Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1812.01311 · Physical review. B./Physical review. B

Quantum criticality on a chiral ladder: An SU(2) infinite density matrix renormalization group study

Philipp Schmoll, Andreas Haller, Matteo Rizzi, Román Orús

Published May 13, 2019• 8 citations
arXivHTMLPDF

Abstract

In this paper we study the ground-state properties of a ladder Hamiltonian with chiral $\text{SU}(2)$-invariant spin interactions, a possible first step toward the construction of truly two-dimensional nontrivial systems with chiral properties starting from quasi-one-dimensional ones. Our analysis uses a recent implementation by us of $\text{SU}(2)$ symmetry in tensor network algorithms, specifically for infinite density matrix renormalization group. After a preliminary analysis with Kadanoff coarse graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground-state properties match those of the universality class of a $c=1$ conformal field theory (CFT) in $(1+1)$ dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground-state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-$\frac{1}{2}$ Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an $\text{SU}{(2)}_{1}$ Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct matrix product operators for $\text{SU}(2)$-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.

DOI
10.1103/physrevb.99.205121

Related papers

Quantum Critical Spin-2 Chain with Emergent SU(3) Symmetry

Pochung Chen, Zhi-Long Xue, Ian P. McCulloch +3 more

We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies and entanglement entropy by exact diagonalization and density-matrix renormalization group methods. From the numerical results of the energy spectra, central charge, and scaling dimension we identify the conformal field theory describing the whole critical phase to be the SU(3)_{1} Wess-Zumino-Witten model. We find that, while the Hamiltonian is only SU(2) invariant, in this critical phase there is an emergent SU(3) symmetry in the thermodynamic limit.

arXiv:1402.5713•Physical Review Letters•Apr 9, 2015•24 citations

Momentum-space entanglement in quantum spin chains

Rex Oliver Lundgren

In this dissertation, the momentum-space entanglement properties of several quantum spin chains are investigated. In the second chapter we re- view important results on many-body entanglement. In the third chapter, we numerically study the momentum-space entanglement spectra of bosonic and fermionic formulations of the spin-1/2 XXZ chain [Phys. Rev. Lett. 113, 256404 (2014)]. We investigate the behavior of the entanglement gaps, present in both formulations, across quantum phase transitions in the XXZ chain. In both cases, finite size scaling suggests that the entanglement gap closure does not occur at the physical transition points. For bosons, we find that the entan- glement gap depends on the scaling dimension of the conformal field theory as varied by the X X Z anisotropy. For fermions, the infinite entanglement gap present at the XX point persists well past the phase transition at the Heisenberg point. We elaborate on how this may support the numerical study of phase transitions. In the third chapter, we advocate that in certain critical spin chains a gap in the momentum-space entanglement spectrum separates the universal part of the spectrum, which is determined by the associated conformal field theory, from the non-universal part, which is specific to the model [arXiv:1512.09030]. We provide affirmative evidence from multicritical spin-1 chains with low energy sectors described by the SU(2)2 or the SU(3)1 Wess-Zumino-Witten model. In chapter four, we study momentum- space en- tanglement in quantum spin-half ladders [Phys. Rev. B 93, 125107 (2016)]. When the system is gapped, we analytically find the momentum-space entan- glement Hamiltonian is described by a chiral conformal field theory with a central charge of two. When the system is gapless, the entanglement Hamil- tonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. We also analytically include the effect of a certain irrelevant terms (in the renormalization group sense) on the entanglement spectrum. In chapter six, we investigate the momentum- space entanglement spectrum after a quantum quench [arXiv:1603.01997]. We show that the momentum-space entanglement spectrum of the XXZ spin- half chain possesses many universal features both in equilibrium and after a quantum quench.

