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The Onsager algebra, invented to solve the two-dimensional Ising model, can be used to construct conserved charges for a family of integrable $N$-state chiral clock models. We show how it naturally gives rise to a "pivot" procedure for this family of chiral Hamiltonians. These Hamiltonians have an anti-unitary CPT symmetry that when combined with the usual $\mathbb{Z}_N$ clock symmetry gives a non-abelian dihedral symmetry group $D_{2N}$. We show that this symmetry gives rise to symmetry-protected topological (SPT) order in this family for all even $N$, and representation-SPT (RSPT) physics for all odd $N$. The simplest such example is a next-nearest-neighbour chain generalising the spin-1/2 cluster model, an SPT phase of matter. We derive a matrix-product state representation of its fixed-point ground state along with the ensuing entanglement spectrum and symmetry fractionalisation. We analyse a rich phase diagram combining this model with the Onsager-integrable chiral Potts chain, and find trivial, symmetry-breaking and (R)SPT orders, as well as extended gapless regions. For odd $N$, the phase transitions are "unnecessarily" critical from the SPT point of view.

Continuous phase transitions where symmetry is spontaneously broken are ubiquitous in physics and often found between `Landau-compatible' phases where residual symmetries of one phase are a subset of the other. However, continuous `deconfined quantum critical' transitions between Landau-incompatible symmetry-breaking phases are known to exist in certain quantum systems, often with anomalous microscopic symmetries. In this paper, we investigate the need for such special conditions. We show that Landau-incompatible transitions can be found in a family of well-known classical statistical mechanical models with anomaly-free on-site microscopic symmetries, introduced by Jos\'{e}, Kadanoff, Kirkpatick and Nelson (Phys. Rev. B 16, 1217). The models are labeled by a positive integer $Q$ and constructed by a deformation of the 2d classical XY model, defined on any lattice, with an on-site potential that preserves a discrete $Q$-fold spin rotation and reflection symmetry. For a range of temperatures, even $Q$ models exhibit two Landau-incompatible partial symmetry-breaking phases and a direct transition between them for $Q \ge 4$. Characteristic features of Landau-incompatible transitions are easily seen, such as enhanced symmetries and melting of charged defects. For odd $Q$, and corresponding temperature ranges, two regions of a single partial symmetry-breaking phase are obtained, split by a stable `unnecessary critical' line. We present quantum models with anomaly-free symmetries that also exhibit similar phase diagrams., Comment: 6+8 pages, 3+7 figures (main + appendices)

The bulk-boundary correspondence relates topologically-protected edge modes to bulk topological invariants, and is well-understood for short-range free-fermion chains. Although case studies have considered long-range Hamiltonians whose couplings decay with a power-law exponent $\alpha$, there has been no systematic study for a free-fermion symmetry class. We introduce a technique for solving gapped, translationally invariant models in the 1D BDI and AIII symmetry classes with $\alpha>1$, linking together the quantized winding invariant, bulk topological string-order parameters and a complete solution of the edge modes. The physics of these chains is elucidated by studying a complex function determined by the couplings of the Hamiltonian: in contrast to the short-range case where edge modes are associated to roots of this function, we find that they are now associated to singularities. A remarkable consequence is that the finite-size splitting of the edge modes depends on the topological winding number, which can be used as a probe of the latter. We furthermore generalise these results by (i) identifying a family of BDI chains with $\alpha<1$ where our results still hold, and (ii) showing that gapless symmetry-protected topological chains can have topological invariants and edge modes when $\alpha -1$ exceeds the dynamical critical exponent., Comment: Simplified treatment of singularities. Additional results and discussion

The symmetry-resolved R\'enyi entanglement entropy is the R\'enyi entanglement entropy of each symmetry sector of a density matrix $\rho$. This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given by $N$ massless Dirac fermions. For the density matrix, $\rho_A$, of subsystems of $L$ neighbouring sites we calculate the leading terms in the large $L$ asymptotic expansion of the symmetry-resolved R\'enyi entanglement entropies. This follows from a large $L$ expansion of the charged moments of $\rho_A$; we derive $tr(e^{i \alpha Q_A} \rho_A^n) = a e^{i \alpha \langle Q_A\rangle} (\sigma L)^{-x}(1+O(L^{-\mu}))$, where $a, x$ and $\mu$ are universal and $\sigma$ depends only on the $N$ Fermi momenta. We show that the exponent $x$ corresponds to the expectation from CFT analysis. The error term $O(L^{-\mu})$ is consistent with but weaker than the field theory prediction $O(L^{-2\mu})$. However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators., Comment: 20 pages + appendix

A key task in the study of networked systems is to derive local and global properties that impact connectivity, synchronizability, and robustness. Computing shortest paths or geodesics in the network yields measures of node centrality and network connectivity that can contribute to explain such phenomena. We derive an analytic distribution of shortest path lengths, on the giant component in the supercritical regime or on small components in the subcritical regime, of any sparse (possibly directed) graph with conditionally independent edges, in the infinite-size limit. We provide specific results for widely used network families like stochastic block models, dot-product graphs, random geometric graphs, and graphons. The survival function of the shortest path length distribution possesses a simple closed-form lower bound which is asymptotically tight for finite lengths, has a natural interpretation of traversing independent geodesics in the network, and delivers novel insight in the above network families. Notably, the shortest path length distribution allows us to derive, for the network families above, important graph properties like the bond percolation threshold, size of the giant component, average shortest path length, and closeness and betweenness centralities. We also provide a corroborative analysis of a set of 20 empirical networks. This unifying framework demonstrates how geodesic statistics for a rich family of random graphs can be computed cheaply without having access to true or simulated networks, especially when they are sparse but prohibitively large., Comment: 32 pages, 12 figures

