Implications of the Duality

Oct 13, 2025 by pinkandtrash.com 139 views

Introduction

The study of condensed matter systems has led to the discovery of various intriguing phenomena, with the transverse-field Ising model (TFIM) being one of the most well-studied models in this context. The TFIM is a statistical mechanics model that exhibits a rich phase diagram, including a quantum phase transition. Recently, there has been a growing interest in the connection between the TFIM and gauge theories, particularly the $\mathbb{Z}_2$ gauge theory. In this article, we will delve into the duality between the 2+1 d transverse-field Ising model and the $\mathbb{Z}_2$ gauge theory, as discussed in McGreevy's Lecture notes "Where do QFTs come from?".

Background

The transverse-field Ising model is a one-dimensional (1D) statistical mechanics model that consists of a chain of spins interacting with each other through a transverse magnetic field. The Hamiltonian of the TFIM is given by:

H = -\sum_{i} J \sigma^z_i \sigma^z_{i+1} - h \sum_{i} \sigma^x_i

where $J$ is the exchange interaction, $h$ is the transverse magnetic field, and $\sigma^z_i$ and $\sigma^x_i$ are the Pauli matrices at site $i$ . The TFIM exhibits a quantum phase transition at a critical value of the transverse magnetic field, $h_c = 3J/4$ , separating a ferromagnetic phase from a paramagnetic phase.

On the other hand, the $\mathbb{Z}_2$ gauge theory is a quantum field theory that describes the dynamics of a gauge field in 2+1 dimensions. The gauge field is a vector field that transforms under the action of the $\mathbb{Z}_2$ group, which is a discrete group with two elements. The $\mathbb{Z}_2$ gauge theory is a topological field theory, meaning that it is invariant under continuous deformations of the spacetime manifold.

Duality between TFIM and $\mathbb{Z}_2$ Gauge Theory

The duality between the TFIM and the $\mathbb{Z}_2$ gauge theory was first proposed by McGreevy in his Lecture notes "Where do QFTs come from?". The duality states that the TFIM in 2+1 dimensions is equivalent to the $\mathbb{Z}_2$ gauge theory in 2+1 dimensions. This duality can be understood as a mapping between the two theories, where the spins in the TFIM are mapped to the gauge fields in the $\mathbb{Z}_2$ gauge theory.

The duality can be formulated as follows:

\mathcal{Z}_{\text{TFIM}}(J, h) = \mathcal{Z}_{\mathbb{Z}_2}(\lambda)

where $\mathcal{Z}_{\text{TFIM}}$ is the partition function of the TFIM, $\mathcal{Z}_{\mathbb{Z}_2}$ is the partition function of the $\mathbb{Z}_2$ gauge theory, $J$ and $h$ are the parameters of the TFIM, and $\lambda$ is the parameter of the $\mathbb{Z}_2$ gauge theory.

The duality can be understood as a mapping between the two theories, where the spins in the TFIM are mapped to the gauge fields in the $\mathbb{Z}_2$ gauge theory. This mapping can be written as:

\sigma^z_i \mapsto \epsilon_{ij} \epsilon_{kl} A_{jk} A_{il}

where $\sigma^z_i$ is the spin at site $i$ in the TFIM, $\epsilon_{ij}$ is the Levi-Civita symbol, and $A_{jk}$ is the gauge field at site $(j, k)$ in the $\mathbb{Z}_2$ gauge theory.

Implications of the Duality

The duality between the TFIM and the $\mathbb{Z}_2$ gauge theory has several implications. Firstly, it provides a new perspective on the phase diagram of the TFIM, which can be understood in terms of the topological properties of the $\mathbb{Z}_2$ gauge theory. Secondly, it provides a new tool for studying the properties of the TFIM, which can be used to compute the partition function and other physical quantities.

The duality also has implications for the study of topological phases of matter. The $\mathbb{Z}_2$ gauge theory is a topological field theory, which means that it is invariant under continuous deformations of the spacetime manifold. This invariance is a key feature of topological phases of matter, which are characterized by the presence of topological order. The duality between the TFIM and the $\mathbb{Z}_2$ gauge theory provides a new way to study topological phases of matter, which can be used to compute the topological invariants of the system.

Conclusion

In conclusion, the duality between the 2+1 d transverse-field Ising model and the $\mathbb{Z}_2$ gauge theory is a fascinating phenomenon that has far-reaching implications for our understanding of condensed matter systems. The duality provides a new perspective on the phase diagram of the TFIM, which can be understood in terms of the topological properties of the $\mathbb{Z}_2$ gauge theory. It also provides a new tool for studying the properties of the TFIM, which can be used to compute the partition function and other physical quantities.

