Large Deviations in Quantum Spin Chains

The dynamical behavior of the quantum state of different quantum spin chains, with designed site-dependent interaction strengths, is analyzed when the initial state belongs to the one excitation subspace. Our results show that the inhomogeneous chains transfer single excitations with near-perfect fidelity for two different spin chain Hamiltonians. The first one is the ferromagnetic Heisenberg Hamiltonian with nearest neighbor interactions. The second spin chain has long-range anisotropic interactions, ferromagnetic in one direction and antiferromagnetic in the orthogonal plane. Both designed chains have in common a partially ordered spectrum and well-localized eigenvectors. This physical trait unifies the description of both kinds of systems.

The dynamical behaviour of the quantum state of different quantum spin chains, with designed site dependent interaction strengths, is analyzed when the initial state belongs to the one excitation subspace. It is shown that the inhomogeneous chains are able to transfer excitations with near perfect fidelity. This behaviour is found for two very different spin chain Hamiltonians. The first one is the ferromagnetic Heisenberg Hamiltonian with nearest neighbor interactions, the second one describes a chain with long range anisotropic interactions which are ferromagnetic in the $z$ direction and antiferromagnetic in the $(x,y)$ plane. It is shown that both designed chains have in common a partially ordered spectrum and well localized eigenvectors. This physical trait unifies the description of both kind of systems.

Field-induced excitation gaps in quantum spin chains are an interesting phenomenon related to confinements of topological excitations. In this paper I present a type of this phenomenon. I show that an effective magnetic field with a fourfold screw symmetry induces the excitation gap accompanied by dimer orders. The dimer order parameter and the excitation gap exhibit characteristic power-law dependence on the fourfold screw-symmetric field. Moreover, the field-induced dimer order and the field-induced N\'eel order coexist when the external uniform magnetic field, the fourfold screw-symmetric field, and the twofold staggered field are applied. This situation is in close connection with a compound $[\mathrm{Cu}(\mathrm{pym}){({\mathrm{H}}_{2}\mathrm{O})}_{4}]{\mathrm{SiF}}_{6}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{H}}_{2}\mathrm{O}$ [J. Liu et al., Phys. Rev. Lett. 122, 057207 (2019)]. In this paper I discuss a mechanism of field-induced dimer orders by using a density-matrix renormalization group method, a perturbation theory, and quantum field theories.

We investigate the inversion phenomena between the XXZ anisotropies of the Hamiltonian and the wave function in quantum spin chains. We focus on the S = 1/2 geometrically frustrated 3-leg ladder system with the XXZ interaction anisotropy. By use of the degenerate perturbation theory from the strong rung coupling limit, we have obtained the ground-state phase diagram. In some parameter regions, the Tomonaga-Luttinger spin liquid state is realized in spite of the Ising-like anisotropy, and the Neel state in spite of the XY-like anisotropy.

In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki transformation ${U}_{\mathrm{KT}}$, which defines a duality between the Haldane phase and the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-breaking phase. In this paper, we find that ${U}_{\mathrm{KT}}$ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a ``hidden symmetry breaking'' interpretation of the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i.e., invariant under ${U}_{\mathrm{KT}}$) is natural to be at a critical or multicritical point. We study concrete examples to demonstrate this argument. In particular, when $H$ is the Hamiltonian of the spin-1 antiferromagnetic Heisenberg chain, we prove that the self-dual model $H+{U}_{\mathrm{KT}}H{U}_{\mathrm{KT}}$ is exactly equivalent to a gapless spin-$1/2$ XY chain, which also implies an emergent quantum anomaly. On the other hand, we show that the topological and trivial Ising critical phases that are dual to each other meet at a multicritical point which is indeed self-dual.

A variational estimate of the free energy of a quantum spin chain based on the stochastic behavior of the quantization axes is proposed. It allows a description of the nonlinear excitations and quantum spin waves within the same framework. For an easy-plane ferromagnet with an external field in the plane it reveals that the nonlinear excitations (solitons) are predominantly described by the classical degrees of freedom. Considerable effort has been devoted to the theoretical description1 and experimental verification of the existence of nonlinear excitations. In this field magnetic chain systems have played an important part.2 Chain systems with an easy-plane anisotropy, such as (C6 H11 NH3 )CuBr3 (CHAB) and CsNiF3 , which are subjected to a symmetry breaking field, were found to be soliton bearing.3 Theoretically the description of the nonlinear effect is based on the mapping of the spin dynamics on a sine-Gordon model. Further research4 indicated that the interpretation of the thermodynamical properties of real systems in terms of the sine-Gordon model required however quantum and anisotropy corrections. Since the dipolar interactions between the spins in the chain are considered to be negligible, the direction of the spin chain in configuration space with respect to any preferred direction in spin space does not enter explicitly in the Hamiltonian. This makes it interesting to define a site-dependent z direction in spin space which makes an angle (Θn ,φn ) with the z direction in configuration space. The model Hamiltonian can then be written in terms of the spin operators defined with respect to the site-dependent quantization axis. The representation with equal quantization axis for all the sites is related to the representation with different quantization axes at each site by a rotation in spin space. The unitary transformation that describes the rotation contains the angles (Θn ,φn ), which we have treated as random variables. Based on this observation we have proposed a new variational approach for the quantum spin chain. We will outline the main results here. A full account is published elsewhere.5 In our approach the free energy of the system can be separated in a quantum part and a classical part. The free energy F of the chain is then approximated by an upper bound F<Fcl +Fqm , where βFcl =−ln[E(exp−βVcl )] and βFqm =−ln Tr(exp−β〈Hqm 〉). The first part is a configurational classical free energy, depending on the choice of the probability distribution for the quantization axes via the expectation E. The second contribution is the free energy of a modified quantum chain, whose parameters are classical averages and can be obtained from the configurational classical free energy. For easy-plane ferromagnetic chains, the nonlinear excitations are comprised in the classical part, whereas the quantum part contains predominantly linear excitations. For those thermodynamical quantities that are insensitive to the contribution of the quantum excitations, the chain is thermodynamically equivalent with a classical sine-Gordon system. The out-of-plane corrections are incorporated in the quantum part of the free energy. This behavior is corroborated experimentally by measurements of the excess heat capacity of CHAB for different magnetic fields with the same in-plane component.6 The validity of our approach is, however, not restricted to easy-plane ferromagnetic chains. Also for more complex systems with, for example, an orthorhombic or helical symmetry, the introduction of stochastic quantization axes can shed new light on some of the thermodynamical properties of these quantum systems in terms of their classical equivalent.

