arXiv:1903.07354 · Physical review. B./Physical review. B

Many-body localization in the presence of long-range interactions and long-range hopping

Sabyasachi Nag, Arti Garg

Published Jun 7, 2019• 75 citations

Abstract

We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of long-range interactions and long-range hopping. Based on perturbative arguments there is a common belief that MBL can exist only in systems with short-range interactions and short-range hopping. We analyze effects of power-law interactions and power-law hopping, separately, on a system which has all the single-particle states localized in the absence of interactions. Since delocalization is driven by proliferation of resonances in the Fock space, we mapped this model to an effective Anderson model on a complex graph in the Fock space and calculated the probability distribution of the number of resonances up to third order. Though the most-probable value of the number of resonances diverge for the system with long-range hopping [$t(r)\ensuremath{\sim}{t}_{0}/{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}<2$], there is no enhancement of the number of resonances as the range of power-law interactions increases. This indicates that the long-range hopping delocalizes the many-body localized system but in contrast to this, there is no signature of delocalization in the presence of long-range interactions. We further provide support in favor of this analysis based on dynamics of the system after a quench starting from a charge density wave ordered state, level spacing statistics, return probability, participation ratio and Shannon entropy in the Fock space. We demonstrate that MBL persists in the presence of long-range interactions though long-range hopping with $1<\ensuremath{\alpha}<2$ delocalizes the system partially, with almost all the states extended for $\ensuremath{\alpha}\ensuremath{\le}1$. Even in a system which has single-particle mobility edges in the noninteracting limit, turning on long-range interactions does not cause delocalization.

DOI: 10.1103/physrevb.99.224203

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Many-body localization in incommensurate models with a mobility edge

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arXiv:1903.07354 · Physical review. B./Physical review. B

Many-body localization in the presence of long-range interactions and long-range hopping

Sabyasachi Nag, Arti Garg

Published Jun 7, 2019• 75 citations

arXiv HTML PDF

Abstract

t(r)\ensuremath{\sim}{t}_{0}/{r}^{\ensuremath{\alpha}}

with

\ensuremath{\alpha}<2

], there is no enhancement of the number of resonances as the range of power-law interactions increases. This indicates that the long-range hopping delocalizes the many-body localized system but in contrast to this, there is no signature of delocalization in the presence of long-range interactions. We further provide support in favor of this analysis based on dynamics of the system after a quench starting from a charge density wave ordered state, level spacing statistics, return probability, participation ratio and Shannon entropy in the Fock space. We demonstrate that MBL persists in the presence of long-range interactions though long-range hopping with

1<\ensuremath{\alpha}<2

delocalizes the system partially, with almost all the states extended for

\ensuremath{\alpha}\ensuremath{\le}1

. Even in a system which has single-particle mobility edges in the noninteracting limit, turning on long-range interactions does not cause delocalization.

DOI: 10.1103/physrevb.99.224203

Related papers

Many-body mobility edges in a one-dimensional system of interacting fermions

Sabyasachi Nag, Arti Garg

We study many-body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold potential $h<{h}_{c}$ , the noninteracting system has single particle mobility edges (MEs) at $\ifmmode\pm\else\textpm\fi{}{E}_{c}$ while for $h>{h}_{c}$ all the single particle states are localized. We demonstrate that even in the presence of single particle MEs, interactions do not always delocalize the system and the interacting system can have MBL. Our numerical calculation of energy level spacing statistics, participation ratio in the Fock space, and Shannon entropy shows that for some regime of particle densities, even for $h<{h}_{c}$ , many-body states at the top (with $E>{E}_{2}$ ) and the bottom of the spectrum (with $E<{E}_{1}$ ) remain localized, though states in the middle of the spectrum are delocalized. Variance of entanglement entropy (EE) also diverges at ${E}_{1,2}$ , indicating a transition from MBL to a delocalized regime, though transitions from volume to area law scaling for EE and from thermal to nonthermal behavior occur inside the MBL regime much below ${E}_{1}$ and above ${E}_{2}$ .We study many-body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold potential $h<{h}_{c}$, the noninteracting system has single particle mobility edges (MEs) at $\ifmmode\pm\else\textpm\fi{}{E}_{c}$ while for $h>{h}_{c}$ all the single particle states are localized. We demonstrate that even in the presence of single particle MEs, interactions do not always delocalize the system and the interacting system can have MBL. Our numerical calculation of energy level spacing statistics, participation ratio in the Fock space, and Shannon entropy shows that for some regime of particle densities, even for $h<{h}_{c}$, many-body states at the top (with $E>{E}_{2}$) and the bottom of the spectrum (with $E<{E}_{1}$) remain localized, though states in the middle of the spectrum are delocalized. Variance of entanglement entropy (EE) also diverges at ${E}_{1,2}$, indicating a transition from MBL to a delocalized regime, though transitions from volume to area law scaling for EE and from thermal to nonthermal behavior occur inside the MBL regime much below ${E}_{1}$ and above ${E}_{2}$.

