arXiv:1804.08974

Exact density matrix for quantum group invariant sector of XXZ model

Fedor Aleksandrovich Smirnov

Published Apr 24, 2018• 1 citation

Abstract

Using the fermionic basis we obtain the expectation values of all $U_q(\slt)$-invariant local operators on 8 sites for the anisotropic six-vertex model on a cylinder with generic Matsubara data. In the case when the $U_q(\slt)$ symmetry is not broken this computation is equivalent to finding the entire density matrix up to 8 sites. As application, we compute the entanglement entropy without and with temperature, and compare the results with CFT predictions.

DOI: 10.48550/arxiv.1804.08974

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Density matrix elements and entanglement entropy for the spin-1/2 XXZ chain at Δ = 1/2

Jun Sato, Masahiro Shiroishi

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

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Hidden Grassmann structure in the XXZ model III: introducing the Matsubara direction

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Fourth-neighbour two-point functions of the XXZ chain and the fermionic basis approach

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Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

Nick G. Jones

Abstract The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix $$\rho $$ ρ . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given byNmassless Dirac fermions. For the density matrix, $$\rho _A$$ ρA , of subsystems ofLneighbouring sites we calculate the leading terms in the largeLasymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a largeLexpansion of the charged moments of $$\rho _A$$ ρA ; we derive $$\mathrm {tr}(\mathrm {e}^{\mathrm {i}\alpha Q_A} \rho _A^n)~=~a \mathrm {e}^{\mathrm {i}\alpha \langle Q_A\rangle } (\sigma L)^{-x}(1+O(L^{-\mu }))$$ tr(eiαQAρAn)=aeiα⟨QA⟩(σL)-x(1+O(L-μ)) , wherea, xand $$\mu $$ μ are universal and $$\sigma $$ σ depends only on theNFermi momenta. We show that the exponentxcorresponds to the expectation from CFT analysis. The error term $$O(L^{-\mu })$$ O(L-μ) is consistent with but weaker than the field theory prediction $$O(L^{-2\mu })$$ O(L-2μ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.

arXiv:2202.11728Journal of Statistical PhysicsJul 8, 202224 citations

arXiv:1804.08974

Exact density matrix for quantum group invariant sector of XXZ model

Fedor Aleksandrovich Smirnov

Published Apr 24, 2018• 1 citation

arXiv HTML PDF

Abstract

Using the fermionic basis we obtain the expectation values of all

U_q(\slt)

-invariant local operators on 8 sites for the anisotropic six-vertex model on a cylinder with generic Matsubara data. In the case when the

U_q(\slt)

symmetry is not broken this computation is equivalent to finding the entire density matrix up to 8 sites. As application, we compute the entanglement entropy without and with temperature, and compare the results with CFT predictions.

DOI: 10.48550/arxiv.1804.08974

Related papers

Density matrix elements and entanglement entropy for the spin-1/2 XXZ chain at Δ = 1/2

Jun Sato, Masahiro Shiroishi

We have analytically obtained all the density matrix elements up to six lattice sites for the spin-1/2 Heisenberg XXZ chain at Δ = 1/2. We use the multiple integral formula of the correlation function for the massless XXZ chain derived by Jimbo and Miwa. As for the spin–spin correlation functions, we have newly obtained the fourth- and fifth-neighbour transverse correlation functions. We have calculated all the eigenvalues of the density matrix and analyse the eigenvalue distribution. Using these results the exact values of the entanglement entropy for the reduced density matrix up to six lattice sites have been obtained. We observe that our exact results agree quite well with the asymptotic formula predicted by the conformal field theory.We have analytically obtained all the density matrix elements up to six lattice sites for the spin-1/2 Heisenberg XXZ chain at Δ = 1/2. We use the multiple integral formula of the correlation function for the massless XXZ chain derived by Jimbo and Miwa. As for the spin–spin correlation functions, we have newly obtained the fourth- and fifth-neighbour transverse correlation functions. We have calculated all the eigenvalues of the density matrix and analyse the eigenvalue distribution. Using these results the exact values of the entanglement entropy for the reduced density matrix up to six lattice sites have been obtained. We observe that our exact results agree quite well with the asymptotic formula predicted by the conformal field theory.

arXiv:0704.0850Journal of Physics A Mathematical and TheoreticalJul 12, 200726 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.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:1802.05482Physical review. B./Physical review. BApr 20, 201836 citations

Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States

Hermann Boos

We generalize the results of [Comm.Math.Phys.299 (2010), 825-866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the sixvertex model acting in the so-called Matsubara direction.We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particlehole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand.As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators.In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987).Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations.On the side of the lattice model we use the excited-state TBA approach.We consider in detail the case of the descendant at level 8.We generalize the results of [Comm.Math.Phys.299 (2010), 825-866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the sixvertex model acting in the so-called Matsubara direction.We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particlehole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand.As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators.In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987).Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations.On the side of the lattice model we use the excited-state TBA approach.We consider in detail the case of the descendant at level 8.

arXiv:1010.0858Symmetry Integrability and Geometry Methods and ApplicationsJan 13, 201114 citations

Hidden Grassmann structure in the XXZ model III: introducing the Matsubara direction

