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arXiv:2408.16531 · Journal of Statistical Mechanics Theory and Experiment

Fermionic logarithmic negativity in the Krawtchouk chain

Gabrielle Blanchet, Gilles Parez, Luc Vinet

Published Nov 6, 2024• 5 citations
arXivHTMLPDF

Abstract

Abstract The entanglement of non-complementary regions is investigated in an inhomogeneous free-fermion chain through the lens of the fermionic logarithmic negativity. Focus is on the Krawtchouk chain, whose relation to the eponymous orthogonal polynomials allows for exact diagonalization and analytical calculations of certain correlation functions. For adjacent regions, the negativity scaling corresponds to that of a conformal field theory with central charge c = 1, in agreement with previous studies on bipartite entanglement in the Krawtchouk chain. For disjoint regions, we focus on the skeletal regime where each region reduces to a single site. This regime is sufficient to extract the leading behaviour at large distances. In the bulk, the negativity decays as d − 4 Δ f with Δ f = 1 / 2 , where d is the separation between the regions. This is in agreement with the homogeneous result of free Dirac fermions in one dimension. Surprisingly, when one site is close to the boundary, this exponent changes and depends on the parity of the boundary site m = 0 , 1 , 2 , … , with Δ f even = 3 / 8 and Δ f odd = 5 / 8 . The results are supported by numerics and analytical calculations.

DOI
10.1088/1742-5468/ad84d8

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Abstract The entanglement of non-complementary regions is investigated in an inhomogeneous free-fermion chain through the lens of the fermionic logarithmic negativity. Focus is on the Krawtchouk chain, whose relation to the eponymous orthogonal polynomials allows for exact diagonalization and analytical calculations of certain correlation functions. For adjacent regions, the negativity scaling corresponds to that of a conformal field theory with central charge c = 1, in agreement with previous studies on bipartite entanglement in the Krawtchouk chain. For disjoint regions, we focus on the skeletal regime where each region reduces to a single site. This regime is sufficient to extract the leading behaviour at large distances. In the bulk, the negativity decays as d − 4 Δ f with Δ f = 1 / 2 , where d is the separation between the regions. This is in agreement with the homogeneous result of free Dirac fermions in one dimension. Surprisingly, when one site is close to the boundary, this exponent changes and depends on the parity of the boundary site m = 0 , 1 , 2 , … , with Δ f even = 3 / 8 and Δ f odd = 5 / 8 . The results are supported by numerics and analytical calculations.
We study the entanglement between disjoint subregions in quantum critical systems through the lens of the logarithmic negativity. We work with systems in arbitrary dimensions, including conformal field theories and their corresponding lattice Hamiltonians, as well as resonating valence-bond states. At small separations, the logarithmic negativity is big and displays universal behavior, but we show nonperturbatively that it decays faster than any power at large separations. This can already be seen in the minimal setting of single-spin subregions. The corresponding absence of distillable entanglement at large separations generalizes the one-dimensional result, and indicates that quantum critical ground states do not possess long-range bipartite entanglement, at least for bosons. For systems with fermions, a more suitable definition of the logarithmic negativity exists that takes into account fermion parity, and we show that it decays algebraically. Along the way we obtain general results for the moments of the partially transposed density matrix. Published by the American Physical Society 2024
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Abstract We introduce a family of inhomogeneous XX spin chains whose squared couplings are a polynomial of degree at most four in the site index. We show how to obtain an asymptotic approximation for the Rényi entanglement entropy of all such chains in a constant magnetic field at half filling by exploiting their connection with the conformal field theory of a massless Dirac fermion in a suitably curved static background. We study the above approximation for three particular chains in the family, two of them related to well-known quasi-exactly solvable quantum models on the line and the third one to classical Krawtchouk polynomials, finding an excellent agreement with the exact value obtained numerically when the Rényi parameter α is less than one. When α ≥ 1 we find parity oscillations, as expected from the homogeneous case, and show that they are very accurately reproduced by a modification of the Fagotti-Calabrese formula. We have also analyzed the asymptotic behavior of the Rényi entanglement entropy in the non-standard situation of arbitrary filling and/or inhomogeneous magnetic field. Our numerical results show that in this case a block of spins at each end of the chain becomes disentangled from the rest. Moreover, the asymptotic approximation for the case of half filling and constant magnetic field, when suitably rescaled to the region of non-vanishing entropy, provides a rough approximation to the entanglement entropy also in this general case.
We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose of the reduced density matrix of a subsystem A = A 1 ∪ A 2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces are obtained. The logarithmic negativity is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ 1 ,ℓ 2 in an infinite system, we derive the result ∼ ( c /4)ln(ℓ 1 ℓ 2 /(ℓ 1 + ℓ 2 )), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n → 1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.
Abstract We study the ground-state entanglement Hamiltonian of several disjoint intervals for the massless Dirac fermion on the half-line. Its structure consists of a local part and a bi-local term that couples each point to another one in each other interval. The bi-local operator can be either diagonal or mixed in the fermionic chiralities and it is sensitive to the boundary conditions. The knowledge of such entanglement Hamiltonian is the starting point to evaluate the negativity Hamiltonian, i.e. the logarithm of the partially transposed reduced density matrix, which is an operatorial characterisation of entanglement of subsystems in mixed states. We find that the negativity Hamiltonian inherits the structure of the corresponding entanglement Hamiltonian. We finally show how the continuum expressions for both these operators can be recovered from exact numerical computations in free-fermion chains.
Abstract We investigate quantum entanglement in a non-relativistic critical system by calculating the logarithmic negativity of a class of mixed states in the quantum Lifshitz model in one and two spatial dimensions. In 1+1 dimensions we employ a correlator approach to obtain analytic results for both open and periodic biharmonic chains. In 2+1 dimensions we use a replica method and consider spherical and toroidal spatial manifolds. In all cases, the universal finite part of the logarithmic negativity vanishes for mixed states defined on two disjoint components. For mixed states defined on adjacent components, we find a non-trivial logarithmic negativity reminiscent of two-dimensional conformal field theories. As a byproduct of our calculations, we obtain exact results for the odd entanglement entropy in 2+1 dimensions.