Quantum Renormalization Group Research Articles

Behavior of quantum coherence in quantum phase transitions of two-dimensional XY and ising models

We investigate the quantum coherence behavior of the ground states of 2D Heisenberg XY model and 2D Ising model with a transverse field on square lattices using the Quantum Renormalization Group (QRG) method. Our analysis focused on the ground state density matrix and its marginal states, revealing non-analytic behavior of quantum coherence (especially two-site coherence) near the critical point. This behavior allowed us to detect quantum phase transitions (QPT) in these models. By examining the scaling behavior of the maximum derivative of quantum coherence with system size, we determined the critical exponent of coherence for both models and the length exponent of the Ising model. Additionally, we investigated the time evolution of coherence in both models. Our results closely align with those obtained from entanglement analysis, that is while quantum coherence requires fewer computational calculations compared to discord and entanglement approaches.

Renormalization of net steered coherence and average coherence in the XXZ model with Dzyaloshinskii–Moriya interaction

The critical behaviors of the net steered coherence (NSC) and average coherence (AC) in the XXZ model with Dzyaloshinskii–Moriya interaction are investigated by adopting the quantum renormalization group method. It is found that they both develop different saturated values in different phases of the model after enough iterations of the renormalization of the coupling constants. In addition, their susceptibilities at the phase transition point are divergent in the thermodynamic limit, and the scaling exponent is determined by the exponent governing the divergence of the correlation length close to the phase transition point. This exemplifies that both the NSC and AC can serve as reliable indicators of quantum phase transitions.

Crossover behaviors of entanglement dynamics at the critical point in XXZ spin chain

In this paper, we investigate the crossover behaviors of entanglement and return probability at the critical point in one-dimensional XXZ model by quantum renormalization group method. Two types of quench protocols (A) adding the interaction term in z spin direction suddenly and (B) rotating all the even spins by π about the z spin direction suddenly are considered to drive the dynamics of the system, respectively. The dynamical behaviors are closely related to the initial state under two types of quench protocols, especially at N→∞. For the left of critical point, return probabilities eventually approach one under both types of quench protocols. For the right of critical point, return probability oscillates between the maximum value and zero for quench A and tends to one for quench B, respectively. In addition, we discuss the evolution period of entanglement (return probability), find that it is positively and negatively correlated with N in the spin-fluid and Néel phases, respectively. By further analyzing the behaviors of entanglement, return probability and evolution period under both types of quench protocols near the critical point, we observe that they exist the same scaling behaviors, in which the exponents are identical to entanglement exponent of the stationary system, which is related to the results of renormalization group.

Effects of quantum quench on entanglement dynamics in antiferromagnetic Ising model

We study the relationship between quench dynamics of entanglement and quantum phase transition in the antiferromagnetic Ising model with the Dzyaloshinskii–Moriya (DM) interaction by using the quantum renormalization-group method and the definition of negativity. Two types of quench protocols (i) adding the DM interaction suddenly and (ii) rotating the spins around x axis are considered to drive the dynamics of the system, respectively. By comparing the behaviors of entanglement in both types of quench protocols, the effects of quench on dynamics of entanglement are studied. It is found that there is the same characteristic time at which the negativity firstly reaches its maximum although the system shows different dynamical behaviors. Especially, the characteristic time can accurately reflect the quantum phase transition from antiferromagnetic to saturated chiral phases in the system. In addition, the correlation length exponent can be obtained by exploring the nonanalytic and scaling behaviors of the derivative of the characteristic time.

Robustness of quantum coherence and quantum criticality in spin-1 many-body system

The tripartite quantum coherence for spin-12 and bipartite quantum coherence for spin-1 many-body systems are studied with the quantum renormalization group method to investigate the quantum criticality behaviour of both systems. We find both spin-12 and spin-1 systems exhibit quantum criticality behaviour after different number of iterations. However, the bipartite quantum coherence for spin-1 system reaches the quantum criticality behaviour in lesser number of iterations than the tripartite quantum coherence for spin-12 system. The quantum criticality behaviour of both systems is also investigated through non-analytic and the scaling behaviour of quantum coherence. Additionally, quantum correlations and quantum phase transition are observed to be more vigorous with higher spin comparatively to the number of parties (i.e., bipartite, tripartite, multipartite) involved in a quantum correlations.