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Many-body localization and enhanced nonergodic subdiffusive regime in the presence of random long-range interactions

Yogeshwar Prasad, Arti Garg

We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of random interactions ${V}_{ij}$ decaying as power-law ${V}_{ij}/{({r}_{ij})}^{\ensuremath{\alpha}}$ with distance ${r}_{ij}$ . We demonstrate that MBL survives even for $\ensuremath{\alpha}<1$ and is preceded by a broad nonergodic subdiffusive phase. Starting from parameters at which the short-range interacting system shows an infinite temperature MBL phase, turning on random power-law interactions results in many-body mobility edges in the spectrum with a larger fraction of ergodic delocalized states for smaller values of $\ensuremath{\alpha}$. Hence, the critical disorder ${h}_{c}^{r}$ , at which ergodic to nonergodic transition takes place, increases with the range of interactions. Time evolution of the density imbalance $I(t)$ , which has power-law decay $I(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}$ in the intermediate to large time regime, shows that the critical disorder ${h}_{c}^{I}$ , above which the system becomes diffusionless (with $\ensuremath{\gamma}\ensuremath{\sim}0$) and transits into the MBL phase, is much larger than ${h}_{c}^{r}$ . In between ${h}_{c}^{r}$ and ${h}_{c}^{I}$ there is a broad nonergodic subdiffusive phase, which is characterized by the Poissonian statistics for the level spacing ratio, multifractal eigenfunctions, and a nonzero dynamical exponent $\ensuremath{\gamma}\ensuremath{\ll}1/2$. The system continues to be subdiffusive even on the ergodic side ( $h<{h}_{c}^{r}$ ) of the MBL transition, where the eigenstates near the mobility edges are multifractal. For $h<{h}_{0}<{h}_{c}^{r}$ , the system is superdiffusive with $\ensuremath{\gamma}>1/2$. The rich phase diagram obtained here is unique to the random nature of long-range interactions. We explain this in terms of the enhanced correlations among local energies of the effective Anderson model induced by random power-law interactions.We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of random interactions ${V}_{ij}$ decaying as power-law ${V}_{ij}/{({r}_{ij})}^{\ensuremath{\alpha}}$ with distance ${r}_{ij}$. We demonstrate that MBL survives even for $\ensuremath{\alpha}<1$ and is preceded by a broad nonergodic subdiffusive phase. Starting from parameters at which the short-range interacting system shows an infinite temperature MBL phase, turning on random power-law interactions results in many-body mobility edges in the spectrum with a larger fraction of ergodic delocalized states for smaller values of $\ensuremath{\alpha}$. Hence, the critical disorder ${h}_{c}^{r}$, at which ergodic to nonergodic transition takes place, increases with the range of interactions. Time evolution of the density imbalance $I(t)$, which has power-law decay $I(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}$ in the intermediate to large time regime, shows that the critical disorder ${h}_{c}^{I}$, above which the system becomes diffusionless (with $\ensuremath{\gamma}\ensuremath{\sim}0$) and transits into the MBL phase, is much larger than ${h}_{c}^{r}$. In between ${h}_{c}^{r}$ and ${h}_{c}^{I}$ there is a broad nonergodic subdiffusive phase, which is characterized by the Poissonian statistics for the level spacing ratio, multifractal eigenfunctions, and a nonzero dynamical exponent $\ensuremath{\gamma}\ensuremath{\ll}1/2$. The system continues to be subdiffusive even on the ergodic side ($h<{h}_{c}^{r}$) of the MBL transition, where the eigenstates near the mobility edges are multifractal. For $h<{h}_{0}<{h}_{c}^{r}$, the system is superdiffusive with $\ensuremath{\gamma}>1/2$. The rich phase diagram obtained here is unique to the random nature of long-range interactions. We explain this in terms of the enhanced correlations among local energies of the effective Anderson model induced by random power-law interactions.

