Science (Faculty of)
http://hdl.handle.net/10012/9936
20200716T04:55:59Z

Tensor networks, quantum spin chains, and quantum field theory
http://hdl.handle.net/10012/16049
Tensor networks, quantum spin chains, and quantum field theory
Zou, Yijian
Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operatorstate correspondence, thus completing the project initiated by Cardy and others in the 80's.
Our method is based on the lowenergy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the KooSaleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the lowenergy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finitesize corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute lowenergy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry.
We test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on lowenergy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.
20200707T00:00:00Z

Renormalization, Entanglement and Continuous Tensor Networks
http://hdl.handle.net/10012/16047
Renormalization, Entanglement and Continuous Tensor Networks
Franco Rubio, Adrián
The study of the ground states of local Hamiltonians in quantum manybody and quantum field theoretic systems is a source of many research problems of great complexity. Almost in its entirety, this thesis deals with constructions related to the field of tensor networks, which are efficient parametrizations of mathematical objects from quantum manybody theory (such as quantum states and operators) that exploit our knowledge of their entanglement structures in order to obtain improvements in our computational abilities. In the past decade, the new research program of continuous tensor networks has arisen with the goal of reproducing, in the setting of quantum field theory, the success that tensor network techniques have enjoyed on the lattice. First, we present a number of results from research performed on noninteracting examples of the continuous Multiscale Entanglement Renormalization Ansatz, or cMERA, a continuous tensor network that provides a variational wavefunctional for a quantum field theoretic ground state. Initially, we study the correlations and entanglement structure of cMERA states, providing evidence that such states can be interpreted as UV regularized versions of the physical states they approximate. After that, we study what modifications of the formalism are necessary in order to consider systems with gauge symmetry, or in the presence of boundaries and defects. We obtain prescriptions for the treatment of these cases which, we expect, may also hold in the interacting setting. We then abandon briefly the area of continuous tensor networks to present a piece of research within a different though related research program: that of the study of manifestations of universal emergent behaviour in critical lattice systems. In particular we bring attention to the existence of an approximate representation of the Virasoro algebra supported on the eigenvectors of the reduced density matrices of critical lattice systems. We call this the entanglement algebra, and study its accuracy in the particular case of the Ising model. Lastly, we present a proposal for a continuous version of the Tensor Network Renormalization (TNR) algorithm, which we dub continuous TNR, or cTNR. cTNR operates on UV regularized classical partition functions or quantum Euclidean path integrals, and generates a realspace renormalization group flow via continuous coarsegraining and rescaling operations. We show that a UV regularized version of the free boson path integral can be made a fixed point of such a renormalization flow, in a way that allows for recovery of the conformal data associated to the free boson conformal field theory.
20200707T00:00:00Z

Reconstructing quantum molecular rotor ground states
http://hdl.handle.net/10012/16033
Reconstructing quantum molecular rotor ground states
De Vlugt, Isaac J. S.; Iouchtchenko, Dmitri; Merali, Ejaaz; Roy, PierreNicholas; Melko, Roger
Nanomolecular assemblies of C60 can be synthesized to enclose dipolar molecules. The lowtemperature states of such endofullerenes are described by quantum mechanical rotors, which are candidates for quantum information devices with higherdimensional local Hilbert spaces. The experimental exploration of endofullerene arrays comes at a time when machine learning techniques are rapidly being adopted to characterize, verify, and reconstruct quantum states from measurement data. In this paper, we develop a strategy for reconstructing the ground state of chains of dipolar rotors using restricted Boltzmann machines (RBMs) adapted to train on data from higherdimensional Hilbert spaces. We demonstrate accurate generation of energy expectation values from an RBM trained on data in the freerotor eigenstate basis and explore the learning resources required for various chain lengths and dipolar interaction strengths. Finally, we show evidence for fundamental limitations in the accuracy achievable by RBMs due to the difficulty in imposing symmetries in the sampling procedure. We discuss possible avenues to overcome this limitation in the future, including the further development of autoregressive models such as recurrent neural networks for the purposes of quantum state reconstruction.
© 2020 American Physical Society
20200706T00:00:00Z

An expanded shale δ98Mo record permits recurrent shallow marine oxygenation during the Neoarchean
http://hdl.handle.net/10012/16023
An expanded shale δ98Mo record permits recurrent shallow marine oxygenation during the Neoarchean
Ostrander, Chadlin; Kendall, Brian; Olson, Stephanie; Lyons, Timothy; Gordon, Gwyneth; Romaniello, Stephen; Zheng, Wang; Reinhard, Christopher; Roy, Moutusi; Anbar, Ariel
Multiple attempts have been made using the ancient shale record to track the molybdenum isotope composition (δ98Mo) of seawater during the final twohundred million years of the Archean Eon (2.7 to 2.5 billionyearsago, or Ga). This seawater δ98Mo value is important because it should have scaled with levels of ocean oxygenation during the runup to the Great Oxidation Event (GOE). Unfortunately, however, it is difficult to tell if the majority of the existing lateArchean shale δ98Mo record tracks an ancient seawater value. Here, we further attempt to track preGOE seawater δ98Mo using an expanded and wellcharacterized shale sample set from Western Australia (Jeerinah, Wittenoom, and Mt. Sylvia formations) and South Africa (Nauga and Klein Naute formations). Most importantly, and in contrast to most previous Mo isotope studies of similarly aged shales, local redox conditions for our shales have been independently constrained using the iron (Fe) speciation proxy and bottom water Mo contents at the time of deposition have been qualitatively estimated using Mo/TOC ratios. Local redox conditions and bottom water Mo availability are important parameters because transfer of the seawater δ98Mo to sediments today is shown to be dependent on these conditions. According to our updated sedimentary δ98Mo record, seawater δ98Mo commonly exceeded 1.0‰ between ~2.69 Ga and 2.50 Ga. In order to drive such a heavy seawater δ98Mo, there must have been a marine sink with a strong preference for lightermass Mo isotopes frequently present over this timeframe. The operation of some anaerobic processes in lateArchean marine settings could theoretically explain the heavier seawater δ98Mo. Such processes are known to promote the preferential retention of lightermass Mo isotopes in marine sediments today (e.g., interactions between Mo and organic matter or the formation of thiocomplexes). Alternatively, or in addition, adsorption of lightermass Mo isotopes to Fe and Mn oxide minerals formed in oxygenated marine environments can explain the heavier
seawater δ98Mo. A compilation of previous work suggests that oxygenated shallow marine environments were fairly common during the lateArchean, and thus Mo adsorption to FeMn oxides formed in these settings probably played an important role in driving heavy seawater δ98Mo over this timeframe.
20200120T00:00:00Z