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| Fig. 1: The one dimensional RG rules of Goremykina et al. are shown here. [4] li labels the original lengths of the blocks and T/I stands for thermal/insulating blocks respectively. The new length of the combined block is a weighted sum of the lengths of the original blocks with weightings αT,αI chosen to reflect the entanglement times of the combined block. (Source: Z. Shi) |
Many body localization (MBL) is a novel dynamical phase of disordered quantum matter where the existence of an extensive number of local conserved quantities (referred to as l-bits) prevent the system from reaching local thermal equilibrium. These emergent conserved quantities are stable under perturbations of the model parameters, in contrast to the often non-local conserved quantities that appear in fine-tuned integrable models. (See Korepin et al. for some examples of quantum integrable models. [1]) Furthermore, the presence of l-bits immediately imply many striking physical properties - preservation of local memories at infinite times, logarithmic growth of subregion entanglement entropy, etc. (See Nandkishore et al. and references therein. [2]) These properties are not only of theoretical interest to the foundations of statistical mechanics but are also important to stable storage and processing of coherent quantum information.
The groundbreaking work of Imbrie in 2014 identified these l-bits in an open subset of the parameter space (labeled by three disorder strength variables h, γ, J) of a disordered Ising model, thus putting one-dimensional MBL on a firm theoretical footing. [3]
One natural question is: what happens in the rest of the parameter space? Numerically, the answer is clear: simulations have confirmed the existence of a thermal phase in Imbrie's model away from the MBL region. But it is an embarrassing fact that a rigorous proof of the more familiar thermal phase is still missing. From a mathematical perspective, this fact is perhaps not surprising, because Imbrie's proof of MBL fundamentally relies on the convergence of perturbation theory when one of the disorder parameters γ is small (along with some additional control over the non-perturbative resonances). On the thermal side, all parameters are comparable to each other and no perturbation theory should be trusted whatsoever. This is one of the fundamental challenges of strongly coupled many-body physics in general.
One additional challenge is that the transition between MBL and thermal phases is fundamentally a transition between systems in and out of equilibrium. Therefore, none of the conventional intuitions about equilibrium phase transitions applies and progress must come from novel phenomenological models.
In this review, we will present a heuristic renormalization group (RG) model first introduced by Goremykina et al., the latest of a series of RG models proposed in the past five years. [4] The earliest model of Vosk et al. is microscopically motivated but inaccessible analytically. [5] A later variation by Zhang et al. obtained more precise analytic results after making an unphysical assumption about the entanglement times of thermal and MBL systems. [6] The work we review finds a sweet spot between these earlier works, restoring some portion of the microscopic structure in Vosk et al. while preserving solvability. [5,6] We now proceed to introduce the rules of the RG flow and analyze its properties.
The general philosophy behind the RG rules is as follows: numerically, the phenomenon of MBL is ubiquitous across models with different microscopic interactions. The theoretical challenge is not to solve all of these models precisely, but rather to identify universal features. The universal feature we focus on in this article is the logarithmic growth of subsystem entanglement entropy in MBL systems to be contrasted with the power law growth in thermal systems.
Taking advantage of this distinction, we can partition a general 1D system unambiguously into spatially alternating thermal and insulating (localized) blocks. We associate to each block of length l an entanglement time τ which is the time it takes for the entanglement entropy of the block to saturate to its thermal entropy upon coupling to a large bath. In a thermal/insulating block, τ scales linearly/exponentially with the length l respectively. The state of the system is then specified by the length distribution of thermal/insulating blocks.
