SPTO in Open Quantum Systems

Caroline de Groot 1 , Norbert Schuch 2 , Alex Turzillo 1,3 .
13

Max Planck Institute of Quantum Optics
2
University of Vienna
Perimeter Institute for Theoretical Physics
arXiv: 2112.04483

November 8, 2022

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Topological phases in open systems

What are the topological phases of open systems? Of mixed states?

In particular, which closed topological phases are robust against dissipation?

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Defining phase equivalence
In closed systems...
paths in the space of gapped Hamiltonians... or equivalently...
H

fast, local Hamiltonian evolution ψ −→ ψ′... approximated by low depth circuits of local gates (e.g. polylog(L) depth of constant size gates)

In open systems... Coser & Pérez-Garcı́a [1810.05092] propose using
LL

1
2
fast, local Lindbladian evolutions ρ −→ρ′ and ρ′ −→ρmotivation: local observables are analytic within phases.

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Defining phase equivalence for SPT phases
In closed systems...
paths in the space of symmetric gapped Hamiltonians
fast, local, symmetric Hamiltonian evolution... approximated by low depth circuits of local symmetric gates

In open systems...?
Not obvious what symmetry condition to impose on Lindbladian evolutions.
Two natural guesses...
weak symmetry: Ug ◦ L ◦ Ug† = L .
strong symmetry: L† (Ug ) = 0 .

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Dynamical and static definitions of phases

Key idea
A “good” definition of phase equivalence is one for which order parameters areconstant within phases...
dynamical definition = static definition

For SPT orders in 1D... we consider string order parameters.
... more on these later

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Proposed SPT phase equivalence
Main claim
SPT order, as defined by string order parameters,
is preserved by a fast, local Lindbladian evolution L
if and only if L satisfies a strong symmetry condition.

Interpretations
1. Simply a rule for determining whether or not SPT order is robust against
certain couplings to the environment.
2. A “good” definition of SPT phase equivalence in open systems.
Two mixed states belong to the same SPT phase if they are related by...
LL

1
2fast, local, strongly symmetric Lindbladian evolutions ρ −→ρ′ and ρ′ −→ρ.

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Weak and Strong Symmetry Conditions

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Weakly symmetric noise destroys 1D SPT order
C&PG constructs a fast, local Lindbladian evolution that does|ψ⟩−→|product⟩

to any 1D state |ψ⟩.
This evolution destroys SPT order.

Solution...? Maybe this Lindbladian is not symmetric.
However it satisfies
Ug ◦ L ◦ Ug† = L .

(weak symmetry)Is there a stronger symmetry condition on L that protects SPT order?
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Weak symmetry allows interference between trajectories
Work with channels E. Have in mind Lindbladian evolutions Et = e tL .
L is weakly symmetric precisely when Et satisfies, at all times t,Ug ◦ Et ◦ Ug† = Et .

(weak symmetry)

Weak symmetry (WS) in terms of Kraus operators:
XX(Ug Ki Ug† ) ρ (Ug Ki Ug† )† =Ki ρ Ki† ,∀g .

⇒ For each g , there is a basis of Kraus operators Kig where
Ug Kig Ug† = e iθi (g ) Kig ,∀i .

Different phases θi ̸= θj give rise to interference between trajectories.
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9 / 25Strong symmetry

Strong symmetry (SS): trajectories transform with the same phase:
Ug Ki Ug† = e iθ(g ) Ki ,∀ i, g .

This condition is basis-independent, so the superscript g can be dropped.
Neglect the weak invariant θ(g ). It vanishes for E = e tL .

Comment: WS and SS are the same for reversible channels ρ 7→ W ρW † .

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Strong symmetry of Lindbladians

Using the relation between the Ki and the jump operators Li , we see that...
...families Et = e tL of SS channels are generated by SS Lindbladians:
Ug H = HUg ,Ug Li = Li Ug ,∀ i, g .

The SS condition on L has appeared previously. [ Buča , Prosen 12 ] [ Albert , Jiang 13 ]Alex Turzillo

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Strong symmetry in the Heisenberg picture
The Heisenberg picture is convenient for studying the evolution of observables:Tr[E(ρ) O] = Tr ρ E † (O) .
Strong symmetry of E means that E † fixes the symmetry operators:
Ug Ki Ug† = Ki ,∀ i, g⇐⇒E † (Ug ) = Ug ,∀g .

... in other words, charge is conserved in the system alone.
In terms of the Lindbladian, strong symmetry means...
L† (Ug ) = 0 ,∀g .

The SPT-destroying Lindbladian of C&PG fails this condition.

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Strong symmetry in terms of purifications
Purification: unitary W on H ⊗ A such that Ki = ⟨ei |W |a⟩.

Claim: If ∃ a W symmetric with respect to some Ug ⊗ UgA , the channel E is WS.

Claim: The channel is SS if and only if there exists a W with UgA = 1.
Strong symmetry means the system and bath couple by symmetric terms:
W = e −itH/ℏ ,H=XHiS ⊗ HiESS==⇒Ug HiS = HiS Ug , ∀ i, g .

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Review: String Order for SPT Phases

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Symmetry fractionalization in one dimension
SPT phases are characterized by patterns of symmetry fractionalization on theedge.

