Small bits of cold, dense matter

Alessandro Roggero (LANL)with: S.Gandolfi & J.Carlson (LANL), J.Lynn (TUD) and S.Reddy (INT)

ArXiv:1712.10236

Nuclear ab initio Theories and Neutrino PhysicsINT - Seattle - 12 March, 2018

The (conjectured) QCD phase diagramfigure from Fukushima & Hatsuda (2011)

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 1 / 19

The (conjectured) QCD phase diagramfigure from Fukushima & Hatsuda (2011)

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 1 / 19

Where to find cold dense QCD matterOptical + X-ray images from NASA/ESA

Neutron Starsborn in the aftermath of core-collapse supernovaemassive and compact objects: R ≈ 10 km M ≈ 1− 2 M�central densities can be several times nuclear density ρ0 ≈ 1014gr/cm3

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 2 / 19

Can measurements of neutron stars help?

Masses and radiiDemorest et al. (2010)

7 8 9 10 11 12 13 14 15Radius (km)

0.0

0.5

1.0

1.5

2.0

2.5

Mass

(so

lar)

AP4

MS0

MS2

MS1

MPA1

ENG

AP3

GM3PAL6

GS1

PAL1

SQM1

SQM3

FSU

GR

P < �

causality

rotation

J1614-2230

J1903+0327

J1909-3744

Double NS Systems

Nucleons Nucleons+ExoticStrange Quark Matter

pure quark stars disfavoredhybrid stars still compatible

Alford et al. (2004)Zdunik & Hansel (2013)Lastowiecki et al. (2015)

Alford & Han (2016)Baym et al. (2017)

Neutrino coolingdirect URCA produce fast cooling in the core (Iwamoto (1980))

d→ u+ e− + νe u+ e− → d+ νe

quark pairing can drastically reduce emissivity (Page et al. (2000))

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 3 / 19

Ab initio calculations with Lattice QCDfigure from Fukushima & Hatsuda (2011)

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 4 / 19

Using Lattice QCD at fixed A and T = 0

NPLQCD,PACS,HAL,CalLatFukushima & Hatsuda (2011)

Sign problem in LQCD

ELQCDG = EG +O(e−(E1−EG)τ

)∆E/E ∼ eA(MN− 3

2mπ)τ

Systems in small volumes with low Aare becoming possible

What can we learn from these small systems?

nucleon interactions by matching to EFT (eg. P.Klos et al. (2016) )signatures of high-density phase transitions (this talk)

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 5 / 19

Detecting change in degrees of freedom in small boxesIDEA

perform simulations in both nucleonic and quark modelsstudy energy spectrum as a function of Areduce box size L to reach higher densities (few times n0)

Features in the spectra determined by the high momenta (short distances)arising in small periodic volumes. In particular we look for:

shell closures (large gaps in single-particle energies)pairing effects in open-shell systems

CAVEATSlarge corrections from long-range physics for small box size L

qualitatively similar to those in LQCD if pions are consistently includedcan study effect by using calculations for larger mπ (easier)

small systems won’t capture critical behavior (only qualitative)

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 6 / 19

Detecting change in degrees of freedom in small boxesIDEA

perform simulations in both nucleonic and quark modelsstudy energy spectrum as a function of Areduce box size L to reach higher densities (few times n0)

Features in the spectra determined by the high momenta (short distances)arising in small periodic volumes. In particular we look for:

shell closures (large gaps in single-particle energies)pairing effects in open-shell systems

CAVEATSlarge corrections from long-range physics for small box size L

qualitatively similar to those in LQCD if pions are consistently includedcan study effect by using calculations for larger mπ (easier)

small systems won’t capture critical behavior (only qualitative)

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 6 / 19

Nucleonic models

for now limited to systems of non-relativistic neutrons only

HN = − ~2

2MN

∑i

∇2i +

∑i<j

Vij

We choose 3 different realistic NN interactions (match exp. phase-shifts)Argonne AV18 ( R.Wiringa, V.Stoks and R.Schiavilla (1995) )Argonne AV8’ ( R.Wiringa & S.Pieper (2002) )local chiral EFT at N2LO ( A.Gezerlis et al. (2014) )

Many-body forces will play a role but:the features we identify depend predominantly on the single-particlestates and some of them are present even for non-interacting nucleonsthe structure of 3-,4- and many-body forces will be very different

