §6 · p. 51

  • Results

    The review connects MPO algebraic structure with algorithms for determining eigenstructure numerically and illustrates applications ranging from exact Bethe-ansatz results to MPS approximations.

    Source: §4.2.1 · p. 25, §5.6.3 · p. 50, §6 · p. 51

  • Takeaways & Limitations

    MPO diagonalization links exact results in statistical physics and integrable systems with computational approaches to many-body eigenvectors.

    Source: §6 · p. 51

    Translational MPO diagonalization remains challenging for the non-ergodic directed-percolation model with a trivial fixed point, and MPO excited states lack supporting theoretical intuition.

    Source: §3.4.1 · p. 18, §5.5 · p. 44

Abstract

from arXiv · show

Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.

1. Introduction

Tensor networks describe many-body systems through entanglement degrees of freedom, offering a real-space framework suited to strongly interacting systems. This review places matrix product operators within that framework and surveys their classic applications and newer theoretical and computational developments.

1. Introduction

  • Tensor networks encode many-body wavefunctions with local tensors linked according to how entanglement and correlations are distributed.Their entanglement degrees of freedom can reduce the effective spatial dimensionality of the description.Source: §1 · pp. 2–3
  • MPS, tree tensor networks, PEPS, and MERA are distinct tensor-network classes that arise naturally from compressing a path-integral representation.MPS and tree networks have no loops; PEPS have loops, while MERA adds a scale dimension.Source: §1 · p. 3
  • MPOs represent operators including Hamiltonians, path-integral transfer matrices, cellular automata, and the virtual operators that encode correlations in PEPS.They also represent density operators such as exp(−βH).Source: §1 · pp. 3–4
  • MPOs connect classic results such as Onsager’s Ising solution with integrable-system algebras, topological-order representations [48] [49] [50], and DMRG.The review aims to relate these established applications to recent theoretical and computational developments, while its lecture origins explain its selective topic choices.Source: §1 · p. 4

2. Matrix Product Operators: Definitions and Normal Forms

The section defines translationally invariant MPOs and explains how gauge freedom and invariant subspaces lead to canonical descriptions. It then presents transfer-matrix tools and the fundamental theorem that characterizes when injective tensor descriptions represent the same infinite-system state.

Definitions

  • A translationally invariant MPO is specified by a local tensor with physical and virtual indices plus a matrix that closes the virtual indices.The review focuses on uniform MPO families acting on N d-dimensional spins.Source: §2 · pp. 4–5

Gauge freedom and injectivity

  • Gauge transformations leave the represented MPO unchanged; injectivity holds when products of its local matrices span the full D^2-dimensional matrix space.When they do not span that space, invariant subspaces can be brought into block-diagonal form by a unitary gauge transformation.Source: §2 · p. 5

Canonical forms

  • For an injective MPO with M = I, the transfer matrix E = Σ_ij A^ij ⊗ overline(A^ij) is the central object for constructing a normal form.The discussion treats the MPO as a state in a doubled Hilbert space when defining this transfer matrix.Source: §2 · p. 6
  • Injectivity makes the largest-eigenvalue eigenvector of E unique and strictly positive, enabling a gauge transform and rescaling that produce an isometry.The resulting right or left normal form is unique up to an additional unitary gauge transformation.Source: §2 · pp. 6–7

Fundamental theorem

  • Two injective translationally invariant MPS represent the same infinite-system state if and only if a gauge transform maps one tensor to the other.The proof uses a mixed transfer matrix whose largest eigenvalue in magnitude must be one.Source: §2 · p. 7

MPO normal forms

  • MPO normal forms separate invariant blocks, remove irrelevant off-diagonal components where possible, and combine repeated injective blocks with appropriate weights.The construction is applied recursively when diagonal blocks remain non-injective; lower blocks can encode sums of local operators.Source: §2 · pp. 6–7

Examples

  • The examples show that squaring the CZX MPO reduces to a bond-dimension-1 normal form, while an MPO for Σ_i X_i cannot be reduced further in the stated form.The squared CZX MPO represents (−1)^N I.Source: §2 · p. 8

3. Matrix product operators and transfer matrices

The section presents transfer matrices and MPOs as tools for counting configurations and describing equilibrium and quantum systems. It connects leading eigenvalues to statistical-mechanical quantities, and path-integral constructions to quantum ground states and sampling methods.

