An introduction to integrable techniques for one-dimensional quantum systems

Abstract

This monograph introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics. The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model. Finally, the low energy physics of these integrable models is mapped into the corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.

137 pages, with several figures. Minor corrections over the published version. Comments are always welcome!

References in corpus (9)

• Quenching the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs Ensemble Predictions
• The Statistics of the Work Done on a Quantum Critical System by Quenching a Control Parameter
• Correlations after quantum quenches in the XXZ spin chain: Failure of the Generalized Gibbs Ensemble
• Dynamical structure factor of one-dimensional Bose gases: experimental signatures of beyond-Luttinger liquid physics
• Fermi-Luttinger liquid: Spectral function of interacting one-dimensional fermions
• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Correlation functions of the one-dimensional attractive Bose gas
• Introduction to the thermodynamic Bethe ansatz
• Entropy of XY Spin Chain and Block Toeplitz Determinants

Cited by in corpus (161)

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• Universal spin dynamics in infinite-temperature one-dimensional quantum magnets
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• Operationally accessible entanglement of one dimensional spinless fermions
• The quantum Ising chain for beginners
• Non-Hermitian Floquet topological superconductors with multiple Majorana edge modes
• Catching Bethe phantoms and quantum many-body scars: Long-lived spin-helix states in Heisenberg magnets
• Machine Learning Phase Transitions with a Quantum Processor
• Fisher information approach to non-equilibrium phase transitions in quantum XXZ spin chain with boundary noise
• Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions
• Detection of quantum phases via out-of-time-order correlators
• Spread Complexity and Topological Transitions in the Kitaev Chain
• Time crystallinity in open quantum systems
• Front dynamics and entanglement in the XXZ chain with a gradient
• Floquet dynamical quantum phase transitions under synchronized periodic driving
• Controlling energy storage crossing quantum phase transitions in an integrable spin quantum battery
• Variational quantum eigensolver for the Heisenberg antiferromagnet on the kagome lattice
• Local and nonlocal order parameters in the Kitaev chain
• Quantum Many-Body Scars and Quantum Criticality
• Ground-state energy and excitation spectrum of the Lieb-Liniger model : accurate analytical results and conjectures about the exact solution
• Complexity of frustration: a new source of non-local non-stabilizerness
• Localization in spin chains with facilitation constraints and disordered interactions
• Predicting Gibbs-State Expectation Values with Pure Thermal Shadows
• Multi-class quantum classifiers with tensor network circuits for quantum phase recognition
• Hydrodynamical phase transition for domain-wall melting in the XY chain
• Magnetization and entanglement after a geometric quench in the XXZ chain
• Approximating many-body quantum states with quantum circuits and measurements
• Quantum Phase Transition induced by Topological Frustration
• Reaction-limited quantum reaction-diffusion dynamics
• Undecidability of the Spectral Gap in One Dimension
• The Frustration of being Odd: Universal Area Law violation in local systems
• Quantum time dynamics of 1D-Heisenberg models employing the Yang-Baxter equation for circuit compression
• Stabilizer entropy in non-integrable quantum evolutions
• Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension
• On the Sample Complexity of Quantum Boltzmann Machine Learning
• Characterizing quantum correlations in spin chains
• Topologically induced prescrambling and dynamical detection of topological phase transitions at infinite temperature
• Local quench spectroscopy of many-body quantum systems
• Unraveling the Excitation Spectrum of Many-Body Systems from Quantum Quenches
• Interactions and integrability in weakly monitored Hamiltonian systems
• Adiabatic formation of bound states in the 1d Bose gas
• Variational Gibbs State Preparation on NISQ devices
• Scale-free non-Hermitian skin effect in a boundary-dissipated spin chain
