Emptiness formation probability and Painlevé V equation in the XY spin chain

Abstract

We reconsider the problem of finding $L$ consecutive down spins in the ground state of the XY chain, a quantity known as the Emptiness Formation Probability. Motivated by new developments in the asymptotics of Toeplitz determinants, we show how the crossover between the critical and off-critical behaviour of the Emptiness Formation Probability is exactly described by a $\tau$ function of a Painlevé V equation. Following a recent proposal, we also provide a power series expansion for the $\tau$ function in terms of irregular conformal blocks of a Conformal Field Theory with central charge $c=1$. Our results are tested against lattice numerical calculations, showing excellent agreement. We finally rediscuss the free fermion case where the Emptiness Formation Probability is characterized by a Gaussian decay for large $L$.

arXiv comment25 pages, 9 figures. References and a footnote added

References in corpus (38)

• Entanglement and alpha entropies for a massive Dirac field in two dimensions
• Symmetry-resolved entanglement in many-body systems
• Inhomogeneous field theory inside the arctic circle
• Symmetry resolved entanglement in free fermionic systems
• Random Matrix Theory and Entanglement in Quantum Spin Chains
• How instanton combinatorics solves Painlevé VI, V and III's
• Correlation Function of the Spin-1/2 XXX Antiferromagnet
• Conformal field theory of Painlevé VI
• Characterizing correlations with full counting statistics: classical Ising and quantum XY spin chains
• The arctic curve of the domain-wall six-vertex model
• Determinant Representations for Correlation Functions of Spin-1/2 XXX and XXZ Heisenberg Magnets
• New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain
• Differential Equations for Sine-Gordon Correlation Functions at the Free Fermion Point
• Emptiness Formation Probability for the One-Dimensional Isotropic XY Model
• Large distance asymptotic behavior of the emptiness formation probability of the XXZ spin-1/2 Heisenberg chain
• Asymptotics of Toeplitz Determinants and the Emptiness Formation Probability for the XY Spin Chain
• Emptiness formation probability of the XXZ spin-1/2 Heisenberg chain at Delta=1/2
• From Form Factors to Correlation Functions: The Ising Model
• Emptiness formation probability in the domain-wall six-vertex model
• Emptiness formation probability, Toeplitz determinants, and conformal field theory
• Irregular conformal blocks and connection formulae for Painlevé V functions
• Asymptotic Behavior of the Emptiness Formation Probability in the Critical Phase of XXZ Spin Chain
• Emergence of a singularity for Toeplitz determinants and Painleve V
• Emptiness Formation Probability for the Anisotropic XY Spin Chain in a Magnetic Field
• Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations
• Two-point correlation functions of scaling fields in the Dirac theory on the Poincare disk
• Confluent conformal blocks and the Teukolsky master equation
• Formation probabilities and Shannon information and their time evolution after quantum quench in transverse-field XY-chain
• Argyres-Douglas theories, Painlevé II and quantum mechanics
• On the probability of ferromagnetic strings in antiferromagnetic spin chains
• Painlevé Functions in Statistical Physics
• A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions
• On the emptiness formation probability of the open XXZ spin-$\tf{1}{2}$ chain
• Finite size emptiness formation probability of the XXZ spin chain at $\Delta =-1/2$
• Formation probabilities in quantum critical chains and Casimir effect
• On the Möbius transformation in the entanglement entropy of fermionic chains
• Finite size corrections to scaling of the formation probabilities and the Casimir effect in the conformal field theories
• Entanglement entropy and Möbius transformations for critical fermionic chains

Cited by in corpus (9)

• Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains
• Formation probabilities and statistics of observables as defect problems in the free fermions and the quantum spin chains
• Exact full counting statistics for the staggered magnetization and the domain walls in the XY spin chain
• Kink correlations, domain size distribution, and the emptiness formation probability after the Kibble-Zurek quench in the quantum Ising chain
• Universal logarithmic correction to Rényi (Shannon) entropy in generic systems of critical quadratic fermions
• Scaling of the Formation Probabilities and Universal Boundary Entropies in the Quantum XY Spin Chain
• Boundary emptiness formation probabilities in the six-vertex model at $\Delta =-\frac{1}{2}$
• Shannon entropy in quasiparticle states of quantum chains
• Entanglement entropies of an interval in the free Schrödinger field theory at finite density