The-long wavelength physics can be extracted by a renormalization group (RG) flow that combines small blocks into larger ones and reduces the total number of thermal/insulating labels required to characterize the system. More specifically, when renormalizing from length scale Γ to Γ + δΓ, we take all insulating/thermal blocks with lengths in the range [Γ,Γ+δΓ] and combine them with the two surrounding blocks as shown in Fig. 1. li labels the original lengths of the blocks and T/I stands for thermal/insulating respectively. Note that importantly, the length of the combined block is not simply the sum of lengths of the constituent blocks. Instead there are weighting factors αT, αI associated to the middle block. This is because we want the new length to reflect the entanglement spreading times of the combined block. When two thermal blocks absorb an insulating block, the combined thermal block has a much larger τ due to the exponentially long time it takes to spread entanglement across the insulating block in the middle. On the other hand, when two insulating blocks absorb a thermal block, the linear entanglement time of the thermal block gives an almost negligible contribution. As a result, on heuristic grounds, we should have αT << 1 << αI.
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| Fig. 2: The line of RG fixed points in the αT = 0 limit are shown in the plot. The flow lines are meant to be schematic. (Source: Z. Shi) |
With these basic RG rules, we can pass to a continuum limit and recast the discrete RG rules as integro-differential flow equations for the probability distribution of thermal/insulating blocks of different lengths (see the extended report for a detailed derivation). In the case αT = αI = -1, these equations reduce to those studied by Fisher in his seminal paper. [7] But we are of course interested in a very different limit, where exact solutions are not possible. As we see in the next section, Goremykina et al. were nevertheless able to extract important physical properties from the flow equations via a clever sequence of asymptotic analyses. [4]
The first step in the analysis is a calculation of the stationary solution to the flow equation (i.e. fixed point of the RG flow equations). Although the physical properties we extract are independent of the precise values of αT, αI, it is analytically convenient to impose αT αI = 1 (physically motivated since it guarantees the asymptotic conservation of the total length of the system as we flow into the longer length scales). Under this additional assumption, the fixed points QT and QI can be solved for exactly for α << 1 and good matching with numerical solutions was obtained by Goremykina et al.. [4] Additional critical properties near the RG fixed point can be approached by a variety of standard techniques which we enumerate below:
As a first pass, one can take small perturbations away from the fixed point parametrized by δQT,I(Γ, η) = Γ1/ν fT,I(η) where η is a length variable over which we define the probability distribution, Γ sets the current RG length scale, ν is the critical exponent that describes universal scaling properties of solutions near the fixed point, and fT,I(η) is an arbitrary function parametrizing deviations from the fixed point. To linear order in f, the flow equations dictate that 1/ν is the maximal eigenvalue of a certain integro-differential operator (explained in more detail in the extended report). In the infinite αI limit, a combination of numerical and analytic techniques give the asymptotic form of the maximal eigenvalue ν = ln(1+αI). The divergence of this critical exponent is in sharp contrast to finite estimates obtained in previous RG studies. [5,6]
Since the critical exponent ν diverges as 1/αI = αT goes to 0, the ansatz δQT,I (Γ, η) = Γ1/ν fT,I(η) becomes a marginal perturbation that doesn't decay to zero in the small Γ regime. To study the direction of RG flows, we must go beyond the linear order RG flow equation in fT,I(η). This seems like a daunting task. But Goremykina et al. circumvented an explicit expansion by a slick resummation trick: they examined the solution to the linearized flow equation and postulated a family of functions that coincide with the linear solution to leading order. [4] This procedure is the most hand-wavy step in the argument. But with this leap of faith, we get a family of ansatz distributions QT and QI parametrized by two functions κ and γ. Remarkably, plugging this ansatz into the exact RG flow equations gives a line of fixed points κ > 0, γ = 0 in the αT = 0 limit as shown schematically in Fig. 2. Solving the RG equations approximately near the fixed point gives an exponential Kosterlitz-Thouless scaling of the correlation length in sharp contrast to previous papers that propose power law scalings (we explain this more precisely in the extended report). [5]
One can study other aspects of the RG flow by including more parameters into the RG rules. Goremykina et al. chose as an example the total length lT of all microscopic thermal/insulating blocks that constitute a given renormalized thermal/insulating block of length l. [4] The RG rules for lT has αI = αT = 0. The asymptotic scaling of l as a function of lT can be extracted. The scaling exponent is called the fractal dimension. We naturally expect that in the MBL limit, insulating blocks should have fractal dimension one because the entanglement times of insulating blocks always dominate over the thermal blocks. This is indeed confirmed numerically and analytically in the supplementary materials of Goremykina et al. [4]
When we try to extend the RG scheme to higher dimensions, some challenges immediately present themselves. First of all, one dimensional systems have a special topology that ensures every block is always surrounded by two blocks. This allows a simple RG rule that always combines three blocks into one. The same is no longer true in higher dimensions (as we will explain in more detail later). Second, the natural additive quantity in higher dimensional RG is the volume of thermal/insulating blocks. However, the entanglement times scale with length rather than volume. The miracle of one dimension is that length coincides with volume. In higher dimensions, this misalignment of scales makes it hard to identify the appropriate renormalization parameters. Nevertheless, we believe that RG is at least possible in two dimensions due to some interesting mathematical results that relate linear and quadratic dimensions in 2D.