• Effects of defects in the XY chain with frustrated boundary conditions
• Critical quantum thermometry and its feasibility in spin systems
• Subsystem distance after a local operator quench
• Dynamical quantum phase transitions following a noisy quench
• HamLib: A library of Hamiltonians for benchmarking quantum algorithms and hardware
• Deviations from Off-Diagonal Long-Range Order in One-Dimensional Quantum Systems
• Universal off-diagonal long-range order behaviour for a trapped Tonks-Girardeau gas
• Return amplitude after a quantum quench in the XY chain
• Universal scaling of spin mixing dynamics in a strongly interacting one-dimensional Fermi gas
• Bound state dynamics in the long-range spin-$\frac{1}{2}$ XXZ model
• Charging free fermion quantum batteries
• Replacing neural networks by optimal analytical predictors for the detection of phase transitions
• Effects of energy extensivity on the quantum phases of long-range interacting systems
• Quantum Computing by Cooling
• Odd thermodynamic limit for the Loschmidt echo
• Rigorous proof for the non-local correlation functions in the antiferromagnetic seamed transverse Ising ring
• Resilience of the topological phases to frustration
• Ring frustration and factorizable correlation functions of critical spin rings
• Functional field integral approach to quantum work
• Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field
• String and conventional order parameters in the solvable modulated quantum chain
• Long-range entanglement and topological excitations
• Magnetic excitations in the one-dimensional Heisenberg-Ising model with external fields and their experimental realizations
• Achieving spin-squeezed states by quench dynamics in a quantum chain
• Magic phase transition and non-local complexity in generalized $W$ State
• Disorder lines, modulation, and partition function zeros in free fermion models
• Nonstabilizerness in the unitary and monitored quantum dynamics of XXZ-staggered and SYK models
• Deep learning Local Reduced Density Matrices for Many-body Hamiltonian Estimation
• Simulating continuous symmetry models with discrete ones
• Topological phases of an interacting Majorana Benalcazar-Bernevig-Hughes model
• Charge-4$e$ Superconductivity in a Hubbard model
• Static and dynamic phases of a Tonks-Girardeau gas in an optical lattice
• Out-of-Time-Order Correlators in One-Dimensional XY model
• Towards a phase diagram of the topologically frustrated XY chain
• Detection of Berezinskii-Kosterlitz-Thouless transition via Generative Adversarial Networks
• Random unitaries, Robustness, and Complexity of Entanglement
• Universality and two-body losses: lessons from the effective non-Hermitian dynamics of two particles
• Anti Kibble-Zurek behavior in the quantum XY spin-1/2 chain driven by correlated noisy magnetic field and anisotropy
• Phase diagram and topological order in the modulated $XYZ$ chain with magnetic fields
• Open quantum dynamics theory for a complex subenvironment system with a quantum thermostat: Application to a spin heat bath
• Edge modes and boundary impurities in the anisotropic Heisenberg spin chain
• Classical origins of Landau-incompatible transitions
• Disentanglement, disorder lines, and Majorana edge states in a solvable quantum chain
• Lower Bounds for the Trotter Error
• A Comprehensive Cross-Model Framework for Benchmarking the Performance of Quantum Hamiltonian Simulations
• Robustness and eventual slow decay of bound states of interacting microwave photons in the Google Quantum AI experiment
• Resilience of topological superconductivity under particle current
• Squashed entanglement in one-dimensional quantum matter
• Spectrum statistics in the integrable Lieb-Liniger model
• Confinement of many-body Bethe strings
• Charging a Dimerized Quantum XY Chain
• Quantum Groups as Global Symmetries
• Linear and Non-Linear Response of Quadratic Lindbladians
• Prethermalization in coupled one-dimensional quantum gases
• Semiclassical simulations predict glassy dynamics for disordered Heisenberg models
• Phase transition in complex-time Loschmidt echo of short and long range spin chain
• Local integrals of motion in dipole-conserving models with Hilbert space fragmentation
• Exact spectral function of the Tonks-Girardeau gas at finite temperature