We derive general constraints on order and disorder parameters in Ising symmetric spin chains. Our main result is a theorem showing that every gapped, translationally invariant, Ising symmetric spin chain has either a nonzero order parameter or a nonzero disorder parameter. We also prove two more constraints on order and disorder parameters: (i) it is not possible for a gapped, Ising symmetric spin chain to have both a nonzero order parameter and a nonzero disorder parameter; and (ii) it is not possible for a spin chain of this kind to have a nonzero disorder parameter that is odd under the symmetry. These constraints have an interesting implication for self-dual Ising symmetric spin chains: every self-dual spin chain is either gapless or has a degenerate ground state in the thermodynamic limit. All of these constraints generalize to spin chains without translational symmetry. Our proofs rely on previously known bounds on entanglement and correlations in one dimensional systems, as well as the Fuchs-van de Graaf inequality from quantum information theory.

Large Deviations of Realized Volatility

We study S=1/2 quantum spin chains with shift-invariant and inversion-symmetric next-nearest-neighbor interaction, also known as zigzag spin chains. We completely classify the integrability and non-integrability of the above class of spin systems. We prove that in this class there are only two integrable models, a classical model and a model solvable by the Bethe ansatz, and all the remaining systems are non-integrable. Our classification theorem confirms that within this class of spin chains, there is no missing integrable model. This theorem also implies the absence of intermediate models with a finite number of local conserved quantities.

Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.

Quantum spin chains with composite spins have been used to approximate conventional chains with higher spins. For instance, a spin 1 (or ) chain was sometimes approximated by a chain with two (or three) spin ’s per site. However, little examination has been given as to whether this approximation, effectively assuming the first Hund rule per site, is valid and why. In this paper, the validity of this approximation is investigated numerically. We diagonalize the Hamiltonians of spin chains with a spin 1 and per site and with two and three spin ’s per site. The low energy excitation spectrum for the spin chain with M spin ’s per site is found to coincide with that of the corresponding conventional chain with one spin per site. In particular, we find that as the system size increases, an increasingly larger block of consecutive lowest energy states with maximal spin per site is observed, robustly supporting the first Hund rule even though the exclusion principle does not apply and the system does not possess Coulomb repulsion. As for why this approximation works, we show that this effective Hund rule emerges as a plausible consequence when applying to composite spin systems the Lieb–Mattis theorem, which is originally for the ground state of ferrimagnetic and antiferromagnetic spin systems.

We consider the classification problem of symmetry protected topological (SPT) phases on quantum spin systems.SPT phases are gapped short-range-entangled quantum phases with a symmetry G.We explain that in one and two-dimensional quantum spin systems, there are H 2 (G, U (1))/ H 3 (G, U (1))-valued invariant, confirming a physicists conjecture. Contents1. Symmetry protected topological phases 41 2. SPT-phases in 1-dimensional systems 63 3. SPT-phases in 2-dimensional systems 78

Bogomolny's transfer operator method plays a significant role in the study of quantum chaos, along with other well known methods like Gutzwiller's trace formula and the dynamical zeta function, which generalize the Einstein-Brillouin-Keller quantization rule from integrable systems to chaotic systems. According to the theory, the Fredholm determinant of the transfer operator, defined on a Poincaré section of a classical physical system, provides a quantization condition to the energy spectrum of the corresponding quantum system. This study presents two factorization formulas, which relate different quantization conditions defined on different classical trajectory segments. These explicit relations answer the question of why all these classical quantization conditions determine exactly the same energy spectrum of the corresponding quantum systems. As an example, these formulas are illustrated in the equilateral triangular billiard.

The aim of this paper is the construction of numerical tools forthe efficient approximation of transport phenomena in non-autonomous dynamical systems.We focus on transfer operator methods which have been developed in the last yearsfor the treatment of non-autonomous dynamical systems.For instance Froyland et al. [11] proposed a methodfor the approximation of so-called coherent pairs -- these pairs of setsrepresent time-dependent slowly mixing structures -- bythresholding singular vectors of a normalized transfer operator over a fixed time-interval.In principle such transfer operator methods involve long term simulationsof trajectories on the whole state space. In our main result we showthat transport phenomena over a fixed (long) time horizonimply the existence of almost invariant sets over shorter time intervalsif the transport process is slow enough.This fact is used to formulate an algorithm that preselects part of state spaceas a candidate for containing one of the sets of a coherent pair.By this we significantly reduce the related numerical effort.

There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism, occuring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2)_k anyons, well-known deformations of SU(2), in which only the first k+1 angular momenta of SU(2) occur. In this manuscript, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin S=1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter k when considering su(2)_k anyonic theories with k>4, as well as for the special case of the su(2)_4 theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-1/2 chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.