Conformal Field Theory and the Hot Phase of the 3D U(1) Gauge Theory

Conformal ﬁeld theory and the hot phase of the 3D U(1) gauge theory

Alessandro Nada1

in collaboration with

Michele Caselle2, Marco Panero2 and Davide Vadacchino3

1NIC, DESY Zeuthen

2Universit`adi Torino & INFN, Sezione di Torino

3INFN, Sezione di Pisa

13th December 2019 The XVIII Workshop on Statistical Mechanics and Non-pertubative Field Theory Bari, 11-13 December 2019

1 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Center symmetry in pure-gauge theories

For Yang-Mills theories with gauge symmetry group G: the deconﬁnement transition is related to the spontaneous breaking of a global symmetry → the “center” symmetry when C(G) is not trivial

Order parameter: Polyakov loop

P → zP z ∈ C(G) htr Pi = 0 in the conﬁned phase and htr Pi 6= 0 in the deconﬁned phase

One can think of integrating out all degrees of freedom except the order parameter itself → spin model with a global symmetry given by the center of G

2 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Center symmetry in pure-gauge theories

For Yang-Mills theories with gauge symmetry group G: the deconﬁnement transition is related to the spontaneous breaking of a global symmetry → the “center” symmetry when C(G) is not trivial

Order parameter: Polyakov loop

P → zP z ∈ C(G) htr Pi = 0 in the conﬁned phase and htr Pi 6= 0 in the deconﬁned phase

One can think of integrating out all degrees of freedom except the order parameter itself → spin model with a global symmetry given by the center of G

2 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 The Svetitsky-Yaﬀe conjecture

Argument of [Svetitsky,Yaﬀe; 1982]:

I at T = Tc

I for a continuous (2nd order) phase transition the critical behaviour of

theory with gauge symmetry group G ⇔ spin model in d dimensions with the in d + 1 spacetime dimensions center of G as global symmetry group is equivalent

Example: universality class in SU(3) in 2+1 dimensions 3-state Potts model in 2 dimensions ⇔ SU(2) in 3+1 dimensions Ising model in 3 dimensions

Phases are “exchanged”: “hot” phase “ordered” phase ⇔ htr Pi 6= 0 hmi 6= 0 and “cold” phase “disordered” phase ⇔ htr Pi = 0 hmi = 0

3 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 The Svetitsky-Yaﬀe conjecture

Argument of [Svetitsky,Yaﬀe; 1982]:

I at T = Tc

I for a continuous (2nd order) phase transition the critical behaviour of

theory with gauge symmetry group G ⇔ spin model in d dimensions with the in d + 1 spacetime dimensions center of G as global symmetry group is equivalent

Example: universality class in SU(3) in 2+1 dimensions 3-state Potts model in 2 dimensions ⇔ SU(2) in 3+1 dimensions Ising model in 3 dimensions

Phases are “exchanged”: “hot” phase “ordered” phase ⇔ htr Pi 6= 0 hmi 6= 0 and “cold” phase “disordered” phase ⇔ htr Pi = 0 hmi = 0

3 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 There is a “special” case:

U(1) gauge theory in (2+1)D ⇔ U(1) spin model in 2 dimensions XY model

Mermin-Wagner theorem ⇒ no spontaneous symmetry breaking ⇒ Kosterlitz-Thouless transition phases are characterized by “topological” order

In addition, the low-temperature phase of the XY model

I is scale-invariant

I has a conformal-ﬁeld-theory description

I is a line of ﬁxed points

What happens in the corresponding (hot, T > Tc ) phase in the U(1) theory? Can we extend the conjecture?

Test: conformal-ﬁeld theory predictions for U(1) observables!

