Symmetry Enriched c-Theorems SPT Transitions Clay C ordova1and Diego Garc a-Sep ulveda2 Kadano Center for Theoretical Physics Enrico Fermi Institute University of Chicago

2025-05-02 0 0 982.48KB 53 页 10玖币
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Symmetry Enriched c-Theorems & SPT Transitions
Clay C´ordova1and Diego Garc´ıa-Sep´ulveda2
Kadanoff Center for Theoretical Physics & Enrico Fermi Institute, University of Chicago
We derive universal constraints on (1 + 1)drational conformal field theories (CFTs) that can arise
as transitions between topological theories protected by a global symmetry. The deformation away
from criticality to the trivially gapped phase is driven by a symmetry preserving relevant deformation
and under renormalization group flow defines a conformal boundary condition of the CFT. When
a CFT can make a transition between distinct trivially gapped phases the spectrum of the CFT
quantized on an interval with the associated boundary conditions has degeneracies at each energy
level. Using techniques from boundary CFT and modular invariance, we derive universal inequalities
on all such degeneracies, including those of the ground state. This establishes a symmetry enriched
c-theorem, effectively a lower bound on the central charge which is strictly positive, for this class
of CFTs and symmetry protected flows. We illustrate our results for the case of flows protected by
SU(M)/ZMsymmetry. In this case, all SPT transitions can arise from the WZW model SU(M)1,
and we develop a dictionary between conformal boundary conditions and relevant operators.
October 2022
1clayc@uchicago.edu
2dgarciasepulveda@uchicago.edu
arXiv:2210.01135v1 [hep-th] 3 Oct 2022
Contents
1 Introduction 2
1.1 Symmetry Enriched Renormalization Group Flows . . . . . . . . . . . . . . . . . . . 2
1.2 Boundary Conformal Field Theory and Inflow . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Constraints from Modular Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 FlowsFromWZWModels................................. 9
2 Boundary CFT and Renormalization Group Flows 9
2.1 ReviewofBoundaryCFT ................................. 10
2.2 Boundary Conditions from Renormalization Group Flows . . . . . . . . . . . . . . . 15
3 Bounds on Partition Functions and the Central Charge 17
3.1 Bounds on Partition Functions in Theories with Extended Chiral Algebras . . . . . . 17
3.2 Bounds on Interval Partition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Inequalities Relating Boundary and Bulk Data . . . . . . . . . . . . . . . . . . . . . 22
3.4 Example: Inequalities for SU(2)kat Large k....................... 25
4 Boundary Conditions for SU(M)1WZW and Bulk Deformations 26
4.1 Bound at Small and Large M............................... 26
4.2 Bulk Deformations and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 27
4.2.1 Deforming the SU (2)1Critical Theory and SPT Phases . . . . . . . . . . . . 28
4.2.2 Generalization to SU (M)1............................. 30
4.2.3 Conformal Boundary Conditions from the Most-Relevant Perturbation . . . . 32
5Z2×Z2SPT Transition via Gauged Ising Models 33
5.1 Bulk Analysis of Relevant Deformations . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Z2×Z2Transition in the Boundary CFT . . . . . . . . . . . . . . . . . . . . . . . . 35
A First Order Transitions 36
B Sine and Cosine Deformations 37
CSU(M)1Deformation: Maxima and Minima of Vw1(ϕ)40
D Comments on Flows for Other Simply-Laced Algebras 42
E A Bound on cfor Irrational Theories? 45
1
1 Introduction
In this paper we explore the properties of two-dimensional conformal field theories that appear as
second order transitions between trivially gapped phases. For rational CFTs, we derive a universal
inequality relating the associated ground state degeneracy of the CFT defined on a strip and the
bulk central charge. We apply this bound to a variety of renormalization group flows enriched
by global symmetries where the non-trivial ground state degeneracy is enforced by a symmetry
protected trivially gapped phase.
1.1 Symmetry Enriched Renormalization Group Flows
One of the foundational results about quantum field theory is the monotonicity of the renormaliza-
tion group flow. In two spacetime dimensions this idea is famously quantified by the c-theorem [1]:
along a renormalization group flow the central charge cdecreases. In particular, for a conformal
field theory which can flow to a gapped phase (which has vanishing central charge) we deduce that
the central charge is non-negative
c≥0.(1.1)
This simple result also follows directly from unitarity and reflects a basic feature of Virasoro repre-
sentation theory. By now, these monotonicity results have been generalized to different spacetime
dimensions [2–11] and may be derived from a variety of disparate theoretical techniques including
anomalies, information theory, and holography—each leading to the same underlying conclusions
about coarse graining in field theory.
In the spirit of the ideas to follow, one may also view the inequality (1.1) as a basic statement
about second order phase transitions. Indeed, intuitively a gapped phase is the long-distance limit
of a generic quantum field theory, and may be thought of generally as a topological field theory
with only long-range correlation functions but no non-trivial local degrees of freedom. Meanwhile,
a conformal field theory, a gapless system, results from tuning parameters to close the gap and
requires non-trivial power law correlation functions measured by c.
In this paper, we will enrich this paradigm by considering flows that preserve an (ordinary)
global symmetry G. One of our main aims is to strengthen the bound (1.1) by tracking the symmetry
action to long distances. We therefore consider a flow triggered by a Ginvariant operator O
δS =λZd2xO(x).(1.2)
In the simplest analysis, we assume that the infrared is trivially gapped, so in particular Gis not
spontaneously broken. In this situation the IR is described by a symmetry protected topological
