Relations between entanglement and multifractality: deviations from ergodicity.
Relations between entanglement and multifractality: deviations from ergodicity.
In interacting setting of non-integrable quantum systems the concept of eigenstate thermalization hypothesis (ETH) comes into play. ETH bridges the gap between the unitary evolution in quantum mechanics and the relaxation in statistical physics of chaotic classical systems and claims that a generic isolated quantum system thermalizes as a result of its own dynamics. The thermalization in this sense is given either in terms of fluctuations of local observables or in terms of entanglement entropy. The main idea behind ETH is the coherent mixture of degrees of freedom leading to the effective relaxation of a local observable at long times and to the volume law scaling of the entanglement entropy of an eigenstate.
In interacting setting of non-integrable quantum systems the concept of eigenstate thermalization hypothesis (ETH) comes into play. ETH bridges the gap between the unitary evolution in quantum mechanics and the relaxation in statistical physics of chaotic classical systems and claims that a generic isolated quantum system thermalizes as a result of its own dynamics. The thermalization in this sense is given either in terms of fluctuations of local observables or in terms of entanglement entropy. The main idea behind ETH is the coherent mixture of degrees of freedom leading to the effective relaxation of a local observable at long times and to the volume law scaling of the entanglement entropy of an eigenstate.
The notion of weak ergodicity mentioned in the research direction "Non-ergodic phases of matter" is also closely related to the emergence of eigenstate thermalization in strongly interacting many-body systems. The deviations from the pure random matrix behavior (right inset in Fig. below) of the wave function moments and their entanglement in such systems may be explained in terms of the presence of special spectral edge states reducing the support set of the mid-spectrum states only to a finite fraction of the total Hilbert space.
The notion of weak ergodicity mentioned in the research direction "Non-ergodic phases of matter" is also closely related to the emergence of eigenstate thermalization in strongly interacting many-body systems. The deviations from the pure random matrix behavior (right inset in Fig. below) of the wave function moments and their entanglement in such systems may be explained in terms of the presence of special spectral edge states reducing the support set of the mid-spectrum states only to a finite fraction of the total Hilbert space.
Indeed, the ground and low excited states of non-integrable quantum many-body systems with local Hamiltonians are known to obey area law entanglement entropy, show residual local correlations, and occupy a zero fraction of the Hilbert space (see the middle bottom inset in the Fig. on the left).
Indeed, the ground and low excited states of non-integrable quantum many-body systems with local Hamiltonians are known to obey area law entanglement entropy, show residual local correlations, and occupy a zero fraction of the Hilbert space (see the middle bottom inset in the Fig. on the left).
In the paper "Entanglement of mid-spectrum eigenstates of chaotic many-body systems—deviation from random ensembles" we show that due to the orthogonality of the eigenbasis, the mid-spectrum (infinite temperature) states are depleted from the Hilbert space sector occupied by the low-entanglement states (see left top inset in Fig.). This so-called "Hilbert space blockade" leads to the additional small eigenstate coefficients |cmid| in their distribution in the computational basis (see panel (a) in Fig. on the left). Moreover the "location" of these small values is correlated with the large coefficients of edge spectral states, |clow| and |chigh| (see panels (b1-b2) in Fig.).
In the paper "Entanglement of mid-spectrum eigenstates of chaotic many-body systems—deviation from random ensembles" we show that due to the orthogonality of the eigenbasis, the mid-spectrum (infinite temperature) states are depleted from the Hilbert space sector occupied by the low-entanglement states (see left top inset in Fig.). This so-called "Hilbert space blockade" leads to the additional small eigenstate coefficients |cmid| in their distribution in the computational basis (see panel (a) in Fig. on the left). Moreover the "location" of these small values is correlated with the large coefficients of edge spectral states, |clow| and |chigh| (see panels (b1-b2) in Fig.).
