Random Transverse Field Ising Model Research Articles

Quantum-critical properties of the one- and two-dimensional random transverse-field Ising model from large-scale quantum Monte Carlo simulations

We study the ferromagnetic transverse-field Ising model with quenched disorder at T = 0T=0 in one and two dimensions by means of stochastic series expansion quantum Monte Carlo simulations using a rigorous zero-temperature scheme. Using a sample-replication method and averaged Binder ratios, we determine the critical shift and width exponents \nu_\mathrm{s}νs and \nu_\mathrm{w}νw as well as unbiased critical points by finite-size scaling. Further, scaling of the disorder-averaged magnetisation at the critical point is used to determine the order-parameter critical exponent \betaβ and the critical exponent \nu_{\mathrm{av}}νav of the average correlation length. The dynamic scaling in the Griffiths phase is investigated by measuring the local susceptibility in the disordered phase and the dynamic exponent z'z′ is extracted. By applying various finite-size scaling protocols, we provide an extensive and comprehensive comparison between the different approaches on equal footing. The emphasis on effective zero-temperature simulations resolves several inconsistencies in existing literature.

Localization of overdamped bosonic modes and transport in strange metals

The strange metal phase of correlated electrons materials was described in a recent theory by a model of a Fermi surface coupled a two-dimensional quantum critical bosonic field with a spatially random Yukawa coupling. With the assumption of self-averaging randomness, similar to that in the Sachdev-Ye-Kitaev model, numerous observed properties of a strange metal were obtained for a wide range of intermediate temperatures, including the linear in temperature resistivity. The Harris criterion implies that spatial fluctuations in the local position of the critical point must dominate at lower temperatures. For an [Formula: see text]-component boson with [Formula: see text], we use multiple graphics processing units (GPUs) to compute the real frequency spectrum of the boson propagator in a self-consistent mean-field treatment of the boson self-interactions, but an exact treatment of multiple realizations of the spatial randomness from the random boson mass. We find that Landau damping from the fermions leads to the emergence of the physics of the random transverse-field Ising model at low temperatures, as has been proposed by Hoyos, Kotabage, and Vojta. This regime is controlled by localized overdamped eigenmodes of the bosonic scalar field, also has a resistivity which is nearly linear-in-temperature, and extends into a "quantum critical phase" away from the quantum critical point, as observed in several cuprates. For the [Formula: see text] Ising scalar, the mean-field treatment is not applicable, and so we use Hybrid Monte Carlo simulations running on multiple GPUs; we find a rounded transition and localization physics, with strange metal behavior in an extended region around the transition.

Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree

We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant (independent of the system volume) and algebraically vanishing (scaling as an inverse power law with the system volume) boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.

Complex quantum network models from spin clusters

In the emerging quantum internet, complex network topology could lead to efficient quantum communication and robustness against failures. However, there are concerns about complexity in quantum communication networks, such as potentially limited end-to-end transmission capacity. These challenges call for model systems in which the impact of complex topology on quantum communication protocols can be explored. Here, we present a theoretical model for complex quantum communication networks on a lattice of spins, wherein entangled spin clusters in interacting quantum spin systems serve as communication links between appropriately selected regions of spins. Specifically, we show that ground state Greenberger-Horne-Zeilinger clusters of the two-dimensional random transverse-field Ising model can be used as communication links between regions of spins. Further, the resulting quantum networks can have complexity comparable to that of the classical internet. Our work provides a generative model for further studies towards determining the network characteristics of the emerging quantum internet.

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.

Superuniversality from disorder at two-dimensional topological phase transitions

We investigate the effects of quenched randomness on topological quantum phase transitions in strongly interacting two-dimensional systems. We focus first on transitions driven by the condensation of a subset of fractionalized quasiparticles (`anyons') identified with `electric charge' excitations of a phase with intrinsic topological order. All other anyons have nontrivial mutual statistics with the condensed subset and hence become confined at the anyon condensation transition. Using a combination of microscopically exact duality transformations and asymptotically exact real-space renormalization group techniques applied to these two-dimensional disordered gauge theories, we argue that the resulting critical scaling behavior is `superuniversal' across a wide range of such condensation transitions, and is controlled by the same infinite-randomness fixed point as that of the 2D random transverse-field Ising model. We validate this claim using large-scale quantum Monte Carlo simulations that allow us to extract zero-temperature critical exponents and correlation functions in (2+1)D disordered interacting systems. We discuss generalizations of these results to a large class of ground-state and excited-state topological transitions in systems with intrinsic topological order as well as those where topological order is either protected or enriched by global symmetries. When the underlying topological order and the symmetry group are Abelian, our results provide prototypes for topological phase transitions between distinct many-body localized phases.