Renormalization of steered coherence and quantum phase transitions in the alternating Ising model

We explore net steered coherence defined with respect to the mutually unbiased bases. By using the quantum renormalization group method, we studied its scaling property for the ground state of two blocks of the alternating Ising model in the renormalized space. The results show that different patterns in the net steered coherence describe different phases of the model, and its susceptibility with respect to the magnetic field is divergent at the critical points of quantum phase transitions. As the net steered coherence exists for almost all bipartite states, it can serve as an alternative for exploring quantum criticality of the many-body systems in the parameter regions without entanglement and quantum discord.

Uncertainty Relation and Quantum Phase Transition in the Two-Dimensional Ising Model

By using quantum renormalization group (QRG) approach, we first derive the effective Hamiltonian and QRG equations of the two-dimensional (2D) Ising models with two different time-dependent transverse magnetic fields analytically. Then we examine the nonanalytic and scaling behaviors of the linear-entropy-based uncertainty relation and quantum entanglement of the models near the critical point through numerical analysis. Moreover, we investigate the relation between the quantum critical point and the external magnetic field. Our results show that both the uncertainty relation and the quantum entanglement are feasible to detect the quantum phase transition (QPT), and the uncertainty relation may be a better indicator of QPT than quantum entanglement. Our findings could shed new light on the observable of the QPTs of the solid-state system with the uncertainty relation.

Renormalization of multipartite entanglement near the critical point of two-dimensional XXZ model with Dzyaloshinskii–Moriya interaction

We investigate the multipartite entanglement and the trace distance for the two-dimensional anisotropic spin-12 XXZ lattice with Dzyaloshinskii–Moriya interaction. It is found that for a many-body quantum system the multipartite entanglement is more advantageous than the bipartite entanglement due to the monogamy property. Both the quantifiers, the multipartite entanglement and the trace distance, decreases with an increase in the anisotropy and the Dzyaloshinskii–Moriya interaction tends to restore the spoiled entanglement. The quantum reormalization group method is used to compute the stable and the unstable fixed points. We observe that the quantum phase transition point is independent of the chosen quantifier as the thermodynamic limit is reached. After sufficient iterations of the quantum renormalization group, we observe two different saturated values of both the quantifiers that represent two separate phases, the spin-fluid phase and the Néel phase. The first derivative and the scaling behavior of the renormalized entanglement quantifiers are calculated. At quantum phase transition point, the non-analytic behavior of the first derivative of the two quantifiers as a function of lattice size is examined and it is found that the universal finite-size scaling law is obeyed. Furthermore, we observe that at the critical point the scaling exponent for the multipartite entanglement and the trace distance can describe the correlation length of the model.

Genuine Tripartite Entanglement as a Probe of Quantum Phase Transitions in a Spin‐1 Heisenberg Chain with Single‐Ion Anisotropy

AbstractThe quantum phase transitions of spin‐1 Heisenberg chains with an easy‐axis anisotropy Δ and a uniaxial single‐ion anisotropy D are studied using a multipartite entanglement approach. The genuine tripartite entanglement between the spin blocks, measured by the tripartite qutrit hyperdeterminant, is calculated within the quantum renormalization group method. Using this approach, the phase boundaries between the topological Haldane, large‐D, and anti‐ferromagnetic Néel phases are determined in the half plane with . When the size of the spin blocks increases, the genuine tripartite entanglement between the blocks exhibits a nonzero plateau in the topological Haldane phase, and experiences abrupt drops at both the phase boundaries between the Haldane–large‐D and Haldane–Néel phases, which suggests the usage of genuine multipartite entanglement as a probe of topological phases in spin systems.

Exploring the Koch fractal lattice with quantum renormalization group method

We investigate quantum criticality of many-body systems in the context of dimensions through the behavior of tripartite entanglement and tripartite quantum coherence. Following the approach of quantum renormalization group (QRG) method, the analysis on one-dimensional Heisenberg Ising model and non-integer Hausdorff dimensional Koch fractal lattice is made with the increasing size of the systems and anisotropy in the spin components. In thermodynamics limit, the two quantities develop three different regions for each system, signaling the existence of two critical points. The presence of quantum criticality is supported by the divergence of the first derivative of both quantities and their scaling with the increasing number of particles. Our study shows that quantum criticality in the non-integer Hausdorff dimension of Koch fractal lattice is more sensitive to the size than in the one-dimensional spin chains.