arXiv:2010.12485Physical review. B./Physical review. BFeb 8, 202117 citations

Effect of long-range hopping and interactions on entanglement dynamics and many-body localization

Rajeev Singh, Roderich Moessner, Dibyendu Roy

We numerically investigate the dynamics of entanglement in a chain of spinless fermions with nonrandom but long-range hopping and interactions, and with random on-site energies. For moderate disorder in the absence of interactions, the chain hosts delocalized states at the top of the band which undergo a delocalization-localization transition with increasing disorder. We find an interesting regime in this noninteracting disordered chain where the long-time entanglement entropy scales as $S(t)\ensuremath{\sim}lnt$ and the saturated entanglement entropy scales with system size $L$ as $S(L,t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\sim}lnL$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.We numerically investigate the dynamics of entanglement in a chain of spinless fermions with nonrandom but long-range hopping and interactions, and with random on-site energies. For moderate disorder in the absence of interactions, the chain hosts delocalized states at the top of the band which undergo a delocalization-localization transition with increasing disorder. We find an interesting regime in this noninteracting disordered chain where the long-time entanglement entropy scales as $S(t)\ensuremath{\sim}lnt$ and the saturated entanglement entropy scales with system size $L$ as $S(L,t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\sim}lnL$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.

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Initial state dependent dynamics across the many-body localization transition

Yogeshwar Prasad, Arti Garg

We investigate quench dynamics across many-body localization (MBL) transition in an interacting one-dimensional system of spinless fermions with aperiodic potential. We consider a large number of initial states characterized by the number of kinks ${N}_{\mathrm{kinks}}$ in the density profile, such that the equal number of sites are occupied between any two consecutive kinks. We show that on the delocalized side of the MBL transition the dynamics becomes faster with increase in ${N}_{\mathrm{kinks}}$ such that the decay exponent $\ensuremath{\gamma}$ in the density imbalance increases with increase in ${N}_{\mathrm{kinks}}$ . The growth exponent of the mean square displacement which shows a power-law behavior $\ensuremath{\langle}{x}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ in the long-time limit is much larger than the exponent $\ensuremath{\gamma}$ for one-kink and other low-kink states though $\ensuremath{\beta}\ensuremath{\sim}2\ensuremath{\gamma}$ for a charge density wave state. As the disorder strength increases ${\ensuremath{\gamma}}_{{N}_{\mathrm{kink}}}\ensuremath{\rightarrow}0$ at some critical disorder, ${h}_{{N}_{\mathrm{kinks}}}$ , which is a monotonically increasing function of ${N}_{\mathrm{kinks}}$ . A one-kink state always underestimates the value of the disorder at which the MBL transition takes place but ${h}_{\text{1}\phantom{\rule{0.16em}{0ex}}\text{kink}}$ coincides with the onset of the subdiffusive phase preceding the MBL phase. This is consistent with the dynamics of interface broadening for the one-kink state. We show that the bipartite entanglement entropy has a logarithmic growth $aln(Vt)$ not only in the MBL phase but also in the delocalized phase and in both the phases the coefficient $a$ increases with ${N}_{\mathrm{kinks}}$ as well as with the interaction strength $V$ . We explain this dependence of dynamics on the number of kinks in terms of the normalized participation ratio of initial states in the eigenbasis of the interacting Hamiltonian.We investigate quench dynamics across many-body localization (MBL) transition in an interacting one-dimensional system of spinless fermions with aperiodic potential. We consider a large number of initial states characterized by the number of kinks ${N}_{\mathrm{kinks}}$ in the density profile, such that the equal number of sites are occupied between any two consecutive kinks. We show that on the delocalized side of the MBL transition the dynamics becomes faster with increase in ${N}_{\mathrm{kinks}}$ such that the decay exponent $\ensuremath{\gamma}$ in the density imbalance increases with increase in ${N}_{\mathrm{kinks}}$. The growth exponent of the mean square displacement which shows a power-law behavior $\ensuremath{\langle}{x}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ in the long-time limit is much larger than the exponent $\ensuremath{\gamma}$ for one-kink and other low-kink states though $\ensuremath{\beta}\ensuremath{\sim}2\ensuremath{\gamma}$ for a charge density wave state. As the disorder strength increases ${\ensuremath{\gamma}}_{{N}_{\mathrm{kink}}}\ensuremath{\rightarrow}0$ at some critical disorder, ${h}_{{N}_{\mathrm{kinks}}}$, which is a monotonically increasing function of ${N}_{\mathrm{kinks}}$. A one-kink state always underestimates the value of the disorder at which the MBL transition takes place but ${h}_{\text{1}\phantom{\rule{0.16em}{0ex}}\text{kink}}$ coincides with the onset of the subdiffusive phase preceding the MBL phase. This is consistent with the dynamics of interface broadening for the one-kink state. We show that the bipartite entanglement entropy has a logarithmic growth $aln(Vt)$ not only in the MBL phase but also in the delocalized phase and in both the phases the coefficient $a$ increases with ${N}_{\mathrm{kinks}}$ as well as with the interaction strength $V$. We explain this dependence of dynamics on the number of kinks in terms of the normalized participation ratio of initial states in the eigenbasis of the interacting Hamiltonian.

