Unconventional Symmetries in Many-Body Physics
Sanjay Moudgalya
Walter Burke Institute of Theoretical Physics
California Institute of Technology

arXiv: 2108.10824 [PRX 12, 011050 (2022)]
SM, Olexei I. Motrunich
arXiv: 2209.03370, arXiv: 2209.03377

14th November 2022

1 / 22

Symmetries in Quantum Many-Body Physics
Lattice
Nsystem with a finite-dimensional tensor product Hilbert space
H = Lj=1 Cd with a Hamiltonian H
Symmetries/conserved quantities {Qα } are defined as operators that
commute with the Hamiltonian, [H, Qα ] = 0.
Conventionally, additional structure is imposed on {Qα }.

Internal symmetries: On-site unitary representations of a (Lie) group G
 iαZ
if G = U(1) eQα = û(g ) ⊗ û(g ) ⊗ · · · ⊗ û(g ), e.g. û(g ) = i α⃗ ·⃗σe
if G = SU(2)Continuous symmetries: Conserved quantities are typically sums of
local operators, e.g. total charge, number of domain walls, etc.
Lattice symmetries: Unitary operators that implement translation,
rotation, reflection, etc.
2 / 22

Symmetries in Quantum Many-Body Physics

Regular Quantum
Number Sectors…

Symmetric Hamiltonians can beblock-diagonalized intosymmetry quantum numbersectors.
Sectors are uniquely labelled byeigenvalues under (a maximallycommuting subset of) the {Qα }.

Various generalizations of conventional symmetries are under active
exploration: Categorical symmetries, MPO symmetries, etc.1
This talk: Different (?) generalization motivated by recent work on thedynamics of certain quantum systems.
1J.McGreevy (2022)
3 / 22

Quantum Many-Body Dynamics and Symmetries

Ergodicity in Isolated Quantum SystemsA quantum Hamiltonian is said to be ergodic if any initial state |ψ(0)⟩evolves into a “thermal” state |ψ(t)⟩ = e −iHt |ψ(0)⟩Reduced density matrix of a thermal
state is the Gibbs density matrix of thesubsystem
ρ = |ψ⟩ ⟨ψ| ,ρA = TrB (ρ) ,ρA ∼ e −βH|A

Eigenstate Thermalization Hypothesis(ETH): Eigenstates |En ⟩ in the middle ofthe spectrum are thermal2
Entanglement entropy obeys a volume
law S ∼ log D ∼ L
Eigenstate properties are a “smooth”
function of energy
2J.M.Deutsch (1991), M. Srednicki (1994)
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Ergodicity in Symmetric Isolated Quantum
Systems
PHYSICALREVIEW LETTE(a) 1

With symmetries: ρA ∼ e −β(H−µN)|AETH should hold for eigenstates withineach symmetry sector
Recent analytical and experimental
progress has identified two new types
of “weak” violations3(a)Hilbert Space Fragmentation

(b0.8
0.6
0.4
0.2
001020(b)(a)

QuantumMany-Body
ScarsFFIG.
2. (a) Entanglement
entropyFIG.
of thewithin theentrop
2. eigenstates(a) Entanglement
zasectorðn¼8;
S¼−2Þ
under
anequibipartitioningofthez
DW in several diagViolations can be seen
sector ðnz DW ¼ 8; S ¼ −2Þ unde0system. Red line, Page value of the ðnDW ; S Þ sector; green line,system. Red line, Page value of su
thnostics, e.g., entanglement entropy of
Page4 value of the largest connected subsector. (b) Entanglement
Pagevalueof
thelargest
connecte
the eigenstates.
gentropy growth (normalized by the Page value) after a quantum
entropygrowth(normalizedbythfrquench starting from random product states, averaged over 200
quench starting from random prosy
initial states.
2FIG.
2. Bipartite
entanglement entropy S of eigenstates ofin
initialstates.
M.Serbyn, D.A.Abanin, Z.Papic (2020); SM, B.A.Bernevig, N.Regnault (2021)
H for d = 1, L = 10 and (h, D, J ) = (1, 0.1, 0.1) with OBC.
4
Z.C.Yang, F.Liu, A.V.Gorshkov, T.Iadecola (2020); M.Schecter, T.Iadecola(2019)
Statesin the zero-magnetization sector (smaller points) aresu
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A

