Renormalization and tensor product states in spin chains and lattices (0910.1130v1)

Abstract: We review different descriptions of many--body quantum systems in terms of tensor product states. We introduce several families of such states in terms of known renormalization procedures, and show that they naturally arise in that context. We concentrate on Matrix Product States, Tree Tensor States, Multiscale Entanglement Renormalization Ansatz, and Projected Entangled Pair States. We highlight some of their properties, and show how they can be used to describe a variety of systems.

• The paper presents a comprehensive analysis of tensor product states that efficiently reduce the exponential complexity inherent in quantum many-body systems.
• It systematically compares key TPS families—including MPS, TTS, MERA, and PEPS—highlighting their application to one- and higher-dimensional systems.
• The study underscores practical implications by detailing numerical algorithms that enhance simulation accuracy in quantum lattice models.

An Overview of Renormalization and Tensor Product States in Spin Chains and Lattices

The paper "Renormalization and Tensor Product States in Spin Chains and Lattices" by J. Ignacio Cirac and Frank Verstraete provides a comprehensive examination of tensor product states (TPS) used to model many-body quantum systems. The authors review several representative families of such states, including Matrix Product States (MPS), Tree Tensor States (TTS), Multiscale Entanglement Renormalization Ansatz (MERA), and Projected Entangled Pair States (PEPS), focusing on their application to spin chains and lattice systems.

Quantum many-body systems are notoriously difficult to describe due to the exponential growth in the number of parameters as the number of particles increases. Traditional approaches struggle to manage this complexity, necessitating the development of new methods like TPS, which provide an efficient alternative by employing a polynomially bounded number of parameters.

Description of Tensor Product States

1. Matrix Product States (MPS): MPS are motivated by real-space renormalization methods such as the Density Matrix Renormalization Group (DMRG). They efficiently describe one-dimensional systems by leveraging a local product structure that captures essential entanglement properties while maintaining polynomial computational resources. The classic example of MPS arises in the translationally invariant case, known as finitely correlated states, which are powerful for describing both ground states and thermal states in one-dimensional quantum systems.
2. Tree Tensor States (TTS): Extending beyond MPS, TTS exploit hierarchical structures to encapsulate correlations more flexibly, albeit sometimes violating area laws. They are particularly applicable to systems exhibiting logarithmic scaling of entanglement entropy and can represent critical systems better than MPS due to their inherently multiscale structure.
3. Multiscale Entanglement Renormalization Ansatz (MERA): Like TTS, MERA models hierarchical structures but further integrates disentangling operations to manage non-local correlations and fulfill area laws effectively. This makes them ideally suited for studying critical phenomena and phase transitions, notably in systems where classical renormalization techniques like those of Fisher are applicable.
4. Projected Entangled Pair States (PEPS): PEPS serve as a multidimensional generalization of MPS, targeting higher-dimensional lattice systems. They naturally comply with the area law, making them suitable for investigating two-dimensional and higher-dimensional systems. Despite the computational intensity involved in calculating expectations, PEPS can efficiently approximate ground states due to their robust structure, which captures essential quantum correlations.

Theoretical and Practical Implications

The theoretical implications of this paper primarily revolve around identifying the "corner of Hilbert space" that encompasses relevant quantum states of interest. The paper highlights that despite quantum states being exponentially complex in nature, the actual states encountered in practice can be efficiently described using TPS due to their bounded entanglement. This insight has profound impacts on quantum computation and simulation.

Practically, TPS provides a framework for developing powerful numerical algorithms. Algorithms based on MPS and PEPS have successfully tackled both ground state problems and dynamic simulations, offering unprecedented accuracy and efficiency in their respective domains. These methods are seminal in numerical studies of quantum lattice models and have broadened the ability to simulate systems beyond traditional computational approaches.

Future Directions

The paper of TPS in quantum lattice systems paves the way for further research in both algorithmic advancement and exploration of complex quantum phenomena. Future research directions may include refining TPS-based algorithms for increased efficiency, especially in two or higher-dimensional systems, addressing current limitations such as computational overhead and challenges in enforcing translational invariance. Additionally, extending the framework to accommodate long-range interactions and dynamic processes without sacrificing accuracy remains a crucial area of exploration.

In summary, the paper by Cirac and Verstraete not only provides a detailed review of tensor product states but also underscores their critical role in bridging the gap between complex quantum systems and tractable computational models. The insights from this work continue to shape developments in quantum physics, both theoretically and in practical applications.