Dynamics of Quantum Correlation and Entropic Uncertainty in Spin‐12$\frac{1}{2}$ Alternating Transverse Ising Model

AbstractBy implementing the quantum renormalization group method, the effective Hamiltonian of the spin‐ Ising model with regularly altering strengths of both the magnetic fields and the exchange couplings are first derived analytically. Then, tripartite quantum discord (QD) and entropic uncertainty (EU) of the Ising model and their quantum phase transitions are investigated. Moreover, the quantum critical points and correlation length exponents of the model are analyzed, and both the block–block EU and the site–site EU are discussed through numerical calculations. It is found that under the system size increasing, the first derivatives of both tripartite QD and block–block EU show nonanalytic and scaling behaviors near the critical point. Furthermore, the strong quantum correlations among multi‐particles do not necessarily lead to the decrease of EU, as the site–site EU in the vicinity of the critical point increases due to competitions among quantum correlations. The findings may offer new insights into utilizing EU as an observable of the quantum phase transitions.

Finite quantum field theory and renormalization group

Renormalization group methods are applied to a scalar field within a finite, nonlocal quantum field theory formulated perturbatively in Euclidean momentum space. It is demonstrated that the triviality problem in scalar field theory, the Higgs boson mass hierarchy problem and the stability of the vacuum do not arise as issues in the theory. The scalar Higgs field has no Landau pole.

Unitary Long-Time Evolution with Quantum Renormalization Groups and Artificial Neural Networks.

In this work, we combine quantum renormalization group approaches with deep artificial neural networks for the description of the real-time evolution in strongly disordered quantum matter. We find that this allows us to accurately compute the long-time coherent dynamics of large many-body localized systems in nonperturbative regimes including the effects of many-body resonances. Concretely, we use this approach to describe the spatiotemporal buildup of many-body localized spin-glass order in random Ising chains. We observe a fundamental difference to a noninteracting Anderson insulating Ising chain, where the order only develops over a finite spatial range. We further apply the approach to strongly disordered two-dimensional Ising models, highlighting that our method can be used also for the description of the real-time dynamics of nonergodic quantum matter in a general context.

Single-ion anisotropy effects on the critical behaviors of quantum entanglement and correlation in the spin-1 Heisenberg chain

Quantum entanglement and correlations in the spin-1 Heisenberg chain with single-ion anisotropy are investigated using the quantum renormalization group method. Negativity and quantum discord (QD) are calculated with various anisotropy parameters △ and single-ion anisotropy parameters D. We focus on the relations between two abovementioned physical quantities and on transitions between the Néel, Haldane, and large-D phases. It is found that both negativity and QD exhibit step-like patterns in different phases as the size of the system increases. Interestingly, the single-ion anisotropy parameter D, which can be modulated using nuclear electric resonance (2020 Nature 579 205), plays an important role in tuning the quantum phase transition (QPT) of the system. Both the first partial derivative of the negativity and QD with respect to D or △ exhibit nonanalytic behavior at the phase transition points, which corresponds directly to the divergence of the correlation length. The quantum correlation critical exponents derived from negativity and QD are equal, and are the reciprocal of the correlation length exponent at each critical point. This work extends the application of quantum entanglement and correlations as tools for depicting QPTs in spin-1 systems.

Quantum quench dynamics in XY spin chain with ferromagnetic and antiferromagnetic interactions

In this manuscript we investigate the one-dimensional anisotropic XY model with ferromagnetic and antiferromagnetic interactions, which gives more interesting phase diagram and dynamic critical behaviors. By using quantum renormalization-group method, we find that there are three phases in the system: antiferromagnetic Ising phase ordered in “x direction”, spin-fluid phase and ferromagnetic Ising phase ordered in “y direction”. In order to study the dynamical critical behaviors of the system, two quantum quenching methods are used. In both cases, the concurrence, a measure of entanglement, oscillates periodically over time. It if found that in the both cases the periods are the same and can be used as a order parameter for quantum phase transitions. It is also found that there is a scaling behavior for the evolution period and a reciprocal relation for the evolution period exponent, θ and correlation length exponent, ν.

