# Random Geometry in Condensed Matter Theory

Lead Research Organisation:
University of Oxford

Department Name: Oxford Physics

### Abstract

A central task for theoretical physics is explaining how the strange choreography of the microscopic world - the quantum-mechanical dance of electrons - produces the multifarious characteristics of macroscopic stuff: metallicity, magnetism, superconductivity, and so on. Equally crucially, theory should reveal new macroscopic phenomena that have not yet been seen because we did not know to look for them. These are hard tasks, since an atom and a magnet (say) are separated by a staggering jump in scale and complexity.

Fortunately, our understanding of quantum mechanics, as applied to assemblages of many interacting particles, is currently exploding. New types of quantum materials are emerging into the light which - unlike a simple magnet - have no analogue in classical physics. (They rely on the "spooky action at a distance" unique to quantum mechanics and ideas from topology - the mathematics of knots, etc.) Separate work has shown that the central assumptions of statistical mechanics fail radically in some strongly disordered (dirty) quantum systems. Simultaneously, we are discovering that phase transitions between different quantum states are far subtler than we thought.

The unifying theme for this proposal is a very general picture of physical systems in terms of fluctuating extended objects - for example vortex lines, or flux lines, or 'worldlines' in space- time. Such geometric descriptions are often more useful than descriptions in terms of electrons. For example, certain exotic states (quantum 'spin liquids' and related 'topological paramagnets') are best viewed as as Schrodinger's-cat-like mixtures of different configurations of loops, representing flux lines in a field which emerges miraculously from the dance of the electrons.

Using pictures like this, I will tackle such questions as: how do we describe the new types of quantum phase transition theoretically? What do they teach us about quantum field theory? How do we realise the theoretically predicted topological states? How robust are they to perturbations and disorder? These are crucial questions for theoretical physics, which we must answer in order to explain the diversity of material behaviours that emerge from the (deceptively simple) laws of quantum mechanics.

Fortunately, our understanding of quantum mechanics, as applied to assemblages of many interacting particles, is currently exploding. New types of quantum materials are emerging into the light which - unlike a simple magnet - have no analogue in classical physics. (They rely on the "spooky action at a distance" unique to quantum mechanics and ideas from topology - the mathematics of knots, etc.) Separate work has shown that the central assumptions of statistical mechanics fail radically in some strongly disordered (dirty) quantum systems. Simultaneously, we are discovering that phase transitions between different quantum states are far subtler than we thought.

The unifying theme for this proposal is a very general picture of physical systems in terms of fluctuating extended objects - for example vortex lines, or flux lines, or 'worldlines' in space- time. Such geometric descriptions are often more useful than descriptions in terms of electrons. For example, certain exotic states (quantum 'spin liquids' and related 'topological paramagnets') are best viewed as as Schrodinger's-cat-like mixtures of different configurations of loops, representing flux lines in a field which emerges miraculously from the dance of the electrons.

Using pictures like this, I will tackle such questions as: how do we describe the new types of quantum phase transition theoretically? What do they teach us about quantum field theory? How do we realise the theoretically predicted topological states? How robust are they to perturbations and disorder? These are crucial questions for theoretical physics, which we must answer in order to explain the diversity of material behaviours that emerge from the (deceptively simple) laws of quantum mechanics.

### Planned Impact

(1) Impact on quantum condensed matter physics. Ninety years after Schrodinger wrote down his famous equation, we are still only beginning to understand the ways in which large numbers of quantum-mechanical particles can organise themselves. We have repeatedly discovered, either through experiments (for example in cuprate superconductors) or through theory (as for topological insulators and their recent generalisations) that our language for describing 'types of stuff' is incomplete. The context for this project is our long struggle to extend this language to describe those materials in which interactions have the most dramatic effects. This theoretical understanding will underpin the revolutionary technological discoveries of the future.

Firstly, this project will help push our understanding of quantum criticality beyond the comfort zone of traditional theory. The new tools developed will impact the large community grappling with quantum criticality in insulators, metals and superconductors. Secondly, the project will provide new heuristics for thinking about topological phases of matter. This will impact researchers working both on how such states behave theoretically and on how to realise them in the laboratory.

The research programme, which pursues topical subjects from a fresh perspective, will also contribute to the vigour of the condensed matter community in the UK. The impact on the local community will include transfer of knowledge to students and other young researchers through personal interactions, seminars and collaborations.

(2) Applications. Ideas from the project will eventually have a downstream impact on applications. Theoretical understanding of strongly correlated electrons will be essential for turning the unconventional physics of critical or topological systems into predictable, controllable technologies. Both critical systems and topological states hold the promise of important applications. The key property of critical systems is their enhanced sensitivity to influence (e.g. pressure, temperature, applied fields, doping, etc.), which could one day be used to create easily 'switchable' devices. A fuller understanding of quantum criticality may also be key to reaching higher temperature superconductivity. Many applications have already been suggested for the best-understood topological states, ranging from spintronic devices to - at the most amibitious end of the scale - topologically protected quantum computing, in which quantum entanglement and topology are used to safeguard quantum information from environmental noise.

(3) Impact on other academic fields. The three parts of this project all make use of a conceptual picture in terms of extended, linelike degrees of freedom, coupled with field theory for performing quantitative calculations. The picture of extended degrees of freedom leads to close connections with other fields, as described in 'Academic Beneficiaries'. The project will help to invigorate these fields (all of which are strongly represented in the UK), and to generate dialogue between these fields and hard condensed matter.

(4) Social impact. Scientific discoveries contribute to our shared culture. The project includes plans for broader dissemination via articles for a general audience, which will communicate the excitement of recent topological ideas in physics through simple examples. Generating enthusiasm for science will help encourage young people to study it, and will lead to greater engagement with science policy.