3. Matrix product operators and transfer matrices

  • MPO transfer matrices arise in classical partition functions, quantum path integrals, tensor-network algorithms, and non-equilibrium statistical physics.For systems with a spectral gap, repeated MPO application leaves the dominant eigenvalue and its eigenspace projector in the large-iteration limit.Source: §3 · p. 9

3.1.1. Combinatorial problems on the lattice.

  • Lattice counting problems can be formulated through transfer-matrix eigenvalues, including bit strings with forbidden adjacent ones, tilings, dimers, and ice configurations.For the bit-string recursion, the leading eigenvalue is (1 + √5)/2 and governs the exponential scaling of the count.Source: §3.1.1 · pp. 9–11
  • Adjusting the occupied-site weight tracks densities and reveals a jamming transition in the lattice model; the triangular-lattice version corresponds to Baxter’s exactly solved hard-hexagon problem.The model forbids adjacent occupied sites.Source: §3.1.1 · p. 10

3.1.2. Equilibrium Statistical physics in 2D.

  • In equilibrium statistical physics, the free energy per site of an infinite local system is proportional to the logarithm of its transfer matrix’s leading eigenvalue.MPS variational methods can find leading eigenvalues and eigenvectors, allowing simulations of two-dimensional local classical systems to almost machine precision.Source: §3.1.2 · p. 11
  • The two-dimensional Ising model’s exact transfer-matrix solution, obtained by Onsager, produced critical exponents that differed from mean-field predictions.The transfer matrix can be diagonalized using either the Bethe ansatz or the Jordan–Wigner transformation.Source: §3.1.2 · p. 12

3.2. Path Integrals in 1+1 D

  • A quantum path integral maps a one-dimensional quantum problem to an effective two-dimensional classical-like partition function, and its transfer-matrix leading eigenvector gives the ground state.The transfer matrix can be approximated with Trotter–Suzuki splitting when the Hamiltonian terms do not commute.Source: §3.2 · p. 12
  • For the Heisenberg chain, a bond-dimension-2 six-vertex transfer matrix has the exact ground state as its leading eigenvector; a suitable spin rotation makes its weights positive for quantum Monte Carlo.The ability to make all weights positive is described as accidental, and is not known for many interesting problems; this is the sign problem.Source: §3.2 · p. 13

3.3. Matrix product operators in PEPS: physics at the edge

PEPS represent two-dimensional quantum states with local tensors, while their transfer matrices connect observable calculations to classical two-dimensional statistical mechanics and make leading eigenvectors central to computation. The section then shows how MPOs describe PEPS entanglement and topological virtual symmetries, and how MPO transfer matrices encode nonequilibrium cellular automata, including directed percolation and ASEP.

3.3. Matrix product operators in PEPS: physics at the edge

  • PEPS parameterize two-dimensional ground-state wavefunctions with local tensors whose physical index is joined by four virtual indices on a square lattice.The virtual indices have dimension D.Source: §3.3 · pp. 13–14

3.3.1. PEPS transfer matrix.

  • Observable expectation values reduce to the leading eigenvector of a double-layer transfer matrix, making its computation the bottleneck in PEPS variational optimization.The transfer matrix acts on a Hilbert space of dimension D^2N and can be viewed as a completely positive map whose leading eigenvector is a positive operator; injectivity ensures a unique fixed point.Source: §3.3.1 · p. 14