Total quantum coherence close to the quantum critical points in the XXZ model with Dzyaloshinskii–Moriya interaction

In this paper, by using the quantum renormalization-group method, we study the relation between total quantum coherence based on norm and quantum phase transition in the XXZ model with Dzyaloshinskii–Moriya (DM) interaction. Our findings reveal a distribution relation of quantum coherence in this model, i.e. the value of the quantum coherence of a three-site block state is equal to the sum of its reduced state coherence, and this characteristic can be generalized to a multipartite system. Meanwhile, because of favoring the alignment of spins, the anisotropy suppresses the quantum coherence; on the other hand, by creating the quantum fluctuations, the DM interaction can effectively enhance the value of quantum coherence. Additionally, after several iterations of the renormalization, quantum coherence of the whole block state can form two saturated values, which are related to two different phases: spin-fluid phase and Néel phase. Furthermore, we discover that the total quantum coherence of the block–block state obeys the nonanalytic and scaling properties with the quantum coherence exponent associated with correlation length.

Renormalization of concurrence and quantum Fisher information in two-dimensional XXZ model

We investigate two-dimensional anisotropic spin- $$\frac{1}{2}$$ lattice for XXZ model using concurrence and quantum Fisher information. The model is solved by using quantum renormalization group method and observed the quantum phase transition in thermodynamic limit. After sufficient iterations of quantum renormalization group, the concurrence and quantum Fisher information both attain two different saturated values that represent two different phases, the Neel phase and the spin fluid phase. The first-order derivative and the scaling behavior of both the concurrence and the quantum Fisher information are studied. At phase transition point, the non-analytic behavior of the derivative as a function of lattice size is discussed. Moreover, we observe that the scaling exponent for the concurrence and the quantum Fisher information describe the correlation length of the model at the critical point.

Coherence and Quantum Phase Transition in Spin Models

Based on quantum renormalization group (QRG) method, we investigated quantum coherence and quantum phase transition (QPT) in XXZ chain and XY chain, respectively. The results show that both the geometric quantum coherence and entropic coherecne can accurately indicate the QPT at critical point after enough iteration steps. Moreover, the increasing anisotropy parameter destroys the coherence in the XXZ chain, while enhances it in the XY chain. In addition, focused on the XXZ chain we analyzed the nonanalytic phenomena and scaling behaviors with different theoretical exponents in detail.

Bipartite and tripartite quantum coherence with entanglement in [formula omitted] Heisenberg spins-1/2 chain under renormalization group method

The behavior of bipartite and tripartite quantum coherence in Heisenberg XXZ spin-1/2 chain, in the presence of an applied magnetic field, are studied through quantum renormalization group (QRG) method. In both cases, the amount of quantum coherence is compared with the amount of corresponding entanglement. Bipartite entanglement exactly equals bipartite quantum coherence and tripartite entanglement is always less than the corresponding quantum coherence. It is shown that in thermodynamic limit, quantum coherence becomes nonanalytic in the vicinity of critical point and can be used to predict quantum phase transition (QPT) in the system. The relative strength of the applied magnetic field to the anisotropy of the system considerably affects the critical point. In a given range of anisotropy, the monogamous and polygamous nature of quantum coherence are conditioned to the size of the system.

Entanglement and multipartite quantum correlations in two-dimensional XY model with Dzyaloshinskii–Moriya interaction

The behavior of the bipartite entanglement and multipartite quantum correlations for two dimensional square lattice of Heisenberg XY model with Dzyaloshinskii–Moriya (DM) interaction is studied in the framework of the quantum renormalization group (QRG) method. The analytical results are produced for the renormalized control parameters and a phase diagram is obtained for the model which shows the stable and unstable fix points corresponding to different phases. Concurrence and global quantum discord (GQD) are the quantifiers for the study of entanglement and quantum correlations respectively which are influenced in a different manner by the control parameters anisotropy γ and the DM interaction D. However, the behavior of the quantifiers at the critical point captures the quantum phase transition (QPT) described by the divergence of their first derivative with respect to anisotropy. It is found that the DM interactions among the spins try to recover the entanglement which is more prominent for the multipartite case. GQD is a more suitable candidate as it can describe the multipartite quantum correlations present among many spins (>2) in higher dimensions. Unlike the monogamy in entanglement the quantum correlations increase with the increasing number of the interacting spins.