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 7 / 19

Quark modelswe consider both free and interacting SU(2)f quarks

Hq =∑i

Ti +∑i<j

Vij + Vc

for free quarks we use both relativistic and non-relativistic dispersionfor interacting quarks we assume χ-symmetry is not restored and usemq = 0.3GeV with non-relativistic dispersion

pair interaction is color antisymmetric and tuned to large a tomaximize pairing: ∆q ∼ 20− 100MeV ( Rajagopal & Wilczek (2000))average confining interaction per baryon Vc depending on nB only

we impose baryon number and color, charge and spin neutrality

DISCLAMER: model is not intended to give realistic description of QCD inthis regime but to be qualitatively correct: ie. to reproducethe 2SC color superconducting state (M.Alford et al. (2008))

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 8 / 19

Quark modelswe consider both free and interacting SU(2)f quarks

Hq =∑i

Ti +∑i<j

Vij + Vc

for free quarks we use both relativistic and non-relativistic dispersionfor interacting quarks we assume χ-symmetry is not restored and usemq = 0.3GeV with non-relativistic dispersionpair interaction is color antisymmetric and tuned to large a tomaximize pairing: ∆q ∼ 20− 100MeV ( Rajagopal & Wilczek (2000))

average confining interaction per baryon Vc depending on nB only

we impose baryon number and color, charge and spin neutrality

DISCLAMER: model is not intended to give realistic description of QCD inthis regime but to be qualitatively correct: ie. to reproducethe 2SC color superconducting state (M.Alford et al. (2008))

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 8 / 19

Quark modelswe consider both free and interacting SU(2)f quarks

Hq =∑i

Ti +∑i<j

Vij + Vc

for free quarks we use both relativistic and non-relativistic dispersionfor interacting quarks we assume χ-symmetry is not restored and usemq = 0.3GeV with non-relativistic dispersionpair interaction is color antisymmetric and tuned to large a tomaximize pairing: ∆q ∼ 20− 100MeV ( Rajagopal & Wilczek (2000))average confining interaction per baryon Vc depending on nB only

we impose baryon number and color, charge and spin neutrality

DISCLAMER: model is not intended to give realistic description of QCD inthis regime but to be qualitatively correct: ie. to reproducethe 2SC color superconducting state (M.Alford et al. (2008))

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 8 / 19

Quark modelswe consider both free and interacting SU(2)f quarks

Hq =∑i

Ti +∑i<j

Vij + Vc

for free quarks we use both relativistic and non-relativistic dispersionfor interacting quarks we assume χ-symmetry is not restored and usemq = 0.3GeV with non-relativistic dispersionpair interaction is color antisymmetric and tuned to large a tomaximize pairing: ∆q ∼ 20− 100MeV ( Rajagopal & Wilczek (2000))average confining interaction per baryon Vc depending on nB only

we impose baryon number and color, charge and spin neutrality

DISCLAMER: model is not intended to give realistic description of QCD inthis regime but to be qualitatively correct: ie. to reproducethe 2SC color superconducting state (M.Alford et al. (2008))

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 8 / 19

Pairing in quark matter

Attractive one-gluon interaction destabilize Fermi surface ⇒ Cooper pairs

intermediate densities: 2SCRapp et al. (1998), Alford et al. (1998)

2-flavor pairing

large densities: CFLAlford et al. (1999)

9 quark pairing: µ�M2s /∆

figures from Reddy (2004)

structure of 2SC ground state: |Ψ2SC〉 =|ΨfreeB 〉⊗|ΨSF

RG〉

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 9 / 19

Quantum Monte Carlo calculationsfor reviews see Foulkes et al. (2001), Carlson et al. (2015)

BASIC IDEAany quantum system relaxes to its ground-state at low-enough temperature

Given hamiltonian H: E(β) =Tr[He−βH ]Tr[e−βH ]

β�1−−−→ E0 +O(e−β(E1−E0)

)map quantum problem to a classical one with more degrees of freedomuse classical Monte Carlo to evaluate E(β) with statistical uncertainty

problem: direct evaluation usually comes with large noisesolution: modify expectation value using good reference state |Φr〉

E(β) =Tr[He−βH |Φr〉〈Φr|

]Tr [e−βH |Φr〉〈Φr|]

NOTE: traces are in a restricted space now

β�1−−−→〈Ψ0|Φr〉〈Φr|H|Ψ0〉+O

(e−β(E1−E0)

)|〈Φr|Ψ0〉|2 +O

(e−β(E1−E0)

) = E0 +O(e−β(E1−E0)

)

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 10 / 19

Quantum Monte Carlo calculationsfor reviews see Foulkes et al. (2001), Carlson et al. (2015)

