Free fermions at the edge of interacting systems

Abstract

We study the edge behavior of inhomogeneous one-dimensional quantum systems, such as Lieb-Liniger models in traps or spin chains in spatially varying magnetic fields. For free systems these fall into several universality classes, the most generic one being governed by the Tracy-Widom distribution. We investigate in this paper the effect of interactions. Using semiclassical arguments, we show that since the density vanishes to leading order, the strong interactions in the bulk are renormalized to zero at the edge, which simply explains the survival of Tracy-Widom scaling in general. For integrable systems, it is possible to push this argument further, and determine exactly the remaining length scale which controls the variance of the edge distribution. This analytical prediction is checked numerically, with excellent agreement. We also study numerically the edge scaling at fronts generated by quantum quenches, which provide new universality classes awaiting theoretical explanation.

arXiv comment31 pages, 10 figures. v2: various minor improvements, references added. v3: minor improvements

References in corpus (65)

• Level-Spacing Distributions and the Airy Kernel
• One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases
• Calculation of reduced density matrices from correlation functions
• Shape Fluctuations and Random Matrices
• Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents
• Emergent hydrodynamics in integrable quantum systems out of equilibrium
• Open XXZ spin chain: Nonequilibrium steady state and strict bound on ballistic transport
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)
• Bosons in cigar-shape traps: Thomas-Fermi regime, Tonks-Girardeau regime, and between
• Hydrodynamic Diffusion in Integrable Systems
• Time-evolving a matrix product state with long-ranged interactions
• Spin diffusion from an inhomogeneous quench in an integrable system
• Finite temperature Drude weight of the one dimensional spin 1/2 Heisenberg model}
• Bethe-Boltzmann Hydrodynamics and Spin Transport in the XXZ Chain
• Asymptotics in ASEP with Step Initial Condition
• Density-Matrix Spectra of Solvable Fermionic Systems
• Real-time dynamics in spin-1/2 chains with adaptive time-dependent DMRG
• Families of quasi-local conservation laws and quantum spin transport
• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases
• Kinetic theory of spin diffusion and superdiffusion in XXZ spin chains
• Transport in the XX chain at zero temperature: Emergence of flat magnetization profiles
• Microscopic origin of ideal conductivity in integrable quantum models
• Analytic solution of the Domain Wall non-equilibrium stationary state
• Full counting statistics in a propagating quantum front and random matrix spectra
• The thermal conductivity of the spin-1/2 XXZ chain at arbitrary temperature
• Super-diffusion in one-dimensional quantum lattice models
• Inhomogeneous field theory inside the arctic circle
• The Pearcey Process
• Universality in the full counting statistics of trapped fermions
• Logarithmic current fluctuations in non-equilibrium quantum spin chains
• Non-equilibrium steady states in the quantum XXZ spin chain
• Relaxation dynamics in the gapped XXZ spin-1/2 chain
• Inhomogeneous quenches in a fermionic chain: exact results
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• Engineering the Dynamics of Effective Spin-Chain Models for Strongly Interacting Atomic Gases
• Dynamical free energy and the Loschmidt-echo for a class of quantum quenches in the Heisenberg spin chain
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Dynamics of the spin-1/2 Heisenberg chain initialized in a domain-wall state
• Local correlations in a strongly interacting 1D Bose gas
• Random matrices and entanglement entropy of trapped Fermi gases
• Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids
• The arctic curve of the domain-wall six-vertex model
• Noninteracting fermions in a trap and random matrix theory
• From the Quantum Transfer Matrix to the Quench Action: The Loschmidt echo in $XXZ$ Heisenberg spin chains
• Non-analytic behavior of the Loschmidt echo in XXZ spin chains: exact results
• Domain wall dynamics in the Landau--Lifshitz magnet and the classical-quantum correspondence for spin transport
• Large deviations of the maximal eigenvalue of random matrices
• Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain
• Front dynamics and entanglement in the XXZ chain with a gradient
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Emptiness formation probability in the domain-wall six-vertex model
• Nonintersecting random walks in the neighborhood of a symmetric tacnode
• The spin Drude weight of the XXZ chain and generalized hydrodynamics
• The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases
• Out-of-equilibrium dynamics in a quantum impurity model: numerics for particle transport and entanglement entropy
• Exact numerical simulations of a one-dimensional, trapped Bose gas
• Subdiffusive front scaling in interacting integrable models
• A class of states supporting diffusive spin dynamics in the isotropic Heisenberg model
• The right tail exponent of the Tracy-Widom-beta distribution
• The periodic Schur process and free fermions at finite temperature
• Surface and bulk entanglement in free-fermion chains
• Searching for the Tracy-Widom distribution in nonequilibrium processes

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• Extreme value statistics of correlated random variables: a pedagogical review
• Superdiffusion in spin chains
• Kardar-Parisi-Zhang universality from soft gauge modes
• Quantum information scrambling after a quantum quench
• Entanglement revivals as a probe of scrambling in finite quantum systems
• Domain wall problem in the quantum XXZ chain and semiclassical behavior close to the isotropic point
• Entanglement entropies of inhomogeneous Luttinger liquids
• Full counting statistics for interacting trapped fermions
• Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 XXZ model
• Exact large deviation function of spin current for the one dimensional XX spin chain with domain wall initial condition
• Front dynamics in the XY chain after local excitations
• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes
• Extreme boundary conditions and random tilings
• Wigner function for noninteracting fermions in hard wall potentials
• Limit shape phase transitions: a merger of arctic circles
• Scaling at the OTOC Wavefront: free versus chaotic models
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Exact time evolution formulae in the XXZ spin chain with domain wall initial state
• Ground-state energy distribution of disordered many-body quantum systems
• Domain wall dynamics in classical spin chains: free propagation, subdiffusive spreading, and soliton emission
• Non-probabilistic fermionic limit shapes
• Multicritical Schur measures and higher-order analogues of the Tracy-Widom distribution
• Interface fluctuations associated with split Fermi seas
• Gradient corrections to the local density approximation in the one-dimensional Bose gas
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Domain-wall melting and entanglement in free-fermion chains with a band structure