DMRG for 2D Quantum Systems

The density matrix renormalization group (DMRG), originally designed to find ground states of one-dimensional (1D) quantum systems, is surprisingly effective for two-dimensional (2D) systems. But applying DMRG to 2D systems requires great care, since the cost grows exponentially with the linear size of one of the two directions (the "y" direction, say).

To use DMRG effectively for 2D systems, one should use information not only from the largest attainable system size (typically only of the order of 10 sites in the y direction), but from also from smaller systems where DMRG has greater control. In one recent project, we studied the scaling of entanglement at 2D quantum critical points using the numerical linked cluster expansion, which uses information from smaller clusters to reduce finite-size effects on the largest 2D cluster.

In another recent study of a 2D lattice model of coupled "anyons", insight from a 1D field-theoretic picture of our model allowed us to see fingerprints of interesting 2D phases on systems as small as two sites wide!

Related Work:

• E.M. Stoudenmire, Peter Gustainis, Ravi Johal, Stefan Wessel, and Roger G. Melko, "Corner Contributions to the Entanglement Entropy of Strongly-Interacting O(2) Quantum Critical Systems in 2+1 Dimensions", Phys. Rev. B, 90: 235106 (2014)
• E.M. Stoudenmire, David J. Clarke, Roger S. K. Mong, and Jason Alicea, "Assembling Fibonacci Anyons From a Z₃ Parafermion Lattice Model", arxiv:1501.05305 (2014)
• E.M. Stoudenmire and Steven R. White, "Real-Space Parallel Density Matrix Renormalization Group", Phys. Rev. B 87: 115137 (2013)
• E.M. Stoudenmire and Steven R. White, "Studying two-dimensional systems with the density matrix renormalization group", Annual Reviews of Condensed Matter Physics 3: 111 (2012)

Minimally Entangled Typical Thermal States (METTS)

What does a snapshot of a thermal quantum state look like?

Quantum Monte Carlo methods have been very successful in simulating thermal quantum systems by mapping them to classical ensembles which can be efficiently sampled. However, this approach can fail dramatically for systems with mobile fermions or frustrated magnetic exchange interactions, in which case the system is said to have a "sign problem".

One way around the sign problem is to represent a thermal quantum system as an ensemble of pure-state wavefunctions instead of virtual classical states. A particularly efficient ensemble to sample is the set of minimally entangled typical thermal states or METTS. These states are descendants of classical product wavefunctions which contain just enough entanglement to capture finite-temperature quantum correlations.

METTS not only provide an efficient method for simulating thermal quantum systems in 1d and 2d, but provide an intuitive picture of thermal quantum states. There is a wealth of systems waiting to be studied with METTS including one- and two-dimensional frustrated spin-lattice models and continuum electronic wires. We are also investigating powerful new extensions to the basic METTS algorithm that will enable us to push further into two dimensions.

More Reading:

• Steven R. White, "Minimally entangled typical quantum states at finite temperature", Phys. Rev. Lett. 102: 190601 (2009)
• E.M. Stoudenmire and Steven R. White, "Minimally entangled typical thermal state algorithms", New J. Phys. 12: 055026 (2010)

Exact Density Functional Theory With Model 1D Systems

At the heart of the enormously successful density functional theory (DFT) method lies a functional that gives the ground state energy of any system provided only its density. Can this exact energy functional be written down? What are its key properties?

Except for a handful of small systems, computing the exact functional in three dimensions is too difficult to allow a systematic study of its properties. However, in one dimension the extreme power of the density matrix renormalization group (DMRG) allows us to calculate the exact densities of a wide variety of continuum, long-range interacting systems resembling long chains of artificial atoms. From these exact densities, it is relatively straightforward to calculate the exact density functional and a host of other quantities of interest in DFT. We can then use our 1d laboratory to study established density-functional approximations and explore improved functionals which could be applied to strongly correlated systems in 3d.

I gave a talk on this subject in November 2012 at the Perimeter Institute, and a slightly more pedagogical talk at National Taiwan University in December 2012.

Related works:

• Lucas O. Wagner, Thomas E. Baker, E.M. Stoudenmire, Kieron Burke, and Steven R. White, "Kohn-Sham Calculations with the Exact Functional", arxiv:1405.0864
• Lucas O. Wagner, E.M. Stoudenmire, Kieron Burke, and Steven R. White, "Guaranteed convergence of the Kohn-Sham equations", Phys. Rev. Lett. 111: 093003 (2013)
• E.M. Stoudenmire, Lucas O. Wagner, Steven R. White and Kieron Burke, "One-dimensional continuum electronic structure with the density matrix renormalization group and its implications for density functional theory", Phys. Rev. Lett. 109: 056402 (2012)
• Lucas O. Wagner, E.M. Stoudenmire, Steven R. White and Kieron Burke, "Reference electronic structure calculations in one dimension", Phys. Chem. Chem. Phys. 14: 8581 (2012)

The Intelligent Tensor (ITensor) C++ Library

What does a DMRG wavefunction look like?

Beginning with the work of Östlund and Rommer, it has become clear that the density matrix renormalization group (DMRG) is just one of a family of powerful methods for simulating quantum systems. The concept linking these methods is the tensor network state, a diagrammatic language for capturing the internal structure of many-body wavefunctions.

The ITensor Library is an open source C++ library for rapidly developing and applying tensor network algorithms. ITensor is competitive with publicly available codes when performing straight DMRG calculations, but its flexibility has enabled us to extend DMRG in ways that would be inconceivable with other approaches. It is especially well suited for developing next-generation tensor network algorithms such as PEPS, MERA, and METTS (see above).

See a list of papers written using ITensor.