Gap Dependency on Half Spaces in Product Vacua and Boundary State Models Michael Bishop University of California at Davis, Department of Mathematics February 15, 2016 Joint work with Bruno Nachtergaele and Amanda Young Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 1 / 19
Introduction Outline Goal: To discuss the existence and non-existence of a spectral gap family of quantum spin Hamiltonians on half-spaces of d-dimensional lattices. Outline: 1. Define PVBS model and preliminaries 2. Theorem statement 3. Sketch of gapless cases 4. Sketch of gapped cases This work follows from ‘Product Vacua and Boundary State Models in d-Dimensions,” Bachmann, Hamza, Nachtergaele, Young, Journal of Statistical Physics, 2015. Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 2 / 19
Introduction Motivation In cold temperature regimes, the existence of a ‘Spectral Gap’ between the ground state energy and the rest of the spectrum of the Hamiltonian operator affects for the behavior of the physical system: Stability Dynamics Technical bounds in many proofs. A transition between ‘gapped’ and ‘gapless’ due to a change in model parameters is often referred to as a ‘quantum phase transition.’ Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 3 / 19
Introduction Undecidability Recently, Cubitt, Perez-Garcia, and Wolf’s “Undecidability of the Spectral Gap 1 ” presents a family of translation-invariant Hamiltonians with nearest-neighbor interactions where the existence of a spectral gap of an infinite-volume Hamiltonian cannot be determined by from finite-volume Hamiltonians. In an interview, Cubitt said, ”It’s possible for particular cases of a problem to be solvable even when the general problem is undecidable...” This talk presents a family of quantum spin systems where the existence and non-existence of a spectral gap is determined by the finite-volume Hamiltonians. i.e. “Not Always” is not equivalent to “Always Not.” 1 Nature has summary or result, full mathematical result at ArXiv 1502:0457v2 . Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 4 / 19
PVBS Single Species Single Species PVBS Model For each x in finite Λ ⊂ Z d , H x := C 2 with basis | 0 � x , | 1 � x . H Λ := ⊗ x ∈ Λ H x For each coordinate j = 1 , . . . , d , let λ j ∈ ( 0 , ∞ ) . For each (oriented) edge connecting x ∈ Λ to x + e j ∈ Λ , we define the operators h x , x + e j := ( | 01 � − λ j | 10 � ) ( � 01 | − λ j � 10 | ) + | 11 �� 11 | 1 + λ 2 j where | a x a x + e j � = | a x � x ⊗ | a x + e j � x + e j , a = 0 , 1 The Hamiltonian is defined as d H Λ = � � h x , x + e j j = 1 x ∈ Λ , x + e j ∈ Λ Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 5 / 19
PVBS Single Species Ground State Space This Hamiltonian in non-negative definite. On bounded domains, the ground state space is two dimensional: Ψ Λ 0 = | 0 � Λ 1 Ψ Λ � λ x | 1 � Λ 1 = x � C (Λ) x ∈ Λ where | 0 � Λ := ⊗ x ∈ Λ | 0 � x , | 1 � Λ x = ( ⊗ y ∈ Λ , y � = x | 0 � y ) ⊗ | 1 � x , d λ x = � x j � � log ( λ j ) · x j , C (Λ) = λ 2 x λ j = e j = 1 x ∈ Λ Frustration-Free: These ground states are also ground states of the local Hamiltonians h x , x + e j . Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 6 / 19
PVBS Single Species Infinite Domains The infinite domains we focus on are half-spaces determined by normalized vectors m , D := { x ∈ Z d : m · x ≥ 0 } The Hilbert space and Hamiltonian are defined using the Gelfand- Naimark- Segal (GNS) construction with the vacuum state. Λ N ր D � Ψ Λ N 0 , · Ψ Λ N ω D 0 ( · ) = 0 � lim on quasi local observables A D := A loc . Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 7 / 19
PVBS Single Species Infinite Domains From this state, the GNS construction defines a Hilbert Space H D 0 , a representation π D 0 mapping the quasi-local algebra to the bounded operators on H D 0 , and a cyclic vector Ω D 0 . For spectral properties, it is sufficient to consider H D as the limit of the representations of finite lattice Hamiltonians H Λ . For local observables A ∈ A D loc , 0 ( H Λ A )Ω D H D π D 0 ( A )Ω D Λ ր D π D 0 = lim 0 Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 8 / 19