4 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Compact QED

U(1) gauge theory (QED) without matter ﬁelds in a three-dimensional spacetime

Note:

I in classical electrodynamics in 2 space dimensions the Coulombic potential is logarithmic

Compact U(1) on the lattice:

1 X X S = − Re Uµν (x) W ae2 x∈Λ 1≤µ<ν≤3 with ? ? Uµν (x) = Uµ(x)Uν (x + aµˆ)Uµ(x + aνˆ)Uν (x) a Uµ(x) = exp ieaAµ x + 2 µˆ on the links → the Aµ are periodic 2πk Aµ → Aµ + ea

Compact formulation allows “topological” ﬁeld conﬁgurations: at a 6= 0

I [Polyakov; 1975,1976]: linearly conﬁning potential for static charges

I [G¨opfert,Mack;1982]: analytic proof of non-zero mass gap and string tension

5 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Compact QED

U(1) gauge theory (QED) without matter ﬁelds in a three-dimensional spacetime

Note:

I in classical electrodynamics in 2 space dimensions the Coulombic potential is logarithmic

Compact U(1) on the lattice:

Compact formulation allows “topological” ﬁeld conﬁgurations: at a 6= 0

I [Polyakov; 1975,1976]: linearly conﬁning potential for static charges

I [G¨opfert,Mack;1982]: analytic proof of non-zero mass gap and string tension

5 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Compact QED

U(1) gauge theory (QED) without matter ﬁelds in a three-dimensional spacetime

Note:

I in classical electrodynamics in 2 space dimensions the Coulombic potential is logarithmic

Compact U(1) on the lattice:

Compact formulation allows “topological” ﬁeld conﬁgurations: at a 6= 0

I [Polyakov; 1975,1976]: linearly conﬁning potential for static charges

I [G¨opfert,Mack;1982]: analytic proof of non-zero mass gap and string tension

5 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 A phase transition at ﬁnite T

Compact U(1) at non-zero temperature T = 1 Nt a

I similar to SU(N) pure-gauge theories

I phase transition at a critical temperature Tc

low-temperature phase, T < Tc high-temperature region, T > Tc nonzero mass gap no mass scale (or Debye screening) linearly conﬁning potential “ logarithmic conﬁnement”

hP?(r)P(0)i ∼ exp(−r/ξ(t)) hP?(r)P(0)i ∼ r −η(T )

Recent results [Borisenko et al.; 2015]:

I U(1) phase transition in the universality class of XY model (η(Tc ) = 1/4)

I inﬁnite-order phase transition (exponentially-diverging correlation length)

ξ(t) = exp(bt−ν )

I diﬃcult to study close to Tc , large volumes required

6 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 A phase transition at ﬁnite T

Compact U(1) at non-zero temperature T = 1 Nt a

I similar to SU(N) pure-gauge theories

I phase transition at a critical temperature Tc

low-temperature phase, T < Tc high-temperature region, T > Tc nonzero mass gap no mass scale (or Debye screening) linearly conﬁning potential “ logarithmic conﬁnement”

hP?(r)P(0)i ∼ exp(−r/ξ(t)) hP?(r)P(0)i ∼ r −η(T )

Recent results [Borisenko et al.; 2015]:

I U(1) phase transition in the universality class of XY model (η(Tc ) = 1/4)

I inﬁnite-order phase transition (exponentially-diverging correlation length)

ξ(t) = exp(bt−ν )

I diﬃcult to study close to Tc , large volumes required

6 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 A conformal description for the low-T phase of the XY model

In the XY model in two dimensions for T < TKT r −η G(r) = h~s(x) · ~s(y)i ∼ L

I the correlation function is scale-invariant!

I i.e. invariant under global dilatation transformations

x → λx

For classical systems with local interactions: invariance under global dilatations + translations + rotations ⇒ conformal invariance x → λ(x)x they leave invariant the relative angle between vectors

I a conformal ﬁeld theory (CFT) description of the XY model in the low-T phase is possible

I free, massless compact bosonic ﬁeld with central charge c = 1

7 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 A conformal description for the low-T phase of the XY model

In the XY model in two dimensions for T < TKT r −η G(r) = h~s(x) · ~s(y)i ∼ L

I the correlation function is scale-invariant!