3

Whereas these models allowcolormobile
to becoded by theexcitations
density of states (warmercolors
5 /imply
22ranL1

Hilbert Space Fragmentation
Dynamics under certain local Hamiltonians
splits the Hilbert space into exponentiallymany dynamically disconnected subspaces{K(H, |Rα ⟩}, |Rα ⟩ being product statesK ∼exp(L)

H=M
j=1

K (H, |Rα ⟩)onK (H, |R⟩) = spant e −iHt |R⟩

14

Different subspaces are not distinguished byobvious symmetry quantum numbers, canshow vastly different properties!5Initial product states never thermalize w.r.t.
the full Hilbert space due to “hidden”
blocks after resolving known symmetries
5

FIG. 2. (Color online) (a) Time-evolution of the expectation value of on-site charge operators for the middle site and a site away
SM, A.Prem, R.Nandkishore, N.Regnault,
B.A.Bernevig (2019)from the middle under the pair hopping Hamiltonian with PBC, starting from an initial state of the form | 0 i = |⇤ · · · ⇤ + ⇤ · · · ⇤iwhere ⇤ = ", # for N = 15 with total spin S z = 0. The horizontal lines show the infinite-temperature expectation values6of/the22z

Hilbert Space Fragmentation
Fragmentation generically occurs
in one dimensional systems
P
conserving dipole moment ( j jSjz with OBC)6,7

Example: spin-1 dipole
Hamiltonian that implements theP conserving−+ 2 −
following rules (H = j (Sj−1 (Sj ) Sj+1 + h.c.))|+ − 0⟩ ↔ |0 + −⟩ , |0 − +⟩ ↔ |− + 0⟩|+ − +⟩ ↔ |0 + 0⟩ , |− + −⟩ ↔ |0 − 0⟩

Exponentially many one-dimensional subspaces (“frozen” eigenstates)|+ + − − · · · + + − −⟩ , |0 + +0 + + · · · 0 + +⟩Subspaces with non-local conserved quantities, e.g. a product state|0 · · · 0 + 0 · · · 0⟩ can only evolve to states with “string-order”|0 · · · 0 + 0 · · · 0 − 0 · · · 0 + · · · 0⟩

6P.Sala, T.Rakovszky, R.Verresen, M.Knap, F.Pollmann (2019)

7V.Khemani, M.Hermele, R.Nandkishore (2019)7 / 22

Quantum Many-Body Scars
Tower of States
Non-integrable models with quasiparticle towers of eigenstates deep inNow consider a state with N of these magnons
on the groundstate
the spectrum have been discovered8|S2N i = P N |G i
P+ 2
9 P =
jAKLT, stateswithj (−1)
Turnsoutspin|S2Nchain:
i is an exactstatewith(SEj =) 2N,s = 2N
forNallquasiparticles
evenL!dispersing with k = π are exact eigenstates for finite system sizes L!
|G i|S2 i = P |G i
|S4 i = P 2 |G i
..
.
..
.

- cm
c- cm
c- cm
c- cm
c- cm
c- cm
c- cm
c- cm
c- cm
c-- cm
c- cm
c- cm
c- cm
c6 c6mc6 c6m
c- cm
c- cm
c- cm
c- cm
c- cm
cc6
m
c6 c6
m
c- · · · - cm
c6 c6
m
c6 c6
m
c- cm
c6

L

|SL i = P 2 |G i = |F i
Implications for ETH?

..
.
..
.

cm
c6 c6
m
cc6
m
c6 c6
m
c6 c6
m
c6 c6
m
c6 c6
m
c6 c6
m
c6 c6
m
c6668

SM, B.A.Bernevig, N.Regnault (2021)

9D.P.Arovas (1989); SM, S.Rachel, B.A.Bernevig, N.Regnault (2017)14 / 33

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Quantum Many-Body Scars
States have entanglement entropy S ∼ log L =⇒ Violation of Strong
ETH!
Equally spaced tower: leads to exact revivals from simple initialstates10

