# A new numerical approach to strongly correlated quantum physics in 2D.

Lead Research Organisation:
University College London

Department Name: London Centre for Nanotechnology

### Abstract

Quantum physics in two dimensions is technologically relevant and fundamentally interesting; it is also difficult - available methods are generally limited to small system sizes or unrealistic approximations. The main problem is the number of degrees of freedom one must consider, which grows exponentially with system size. Methods for solving anything but the most trivial problems must select which of these degrees of freedom really matter and discard the rest.

The technique known as the 'density matrix renormalisation group' (DMRG) has revolutionised numerical studies of quantum systems by selecting the essential degrees of freedom in a remarkably efficient manner. DMRG has proven to be a highly accurate and robust tool for calculating properties of materials that are quasi-1D: systems that can be described as a one dimensional lattice or 'chain' of sites. Unfortunately this method stumbles in two spatial dimensions and above - it quickly becomes inefficient as system size grows.

A result from quantum information theory, known as the 'area law' shows the failure of conventional 2D DMRG is linked to the enhanced growth of quantum entanglement above 1D. Too much entanglement between different subregions of the system causes DMRG to grind to a halt. The area law states that entanglement scales with the boundary or 'area' between two subregions. In 1D the boundary can only be one or two points, independent of the total system size. In 2D the area will in fact scale like the perimeter of a subregion. In other words, it will increase linearly as the system gets bigger, until the DMRG approach is too inefficient to be useful.

Effective numerical techniques are vital because analytic methods based on simple approximations fail for the most interesting problems, where quantum fluctuations are dominant, while more sophisticated exact approaches available in 1D do not have analogues in higher dimensions.

By considering the anisotropic 2D case of a coupled array of exactly solvable chains we can minimise the relevant boundary area, thus minimising the entanglement problem. Combining the properties of the exactly solvable subunits with the power of DMRG, leads to an algorithm that can efficiently perform larger scale simulations than competing techniques.

Using an anisotropic representation does not prohibit us from the applying the results to isotropic systems as we are generally interested in 'universal' quantities that are independent of such microscopic details.

I intend to develop this algorithm beyond its proof-of-concept implementation into a general tool for studying the properties of two dimensional quantum systems, including their quantum information content.

In so doing, I will apply it to an important benchmark problem, relevant to the cuprate high temperature superconductors (materials that conduct electricity without resistance). I will also take advantage of the underlying 'matrix product state' structure of the technique to extend it to the emerging field of out-of-equilibrium quantum problems, where a system is 'quenched' by suddenly changing one of its properties (for example the strength of interactions).

The technique known as the 'density matrix renormalisation group' (DMRG) has revolutionised numerical studies of quantum systems by selecting the essential degrees of freedom in a remarkably efficient manner. DMRG has proven to be a highly accurate and robust tool for calculating properties of materials that are quasi-1D: systems that can be described as a one dimensional lattice or 'chain' of sites. Unfortunately this method stumbles in two spatial dimensions and above - it quickly becomes inefficient as system size grows.

A result from quantum information theory, known as the 'area law' shows the failure of conventional 2D DMRG is linked to the enhanced growth of quantum entanglement above 1D. Too much entanglement between different subregions of the system causes DMRG to grind to a halt. The area law states that entanglement scales with the boundary or 'area' between two subregions. In 1D the boundary can only be one or two points, independent of the total system size. In 2D the area will in fact scale like the perimeter of a subregion. In other words, it will increase linearly as the system gets bigger, until the DMRG approach is too inefficient to be useful.

Effective numerical techniques are vital because analytic methods based on simple approximations fail for the most interesting problems, where quantum fluctuations are dominant, while more sophisticated exact approaches available in 1D do not have analogues in higher dimensions.

By considering the anisotropic 2D case of a coupled array of exactly solvable chains we can minimise the relevant boundary area, thus minimising the entanglement problem. Combining the properties of the exactly solvable subunits with the power of DMRG, leads to an algorithm that can efficiently perform larger scale simulations than competing techniques.

Using an anisotropic representation does not prohibit us from the applying the results to isotropic systems as we are generally interested in 'universal' quantities that are independent of such microscopic details.

I intend to develop this algorithm beyond its proof-of-concept implementation into a general tool for studying the properties of two dimensional quantum systems, including their quantum information content.

In so doing, I will apply it to an important benchmark problem, relevant to the cuprate high temperature superconductors (materials that conduct electricity without resistance). I will also take advantage of the underlying 'matrix product state' structure of the technique to extend it to the emerging field of out-of-equilibrium quantum problems, where a system is 'quenched' by suddenly changing one of its properties (for example the strength of interactions).

### Planned Impact

Quantum systems that are effectively two dimensional exist in nature - either as thin films or as important subunits of otherwise 3D materials - making them experimentally and technologically relevant. However theoretically it is very difficult to understand their properties. The primary impact of this proposal is then the production of knowledge, to benefit the physics, quantum information and nanotechnology communities in understanding fundamental physical concepts. In time, our understanding of these concepts will shape the development of future materials and their applications.