Asymptotically Exact Theory for Nonlinear Spectroscopy of Random Quantum Magnets.

We study nonlinear response in quantum spin systems near infinite-randomness critical points. Nonlinear dynamical probes, such as two-dimensional (2D) coherent spectroscopy, can diagnose the nearly localized character of excitations in such systems. We present exact results for nonlinear response in the 1D random transverse-field Ising model, from which we extract information about critical behavior that is absent in linear response. Our analysis yields exact scaling forms for the distribution functions of relaxation times that result from realistic channels for dissipation in random magnets. We argue that our results capture the scaling of relaxation times and nonlinear response in generic random quantum magnets in any spatial dimension.

Confinement, reduced entanglement, and spin-glass order in a random quantum spin-ice model

We study an effective spin model derived perturbatively from random transverse-field Ising model on the pyrochlore lattice. The model consists of spin-configurations on the pyrochlore lattice, restricted to the spin-ice subspace, with spins interacting with random Ising exchange couplings as well as ring exchanges along the hexagons of the lattice. This model is studied by exact diagonalization upto N=64 site systems. We calculate spin-glass correlation functions and local entanglement entropy $S_T$ between spins in a single tetrahedron and the rest of the system. We find that the model undergoes two phase transitions. At weak randomness the model is in a quantum spin-ice phase where $S_T=\ln{6}$. Increasing randomness first leads to a frozen phase, with long-range spin-glass order and $S_T=\ln{2}$ corresponding to the Cat states associated with Ising order. Further increase in randomness leads to a random resonating-hexagon phase with a frozen backbone of spins and a broad distribution of entanglement entropies. The implications of these studies for non-Kramers rare-earth pyrochlores are discussed.

Local entanglement and confinement transitions in the random transverse-field Ising model on the pyrochlore lattice

We use numerical linked cluster expansions (NLC) and exact diagonalization to study confinement transitions out of the quantum spin liquid phase in the pyrochlore-lattice Ising antiferromagnet with random transverse fields. We calculate entanglement entropies associated with local regions defined by single tetrahedron to observe these transitions. The randomness-induced confinement transition is marked by a sharp reduction in the local entanglement and a concomitant increase in Ising correlations. In NLC, it is studied through the destruction of loop resonances due to random transverse-fields. The confining phase is characterized by a distribution of local entanglement entropies, which persists to large random fields.

Quantum Superconductor-Metal Transitions in the Presence of Quenched Disorder

InOx films that are less disordered than those exhibiting direct quantum superconductor-insulator transitions feature quantum superconductor-metal transitions tuned by magnetic field. Resistance data across this superconductor-metal transition obey activated scaling, with critical exponents suggesting that the transition is governed by an infinite-randomness critical point in the universality class of the random transverse-field Ising model in two dimensions. The transition is accompanied by quantum Griffiths effects. This unusual behavior is expected for systems with quenched disorder in the presence of ohmic dissipation. Disorder leads to the formation of large rare regions which are locally ordered superconducting puddles dispersed in a metallic matrix. Their dissipative dynamics causes the activated scaling, as predicted by a renormalization group theory.

Infinite-randomness fixed point of the quantum superconductor-metal transitions in amorphous thin films

The magnetic-field-tuned quantum superconductor-insulator transitions of disordered amorphous indium oxide films are a paradigm in the study of quantum phase transitions, and exhibit power-law scaling behavior. For superconducting indium oxide films with low disorder, such as the ones reported on here, the high-field state appears to be a quantum-corrected metal. Resistance data across the superconductor-metal transition in these films are shown here to obey an activated scaling form appropriate to a quantum phase transition controlled by an infinite randomness fixed point in the universality class of the random transverse-field Ising model. Collapse of the field-dependent resistance vs. temperature data is obtained using an activated scaling form appropriate to this universality class, using values determined through a modified form of power-law scaling analysis. This exotic behavior of films exhibiting a superconductor-metal transition is caused by the dissipative dynamics of superconducting rare regions immersed in a metallic matrix, as predicted by a recent renormalization group theory. The smeared crossing points of isotherms observed are due to corrections to scaling which are expected near an infinite randomness critical point, where the inverse disorder strength acts as an irrelevant scaling variable.

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.

