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| Fig. 1: a) Tensor network representation of WWF. The total unitary transformation is a sum of an infinite stack of infinitesimal unitary tensor strings. b) Tensor network representation of a matrix product eigenstate. (Source: T. Nebabu, following Pekker et al. [5]) |
According to the eigenstate thermalization hypothesis (ETH), a large class of isolated quantum many-body systems are capable of reaching thermal equilibrium. This means that the expectation values of few-body observables at late times are well predicted by the energy of the system rather than by the microscopic degrees of freedom, just as in the classical Gibbs ensemble. [1] Thermalization is known to occur for generic non-integrable systems with a large amount of entanglement in their eigenstates, since the scrambling of information about the system's initial state results in thermalizing behavior at late times. In contrast, in the presence of disorder, non-integrable systems can fail to thermalize - a phenomenon known as many-body localization (MBL). [2] In this case, the presence of disorder dramatically slows the spread of entanglement, resulting in a failure to come to thermal equilibrium. [3]
Although MBL is known to survive to all orders in perturbation theory even with power-law interactions, it is an open question whether the MBL phase is stable to non-perturbative effects. [4] Incidentally, the most compelling evidence for the survival of the MBL phase relies on the existence of a local unitary transformation that recasts MBL Hamiltonian into an integrable form in terms of emergent local integrals of motion. Then, the lack of thermalization in the MBL regime can be attributed to an extensive number of conserved quantities. For instance, one expects that deeply within the localized phase, the Hamiltonian written originally in terms of "p-bits," i.e. Pauli operators can be rewritten as a local Hamiltonian in terms of "l-bits" (localized bits) with pseudospin operators.
where the τiz operators are quasilocal and the coupling constants εi1,i2,... decay exponentially over longer ranges.
Finding such a local transformation that recasts the Hamiltonian in this way is challenging. This stems from the fact that there are infitely many choices for the LIOMs (any linear combination of them is also conserved), and only a restricted set of unitary transformations will result in maximally-localized "l-bits." A promising approach is to define a type of renormalization group flow that performs a succession of local similarity transformations that eventually diagonalizes the Hamiltonian, putting it into the above form.
In the paper, we discuss two different methods that have recently been implemented in MBL systems: 1) Wilson-Wegner Flow (WWF) and 2) Spectrum Bifurcation RG (SBRG). In WWF, one defines a continuous flow parameter that characterizes the energy scale one is operating at in the RG step. Meanwhile, spectrum bifurcation involves a discrete RG flow with a finite total number of steps. In both cases, one applies a sequence of unitary transformations that "removes" higher energy scales and disentangle the physical degrees of freedom.
In both the WWF and SBRG, the total unitary transformation implementing the diagonalization may be visualized as a tensor network. Viewing the RG protocol in this way provides two main advantages: 1) for a large class of MBL Hamiltonians, the RG flow may be implemented entirely using matrix product operators, which can speed up computation 2) the tensor-network picture admits a holographic interpretation, which can give physical insight into the entanglement properties of the MBL system. By relating the network to a physical geometry connecting the boundary p-bits to the bulk l-bits, one can diagnose the localization of the Hamiltonian using geometric features.
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| Fig. 2: Visualization of the SBRG procedure. (Source: T. Nebabu, after You et al. [6]) |
The general WWF protocol, first developed for non-perturbative QCD, is a way to continuously flow the Hamiltonian towards its diagonalized form by applying a continuously-varying unitary transformation. In typical real-space RG schemes, one coarse-grains the degrees of freedom to obtain an effective low-energy Hamiltonian. In contrast, the WWF preserves the number of degrees of freedom while decoupling those that are separated by large energy scales. Then, the rules for the flow are governed by a differential equation for the elements of the Hamiltonian that reduces the magnitude of the off-diagonal elements. Roughly speaking, one can think of WWF as introducing an energy scale β, and then performing a similarity transformation H(β) = U(β) H U†(β) so that the "dressed" Hamiltonian H(β) describes only processes that correspond to an energy transfer of β or less. The flow towards higher β then moves towards progressively smaller energy scales. The results of this method reveals 1) operator spreading in the RG flow that mimics real-space operator growth 2) a segmented tensor network geometry in the MBL phase with a size that shrinks exponentially. A schematic of the tensor network that implements the similarity transformation of the WWF is shown in Fig. 1.
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| Fig. 3: Fragmented holographic geometry of an MBL system under the SBRG. Yellow blocks denote the Clifford gates, blue dots denote the physical qubits, and red arrows denote the emergent qubits. (Source: T. Nebabu, after You et al.[6]) |
The spectrum-bifurcation RG method (SBRG), developed in You et al., can be seen as a discrete approximation to the regular WWF flow. [2] Rather than defining a unitary transformation as a function of a continuous parameter, a discrete sequence of block-diagonalizations is executed. At each step, the largest energy scale is targeted and block diagonalized using Clifford gates. This is repeated for the next highest energy scale until the block sizes reduce to 1, as shown in Fig 2. The SBRG method provides an efficient method for finding emergent integrals of motion for large system sizes and exploring transitions between MBL phases.
One can interpret the Clifford circuit used in the SBRG geometrically. In our geometric model, we will plant the physical degrees of freedom on the boundary of some circular geometry and the emergent degrees of freedom within the bulk interior. The energy scale of the emergent qubit is characterized by the Clifford gate's depth within the bulk of the circuit, which one associates to a radial position in the bulk. Then the Clifford circuit manifests as a disentangler network mapping the boundary eigenstates (IR) to an emergent bulk product state (UV). The result is a geometry that appears largely fragmented, with nearby boundary degrees of freedom becoming mixed by transformations but far-separated qubits remaining disconnected, as shown schematically in Fig 3.
© Tamra Nebabu. 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] J. M. Deutsch, "Eigenstate Thermalization Hypothesis," Rep. Prog. Phys. 81, 082001 (2018).
[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] J. H. Bardarson, F. Pollmann, and J. E. Moore, "Unbounded Growth of Entanglement in Models of Many-Body Localization," Phys. Rev. Lett. 109, 017202 (2012).
[4] D. M. Basko, I.L. Aleiner, and B. L. Altshuler, "Metal-Insulator Transition in a Weakly Interacting Many-Electron System with Localized Single-Particle States," Ann. Phys. (USA) 321, 1126 (2006).
[5] D. Pekker et al., "Fixed Points of Wegner-Wilson Flows and Many-Body Localization," Phys. Rev. Lett. 119, 075701 (2017).
[6] Y.-Z. You, X.-L. Qi, and C. Xu, "Entanglement Holographic Mapping of Many-Body Localized System by Spectrum Bifurcation Renormalization Group," Phys. Rev. B 93, 104205 (2016).