3.3.2. Entanglement Hamiltonians.

  • The entanglement spectrum of a PEPS boundary can be obtained from the MPO product of its left and right transfer-matrix eigenoperators, and MPS methods can find its eigenvalues and eigenvectors.The logarithm of the operator √ρLρR√ρL is the entanglement Hamiltonian; its eigenvalues give the entanglement spectrum.Source: §3.3.2 · p. 15

3.3.3. Matrix product operators and topological order.

  • For topological PEPS, hidden virtual MPO symmetries generate an algebra that encodes anyon properties, including statistics and braiding.The associated constant correction to boundary entanglement entropy is related to the PEPS non-injectivity and its hidden virtual symmetry.Source: §3.3.3 · p. 16
  • Closed MPO algebras can be used to construct PEPS with those MPOs as symmetries, yielding quantum-double and string-net classes as well as broader fermionic and symmetry-protected or symmetry-enhanced phases.Breaking the virtual symmetries in leading eigenvectors characterizes different anyon-condensation patterns.Source: §3.3.3 · p. 17

3.4. Nonequilibrium statistical physics and cellular automata in 1D

  • In one-dimensional nonequilibrium physics, an MPO can represent a stochastic update matrix whose eigenvector with eigenvalue 1 is the stationary distribution when the matrix is ergodic.Source: §3.4 · p. 17

3.4.1. Directed Percolation.

  • Directed percolation has second-order transitions, but its non-ergodic stochastic MPO has multiple fixed points, leaving thermodynamic-limit MPS diagonalization challenging because a trivial zero-entanglement fixed point exists.The continuous-time version has had leading eigenvectors found with MPS methods.Source: §3.4.1 · pp. 17–18, Figure 9 · p. 18

3.4.2. Asymmetric Exclusion Processes.

  • For fully parallel ASEP, the MPO's leading eigenvector has an analytical MPS form built using an explicit matrix representation of an operator algebra.The process includes rightward hopping with probability p, inflow probability pin, and outflow probability pout.Source: §3.4.2 · p. 19, Figure 10 · p. 19

4. Exact methods for diagonalizing matrix product operators

This section surveys exact methods for diagonalizing matrix product operators, including transfer-matrix mappings to free fermions, Bethe-ansatz algebras, and MPS representations of eigenstates. It also covers discrete MPO algebras and their fusion-category data, alongside exact MPS representations of stochastic steady states, including the parallel-update ASEP fixed point.

4.1. Mapping to free fermions

  • The free-fermion method gives the square-lattice dimer growth rate (1.33851...)^NM and all transfer-matrix eigenvectors.The method follows a mapping to free fermions, as in Lieb; the growth result was also proven by Kasteleyn using Pfaffians.Source: §4 · pp. 19–20, §4.1 · p. 22
  • The classical Ising model’s exact solution is obtained along similar lines; in the symmetry-broken regime, its leading eigenvector is twofold degenerate, and external-field perturbation within that subspace yields the Onsager–Yang magnetization formula.M ∼ |Tc − T|1/8.Source: §4.1 · p. 22

4.2.1. The algebraic Bethe ansatz.

  • Associativity of the gauge tensors that exchange commuting MPOs yields the Yang–Baxter equation and a framework for constructing integrable models [85, 86].The fundamental theorem of MPOs identifies commuting MPOs through a virtual gauge transformation.Source: §4.2.1 · pp. 22–23
  • Bethe-equation solutions produce orthogonal eigenstates of commuting MPO transfer matrices, and the algebraic Bethe ansatz represents all eigenstates as matrix-product states, with bond dimension scaling exponentially in particle number or B-operator applications.Source: §4.2.1 · p. 25

4.2.2. The coordinate Bethe ansatz.

  • Coordinate Bethe wavefunctions become MPS of bond dimension 2^M, with their permutation phases encoded by matrices and their algebraic-ansatz form related by a gauge transformation.Setting the scattering phase to π recovers free-fermion antisymmetry.Source: §4.2.2 · pp. 25–27
  • The square-ice MPO can be embedded in an integrable Bethe-ansatz family, giving exact eigenvectors and an eigenvalue per site identified with Lieb’s square-ice constant.The supplied passage truncates the constant after “8,” so no further value is reported.Source: §4.2.2 · p. 27