BASIC IDEAany quantum system relaxes to its ground-state at low-enough temperature

Given hamiltonian H: E(β) =Tr[He−βH ]Tr[e−βH ]

β�1−−−→ E0 +O(e−β(E1−E0)

)map quantum problem to a classical one with more degrees of freedomuse classical Monte Carlo to evaluate E(β) with statistical uncertainty

problem: direct evaluation usually comes with large noisesolution: modify expectation value using good reference state |Φr〉

E(β) =Tr[He−βH |Φr〉〈Φr|

]Tr [e−βH |Φr〉〈Φr|]

NOTE: traces are in a restricted space now

β�1−−−→〈Ψ0|Φr〉〈Φr|H|Ψ0〉+O

(e−β(E1−E0)

)|〈Φr|Ψ0〉|2 +O

(e−β(E1−E0)

) = E0 +O(e−β(E1−E0)

)

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 10 / 19

Quantum Monte Carlo calculationsfor reviews see Foulkes et al. (2001), Carlson et al. (2015)

BASIC IDEAany quantum system relaxes to its ground-state at low-enough temperature

Given hamiltonian H: E(β) =Tr[He−βH ]Tr[e−βH ]

β�1−−−→ E0 +O(e−β(E1−E0)

)map quantum problem to a classical one with more degrees of freedomuse classical Monte Carlo to evaluate E(β) with statistical uncertainty

problem: direct evaluation usually comes with large noisesolution: modify expectation value using good reference state |Φr〉

E(β) =Tr[He−βH |Φr〉〈Φr|

]Tr [e−βH |Φr〉〈Φr|]

NOTE: traces are in a restricted space now

β�1−−−→〈Ψ0|Φr〉〈Φr|H|Ψ0〉+O

(e−β(E1−E0)

)|〈Φr|Ψ0〉|2 +O

(e−β(E1−E0)

) = E0 +O(e−β(E1−E0)

)Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 10 / 19

Quantum Monte Carlo calculations: fermions and pairing

Fermionic states have non trivial phases (entanglement)sign problem: statistical noise grows exponentially with β (cf. LQCD)

E(β)→ E0 +O(e−β(E1−E0)

)∆E(β)→ O

(eβ(E

F0 −EB0 )

)

phase constrained Quantum Monte Carlo (Ortiz et al. (1993))

E(β) =Tr[He−βH |Φr〉〈Φr|

]Tr [e−βH |Φr〉〈Φr|]

→ EC(β) =TrC

[He−βH |Φr〉〈Φr|

]TrC [e−βH |Φr〉〈Φr|]

Restricted traces over states |ψ〉 such that: 〈ψ|Φr〉 > 0⇒|ψ〉 fermionicSystems with superfluid pairing can be described by appropriate |Ψr〉

|ΦSFr 〉 = A|ΨA

12〉⊗|ΨB34〉⊗|ΨC

56〉 ⊗ · · · (eg. Carlson et al. (2003))

By choosing the same (s–wave) state for every pair |ΦSFr 〉 =|ΦBCS

r 〉

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 11 / 19

Quantum Monte Carlo calculations: fermions and pairing

Fermionic states have non trivial phases (entanglement)sign problem: statistical noise grows exponentially with β (cf. LQCD)

E(β)→ E0 +O(e−β(E1−E0)

)∆E(β)→ O

(eβ(E

F0 −EB0 )

)phase constrained Quantum Monte Carlo (Ortiz et al. (1993))

E(β) =Tr[He−βH |Φr〉〈Φr|

]Tr [e−βH |Φr〉〈Φr|]

→ EC(β) =TrC

[He−βH |Φr〉〈Φr|

]TrC [e−βH |Φr〉〈Φr|]

Restricted traces over states |ψ〉 such that: 〈ψ|Φr〉 > 0⇒|ψ〉 fermionic

Systems with superfluid pairing can be described by appropriate |Ψr〉

|ΦSFr 〉 = A|ΨA

12〉⊗|ΨB34〉⊗|ΨC

56〉 ⊗ · · · (eg. Carlson et al. (2003))

By choosing the same (s–wave) state for every pair |ΦSFr 〉 =|ΦBCS

r 〉

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 11 / 19

Quantum Monte Carlo calculations: fermions and pairing

Fermionic states have non trivial phases (entanglement)sign problem: statistical noise grows exponentially with β (cf. LQCD)

E(β)→ E0 +O(e−β(E1−E0)

)∆E(β)→ O

(eβ(E

F0 −EB0 )

