Entanglement entropy in random quantum spin-Schains

Abstract

We discuss the scaling of entanglement entropy in the random singlet phase (RSP) of disordered quantum magnetic chains of general spin-S. Through an analysis of the general structure of the RSP, we show that the entanglement entropy scales logarithmically with the size of a block and we provide a closed expression for this scaling. This result is applicable for arbitrary quantum spin chains in the RSP, being dependent only on the magnitude S of the spin. Remarkably, the logarithmic scaling holds for the disordered chain even if the pure chain with no disorder does not exhibit conformal invariance, as is the case for Heisenberg integer spin chains. Our conclusions are supported by explicit evaluations of the entanglement entropy for random spin-1 and spin-3/2 chains using an asymptotically exact real-space renormalization group approach. I -INTRODUCTIONQuantum information science provides fruitful connections between different branches of physics. In this context, the relationship between entanglement, which is a fundamental resource for quantum information applications [1], and the theory of quantum critical phenomena [2,3] has been a focus of intensive research. Specifically, entanglement measures have been proposed as a tool to characterize quantum phase transitions (See, e.g., Refs. [4,5,6,7,8]). In this direction, a successful approach has been the analysis of bipartite entanglement in quantum systems as measured by the von Neumann entropy. Given a quantum system in a pure state |ψ and a bipartition of the system into two blocks A and B, entanglement between A and B can be measured by the von Neumann entropy S of the reduced density matrix of either block, i.e.,where ρ A = Tr B ρ and ρ B = Tr A ρ denote the reduced density matrices of blocks A and B, respectively, with ρ = |ψ ψ|. By evaluating S for quantum spin systems, Ref.[6] numerically found that entanglement displays a logarithmic scaling with the size of the block in critical (gapless) chains and saturates to a constant value in chains with a gap for excitations. The logarithmic scaling was proven in general for one-dimensional quantum models exhibiting conformal invariance [9,10], with the scaling governed by a universal factor, given by the central charge of the associated conformal field theory. Indeed, for a block of spins of length L in a quantum chain, von Neumann entropy S(L) scales aswhere c is the central charge and k is a non-universal constant. Recently, the behavior of entanglement entropy in critical spin chains has also been discussed in presence of disorder [11,12,13,14,15]. Disorder appears as an essential feature in a number of condensed matter systems, motivating a great deal of theoretical and experimental research (e.g., see [16,17]). Moreover, disorder usually introduces a further effect, namely, the breaking of conformal symmetry in a critical model. Remarkably, Refael and Moore [11] have shown that, even in the absence of conformal invariance, due to disorder, the logarithmic scaling given by Eq. (2) holds for the spin-1/2 rand...

Increasing entanglement through engineered disorder in the random Ising chain

“…For general nonhomogeneous models, as said before, the entropy still grows logarithmically but with a different prefactor c. 17,19,20 The d = 2 model can be solved analytically in both the homogeneous and disordered case as it reduces, as noticed earlier, to the transverse field Ising model: 26,27 one first performs a Jordan-Wigner transformation that maps the spins into fermions, then using a Bogoliubov transformation, whose coefficients are found via an exact diagonalization, the system is mapped into noninteracting fermions. Finally, the eigenvalues of the reduced density matrix l and thus the entanglement can be determined from the two-point regular and anomalous correlation functions as shown in Ref.…”

“…16 The average block entropy has been also studied in the context of random critical spin chains. 12,[17][18][19][20][21][22] In this case, on the one hand works on random Ising and XXZ spin-1/2 chains 12,17,18 suggest that the block entropy still grows logarithmically with the size of the block but with a smaller prefactor of the logarithm with respect to the ordered model, being "renormalized" by a factor ln 2. On the other hand, the block entropy in random quantum Potts chains with spin dimension d Ն 2 has been studied in Ref.…”

Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

“…In these systems the off-critical regions are also gapless and the excitation energies in these socalled Griffiths phases scale as ǫ ∼ L −z with a nonuniversal dynamical exponent z < ∞. Even so, certain random critical points in 1D are shown to have logarithmic divergences of entanglement entropy with universal coefficients, as in the pure case; these include infiniterandomness fixed points in the random-singlet universality class [12,13,14,15,16] and a class of aperiodic singlet phases [17].…”

“…In these systems the off-critical regions are also gapless and the excitation energies in these socalled Griffiths phases scale as ǫ ∼ L −z with a nonuniversal dynamical exponent z < ∞. Even so, certain random critical points in 1D are shown to have logarithmic divergences of entanglement entropy with universal coefficients, as in the pure case; these include infiniterandomness fixed points in the random-singlet universality class [12,13,14,15,16] and a class of aperiodic singlet phases [17].In this paper we consider the random quantum Ising model in two dimensions (2D), and examine the disorderaveraged entanglement entropy. The critical behavior of this system is governed by an IRFP [10,11] implying that the disorder strength grows without limit as the system is coarse grained in the renormalization group (RG) sense.…”

Universal and nonuniversal contributions to block-block entanglement in many-fermion systems

“…(3), depending on x and L only, and the second is the systemspecific term s 1 , depending on n and U , which we extract from the thermodynamic limit (6). The third by definition comprises all possible finite-size corrections not resulting from the CC analysis and neither contained in the infinite-size limit leading to the identification s 1 (n, U ) = S(x = 1, L → ∞; n, U ).…”