4.3. Discrete MPO algebras and tensor fusion categories

  • Associativity constraints on discrete MPO fusion produce F-symbols obeying the pentagon equation; for given fusion rules N^c_ab, only finitely many essentially different solutions exist.These constructions connect discrete MPO algebras to tensor fusion categories, the mathematical basis for describing topological order and anyons in two dimensions [92].Source: §4.3 · pp. 28–29
  • Pentagon solutions define fusion tensors, MPOs, and PEPS, enabling string nets and quantum doubles on arbitrary lattices, with anyonic sectors and braiding encoded by central idempotents [50].The MPO formalism gives explicit expressions for elementary excitations and their braiding properties.Source: §4.3 · p. 29

4.4. Non-equilibrium steady states as matrix product states

  • Derrida et al. reduced open-boundary, translation-invariant stochastic master equations to algebraic equations for MPS matrices, yielding solutions with bond dimension linear in system size, unlike the exponential scaling of Alcaraz and Lazo.Source: §4.4 · p. 30
  • For fully parallel ASEP, an MPS depending on α, β, and p is sought, and a zipper tensor satisfying local equations is sufficient for an eigenvalue-1 fixed point.The parameters are the entry probability α, conditional forward-hop probability p, and exit probability β.Source: §4.4 · p. 30
  • For finite N-site ASEP, the steady state is an exact MPS of bond dimension N, with rank N across a half-system cut rather than 2^(N/2), giving exponential compression.The corresponding virtual correlations have no clear operational meaning in classical stochastic MPS, though attempts one.Source: §4.4 · pp. 30, 32
  • For continuous-time ASEP, a tensor-product similarity transform maps the master equation to XXZ with non-Hermitian boundary conditions, making all excited states Bethe-ansatz-solvable but yielding exponentially scaling MPS bond dimension [96, 97].Source: §4.4 · p. 32

5. Numerical methods for diagonalizing matrix product operators

This section develops MPS-based numerical methods for extremal eigenvectors of MPOs, treating the leading eigenvector as a fixed point and also targeting excitations around it. The discussion assumes translation invariance in the thermodynamic limit, while noting that finite-lattice and larger-unit-cell versions can be formulated.

5. Numerical methods for diagonalizing matrix product operators

  • Leading MPO eigenvectors lack the rigorous approximation bounds available for gapped local Hamiltonian ground states, yet MPS-based algorithms can be formulated for finite systems and the thermodynamic limit.The section formulates the eigenvalue problem using the differential geometry of MPS and discusses algorithms for fixed points and excitations.Source: §5 · p. 32
  • The eigenvector of an MPO eigenvalue of largest magnitude represents its fixed point under infinitely repeated MPO application; the section also targets excitations around that fixed point.Source: §5 · p. 32
  • The numerical discussion assumes translation-invariant MPOs and dominant eigenvectors in the thermodynamic limit, with dominant eigenvalue scaling as λ^N = exp(−fN).Here f = −log λ is called the free energy per site; the dominant eigenvector is assumed not to break translation invariance.Source: §5 · p. 33
  • The authors outline MPS normal-form algorithms, an MPS-manifold eigenvalue formulation, and a strategy for targeting excited MPO eigenstates.The stated setting can be generalized to p-site unit cells or finite lattices.Source: §5 · pp. 32–33

5.1. Manifold of uniform matrix product states and its tangent bundle

Uniform MPS are parameterized by a site tensor and boundary data, with injectivity and gauge freedom shaping their normal forms. Their manifold structure provides tangent spaces and projectors used in the numerical treatment.

