# Emerging correlations from strong driving: a tensor network projection variational Monte Carlo approach to 2D quantum lattice systems

Lead Research Organisation:
University of Bristol

Department Name: Physics

### Abstract

Much of the technology we have is based on exploiting special materials like semiconductors. The next revolution is likely to emerge from so-called quantum materials. However, while their behaviour has the potential to be extremely useful, it is also complex to understand and control. Insights gained from this research will help determine the viability of controlling quantum materials with light and the possible exploitation of dynamical non-equilibrium properties in future nano-devices.

Controlling materials with light is interesting because it is well known that driven systems can exhibit behaviour not seen when stationary. There are two simple examples of this. The first is a so-called Kapitza pendulum. This is a normal pendulum whose pivot point undergoes vertical oscillations that are rapid but small in amplitude. What is striking about this pendulum is that the inverted position, normally unstable to gravity, is dynamically stabilised by the periodic driving. The second is a ball on a rotating saddle. The ball cannot be stably positioned at the inflection point when the saddle is stationary. However, if the saddle is rotated above some threshold angular velocity then the ball can be balanced in the time-averaged bowl swept out by the saddle. The same ideas apply to many-body systems like materials and it is becoming increasingly relevant to study their behaviour.

An important class of many-body systems are those that exhibit strong correlations due to interactions between their constituents. The everyday world is full of such systems. For example traffic jams form along roads due to a combination of many vehicles and a strong repulsion between them to avoid occupying the same piece of road. However, ants marching in a line never suffer from such traffic jams despite facing very similar restrictions because they don't overtake one another. These two examples demonstrate how subtle differences in the precise microscopic nature of interactions may lead to qualitatively different macroscopic properties. Describing such correlations poses major challenges for the theoretical study of interacting systems, and no more so than for the case of quantum systems. In the quantum case strong interactions lead to some of the least well-understood phenomena of condensed matter, like high-Tc superconductivity, frustration, and topological phases such as fractional quantum Hall physics. These effects only appear at low temperatures and typically in materials with a dominant two-dimensional character.

Since quantum materials exhibit functional properties there is a major research effort to stabilise and optimise them at higher temperatures for future technological applications. A recent approach to this is to periodically drive a many-body quantum system to "dynamically stabilise" macroscopic quantum effects beyond where they occur in equilibrium. The question is made even more compelling by spectacular advances in high-field THz generation technology. This allows selective driving of low-energy excitations of real solids, like vibrations, enabling a crystal lattice to be shaken, modulated or distorted in controlled ways. This has created an exciting interface between driven systems and many-body physics engaging a large body of researchers worldwide.

A crucial issue hampering the use of periodic driving in engineering materials is heating that might wash out the desired effects. This project examines this problem within the context of one of the most important model Hamiltonians, the Hubbard model, which captures the essential physics of strong correlations. Current numerical methods struggle to give a conclusive answer to this issue. A unique feature of this project will be the development of a combined Monte Carlo and tensor network approach potentially rich enough to accurately describe the dynamical behaviour of the driven Hubbard model. The resulting high performance software will be publically available.

Controlling materials with light is interesting because it is well known that driven systems can exhibit behaviour not seen when stationary. There are two simple examples of this. The first is a so-called Kapitza pendulum. This is a normal pendulum whose pivot point undergoes vertical oscillations that are rapid but small in amplitude. What is striking about this pendulum is that the inverted position, normally unstable to gravity, is dynamically stabilised by the periodic driving. The second is a ball on a rotating saddle. The ball cannot be stably positioned at the inflection point when the saddle is stationary. However, if the saddle is rotated above some threshold angular velocity then the ball can be balanced in the time-averaged bowl swept out by the saddle. The same ideas apply to many-body systems like materials and it is becoming increasingly relevant to study their behaviour.

