Strong Disorder Renormalization Group Method Research Articles

Multipartite entanglement in the random Ising chain

Quantifying entanglement of multiple subsystems is a challenging open problem in interacting quantum systems. Here, we focus on two subsystems of length $\ensuremath{\ell}$ separated by a distance $r=\ensuremath{\alpha}\ensuremath{\ell}$ and quantify their entanglement negativity ($\mathcal{E}$) and mutual information ($\mathcal{I}$) in critical random Ising chains. We find universal constant $\mathcal{E}(\ensuremath{\alpha})$ and $\mathcal{I}(\ensuremath{\alpha})$ over any distances, using the asymptotically exact strong disorder renormalization group method. Our results are qualitatively different from both those in the clean Ising model and random spin chains of a singlet ground state, like the spin-$\frac{1}{2}$ random Heisenberg chain and the random XX chain. While for random singlet states $\mathcal{I}(\ensuremath{\alpha})/\mathcal{E}(\ensuremath{\alpha})=2$, in the random Ising chain this universal ratio is strongly $\ensuremath{\alpha}$ dependent. This deviation between systems contrasts with the behavior of the entanglement entropy of a single subsystem, for which the various random critical chains and clean models give the same qualitative behavior. The reason is that $\mathcal{E}$ and $\mathcal{I}$ are sensitive to higher order correlations in the ground-state structure. Therefore, studying multipartite entanglement provides additional universal information in random quantum systems, beyond what we can learn from a single subsystem.

Testing the validity of random-singlet state for long-range hopping models through the scaling of entanglement entropy

We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with random hopping amplitudes decaying with the distance as $|t_l|\sim l^{-\alpha}$, and address the question how far an analogue of the random-singlet state (RSS) conceived originally for describing the ground state of certain random spin chains is valid for this model. For this purpose, we study the effective central charge characterizing the logarithmic divergence of the entanglement entropy (EE) and the prefactor of the distribution of distances between localization centers on the two sublattices, which must fulfill a consistency relation for a RSS. For $\alpha>1$, we find by exact diagonalization an overall logarithmic divergence of the entanglement entropy with an effective central charge varying with $\alpha$. The large $\alpha$ limit of the effective central charge is found to be different from that of the nearest-neighbor hopping model. The consistency relation of RSS is violated for $\alpha\le 2$, while for $\alpha>2$ it is possibly valid, but this conclusion is hampered by a crossover induced by the short-range fixed point. The EE is also calculated by the strong-disorder renormalization group (SDRG) method numerically. Besides the traditional scheme, we construct and apply a more efficient minimal SDRG scheme having a linear (nearest-neighbor) structure, which turns out to be an accurate approximation of the full SDRG scheme for not too small $\alpha$. The SDRG method is found to provide systematically lower effective central charges than exact diagonalization does, nevertheless it becomes more and more accurate for increasing $\alpha$. Furthermore, as opposed to nearest-neighbor models, it indicates a weak dependence on the disorder distribution.

Quantum multicritical point in the two- and three-dimensional random transverse-field Ising model

Quantum multicritical points (QMCPs) emerge at the junction of two or more quantum phase transitions due to the interplay of disparate fluctuations, leading to novel universality classes. While quantum critical points have been well characterized, our understanding of QMCPs is much more limited, even though they might be less elusive to study experimentally than quantum critical points. Here, we characterize the QMCP of an interacting heterogeneous quantum system in two and three dimensions, the ferromagnetic random transverse-field Ising model (RTIM). The QMCP of the RTIM emerges due to both geometric and quantum fluctuations, studied here numerically by the strong disorder renormalization group method. The QMCP of the RTIM is found to exhibit ultraslow, activated dynamic scaling, governed by an infinite disorder fixed point. This ensures that the obtained multicritical exponents tend to the exact values at large scales, while also being universal---i.e., independent of the form of disorder---providing a solid theoretical basis for future experiments.