Y-BODY SCAR
STATES WITH
EMERGENT… of small dynamically
PHYSICALREVIEW B 101, subspace
024306 (2020)11
Alternateview:
Existence
disconnected

ained from !
Eq. (4) of Ref. [23] by applying
ormation i even σiz . Thus, to prepare thees to prepare the ground state of Hξ , e.g.,c ramps, similar to how the Néel state was8].
can be viewed as containing a distribution
magnons parameterized by |ξ |. In particular,factor Z (|ξ |2 ), Eq. (4.2b), can be interpreted
cal partition function for a gas of π magnons
ature and finite fugacity |ξ |2 . It immediately
expansion coefficients |cn |2 , Eq. (4.3), canthe classical grand-canonical probability ofying a state with n π magnons. Thus, by10T.Iadecola,M.Schecter(2019)n effectively
tunethe initial
distribution of11
M.Serbyn,
D.A.Abanin,
Z.Papic (2020)
ubsequently
follow
the quantum
evolutionn.

9 / 22

Dynamically Disconnected Subspaces
Weak ergodicity breaking = existence of unexpected “dynamicallydisconnected subspaces”
(a)

(b)

(c)

Regular Quantum
Number Sectors

Thermal…

…

QMBS

While on-site or other conventional symmetries do not explain theseblocks, allowing arbitrary operators to be conserved quantities isproblematic[H, |En ⟩ ⟨En |] = 0=⇒ exponentially many conserved quantities?!

What is an appropriate definition of a conserved quantity?1212

Similar problems exist in defining integrability in finite-dimensional systems: E.A.Yuzbashyan, B.S.Shastry (2013)10 / 22

Symmetries and Commutant Algebras

Commutant algebras
Key observation:
P Same block structure appears for entire classes ofHamiltonians { j Jj hj,j+1 }Natural to look for operators that commute with this entire family.
b[O,X
Jj hj,j+1 ] = 0 ∀{Jj }.
jb (not necessarily local)
Commutant Algebra C: algebra of operators Ob = 0 ∀jsuch that [hj,j+1 , O](b1 + α2 O
b2 ∈ C for any α1 , α2 ∈ Cα1 O
b1 ∈ C, O
b2 ∈ C =⇒Obbbb
O1 O2 , O2 O1 ∈ C
C commutes with the full “bond algebra” A generated by {hj,j+1 }(A = ⟨⟨{hj,j+1 }⟩⟩).

11 / 22

Commutant Algebras
A and C are unital †-closed (vonNeumann) algebras, centralizersof each other (Double
Commutant Theorem)

𝒜

𝒵

𝒞…

Representation theory: Can
unitarily transform into a basis inbA ∈ A and hbC ∈ C have
which hthe matrix representations
MbA W =W †h(MDλ ⊗ 1dλ )ℒ(ℋ)λλ(1Dλ ⊗ Ndλ )…

{Dλ } and {dλ }: dimensions ofirreducible representations of Aand C.

…

M…

bC W =W †h12 / 22

Dynamically Disconnected Subspaces
Equivalently: Basis in which all elements of A are maximally block
diagonal
P
Hamiltonian H = j Jj hj,j+1 ∈ A, block diagonal form defines
quantum number sectors/dynamically disconnected “Krylov subspaces”

For each λ: dλ number ofP degenerate Dλ -dimensional blocks, total
number of blocks: K = λ dλP
K can be bounded using dim(C) = λ dλ2 , the number of linearly
independent operators in C, given by
1log(dim(C)) ≤ log K ≤ log(dim(C))
2log(dim(C))∼ O(1)∼ log L
∼LExample
Discrete Global Symmetry
Continuous Global Symmetry
Fragmentation
13 / 22

Conventional Symmetries

Regular Quantum
Number Sectors…

Simple Examples: Abelian C…

Abelian C =⇒ dλ = 1, K = dim(C)P
Generic Hamiltonians j Jj hj,j+1 with nob =0
symmetries, solve for [hj,j+1 , O]
C = {1}, K = dim(C) = 1

Ising models H =b =0and [Zj , O]PLb

j=1 [Jj Xj Xj+1 + hj Zj ], solve for [Xj Xj+1 , O] = 0

C = span{1,YjZj }, K = dim(C) = 2.

PSpin- 12 XX models H = Lj=1 [Jj (Xj Xj+1 + Yj Yj+1 ) + hj Zj ], solve forb = 0 and [Zj , O]
b =0
[Xj Xj+1 + Yj Yj+1 , O]C = ⟨⟨Ztot ⟩⟩ = span{1, Ztot , (Ztot )2 , · · · , (Ztot )L }, Ztot =K = dim(C) = L + 1.