An important class of many-body systems are those that exhibit strong correlations due to interactions between their constituents. The everyday world is full of such systems. For example traffic jams form along roads due to a combination of many vehicles and a strong repulsion between them to avoid occupying the same piece of road. However, ants marching in a line never suffer from such traffic jams despite facing very similar restrictions because they don't overtake one another. These two examples demonstrate how subtle differences in the precise microscopic nature of interactions may lead to qualitatively different macroscopic properties. Describing such correlations poses major challenges for the theoretical study of interacting systems, and no more so than for the case of quantum systems. In the quantum case strong interactions lead to some of the least well-understood phenomena of condensed matter, like high-Tc superconductivity, frustration, and topological phases such as fractional quantum Hall physics. These effects only appear at low temperatures and typically in materials with a dominant two-dimensional character.

Since quantum materials exhibit functional properties there is a major research effort to stabilise and optimise them at higher temperatures for future technological applications. A recent approach to this is to periodically drive a many-body quantum system to "dynamically stabilise" macroscopic quantum effects beyond where they occur in equilibrium. The question is made even more compelling by spectacular advances in high-field THz generation technology. This allows selective driving of low-energy excitations of real solids, like vibrations, enabling a crystal lattice to be shaken, modulated or distorted in controlled ways. This has created an exciting interface between driven systems and many-body physics engaging a large body of researchers worldwide.

A crucial issue hampering the use of periodic driving in engineering materials is heating that might wash out the desired effects. This project examines this problem within the context of one of the most important model Hamiltonians, the Hubbard model, which captures the essential physics of strong correlations. Current numerical methods struggle to give a conclusive answer to this issue. A unique feature of this project will be the development of a combined Monte Carlo and tensor network approach potentially rich enough to accurately describe the dynamical behaviour of the driven Hubbard model. The resulting high performance software will be publically available.

### Planned Impact

Next-generation technologies exploiting so-called quantum materials for ultra-fast switches, memory and processing devices may involve integration with THz opto-electronics. This opens up new vistas of opportunities to probe, drive and control the functional behaviour of interacting quantum many-body systems with tailored laser pulses. It is against this overarching backdrop that this project has two major themes for potential long-term impact. The first is through the scientific output I will produce that is aimed at answering fundamental questions about the behaviour of an archetypal model of a quantum material when it is strongly driven. The second is through the novel methodologies I will develop in this project to answer those questions.

To fully realise this promise more advanced methods of simulating the dynamical behaviour of these systems is needed to guide development and test feasibility. As such the scientific findings of this work not only answer some important questions in this direction but also lay the foundations of a powerful numerical technique that can provide these much sought-after predictive capabilities to these problems. Long lasting impact in this growing field will be fostered by giving access to these methods via an easy to use online interface, www.tntgo.org, and by making all codes freely available within the tensor network theory library (TNT) project, http://ccpforge.cse.rl.ac.uk/gf/project/tntlibrary/. This enables both theorists and experimentalists to utilise the deliverables of this project to help realise these long-term technological advances. The workshop bolt on planned will also advertise and introduce the outputs of this project to influential individuals and key beneficiaries in the community.

Beyond this direct application I further anticipate that the inclusion of advanced stochastic methods into the TNT library will have far-reaching impact for its use in the field of Big Data analysis. Owing to the generality of tensors they can serve as the foundation for the development of new algorithms widely believed to provide the much sought after extension of standard linear algebra algorithms to multidimensional data analysis. Given the huge importance of complex classical systems in natural sciences, operations research and industry as yet unforeseen applications of hybrid TNT methods could emerge in the near future. The exploitation of such applications will be of immediate relevance to many scientists working in the UK and further afield. To aid this, industrial stakeholders like MathWorks, Wolfram and NAG will be invited to the workshop bolt on making them aware of developments and fostering on going contacts during and after this project. Finally the UK academic community and beyond will benefit from the build-up of a cohort of scientists engaged in the project with skills in developing high-quality HPC software. Further work on algorithms and numerical methods underpinning this project will carry benefits for other fields of Science, Engineering and Medicine wishing to engage HPC effectively.

To fully realise this promise more advanced methods of simulating the dynamical behaviour of these systems is needed to guide development and test feasibility. As such the scientific findings of this work not only answer some important questions in this direction but also lay the foundations of a powerful numerical technique that can provide these much sought-after predictive capabilities to these problems. Long lasting impact in this growing field will be fostered by giving access to these methods via an easy to use online interface, www.tntgo.org, and by making all codes freely available within the tensor network theory library (TNT) project, http://ccpforge.cse.rl.ac.uk/gf/project/tntlibrary/. This enables both theorists and experimentalists to utilise the deliverables of this project to help realise these long-term technological advances. The workshop bolt on planned will also advertise and introduce the outputs of this project to influential individuals and key beneficiaries in the community.