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| Fig. 3: Combination of thermal/insulating (T/I) blocks in the 2D case. We show four cases where an insulating block is surrounded by 1,2,3,4 thermal blocks respectively. This is meant to demonstrate that any integer number of surrounding blocks is allowed. (Source: Z. Shi) |
Consider a 2D domain of finite extent. When we partition the domain into blocks, we are giving the domain the structure of a graph. Every vertex of the graph can be assigned a degree which counts the number of blocks that intersects with the vertex. The assignment of thermal/insulating labels to these blocks is the simply the two-coloring problem. Elementary graph theory tells us that such an assignment is possible iff every vertex has even degree. Under the same assumption, we can define the RG procedure to be the combination of a central block with all blocks that share an edge with itself. This is shown pictorially in Fig. 3.
Remarkably, the even degree condition is preserved under the RG procedure and we get a well-defined flow. Next we must identify the variable that keeps track of changes in entanglement times. Area is not a good candidate because a thin ribbon and a round disc can have comparable area but drastically different linear dimensions. It would be unphysical to treat them as equal objects under RG flow. We therefore need some way to quantify the discrepancy between area and length. This link is provided by the isoperimetric inequality in differential geometry. For two dimensions, it roughly says that for any closed curve C, L2 ≥ 4 π A where equality is achieved by any circle. Deviations from the perfectly symmetric circle can be encoded in the mean squared curvature Δκ = ∫ ds [κ(s) - <κ>]2 which is a coordinate invariant measure of the degree of irregularity in the curve C bounding a given block (here, <κ> is the mean curvature). Therefore, a preliminary proposal would be to work out RG rules for A with a Δκ-dependent compensation factor that controls the change in entanglement time upon block combination. In this review we don't have time to flesh out the details of this proposal, but it would be interesting to see whether the techniques we learn from the 1D RG flow can be adapted to extract some properties of MBL transitions (if they exist at all) in higher dimensions.
© Zhengyan Shi. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, 1997).
[2] R. Nandkishore and D. A. Huse, "Many-Body Localization and Thermalization in Quantum Statistical Mechanics," Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
[3] John Z. Imbrie, "On Many-Body Localization for Quantum Spin Chains," J. Stat. Phys. 163, 998 (2016).
[4] A. Goremykina, R. Vasseur, and M. Serbyn, "Analytically Solvable Renormalization Group for the Many-Body Localization Transition," Phys. Rev. Lett. 122, 040601, (2019).
[5] R. Vosk, D. A. Huse, and E. Altman, "Theory of the Many-Body Localization Transition in One-Dimensional Systems," Phys. Rev. X 5, 031032 (2015).
[6] L. Zhang et al., "Many-Body Localization Phase Transition: A Simplified Strong-Randomness Approximate Renormalization Group," Phys. Rev. B 93, 224201 (2016).
[7] D. S. Fisher, "Critical Behavior of Random Transverse-Field Ising Spin Chains," Phys. Rev. B 51, 6411 (1995).