The knowledge takes two forms: firstly knowledge of the properties of particular 2D quantum systems, discerned by application of the algorithm. For example, I intend to explore the nature of the mysterious 'pseudogap' phase of cuprate high temperature superconductors, by applying the algorithm to the 2D Hubbard model near half filling. This also includes more general knowledge of the behaviour of quantum entanglement measures in many-body systems.

Secondly, developing the method constitutes knowledge in itself that will allow it to be extended to new situations, such as non-equilibrium 2D systems, and provide clues to increasingly effective numerical treatments of problems in 2D and 3D. This impact also lies in producing a general algorithm, into which one can 'drop' a suitable 2D model. This will allow non-specialists to take advantage of its power in their own investigations.

The algorithmic work is expected to increase the role of the UK in the density matrix renormalisation group and matrix product state communities, within physics.

The results for specific 2D models will be of benefit to scientists (both theoretical and experimental workers) and quantum chemists in the UK and internationally. Strong efforts will be made to foster international collaboration in order to maximise the impact of the work beyond the UK.

By having an early role in this science, the UK has a better chance of participating in the development of related numerical methods and their resulting applications. Because these methods are based on both mathematical and computational techniques of some sophistication, their use by young scientists at the graduate and undergraduate level will provide training in analytical and computational skills.

The participation of Dr. Robert Konik as a visiting researcher (estimated by Dr. Konik as an effective contribution of $12500 per year by Brookhaven National Laboratory) will provide impact via new collaborative connections between UK and US research, particularly with the experimental and theory groups at his host institution. It will also allow for access to powerful computing clusters at BNL including the Bluegene supercomputing facility.

The knowledge takes two forms: firstly knowledge of the properties of particular 2D quantum systems, discerned by application of the algorithm. For example, I intend to explore the nature of the mysterious 'pseudogap' phase of cuprate high temperature superconductors, by applying the algorithm to the 2D Hubbard model near half filling. This also includes more general knowledge of the behaviour of quantum entanglement measures in many-body systems.

Secondly, developing the method constitutes knowledge in itself that will allow it to be extended to new situations, such as non-equilibrium 2D systems, and provide clues to increasingly effective numerical treatments of problems in 2D and 3D. This impact also lies in producing a general algorithm, into which one can 'drop' a suitable 2D model. This will allow non-specialists to take advantage of its power in their own investigations.

The algorithmic work is expected to increase the role of the UK in the density matrix renormalisation group and matrix product state communities, within physics.

The results for specific 2D models will be of benefit to scientists (both theoretical and experimental workers) and quantum chemists in the UK and internationally. Strong efforts will be made to foster international collaboration in order to maximise the impact of the work beyond the UK.

By having an early role in this science, the UK has a better chance of participating in the development of related numerical methods and their resulting applications. Because these methods are based on both mathematical and computational techniques of some sophistication, their use by young scientists at the graduate and undergraduate level will provide training in analytical and computational skills.

The participation of Dr. Robert Konik as a visiting researcher (estimated by Dr. Konik as an effective contribution of $12500 per year by Brookhaven National Laboratory) will provide impact via new collaborative connections between UK and US research, particularly with the experimental and theory groups at his host institution. It will also allow for access to powerful computing clusters at BNL including the Bluegene supercomputing facility.

## People |
## ORCID iD |

Andrew John James (Principal Investigator / Fellow) |

### Publications

Dean M
(2014)

*Itinerant effects and enhanced magnetic interactions in Bi-based multilayer cuprates*in Physical Review B
James A
(2015)

*Quantum quenches in two spatial dimensions using chain array matrix product states*in Physical Review B
James AJA
(2019)

*Nonthermal States Arising from Confinement in One and Two Dimensions.*in Physical review letters
James AJA
(2018)

*Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-Abelian bosonization to truncated spectrum methods.*in Reports on progress in physics. Physical Society (Great Britain)
Robinson N
(2019)

*Signatures of rare states and thermalization in a theory with confinement*in Physical Review B
Shi Y
(2017)

*Auxiliary fermion approach to the resonant inelastic x-ray scattering response in an underdoped cuprate*in Physical Review BDescription | Quantum systems that are effectively two dimensional (layered) are manifestly harder to study than those that are one dimensional (chain like), because methods that work well in 1D do not extend in an obvious way to higher dimensions. We have shown that by combining analytical and numerical techniques from 1D in a non-standard way, one can usefully simulate 2D quantum systems, including properties that change with time. In doing so we have produced an open source software tool for theorists that is flexible enough that it can be used to study a large number of complicated 'many-body' quantum systems in 2D. These systems are relevant to real world physical phenomena such as high temperature superconductivity, and quantum magnetism. The software realises 2D quantum systems as arrays of 1D quantum chains. Arrays of infinitely many chains can be formed, and a variety of static and dynamic (changing with time) properties can be calculated. Users can study predefined models, or define their own. The software is being used by my collaborators in several different projects. Most recently ChainAMPS has been used to investigate how quantum systems reach thermal equilibrium, in tandem with analytical methods. This has revealed that in systems with confined excitations there can be 'rare' states that do not thermalise. An understanding of the origins and properties of such athermal states could be important to realising robust systems for quantum computation. This work has led to two preprints which are currently with journal referees. |