Beyond this direct application I further anticipate that the inclusion of advanced stochastic methods into the TNT library will have far-reaching impact for its use in the field of Big Data analysis. Owing to the generality of tensors they can serve as the foundation for the development of new algorithms widely believed to provide the much sought after extension of standard linear algebra algorithms to multidimensional data analysis. Given the huge importance of complex classical systems in natural sciences, operations research and industry as yet unforeseen applications of hybrid TNT methods could emerge in the near future. The exploitation of such applications will be of immediate relevance to many scientists working in the UK and further afield. To aid this, industrial stakeholders like MathWorks, Wolfram and NAG will be invited to the workshop bolt on making them aware of developments and fostering on going contacts during and after this project. Finally the UK academic community and beyond will benefit from the build-up of a cohort of scientists engaged in the project with skills in developing high-quality HPC software. Further work on algorithms and numerical methods underpinning this project will carry benefits for other fields of Science, Engineering and Medicine wishing to engage HPC effectively.

## People |
## ORCID iD |

Stephen Richard Clark (Principal Investigator) |

### Publications

Balachandran V
(2019)

*Energy Current Rectification and Mobility Edges.*in Physical review letters
Brenes M
(2020)

*Tensor-network method to simulate strongly interacting quantum thermal machines*in Under review at Physical Review X
Clark S
(2018)

*Unifying neural-network quantum states and correlator product states via tensor networks*in Journal of Physics A: Mathematical and Theoretical
Cook M
(2020)

*Controllable Finite-Momenta Dynamical Quasicondensation in the Periodically Driven One-Dimensional Fermi-Hubbard Model*in To appear in Physical Review A
Coulthard J
(2018)

*Ground-state phase diagram of the one-dimensional t - J model with pair hopping terms*in Physical Review B
Guarnieri G
(2019)

*Thermodynamics of precision in quantum nonequilibrium steady states*in Physical Review Research
Hedayat H
(2019)

*Excitonic and lattice contributions to the charge density wave in 1 T - TiS e 2 revealed by a phonon bottleneck*in Physical Review Research
Mendoza-Arenas J
(2019)

*Asymmetry in energy versus spin transport in certain interacting disordered systems*in Physical Review B
Secular P
(2020)

*Parallel time-dependent variational principle algorithm for matrix product states*in Under review in Physical Review B
Tangpanitanon J
(2019)

*Hidden order in quantum many-body dynamics of driven-dissipative nonlinear photonic lattices*in Physical Review A### Related Projects

Project Reference | Relationship | Related To | Start | End | Award Value |
---|---|---|---|---|---|