M Jimbo, T Miwa, Fedor Aleksandrovich Smirnov

We address the problem of computing temperature correlation functions of the XXZ chain, within the approach developed in our previous works. In this paper we calculate the expected values of a fermionic basis of quasi-local operators, in the infinite volume limit while keeping the Matsubara (or Trotter) direction finite. The result is expressed in terms of two basic quantities: a ratio ρ(ζ) of transfer matrix eigenvalues and a nearest neighbour correlator ω(ζ, ξ). We explain that the latter is interpreted as the canonical second kind differential in the theory of deformed Abelian integrals.We address the problem of computing temperature correlation functions of the XXZ chain, within the approach developed in our previous works. In this paper we calculate the expected values of a fermionic basis of quasi-local operators, in the infinite volume limit while keeping the Matsubara (or Trotter) direction finite. The result is expressed in terms of two basic quantities: a ratio ρ(ζ) of transfer matrix eigenvalues and a nearest neighbour correlator ω(ζ, ξ). We explain that the latter is interpreted as the canonical second kind differential in the theory of deformed Abelian integrals.

arXiv:0811.0439Journal of Physics A Mathematical and TheoreticalJul 14, 200979 citations

Fourth-neighbour two-point functions of the XXZ chain and the fermionic basis approach

Frank Göhmann, Raphael Kleinemühl, Alexander Weiße

Abstract We give a descriptive review of the fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions ρ and ω and their derivatives with coefficients that are rational in the deformation parameter q of the model. These coefficients are of ‘algebraic origin’. They do not depend on the choice of the density matrix, which only impacts the form of ρ and ω . As an example we work out in detail the case of the grand canonical ensemble at temperature T and magnetic field h for q in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for h , T = 0 and for finite h and T .Abstract We give a descriptive review of the fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions ρ and ω and their derivatives with coefficients that are rational in the deformation parameter q of the model. These coefficients are of ‘algebraic origin’. They do not depend on the choice of the density matrix, which only impacts the form of ρ and ω . As an example we work out in detail the case of the grand canonical ensemble at temperature T and magnetic field h for q in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for h , T = 0 and for finite h and T .

arXiv:2104.10071Journal of Physics A Mathematical and TheoreticalAug 23, 20214 citations

Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

Nick G. Jones

Abstract The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix $\rho$ ρ . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given byNmassless Dirac fermions. For the density matrix, $\rho _A$ ρA , of subsystems ofLneighbouring sites we calculate the leading terms in the largeLasymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a largeLexpansion of the charged moments of $\rho _A$ ρA ; we derive $\mathrm {tr}(\mathrm {e}^{\mathrm {i}\alpha Q_A} \rho _A^n)~=~a \mathrm {e}^{\mathrm {i}\alpha \langle Q_A\rangle } (\sigma L)^{-x}(1+O(L^{-\mu }))$ tr(eiαQAρAn)=aeiα⟨QA⟩(σL)-x(1+O(L-μ)) , wherea, xand $\mu$ μ are universal and $\sigma$ σ depends only on theNFermi momenta. We show that the exponentxcorresponds to the expectation from CFT analysis. The error term $O(L^{-\mu })$ O(L-μ) is consistent with but weaker than the field theory prediction $O(L^{-2\mu })$ O(L-2μ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.Abstract The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix $$\rho $$ ρ . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given byNmassless Dirac fermions. For the density matrix, $$\rho _A$$ ρA , of subsystems ofLneighbouring sites we calculate the leading terms in the largeLasymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a largeLexpansion of the charged moments of $$\rho _A$$ ρA ; we derive $$\mathrm {tr}(\mathrm {e}^{\mathrm {i}\alpha Q_A} \rho _A^n)~=~a \mathrm {e}^{\mathrm {i}\alpha \langle Q_A\rangle } (\sigma L)^{-x}(1+O(L^{-\mu }))$$ tr(eiαQAρAn)=aeiα⟨QA⟩(σL)-x(1+O(L-μ)) , wherea, xand $$\mu $$ μ are universal and $$\sigma $$ σ depends only on theNFermi momenta. We show that the exponentxcorresponds to the expectation from CFT analysis. The error term $$O(L^{-\mu })$$ O(L-μ) is consistent with but weaker than the field theory prediction $$O(L^{-2\mu })$$ O(L-2μ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.

arXiv:2202.11728Journal of Statistical PhysicsJul 8, 202224 citations

Using the fermionic basis we obtain the expectation values of all

U_q(\slt)

-invariant local operators on 8 sites for the anisotropic six-vertex model on a cylinder with generic Matsubara data. In the case when the

U_q(\slt)

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.

Abstract The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix

\rho

ρ . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given byNmassless Dirac fermions. For the density matrix,

\rho _A

ρA , of subsystems ofLneighbouring sites we calculate the leading terms in the largeLasymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a largeLexpansion of the charged moments of

\rho _A

ρA ; we derive

\mathrm {tr}(\mathrm {e}^{\mathrm {i}\alpha Q_A} \rho _A^n)~=~a \mathrm {e}^{\mathrm {i}\alpha \langle Q_A\rangle } (\sigma L)^{-x}(1+O(L^{-\mu }))

tr(eiαQAρAn)=aeiα⟨QA⟩(σL)-x(1+O(L-μ)) , wherea, xand

\mu

μ are universal and

\sigma

σ depends only on theNFermi momenta. We show that the exponentxcorresponds to the expectation from CFT analysis. The error term

O(L^{-\mu })

O(L-μ) is consistent with but weaker than the field theory prediction

O(L^{-2\mu })

O(L-2μ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.