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| Fig. 1: The VHA model. Each block represents a finite segment of spins and is associated with a rate Γ which is proportional to the inverse thermalization time and a coupling strength g that characterizes the thermalizing/localization behavior of the block. The interblock rate Γij is defined by the inverse time for the two blocks to fully relax. Each coarse graining step involves combining the two blocks with the strongest two-block rate together, with the new interblock rate ΓL defined according to the RG rule (see text). For this example the two-block rate Γ23 is assumed to be the largest among Γij's, so block 2 and block 3 are decimated into a larger block in this step of RG. (Source: K.-Y. Lin, after Vosk et al. [3]) |
The ability to succinctly describe the behavior of a macroscopic interacting system using only a handful of parameters lies at the heart of many-body physics. However, formulating simple theories in the vicinity of phase transitions can be challenging due to fluctuations at all length scales up to the macroscopic system size. In his seminal paper, Wilson conceived of scaling invariance at critical points due to the absence/divergence of length scales and put forward a elegant theory of momentum space renormalization group (RG). [1] Applying the essence of the RG method, namely the invariance of the theory under coarse graining and integrating out degrees of freedom, to real space Hamiltonians, Kadanoff constructed a powerful tool to tackle the critical behavior of Ising-like systems. [2]
The many-body localized (MBL) phase has attracted substantial interests, both theoretically and experimentally due to its potential application to quantum computing. Such a phase is a class of many-body interacting system in which, contrary to conventional wisdom from classical thermodynamics, thermalization behavior is absent and memory of the initial configuration is retained at late times. Contrary to equilibrium (ground state) phases, the MBL phase is distinct in its dynamical behavior, for instance the vanishing of DC conductance, logarithmic growth of entanglement entropy and the Poisson level statistics signifying a emergent integrability.
Characterizing the nature of the MBL-thermalization transition is both illuminating and challenging since the conventional paradigm that focuses on the absence/presence of gaps does not work. The microscopic picture of how fluctuations drive the system across the transition is also vastly different.
The application of RG to the MBL transitions, nevertheless, is not necessarily hindered by aforementioned technical challenges as it only relies on the scaling invariance of the system. In fact, several RG approaches have been proposed to study the property of the MBL transition. [3-8] While each of them differs in the RG rules which lead to different scaling behavior and falls short on microscopic justifications, there are some overall agreements among these approaches, and a convergence trend towards reaching a final consensus is apparent. This report serves as a review and comparison of these approaches.
The earlier works focus on joining local degrees of freedom (for instance spin-1/2) into bigger blocks. [3-5] It is not necessary for one to start with a particular microscopic Hamiltonian, but one can picture a disordered Heisenberg chain to help making the process more intuitive:
where hi is distributed uniformly in [-W,W], and W is the single particle disorder strength.
In the block decimation process, we only keep track of the inter/intrablock coupling strengths Γij/Γi and the average energy spacings Δi. Depending the strength of the dimensionless coupling parameter gi ~ Γi/Δi, the blocks can be classified to be in the thermalizing (ETH) regime when g >> 1 or in the insulating (MBL) regime when g << 1. Since the entanglement dynamics of thermal and insulating blocks are qualitatively different, it is necessary to take different approaches to derive block decimation rules for thermal blocks and insulating blocks. (See Fig. 1 for the definition of the parameters g's and Γ's.)