EP/P025110/1 | 01/08/2017 | 23/09/2018 | £101,215 | ||

EP/P025110/2 | Transfer | EP/P025110/1 | 24/09/2018 | 23/10/2019 | £45,369 |

Description | This project aim was to develop and use variational descriptions of many-body quantum systems relevant to both cold-atom systems and quantum materials, in particular when driven out of equilibrium. The project is still active, but there have been several major accomplishments towards the objectives: (1) In 2017 a novel type of variational ansatz, called Neural-network Quantum States (NQS), was proposed by researchers Carleo and Troyer from ETH Zurich. This formalism exploits connections to the growing field of machine learning. Specifically the structure of NQS mimics those used in neural networks tasked with "learning" complex data. The essential idea was to use the same approach to "learn" the structure of quantum states occurring in many-body systems. For simple model systems this has turned out to be successful, but understanding why and how it may be generalised to more complex problems is an open question. The first major accomplishment of this project was to unify NQS with the tensor network formalism showing that they are equivalent and shedding light on the power of the NQS formalism. This has resulted in a publication: Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks Stephen R. Clark J. Phys. A: Math. Theor. 51 135301 (2018), which has already been cited 30 times and is guiding the field on how best to exploit NQS in future variational Monte Carlo calculations. Follow up work has also developed the parallelised code for the time-dependent variation principle using tensor networks: Parallel time-dependent variational principle algorithm for matrix product states Paul Secular, Nikita Gourianov, Michael Lubasch, Sergey Dolgov, Stephen R. Clark, Dieter Jaksch arXiv:1912.06127, currently under review in Physical Review B (2) A very recent piece of work that is part of this project has unravelled some fundamental features of non-equilibrium steady states (NESS) of quantum systems. Specifically, we envisage a small quantum system coupled to two external large systems (e.g. leads) which may have different temperatures and/or chemical potentials. Left for a long time the combined leads + system reach a NESS with energy and charge current flowing across the quantum system. This DC driven quantum system setup is crucial to autonomous engines operating at the nano-scale. However, they can be prone to deleterious fluctuations in the currents which increase, for fixed power output, the more reversible the operation regime is. This fundamental trade-off between current fluctuations and entropy production forms the basis of the recently formulated thermodynamic uncertainty relations (TURs). However, these relations have so far only been derived for classical Markovian systems and can be violated in the quantum regime. A major accomplishment of this project was to show that the geometry of quantum non-equilibrium steady-states alone, already directly implies the existence of a TUR, but with a looser bound. This result helps to shed light on the delicate relationship between quantum effects and current fluctuations in autonomous machines. This work has been published: Thermodynamics of precision in quantum non-equilibrium steady states Giacomo Guarnieri, Gabriel T. Landi, Stephen R. Clark, John Goold Phys. Rev. Research 1, 033021 (2019) Further followup working building a Tensor Network package for numerical calculations for quantum thermal machines has the following preprint: Tensor-network method to simulate strongly interacting quantum thermal machines Marlon Brenes, Juan Jose Mendoza-Arenas, Archak Purkayastha, Mark T. Mitchison, Stephen R. Clark, John Goold arXiv:1912.02053, currently under review inn Physical Review X. (3) In this project we have also studied nonlinear photonic lattices driven by two-photon parametric processes. We have shown that this type of driving can generate long-range hidden order from the vacuum even in the presence of photon losses and dissipation. This order is shown to resembles that found in topological systems like a Haldane insulator. This work highlights how driving and dissipation can work in unison to create novel types of quantum ordering. This work was published: Hidden Order in Quantum Many-body Dynamics of Driven-Dissipative Nonlinear Photonic Lattices Jirawat Tangpanitanon, Stephen R. Clark, V. M. Bastidas, Rosario Fazio, Dieter Jaksch, Dimitris G. Angelakis Phys. Rev. A 99, 043808 (2019) (4) In this project we have studied energy transport in a boundary driven quantum wire. We showed that for strongly interacting quantum wires the energy transport is diffusive in the presence of disorder - a result which contrasts to particle currents which are subdiffusive in the same regime. This is a surprising result and has implications for the use of quantum wires for thermoelectric devices. This culminated in two publications: Asymmetry in energy versus spin transport in certain interacting, disordered systems Juan Jose Mendoza-Arenas, Marko Znidaric, Vipin Kerala Varma, John Goold, Stephen R. Clark, Antonello Scardicchio Phys. Rev. B 99, 094435 (2019) Heat current rectification and mobility edges Vinitha Balachandran, Stephen R. Clark, John Goold, Dario Poletti Phys. Rev. Lett. 123, 020603 (2019) (5) Driven quantum systems often exhibit Hamiltonians that are different from the undriven system. This is called Floquet engineering. In this project we have performed some crucial work looking at the kind of states that are stabilised in novel Hamiltonians only reachable via driving. This has culminated in the following papers where we show that superconducting order can appear for sufficiently strongly driven systems: Ground state phase diagram of the one-dimensional t-J model with pair hopping terms J. R. Coulthard, S. R. Clark, D. Jaksch Phys. Rev. B 98, 035116 (2018) Controllable Finite-Momenta Dynamical Quasicondensation in the Periodically Driven One-Dimensional Fermi-Hubbard Model Matthew W Cook, Stephen R Clark arXiv:1906.05412, to appear shortly in Physical Review A (6) A final outcome of this project has been a collaboration with experimentalists who have used ultra-fast probing techniques on a specific class of 2D materials (layered transition metal dichalcogenide 1T-TiSe2) that display intriguing charge-density wave order that is not fully understood. I contributed to the theory analysis underpinning this work and using techniques from this project on driven systems to unravel a bottleneck in the relaxation dynamics associated to electronically excited phonon modes. This work resulted in the following publication: Excitonic and lattice contributions to the charge density wave in 1T-TiSe2 revealed by a phonon bottleneck Hamoon Hedayat, Charles J. Sayers, Davide Bugini, Claudia Dallera, Daniel Wolverson, Tim Batten, Sara Karbassi, Sven Friedemann, Giulio Cerullo, Jasper van Wezel, Stephen R. Clark, Ettore Carpene, Enrico Da Como Phys. Rev. Research 1, 023029 (2019) |