• Real space circuit complexity as a probe of phase diagrams
• From Ising model to Kitaev Chain -- An introduction to topological phase transitions
• Hidden time-reversal in driven XXZ spin chains: exact solutions and new dissipative phase transitions
• Super-Tonks-Girardeau Quench in the Extended Bose-Hubbard Model
• One-body dynamical correlation function of Lieb-Liniger model at finite temperature
• Exact volume-law entangled zero-energy eigenstates in a large class of spin models
• Quantum Coherence of Topologically Frustrated Spin Chains
• Dynamics of one-dimensional quantum many-body systems in time-periodic linear potentials
• Exactly Solvable 1D Quantum Models with Gamma Matrices
• Quantum state transfer using 1D Heisenberg Hamiltonian on quasi-1D lattices
• Self-Supervised Ensemble Learning: A Universal Method for Phase Transition Classification of Many-Body Systems
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• Separability and parity transitions in $XYZ$ spin systems under nonuniform fields
• Entanglement generation and scaling from noisy quenches across a quantum critical point
• Assessing bound states in a one-dimensional topological superconductor: Majorana versus Tamm
• Emergent Carroll symmetry at phase separation in one-dimensional lattice systems
• Observation of a generalized Gibbs ensemble in photonics
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• Strong interaction induced dimensional crossover in 1D quantum gas
• Estimating entanglement in 2D Heisenberg model in the strong rung-coupling limit
• The Hilbert-space structure of free fermions in disguise
• An exact local mapping from clock-spins to fermions
• Long-distance genuine multipartite Entanglement between Magnetic Defects in Spin Chains
• Universal quasi-Fermi liquid physics of one-dimensional interacting fermions
• Few-body precursors of topological frustration
• Subsystem Evolution Speed as Indicator of Relaxation
• Bethe Ansatz Solutions for Certain Periodic Quantum Circuits
• Berry connection and quantum geometry in time-dependent systems with instantaneous quantum integrable field theory
• Universal efficiency boost in prethermal quantum heat engines at negative temperature
• The fate of local order in topologically frustrated spin chains
• Dynamics of quantum Fisher and Wigner-Yanase skew information following a noisy quench
• Harnessing spin-qubit decoherence to probe strongly-interacting quantum systems
• Logarithmic growth of peripheral entanglement concentrated via noisy measurements in a star network of spins
• Dynamic structure factor of quantum hard rods from exact form-factors
• Self-congruent point in critical matrix product states: An effective field theory for finite-entanglement scaling
• Fractal decompositions and tensor network representations of Bethe wavefunctions
• Finite time path field theory perturbative methods for local quantum spin chain quenches
• Entanglement entropy of two disjoint intervals and spin structures in interacting chains in and out of equilibrium
• Finite time path field theory and a new type of universal quantum spin chain quench behavior
• The quantum XY chain with boundary fields: finite-size gap and phase behavior
• Symmetry oscillations sensitivity to SU(2)-symmetry breaking in quantum mixtures
• Ultrafast light-induced long range antiferromagnetic correlations in paramagnets
• Entanglement transition in a cluster spin chain coupled with free spins
• Separation of quadrupole, spin, and charge across the magnetic phases of a one-dimensional interacting spin-1 gas
• Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors
• Necklace Ansatz for strongly repulsive spin mixtures on a ring
• Efficient computation of average subsystem Bures distance between fermionic Gaussian states
• Detecting entanglement with transport measurement in weakly interacting and fluctuating systems
• Resource complexity of Symmetry Protected Topological phases
• Loop Charges and Fragmentation in Pairwise Difference Conserving Circuits
• Ground state energy and phase transitions of Long-range XXZ using VQE
• Cluster Ising quantum batteries can mimic super-extensive charging power
• Effective metal-insulator non-equilibrium quantum phase transition
• Universal features of non-analytical energy storage in quantum critical quantum batteries
• Many-body thermal states on a quantum computer: a variational approach