# Simulating cold interacting atoms on disordered optical lattices

Lead Research Organisation:
University of Warwick

Department Name: Physics

### Abstract

Developing a theory of disorder in interacting systems is one of the most important challenges in condensed matter physics. Static disorder affects the conductivity of materials and can drive a metal-insulator transition and, while a lot is known about disorder in non-interacting systems, far less is known about its effects in systems of interacting particles. Our proposal is to develop a numerical tool to model accurately the systems of interacting cold atoms being probed experimentally. Recent developments in theoretical modeling, together with the extraordinary level of experimental control offered by cold atom systems, mean that for the first time we are in a position to compute properties of the actual disordered systems which are being generated in the laboratory.

We will use density matrix renormalization group (DMRG) calculations (and its more modern forms such as tensor network states) to compute the ground state properties of a large system of interacting particles trapped in a lattice. Although the DMRG approach was originally developed to build up a system piece by piece by embedding it in an effective environment which assumed no disorder, the algorithm now repeats this growth process on the same system many times until convergence is reached. This way both the system and the environment can include the effects of disorder correctly. That this might work has been known for a while, but previous attempts were severely restricted by the large computational cost. With recent algorithmic advances in large matrix diagonalisation by the investigators and with the available expertise in our High Performance Computing Centre we are ideally placed to perform these calculations, which we believe could have huge implications for the study of low-dimensional disordered systems of interacting particles. We will characterize the properties of the system for different strengths of interactions between particles and different strengths of disorder potential.

As well as computing the ground-state properties of interacting bosons in low dimensions, we will directly compare the results of the calculations with the results from an experimental group in Birmingham working on cold lattice systems. Because the systems studied in the laboratory are now so close to the model systems and so well-controlled, we will be in a position to work on the same systems theoretically as those being studied experimentally. One issue of importance to the interpretation of experimental data is the effect of small time-dependent variations of the equation of motion of the experimental systems, which is intrinsic to the use of laser patterns to generate interaction potentials. We will model the effect of such potentials and devise ways of separating out their effects from the intrinsic low temperature properties of the cold atoms. We will also be able to study experimental density profile data to look for evidence of wavefunction multifractality close to the localization-delocalisation transition. Evidence from computations suggest that the single-particle wavefunctions have very unusual internal correlations close to this localization transition. The direct access to the experimental data from Birmingham, will allow us to look for these correlations directly.

We will use density matrix renormalization group (DMRG) calculations (and its more modern forms such as tensor network states) to compute the ground state properties of a large system of interacting particles trapped in a lattice. Although the DMRG approach was originally developed to build up a system piece by piece by embedding it in an effective environment which assumed no disorder, the algorithm now repeats this growth process on the same system many times until convergence is reached. This way both the system and the environment can include the effects of disorder correctly. That this might work has been known for a while, but previous attempts were severely restricted by the large computational cost. With recent algorithmic advances in large matrix diagonalisation by the investigators and with the available expertise in our High Performance Computing Centre we are ideally placed to perform these calculations, which we believe could have huge implications for the study of low-dimensional disordered systems of interacting particles. We will characterize the properties of the system for different strengths of interactions between particles and different strengths of disorder potential.

As well as computing the ground-state properties of interacting bosons in low dimensions, we will directly compare the results of the calculations with the results from an experimental group in Birmingham working on cold lattice systems. Because the systems studied in the laboratory are now so close to the model systems and so well-controlled, we will be in a position to work on the same systems theoretically as those being studied experimentally. One issue of importance to the interpretation of experimental data is the effect of small time-dependent variations of the equation of motion of the experimental systems, which is intrinsic to the use of laser patterns to generate interaction potentials. We will model the effect of such potentials and devise ways of separating out their effects from the intrinsic low temperature properties of the cold atoms. We will also be able to study experimental density profile data to look for evidence of wavefunction multifractality close to the localization-delocalisation transition. Evidence from computations suggest that the single-particle wavefunctions have very unusual internal correlations close to this localization transition. The direct access to the experimental data from Birmingham, will allow us to look for these correlations directly.