We construct new families of spin chain Hamiltonians that are local, integrable and translationally invariant. To do so, we make use of the Clifford group that arises in quantum information theory. We consider translation invariant Clifford group transformations that can be described by matrix product operators (MPOs). We classify the translation invariant Clifford group transformations that consist of a shift operator and an MPO of bond dimension two -- this includes transformations that preserve locality of all Hamiltonians; as well as those that lead to non-local images of particular operators but nevertheless preserve locality of certain Hamiltonians. We characterise the translation invariant Clifford group transformations that take single-site Pauli operators to local operators on at most five sites -- examples of Quantum Cellular Automata -- leading to a discrete family of Hamiltonians that are equivalent to the canonical XXZ model under such transformations. For spin chains solvable by algebraic Bethe Ansatz, we explain how conjugating by a matrix product operator affects the underlying integrable structure. This allows us to relate our results to the usual classifications of integrable Hamiltonians. We also treat the case of spin chains solvable by free fermions., Comment: Revised discussion of R-matrix and Yang-Baxter algebra in transformed models. Other minor changes and additional references added. 23 pages + appendix

Although free-fermion systems are considered exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or entanglement measures. We derive closed expressions for such quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains called generalised cluster models. While there is a bijection between general models in these classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results: (1) we derive closed expressions for the string correlation functions - the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into ground state entanglement; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit - an independent and explicit construction for the BDI class is given in a concurrent work [Phys. Rev. Res. 3 (2021), 033265, 26 pages, arXiv:2105.12143]; (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. General models in these classes can be obtained by taking limits of the models we analyse, giving a further application of our results. To the best of our knowledge, these results constitute the first application of Day's formula and Gorodetsky's formula for Toeplitz determinants to many-body quantum physics.

Models whose ground states can be written as an exact matrix product state (MPS) provide valuable insights into phases of matter. While MPS-solvable models are typically studied as isolated points in a phase diagram, they can belong to a connected network of MPS-solvable models, which we call the MPS skeleton. As a case study where we can completely unearth this skeleton, we focus on the one-dimensional BDI class -- non-interacting spinless fermions with time-reversal symmetry. This class, labelled by a topological winding number, contains the Kitaev chain and is Jordan-Wigner-dual to various symmetry-breaking and symmetry-protected topological (SPT) spin chains. We show that one can read off from the Hamiltonian whether its ground state is an MPS: defining a polynomial whose coefficients are the Hamiltonian parameters, MPS-solvability corresponds to this polynomial being a perfect square. We provide an explicit construction of the ground state MPS, its bond dimension growing exponentially with the range of the Hamiltonian. This complete characterization of the MPS skeleton in parameter space has three significant consequences: (i) any two topologically distinct phases in this class admit a path of MPS-solvable models between them, including the phase transition which obeys an area law for its entanglement entropy; (ii) we illustrate that the subset of MPS-solvable models is dense in this class by constructing a sequence of MPS-solvable models which converge to the Kitaev chain (equivalently, the quantum Ising chain in a transverse field); (iii) a subset of these MPS states can be particularly efficiently processed on a noisy intermediate-scale quantum computer., Comment: 21 pages + appendix

We outline the construction of a molecular system that could, in principle, implement a thermodynamically reversible Universal Turing Machine (UTM). By proposing a concrete-albeit idealised-design and operational protocol, we reveal fundamental challenges that arise when attempting to implement arbitrary computations reversibly. Firstly, the requirements of thermodynamic reversibility inevitably lead to an intricate design. Secondly, thermodynamically reversible UTMs, unlike simpler devices, must also be logically reversible. Finally, implementing multiple distinct computations in parallel is necessary to take the cost of external control per computation to zero, but this approach is complicated the distinct halting times of different computations., Comment: 16 pages, 8 figures

Networks are a widely-used tool to investigate the large-scale connectivity structure in complex systems and graphons have been proposed as an infinite size limit of dense networks. The detection of communities or other meso-scale structures is a prominent topic in network science as it allows the identification of functional building blocks in complex systems. When such building blocks may be present in graphons is an open question. In this paper, we define a graphon-modularity and demonstrate that it can be maximised to detect communities in graphons. We then investigate specific synthetic graphons and show that they may show a wide range of different community structures. We also reformulate the graphon-modularity maximisation as a continuous optimisation problem and so prove the optimal community structure or lack thereof for some graphons, something that is usually not possible for networks. Furthermore, we demonstrate that estimating a graphon from network data as an intermediate step can improve the detection of communities, in comparison with exclusively maximising the modularity of the network. While the choice of graphon-estimator may strongly influence the accord between the community structure of a network and its estimated graphon, we find that there is a substantial overlap if an appropriate estimator is used. Our study demonstrates that community detection for graphons is possible and may serve as a privacy-preserving way to cluster network data.