I i.e. invariant under global dilatation transformations

x → λx

I a conformal ﬁeld theory (CFT) description of the XY model in the low-T phase is possible

I free, massless compact bosonic ﬁeld with central charge c = 1

7 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 CFT and the U(1) hot phase - P correlator

Conformal invariance → constraints on correlation functions

We consider quasi-primary ﬁelds Qi that transform under general conformal mapping as

0 ∆i /D ∂x 0 Qi (x) = Qi (x ) ∂x

∆i = “scaling dimension” For the two-point function conformal invariance tells us δ hQ (x )Q (x )i = ab a 1 b 2 2∆ |x1 − x2| a E.g. for the spin ﬁeld ~s 1 h~s(r) · ~s(0)i ∼ r η with η = 2∆s

Identiﬁcation of corresponding operators:

~s −→ P implies 1 hP?(r)P(0)i ∼ r η

8 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 CFT and the U(1) hot phase - P correlator

Conformal invariance → constraints on correlation functions

We consider quasi-primary ﬁelds Qi that transform under general conformal mapping as

0 ∆i /D ∂x 0 Qi (x) = Qi (x ) ∂x

Identiﬁcation of corresponding operators:

~s −→ P implies 1 hP?(r)P(0)i ∼ r η

8 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 In U(1) we investigate the Polyakov loop correlator ratio with

hP?(r + a)P(0)i 1 −η H(r) = = 1 + hP?(r)P(0)i r/a

0.95 0.95 0.95

0.9 0.9 0.9

0.85 0.85 0.85 ) r (

H β = 3.01 β = 3.25 β = 4.1 0.8 0.8 0.8

0.75 0.75 0.75

0.7 0.7 0.7

0.65 0 25 50 0 25 50 0 25 50 r / a η = 0.2469(12) η = 0.2299(10) η = 0.1750(9)

2 Note: β = 1/ag inverse coupling, controls T = 1/aNt via lattice spacing

9 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Behaviour of η(T )

We recall that η(Tc ) = 1/4 A (very) crude ansatz for η(T ) βc η = a1 + a0 β works surprisingly well!

0.29

0.28 0.25

0.27

0.26

η 0.2 0.25

0.24

η 0.23 0.15 0.7 0.8 0.9 1 / 0.22 βcr β 0.21

0.2

0.19

0.18

0.17 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 β

10 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 CFT and the U(1) hot phase - Flux tube proﬁle

Another crucial observable: the proﬁle of the ﬂux tube between a pair of static electric sources

I Connected correlator of the ﬁeld-strength E(x ) in a x t L t background of two Polyakov lines (sources)

? hP (r)P(0)E(xt )i W (r, xt ) = − hE(xt )i hP?(r)P(0)i P(x) P(y)

I E(xt ) is placed at a transverse distance xt from the d temporal plane of the sources Image from [EPJ Web Conf. 175 (2018) 12006]

11 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 CFT and the U(1) hot phase - Flux tube proﬁle

For the three point-function conformal invariance tells us that

cabc hQa(x1)Qb(x2)Qc (x3)i = ∆a−∆b −∆c ∆b −∆a−∆c ∆c −∆a−∆b x12 x13 x23 cabc = structure constant of the algebra of the ﬁelds

For the ﬂux-tube proﬁle we have that

h~s(r)~s(0)φ(x)i c 4x2 −∆φ 4x2 −∆φ W (r, x) → − hφi = ssφ 1 + = C(r) 1 + h~s(r)~s(0)i (r/4)∆φ r 2 r 2 where

I new operator φ (ﬁeld-strength) with (unknown) scaling dimension ∆φ

I φ(x) is placed in the perpendicular line that goes through the midpoint between the two charges

I hφi = 0 due to scale invariance

12 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 CFT and the U(1) hot phase - Flux tube proﬁle

For the three point-function conformal invariance tells us that

cabc hQa(x1)Qb(x2)Qc (x3)i = ∆a−∆b −∆c ∆b −∆a−∆c ∆c −∆a−∆b x12 x13 x23 cabc = structure constant of the algebra of the ﬁelds

