Free-fermion Chain Research Articles

Boundary effects on symmetry resolved entanglement

We study the symmetry resolved entanglement entropies in one-dimensional systems with boundaries. We provide some general results for conformal invariant theories and then move to a semi-infinite chain of free fermions. We consider both an interval starting from the boundary and away from it. We derive exact formulas for the charged and symmetry resolved entropies based on theorems and conjectures about the spectra of Toeplitz+Hankel matrices. En route to characterise the interval away from the boundary, we prove a general relation between the eigenvalues of Toeplitz+Hankel matrices and block Toeplitz ones. An important aspect is that the saddle-point approximation from charged to symmetry resolved entropies introduces algebraic corrections to the scaling that are much more severe than in systems without boundaries.

Non-Hermitian quantum impurity systems in and out of equilibrium: Noninteracting case

We provide systematic analysis on a non-Hermitian PT -symmetric quantum impurity system both in and out of equilibrium, based on exact computations. In order to understand the interplay between non-Hermiticity and Kondo physics, we focus on a prototypical noninteracting impurity system, the resonant level model, with complex coupling constants. Explicitly constructing biorthogonal basis, we study its thermodynamic properties as well as the Loschmidt echo starting from the initially disconnected two free fermion chains. Remarkably, we observe the universal crossover physics in the Loschmidt echo, both in the PT broken and unbroken regimes. We also find that the ground state quantities we compute in the PT broken regime can be obtained by analytic continuation. It turns out that Kondo screening ceases to exist in the PT broken regime, which was also previously predicted in the non-hermitian Kondo model. All the analytical results are corroborated against biorthogonal free fermion numerics.

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion**Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF-2018-125), and by the Hungarian National Research, Development and Innovation Office (Grant No. K115383). Its was supported by the NSF grant DMS-1700261. Virtanen was supported in part by EPSRC grants EP/M024784/1 and EP/T008636/1. Gehér and

We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv:1104.1004, for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.

Entanglement and boundary entropy in quantum spin chains with arbitrary direction of the boundary magnetic fields

We calculate the entanglement and the universal boundary entropy (BE) in the critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory (BCFT) depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to the simple fact that the boundary field breaks the bulk symmetry or does not. We present a procedure to estimate the universal BE, based on the finite-size corrections of the entanglement entropy, that apply to the ADBMF. To calculate the universal BE in the XXZ chain, we use the density matrix renormalization group (DMRG). The transverse field XY chain with ADBMF after Jordan-Wigner (JW) transformation is not a quadratic free fermion Hamiltonian. We map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem {\it{integrable}}. The eigenstates of the transverse field XY chain can be obtained by proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.

Time evolution of entanglement negativity across a defect

We consider a quench in a free-fermion chain by joining two homogeneous half-chains via a defect. The time evolution of the entanglement negativity is studied between adjacent segments surrounding the defect. In case of equal initial fillings, the negativity grows logarithmically in time and essentially equals one-half of the Rényi mutual information with index α = 1/2 in the limit of large segments. In sharp contrast, in the biased case one finds a linear increase followed by the saturation at an extensive value for both quantities, which is due to the backscattering from the defect and can be reproduced in a quasiparticle picture. Furthermore, a closer inspection of the subleading corrections reveals that the negativity and the mutual information have a small but finite difference in the steady state. Finally, we also study a similar quench in the XXZ spin chain via density-matrix renormalization group methods and compare the results for the negativity to the fermionic case.

Entanglement Hamiltonians in 1D free lattice models after a global quantum quench

We study the temporal evolution of the entanglement Hamiltonian of an interval after a global quantum quench in free lattice models in one spatial dimension. In a harmonic chain we explore a quench of the frequency parameter. In a chain of free fermions at half filling we consider the evolution of the ground state of a fully dimerised chain through the homogeneous Hamiltonian. We focus on critical evolution Hamiltonians. The temporal evolutions of the gaps in the entanglement spectrum are analysed. The entanglement Hamiltonians in these models are characterised by matrices that provide also contours for the entanglement entropies. The temporal evolution of these contours for the entanglement entropy is studied, also by employing existing conformal field theory results for the semi-infinite line and the quasi-particle picture for the global quench.

Free-Fermion entanglement and orthogonal polynomials

We present a simple construction for a tridiagonal matrix T that commutes with the hopping matrix for the entanglement Hamiltonian of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since T is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of , and therefore the entanglement entropy.

Entanglement in a fermion chain under continuous monitoring

We study the entanglement entropy of the quantum trajectories of a free fermion chain under continuous monitoring of local occupation numbers. We propose a simple theory for entanglement entropy evolution from disentangled and highly excited initial states. It is based on generalized hydrodynamics and the quasi-particle pair approach to entanglement in integrable systems. We test several quantitative predictions of the theory against extensive numerics and find good agreement. In particular, the volume law entanglement is destroyed by the presence of arbitrarily weak measurement.

On the continuum limit of the entanglement Hamiltonian

We consider the entanglement Hamiltonian for an interval in a chain of free fermions in its ground state and show that the lattice expression goes over into the conformal one if one includes the hopping to distant neighbours in the continuum limit. For an infinite chain, this can be done analytically for arbitrary fillings and is shown to be the consequence of the particular structure of the entanglement Hamiltonian, while for finite rings or temperatures the result is based on numerical calculations.