### Planned Impact

Support for this project will bring computational modelling much closer to experiment. An exciting aspect of cold atom physics we are proposing to study is that the optical lattice potentials trapping the atoms as well as the disorder are under complete experimental control and are very accurately described by the simple models studied theoretically. It is possible for both theory and experiment to vary parameters (one-body or two-body) essentially at will. At the local level our project will strengthen the link between the computational and theoretical physics at Warwick and RHUL and the cold atom experimental group in Birmingham. However, we also expect that it will lead to many collaborations within the UK and beyond.

The main training component of the project will be the career development of the PhD student at Warwick. S/he will be well-placed not only to develop their own career through their research and through the planned interaction with the group at Birmingham but will also participate in the physics and high-performance computing courses at the Midlands Physics Alliance Graduate School as well as the Warwick Center for Scientific Computing (CSC), respectively. S/he will also benefit from interaction and collaboration with the people from the other CSC-affiliated groups working in biology, chemistry, computer science, mathematics and statistics. The CSC is a national center for training in High Performance Computing and is very well-equipped both in terms of hardware and support for the numerical work proposed.

The project will be managed from Warwick. The collaboration with the experimental group in Birmingham is a natural one, as we are in regular contact in the context of the Midlands Physics Alliance and our joint graduate school. Particularly in the second and third year there will be frequent visits to and from Birmingham when we compare our models with experimental results.

The main training component of the project will be the career development of the PhD student at Warwick. S/he will be well-placed not only to develop their own career through their research and through the planned interaction with the group at Birmingham but will also participate in the physics and high-performance computing courses at the Midlands Physics Alliance Graduate School as well as the Warwick Center for Scientific Computing (CSC), respectively. S/he will also benefit from interaction and collaboration with the people from the other CSC-affiliated groups working in biology, chemistry, computer science, mathematics and statistics. The CSC is a national center for training in High Performance Computing and is very well-equipped both in terms of hardware and support for the numerical work proposed.

The project will be managed from Warwick. The collaboration with the experimental group in Birmingham is a natural one, as we are in regular contact in the context of the Midlands Physics Alliance and our joint graduate school. Particularly in the second and third year there will be frequent visits to and from Birmingham when we compare our models with experimental results.

### Publications

Goldsborough A
(2014)

*Self-assembling tensor networks and holography in disordered spin chains*in Physical Review B
Goldsborough A
(2015)

*Using entanglement to discern phases in the disordered one-dimensional Bose-Hubbard model*in EPL (Europhysics Letters)
Goldsborough AM
(2015)

*Leaf-to-leaf distances and their moments in finite and infinite ordered m-ary tree graphs.*in Physical review. E, Statistical, nonlinear, and soft matter physics
Goldsborough Andrew M.
(2015)

*Tensor networks and geometry for the modelling of disordered quantum many-body systems*
Goldsborough Andrew M.
(2015)

*Leaf-to-leaf distances in Catalan trees*in arXiv e-prints
Goldsborough, AM
(2019)

*DIAGRAMMATIC APROACH TO LEAF-TO-LEAF DISTANCES IN CATALAN TREES A.*in Journal of Pure and Applied Mathematics: Advances and Applications
Núñez C
(2016)

*Silicene-based spin-filter device: impact of random vacancies*in 2D Materials
Oswald J
(2015)

*Imaging of Condensed Quantum States in the Quantum Hall Effect Regime*in Physics Procedia
Oswald J
(2017)

*Exchange-mediated dynamic screening in the integer quantum Hall effect regime*in EPL (Europhysics Letters)Description | Tensor networks provide a powerful and elegant approach to quantum many-body simulation. The simplest example is the density matrix renormalisation group (DMRG), which is based on the variational update of a matrix product state (MPS). It has proved to be the most accurate approach for the numerical study of strongly correlated one dimensional systems. It has recently been shown that there is a connection between the geometry of tensor networks and the entanglement and correlation properties that it can encode, which is a generalisation of the so called area law for entanglement entropy. This suggests that whilst gapped quantum systems can be accurately modeled using an MPS, a tensor network with a holographic geometry is natural to capture the logarithmic entanglement scaling and power law decaying correlation functions of critical systems. We create an algorithm for the disordered Heisenberg Hamiltonian that self assembles a tensor network based on the disorder in the couplings. The geometry created is that of a disordered tree tensor network (TTN) that when averaged has the holographic properties characteristic of critical systems. |

