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2002 Spectrum of a duality–twisted Ising quantum chain

Uwe Grimm

Published Nov 19, 2001• 14 citations
OpenAlex

Abstract

Abstract. The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function. PACS numbers: 05.50.+q, 75.10.Jm, 71.10.Fd, 11.25.Hf Recently, there has been a lot of interest in generalised twisted partition functions of conformal field theory [1, 2, 3, 4] and the corresponding conformal twisted boundary conditions in solvable lattice models of statistical mechanics [5]. Examples of such boundary conditions leading to the appearance of “exotic ” spinor operators in the

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The Ising model in two dimensions with special toroidal boundary conditions is analyzed. These boundary conditions, which we call duality-twisted boundary conditions, may be interpreted as inserting a specific defect line ("seam") in the system, along noncontractible circles of the cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of a transfer matrix for the critical ferromagnetic Ising model on the M×N square lattice wrapped on the torus with a specific defect line. As a result we have obtained analytically the partition function for the Ising model with such boundary conditions. In the case of infinitely long cylinders of circumference L with duality-twisted boundary conditions we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find that the ratio of subdominant finite-size correction terms in the asymptotic expansion of the free energy and the inverse correlation lengths should be universal. We verify such universal behavior in the framework of a perturbating conformal approach by calculating the universal structure constant C_{n1n} for descendent states generated by the operator product expansion of the primary fields. For such states the calculations of an universal structure constants is a difficult task, since it involves knowledge of the four-point correlation function, which in general is not fixed by conformal invariance except for some particular cases, including the Ising model.

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The spectrum of the quantum Ising chain is studied in the finite-size scaling limit for periodic and antiperiodic boundary conditions. The finite-size corrections are computed for all energy gaps. The results of the model are used to test the Privman-Fisher universality hypothesis and conformal theory. At the critical point, it is shown how to build the spectrum from the irreducible representations of the Virasoro algebra. In addition, the author finds evidence for an unsuspected symmetry for systems both non-critical and of finite size.

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Topological conformal defects with tensor networks

Markus Hauru, Glen Evenbly, Wen Wei Ho +2 more

The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the ${\mathbb{Z}}_{2}$ symmetry defect ${D}_{\ensuremath{\epsilon}}$ and the Kramers-Wannier duality defect ${D}_{\ensuremath{\sigma}}$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function ${Z}_{D}$ of the critical Ising model in the presence of a topological conformal defect $D$ is expressed in terms of the scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$ and conformal spins ${s}_{\ensuremath{\alpha}}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix ${M}_{D}$ for the partition function ${Z}_{D}$. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ${Z}_{{D}_{\ensuremath{\epsilon}}}$ and ${Z}_{{D}_{\ensuremath{\sigma}}}$ of the critical Ising model with either a symmetry defect ${D}_{\ensuremath{\epsilon}}$ or a duality defect ${D}_{\ensuremath{\sigma}}$. We also explain how to coarse grain the corresponding transfer matrices ${M}_{{D}_{\ensuremath{\epsilon}}}$ and ${M}_{{D}_{\ensuremath{\sigma}}}$, from which we can extract accurate numerical estimates of ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\epsilon}}}$ and ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\sigma}}}$. Two key ingredients of our approach are (i) coarse graining of the defect $D$, which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins ${s}_{\ensuremath{\alpha}}$.