5.1. Manifold of uniform matrix product states and its tangent bundle

  • A uniform MPS is specified by a single three-index tensor and boundary matrix M, which must commute with the site tensors to ensure translation invariance.Source: §5.1 · p. 33
  • For injective MPS, bulk expectation values become independent of M in the thermodynamic limit, and the only commuting M is proportional to the identity.Source: §5.1 · p. 33
  • Gauge transformation and normalization bring an injective MPS tensor into left or right orthonormal form, related by C.Figure 20 depicts the tensor and both normal forms through their isometric constraints.Source: §5.1 · p. 33, Figure 20 · p. 34
  • The singular value decomposition C = USV† yields the Schmidt decomposition and left and right canonical forms, unique up to a diagonal unitary; normalization requires tr(CC†) = 1.Source: §5.1 · p. 34
  • Injective MPS of fixed bond dimension form a smooth complex manifold with a linear tangent space, whose tensor parameterization has gauge redundancy.Adding AsL X − XAsR to Bs leaves the tangent-vector definition invariant.Source: §5.1 · p. 34
  • A tangent-space projector for finite MPS generalizes to uniform MPS, and the half-infinite-region projectors select supports of the corresponding reduced density matrices.Source: §5.1 · p. 35

5.2. Algorithms for obtaining normal forms of uniform matrix product states

The section contrasts a fixed-point eigensolver for putting a uniform MPS into normal form with an iterative QR-based approach. The latter can be accelerated with an in-loop Arnoldi step, while the standard fixed-point route can lose accuracy when small Schmidt coefficients are present.

Standard fixed-point approach

  • The fixed-point approach can resolve small singular values of C only to order √ϵ, and small eigenvalues of ρL may be O(ϵ) and negative.Because C also has large singular values, its condition number is large; using C−1 introduces another loss of accuracy, so the isometric constraint can be violated above machine precision.Source: §5.2 · p. 35

QR iteration

  • The QR iteration converges to a left isometry AL and an upper triangular C, but its rate matches the power method for finding ρL.That rate is typically insufficient when the transfer matrix has a small gap, corresponding to long correlation lengths.Source: §5.2 · p. 36

Arnoldi improvement

  • An in-loop Arnoldi solve for an improved C guess can make the algorithm equally fast as the standard approach but more accurate.Because the fixed point need not be solved to machine precision while AL is still changing, a small Krylov subspace is typically sufficient at each iteration.Source: §5.2 · p. 36

Generality

  • The algorithm applies to any input tensor A; for a generic tensor it finds CAi = λAiL C, with λ giving the norm per site in the thermodynamic limit.For A = AR, λ is automatically 1 at every step.Source: §5.2 · p. 37

5.3. Approximating the MPO eigenvalue problem in the MPS manifold

Repeated MPO application makes the dominant eigenvalue and eigenspace projector central, but standard power-method approaches can be slow or costly for MPS. The paper generalizes projection methods to the MPS manifold, imposing a tangent-space Galerkin condition; at a stationary point, the residual cannot be reduced further without leaving the fixed-bond-dimension manifold.

Motivation

  • In the gapped-spectrum limit, repeated MPO application leaves the dominant eigenvalue and eigenspace projector, irrespective of boundary conditions.This is why many applications seek the largest-magnitude eigenvalue and its leading eigenvector.Source: §5.3 · p. 37

Challenges

  • Applying an MPO to an MPS multiplies its bond dimension by D′, while the power method can also converge slowly when the leading eigenvalue gap is small.Linear combinations of injective MPS do not solve this problem in the thermodynamic limit, because any two are equal or orthogonal.Source: §5.3 · p. 38

Manifold projection

  • The proposed approach seeks the best leading-eigenvector approximation as an injective MPS by requiring the residual to be orthogonal to the MPS tangent space.The paper reports that this orthogonal projection is more stable in practice than an oblique projection onto a different tangent space.Source: §5.3 · p. 38

Stationarity

  • At a stationary point, the uniform MPS is a solution of the MPO eigenvalue problem within the MPS manifold, with no further residual decrease possible at the given bond dimension.The graphical conditions comprise four eigenvalue problems; the second condition in Figure 21(c) follows from the first together with the conditions in (b).Source: §5.3 · p. 39, Figure 21 · p. 40