Exploitation Route | The use of tensor networks within the fields of condensed matter physics and quantum information theory is becoming ever more common. MPS-based DMRG is widely believed to be the most accurate method of numerically modelling one dimensional systems and it is being applied in increasingly complicated scenarios. Projected entangled pair states (PEPS) are being used both as a numerical method and as an analytic platform to uncover topological properties of matter. MERA and holographic tensor networks have become a useful tool within high energy physics and are being applied in linking entanglement with gravity in string theory. There are many ways that tensor networks can aid the study of disordered systems. Although DMRG is in some ways imperfect for the modelling of disorder, it is so efficient that much can still be learned by applying it. Beyond the Heisenberg and Bose-Hubbard models, there are still a myriad of possible Hamiltonians that can be examined with DMRG. A current area of intense research is many-body localisation (MBL), the generalisation of Anderson localisation to interacting many-body systems. It is believed that the area law holds for all excited states in systems with MBL up to some mobility energy, unlike gapped quantum systems where only the ground state is area law satisfying. This in principle should allow for an efficient MPS representation, and therefore accurate DMRG simulation, of any state in a one-dimensional MBL spectrum. Strong disorder renormalisation techniques such as tSDRG can be used as high precision methods when disorder is strong. The method should be accurate for use with the FM/AFM disordered spin-1/2 Heisenberg model where large effective spins would be created as the renormalisation progresses. Beyond spin-1/2 there have been exciting discoveries in disordered spin-3/2 Heisenberg systems, where the rich phase diagram contains topological phases as well as spin doublet and triplet phases. It would be fascinating to uncover the optimal tensor network geometries in these situations. More generally we would like to be able to construct an algorithm that can decide on the best network geometry for any system under consideration. Currently the geometry in most tensor network approaches is set by hand using prior knowledge of the model. In a network that can self optimise the structure, the final geometry can become a resource for learning about the properties of the wavefunction. Perhaps with these ideas, truly scalable two and three dimensional tensor network algorithms may be a possibility with and without disorder. |

Sectors | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Education |

Description | Findings have been used in outreach activities to local schools. The University of Warwick Department of Physics has a proven track record of engaging young people in its outreach activities. Hundreds of Brownies visited the university to take part in a physics activity day in order to earn a custom made 'physics badge' at the event (October 2018). Thousands of school children, families, youth groups and individuals attended the Warwick Christmas Lecture series (November/December 2019), organised by and mostly delivered by the physics department (4 out of 6 talks). Each spring a Science Gala is held in the department, welcoming over 700 visitors for an evening of engaging hands-on science activities and talks. These are simply a few examples of the breadth of activities undertaken by the department. PI Roemer is a regular contributor to these events, speaking about his recent research activities. During the duration of the grant and afterwards, Roemer used some of the findings in the grant to explain the motivation of modern theoretical physicists to this diverse audience on a number of occasions. |

First Year Of Impact | 2015 |

Sector | Education |

Impact Types | Cultural,Societal |

Description | Senior Visiting Professor of the Chinese Academy of Sciences |

Amount | ￥50,000 (CNY) |

Organisation | Chinese Academy of Sciences |

Sector | Public |

Country | China |

Start | 01/2016 |

End | 12/2016 |

Title | Dataset for "Using entanglement to discern phases in the disordered one-dimensional Bose-Hubbard model" |

Description | Dataset for "Using entanglement to discern phases in the disordered one-dimensional Bose-Hubbard model" A. M. Goldsborough, R. A. Römer EPL 111, 26004-6 (2015) http://dx.doi.org/10.1209/0295-5075/111/26004 |

Type Of Material | Database/Collection of data |

Year Produced | 2015 |

Provided To Others? | Yes |

Impact | more than 200 downloads since July 2015 |

URL | http://wrap.warwick.ac.uk/71189 |

Title | Fork of the ITensor code used for simulating the disordered Bose Hubbard model |