arXiv:2202.05217Physical review. B./Physical review. BJun 6, 20229 citations

Localization in a random XY model with long-range interactions: Intermediate case between single-particle and many-body problems

Alexander L. Burin

Many-body localization in an XY model with a long-range interaction is investigated. We show that in the regime of a high strength of disordering compared to the interaction an off-resonant flip-flop spin-spin interaction (hopping) generates the effective Ising interactions of spins in the third order of perturbation theory in a hopping. The combination of hopping and induced Ising interactions for the power-law distance dependent hopping $V(R)\ensuremath{\propto}{R}^{\ensuremath{-}\ensuremath{\alpha}}$ always leads to the localization breakdown in a thermodynamic limit of an infinite system at $\ensuremath{\alpha}<3d/2$ where $d$ is a system dimension. The delocalization takes place due to the induced Ising interactions $U(R)\ensuremath{\propto}{R}^{\ensuremath{-}2\ensuremath{\alpha}}$ of ``extended'' resonant pairs. This prediction is consistent with the numerical finite size scaling in one-dimensional systems. Many-body localization in an XY model is more stable with respect to the long-range interaction compared to a many-body problem with similar Ising and Heisenberg interactions requiring $\ensuremath{\alpha}\ensuremath{\ge}2d$ which makes the practical implementations of this model more attractive for quantum information applications. The full summary of dimension constraints and localization threshold size dependencies for many-body localization in the case of combined Ising and hopping interactions is obtained using this and previous work and it is the subject for the future experimental verification using cold atomic systems.Many-body localization in an XY model with a long-range interaction is investigated. We show that in the regime of a high strength of disordering compared to the interaction an off-resonant flip-flop spin-spin interaction (hopping) generates the effective Ising interactions of spins in the third order of perturbation theory in a hopping. The combination of hopping and induced Ising interactions for the power-law distance dependent hopping $V(R)\ensuremath{\propto}{R}^{\ensuremath{-}\ensuremath{\alpha}}$ always leads to the localization breakdown in a thermodynamic limit of an infinite system at $\ensuremath{\alpha}<3d/2$ where $d$ is a system dimension. The delocalization takes place due to the induced Ising interactions $U(R)\ensuremath{\propto}{R}^{\ensuremath{-}2\ensuremath{\alpha}}$ of ``extended'' resonant pairs. This prediction is consistent with the numerical finite size scaling in one-dimensional systems. Many-body localization in an XY model is more stable with respect to the long-range interaction compared to a many-body problem with similar Ising and Heisenberg interactions requiring $\ensuremath{\alpha}\ensuremath{\ge}2d$ which makes the practical implementations of this model more attractive for quantum information applications. The full summary of dimension constraints and localization threshold size dependencies for many-body localization in the case of combined Ising and hopping interactions is obtained using this and previous work and it is the subject for the future experimental verification using cold atomic systems.