arXiv:1512.03846•Physical review. B./Physical review. B•Sep 9, 2016•70 citations
Abstract. The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function. PACS numbers: 05.50.+q, 75.10.Jm, 71.10.Fd, 11.25.Hf Recently, there has been a lot of interest in generalised twisted partition functions of conformal field theory [1, 2, 3, 4] and the corresponding conformal twisted boundary conditions in solvable lattice models of statistical mechanics [5]. Examples of such boundary conditions leading to the appearance of “exotic ” spinor operators in the
The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalized twisted partition function.
The author discusses a new class of toroidal boundary conditions for one-dimensional quantum Hamiltonians with S n symmetry which are related to two-dimensional n-state Potts models in the extreme anisotropic Hamiltonian limit. At their self-dual point (a point where a second-order phase transition occurs for n=2, 3, 4) the duality transformation is shown to be an additional symmetry giving rise to a new class of 'duality twisted' toroidal boundary conditions. The corresponding Hamiltonians are given in terms of generators of the periodic Temperley-Lieb algebra with an odd number of generators. The author discusses as an example the critical Ising model. Here the complete spectrum for the new boundary conditions can be obtained from a projection mechanism in the spin-1/2 XXZ Heisenberg chain.
The Ising model in two dimensions with special toroidal boundary conditions is analyzed. These boundary conditions, which we call duality-twisted boundary conditions, may be interpreted as inserting a specific defect line ("seam") in the system, along noncontractible circles of the cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of a transfer matrix for the critical ferromagnetic Ising model on the M×N square lattice wrapped on the torus with a specific defect line. As a result we have obtained analytically the partition function for the Ising model with such boundary conditions. In the case of infinitely long cylinders of circumference L with duality-twisted boundary conditions we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find that the ratio of subdominant finite-size correction terms in the asymptotic expansion of the free energy and the inverse correlation lengths should be universal. We verify such universal behavior in the framework of a perturbating conformal approach by calculating the universal structure constant C_{n1n} for descendent states generated by the operator product expansion of the primary fields. For such states the calculations of an universal structure constants is a difficult task, since it involves knowledge of the four-point correlation function, which in general is not fixed by conformal invariance except for some particular cases, including the Ising model.
The spectrum of the quantum Ising chain is studied in the finite-size scaling limit for periodic and antiperiodic boundary conditions. The finite-size corrections are computed for all energy gaps. The results of the model are used to test the Privman-Fisher universality hypothesis and conformal theory. At the critical point, it is shown how to build the spectrum from the irreducible representations of the Virasoro algebra. In addition, the author finds evidence for an unsuspected symmetry for systems both non-critical and of finite size.
Universal properties of a critical quantum spin chain are encoded in the underlying conformal field theory (CFT). This underlying CFT is fully characterized by its conformal data. We propose a method to extract the conformal data from a critical quantum spin chain with both periodic and antiperiodic boundary conditions (PBC and APBC) based on low-energy eigenstates, generalizing previous work on spin chains with only PBC. First, scaling dimensions and conformal spins are extracted from the energies and momenta of the eigenstates. Second, the Koo-Saleur formula of lattice Virasoro generators is generalized to APBC and used to identify conformal towers. Third, local and string operators on the lattice are identified with CFT operators with PBC and APBC, respectively. Finally, operator product expansion coefficients are extracted by computing matrix elements of lattice primary operators in the low-energy subspaces with PBC and APBC. To go beyond exact diagonalization, tensor network methods based on periodic uniform matrix product states are used. We illustrate our approach with critical and tricritical Ising quantum spin chains. In the latter case, we propose lattice operators that correspond to super-Virasoro generators and verify their action on low-energy eigenstates. In this way, we explore the emergence of superconformal symmetry in the quantum spin chain.
The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the Z2{\mathbb{Z}}_{2}Z2​ symmetry defect ${D}_{\ensuremath{\epsilon}}$ and the Kramers-Wannier duality defect ${D}_{\ensuremath{\sigma}}$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function ZD{Z}_{D}ZD​ of the critical Ising model in the presence of a topological conformal defect DDD is expressed in terms of the scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$ and conformal spins ${s}_{\ensuremath{\alpha}}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix MD{M}_{D}MD​ for the partition function ZD{Z}_{D}ZD​. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ${Z}_{{D}_{\ensuremath{\epsilon}}}$ and ${Z}_{{D}_{\ensuremath{\sigma}}}$ of the critical Ising model with either a symmetry defect ${D}_{\ensuremath{\epsilon}}$ or a duality defect ${D}_{\ensuremath{\sigma}}$. We also explain how to coarse grain the corresponding transfer matrices ${M}_{{D}_{\ensuremath{\epsilon}}}$ and ${M}_{{D}_{\ensuremath{\sigma}}}$, from which we can extract accurate numerical estimates of ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\epsilon}}}$ and ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\sigma}}}$. Two key ingredients of our approach are (i) coarse graining of the defect DDD, which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins ${s}_{\ensuremath{\alpha}}$.