Powerful method to evaluate the mass gaps of free-particle quantum critical systems

We present a numerical method for the evaluation of the mass gap, and the low-lying energy gaps, of a large family of free-fermionic and free-parafermionic quantum chains. The method is suitable for some generalizations of the quantum Ising and XY models with multispin interactions. We illustrate the method by considering the Ising quantum chains with uniform and random coupling constants. The mass gaps of these quantum chains are obtained from the largest root of a characteristic polynomial. We also show that the Laguerre bound, for the largest root of a polynomial, used as an initial guess for the largest root in the method, gives us estimates for the mass gaps sharing the same leading finite-size behavior as the exact results. This opens an interesting possibility of obtaining precise critical properties very efficiently which we explore by studying the critical point and the paramagnetic Griffiths phase of the quantum Ising chain with random couplings. In this last phase, we obtain the effective dynamical critical exponent as a function of the distance-to-criticality. Finally, we compare the mass gap estimates derived from the Laguerre bound and the strong-disorder renormalization-group method. Both estimates require comparable computational efforts, with the former having the advantage of being more accurate and also being applicable away from infinite-randomness fixed points. We believe this method is a relevant tool for tackling critical quantum chains with and without quenched disorder.

Entanglement-based tensor-network strong-disorder renormalization group

We propose an entanglement-based algorithm of the tensor-network strong-disorder renormalization group (tSDRG) method for quantum spin systems with quenched randomness. In contrast to the previous tSDRG algorithm based on the energy spectrum of renormalized block Hamiltonians, we directly utilizes the entanglement structure associated with the blocks to be renormalized. We examine accuracy of the new algorithm for the random antiferromagnetic Heisenberg models on the one-dimensional, triangular, and square lattices. We then find that the entanglement-based tSDRG achieves better accuracy than the previous one for the square lattice model with weak randomness, while it is less efficient for the one-dimensional and triangular lattice models particularly in the strong randomness region. The theoretical background and possible improvements of the algorithm are also discussed.

Corrections to the singlet-length distribution and the entanglement entropy in random singlet phases

We consider random singlet phases of spin-$\frac{1}{2}$, random, antiferromagnetic spin chains, in which the universal leading-order divergence $\frac{\ln 2}{3}\ln\ell$ of the average entanglement entropy of a block of $\ell$ spins, as well as the closely related leading term $\frac{2}{3}l^{-2}$ in the distribution of singlet lengths are well known by the strong-disorder renormalization group (SDRG) method. Here, we address the question of how large the subleading terms of the above quantities are. By an analytical calculation performed along a special SDRG trajectory of the random XX chain, we identify a series of integer powers of $1/l$ in the singlet-length distribution with the subleading term $\frac{4}{3}l^{-3}$. Our numerical SDRG analysis shows that, for the XX fixed point, the subleading term is generally $O(l^{-3})$ with a non-universal coefficient and also reveals terms with half-integer powers: $l^{-7/2}$ and $l^{-5/2}$ for the XX and XXX fixed points, respectively. We also present how the singlet lengths originating in the SDRG approach can be interpreted and calculated in the XX chain from the one-particle states of the equivalent free-fermion model. These results imply that the subleading term next to the logarithmic one in the entanglement entropy is $O(\ell^{-1})$ for the XX fixed point and $O(\ell^{-1/2})$ for the XXX fixed point with non-universal coefficients. For the XX model, where a comparison with exact diagonalization is possible, the order of the subleading term is confirmed but we find that the SDRG fails to provide the correct non-universal coefficient.

Extreme statistics of the excitations in the random transverse Ising chain

In random quantum magnets, like the random transverse Ising chain, low energy excitations are localized in rare regions and there are only weak correlations between them. It is an open question whether these correlations are relevant in the sense of the renormalization group. To answer this question, we calculate the distribution of the excitation energy of the random transverse Ising chain in the disordered Griffiths phase with high numerical precision by the strong disorder renormalization group method and---for shorter chains---by free-fermion techniques. Asymptotically, the two methods give identical results, which are well fitted by the Fr\'echet limit law of the extremes of independent and identically distributed random numbers. Considering the finite size corrections, the two numerical methods give very similar results, but these differ from the correction term for uncorrelated random variables, indicating that the weak correlations between low-energy excitations in random quantum magnets are relevant.

Persistence discontinuity in disordered contact processes with long-range interactions

We study the local persistence probability during non-stationary time evolutions in disordered contact processes with long-range interactions by a combination of the strong-disorder renormalization group method, a phenomenological theory of rare regions, and numerical simulations. We find that, for interactions decaying as an inverse power of the distance, the persistence probability tends to a non-zero limit not only in the inactive phase but also in the critical point. Thus, unlike in the contact process with short-range interactions, the persistence in the limit t → ∞ is a discontinuous function of the control parameter. For stretched exponentially decaying interactions, the limiting value of the persistence is found to remain continuous, similar to the model with short-range interactions.