Away from stationarity

  • Away from a stationary point, AC and C need not be eigenvectors of their respective maps, and the maps’ dominant eigenvalues need not equal λ or 1.The AC-map eigenvalue need not be λ, the largest-magnitude eigenvalue of E, and the C-map eigenvalue need not be 1.Source: §5.3 · p. 40

5.4. Algorithms for the leading eigenvector of a matrix product operator

The section develops variational MPS methods for leading MPO eigenvectors, relating them to CTM approaches and exact matrix algebras. It also gives QR-based and Hermitian-case formulations that avoid explicit inverses.

5.4. Algorithms for the leading eigenvector of a matrix product operator

  • The leading MPO eigenvector is sought as an MPS fixed point by iteratively optimizing coupled eigenvalue problems, using Arnoldi to improve on the power method.The approach is compared with specific CTM formulations and related to the exact matrix algebra used for ASEP.Source: §5.4 · p. 40
  • QR decompositions provide an update that avoids squaring small singular values, which can fall below machine precision in the direct SVD approach.The mismatch between the QR factors supplies an error measure for convergence.Source: §5.4 · p. 42
  • For Hermitian MPO tensors, the algorithm can be reformulated using left and right orthonormalization instead of explicit inverses.This simplification yields Algorithm 4.Source: §5.4 · p. 43
  • When the CTM truncation matrix X vanishes, the method is exact and matches the algebra used to solve ASEP, up to reinterpretation of the MPO.The same construction identifies the two-directional CTM renormalization group and permits increasing bond dimension by retaining more singular values.Source: §5.4 · p. 44
  • The algorithms can be extended to finite systems, repeated unit cells, dynamically enlarged bond dimensions, and simultaneous fixed-point projection in four directions.The latter is described as similar to the fully general CTM renormalization group.Source: §5.4 · p. 44

5.5. Algorithms for the excited states of a matrix product operator

The section approximates MPO excitations by projecting onto momentum-resolved tangent vectors around an MPS fixed point. The resulting eigenvalues are used as complex excitation energies, although the ansatz lacks a theoretical justification for MPOs.

5.5. Algorithms for the excited states of a matrix product operator

  • The projected excitation problem yields eigenvalues µ(k)λ^N = exp(−Nf −e(k)), with f the complex free-energy density and e(k) a complex excitation energy.The authors report that this ansatz seems to work well in practice.Source: §5.5 · p. 44
  • MPO excitations are approximated by an orthogonal projection onto momentum-k tangent vectors around the MPS fixed point.The tangent-space intuition used for Hamiltonian excitations has no corresponding theoretical argument for MPOs.Source: §5.5 · pp. 44–45
  • The effective eigenvalue equation can be applied efficiently when its inverses are computed with an iterative linear solver.The text gives generalized minimal residual and stabilized biconjugate gradient methods as examples.Source: §5.5 · p. 45

5.6. Examples

The examples apply fixed-point and excitation algorithms to hard squares, Potts, Heisenberg, and PEPS transfer matrices. They report numerical precision and spectral features, including an entanglement spectrum with chiral edge-mode signatures.

5.6.1. Hard squares.

  • 58 digits of hard-squares free-energy precision were obtained at chemical potential zero with bond dimension D = 80.The precision improves by a power law with bond dimension; the reported magnetization and compressibility use D = 128, without finite-entanglement scaling of critical exponents.Source: Table 1 · p. 46, §5.6.1 · p. 46

5.6.2. Transfer matrix spectrum of the q-level Potts model.

  • For q = 2, the topological sector has an isolated kink branch, while the trivial sector has continuous even-kink bands; higher q brings additional sector-specific excitations and bound states.Spectra across nontrivial sectors are reported equivalent for q ≥ 3; new isolated excitations appear near criticality, alternating between topological sectors for odd q and trivial sectors for even q.Source: §5.6.2 · p. 49
  • Near the ordered-phase Potts transition, extrapolated trivial- and topological-sector gaps appear to become equal and match the disordered-phase analytical result.For q ≳ 5, large Schmidt-spectrum degeneracies accompany weak first-order transitions and explain the large bond dimensions required in this regime.Source: §5.6.2 · pp. 49–50