Description | AMGoldsborough fork of the ITensor libraries (Homepage: http://itensor.org/). This code was used to perform the simulation for the paper "Using entanglement to discern phases in the disordered one-dimensional Bose-Hubbard model" by Andrew M. Goldsborough and Rudolf A. Roemer (http://dx.doi.org/10.1209/0295-5075/111/26004). The base code is ITensor version 0.2.3 but with additional code written by AMG to support the disordered Bose-Hubbard model amongst other things. The additions to the library are: /itensor/model/bosehubbard.h - Bose-Hubbard basis with max. 2 bosons per site /itensor/model/bosehubbard5.h - Bose-Hubbard basis with max. 5 bosons per site /itensor/hams/BoseHubbardChain.h - Bose-Hubbard Hamiltonian with max. 2 bosons per site /itensor/hams/BoseHubbardChain5.h - Bose-Hubbard Hamiltonian with max. 5 bosons per site /itensor/hams/BoseHubbardChain5pbc.h - Bose-Hubbard Hamiltonian with max. 5 bosons per site with periodic boundaries /itensor/hams/DisorderedBH.h - BH Hamiltonian with vector variables for use with disorder /itensor/hams/DisorderedBH5.h - BH5 Hamiltonian with vector variables for use with disorder /itensor/hams/DisorderedBH5pbc.h - BH5 Hamiltonian with vector variables for use with disorder with PBC /itensor/hams/DisorderedHeisHalf.h - Spin-half Heisenberg Hamiltonian with vector variables for use with disorder /itensor/hams/DisorderedHeisHalfpbc.h - Spin-half Heisenberg Hamiltonian with vector variables and PBC /itensor/hams/DisorderedHub.h - Hubbard Hamiltonian with vector variables for use with disorder /bose_hubbard/disorderedBH5.cc - code for running diordered BH5 DMRG with filling n/L=1 and outputing observables /bose_hubbard/disorderedBH5_half.cc - disordered BH5 DMRG with filling n/L=0.5 /bose_hubbard/disorderedBH5_two.cc - disordered BH5 DMRG with filling n/L=2 /bose_hubbard/disorderedBH5apbc_half.cc - disordered BH5 DMRG with filling n/L=0.5 and anti-periodic boundaries /bose_hubbard/disorderedBH5apbc_two.cc - disordered BH5 DMRG with filling n/L=2 and anti-periodic boundaries /bose_hubbard/disorderedBH5apc.cc - disordered BH5 DMRG with filling n/L=1 and anti-periodic boundaries /bose_hubbard/disorderedBH5pbc.cc - disordered BH5 DMRG with filling n/L=1 and PBC /bose_hubbard/disorderedBH5pbc_half.cc - disordered BH5 DMRG with filling n/L=0.5 and PBC /bose_hubbard/disorderedBH5pbc_two.cc - disordered BH5 DMRG with filling n/L=2 and PBC /bose_hubbard/disorderedHub.cc - disordered Hubbard DMRG /disordered_spinhalf/disordered_spinhalf.cc - code for running disordered spin-half Heisenberg DMRG /disordered_spinhalf/disordered_spinhalfQ.cc - disordered spin-half Heisenberg DMRG keeping U(1) symmertries /disordered_spinhalf/sdrg.cc - recreating the tSDRG code used in http://dx.doi.org/10.1103/PhysRevB.89.214203 /disordered_spinhalf/spinhalfssd_test.cc - testing the sine-squared deformation (http://dx.doi.org/10.1103/PhysRevB.83.060414) You are welcome to download and use the code under the ITensor licence. If you write a paper using this Bose-Hubbard code please cite "Andrew M. Goldsborough, Rudolf A. Römer, EPL 111, 26004-6 (2015)". |

Type Of Technology | Software |

Year Produced | 2015 |

Open Source License? | Yes |

Impact | to early to say at the moment |

URL | https://github.com/AMGoldsborough/ITensor |

Description | Presentation during various visits of local schools |

Form Of Engagement Activity | Participation in an open day or visit at my research institution |

Part Of Official Scheme? | No |

Geographic Reach | National |

Primary Audience | Schools |

Results and Impact | RA Roemer presented at various open days and school visits including the XMas science gala in early 2016 |

Year(s) Of Engagement Activity | 2015,2016 |