Pj Zj14 / 22

Simple Examples: Non-Abelian Cyxz
C = ⟨⟨Stot, Stot, Stot⟩⟩…
…

Block-diagonal form (Schur-Weylduality):0 ≤ λ ≤ L/2: S 2 eigenvalues
dλ = 2λ + 1: irreps of su(2)Dλ : irreps of SL…

x α y βz γ
= spanα,β,γ {(Stot) (Stot ) (Stot) }…

Non-Abelian C =⇒ some dλ > 1 =⇒ degeneracies
Example: spin- 21 Heisenberg modelP
H = j Jj S⃗j · S⃗j+1 , A = ⟨⟨{S⃗j · S⃗j+1 }⟩⟩Example: Stabilizer codes – A is the group algebra of the stabilizer
group, C consists of A and the non-trivial logical operators.
12SM, O.I.Motrunich (2021)
15 / 22

New View on Symmetries
Symmetries well defined forfamilies of Hamiltonians, pair ofalgebras A and C associatedwith any symmetry.

ℒ(ℋ)

𝒜𝒵

𝒞

A is generated by a set of local
operators, C is its centralizer.
Symmetries of several standard Hamiltonians can be understood thisway, including free-fermion models, Hubbard models13,14
Conventional commutants C: Full commutant generated by“conventional” conserved quantities, dim(C) scales sub-exponentiallywith system size.
In general: Start with any set of non-commuting local operators,
generate their algebra A, then determine commutant C – gives rise tonovel unconventional symmetries!
13

SM, O.I.Motrunich (2022)

14

Some of them have mildly non-standard symmetries
16 / 22

Unconventional Symmetries(b) (b)(c) (c)RegularRegular
Quantum
Quantum
NumberNumber
SectorsSectors

ThermalThermal…
…

…
…

QMBS QMBS

“Classical” Fragmentation
Fragmentation occurs when dim(C) ∼ exp(L)Consider the t − Jz Hamiltonian: hopping with two species of particles|↑ 0⟩ ↔ |0 ↑⟩, |↓ 0⟩ ↔ |0 ↓⟩PP †
zc̃i,σ c̃j,σ+ h.c. + Jj,j+1
Siz Sjz )Ht−Jz = (−tj,j+1jσ∈{↑,↓}†c̃j,σ = cj,σ 1 − cj,−σcj,−σHas two U(1) symmetries N ↑ =PP ↓↑↓j Nj and N =j Nj

Full pattern of spins (↑ or ↓) preserved
P in one dimension with OBC,
number of Krylov subspaces K = Lj=0 2j = 2L+1 − 1|0 ↑↓ 0 ↓↑ 0⟩ ←↛|0 ↑↑ 0 ↓↓ 0⟩

Fragmentation in the product state basis =⇒ essentially classical1515D.Dhar, M.Barma (1993); G.I.Menon, M.Barma, D.Dhar (1997)
17 / 22

“Classical” Fragmentation
Local operators Nj↑ and Nj↓ satisfy the relations
βα[hj,j+1 , Njα + Nj+1] = 0, [hj,j+1 , Njα Nj+1] = 0, α, β ∈ {↑, ↓}The full commutant algebra C can be explicitly constructed,dim(C) = 2L+1 − 1 ∼ exp(L)N σ1 σ2 ···σk =X

j1 <j2 <···<jkNjσ11 Njσ22 · · · Njσk k , σj ∈ {↑, ↓}Most of these are functionally independent from the conventional
conserved quantities N ↑ and N ↓ =⇒ new dynamically disconnectedsubspaces
Classical fragmentation: All conserved quantities diagonal,
Hamiltonian block-diagonal in product state basis
Similar construction works for dipole-conserving
models, exact results√in some cases (e.g. dim(C) ∼ (1 + 2)L for range-3 spin-1 model)

18 / 22

“Quantum” Fragmentation

j=1104103

Degeneracy

Disordered SU(3)-symmetric spin-1
biquadratic model, eigenstatedegeneracies grow exponentially withL =⇒ hidden symmetries
LX
H=Jj (S⃗j · S⃗j+1 )2102101
10002468Energy

10121416√

A = ⟨⟨(S⃗j · S⃗j+1 )2 ⟩⟩ = TLL (q = 3+2 5 ), commutant C can be explicitly
constructed,16 dim(C) ∼ exp(L)[(S⃗j · S⃗j+1 )2 , (Mβα )j + (Mβα )j+1 ] = 0, [(S⃗j · S⃗j+1 )2 , (Mβα )j (Mδγ )j+1 ] = 0,···αkMβα11βα22···β=kX

j1 <j2 <···<jk(Mβα11 )j1 (Mβα22 )j2 · · · (Mβαkk )jk .