5.6.3. Schmidt spectrum of the Heisenberg model.

  • The Heisenberg-model fixed point is the ground state of the 6-vertex transfer matrix, and its MPS Schmidt spectrum has exact two-fold degeneracy.The reported approximation uses D = 512 or larger without symmetries or other techniques.Source: §5.6.3 · p. 50

5.6.4. Spectrum of the corner transfer matrix for the filtered Fibonacci string-net model.

  • For the filtered Fibonacci string-net PEPS, the CTM spectrum raised to the 3/2 power matches the fixed-point Schmidt spectrum after normalization.The transfer MPO has bond dimension D′ = 25, and its hexagonal geometry gives a 120° corner without reflection symmetry.Source: §5.6.4 · p. 50, Figure 26 · p. 51
  • The lowest entanglement-Hamiltonian excitations of a chiral-spin-liquid PEPS show signatures of chiral edge modes in the thermodynamic limit.The calculation constructs the half-plane reduced-density-matrix spectrum from left and right transfer-matrix fixed points, then applies the excitation ansatz across k ∈ [0, π].Source: §5.6.4 · p. 51

6. Outlook and conclusion

Transfer-matrix and MPO diagonalization has advanced understanding of many-body phases and transitions, while MPO algebra also guides numerical eigenstructure algorithms. Extending these ideas to PEPOs remains a challenge for studying higher-dimensional systems.

6. Outlook and conclusion

  • Diagonalizing transfer matrices and MPOs has advanced understanding of quantum and classical many-body phases and phase transitions, while MPO algebra guides numerical eigenstructure algorithms.The paper also highlights algebraic Bethe ansatz and MPO representations of tensor fusion categories as examples of this structure.Source: §6 · p. 51
  • A central challenge is building an algebraic and computational framework for PEPOs to study 3D classical statistical mechanics and 2D quantum phases.The paper notes tetrahedral-equation approaches [135] and PEPO algebras for Walker and Wang theories, but says nontrivial tetrahedral solutions have been difficult to find.Source: §6 · pp. 51–52
  • Figure 27 presents the thermodynamic-limit energy-momentum spectrum of the entanglement Hamiltonian for a chiral spin liquid PEPS.It was computed using the section's MPS methods.Source: Figure 27 · p. 52
  • To the authors' knowledge, exact diagonalizations have not been reported for a PEPO family exhibiting phase transitions, such as the 3D Ising model.They suggest numerical approaches may therefore be the only recourse, with algebra guiding fast and reliable methods.Source: §6 · p. 52

A. Explicit representation of the matrix product state steady state for the asymmetric exclusion process with parallel updates

The section gives an explicit MPS solution for the ASEP with parallel updates and presents a simpler proof based on zipper conditions.

A. Explicit representation of the matrix product state steady state for the asymmetric exclusion process with parallel updates

  • The section represents the ASEP steady-state MPS solution for parallel updates using the tensors Ai, which are the same as those obtained in.The explicit solution is described as involving infinite-dimensional matrices, found by guesswork.Source: A. Explicit representation of the… · pp. 52–53
  • Introducing tensors Bi that satisfy the zipper conditions provides a new and simpler proof of the solution's correctness.The authors contrast this proof with the original proofs in.Source: A. Explicit representation of the… · p. 52
  • The explicit solution is presented through infinite-dimensional matrices obtained by guesswork.The supplied passages do not specify further properties of those matrices.Source: A. Explicit representation of the… · p. 53

DISCLOSURE STATEMENT

The authors report no affiliations, memberships, funding, or financial holdings that might affect the review's objectivity.

DISCLOSURE STATEMENT

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