The duality between the TFIM and the $\mathbb{Z}_2$ gauge theory is a testament to the power of theoretical physics in uncovering the underlying structure of complex systems. It highlights the importance of interdisciplinary research, which can lead to new insights and discoveries in various fields of physics.

References

McGreevy, J. (2016). Where do QFTs come from? Lecture notes.
Itzykson, C., & Drouffe, J. (1989). Statistical field theory. Cambridge University Press.
Polyakov, A. M. (1987). Gauge fields and strings. Harwood Academic Publishers.
Wen, X. G. (2004). Quantum field theory of many-body systems: From the origin of sound to an origin of light and electrons. Oxford University Press.
Q&A: Unveiling the Duality between 2+1 d Transverse-Field Ising Model and $\mathbb{Z}_2$ Gauge Theory =====================================================================================

Introduction

In our previous article, we explored the duality between the 2+1 d transverse-field Ising model (TFIM) and the $\mathbb{Z}_2$ gauge theory. This duality has far-reaching implications for our understanding of condensed matter systems and has sparked a lot of interest in the physics community. In this article, we will answer some of the most frequently asked questions about the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory.

Q: What is the significance of the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory?

A: The duality between the TFIM and the $\mathbb{Z}_2$ gauge theory is significant because it provides a new perspective on the phase diagram of the TFIM. It also provides a new tool for studying the properties of the TFIM, which can be used to compute the partition function and other physical quantities.

Q: How does the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory work?

A: The duality between the TFIM and the $\mathbb{Z}_2$ gauge theory can be understood as a mapping between the two theories. The spins in the TFIM are mapped to the gauge fields in the $\mathbb{Z}_2$ gauge theory. This mapping can be written as:

\sigma^z_i \mapsto \epsilon_{ij} \epsilon_{kl} A_{jk} A_{il}

where $\sigma^z_i$ is the spin at site $i$ in the TFIM, $\epsilon_{ij}$ is the Levi-Civita symbol, and $A_{jk}$ is the gauge field at site $(j, k)$ in the $\mathbb{Z}_2$ gauge theory.

Q: What are the implications of the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory for the study of topological phases of matter?

A: The duality between the TFIM and the $\mathbb{Z}_2$ gauge theory has implications for the study of topological phases of matter. The $\mathbb{Z}_2$ gauge theory is a topological field theory, which means that it is invariant under continuous deformations of the spacetime manifold. This invariance is a key feature of topological phases of matter, which are characterized by the presence of topological order. The duality between the TFIM and the $\mathbb{Z}_2$ gauge theory provides a new way to study topological phases of matter, which can be used to compute the topological invariants of the system.

Q: Can the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory be used to study other systems?

A: Yes, the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory can be used to study other systems. The duality is a general concept that can be applied to other systems that exhibit similar properties. For example, the duality can be used to study the properties of other topological phases of matter, such as the topological insulators and superconductors.

Q: What are the challenges associated with the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory?

A: One of the challenges associated with the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory is the difficulty in computing the partition function and other physical quantities. The duality requires a deep understanding of the underlying physics of the system, which can be challenging to obtain. Additionally, the duality is a non-trivial mapping between the two theories, which can make it difficult to compute the physical quantities of interest.

Q: What are the future directions for research on the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory?

A: One of the future directions for research on the duality between the TFIM and the $\mathbb{Z}_2$ gauge theory is to study the properties of other topological phases of matter. The duality can be used to study the properties of other topological phases of matter, such as the topological insulators and superconductors. Additionally, the duality can be used to study the properties of other systems that exhibit similar properties, such as the quantum Hall effect and the fractional quantum Hall effect.

Conclusion

In conclusion, the duality between the 2+1 d transverse-field Ising model and the $\mathbb{Z}_2$ gauge theory is a fascinating phenomenon that has far-reaching implications for our understanding of condensed matter systems. The duality provides a new perspective on the phase diagram of the TFIM and provides a new tool for studying the properties of the TFIM. The duality also has implications for the study of topological phases of matter and can be used to study other systems that exhibit similar properties.

References

McGreevy, J. (2016). Where do QFTs come from? Lecture notes.
Itzykson, C., & Drouffe, J. (1989). Statistical field theory. Cambridge University Press.
Polyakov, A. M. (1987). Gauge fields and strings. Harwood Academic Publishers.
Wen, X. G. (2004). Quantum field theory of many-body systems: From the origin of sound to an origin of light and electrons. Oxford University Press.