)phase constrained Quantum Monte Carlo (Ortiz et al. (1993))

E(β) =Tr[He−βH |Φr〉〈Φr|

]Tr [e−βH |Φr〉〈Φr|]

→ EC(β) =TrC

[He−βH |Φr〉〈Φr|

]TrC [e−βH |Φr〉〈Φr|]

Restricted traces over states |ψ〉 such that: 〈ψ|Φr〉 > 0⇒|ψ〉 fermionicSystems with superfluid pairing can be described by appropriate |Ψr〉

|ΦSFr 〉 = A|ΨA

12〉⊗|ΨB34〉⊗|ΨC

56〉 ⊗ · · · (eg. Carlson et al. (2003))

By choosing the same (s–wave) state for every pair |ΦSFr 〉 =|ΦBCS

r 〉

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 11 / 19

4 neutrons in a box: free caseGandolfi, Carlson, Roggero, Lynn, Reddy (2017)

Particle mass (GeV) N ρ (fm−3) E (k=1, GeV)Nucleon 0.94 4 0.16 0.096rel q 0.0 4 0.16 0.424rel q 0.3 4 0.16 0.219non-rel q 0.3 4 0.16 0.299Nucleon 0.94 4 0.32 0.152rel q 0.0 4 0.32 0.534rel q 0.3 4 0.32 0.313non-rel q 0.3 4 0.32 0.476

nuclear case: two nucleons in the k = 0 shell and two in |k| = 1 shellquark case: all quarks in k = 0 shell costs twice MN −M∆ (≈ 300MeV )

at low density (large volumes) 4 neutron system is favoredat large density (small volume) all quarks in k = 0 shell preferable

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 12 / 19

4 neutrons in a box: surprise in the interacting caseref. states with different symmetry: |Φ4N

r 〉 = A|ψS,k=012 〉⊗|ψX,|k|=1

34 〉

NN interaction favor d-wave pairing for the 2 neutrons in |k| = 1 state

Gandolfi, Carlson, Roggero, Lynn, Reddy (2017)

0 0.08 0.16 0.24 0.32 0.4 (fm­3)

0

50

100

150

200

250

E/N

 (MeV

)

N2LO d­wave N2LO p­wave N2LO s­wave AV18 d­waveAV18 p­waveAV18 s­wave

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 13 / 19

4 neutrons in a box: surprise in the interacting caseref. states with different symmetry: |Φ4N

r 〉 = A|ψS,k=012 〉⊗|ψX,|k|=1

34 〉

NN interaction favor d-wave pairing for the 2 neutrons in |k| = 1 state

Gandolfi, Carlson, Roggero, Lynn, Reddy (2017)

0 0.08 0.16 0.24 0.32 0.4 (fm­3)

0

50

100

150

200

250

E/N

 (MeV

)

N2LO d­wave N2LO p­wave N2LO s­wave AV18 d­waveAV18 p­waveAV18 s­wave

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 13 / 19

4 neutrons in a box: interacting case (phase shifts)Gandolfi, Carlson, Roggero, Lynn, Reddy (2017)

0 50 100 150 200 250 300

ECM

(MeV)

-40

-20

0

20

40

600 .08 0.16 0.24 0.32

ρ (N=4)

1S0

3P2

1D2

AV18AV8’N2LO

s-wave interaction turns repulsive at large momenta (small volumes)d-wave state favored over p-wave due to spatial symmetry of wf

d-wave state is symmetric across periodic boundaries → attractiveinterference between periodic images for p-wave state

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 14 / 19

4 neutrons in a box: comparison with quarks modelsGandolfi, Carlson, Roggero, Lynn, Reddy (2017)

0 0.08 0.16 0.24 0.32 0.4 (fm­3)

0

50

100

150

200

250

E/N

 (MeV

)

N2LO d­wave N2LO p­wave N2LO s­wave AV18 d­waveAV18 p­waveAV18 s­wave

nuclear case: at ρ = 2ρ0 s-d gap is ' 75 MeV for both interactionsquark case: at ρ = 2− 3ρ0 s-wave is still ≈ 30− 40 MeV lower in energy

conclusions are not altered by inclusion of spin dependent gluonexchange interactions (Carlson, Kogut and Pandharipande (1983))

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 15 / 19

Evolution of the spectrum with particle numberGandolfi, Carlson, Roggero, Lynn, Reddy (2017)

2 4 6 8 10 12 14N

0

20

40

60

80

100

120

E/N

(M

eV

)