Schmidt gap in random spin chains

We numerically investigate the low-lying entanglement spectrum of the ground state of random one-dimensional spin chains obtained after partition of the chain into two equal halves. We consider two paradigmatic models: the spin-1/2 random transverse field Ising model, solved exactly, and the spin-1 random Heisenberg model, simulated using the density matrix renormalization group. In both cases we analyze the mean Schmidt gap, defined as the difference between the two largest eigenvalues of the reduced density matrix of one of the two partitions, averaged over many disorder realizations. We find that the Schmidt gap detects the critical point very well and scales with universal critical exponents.

Instabilities of a U(1) Quantum Spin Liquid in Disordered Non-Kramers Pyrochlores.

Quantum spin liquids (QSLs) are exotic phases of matter exhibiting long-range entanglement and supporting emergent gauge fields. A vigorous search for experimental realizations of these states has identified several materials with properties hinting at QSL physics. A key issue in understanding these QSL candidates is often the interplay of weak disorder of the crystal structure with the spin liquid state. It has recently been pointed out that in at least one important class of candidate QSLs-pyrochlore magnets based on non-Kramers ions such as Pr^{3+} or Tb^{3+}-structural disorder can actually promote a U(1) QSL ground state. Here we set this proposal on a quantitative footing by analyzing the stability of the QSL state in the minimal model for these systems: a random transverse field Ising model. We consider two kinds of instability, which are relevant in different limits of the phase diagram: condensation of spinons and confinement of the U(1) gauge fields. Having obtained stability bounds on the QSL state, we apply our results directly to the disordered candidate QSL Pr_{2}Zr_{2}O_{7}. We find that the available data for currently studied samples of Pr_{2}Zr_{2}O_{7} are most consistent with it a ground state outside the spin liquid regime, in a paramagnetic phase with quadrupole moments near saturation due to the influence of structural disorder.

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.

Critical properties of the contact process with quenched dilution

We have studied the critical properties of the contact process on a square lattice with quenched site dilution by Monte Carlo simulations. This was achieved by generating in advance the percolating cluster, through the use of an appropriate epidemic model, and then by the simulation of the contact process on the top of the percolating cluster. The dynamic critical exponents were calculated by assuming an activated scaling relation and the static exponents by the usual power law behavior. Our results are in agreement with the prediction that the quenched diluted contact process belongs to the universality class of the random transverse-field Ising model. We have also analyzed the model and determined the phase diagram by the use of a mean-field theory that takes into account the correlation between neighboring sites.

Probability distribution of the entanglement across a cut at an infinite-randomness fixed point

We calculate the probability distribution of entanglement entropy $S$ across a cut of a finite one-dimensional spin chain of length $L$ at an infinite-randomness fixed point using Fisher's strong randomness renormalization group (RG). Using the random transverse-field Ising model as an example, the distribution is shown to take the form $p(S|L)\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\psi}(k)}$, where $k\ensuremath{\equiv}S/ln\left[L/{L}_{0}\right]$, the large deviation function $\ensuremath{\psi}(k)$ is found explicitly, and ${L}_{0}$ is a nonuniversal microscopic length. We discuss the implications of such a distribution on numerical techniques that rely on entanglement, such as matrix-product-state-based techniques. Our results are verified with numerical RG simulations, as well as the actual entanglement entropy distribution for the random transverse-field Ising model which we calculate for large $L$ via a mapping to Majorana fermions.

Monte Carlo simulations of the disordered three-color quantum Ashkin-Teller chain

We investigate the zero-temperature quantum phase transitions of the disordered three-color quantum Ashkin-Teller spin chain by means of large-scale Monte Carlo simulations. We find that the first-order phase transitions of the clean system are rounded by the quenched disorder. For weak inter-color coupling, the resulting emergent quantum critical point between the paramagnetic phase and the magnetically ordered Baxter phase is of infinite-randomness type and belongs to the universality class of the random transverse-field Ising model, as predicted by recent strong-disorder renormalization group calculations. We also find evidence for unconventional critical behavior in the case of strong inter-color coupling, even though an unequivocal determination of the universality class is beyond our numerical capabilities. We compare our results to earlier simulations, and we discuss implications for the classification of phase transitions in the presence of disorder.

Activated scaling in disorder-rounded first-order quantum phase transitions

First-order phase transitions, classical or quantum, subject to randomness coupled to energy-like variables (bond randomness) can be rounded, resulting in continuous transitions (emergent criticality). We study perhaps the simplest such model, quantum three-color Ashkin-Teller model and show that the quantum critical point in $(1+1)$ dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. The behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case. We believe that this fact must be attended to when discussing quantum critical points in numerous physical systems, which may be first-order transitions in disguise.

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.