Detection of topology via entanglement oscillations

We introduce a characterization of topological order based on bulk oscillations of the entanglement entropy and the definition of an `entanglement gap', showing that it is generally applicable to pure and disordered quantum systems. Using exact diagonalization and the strong disorder renormalization group method, we demonstrate that this approach gives results in agreement with the use of traditional topological invariants, especially in cases where topological order is known to persist in the presence of off-diagonal bond disorder. The entanglement gap is then used to analyze classes of quantum systems with alternating bond types, allowing us to construct their topological phase diagrams. The validity of these phase diagrams is verified in limiting cases of dominant bond types, where the solution is known.

Tensor network renormalization group study of spin-1 random Heisenberg chains

We use a tensor network strong-disorder renormalization group (tSDRG) method to study spin-1 random Heisenberg antiferromagnetic chains. The ground state of the clean spin-1 Heisenberg chain with uniform nearest-neighbor couplings is a gapped phase known as the Haldane phase. Here we consider disordered chains with random couplings, in which the Haldane gap closes in the strong disorder regime. As the randomness strength is increased further and exceeds a certain threshold, the random chain undergoes a phase transition to a critical random-singlet phase. The strong-disorder renormalization group method formulated in terms of a tree tensor network provides an efficient tool for exploring ground-state properties of disordered quantum many-body systems. Using this method we detect the quantum critical point between the gapless Haldane phase and the random-singlet phase via the disorder-averaged string order parameter. We determine the critical exponents related to the average string order parameter, the average end-to-end correlation function and the average bulk spin-spin correlation function, both at the critical point and in the random-singlet phase. Furthermore, we study energy-length scaling properties through the distribution of energy gaps for a finite chain. Our results are in closer agreement with the theoretical predictions than what was found in previous numerical studies. As a benchmark, a comparison between tSDRG results for the average spin correlations of the spin-1/2 random Heisenberg chain with those obtained by using unbiased zero-temperature QMC method is also provided.

Population boundary across an environmental gradient: Effects of quenched disorder

Population boundary is a classic indicator of climatic response in ecology. In addition to known challenges, the spatial and dynamical characteristics of the boundary are not only affected by the spatial gradient in the environmental factors, but also by local heterogeneities in the regional characteristics. Here, we capture the effects of quenched heterogeneities on the ecological boundary with the disordered contact process in one and two dimensions with a linear spatial trend in the local control parameter. We apply the strong-disorder renormalization group method to calculate the sites occupied with an $O(1)$ probability in the stationary state, readily yielding the population front's position as the outermost site locally as well as globally for the entire boundary. We show that under a quasistatic change of the global environment, mimicking climate change, the front advances intermittently: long quiescent periods are interrupted by rare but long jumps. The characteristics of this intermittent dynamics are found to obey universal scaling laws in terms of the gradient, conjectured to be related to the correlation-length exponent of the model. Our results suggest that current observations might misleadingly show little to no climate response for an extended period of time, concealing the long-term effects of climate change.

Entanglement entropy of disordered quantum wire junctions

We consider different disordered lattice models composed of M linear chains glued together in a star-like manner, and study the scaling of the entanglement between one arm and the rest of the system using a numerical strong-disorder renormalization group method. For all studied models, the random transverse-field Ising model (RTIM), the random XX spin model, and the free-fermion model with random nearest-neighbor hopping terms, the average entanglement entropy is found to increase with the length L of the arms according to the form . For the RTIM and the XX model, the effective central charge is universal with respect to the details of junction, and only depends on the number M of arms. Interestingly, for the RTIM, decreases with M, whereas for the XX model it increases. For the free-fermion model, depends also on the details of the junction, which is related to the sublattice symmetry of the model. In this case, both increasing and decreasing tendency with M can be realized with appropriate junction geometries.

Transverse spin correlations of the random transverse-field Ising model

The critical behavior of the random transverse-field Ising model in finite dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in $d=1,2$ and $3$ dimensions. At the critical point an algebraic decay of the form $\sim r^{-\eta_t}$ is found, with a decay exponent being approximately $\eta_t \approx 2+2d$. In $d=1$ the results are related to dimer-dimer correlations in the random AF XX-chain and have been tested by numerical calculations using free-fermionic techniques.

Fractal Dimension of Interfaces in Edwards-Anderson and Long-range Ising Spin Glasses: Determining the Applicability of Different Theoretical Descriptions.

The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015)FRACEG0218-348X10.1142/S0218348X15500425] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.

