Gap Dependency on Half Spaces in Product Vacua and Boundary State - - PowerPoint PPT Presentation

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

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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.

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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.’

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Introduction

Undecidability

Recently, Cubitt, Perez-Garcia, and Wolf’s “Undecidability of the Spectral Gap1” 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.”

1Nature 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 ⊂ Zd, Hx := C2 with basis |0x, |1x.

HΛ := ⊗x∈ΛHx

For each coordinate j = 1, . . . , d, let λj ∈ (0, ∞). For each (oriented) edge connecting x ∈ Λ to x + ej ∈ Λ, we define the operators hx,x+ej := (|01 − λj|10) (01| − λj10|) 1 + λ2

j

+ |1111|

where

|axax+ej = |axx ⊗ |ax+ejx+ej , a = 0, 1

The Hamiltonian is defined as HΛ =

d

• j=1
• x∈Λ, x+ej∈Λ

hx,x+ej

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PVBS Single Species

Ground State Space

This Hamiltonian in non-negative definite. On bounded domains, the ground state space is two dimensional:

ΨΛ

0 = |0Λ

ΨΛ

1 =

1

• C(Λ)
• x∈Λ

λx|1Λ

x

where

|0Λ := ⊗x∈Λ|0x, |1Λ

x = (⊗y∈Λ, y=x|0y) ⊗ |1x,

λx =

d

• j=1

λ

xj j = e

• log(λj)·xj, C(Λ) =
• x∈Λ

λ2x

Frustration-Free: These ground states are also ground states of the local Hamiltonians hx,x+ej.

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PVBS Single Species

Infinite Domains

The infinite domains we focus on are half-spaces determined by normalized vectors m, D := {x ∈ Zd : m · x ≥ 0} The Hilbert space and Hamiltonian are defined using the Gelfand- Naimark- Segal (GNS) construction with the vacuum state.

ωD

0 (·) =

lim

ΛNրDΨΛN

0 , ·ΨΛN

• n quasi local observables AD := Aloc.

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PVBS Single Species

Infinite Domains

From this state, the GNS construction defines a Hilbert Space HD

0 , a

representation πD

0 mapping the quasi-local algebra to the bounded operators

• n HD

0 , and a cyclic vector ΩD 0 .

For spectral properties, it is sufficient to consider HD as the limit of the representations of finite lattice Hamiltonians HΛ. For local observables A ∈ AD

loc,

HDπD

0 (A)ΩD 0 = lim

ΛրD πD

0 (HΛA)ΩD

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PVBS Single Species

Gap Definitions

Definition We say the GNS Hamiltonian HD is gapped if there exists δ > 0 such that the spectrum of HD is empty between 0 and δ spec(HD) ∩ (0, δ) = ∅ (1) If there does exist such a δ, then the spectral gap of HD is defined by

γ(HD) := sup{δ > 0 : spec(HD) ∩ (0, δ) = ∅}

(2) If there does not exist such a δ, we say the GNS Hamiltonian HD is gapless. For simplicity of notation, we will rewrite γ(HΛ) and γ(HD) as γ(Λ) and γ(D), respectively.

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Theorem

Theorem

Theorem Given a half-space D with inward normal vector m and any

λ1, . . . , λd ∈ (0, ∞), define log λ := (log λ1, . . . , log λd):

Existence of Spectral Gap: If log

λ = − log λm, then γ(HD) > 0.

Non-existence of Spectral Gap: If log

λ = − log λm, then γ(HD) = 0.

Essentially, the gap is determined by whether log

λ is an outward normal to D.

The proofs generalize to Zd: Corollary If λ1 = · · · = λd = 1, then HZd is gapless. Otherwise, HZd is gapped.

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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, λ1, . . . , λd ∈ (0, ∞), and unit vectors m ∈ Rd such that m · log

λ < 0, the spectral gap has the following upper bound: γ(HD) ≤ C log λ| sin(θ)|,

where bθ 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.

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Gapped Cases

Sketch of Gapped Cases

When log

λ = − log λm, then HD is gapped.

