The theory of a massless two-dimensional scalar field with a periodic boundary interaction is considered. At a critical value of the period this system defines a conformal field theory and can be reexpressed in terms of free fermions, which provide a simple realization of a hidden SU(2) symmetry of the original theory. The partition function and the boundary S matrix can be computed exactly as a function of the strength of the boundary interaction. We first consider open strings with one interacting and one Dirichlet boundary, and then with two interacting boundaries. The latter corresponds to motion in a periodic tachyon background, and the spectrum exhibits an interesting band structure which interpolates between free propagation and tight binding as the interaction strength is varied.The theory of a massless two-dimensional scalar field with a periodic boundary interaction is considered. At a critical value of the period this system defines a conformal field theory and can be reexpressed in terms of free fermions, which provide a simple realization of a hidden SU(2) symmetry of the original theory. The partition function and the boundary S matrix can be computed exactly as a function of the strength of the boundary interaction. We first consider open strings with one interacting and one Dirichlet boundary, and then with two interacting boundaries. The latter corresponds to motion in a periodic tachyon background, and the spectrum exhibits an interesting band structure which interpolates between free propagation and tight binding as the interaction strength is varied.

arXiv:hep-th/9404008Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJul 15, 1994154 citations

Embedding Space Construction of Conformal Field Theory From First Principles

Sutirtha Mukherjee

This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to $\mathrm{SO}(d+1,1)$ , and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra $[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$ from infinitesimal generators $\ell_n = -z^{n+1}\partial_z$ , computing the central charge explicitly for free theories (boson $c=1$ , Majorana fermion $c=1/2$ , $bc$ -ghosts $c = 1-3(2\lambda-1)^2$ ) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states $|h,\bar{h}\rangle$ , deriving the transformation law $T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]$ that reveals the geometric origin of the conformal anomaly.This article presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to $\mathrm{SO}(d+1,1)$, and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra $[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$ from infinitesimal generators $\ell_n = -z^{n+1}\partial_z$, computing the central charge explicitly for free theories (boson $c=1$, Majorana fermion $c=1/2$, $bc$-ghosts $c = 1-3(2\lambda-1)^2$) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states $|h,\bar{h}\rangle$, deriving the transformation law $T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]$ that reveals the geometric origin of the conformal anomaly.

Feb 2, 20260 citations

Symmetry Resolution of Entanglement Measures in Free Fermionic Systems and Conformal Field Theories

Riccarda Bonsignori

Entanglement and symmetries are two pillars of our understanding of modern Physics, but surprisingly the study of the interplay between these two fundamental concepts has become the subject of an intense research area only in the last few years. In particular, the resolution of entanglement in the various symmetry sectors of a quantum many-body system with a global conserved charge gained a lot of interest within the entanglement community. As shown in some experimental papers, the study of this symmetry resolved entanglement is necessary for a better understanding of the non-equilibrium dynamics of quantum many-body systems. This Thesis is devoted to the study of the symmetry resolution of the entanglement measures in free fermionic systems and Conformal Field Theories. It is organised as follows. In Chap. 1 we give an overview of the main definitions and concepts related to the entanglement and its measures in quantum mechanics and quantum field theory. Then, we specialize to the case of conformal field theories, to derive expressions of the entanglement entropy and negativity in all the cases that will be examined in the second part of the Thesis. We finally illustrate the main free-fermionic techniques that will be used to perform the analytical as well as numerical lattice calculations. In Chap. 2, we consider a $1$ -dimensional system of free fermions hopping on a lattice, in which the conserved $U(1)$ charge is the total particle number. We evaluate the symmetry resolved R'enyi and entanglement entropies determining exactly all the non-universal constants and logarithmic corrections to the scaling, thanks to the so-called Fisher-Hartwig conjecture. In Chap. 3, we study the symmetry resolved entanglement entropies in $1$ -dimensional systems in presence of boundaries, considering the cases of an interval starting from the boundary and disconnected from it. We evaluate the charged and symmetry resolved entropies for conformal invariant theories and a semi-infinite chain of free fermions, finding entanglement equipartition. Using a generalization of the lattice calculations performed in Chap. 2, we also evaluate the first term breaking equipartition for the free fermionic chain. In Chap. 4, we study the symmetry resolution of the entanglement negativity in free fermionic systems, which admits a decomposition in terms of the charge imbalance between the two subsystems. We use conformal field theory to derive the universal behavior of the charge imbalance resolved negativity for a free Dirac field at finite temperature and size. For the lattice version of the Dirac filed, we also determine exactly the model-dependent terms using the Fisher-Hartwig conjecture. We find that at leading order the charge imbalance resolved negativity verifies equipartition. For the lattice model, we also evaluate the first corrections to this equipartition. In Chap. 5, we study the time evolution of the symmetry resolved entanglement entropy and a suitably defined symmetry resolved mutual information in a system of free fermions after a quench. In the context of free fermions and conformal field theories, we are able to provide analytical results for the time-dependent charged and resolved entropies, and shed light on two new physical phenomena. First, the symmetry resolved R'enyi entropies start evolving with a calculable time delay that depends on the charge sector. Second, there is an effective equipartition of the entanglement in the limit of large subsystem size. For free fermions, we also find that effective equipartition is verified by the mutual information. We expect that these phenomena occur in more general quantum systems, as they can be understood from the quasiparticle picture for the spreading of entanglement. In the Appendix, we give an overview of conformal field theory, reporting the main concepts and formulas that are needed for a more complete understanding of the previous Chapters.Entanglement and symmetries are two pillars of our understanding of modern Physics, but surprisingly the study of the interplay between these two fundamental concepts has become the subject of an intense research area only in the last few years. In particular, the resolution of entanglement in the various symmetry sectors of a quantum many-body system with a global conserved charge gained a lot of interest within the entanglement community. As shown in some experimental papers, the study of this symmetry resolved entanglement is necessary for a better understanding of the non-equilibrium dynamics of quantum many-body systems. This Thesis is devoted to the study of the symmetry resolution of the entanglement measures in free fermionic systems and Conformal Field Theories. It is organised as follows. In Chap. 1 we give an overview of the main definitions and concepts related to the entanglement and its measures in quantum mechanics and quantum field theory. Then, we specialize to the case of conformal field theories, to derive expressions of the entanglement entropy and negativity in all the cases that will be examined in the second part of the Thesis. We finally illustrate the main free-fermionic techniques that will be used to perform the analytical as well as numerical lattice calculations. In Chap. 2, we consider a $1$-dimensional system of free fermions hopping on a lattice, in which the conserved $U(1)$ charge is the total particle number. We evaluate the symmetry resolved R'enyi and entanglement entropies determining exactly all the non-universal constants and logarithmic corrections to the scaling, thanks to the so-called Fisher-Hartwig conjecture. In Chap. 3, we study the symmetry resolved entanglement entropies in $1$-dimensional systems in presence of boundaries, considering the cases of an interval starting from the boundary and disconnected from it. We evaluate the charged and symmetry resolved entropies for conformal invariant theories and a semi-infinite chain of free fermions, finding entanglement equipartition. Using a generalization of the lattice calculations performed in Chap. 2, we also evaluate the first term breaking equipartition for the free fermionic chain. In Chap. 4, we study the symmetry resolution of the entanglement negativity in free fermionic systems, which admits a decomposition in terms of the charge imbalance between the two subsystems. We use conformal field theory to derive the universal behavior of the charge imbalance resolved negativity for a free Dirac field at finite temperature and size. For the lattice version of the Dirac filed, we also determine exactly the model-dependent terms using the Fisher-Hartwig conjecture. We find that at leading order the charge imbalance resolved negativity verifies equipartition. For the lattice model, we also evaluate the first corrections to this equipartition. In Chap. 5, we study the time evolution of the symmetry resolved entanglement entropy and a suitably defined symmetry resolved mutual information in a system of free fermions after a quench. In the context of free fermions and conformal field theories, we are able to provide analytical results for the time-dependent charged and resolved entropies, and shed light on two new physical phenomena. First, the symmetry resolved R'enyi entropies start evolving with a calculable time delay that depends on the charge sector. Second, there is an effective equipartition of the entanglement in the limit of large subsystem size. For free fermions, we also find that effective equipartition is verified by the mutual information. We expect that these phenomena occur in more general quantum systems, as they can be understood from the quasiparticle picture for the spreading of entanglement. In the Appendix, we give an overview of conformal field theory, reporting the main concepts and formulas that are needed for a more complete understanding of the previous Chapters.