For the ﬂux-tube proﬁle we have that

h~s(r)~s(0)φ(x)i c 4x2 −∆φ 4x2 −∆φ W (r, x) → − hφi = ssφ 1 + = C(r) 1 + h~s(r)~s(0)i (r/4)∆φ r 2 r 2 where

I new operator φ (ﬁeld-strength) with (unknown) scaling dimension ∆φ

I φ(x) is placed in the perpendicular line that goes through the midpoint between the two charges

I hφi = 0 due to scale invariance

12 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Field-strength correlator

However:

I ∆φ unknown

I 2-parameter ﬁts of W (x, r) not precise at large r

Direct way of getting ∆φ: looking at the ﬁeld-strength correlator

Y (x) = hE(x)E(0)i to extract ∆φ with " # 1 1 1 1 Y (x) = b0 · + + b1 · + x2∆φ (L − x)2∆φ x4 (L − x)4 contributions from

I the operator φ (scaling dimension ∆φ)

I the marginal operator associated to action density (exact scaling dimension ∆m = 2)

I ﬁnite spatial size L

13 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 β = 4.0, Ns = 192, Nt = 4 1

0.1

0.01 ) x ( Y

0.001

0.0001

1e-05 0 10 20 30 40 50 60 70 80 90 100 x / a

ﬁt with CFT yields ∆φ = 0.946(5)

14 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Proﬁle of the ﬂux-tube (β = 4.0, Nt = 4)

2 −∆φ W (r, x) = C(r) 1 + 4x with input ∆ = 0.946(5) r 2 φ 0.012

L / a = 128 L / a = 192 0.01 L / a = 256

a 0.008 = 32 r 0.006 ), for r , x (

W 0.004

0.002

0 0 20 40 60 80 100 x / a

I extract C(r) for several values of r

I ﬁt C(r) again in ∆φ and cssφ: c C(r) = ssφ (r/4)∆φ

I the ﬁt yields ∆φ = 0.948(6): self-consistent result!

15 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Conclusions

I “expanding” the Svetitsky-Yaﬀe conjecture: the whole hot phase of the U(1) pure gauge theory in 3D can be mapped to the low-T phase of the XY model in 2D

I CFT constraints valid in XY model → functional forms for U(1) correlation functions

I Polyakov loop (static charges) correlator: the critical index η behaves as expected when the moving away from the critical point at T > Tc

I the scaling dimension ∆φ can be extracted independently from the ﬁeld-strength correlator

I when used for the ﬂux-tube proﬁle: consistent picture

16 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Conclusions

I “expanding” the Svetitsky-Yaﬀe conjecture: the whole hot phase of the U(1) pure gauge theory in 3D can be mapped to the low-T phase of the XY model in 2D

I CFT constraints valid in XY model → functional forms for U(1) correlation functions

I Polyakov loop (static charges) correlator: the critical index η behaves as expected when the moving away from the critical point at T > Tc

I the scaling dimension ∆φ can be extracted independently from the ﬁeld-strength correlator

I when used for the ﬂux-tube proﬁle: consistent picture

Thank you for the attention!

16 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 U(1) in 3 dimensions in the literature

Field conﬁgurations admit topological defects (“magnetic monopoles”) [Polyakov; 1975,1976]

I the ground state can be thought as a plasma of (anti)monopoles this condensation implies conﬁnement of electric charges for all values of β = 1 as a dual I ae2 Meissner eﬀect

I dual superconductor model [Nambu; 1974],[Mandelstam; 1976],[’t Hooft; 1979] Interpretation for deconﬁning transition: monopole-antimonopole conﬁning plasma turns into a dipole plasma at Tc [Chernodub et al.; 2001]

17 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 How is Maxwell theory recovered?

Analytical proof [G¨opfert,Mack;1982] (in the Villain formulation) of the existence of

I a non-zero mass gap

p mD a ' k1 β exp(−k2β) for β 1

I a ﬁnite string tension

2 k3 σa ≥ √ exp(−k2β) for β 1 β the model is conﬁning for any value of β = 1 ! ae2 2 I in the continuum limit( a → 0 means β → ∞ at ﬁxed e ) the Maxwell theory of non-interacting photons is recovered!