Proof Approach: Choose a sequence of finite volumes ΛN with converge to D. Apply the martingale method to the finite volumes ΛN to find a uniform in N lower bound on γ(HΛN). Apply a limit theorem to bound the gap for the GNS Hamiltonian:

γ(HD) ≥ lim sup

ΛNրD

γ(HΛN)

The difficulty is finding appropriate ΛL such that the martingale method generates a uniform lower bound.

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Gapped Cases

Martingale Method

Theorem (Martingale Method, Nachtergaele ‘96) For a finite sequence of volumes Λn which increase to ΛN, if there exists ℓ ≥ 2 such that following three conditions hold for the local Hamiltonians: Condition (1) For some positive constant dℓ,

N

• n=ℓ

HΛn\Λn−ℓ ≤ dℓHΛN Condition (2) For some positive constant γℓ, HΛn\Λn−ℓ ≥ γℓ(I − GΛn\Λn−ℓ)

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Gapped Cases

Martingale Method Continued

Theorem (Martingale Method Continued) Condition (3) There exists a constant ǫℓ <

1

√ ℓ and nℓ such that

nℓ ≤ n ≤ N − 1

GΛn+1\Λn+1−ℓEn ≤ ǫℓ

where En = GΛn − GΛn+1, the projection onto GΛn ∩ G⊥

Λn+1.

Then for any Ψ such that GΛNΨ = 0,

Ψ, HΛNΨ ≥ γℓ

dℓ

(1 − ǫℓ √ ℓ)2Ψ2

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Gapped Cases

Martingale Method Applied

A example volume in two dimensions in the parallelogram

ΛL = {x ∈ Z2 : 0 ≤ m · x

m1

< L, −L ≤ x2 < L}

We apply the martingale method for each coordinate direction:

ΛL Λ(2)

n

Λ(2)

n−ℓ2

Λ(2)

n \Λ(2) n−ℓ2

Λ(2)

Λ(2)

b2 \Λ(2) b2−ℓ2

Λ(1)

n

Λ(1)

n−ℓ1

Λ(1)

n \Λ(1) n−ℓ1

Λ(1)

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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 Λn ր Λ, n ≥ 1, such that Λn and Λn\Λn−ℓ are connected for all n, the operator norm in Condition (3) of the martingale method applied to the PVBS model is given by

GΛn+1\Λn+1−ℓEn2 = C(Λn+1−ℓ)C(Λn+1\Λn)

C(Λn)C(Λn+1\Λn+1−ℓ), (3) where C(Λ) denotes the normalization coefficient.

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Gapped Cases

Condition (3) Heuristic

For Condition (3) to be satisfied, this ratio must be bounded above by ǫ2

ℓ < 1/ℓ

for all n. This is satisfied when λx = exp[log

λ · x] is maximized in the corner for each Λ(j)

n \Λ(j) n−1.

Then C(Λ(j)

n \Λ(j) n−1) is approximately exponential in n. This exponential is

given by ˜

λj := λ−mj/m1

1

λj (or λ1) and the corresponding ratio is bounded

above by an exponential of the form

ǫ2

j = constant ∗ ˜

λ±2(ℓj−1)

j

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Gapped Cases

Spectral Gap bounds

Volumes are chosen case by case so log

λ is maximized in a corner.

The spectral gap for the parallelogram is bounded below by:

γ(ΛL) ≥ min

b2 γ(Λ(2) b2 \Λ(2) b2−ℓ2)(1 − ǫ2

√ℓ2)2 ℓ2 ≥ min

b1, b2 γ(Λ(1) b1 \Λ(1) b1−ℓ1)(1 − ǫ1

√ℓ1)2 ℓ1 (1 − ǫ2 √ℓ2)2 ℓ2

which is positive. The d-dimensional case is iterated d times.

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Gapped Cases

Preprint

M.Bishop, B. Nachtergaele, A.Young. ”Spectral Gap and Edge Excitations of d-dimensional PVBS models on half-spaces.” Journal of Statistical Physics, (), 1-37. DOI :10.1007/s10955-016-1457-4(2016). Thanks for your attention!

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