Quantum fragmentation: Block-diagonal structure of the Hamiltonian
understood in the spin-1 singlet basis, not product state basis
16N.Read, H.Saleur (2007)19 / 22

Quantum Many Body Scars
Aim: Given QMBS eigenstates {|Sn ⟩}, find a locally-generated algebra
Ascar that Cscar = ⟨⟨{|Sn ⟩ ⟨Sn |}⟩⟩

Example: Spin- 12 ferromagnetic multiplet− n{|Sn ⟩ = (Stot) |F ⟩}, |F ⟩ = |↑ · · · ↑⟩ – Start with SU(2) symmetry andsystematically break it17
x , S y , S z ⟩⟩Asym = ⟨⟨{S⃗j · S⃗j+1 }⟩⟩Csym = ⟨⟨Stottot totz ⟩⟩z ⟩⟩Adyn = ⟨⟨{S⃗j · S⃗j+1 }, StotCdyn = ⟨⟨S⃗ 2 , Stotz , {D αAscar = ⟨⟨{S⃗j · S⃗j+1 }, Stotj−1,j,j+1 }⟩⟩ Cscar = ⟨⟨{|Sn ⟩⟨Sn |}⟩⟩Ascar can be explicitly constructed for several known examples ofQMBS18Generators of Ascar are building blocks for constructing quantum
scarred Hamiltonians =⇒ Lots of local perturbations that exactly
preserve the QMBS!
17

D.K.Mark, O.I.Motrunich (2020); N.O’Dea, F.J.Burnell, A.Chandran, V.Khemani (2020)

18

SM, O.I.Motrunich (2022)
20 / 22

Constraints on Realizable Symmetries
Locality of generators of A restricts realizable commutants CPP
No-go result: No locally generated A with C = ⟨⟨ j Sjz , j jSjz ⟩⟩19
Can systematically search for symmetries
realizable using, e.g., spin-1/2 n.n.
z -conserving terms20
Stot
2.0

3.4

1.5

3.2

1.0

A=⟨⟨{Xj Xj+1 +Yj Yj+1 +∆Zj Zj+1 +h− (Zj −Zj+1 )⟩⟩Detects presence of an unconventional
SU(2)q symmetry for−1−1(∆, h− ) = ( q+q2 , q−q2 )Also leads to discovery of non-integrablemodels with Strong Zero Modes21

3.0

0.5

2.8

0.0

2.6

0.52.4

1.0

2.2

1.5
2.0

2.0
2.0

1.5

1.0

0.5

0.0h

0.5

1.0

19

P.Sala, T.Rakovszky, R.Verresen, M.Knap, F.Pollmann (2019); V.Khemani, M.Hermele, R.Nandkishore (2020)

21SM, O.I.Motrunich (in preparation)

21

P.Fendley (2016); D.V.Else, P.Fendley, J.Kemp, C.Nayak (2017)

1.5

2.0

21 / 22

Summary & Outlook
Symmetry ⇐⇒ Pair of (A, C)
A: Local algebra C: Commutant algebra

Conventional symmetries: C generated byconventional conserved quantities,dim(C) ∼ O(1) or dim(C) ∼ poly(L)Concrete definitions:

ℒ(ℋ)

𝒜𝒵

𝒞Fragmentation: dim(C) ∼ exp(L)QMBS: Simultaneous eigenstates ofmultiple non-commuting local operators∗…

Double Commutant Theorem: Buildingblocks for all symmetric local Hamiltonians…
…

Approximate Commutants? PXP Model?

…

Interesting C? Connections tocategorical/MPO symmetries?
Implications for equilibrium physics?
Non-interacting models?

22 / 22