Oct 18, 20210 citations

\mathrm{SO}(d+1,1)

, and developing the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra

[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}

from infinitesimal generators

\ell_n = -z^{n+1}\partial_z

, computing the central charge explicitly for free theories (boson

c=1

, Majorana fermion

c=1/2

bc

-ghosts

c = 1-3(2\lambda-1)^2

) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states

|h,\bar{h}\rangle

, deriving the transformation law

T'(w) = (dw/dz)^{-2}[T(z) - (c/12)\{w;z\}]

that reveals the geometric origin of the conformal anomaly.

1

-dimensional system of free fermions hopping on a lattice, in which the conserved

U(1)

charge is the total particle number. We evaluate the symmetry resolved R'enyi and entanglement entropies determining exactly all the non-universal constants and logarithmic corrections to the scaling, thanks to the so-called Fisher-Hartwig conjecture. In Chap. 3, we study the symmetry resolved entanglement entropies in

1

-dimensional systems in presence of boundaries, considering the cases of an interval starting from the boundary and disconnected from it. We evaluate the charged and symmetry resolved entropies for conformal invariant theories and a semi-infinite chain of free fermions, finding entanglement equipartition. Using a generalization of the lattice calculations performed in Chap. 2, we also evaluate the first term breaking equipartition for the free fermionic chain. In Chap. 4, we study the symmetry resolution of the entanglement negativity in free fermionic systems, which admits a decomposition in terms of the charge imbalance between the two subsystems. We use conformal field theory to derive the universal behavior of the charge imbalance resolved negativity for a free Dirac field at finite temperature and size. For the lattice version of the Dirac filed, we also determine exactly the model-dependent terms using the Fisher-Hartwig conjecture. We find that at leading order the charge imbalance resolved negativity verifies equipartition. For the lattice model, we also evaluate the first corrections to this equipartition. In Chap. 5, we study the time evolution of the symmetry resolved entanglement entropy and a suitably defined symmetry resolved mutual information in a system of free fermions after a quench. In the context of free fermions and conformal field theories, we are able to provide analytical results for the time-dependent charged and resolved entropies, and shed light on two new physical phenomena. First, the symmetry resolved R'enyi entropies start evolving with a calculable time delay that depends on the charge sector. Second, there is an effective equipartition of the entanglement in the limit of large subsystem size. For free fermions, we also find that effective equipartition is verified by the mutual information. We expect that these phenomena occur in more general quantum systems, as they can be understood from the quasiparticle picture for the spreading of entanglement. In the Appendix, we give an overview of conformal field theory, reporting the main concepts and formulas that are needed for a more complete understanding of the previous Chapters.