Firstly, this project will help push our understanding of quantum criticality beyond the comfort zone of traditional theory. The new tools developed will impact the large community grappling with quantum criticality in insulators, metals and superconductors. Secondly, the project will provide new heuristics for thinking about topological phases of matter. This will impact researchers working both on how such states behave theoretically and on how to realise them in the laboratory.

The research programme, which pursues topical subjects from a fresh perspective, will also contribute to the vigour of the condensed matter community in the UK. The impact on the local community will include transfer of knowledge to students and other young researchers through personal interactions, seminars and collaborations.

(2) Applications. Ideas from the project will eventually have a downstream impact on applications. Theoretical understanding of strongly correlated electrons will be essential for turning the unconventional physics of critical or topological systems into predictable, controllable technologies. Both critical systems and topological states hold the promise of important applications. The key property of critical systems is their enhanced sensitivity to influence (e.g. pressure, temperature, applied fields, doping, etc.), which could one day be used to create easily 'switchable' devices. A fuller understanding of quantum criticality may also be key to reaching higher temperature superconductivity. Many applications have already been suggested for the best-understood topological states, ranging from spintronic devices to - at the most amibitious end of the scale - topologically protected quantum computing, in which quantum entanglement and topology are used to safeguard quantum information from environmental noise.

(3) Impact on other academic fields. The three parts of this project all make use of a conceptual picture in terms of extended, linelike degrees of freedom, coupled with field theory for performing quantitative calculations. The picture of extended degrees of freedom leads to close connections with other fields, as described in 'Academic Beneficiaries'. The project will help to invigorate these fields (all of which are strongly represented in the UK), and to generate dialogue between these fields and hard condensed matter.

(4) Social impact. Scientific discoveries contribute to our shared culture. The project includes plans for broader dissemination via articles for a general audience, which will communicate the excitement of recent topological ideas in physics through simple examples. Generating enthusiasm for science will help encourage young people to study it, and will lead to greater engagement with science policy.

## People |
## ORCID iD |

Adam William Nahum (Principal Investigator / Fellow) |

### Publications

Dai Z
(2020)

*Quantum criticality of loops with topologically constrained dynamics*in Physical Review Research
Dai Zhehao
(2019)

*Quantum criticality of loops with topologically constrained dynamics*in arXiv e-prints
Jonay Cheryne
(2018)

*Coarse-grained dynamics of operator and state entanglement*in arXiv e-prints
Khemani V
(2018)

*Velocity-dependent Lyapunov exponents in many-body quantum, semiclassical, and classical chaos*in Physical Review B
Kimchi I
(2018)

*Valence Bonds in Random Quantum Magnets: Theory and Application to YbMgGaO 4*in Physical Review X
Nahum A
(2017)

*Quantum Entanglement Growth under Random Unitary Dynamics*in Physical Review X
Nahum A
(2020)

*Entanglement and dynamics of diffusion-annihilation processes with Majorana defects*in Physical Review Research
Nahum A
(2018)

*Operator Spreading in Random Unitary Circuits*in Physical Review X
Nahum A
(2018)

*Dynamics of entanglement and transport in one-dimensional systems with quenched randomness*in Physical Review B
Nahum Adam
(2020)

*Entanglement and dynamics of diffusion-annihilation processes with Majorana defects*in Physical Review ResearchDescription | (1) Many-body quantum chaos: We have developed a set of new models and concepts for the dynamics of chaotic many-body quantum systems, revealing universal structures (many of them related to the 'random geometry' of strings and membranes in spacetime) underlying the spreading of quantum information through many body systems. This includes detailed theories of how quantum entanglement is produced by many-body systems under their own local dynamics, and how the effect of a local disturbance spreads through the system (the quantum 'butterfly effect'). (2) We discovered a new kind of phase transition in quantum dynamics which is caused by repeated measurement, and is manifested in the entanglement structure of an evolving state, opening a new direction in quantum critical phenomena. (3) Deconfined quantum criticality: We revealed new dualities between a set of field theories that are important for critical phenomena in two-dimensional quantum magnets, explaining a striking emergent symmetry that we observed previously, and leading to exciting conjectures for a range of quantum and classical phase transitions. (4) We introduced a new class of interesting weakly first-order transitions with approximate emergent symmetry. This is pertinent to many key models in field theory and statistical mechanics. (5) We demonstrated a surprising symmetry in a paradigmatic statistical mechanics model, the classical dimer model. (6) Random quantum magnets: We analysed how quenched disorder (impurities) competes with valence bond formation in two-dimensional quantum magnets. This led to surprising theoretical conclusions, including new general constraints on ground states of disordered magnets, and a proposal for the phenomenology of a topical material, YbMgGaO4. (7) We have uncovered new features of quantum criticality in loop gases. |

Exploitation Route | The universal structures which we found to underlie quantum information spreading are a general lens for viewing chaotic many-body dynamics and have inspired follow-on work both using the same tools ("random circuits") and using complementary tools. Our work on field-theory dualities has renewed interest in these field theories and phase transitions and new ways of looking at them. Our work on disordered magnets led to conjectures which we expect to be a fruitful line of investigation in mathematical physics, and elucidated theoretical mechanisms which will be useful in interpreting a wide range of experimental systems. Our work on the emergent statistical mechanics of entanglement and on the effect of measurement on entanglement opens up a range of exciting questions in quantum statistical mechanics. Our work on weak first-order transitions is relevant to the phenomenology of a broad class of systems currently being studied numerically. |

Sectors | Other |

URL | https://arxiv.org/find/cond-mat/1/au:+Nahum_A/0/1/0/all/0/1 |