PVBS Single Species Gap Definitions Definition We say the GNS Hamiltonian H D is gapped if there exists δ > 0 such that the spectrum of H D is empty between 0 and δ spec ( H D ) ∩ ( 0 , δ ) = ∅ (1) If there does exist such a δ , then the spectral gap of H D is defined by γ ( H D ) := sup { δ > 0 : spec ( H D ) ∩ ( 0 , δ ) = ∅} (2) If there does not exist such a δ , we say the GNS Hamiltonian H D is gapless . For simplicity of notation, we will rewrite γ ( H Λ ) and γ ( H D ) as γ (Λ) and γ ( D ) , respectively. Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 9 / 19
Theorem Theorem Theorem Given a half-space D with inward normal vector m and any λ 1 , . . . , λ d ∈ ( 0 , ∞ ) , define log � λ := ( log λ 1 , . . . , log λ d ) : Existence of Spectral Gap : If log � λ � = −� log � λ � m, then γ ( H D ) > 0 . Non-existence of Spectral Gap : If log � λ = −� log � λ � m, then γ ( H D ) = 0 . Essentially, the gap is determined by whether log � λ is an outward normal to D . The proofs generalize to Z d : Corollary If λ 1 = · · · = λ d = 1 , then H Z d is gapless. Otherwise, H Z d is gapped. Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 10 / 19
Gapless Phases on Half-Spaces Angle Dependence of Spectral Gap In fact, the gap closes at a rate at least linear in the angle between the outward normal − m and log � λ : Theorem (Upper Bound) For all d ≥ 2 , λ 1 , . . . , λ d ∈ ( 0 , ∞ ) , and unit vectors m ∈ R d such that m · log � λ < 0 , the spectral gap has the following upper bound: γ ( H D ) ≤ C � log � λ �| sin ( θ ) | , where b θ is the angle between the vectors − m and log � λ . In particular, the gap vanishes if θ = 0 . Proof is straight-forward: calculate the energy of finite volume one-particle ground state in infinite volume. Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 11 / 19
Gapped Cases Sketch of Gapped Cases λ � m , then H D is gapped. When log � λ � = −� log � Proof Approach: Choose a sequence of finite volumes Λ N with converge to D . Apply the martingale method to the finite volumes Λ N to find a uniform in N lower bound on γ ( H Λ N ) . Apply a limit theorem to bound the gap for the GNS Hamiltonian: γ ( H D ) ≥ lim sup γ ( H Λ N ) Λ N ր D The difficulty is finding appropriate Λ L such that the martingale method generates a uniform lower bound. Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 12 / 19
Gapped Cases Martingale Method Theorem (Martingale Method, Nachtergaele ‘96) For a finite sequence of volumes Λ n which increase to Λ N , if there exists ℓ ≥ 2 such that following three conditions hold for the local Hamiltonians: Condition (1) For some positive constant d ℓ , N H Λ n \ Λ n − ℓ ≤ d ℓ H Λ N � n = ℓ Condition (2) For some positive constant γ ℓ , H Λ n \ Λ n − ℓ ≥ γ ℓ ( I − G Λ n \ Λ n − ℓ ) Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 13 / 19
Gapped Cases Martingale Method Continued Theorem (Martingale Method Continued) 1 Condition (3) There exists a constant ǫ ℓ < √ ℓ and n ℓ such that n ℓ ≤ n ≤ N − 1 � G Λ n + 1 \ Λ n + 1 − ℓ E n � ≤ ǫ ℓ where E n = G Λ n − G Λ n + 1 , the projection onto G Λ n ∩ G ⊥ Λ n + 1 . Then for any Ψ such that G Λ N Ψ = 0 , √ � Ψ , H Λ N Ψ � ≥ γ ℓ ℓ ) 2 � Ψ � 2 ( 1 − ǫ ℓ d ℓ Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 14 / 19
Gapped Cases Martingale Method Applied A example volume in two dimensions in the parallelogram Λ L = { x ∈ Z 2 : 0 ≤ m · x < L , − L ≤ x 2 < L } m 1 We apply the martingale method for each coordinate direction: Λ L Λ ( 2 ) n Λ ( 2 ) n \ Λ ( 2 ) n − ℓ 2 Λ ( 2 ) Λ ( 1 ) n \ Λ ( 1 ) n − ℓ 2 n − ℓ 1 Λ ( 1 ) Λ ( 2 ) b 2 \ Λ ( 2 ) Λ ( 1 ) Λ ( 1 ) n − ℓ 1 b 2 − ℓ 2 0 n Λ ( 2 ) 0 Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 15 / 19
Gapped Cases Condition (3) Lemma Condition (3) is exactly calculated for this model. Lemma For each ℓ ≥ 2 , and a sequence of increasing finite volumes Λ n ր Λ , n ≥ 1 , such that Λ n and Λ n \ Λ n − ℓ are connected for all n, the operator norm in Condition (3) of the martingale method applied to the PVBS model is given by � G Λ n + 1 \ Λ n + 1 − ℓ E n � 2 = C (Λ n + 1 − ℓ ) C (Λ n + 1 \ Λ n ) C (Λ n ) C (Λ n + 1 \ Λ n + 1 − ℓ ) , (3) where C (Λ) denotes the normalization coefficient. Michael Bishop (UCDavis) Gaps for Slanted PVBS 2/15/2016 16 / 19
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