arXiv:1503.06508Physical Review BSep 25, 201597 citations

Many-body localization in incommensurate models with a mobility edge

Dong‐Ling Deng, Sriram Ganeshan, Xiaopeng Li +3 more

We review the physics of many-body localization in models with incommensurate potentials. In particular, we consider one-dimensional quasiperiodic models with single-particle mobility edges. A conventional perspective suggests that delocalized states act as a thermalizing bath for the localized states in the presence of of interactions. However, contrary to this intuition there is evidence that such systems can display non-ergodicity. This is in part due to the fact that the delocalized states do not have any kind of protection due to symmetry or topology and are thus susceptible to localization. A study of such incommensurate models, in the non-interacting limit, shows that they admit extended, partially extended, and fully localized many-body states. Non-interacting incommensurate models cannot thermalize dynamically and remain localized upon the introduction of interactions. In particular, for a certain range of energy, the system can host a non-ergodic extended (i.e. metallic) phase in which the energy eigenstates violate the eigenstate thermalization hypothesis (ETH) but the entanglement entropy obeys volume-law scaling. The level statistics and entanglement growth also indicate the lack of ergodicity in these models. The phenomenon of localization and non-ergodicity in a system with interactions despite the presence of single-particle delocalized states is closely related to the so-called many-body proximity effect and can also be observed in models with disorder coupled to systems with delocalized degrees of freedom. Many-body localization in systems with incommensurate potentials (without single-particle mobility edges) have been realized experimentally, and we show how this can be modified to study the the effects of such mobility edges. Demonstrating the failure of thermalization in the presence of a single-particle mobility edge in the thermodynamic limit would indicate a more robust violation of the ETH.We review the physics of many-body localization in models with incommensurate potentials. In particular, we consider one-dimensional quasiperiodic models with single-particle mobility edges. A conventional perspective suggests that delocalized states act as a thermalizing bath for the localized states in the presence of of interactions. However, contrary to this intuition there is evidence that such systems can display non-ergodicity. This is in part due to the fact that the delocalized states do not have any kind of protection due to symmetry or topology and are thus susceptible to localization. A study of such incommensurate models, in the non-interacting limit, shows that they admit extended, partially extended, and fully localized many-body states. Non-interacting incommensurate models cannot thermalize dynamically and remain localized upon the introduction of interactions. In particular, for a certain range of energy, the system can host a non-ergodic extended (i.e. metallic) phase in which the energy eigenstates violate the eigenstate thermalization hypothesis (ETH) but the entanglement entropy obeys volume-law scaling. The level statistics and entanglement growth also indicate the lack of ergodicity in these models. The phenomenon of localization and non-ergodicity in a system with interactions despite the presence of single-particle delocalized states is closely related to the so-called many-body proximity effect and can also be observed in models with disorder coupled to systems with delocalized degrees of freedom. Many-body localization in systems with incommensurate potentials (without single-particle mobility edges) have been realized experimentally, and we show how this can be modified to study the the effects of such mobility edges. Demonstrating the failure of thermalization in the presence of a single-particle mobility edge in the thermodynamic limit would indicate a more robust violation of the ETH.

Jan 1, 201721 citations

t(r)\ensuremath{\sim}{t}_{0}/{r}^{\ensuremath{\alpha}}

with

\ensuremath{\alpha}<2

1<\ensuremath{\alpha}<2

delocalizes the system partially, with almost all the states extended for

\ensuremath{\alpha}\ensuremath{\le}1

. Even in a system which has single-particle mobility edges in the noninteracting limit, turning on long-range interactions does not cause delocalization.

We study many-body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold potential

h<{h}_{c}

, the noninteracting system has single particle mobility edges (MEs) at $\ifmmode\pm\else\textpm\fi{}{E}_{c}$ while for

h>{h}_{c}

all the single particle states are localized. We demonstrate that even in the presence of single particle MEs, interactions do not always delocalize the system and the interacting system can have MBL. Our numerical calculation of energy level spacing statistics, participation ratio in the Fock space, and Shannon entropy shows that for some regime of particle densities, even for

h<{h}_{c}

, many-body states at the top (with

E>{E}_{2}

) and the bottom of the spectrum (with

E<{E}_{1}

) remain localized, though states in the middle of the spectrum are delocalized. Variance of entanglement entropy (EE) also diverges at

{E}_{1,2}

, indicating a transition from MBL to a delocalized regime, though transitions from volume to area law scaling for EE and from thermal to nonthermal behavior occur inside the MBL regime much below

{E}_{1}

and above

{E}_{2}

We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of random interactions

{V}_{ij}

decaying as power-law ${V}_{ij}/{({r}_{ij})}^{\ensuremath{\alpha}}$ with distance

