Skeleton of matrix-product-state-solvable models connecting topological phases of matter

Jones, Nicholas Blurton; Bibo, Julian; Jobst, B.; Pollmann, Frank; Smith, Adam; Verresen, Ruben

doi:10.1103/physrevresearch.3.033265

Cited by 21 publications

(52 citation statements)

References 77 publications

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“…In the center of the phase diagram, we observe a multicritical point with dynamical critical exponent z = 2; this can be perturbed into two topologically distinct symmetry-enriched versions of the Ising criticality [39]. The dotted line is an exactly-solvable frustration-free line which tunes through the multicritical point [36][37][38]. It is instructive to compare this 1D phase diagram to a 2D analogue in Fig.…”

Section: Adding the Pivot: Bec Multicriticalitymentioning

confidence: 92%

“…On the other hand, it is clear that for large h → ∞, we have a gapped phase with two degenerate ground states with a four-site unit cell of the pattern 0011 or 1100. These two regimes are separated by a multicritical point with dynamical critical exponent z = 2 [36][37][38][40][41][42][43][44][45], denoted by a red point in Fig. 1.…”

Section: Adding the Pivot: Bec Multicriticalitymentioning

confidence: 99%

“…5 and which are connected at a multicritical point, constitute a two-dimensional generalization of the concept of the 'tensor network skeleton' introduced recently in Ref. [38] (which studied it in the onedimensional setting with matrix product states).…”

Section: B Direct Interpolation Between Spt Phases: U (1) Pivot Super...mentioning

confidence: 99%

“…FIG.1. Phase diagram of the 1D cluster SPT transition perturbed by the Ising pivot: Hamiltonian (14)[36][37][38]. The central vertical axis has an explicit U (1) symmetry generated by Hpivot, which rotates the trivial and SPT phases into one another.…”

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confidence: 99%

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Pivot Hamiltonians as generators of symmetry and entanglement

Tantivasadakarn,

Thorngren,

Vishwanath

et al. 2021

Preprint

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It is well-known that symmetry-protected topological (SPT) phases can be obtained from the trivial phase by an entangler, a finite-depth unitary operator U . Here, we consider obtaining the entangler from a local 'pivot' Hamiltonian Hpivot such that U = e iπH pivot . This perspective of Hamiltonians pivoting between the trivial and SPT phase opens up two new directions which we explore here. (i) Since SPT Hamiltonians and entanglers are now on the same footing, can we iterate this process to create other interesting states? (ii) Since entanglers are known to arise as discrete symmetries at SPT transitions, under what conditions can this be enhanced to U (1) 'pivot' symmetry generated by Hpivot? In this work we explore both of these questions. With regard to the first, we give examples of a rich web of dualities obtained by iteratively using an SPT model as a pivot to generate the next one. For the second question, we derive a simple criterion guaranteeing that the direct interpolation between the trivial and SPT Hamiltonian has a U (1) pivot symmetry. We illustrate this in a variety of examples, assuming various forms for Hpivot, including the Ising chain, and the toric code Hamiltonian. A remarkable property of such a U (1) pivot symmetry is that it shares a mutual anomaly with the symmetry protecting the nearby SPT phase. We discuss how such anomalous and non-onsite U (1) symmetries explain the exotic phase diagrams that can appear, including an SPT multicritical point where the gapless ground state is given by the fixed-point toric code state. CONTENTSHere we have included the prefactor and sign alternation 1 for 1 Note that the alternating sign (−1) n can be eliminated by conjugating with n X 4n X 4n+1 .

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Section: Adding the Pivot: Bec Multicriticalitymentioning

confidence: 92%

Section: Adding the Pivot: Bec Multicriticalitymentioning

confidence: 99%

Section: B Direct Interpolation Between Spt Phases: U (1) Pivot Super...mentioning

confidence: 99%

mentioning

confidence: 99%

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Pivot Hamiltonians as generators of symmetry and entanglement

Tantivasadakarn,

Thorngren,

Vishwanath

et al. 2021

Preprint

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“…These zeros are 'removable'-for further discussion see Refs. [41,42,49]. 2 In the case N = 1 we will also use the notation kF for the Fermi momentum.…”

Section: 1mentioning

confidence: 99%

Symmetry-resolved entanglement entropy in critical free-fermion chains

Jones

2022

Preprint

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The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix 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 by N massless Dirac fermions. For the density matrix, ρA, of subsystems of L neighbouring sites we calculate the leading terms in the large L asymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a large L expansion of the charged moments of ρA; we derive tr(e, where a, x and µ are universal and σ depends only on the N Fermi momenta. We show that the exponent x corresponds to the expectation from CFT analysis. The error term O(L −µ ) is consistent with but weaker than the field theory prediction O(L −2µ ). However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.

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Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

Jones

2022

J Stat Phys

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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 by N massless Dirac fermions. For the density matrix, $$\rho _A$$ ρ A , of subsystems of L neighbouring sites we calculate the leading terms in the large L asymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a large L expansion of the charged moments of $$\rho _A$$ ρ A ; 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 ( e i α Q A ρ A n ) = a e i α ⟨ Q A ⟩ ( σ L ) - x ( 1 + O ( L - μ ) ) , where a, x and $$\mu $$ μ are universal and $$\sigma $$ σ depends only on the N Fermi momenta. We show that the exponent x corresponds 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 - 2 μ ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.

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Skeleton of matrix-product-state-solvable models connecting topological phases of matter

Cited by 21 publications

References 77 publications

Pivot Hamiltonians as generators of symmetry and entanglement

Pivot Hamiltonians as generators of symmetry and entanglement

Symmetry-resolved entanglement entropy in critical free-fermion chains

Symmetry-Resolved Entanglement Entropy in Critical Free-Fermion Chains

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