The orthogonalization of a random pure state to the low-energy states leads also to the decrease of its entanglement entropy (see solid curved lines in the top and panel (c) in Fig. on the left). This deviation goes beyond residual local correlations as its persist not only for block, but also for so-called comb partitioning (see corresponding insets in the top of Fig.).
The orthogonalization of a random pure state to the low-energy states leads also to the decrease of its entanglement entropy (see solid curved lines in the top and panel (c) in Fig. on the left). This deviation goes beyond residual local correlations as its persist not only for block, but also for so-called comb partitioning (see corresponding insets in the top of Fig.).
Although the question of thermalization in isolated quantum systems is in focus of research interests for several decades, there are still gaps in understanding of the Eigenstate thermalization hypothesis. In the paper "Eigenstate Thermalization, Random Matrix Theory and Behemoths" we address the thermalization of highly non-local operators, Behemoths, in a generic non-integrable quantum mechanical system. We have shown that ETH for local operators can be understood in terms of the operator expansion into a superposition of operator building blocks (Behemoths), which follow super-ETH behavior, and the central-limit theorem for Behemoths.
Although the question of thermalization in isolated quantum systems is in focus of research interests for several decades, there are still gaps in understanding of the Eigenstate thermalization hypothesis. In the paper "Eigenstate Thermalization, Random Matrix Theory and Behemoths" we address the thermalization of highly non-local operators, Behemoths, in a generic non-integrable quantum mechanical system. We have shown that ETH for local operators can be understood in terms of the operator expansion into a superposition of operator building blocks (Behemoths), which follow super-ETH behavior, and the central-limit theorem for Behemoths.
From the fundamental point of view multifractality provides a way of ergodicity breaking in terms of chaotization and equipartitioning. However, in quantum information theory the entanglement is the main measure of ergodicity and thermalization. Thus, it is of particular interest to consider the role of entanglement and multifractality in quantum dynamics, ergodicity and thermalization and their mutual relations in many-body disordered systems in MBL and thermalizing parameter range.
From the fundamental point of view multifractality provides a way of ergodicity breaking in terms of chaotization and equipartitioning. However, in quantum information theory the entanglement is the main measure of ergodicity and thermalization. Thus, it is of particular interest to consider the role of entanglement and multifractality in quantum dynamics, ergodicity and thermalization and their mutual relations in many-body disordered systems in MBL and thermalizing parameter range.
In the paper "Multifractality meets entanglement: relation for non-ergodic extended states" we have found an exact relation between the entanglement entropy and fractal dimensions giving the upper bound for the entanglement entropy for any eigenstate with a given fractal dimension (see upper panels in the left). In addition, we provide an explicit example demonstrating that the entanglement entropy may reach its ergodic (Page) value when the wave function is still highly non-ergodic and occupies a zero fraction of the total Hilbert space.
In the paper "Multifractality meets entanglement: relation for non-ergodic extended states" we have found an exact relation between the entanglement entropy and fractal dimensions giving the upper bound for the entanglement entropy for any eigenstate with a given fractal dimension (see upper panels in the left). In addition, we provide an explicit example demonstrating that the entanglement entropy may reach its ergodic (Page) value when the wave function is still highly non-ergodic and occupies a zero fraction of the total Hilbert space.
The explanation of this phenomenon can be easily given by the structure of the reduced density matrix (see bottom panels in the left). Indeed, for fractal support sets ND smaller than the subsystem size there are only few non-zero elements in the density matrix dominated by the diagonal contribution. As soon as fractal dimension is larger D>0.5 the diagonal of this density matrix is filled leading to the saturation of the entanglement entropy at the Page value.
The explanation of this phenomenon can be easily given by the structure of the reduced density matrix (see bottom panels in the left). Indeed, for fractal support sets ND smaller than the subsystem size there are only few non-zero elements in the density matrix dominated by the diagonal contribution. As soon as fractal dimension is larger D>0.5 the diagonal of this density matrix is filled leading to the saturation of the entanglement entropy at the Page value.