Universal features of higher-form symmetries at phase transitions
We investigate the behavior of higher-form symmetries at various quantum phase transitions. We consider discrete 1-form symmetries, which can be either part of the generalized concept “categorical symmetry” (labelled as \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) ) introduced recently, or an explicit Z_N^{(1)} Z...
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Published in | SciPost physics Vol. 11; no. 2; p. 033 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
SciPost
01.08.2021
|
Online Access | Get full text |
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Summary: | We investigate the behavior of higher-form symmetries at various
quantum phase transitions. We consider discrete 1-form symmetries, which
can be either part of the generalized concept “categorical symmetry”
(labelled as
\tilde{Z}_N^{(1)}
Z
̃
N
(
1
)
)
introduced recently, or an explicit
Z_N^{(1)}
Z
N
(
1
)
1-form symmetry. We demonstrate that for many quantum phase transitions
involving a
Z_N^{(1)}
Z
N
(
1
)
or
\tilde{Z}_N^{(1)}
Z
̃
N
(
1
)
symmetry, the following expectation value
\langle \left( O_\mathcal{C}\right)^2 \rangle
⟨
(
O
)
2
⟩
takes the form
\langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P
⟨
(
log
O
)
2
⟩
∼
−
A
ϵ
P
+
b
log
P
, where
O_\mathcal{C}
O
is an operator defined associated with loop
\mathcal{C}
(or its interior
\mathcal{A}
),
which reduces to the Wilson loop operator for cases with an explicit
Z_N^{(1)}
Z
N
(
1
)
1-form symmetry.
P
P
is the perimeter of
\mathcal{C}
,
and the
b \log P
b
log
P
term arises from the sharp corners of the loop
\mathcal{C}
,
which is consistent with recent numerics on a particular example.
b
b
is a universal microscopic-independent number, which in
(2+1)d
(
2
+
1
)
d
is related to the universal conductivity at the quantum phase
transition.
b
b
can be computed exactly for certain transitions using the dualities
between
(2+1)d
(
2
+
1
)
d
conformal field theories developed in recent years. We also compute the
"strange correlator" of
O_\mathcal{C}
O
:
S_{\mathcal{C}} = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangle
S
=
⟨
0
|
O
|
1
⟩
/
⟨
0
|
1
⟩
where
|0\rangle
|
0
⟩
and
|1\rangle
|
1
⟩
are many-body states with different topological nature. |
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ISSN: | 2542-4653 2542-4653 |
DOI: | 10.21468/SciPostPhys.11.2.033 |