Boundary action Vg may be projective
Vg Vh = ω(g , h)Vgh ,ω : G × G → U(1)[ Chen, Gu, Wen 10 ] [ Schuch, Perez-Garcia, Cirac 11 ] [ Else, Nayak 14 ][ Pollmann, Turner, Berg, Oshikawa 10 ]ω satisfies a cocycle condition and is defined up to coboundaries=⇒ patterns of symmetry fractionalization are classified by group cohomology[ω] ∈ H 2 (G , U(1))
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Fractionalization in MPS

Symmetry means UgN |ψ⟩ = |ψ⟩. Represent each side as an MPS.
Tensors Ai and (Ug )ij Aj define the same state ⇒ related by a gauge trans.:
Ugij Aj = e iϕg Vg Ai Vg† .

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String operators
For simplicity... assume the symmetry G is a finite abelian group.

String operators: long strings of symmetry operators capped by end operators:[ den Nijs, Rommelse 89 ] [ Perez-Garcia, Wolf, Sans, Verstraete, Cirac 08 ] [ Pollmann, Turner 12 ]End operators Oαl and Oαr are charged under G according to the irreps α and α∗
Expectation values on MPS display a selection rule, aka “pattern of zeros”⟨s(g , Oα )⟩ = 0

unless χα (h) =ω(h, g )for all h .
ω(g , h)The pattern determines the ratios ω/ω and therefore the class [ω].
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Patterns of zeros of MPS states
To evaluate the string order parameter ⟨s(g , α)⟩, sandwich the string operator
between ⟨ψ| and |ψ⟩, then apply some relations (injectivity) of the MPS tensor:

Each end evaluates to Tr[Ng Oα ].
Oα transforms as α, while Ng transforms as ω/ω.
Vanishes unless these representations are equal:⟨s(g , Oα )⟩ = 0

unless χα (h) =ω(h, g )for all h .
ω(g , h)If they are equal, generically Tr[Ng Oα ] ̸= 0.
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Reconstructing the SPT invariant
Represent the pattern as an array with columns g , rows α.
For each g , there is a unique α with ⟨s(g , Oα )⟩ =
̸ 0.
⋆ ⋆ ⋆ ⋆⋆
 0 0 0 0  0⟨s(g , Oα )⟩trivial = , ⟨s(g , Oα )⟩Haldane = 0 0 0 0 00 0 0 00
0 0 0⋆ 0 0 0 ⋆ 0 0 0 ⋆Knowledge of all values ⟨s(g , Oα )⟩ determines the ratios ω/ω.
The ratios ω/ω completely determine the cohomology class [ω].
Follows from Schur’s lemma.

Comment: If G is non-abelian, other nonlocal order parameters may be needed
to fully reconstruct [ω]. [ Pollmann, Turner 12 ]Alex Turzillo

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Strong Symmetry and String Order(three slides of technical argument)

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Strongly symmetric uncorrelated noise on string operators
Consider uncorrelated noise
E = E1 ⊗ · · · ⊗ EL .
E is strongly symmetric if and only if the Es are.
The labels (g , α) of the string operator are preserved
E † (s(g , Oα )) = s(g , Oα′ )since the bulk and end parts of the string transform as
Es† (Ug ) = Ug ,Ug† Es† (Oα )Ug = χα (g )Es† (Oα ) .

Conversely, if a channel preserves all string operators, it must be SS.
Next... what about the expectation values of these string operators?
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Strong symmetry of L is necessary and sufficientFix any SPT phase, as defined by its string order.
A semigroup preserves the phase at all finite times
if and only if
it is generated by a strongly symmetric Lindbladian.

‘If’ direction
Does SS protect the pattern of zeros in the expectation values ⟨s(g , Oα )⟩?
Yes. String operators s(g , Oα ) and E † (s(g , Oα )) have the same labels, so they
have the same pattern for generic choices of end operators...
...i.e. if Oα is not annihilated and is orthogonal to neither Ng nor E(Ng ).
Note: at t → ∞, Et may annihilate some end operators and spoil the pattern.
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Transfer matrix argument
Conversely... suppose string order is preserved and show strong symmetry.
String order vanishes unless the following transfer matrix has λmax = 1.

λmax = 1 implies the insertion is a symmetry. [ Bridgeman, Chubb 17 ]=⇒E † (Ug ) = Uσ(g ) ,The family Et defines a continuous path σt from 1 to σ.
For finite G , this implies that σ = 1, which is the strong symmetry condition:Et† (Ug ) = Ug , ∀ t, g=⇒

L is SS

Beyond finite G ... σ is an inner automorphism, and so L can also involve
rotation by a generator of a continuous symmetry under which ω is invariant.
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Interpretation and Related Results

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Recent related results
Ma & Wang [2209.02723] consider “Average SPT order” (ASPTO).
Systems evolve with subgroup H “exact symmetry” (strong symmetry) and G /H“average symmetry” (weak symmetry).
Nevertheless, some SPT order is robust.

Lee, You, Xu [2210.16323] and Zhang, Qi, Bi [2210.17485] detect ASPTO withstrange correlators.

How can non-SS evolutions preserve SPTO?
Resolution: these channels destroy string order but are nevertheless preserve
SPTO in some other sense.

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