•Jan 1, 2016•0 citations

Emergent universality in critical quantum spin chains: entanglement Virasoro algebra

Hu, Qi, Adrián Franco-Rubio, Guifré Vidal

Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. Given a pure state of the system and a division into regions $A$ and $B$, they can be obtained in terms of the $Schmidt~ values$, or eigenvalues $λ_α$ of the reduced density matrix $ρ_A$ for region $A$. In this paper we draw attention instead to the $Schmidt~ vectors$, or eigenvectors $|v_α\rangle$ of $ρ_A$. We consider the ground state of critical quantum spin chains whose low energy/long distance physics is described by an emergent conformal field theory (CFT). We show that the Schmidt vectors $|v_α\rangle$ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT (a chiral version of the original CFT). Indeed, we build weighted sums $H_n$ of the lattice Hamiltonian density $h_{j,j+1}$ over region $A$ and show that the matrix elements $\langle v_αH_n |v_{α'}\rangle$ are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for $H_n^{\tiny \text{CFT}} = \frac 1 2 (L_n + L_{-n})$ in the boundary CFT, where $L_n$'s are (one copy of) the Virasoro generators. We numerically confirm our results using the critical Ising quantum spin chain and other (free-fermion equivalent) models.

arXiv:2009.11383•Sep 23, 2020•5 citations

Density matrix renormalization group simulations of SU( N ) Heisenberg chains using standard Young tableaus: Fundamental representation and comparison with a finite-size Bethe ansatz

Pierre Nataf, Frédéric Mila

We develop an efficient method to perform density matrix renormalization group simulations of the SU($N$) Heisenberg chain with open boundary conditions taking full advantage of the SU($N$) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e., one particle per site in the fermionic formulation), we have benchmarked our results for the ground-state energy up to $N=8$ and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground-state energy is excellent for SU(3) (12 digits). It decreases with $N$, but it is still satisfactory for $N=8$ (six digits). Central charges $c$ are also extracted from the entanglement entropy using the Calabrese-Cardy formula and agree with the theoretical values expected from the $\mathrm{SU}{(N)}_{1}$ Wess-Zumino-Witten conformal field theories.

arXiv:1802.05482•Physical review. B./Physical review. B•Apr 20, 2018•36 citations

Dimerization and exotic criticality in spin-S chains

Natalia Chepiga

In this thesis we have studied the emergence of spontaneously dimerized phases in frustrated spin-S chains, with emphasis on the nature of the critical lines between the dimerized and non-dimerized phases. The main numerical method used in this thesis is the Density Matrix Renormalization Group (DMRG). The DMRG algorithm is a relatively old and well established method for the investigation of the ground-state. In this thesis, we show how to use this algorithm to calculate the excitation spectra of one-dimensional critical systems, known in the context of conformal field theory as conformal towers of states. We have demonstrated that the method works very well for two simple minimal models (the transverse-field Ising model and the three-state Potts model), and we have used it systematically to identify the universality classes and the underlying conformal field theories of various one-dimensional spin systems. It has been known for a long time that the transition to a spontaneously dimerized phase in a spin-1 chain can be either continuous, in the Wess-Zumino-Witten (WZW) SU(2) level 2 universality class, or first order. By combining a careful numerical investigation with a conformal field theory analysis, we were able to detect in a frustrated spin-1 chain with competing next-nearest-neighbor and three-site interactions the presence of yet another type of continuous phase transition that belongs to the Ising universality class. In contrast to the WZW SU(2) level 2 critical line, at which the singlet-triplet gap closes, the Ising transition occurs entirely in the singlet sector, while the singlet-triplet gap remains open. The use of the standard DMRG approach, along the lines mentioned above, has allowed us to provide explicit numerical evidence for the presence of a conformal tower of singlets inside the spin gap. Moreover, according to field theory, a WZW SU(2) level k critical line can turn into a first order transition due to the presence of a marginal operator in the WZW SU(2) level k model. A careful investigation of the conformal towers along the critical lines has allowed us to find the precise location of this point in both S=1 and S=3/2 chains. We have also shown that the nature of the continuous dimerization transitions is related to the topological properties of the corresponding phases, and that the phase diagrams of various frustrated spin chains can be effectively extracted by looking at the local topological order parameter - the degeneracy of the lowest state in the entanglement spectrum. When coupled with the conformal field theory of open systems, DMRG appears to be an extremely powerful tool to characterize not only the phase diagram and the ground-state correlations of quantum one-dimensional systems, but also the excitation spectrum and the conformal structure along critical lines.