Exploitation Route | The results of this project will be of interest to the academic community working on driven quantum many-body systems, including experimentalists in cold-atoms and quantum materials. In the longer term technological applications exploiting the results might be expected in the computing and materials sectors. |

Sectors | Digital/Communication/Information Technologies (including Software),Education |

URL | https://github.com/StephenRClark/NQS-VMC/tree/master |

Description | Dynamics of Complex Quantum Systems - CCPQ Windsor 2019 |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | On Aug 5th 2019, I was again lead organiser of the CCPQ's Dynamics of Complex Quantum Systems held at Cumberland Lodge in Windsor. Funding from my First Grant helped to support this 3 day workshop in which over 60 participants were involved. The bulk of the 20 invited speakers where from across the UK, however we also had a 4 speakers from Europe. The purpose of the workshop was to explore the timely topic of dynamics of complex quantum systems and in particular to discuss numerical methodologies for understanding this. It was an excellent opportunity to advertise the activities of this First Grant, and engage other academics in its objectives. |

Year(s) Of Engagement Activity | 2019 |

URL | http://www.ccpqwindsor.org |

Description | Invited talk at Emergence and Non-Equilibrium Physics: Algorithmic Perspectives on Complex Matter EMNEQ workshop |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | I delivered an invited talk "Controllable dynamical quasicondensation in the periodically driven Hubbard model" at the Emergence and Non-Equilibrium Physics: Algorithmic Perspectives on Complex Matter EMNEQ workshop hosted at the Department of Physics, University of Kent, 30 April - 1 May 2019. |

Year(s) Of Engagement Activity | 2019 |

URL | https://research.kent.ac.uk/theoryandsimulation/events/emneq-algorithms/ |

Description | Invited talk at Machine learning and quantum physics workshop |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | I delivered an invited talk "Neural-network quantum states from a tensor networks perspective" at the Machine learning and quantum physics workshop hosted at the Simon's foundation Flatiron Institute, Center for Computational Quantum Physics, New York, 26 - 27 April 2018. |

Year(s) Of Engagement Activity | 2018 |

Description | Invited talk at Quantum Information Processing in Non-Markovian Quantum Complex Systems workshop |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | I delivered an invited talk "Modelling autonomous quantum thermal machines using mesoscopic reservoirs" at the international workshop on Quantum Information Processing in Non-Markovian Quantum Complex Systems hosted at the University of Nagoya, Japan, 9 - 11 December 2019. |

Year(s) Of Engagement Activity | 2019 |

URL | https://sites.google.com/view/qipcq19/home |

Description | Invited talk at Quantum Materials Control symposium |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | I delivered an invited talk "Dynamical quasicondensation in the periodically driven Hubbard model" at the Quantum Materials Control symposium hosted at the Department of Physics, University of Oxford, 5 July 2019. |

Year(s) Of Engagement Activity | 2019 |

Description | Invited talk at Quantum Materials workshop |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | National |

Primary Audience | Professional Practitioners |

Results and Impact | I delivered an invited talk "Controllable dynamical quasicondensation in the periodically driven Hubbard model" at the Quantum Materials workshop hosted at the University of Bath, 23 July 2018. |

Year(s) Of Engagement Activity | 2018 |