Interesting note: the ratio behaves differently with respect to other confining gauge theories (where it is fixed up to a effects)

m2 D p 3 ∝ β exp(−k2β) σ

→ one can also take the continuum limit on a “line of constant physics” (i.e. keeping mD or σ ﬁxed) ...

18 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Dual formulation - static potential

Dual formulation of the theory: spin model with global Z symmetry → integer-valued variables s placed on the sites of the dual lattice X Y Z = I|s(y)−s(y+aνˆ)|(β) {s} y,ν

I Iα = modiﬁed Bessel function of 1st kind I s(y) − s(y + aνˆ) diﬀerence of Z variables at the ends of each bond Q I taken on bonds of the dual lattice

Polyakov-loop correlation function rewritten as

Z hP?(r)P(0)i = Nt ×Nr Z

ZNt ×Nr → system with a frustration on the surface bounded by Polyakov loops The ratio hP?(r + a)P(0)i hP?(r)P(0)i can be computed to very high precision → combination of snake algorithm [De Forcrand et al.; 2001] and hierarchical updates [Caselle et al.; 2003]

19 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 Dual formulation - ﬂux-tube proﬁle

In the dual formulation it becomes simply

hs(x) − s(x + aνˆ) + nν (x)i W (r, x) = √ Nt ×Nr β

Note that at T = 0 the proﬁle is exponential [Caselle et al.; 2016] in agreement with the dual superconductor model

20 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 The XY model in two dimensions

The well-known Hamiltonian of the XY model is X X H = −J ~s(x) · ~s(y) = −J cos[θ(x) − θ(y)] hx,yi hx,yi where ~s(x) are two-component real vectors of unit length and θ is the angle with a respect to a predeﬁned direction

It has an O(2) internal global symmetry and is ferromagnetic for J > 0

I for T > 0 the system is in a disordered phase, for which h~si = 0 → Mermin-Wagner theorem systems with continuous symmetries are always disordered by thermal ﬂuctuations in 2D

I “conventional” long-range order not allowed at low-T

I however, for T → 0 a non-conventional “quasi-long-range” order emerges, where the lowest-energy states are spin-waves

I signal of a phase transition at T = TKT

21 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 The Kosterlitz-Thouless transition

One can distinguish two diﬀerent phases by analyzing the behaviour of the spin-spin correlation function G(r) = h~s(x) · ~s(y)i, with r = |x − y|

I in the high-T phase we have r G(r) ∼ exp − ξ

where ξ is a temperature-dependent correlation length that for T → TKT ξ b T ∼ exp √ , τ = − 1 a τ TKT → inﬁnite-order phase transition!

I while in the low-T phase r −η G(r) ∼ L where η depends on T as well

T 1 η(T ) = = η(T ) = 1/4 2πJ 2πK KT

22 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 The Kosterlitz-Thouless transition

Where does this behaviour come from? → “vortex” solutions: when the θ ﬁeld winds around a point (vortex center) n times I ∇θ · d l = 2πn

and to each vortex we associate L energy ∼ πn2J ln a If the entropy goes just like S ∼ ln (L/a)2 then the free energy for an isolated vortex is

L Fv ∼ (πJ − 2T ) ln a and so the creation of a free vortex becomes possible when T π KT = J 2 more complicated with more than one vortex: the actual value is T KT = 0.89294(8) J

23 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019 The Kosterlitz-Thouless transition

Image from [J Phys Condens Matter 25, 065501 (2013)]

at high temperatures: the entropy increase at low temperatures: only compensates the energy for the creation of vortex-antivortex pairs are energetically an isolated vortex → free vortices possible in the thermodynamic limit proliferate and interact with each other with a logarithmic Coulomb potential

24 Alessandro Nada (DESY) CFT in 3D U(1) gauge theory 13/12/2019