Theory of the Many-Body Localization Transition in One-Dimensional Systems

We formulate a theory of the many-body localization transition based on a novel real-space renormalization group (RG) approach. The results of this theory are corroborated and intuitively explained with a phenomenological effective description of the critical point and of the “badly conducting” state found near the critical point on the delocalized side. The theory leads to the following sharp predictions: (i) The delocalized state established near the transition is a Griffiths phase, which exhibits subdiffusive transport of conserved quantities and sub-ballistic spreading of entanglement. The anomalous diffusion exponent $\alpha <1/2$ vanishes continuously at the critical point. The system does thermalize in this Griffiths phase. (ii) The many-body localization transition is controlled by a new kind of infinite-randomness RG fixed point, where the broadly distributed scaling variable is closely related to the eigenstate entanglement entropy. Dynamically, the entanglement grows as $\sim \log t$ at the critical point, as it does in the localized phase. (iii) In the vicinity of the critical point, the ratio of the entanglement entropy to the thermal entropy and its variance (and, in fact, all moments) are scaling functions of $L/\xi$, where $L$ is the length of the system and $\xi$ is the correlation length, which has a power-law divergence at the critical point.

Popular Summary

Experimental studies of ultracold atomic systems and considerations of possible frameworks for quantum information processing have motivated a growing interest in the quantum dynamics of many coupled degrees of freedom isolated from their external environment. Such closed quantum systems can have, broadly speaking, two regimes of dynamical behavior: They may approach thermal equilibrium under their own dynamics, or they may fail to do so because they are many-body Anderson localized. In the first regime, quantum correlations that may have been present in the initial state rapidly become inaccessible, and their dynamics become effectively classical. In the localized regime, on the other hand, local quantum coherence persists indefinitely. The phase transition between these two regimes therefore constitutes a sharp boundary between quantum and classical large-scale behavior that remains poorly understood.

Here, we present a novel renormalization group approach that aims to capture the universal features of the phase transition from many-body localized states to thermal states in one-dimensional systems. Our scheme describes the interplay between different regions of the system that locally appear either localized or thermalizing, and how they ultimately either thermalize or insulate each other. The quantum entanglement entropy emerges as a natural scaling variable in this picture, showing a universal change from an area law in localized states to a thermodynamic volume law via a critical point characterized by a broad distribution of entanglement. The important role played by randomness at the critical point leads to the anomalously slow propagation of information within the thermal phase near the transition.

We expect that our predictions may be tested in future experiments with ultracold atoms trapped in a one-dimensional random optical lattice potential.

Figure 1

(a) Coarse-grained model for the many-body localization transition. Each block represents a finite segment of the one-dimensional system. The coupling parameters $g$ and thermalization rates $\mathrm{\Gamma }$, which characterize single blocks and pairs of adjacent blocks, are explained in the text. (b) The model after the basic RG step of joining the pair of blocks with the fastest interblock thermalization rate ${\mathrm{\Gamma }}_{ij}$.

Figure 2

The many-body localization transition tuned by the bare coupling $g_0$ as seen in the RG flow of the dimensionless coupling $g=\mathrm{\Gamma }/\mathrm{\Delta }$. In the thermal phase, $g$ grows exponentially with block length $l$ (linearly with the dimension of the block’s Hilbert space), whereas in the localized phase, $g$ decreases exponentially. (a) Plot of the “typical” variable $⟨\log g⟩$ versus $l$. The data in the thermal side show good scaling collapse with the critical exponent $\nu \approx 3.1$ (inset). On the insulating side, on the other hand, $⟨\log g⟩$ decreases linearly with a nonuniversal slope that depends weakly on the value of $\log g_0$. (b) Plot of the “average” variable $\log ⟨g⟩$ versus $l$. This variable does collapse to a nontrivial scaling function on the insulating side of the transition with the same critical exponent obtained above on the metallic side. The fact that the critical fluctuations affect only the average quantity $\log ⟨g⟩$ suggests that they consist of exponentially rare, long thermal clusters inside the insulator. These fluctuations do not contribute to $⟨\log g⟩$.

Figure 3

Scaling of the coupling constants (averages and standard deviations) with block length at the critical point. The linear growth of the scaling variables with the length indicates a flow to infinite randomness.

Figure 4

(a) Entanglement entropy of eigenstates (solid lines) and its fluctuations (dashed lines) near the many-body localization transition computed from the RG as described in the text for different chain lengths $L$. Both the entropy and its fluctuation are normalized by the expected thermal entropy of $L$ elementary blocks $S_T=L{l}_0\log 2$. (b) Data collapse of $⟨S⟩/S_T$ and $\delta S/S_T$ obtained with the fitted correlation length exponent $\nu =3.1$ ($1/\nu =0.32$).

Figure 5

(a) The relation between length scale and time obtained from the RG flow in the many-body localized phase is logarithmic, i.e., $l\sim S\sim \log t$. (b) In the delocalized phase, the RG flows give a power-law scaling between length and entanglement thermalization time, $l\sim t^{{\alpha }_{\mathrm{ent}}}$. As explained in the text, the corresponding transport exponent $\alpha ={\alpha }_{\mathrm{ent}}/(1+{\alpha }_{\mathrm{ent}})$. (c) The transport exponent $\alpha$ as a function of the tuning parameter reveals a continuous transition from a localized phase to a subdiffusive but thermal Griffiths regime, i.e., with $0<\alpha <1/2$. The transport becomes diffusive, $\alpha =1/2$, deep in the delocalized phase. (d) Vanishing of ${\alpha }_{\mathrm{ent}}$ near the critical point plotted on a log-log plot is consistent with the anticipated scaling ${\alpha }_{\mathrm{ent}}\sim {\xi }_0/\xi \sim (⟨\log g_0⟩-⟨\log g_{0c}⟩{)}^{\nu }$.

Figure 6

Distributions of the entanglement entropy divided by its thermal value for systems of length $L/l_0=10000$ calculated in the different regimes: the localized phase (top-left panel), critical point (top-right panel), Griffiths phase (bottom-left panel), and diffusive regime (bottom-right panel).