For insulating blocks, the couplings are weak and one can take perturbative approach and obtain the relaxation rates using Fermi's golden rule:
For thermalizing blocks, one assumes diffusive operator spread and obtains the relaxation rates using Ohm's law:
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| Fig. 2: The RG rule in Goremykina et al. [7] Within each RG step, blocks with length below the cutoff length scale Ω get combined with its neighboring blocks of the opposite type into a bigger block. The difference of the "effective conductance" gives rise to the parameter α and β. (Source: K.-Y. Lin, after Goremykina et al. [7]) |
While the actual RG rules are slightly different among the earlier works, the numerically obtained universal behavior is similar in terms of correlation length ξ~1/(W-WL)ν with a critical exponent ν~3.5. [3-5]
The previous block decimation approaches suffers from the shortcomings that no analytic tools are readily existent to analyze the RG flow of the parameters. Therefore, the scaling behaviors are only extracted from numerical simulations, which only extend up to system size of ~106. In light of this, several later attempts seek to bypass matrix element calculation in the previous block decimation framework and instead construct phenomenological RG flows based on minimal models that incorporate the past intuition with the MBL transition, such as quantum avalanches and Griffiths effects, which respectively depict the behavior of thermal bubbles destroying MBL phases and insulating bubbles causing sub-diffusive transport in thermal phases. [9] These models hence abstract away the use of Hamiltonians to describe the system. One such analytically solvable model proposed in Zhang et al. and generalized in Goremykina et al. is to take the length distribution of thermal and insulating blocks as the only parameters. [6,7] As shown schematically in Fig. 2, for each coarse graining step, the blocks with length equal to the cutoff length scale gets combined with its neighboring block with the RG law similar to Ohm's law:
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| Fig. 3: The RG flow map derived in Dumitrescu et al. [8] The line of fixed points corresponds to the MBL phase. The two axes ξ and ρ represent the bare localization length and the thermalizing block density respectively. A different set of parameters is used to obtain the RG flow equation in Goremyhkine et al., but they qualitatively describe the same physical quantities. [7] (Source: K.-Y. Lin, after Dumitrescu et al. [8]) |
| lnewI | = | ln-1I + α ln T+ln+1I |
| lnewT | = | ln-1T + β lnI + ln+1T |
Under the strong asymmetry condition β >> 1 >> α(which is realistic), it can then be shown that the probability distribution functions of thermal block lengths and insulating block lengths are characterized by two parameters with solvable flow equations. The RG flow map is shown in Fig. 3, where a line of stable fixed points is apparent on the +x axis. Surprisingly, the RG equation of this model can be directly mapped to the RG equation that describes the Kosterlitz- Thouless transition of the 2D XY model. Contrary to what is shown in the earlier works, the divergence of length scale is of the form η ~ exp(c/(|W-W_c|)1/2), which actually corresponds to a critical exponent ν = ∞. [7-8]
The discrepancy between this work and the earlier work could perhaps be explained by the difficulty to see the KT scaling in a finite size system. [3-5] A later work by Dumitrescu et al. reconciles the earlier results with the KT prediction by examining the scaling behavior of the probability distribution of the thermal block length around the critical point pl ~ l-2. [3-5,8] Dumitrescu et al. also give an elegant reasoning of why the RG equations should be KT-like without assuming any underlying coarse graining rules. [8]
The paper serves to extend the discussions of two frameworks outlined above. In the first half of the paper, we will sketch out the derivation of the block decimation RG rule and explain the results obtained in the earlier works. [3-5] The second part of the paper we will motivate the RG rule of the toy model in Goremykina et al. and demonstrate how the RG rule gives rise to the integral-differential equation that results in the final RG flow equation. [7] We also derive the scaling behavior of the length scale according to the RG equation. Finally, we go through Dumitrescu et al., which touches upon results from both approaches. [8]
© Kuan-Yu Lin. 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] K. G. Wilson, "The Renormalization Group: Critical Phenomena and the Kondo problem," Rev. Mod. Phys. 47, 8773 (1975).
[2] L.P. Kadanoff, "Scaling Laws for Ising Models Near Tc," Physics Physique Fizika 2, 263 (1966).
[3] 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).
[4] A. C. Potter, R. Vasseur, and S. A. Parameswaran, "Universal Properties of Many-Body Delocalization Transitions," Phys. Rev. X 5, 031033 (2015).
[5] P. T. Dumitrescu, R. Vasseur, and A. C. Potter, "Scaling Theory of Entanglement at the Many-Body Localization Transition," Phys. Rev. Lett. 119, 110604 (2017).
[6] L. Zhang et al., "Many-Body Localization Phase Transition: A Simplified Strong-Randomness Approximate Renormalization Group," Phys. Rev. B 93, 224201 (2016).
[7] A. Goremykina, R. Vasseur, and M. Serbyn, "Analytically Solvable Renormalization Group for the Many-Body Localization Transition," Phys. Rev. Lett. 122, 040601 (2019).
[8] P. T. Dumitrescu et al., "Kosterlitz-Thouless Scaling at Many-Body Localization Phase Transitions," Phys. Rev. B 99, 094205 (2019).
[9] W. De Roeck et al., "Stability and Instability Towards Delocalization in Many-Body Localization Systems," Phys. Rev. B 95, 155129 (2017).