phase (SPT). In field theoretic language such a model describes a theory of local contact terms for
the operators defining the Gsymmetry. If we introduce background gauge fields Asourcing G, the
2
IR partition function is then a phase, which is specified by a local action
ZIR[A] = exp 2πi ZX
ω(A),(1.3)
where above, ω(A) may be viewed as the Lagrangian for the SPT and Xis the two-dimensional
spacetime manifold. The action (1.3) is non-trivial because it may not be continuously deformed to
the trivial action while preserving the energy gap above the unique ground state.
Symmetry preserving renormalization group flows to trivially gapped phases are common in the
study of field theory; however, considered in isolation they cannot strengthen the basic inequality
(1.1). The reason is instructive: in defining the UV field theory, we are generally agnostic to the
choice of scheme. The difference between any two schemes is given by a local classical action for the
sources, of which (1.3) is an example. Therefore, by adjusting the ultraviolet scheme we are free to
assume that the IR partition function resulting from any fixed flow to a trivially gapped phase is
described by a completely trivial G-SPT.
The problem becomes more interesting when we have two distinct Ginvariant flows (labelled
±) which each result in a trivially gapped phase. In this case there is an invariant (i.e. scheme
independent) meaning to the difference in the SPTs resulting from the two flows. In other words,
the ratio of infrared actions
ZIR,+[A]
ZIR,−[A]= exp 2πi ZX
ω+−(A),(1.4)
is an invariant feature of the conformal field theory.3This setup has a natural interpretation as a
second order phase transition between G-invariant trivially gapped phases: by deforming parameters
the gap closes and the dynamics of the CFT can provide a transition to a new distinct trivially
gapped phase (See Figure 1). Such SPT transitions have been previously explored in [15–30].
Of course, it is also possible to have a first order transition, with multiple gapped ground states
that achieves a transition between SPTs. We describe an example of this in more detail in Appendix
A. Throughout the remainder of this work, we assume that the transition theory is second order
with a unique vacuum and hence a conformal fixed point. A motivating question for the analysis to
follow is thus, given a Ginvariant conformal field theory with a pair of Ginvariant flows to trivially
gapped phases and a non-vanishing transition action ω+−(A),can we strengthen the bound (1.1)?
It is intuitively clear that the answer to this question must be affirmative. The trivial CFT (with
vanishing central charge c) cannot make a transition between distinct SPT phases, and thus the
non-vanishing transition action ω+−(A) necessitates local degrees of freedom with strictly positive
c. Our goal is to quantify this idea.
A simple example illustrating the ideas described above is given by the group G∼
=Z2. For
bosonic theories there are no possible SPT transitions to consider. Indeed, such phases are classified
3The fact that the CFT can form a transition between distinct SPTs also signals an anomaly in the coupling
constant space of the theory [12–14].
3
by H2(G, U(1)) which vanishes with G∼
=Z2. However, for fermionic theories there is a unique
possible non-trivial transition specified by the Z2-SPT defined by the non-trivial phase of the Kitaev
chain [31]. Formally this action may be written as:
ZIR,+[A]
ZIR,−[A]= exp πi ZX
qρ(A),(1.5)
where qρ(A) is a quadratic function of the Z2background gauge field A, and ρis a fixed reference
spin structure.4Famously, the Z2-SPT transition defined by (1.5) may be achieved through a
massless Majorana fermion χwhich is odd under the Z2symmetry. The mass deformation of this
model deforms the Lagrangian by a term mχ2where mis a real parameter which may have either
sign. Both signs lead to trivially gapped physics preserving the Z2symmetry, but differ by an SPT
(1.5):
lim
m→∞
Zχ[A, +m]
Zχ[A, −m]= exp πi ZX
qρ(A).(1.6)
In fact, it is straightforward to see that the theory of the massless Majorana fermion is the CFT
with smallest possible central charge cthat can achieve the transition (1.5) since all other non-trivial
unitary CFTs have strictly greater central charge.
Below we will derive a general bound on cfor conformal field theories that can transition
between G-SPTs for general group G. For simplicity we focus on the case of bosonic theories where
the SPTs may be classified by group cohomology classes:
ω+−(A)∈H2(G, U(1)) .(1.7)
Working with rational CFTs, those with a finite number of current algebra primaries, we derive
a rigorous inequality relating the central charge, the SPT transition class and the spectrum of
light operators in the CFT. We subsequently illustrate this bound in several concrete examples of
symmetry protected flows.
1.2 Boundary Conformal Field Theory and Inflow
The main conceptual insight that enables our derivation is to relate the analysis of symmetry
protected renormalization group flows to a problem in boundary conformal field theory [25–29].
Conformal boundary conditions arise naturally by activating the deformation (1.2) in only half of
spacetime:
δS =λZx>0
dydx O(y, x).(1.8)
Macroscopically, such a deformation results in a conformal interface between the original CFT (in
the region x0) and the long distance limit of the relevant perturbation (1.2) (in the region
x0). Such interfaces have been previously investigated in [32–38]. In general, a precise definition
4Alternatively, the quantity qρ(A) in (1.5) is the Arf invariant of the spin structure ρ+A.
4
摘要：

SymmetryEnrichedc-Theorems&SPTTransitionsClayCordova1andDiegoGarca-Sepulveda2Kadano CenterforTheoreticalPhysics&EnricoFermiInstitute,UniversityofChicagoWederiveuniversalconstraintson(1+1)drationalconformal eldtheories(CFTs)thatcanariseastransitionsbetweentopologicaltheoriesprotectedbyaglobalsymm...

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