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arXiv:1307.1781 · Physical Review B

Block entanglement and fluctuations in finite-size correlated electron systems

Archak Purkayastha, V. Subrahmanyam

Published May 19, 2014• 5 citations
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Abstract

The block entanglement entropy and fluctuations are investigated in one dimension in finite-size correlated electron systems using the Gutzwiller wave function as a prototype correlated electron state. Entanglement entropy shows logarithmic divergence for all values of the correlation projection parameter $g$, as predicted by conformal field theories for critical systems, but the central charge requires finite-size corrections. There is an infinite correlation length corresponding to correlation between the same kinds of spins, for all values of $g$. A scaling form for the block entropy, as a function of $g$ and the system size $N$, is proposed which predicts a metal-insulator crossover at ${N}^{1/3}g\ensuremath{\approx}0.24$ for $N<{10}^{5}$. Bipartite fluctuations in the number of particles in a block, and the spin fluctuations also obey an approximate scaling. A relation is found between the block entropy and the bipartite spin fluctuations. Our results show some correspondence with an experiment on Ni nanochains.

DOI
10.1103/physrevb.89.195125

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The block entanglement entropy and fluctuations are investigated in one dimension in finite-size correlated electron systems using the Gutzwiller wave function as a prototype correlated electron state. Entanglement entropy shows logarithmic divergence for all values of the correlation projection parameter ggg, as predicted by conformal field theories for critical systems, but the central charge requires finite-size corrections. There is an infinite correlation length corresponding to correlation between the same kinds of spins, for all values of ggg. A scaling form for the block entropy, as a function of ggg and the system size NNN, is proposed which predicts a metal-insulator crossover at N1/3g\ensuremath≈0.24{N}^{1/3}g\ensuremath{\approx}0.24N1/3g\ensuremath≈0.24 for N<105N<{10}^{5}N<105. Bipartite fluctuations in the number of particles in a block, and the spin fluctuations also obey an approximate scaling. A relation is found between the block entropy and the bipartite spin fluctuations. Our results show some correspondence with an experiment on Ni nanochains.
Entanglement spectrum of finite-size correlated electron systems are investigated using the Gutzwiller projection technique. The product of largest eigenvalue and rank of the block reduced density matrix, which is a measure of distance of the state from the maximally entangled state of the corresponding rank, is seen to characterise the insulator to metal crossover in the state. The fraction of distinct eigenvalues exhibits a `chaotic' behaviour in the crossover region, and it shows a `integrable' behaviour at both insulating and metallic ends. The integrated entanglement spectrum obeys conformal field theory (CFT) prediction at the metal and insulator ends, but shows a noticeable deviation from CFT prediction in the crossover regime, thus it can also track a metal-insulator crossover. A modification of the CFT result for the entanglement spectrum for finite size is proposed which holds in the crossover regime also. The adjacent level spacing distribution of unfolded non-zero eigenvalues for intermediate values of Gutzwiller projection parameter ggg is the same as that of an ensemble of random matrices obtained by replacing each block of reduced density matrix by a random real symmetric Toeplitz matrix. It is strongly peaked at zero, with an exponential tail proportional to e−(n/R)se^{-(n/R)s}e−(n/R)s, where sss is the adjacent level spacing, nnn is number of distinct eigenvalues and RRR is the rank of the reduced density matrix.
We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge ccc in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of ccc. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.
In one dimension very general results from conformal field theory and exact calculations for quantum spin chains have established universal scaling properties of the entanglement entropy between two parts of a critical system. Using both analytical and numerical methods, we show that if particle number or spin is conserved, fluctuations in a subsystem obey identical scaling as a function of subsystem size in one dimension. We investigate the effects of boundaries and subleading corrections for critical spin and bosonic chains.
We analyze the finite-size corrections to entanglement in quantum critical systems. By using conformal symmetry and density functional theory, we discuss the structure of the finite-size contributions to a general measure of ground state entanglement, which are ruled by the central charge of the underlying conformal field theory. More generally, we show that all conformal towers formed by an infinite number of excited states (as the size of the system $L\ensuremath{\rightarrow}\ensuremath{\infty}$) exhibit a unique pattern of entanglement, which differ only at leading order (1∕L)2{(1∕L)}^{2}(1∕L)2. In this case, entanglement is also shown to obey a universal structure, given by the anomalous dimensions of the primary operators of the theory. As an illustration, we discuss the behavior of pairwise entanglement for the eigenspectrum of the spin-1∕21∕21∕2 XXZXXZXXZ chain with an arbitrary length LLL for both periodic and twisted boundary conditions.
The entanglement entropy is related to quantum phase transitions. It behaves differently when a quantum system is in a phase with a finite correlation length (such as an insulator phase), compared to a system in a critical phase in which the correlation length is infinite such as a metallic phase. The entanglement entropy of disordered systems does not depend only on the length of the relevant segment and the size of the system but also on the specific realization of disorder. Thus, the entanglement entropy of a disordered system exhibits mesoscopic fluctuations. Here we shall show numerical results that exhibit a crossover in the distribution of the entanglement entropy from a normal distribution at the weak localization regime to a log-normal distribution at the strong localization regime.
The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size $\ensuremath{\chi}$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\ensuremath{\sim}\frac{1}{6}\text{log}\text{ }\ensuremath{\chi}$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\ensuremath{\xi}}_{\ensuremath{\chi}}$ ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-$\ensuremath{\chi}$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ${\ensuremath{\xi}}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\kappa}}$, with $\ensuremath{\kappa}=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\ensuremath{\kappa}\ensuremath{\sim}1.37$. We also show how finite-$\ensuremath{\chi}$ scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.