We study effects of disorder (quenched randomness) in a two-dimensional square-lattice S = 1/2 quantum spin system, the J-Q model with a multi-spin interaction Q supplementing the Heisenberg exchange J. In the absence of disorder the system hosts antiferromagnetic (AFM) and columnar valence-bond-solid (VBS) ground states. The VBS breaks Z4 symmetry spontaneously, and in the presence of arbitrarily weak disorder it forms domains. Using quantum Monte Carlo simulations, we demonstrate two different kinds of such disordered VBS states. Upon dilution, a removed site in one sublattice forces a left-over localized spin in the opposite sublattice. These spins interact through the host system and always form AFM order. In the case of random J or Q interactions in the intact lattice, we find a different, spin-liquid-like state with no magnetic or VBS order but with algebraically decaying mean correlations. Here we identify localized spinons at the nexus of domain walls separating regions with the four different VBS patterns. These spinons form correlated groups with the same number of spinons and antispinons. Within such a group, we argue that there is a strong tendency to singlet formation, because of the native pairing and relatively strong spinonspinon interactions mediated by the domain walls. Thus, the spinon groups are effectively isolated from each other and no long-range AFM order forms. The mean spin correlations decay as r −2 as a function of distance r. We propose that this state is a two-dimensional analogue of the well-known random singlet (RS) state in one dimension, though, in contrast to the latter, the dynamic exponent z here is finite. By studying quantum-critical scaling of the magnetic susceptibility, we find that z varies, taking the value z = 2 at the AFM-RS phase boundary and growing upon moving into the RS phase (thus causing a power-law divergent susceptibility). The RS state discovered here in a system without geometric frustration may correspond to the same fixed point as the RS state recently proposed for frustrated systems, and the ability to study it without Monte Carlo sign problems opens up opportunities for further detailed characterization of its static and dynamic properties. We also discuss experimental evidence of the RS phase in the quasi-two-dimensional square-lattice random-exchange quantum magnets Sr2CuTe1−xWxO6 for x in the range 0.2 − 0.5. *

The quantum Ising chain of length, L, which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is non-universal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time t > 0. For the localized defect the increase of the entropy with log L or with log t involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extended defect the equilibrium entropy is saturated, but the non-equilibrium entropy has a logarithmic time-dependence the prefactor of which depends on the strength of the defect.

By means of free fermionic techniques we study the time evolution of the entanglement entropy, S(t), of a block of spins in the random transverse-field Ising chain after a sudden change of the parameters of the Hamiltonian. We consider global quenches, when the parameters are modified uniformly in space, as well as local quenches, when two disconnected blocks are suddenly joined together. For a non-critical final state, the dynamical entanglement entropy is found to approach a finite limiting value for both types of quenches. If the quench is performed to the critical state, the entropy grows for an infinite block as S(t) \sim ln ln t. This type of ultraslow increase is explained through the strong disorder renormalization group method.Comment: 8 pages, 8(+1) figures; published versio

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The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite randomness fixed point in the diluted transverse Ising model in higher dimensions. PACS numbers: Valid PACS appear hereExtensive studies have been devoted recently to understand ground state entanglement in quantum many-body systems [1]. In particular, the behavior of various entanglement measures at/near quantum phase transitions has been of special interest. One of the widely used entanglement measures is the von Neumann entropy, which quantifies entanglement of a pure quantum state in a bipartite system. Critical ground states in one dimension (1D) are known to have entanglement entropy that diverges logarithmically in the subsystem size with a universal coefficient determined by the central charge of the associated conformal field theory [2]. Away from the critical point, the entanglement entropy saturates to a finite value, which is related to the finite correlation length.In higher dimensions, the scaling behavior of the entanglement entropy is far less clear. A standard expectation is that non-critical entanglement entropy scales as the area of the boundary between the subsystems, known as the "area law" [3,4]. This area-relationship is known to be violated for gapless fermionic systems [5] in which a logarithmic multiplicative correction is found. One might suspect that whether the area law holds or not depends on whether the correlation length is finite or diverges. However, it has turned out that the situation is more complex: numerical findings [7] and a recent analytical study [8] have shown that the area law holds even for critical bosonic systems, despite a divergent correlation length. This indicates that the length scale associated with entanglement may differ from the correlation length. Another ongoing research activity for entanglement in higher spatial dimensions is to understand topological contributions to the entanglement entropy [9].The nature of quantum phase transitions with quenched randomness is in many systems quite different from the pure case. For instance, in a class of systems the critical behavior is governed by a so-called infiniterandomness fixed point (IRFP), at which the energy scale ǫ and the length scale L are related as: ln ǫ ∼ L ψ with 0 < ψ < 1. 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...