•Jan 1, 2017•3 citations

Scaling of entanglement support for matrix product states

Luca Tagliacozzo, Thiago Rodrigues de Oliveira, S. Iblisdir +1 more

The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size $\ensuremath{\chi}$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\ensuremath{\sim}\frac{1}{6}\text{log}\text{ }\ensuremath{\chi}$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\ensuremath{\xi}}_{\ensuremath{\chi}}$ ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-$\ensuremath{\chi}$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ${\ensuremath{\xi}}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\kappa}}$, with $\ensuremath{\kappa}=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\ensuremath{\kappa}\ensuremath{\sim}1.37$. We also show how finite-$\ensuremath{\chi}$ scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.

arXiv:0712.1976•Physical Review B•Jul 14, 2008•314 citations
In this paper we study the ground-state properties of a ladder Hamiltonian with chiral SU(2)\text{SU}(2)SU(2)-invariant spin interactions, a possible first step toward the construction of truly two-dimensional nontrivial systems with chiral properties starting from quasi-one-dimensional ones. Our analysis uses a recent implementation by us of SU(2)\text{SU}(2)SU(2) symmetry in tensor network algorithms, specifically for infinite density matrix renormalization group. After a preliminary analysis with Kadanoff coarse graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground-state properties match those of the universality class of a c=1c=1c=1 conformal field theory (CFT) in (1+1)(1+1)(1+1) dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground-state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-12\frac{1}{2}21​ Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an SU(2)1\text{SU}{(2)}_{1}SU(2)1​ Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct matrix product operators for SU(2)\text{SU}(2)SU(2)-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.
We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained from spin-2 bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies and entanglement entropy by exact diagonalization and density-matrix renormalization group methods. From the numerical results of the energy spectra, central charge, and scaling dimension we identify the conformal field theory describing the whole critical phase to be the SU(3)_{1} Wess-Zumino-Witten model. We find that, while the Hamiltonian is only SU(2) invariant, in this critical phase there is an emergent SU(3) symmetry in the thermodynamic limit.
In this dissertation, the momentum-space entanglement properties of several quantum spin chains are investigated. In the second chapter we re- view important results on many-body entanglement. In the third chapter, we numerically study the momentum-space entanglement spectra of bosonic and fermionic formulations of the spin-1/2 XXZ chain [Phys. Rev. Lett. 113, 256404 (2014)]. We investigate the behavior of the entanglement gaps, present in both formulations, across quantum phase transitions in the XXZ chain. In both cases, finite size scaling suggests that the entanglement gap closure does not occur at the physical transition points. For bosons, we find that the entan- glement gap depends on the scaling dimension of the conformal field theory as varied by the X X Z anisotropy. For fermions, the infinite entanglement gap present at the XX point persists well past the phase transition at the Heisenberg point. We elaborate on how this may support the numerical study of phase transitions. In the third chapter, we advocate that in certain critical spin chains a gap in the momentum-space entanglement spectrum separates the universal part of the spectrum, which is determined by the associated conformal field theory, from the non-universal part, which is specific to the model [arXiv:1512.09030]. We provide affirmative evidence from multicritical spin-1 chains with low energy sectors described by the SU(2)2 or the SU(3)1 Wess-Zumino-Witten model. In chapter four, we study momentum- space en- tanglement in quantum spin-half ladders [Phys. Rev. B 93, 125107 (2016)]. When the system is gapped, we analytically find the momentum-space entan- glement Hamiltonian is described by a chiral conformal field theory with a central charge of two. When the system is gapless, the entanglement Hamil- tonian consists of one gapless mode that is linear in subsystem momentum and one mode with a flat dispersion relation. We also analytically include the effect of a certain irrelevant terms (in the renormalization group sense) on the entanglement spectrum. In chapter six, we investigate the momentum- space entanglement spectrum after a quantum quench [arXiv:1603.01997]. We show that the momentum-space entanglement spectrum of the XXZ spin- half chain possesses many universal features both in equilibrium and after a quantum quench.
Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. Given a pure state of the system and a division into regions AAA and BBB, they can be obtained in terms of the Schmidt valuesSchmidt~ valuesSchmidt values, or eigenvalues λαλ_αλα​ of the reduced density matrix ρAρ_AρA​ for region AAA. In this paper we draw attention instead to the Schmidt vectorsSchmidt~ vectorsSchmidt vectors, or eigenvectors ∣vα⟩|v_α\rangle∣vα​⟩ of ρAρ_AρA​. We consider the ground state of critical quantum spin chains whose low energy/long distance physics is described by an emergent conformal field theory (CFT). We show that the Schmidt vectors ∣vα⟩|v_α\rangle∣vα​⟩ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT (a chiral version of the original CFT). Indeed, we build weighted sums HnH_nHn​ of the lattice Hamiltonian density hj,j+1h_{j,j+1}hj,j+1​ over region AAA and show that the matrix elements ⟨vαHn∣vα′⟩\langle v_αH_n |v_{α'}\rangle⟨vα​Hn​∣vα′​⟩ are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for HnCFT=12(Ln+L−n)H_n^{\tiny \text{CFT}} = \frac 1 2 (L_n + L_{-n})HnCFT​=21​(Ln​+L−n​) in the boundary CFT, where LnL_nLn​'s are (one copy of) the Virasoro generators. We numerically confirm our results using the critical Ising quantum spin chain and other (free-fermion equivalent) models.
We develop an efficient method to perform density matrix renormalization group simulations of the SU(NNN) Heisenberg chain with open boundary conditions taking full advantage of the SU(NNN) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e., one particle per site in the fermionic formulation), we have benchmarked our results for the ground-state energy up to N=8N=8N=8 and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground-state energy is excellent for SU(3) (12 digits). It decreases with NNN, but it is still satisfactory for N=8N=8N=8 (six digits). Central charges ccc are also extracted from the entanglement entropy using the Calabrese-Cardy formula and agree with the theoretical values expected from the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ Wess-Zumino-Witten conformal field theories.
In this thesis we have studied the emergence of spontaneously dimerized phases in frustrated spin-S chains, with emphasis on the nature of the critical lines between the dimerized and non-dimerized phases. The main numerical method used in this thesis is the Density Matrix Renormalization Group (DMRG). The DMRG algorithm is a relatively old and well established method for the investigation of the ground-state. In this thesis, we show how to use this algorithm to calculate the excitation spectra of one-dimensional critical systems, known in the context of conformal field theory as conformal towers of states. We have demonstrated that the method works very well for two simple minimal models (the transverse-field Ising model and the three-state Potts model), and we have used it systematically to identify the universality classes and the underlying conformal field theories of various one-dimensional spin systems. It has been known for a long time that the transition to a spontaneously dimerized phase in a spin-1 chain can be either continuous, in the Wess-Zumino-Witten (WZW) SU(2) level 2 universality class, or first order. By combining a careful numerical investigation with a conformal field theory analysis, we were able to detect in a frustrated spin-1 chain with competing next-nearest-neighbor and three-site interactions the presence of yet another type of continuous phase transition that belongs to the Ising universality class. In contrast to the WZW SU(2) level 2 critical line, at which the singlet-triplet gap closes, the Ising transition occurs entirely in the singlet sector, while the singlet-triplet gap remains open. The use of the standard DMRG approach, along the lines mentioned above, has allowed us to provide explicit numerical evidence for the presence of a conformal tower of singlets inside the spin gap. Moreover, according to field theory, a WZW SU(2) level k critical line can turn into a first order transition due to the presence of a marginal operator in the WZW SU(2) level k model. A careful investigation of the conformal towers along the critical lines has allowed us to find the precise location of this point in both S=1 and S=3/2 chains. We have also shown that the nature of the continuous dimerization transitions is related to the topological properties of the corresponding phases, and that the phase diagrams of various frustrated spin chains can be effectively extracted by looking at the local topological order parameter - the degeneracy of the lowest state in the entanglement spectrum. When coupled with the conformal field theory of open systems, DMRG appears to be an extremely powerful tool to characterize not only the phase diagram and the ground-state correlations of quantum one-dimensional systems, but also the excitation spectrum and the conformal structure along critical lines.
The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size $\ensuremath{\chi}$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\ensuremath{\sim}\frac{1}{6}\text{log}\text{ }\ensuremath{\chi}$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\ensuremath{\xi}}_{\ensuremath{\chi}}$ ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-$\ensuremath{\chi}$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ${\ensuremath{\xi}}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\kappa}}$, with $\ensuremath{\kappa}=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\ensuremath{\kappa}\ensuremath{\sim}1.37$. We also show how finite-$\ensuremath{\chi}$ scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.