Properties of the entanglement Hamiltonian for finite free-fermion chains**This paper is dedicated to the memory of Vladimir Rittenberg, inspiring colleague and founding editor of this journal.

We study the entanglement Hamiltonian for fermionic hopping models on rings and open chains and determine single-particle spectra, eigenfunctions and the form in real space. For the chain, we find a commuting operator as for the ring and compare with its properties in both cases. In particular, a scaling relation between the eigenvalues is found for large systems. We also show how the commutation property carries over to the critical transverse Ising model.

Analytical results for the entanglement Hamiltonian of a free-fermion chain**Dedicated to John Cardy on the occasion of his 70th birthday.

We study the ground-state entanglement Hamiltonian for an interval of N sites in a free-fermion chain with arbitrary filling. By relating it to a commuting operator, we find explicit expressions for its matrix elements in the large-N limit. The results agree with numerical calculations and show that deviations from the conformal prediction persist even for large systems.

Supersymmetry breaking and Nambu-Goldstone fermions in an extended Nicolai model

We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anti-commutator of two supercharges $Q$ and $Q^\dagger$, each of which is comprised solely of fermion operators and possesses one adjustable parameter $g$. When the parameter $g$ vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A: Math. Gen. \textbf{9}, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-$g$ limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when $g > 0$. We also rigorously prove that for sufficiently large $g$ the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.

Surface and bulk entanglement in free-fermion chains

We consider free-fermion chains where full and empty parts are connected by a transition region with narrow surfaces. This can be caused by a linear potential or by time evolution from a step-like initial state. Entanglement spectra, entanglement entropies and fluctuations are determined for subsystems either in the surface region or extending into the bulk. In all cases there is logarithmic behaviour in the subsystem size, but the prefactors in the surface differ from those in the bulk by 3/2. A previous fluctuation result is corrected and a general scaling formula is inferred from the data.

Area-law violation for the mutual information in a nonequilibrium steady state

We study the nonequilibrium steady state of an infinite chain of free fermions, resulting from an initial state where the two sides of the system are prepared at different temperatures. The mutual information is calculated between two adjacent segments of the chain and is found to scale logarithmically in the subsystem size. This provides an example of the violation of the area law in a quantum many-body system outside a zero-temperature regime. The prefactor of the logarithm is obtained analytically and, furthermore, the same prefactor is shown to govern the logarithmic increase of mutual information in time, before the system relaxes locally to the steady state.

Magnetic properties of the quantum spin- $\frac{1}{2}$ XX diamond chain: the Jordan-Wigner approach

The Jordan-Wigner transformation is applied to study magnetic properties of the quantum spin- $\frac{1}{2}$ XX model on the diamond chain. Generally, the Hamiltonian of this quantum spin system can be represented in terms of spinless fermions in the presence of a gauge field and different gauge-invariant ways of assigning the spin-fermion transformation are considered. Additionally, we analyze general properties of a free-fermion chain, where all gauge terms are neglected and discuss their relevance for the quantum spin system. A consideration of interaction terms in the fermionic Hamiltonian rests upon the Hartree-Fock procedure after fixing the appropriate gauge. Finally, we discuss the magnetic properties of this quantum spin model at zero as well as non-zero temperatures and analyze the validity of the approximation used through a comparison with the results of the exact diagonalization method for finite (up to 36 spins) chains. Besides the m = 1/3 plateau the most prominent feature of the magnetization curve is a jump at intermediate field present for certain values of the frustrating vertical bond.

Fano resonances and entanglement entropy

We study the entanglement in the ground state of a chain of free spinless fermions with a single side-coupled impurity. We find a logarithmic scaling for the entanglement entropy of a segment neighboring the impurity. The prefactor of the logarithm varies continuously and contains an impurity contribution described by a one-parameter function while the contribution of the unmodified boundary enters additively. The coefficient is found explicitly by pointing out similarities with other models involving interface defects. The proposed formula gives excellent agreement with our numerical data. If the segment has an open boundary, one finds a rapidly oscillating subleading term in the entropy that persists in the limit of large block sizes. The particle-number fluctuation inside the subsystem is also reported. It is analogous with the expression for the entropy scaling, however, with a simpler functional form for the coefficient.

Cooper pairs and exclusion statistics from coupled free-fermion chains

We show how to couple two free-fermion chains so that the excitations consist of Cooperpairs with zero energy, and free particles obeying (mutual) exclusion statistics. Thisbehaviour is reminiscent of anyonic superconductivity, and of a ferromagnetic version of theHaldane–Shastry spin chain, although here the interactions are local. We solve this modelusing the nested Bethe ansatz, and find all the eigenstates; the Cooper pairscorrespond to exact-string or ‘0/0’ solutions of the Bethe equations. We showhow the model possesses an infinite-dimensional symmetry algebra, which is asupersymmetric version of the Yangian symmetry algebra for the Haldane–Shastry model.

Renormalization Group Studies of the Fibonacci Quantum Spin Chains

Renormalization group studies of the ground state properties of the S =1/2 Fibonacci antiferromagnetic XXZ chain are reviewed. In the XY limit, this model can be mapped onto the free fermion chain which has been extensively studied using the renormalization group approach for the transfer matrix. For the XXZ chain with weak Fibonacci modulation, the investigation by means of the bosonization and perturbational renormalization group method is reviewed. Finally, the recent results of the real space renormalization group approach in the strong modulation limit is presented. Combining the real space renormalization group and the weak modulation renormalization group results, the schematic ground state phase diagram is presented.