Griffiths singularities in the random quantum Ising antiferromagnet: A tree tensor network renormalization group study

The antiferromagnetic Ising chain in both transverse and longitudinal magnetic fields is one of the paradigmatic models of a quantum phase transition. The antiferromagnetic system exhibits a zero-temperature critical line separating an antiferromagnetic phase and a paramagnetic phase; the critical line connects an integrable quantum critical point at zero longitudinal field and a classical first-order transition point at zero transverse field. Using a strong-disorder renormalization group method formulated as a tree tensor network, we study the zero-temperature phase of the quantum Ising chain with bond randomness. We introduce a new matrix product operator representation of high-order moments, which provides an efficient and accurate tool for determining quantum phase transitions via the Binder cumulant of the order parameter. Our results demonstrate an infinite-randomness quantum critical point in zero longitudinal field accompanied by pronounced quantum Griffiths singularities, arising from rare ordered regions with anomalously slow fluctuations inside the paramagnetic phase. The strong Griffiths effects are signaled by a large dynamical exponent $z>1$, which characterizes a power-law density of low-energy states of the localized rare regions and becomes infinite at the quantum critical point. Upon application of a longitudinal field, the quantum phase transition between the paramagnetic phase and the antiferromagnetic phase is completely destroyed. Furthermore, quantum Griffiths effects are suppressed, showing $z<1$, when the dynamics of the rare regions is hampered by the longitudinal field.

Entanglement across extended random defects in the XX spin chain

We study the half-chain entanglement entropy in the ground state of the spin-1/2 XX chain across an extended random defect, where the strength of disorder decays with the distance from the interface algebraically as . In the whole regime , the average entanglement entropy is found to increase logarithmically with the system size L as , where the effective central charge depends on κ. In the regime , where the extended defect is a relevant perturbation, the strong-disorder renormalization group method gives , while, in the regime , where the extended defect is irrelevant in the bulk, numerical results indicate a non-zero effective central charge, which increases with κ. The variation of is thus found to be non-monotonic and discontinuous at .

Properties of the random-singlet phase: From the disordered Heisenberg chain to an amorphous valence-bond solid

We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study S=1/2 spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multi-spin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that can not be generated by the SDRG scheme. In the model with multi-spin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.

Long-range random transverse-field Ising model in three dimensions

We consider the random transverse-field Ising model in $d=3$ dimensions with long-range ferromagnetic interactions which decay as a power $\alpha > d$ with the distance. Using a variant of the strong disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. The distribution of the (sample dependent) pseudo-critical points is found to scale with $1/\ln L$, $L$ being the linear size of the sample. Similarly, the critical magnetization scales with $(\ln L)^{\chi}/L^d$ and the excitation energy behaves as $L^{-\alpha}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed-order.

Strong-disorder renormalization-group study of the one-dimensional tight-binding model

We formulate a strong-disorder renormalization-group (SDRG) approach to study the beta function of the tight-binding model in one dimension with both diagonal and off-diagonal disorder for states at the band center. We show that the SDRG method, when used to compute transport properties, yields exact results since it is identical to the transfer matrix method. The beta function is shown to be universal when only off-diagonal disorder is present even though single-parameter scaling is known to be violated. A different single-parameter scaling theory is formulated for this particular (particle-hole symmetric) case. Upon breaking particle-hole symmetry (by adding diagonal disorder), the beta function is shown to crossover from the universal behavior of the particle-hole symmetric case to the conventional non-universal one in agreement with the two-parameter scaling theory. We finally draw an analogy with the random transverse-field Ising chain in the paramagnetic phase. The particle-hole symmetric case corresponds to the critical point of the quantum Ising model while the generic case corresponds to the Griffiths paramagnetic phase.

Critical behavior of models with infinite disorder at a star junction of chains

We study two models having an infinite-disorder critical point—the zero temperature random transverse-field Ising model and the random contact process—on a star-like network composed of M semi-infinite chains connected to a common central site. By the strong disorder renormalization group method, the scaling dimension xM of the local order parameter at the junction is calculated. It is found to decrease rapidly with the number M of arms, but remains positive for any finite M. This means that, in contrast with the pure transverse-field Ising model, where the transition becomes of first order for M > 2, it remains continuous in the disordered models, although, for not too small M, it is hardly distinguishable from a discontinuous one owing to a close-to-zero xM. The scaling behavior of the order parameter in the Griffiths–McCoy phase is also analyzed.