Exploitation Route | The software has been released as an open source research tool for the academic community, adding to the small set of existing numerical techniques for two dimensional quantum models. Collaborators are using it to study problems in non-equilibrium quantum matter (currently a rapidly expanding field) as well as quantum magnetism. One interesting collaboration hopes to use the software to interpret the results of planned X-ray 'pump-probe' experiments on quantum magnets. This should aid our understanding of what relaxation processes occur in strongly correlated quantum matter when it is pushed far from its equilibrium state. |

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

URL | https://bitbucket.org/chainamps/chainamps-public |

Description | 2D quantum magnets via coupled Heisenberg chains |

Organisation | University of Amsterdam |

Department | Institute of Physics (IoP) |

Country | Netherlands |

Sector | Academic/University |

PI Contribution | I have produced code that allows for simulation of two dimensional quantum materials, both in and out of equilibrium, by coupling together one dimensional chains using matrix product state techniques. This code has been written specifically to take advantage of results such as those generated by the group in Amsterdam. |

Collaborator Contribution | J.S. Caux and his group at UvA have provided knowledge and expertise on exact solutions of 1D Heisenberg chains and code for generating these solutions. These results are a vital ingredient in simulating 2D quantum magnets by coupling together many (in some cases infinitely many) such chains. |

Impact | Currently we are trying to understand the role of certain excitations of the 1D chains and how they affect the 2D material. We have produced working code that can simulate the isotropic spin 1/2 Heisenberg quantum magnet in 2D (including on an infinitely long cylinder) and probe its properties in equilibrium or after a sudden 'quantum quench' change to the system's parameters. This code has been incorporated into the larger code base (of the project described funded by my grant) with other chain systems, to allow for simulation and comparison of multiple different paradigmatic quantum systems in 2D. |

Start Year | 2015 |

Description | Predicting RIXS spectra for high-TC cuprates. |

Organisation | Brookhaven National Laboratory |

Country | United States |

Sector | Public |

PI Contribution | Calculating predictions for the magnetic correlations in high-TC cuprates, using a renormalised mean field approach. |

Collaborator Contribution | Experimental measurement of the RIXS spectra of cuprate high temperature superconductors. |

Impact | Over course of collaboration has led to two published papers and a conference proceedings, one jointly with experimental group. Another joint paper has just been accepted: https://journals.aps.org/prb/accepted/03072YcaI4512e4092674a7915c78e326ef663a8f |

Start Year | 2010 |

Description | Quantum Quenches for coupled chain systems |

Organisation | Brookhaven National Laboratory |

Country | United States |

Sector | Public |

PI Contribution | Developing code for time evolution of coupled chain systems using matrix product states. Computation time using this code. Analytical work |

Collaborator Contribution | Analytical calculations to test against predictions of the code. Knowledge and expertise on one dimensional Luttinger liquids and other integrable systems. Use of cluster computer resources at BNL. |

Impact | Initial work was reported at the workshop 'Emergent Phenomena in the Dynamics of Quantum Matter: Disorder, Quenches, Simulations and Experiments'; in New York in April 2014. This has led to discussion of a further collaboration with the theoretical condensed matter group at the University of Amsterdam. An initial paper was published in Physical Review B in 2015. Working code now exists to study quenches in various types of coupled chain systems and this is being used by a graduate student (Andrew Hallam) at UCL as part of his doctoral research. |

Start Year | 2013 |

Title | ChainAMPS |

Description | Open source software for working with 'chain array matrix product states'. These realise 2D quantum systems as arrays of coupled quantum chains. Includes drivers for finding the spectrum of the 2D quantum system, for time evolution and for post-processing data files to extract measurements such as local moments and correlation functions. Several predefined models are included, but it is flexible enough that user defined models can be studied. This software is more fully featured, and mature than the previous close sourced tools developed as part of this grant. |

Type Of Technology | Physical Model/Kit |

Year Produced | 2016 |

Impact | The software is in use with a graduate student at UCL, and with three collaborators at Brookhaven National Laboratory (US). Interest in the abilities and results has been shown by experimentalists in the Schmiedmayer group at TU Wien (Vienna) and at Brookhaven. It will allow for simulations of experiments that currently cannot be performed by other means. |

URL | https://bitbucket.org/chainamps/chainamps-public |