ρ = 0.48

ρ = 0.32

ρ = 0.24

ρ = 0.16

ρ = 0.08

Model dependence

0 0.1 0.2 0.3 0.4 0.5

ρ (fm-3

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

E/E

FG

N=14 (AV8’)

N=14 (AV18)

N=10N=8N=6

dashed lines free neutronspoints AV18 interaction

N = 14 corresponds to closed neutron shell with |k| = 0, 1

maximum at N = 6 always presentat high density large ≈ 10 MeV differences among neighboring Nvery large gap of ≈ 50 MeV between N = 6 and N = 14

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 16 / 19

Spectra for nucleons and quarks at ρ = 2ρ0

full energiescontaining therest massconfinementenergy (assumedconstant with ρ)to match resultfor N = 14neutrons

Gandolfi, Carlson, Roggero, Lynn, Reddy (2017)

2 4 6 8 10 12 14 16 18# Baryons

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

E/N

(G

eV)

AV18 neutronsFree neutronsPaired quarks (m=0.3 NR)

SU(2) free quarks (m=0.3 NR)

SU(2) free quarks (m=0.3 R)

SU(2) free quarks (m=0)

for both quarks and nucleons no qualitative change from interactionsexception: interacting quarks show shell closure at N = 14 (blue-q)

sizable N = 4− 6 gap comes from pairs filling |k| = 1 quark shell

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 17 / 19

Spectra for nucleons and quarks at ρ = 2ρ0

full energiescontaining therest massconfinementenergy (assumedconstant with ρ)to match resultfor N = 14neutrons

Gandolfi, Carlson, Roggero, Lynn, Reddy (2017)

2 4 6 8 10 12 14 16 18# Baryons

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

E/N

(G

eV)

AV18 neutronsFree neutronsPaired quarks (m=0.3 NR)

SU(2) free quarks (m=0.3 NR)

SU(2) free quarks (m=0.3 R)

SU(2) free quarks (m=0)

for both quarks and nucleons no qualitative change from interactionsexception: interacting quarks show shell closure at N = 14 (blue-q)

sizable N = 4− 6 gap comes from pairs filling |k| = 1 quark shell

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 17 / 19

Spectra for nucleons and quarks at ρ = 3ρ0

Gandolfi, Carlson, Roggero, Lynn, Reddy (2017)

2 4 6 8 10 12 14 16 18# Baryons

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

E/N

(G

eV)

AV18 neutronsFree neutronsPaired quarks (m=0.3 NR)

SU(2) free quarks (m=0.3 NR)

SU(2) free quarks (m=0.3 R)

SU(2) free quarks (m=0)

even larger effect for N = 6 due to larger energy of |k| = 1 statesquarks nucleons

free interacting free interactingE46(2ρ0) 120 MeV 130 MeV 3 MeV 4 MeVE46(3ρ0) 170 MeV 310 MeV 2 MeV 15 MeV

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 18 / 19

Summary

Useful signatures to tell apart relevant degrees of freedompairing pattern for N = 4: EN (s) > EN (d) vs. Eq(s) ≤ Eq(d)

N = 4− 6 energy gap much larger for quarks (seems robust)shell closure at N = 14 if quarks are free (difficult for near term)

Future directionsexplore effects of heavier pion masses

LQCD can be extended to larger A (much) more easilynuclear interactions become (much) more controllable with EFT(/π)

add protons and/or hyperons to the system

Thanks for your attention

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 19 / 19

Summary

Useful signatures to tell apart relevant degrees of freedompairing pattern for N = 4: EN (s) > EN (d) vs. Eq(s) ≤ Eq(d)

N = 4− 6 energy gap much larger for quarks (seems robust)shell closure at N = 14 if quarks are free (difficult for near term)

Future directionsexplore effects of heavier pion masses

LQCD can be extended to larger A (much) more easilynuclear interactions become (much) more controllable with EFT(/π)

add protons and/or hyperons to the system

Thanks for your attention

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 19 / 19

Summary

Useful signatures to tell apart relevant degrees of freedompairing pattern for N = 4: EN (s) > EN (d) vs. Eq(s) ≤ Eq(d)

N = 4− 6 energy gap much larger for quarks (seems robust)shell closure at N = 14 if quarks are free (difficult for near term)

Future directionsexplore effects of heavier pion masses

LQCD can be extended to larger A (much) more easilynuclear interactions become (much) more controllable with EFT(/π)

add protons and/or hyperons to the system

Thanks for your attention

Alessandro Roggero Small bits of cold, dense matter INT - 12 Mar, 2018 19 / 19