{r}_{ij}

. We demonstrate that MBL survives even for $\ensuremath{\alpha}<1$ and is preceded by a broad nonergodic subdiffusive phase. Starting from parameters at which the short-range interacting system shows an infinite temperature MBL phase, turning on random power-law interactions results in many-body mobility edges in the spectrum with a larger fraction of ergodic delocalized states for smaller values of $\ensuremath{\alpha}$. Hence, the critical disorder

{h}_{c}^{r}

, at which ergodic to nonergodic transition takes place, increases with the range of interactions. Time evolution of the density imbalance

I(t)

, which has power-law decay $I(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}$ in the intermediate to large time regime, shows that the critical disorder

{h}_{c}^{I}

, above which the system becomes diffusionless (with $\ensuremath{\gamma}\ensuremath{\sim}0$) and transits into the MBL phase, is much larger than

{h}_{c}^{r}

. In between

{h}_{c}^{r}

and

{h}_{c}^{I}

there is a broad nonergodic subdiffusive phase, which is characterized by the Poissonian statistics for the level spacing ratio, multifractal eigenfunctions, and a nonzero dynamical exponent $\ensuremath{\gamma}\ensuremath{\ll}1/2$. The system continues to be subdiffusive even on the ergodic side (

h<{h}_{c}^{r}

) of the MBL transition, where the eigenstates near the mobility edges are multifractal. For

h<{h}_{0}<{h}_{c}^{r}

, the system is superdiffusive with $\ensuremath{\gamma}>1/2$. The rich phase diagram obtained here is unique to the random nature of long-range interactions. We explain this in terms of the enhanced correlations among local energies of the effective Anderson model induced by random power-law interactions.

L

as $S(L,t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\sim}lnL$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.

{N}_{\mathrm{kinks}}

in the density profile, such that the equal number of sites are occupied between any two consecutive kinks. We show that on the delocalized side of the MBL transition the dynamics becomes faster with increase in

{N}_{\mathrm{kinks}}

such that the decay exponent $\ensuremath{\gamma}$ in the density imbalance increases with increase in

{N}_{\mathrm{kinks}}

. The growth exponent of the mean square displacement which shows a power-law behavior $\ensuremath{\langle}{x}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ in the long-time limit is much larger than the exponent $\ensuremath{\gamma}$ for one-kink and other low-kink states though $\ensuremath{\beta}\ensuremath{\sim}2\ensuremath{\gamma}$ for a charge density wave state. As the disorder strength increases ${\ensuremath{\gamma}}_{{N}_{\mathrm{kink}}}\ensuremath{\rightarrow}0$ at some critical disorder,

{h}_{{N}_{\mathrm{kinks}}}

, which is a monotonically increasing function of

{N}_{\mathrm{kinks}}

. A one-kink state always underestimates the value of the disorder at which the MBL transition takes place but

{h}_{\text{1}\phantom{\rule{0.16em}{0ex}}\text{kink}}

coincides with the onset of the subdiffusive phase preceding the MBL phase. This is consistent with the dynamics of interface broadening for the one-kink state. We show that the bipartite entanglement entropy has a logarithmic growth

aln(Vt)

not only in the MBL phase but also in the delocalized phase and in both the phases the coefficient

a

increases with

{N}_{\mathrm{kinks}}

as well as with the interaction strength

V

. We explain this dependence of dynamics on the number of kinks in terms of the normalized participation ratio of initial states in the eigenbasis of the interacting Hamiltonian.

d

is a system dimension. The delocalization takes place due to the induced Ising interactions $U(R)\ensuremath{\propto}{R}^{\ensuremath{-}2\ensuremath{\alpha}}$ of ``extended'' resonant pairs. This prediction is consistent with the numerical finite size scaling in one-dimensional systems. Many-body localization in an XY model is more stable with respect to the long-range interaction compared to a many-body problem with similar Ising and Heisenberg interactions requiring $\ensuremath{\alpha}\ensuremath{\ge}2d$ which makes the practical implementations of this model more attractive for quantum information applications. The full summary of dimension constraints and localization threshold size dependencies for many-body localization in the case of combined Ising and hopping interactions is obtained using this and previous work and it is the subject for the future experimental verification using cold atomic systems.