Fermion Bag Approach to Hamiltonian Lattice Field Theories

Shailesh Chandrasekharan (Duke University)

work done in collaboration with Emilie Huffman

XQCD 2017, Pisa Italy

Supported by: US Department of Energy, Nuclear Physics Division

Motivation Motivation

Physics of massless Dirac fermions in 2+1 dimensions is becoming exciting in condensed matter physics. Motivation

Physics of massless Dirac fermions in 2+1 dimensions is becoming exciting in condensed matter physics.

Many strongly correlated materials seem to contain such fermions at low energies. Graphene, Topological Insulators, …. Reviews by: Moore, Vishwanath, … Motivation

Physics of massless Dirac fermions in 2+1 dimensions is becoming exciting in condensed matter physics.

Many strongly correlated materials seem to contain such fermions at low energies. Graphene, Topological Insulators, …. Reviews by: Moore, Vishwanath, …

The physics is described by lattice model Hamiltonians, especially with four-fermion interactions. Motivation

Physics of massless Dirac fermions in 2+1 dimensions is becoming exciting in condensed matter physics.

Many strongly correlated materials seem to contain such fermions at low energies. Graphene, Topological Insulators, …. Reviews by: Moore, Vishwanath, …

The physics is described by lattice model Hamiltonians, especially with four-fermion interactions.

They can contain rich phase diagrams, with many interesting “quantum critical points.” Many recent papers on the subject: Herbut, Gies,…. New Exotic Critical Points: V. Ayyar and S.C, 2014 Motivation

Physics of massless Dirac fermions in 2+1 dimensions is becoming exciting in condensed matter physics.

Many strongly correlated materials seem to contain such fermions at low energies. Graphene, Topological Insulators, …. Reviews by: Moore, Vishwanath, …

The physics is described by lattice model Hamiltonians, especially with four-fermion interactions.

They can contain rich phase diagrams, with many interesting “quantum critical points.” Many recent papers on the subject: Herbut, Gies,…. New Exotic Critical Points: V. Ayyar and S.C, 2014

Develop non-perturbative approaches to study them!

But haven’t they been studied earlier by the lattice community? But haven’t they been studied earlier by the lattice community?

In some cases yes, but usually on small lattices and not in the massless limit!

Our goal is to be able to go to larger systems and with exactly massless fermions. Also Narayanan at this conference, …. But haven’t they been studied earlier by the lattice community?

In some cases yes, but usually on small lattices and not in the massless limit!

Our goal is to be able to go to larger systems and with exactly massless fermions. Also Narayanan at this conference, ….

In some cases no, due to sign problems! New solutions have emerged recently! Huffman and S.C, 2013 Li, Jiang and Yao, 2014 Wei, Wu, Li, Zhang and Xiang, 2016 But haven’t they been studied earlier by the lattice community?

In some cases yes, but usually on small lattices and not in the massless limit!

Our goal is to be able to go to larger systems and with exactly massless fermions. Also Narayanan at this conference, ….

In some cases no, due to sign problems! New solutions have emerged recently! Huffman and S.C, 2013 Li, Jiang and Yao, 2014 Wei, Wu, Li, Zhang and Xiang, 2016

So new models have become accessible and some old models can now be solved more efficiently!

Previous work using the Lagrangian Lattice Field Theory Debbio, Hands, Kogut, Kocic,Kim, Strouthos, Sinclair,…(1990’s, 2000’s) Drut and Lahde, 2009, (Staggered fermions, Coulomb Interactions) Typical lattice sizes used less than 2500 spatial sites, but non-zero mass! Previous work using the Lagrangian Lattice Field Theory Debbio, Hands, Kogut, Kocic,Kim, Strouthos, Sinclair,…(1990’s, 2000’s) Drut and Lahde, 2009, (Staggered fermions, Coulomb Interactions) Typical lattice sizes used less than 2500 spatial sites, but non-zero mass!

Most previous work involved staggered or Wilson fermions DWF, Hands (2016) SLAC fermions, Wipf (2017) Previous work using the Lagrangian Lattice Field Theory Debbio, Hands, Kogut, Kocic,Kim, Strouthos, Sinclair,…(1990’s, 2000’s) Drut and Lahde, 2009, (Staggered fermions, Coulomb Interactions) Typical lattice sizes used less than 2500 spatial sites, but non-zero mass!

Most previous work involved staggered or Wilson fermions DWF, Hands (2016) SLAC fermions, Wipf (2017)

Limitations: - Wilson and staggered fermions break important symmetries. - In some cases suffer from sign problems. - HMC method allows one to go to large lattices, but needs extrapolations to m=0. Previous work using the Lagrangian Lattice Field Theory Debbio, Hands, Kogut, Kocic,Kim, Strouthos, Sinclair,…(1990’s, 2000’s) Drut and Lahde, 2009, (Staggered fermions, Coulomb Interactions) Typical lattice sizes used less than 2500 spatial sites, but non-zero mass!

Most previous work involved staggered or Wilson fermions DWF, Hands (2016) SLAC fermions, Wipf (2017)

Limitations: - Wilson and staggered fermions break important symmetries. - In some cases suffer from sign problems. - HMC method allows one to go to large lattices, but needs extrapolations to m=0.

Determination of critical exponents have remained unsatisfactory. Room for improvement.

Previous work using Hamiltonian Lattice Field Theory Assaad, Hirsh, Scalapino, Scalettar, Troyer,….. Previous work using Hamiltonian Lattice Field Theory Assaad, Hirsh, Scalapino, Scalettar, Troyer,….. Most condensed matter theorists use this framework, since problems are naturally formulated as a lattice Hamiltonian. Previous work using Hamiltonian Lattice Field Theory Assaad, Hirsh, Scalapino, Scalettar, Troyer,….. Most condensed matter theorists use this framework, since problems are naturally formulated as a lattice Hamiltonian. The current state of the art algorithm: “auxiliary field determinantal Monte Carlo methods”. Review Article: Assaad,…. Previous work using Hamiltonian Lattice Field Theory Assaad, Hirsh, Scalapino, Scalettar, Troyer,….. Most condensed matter theorists use this framework, since problems are naturally formulated as a lattice Hamiltonian. The current state of the art algorithm: “auxiliary field determinantal Monte Carlo methods”. Review Article: Assaad,….

Most calculations are done with massless fermions Continuous time limit is considered sacred! It avoids an extra fermion doubling, but may not be necessary. Algorithms scale as V3/T Previous work using Hamiltonian Lattice Field Theory Assaad, Hirsh, Scalapino, Scalettar, Troyer,….. Most condensed matter theorists use this framework, since problems are naturally formulated as a lattice Hamiltonian. The current state of the art algorithm: “auxiliary field determinantal Monte Carlo methods”. Review Article: Assaad,….

Most calculations are done with massless fermions Continuous time limit is considered sacred! It avoids an extra fermion doubling, but may not be necessary. Algorithms scale as V3/T

Biggest spatial lattices studied: Otsuka, Yunoki, Sorella, (2016) - 2600 sites (honeycomb lattice) - 1600 sites (square lattice) Are there examples of unsolved problems, where progress would be useful?

Classic Example Classic Example

“Repulsive Hubbard (t-U) Model” for graphene

1 1 H = t c† ci, + U ni, ni, i, " 2 # 2 ij , i hXi X ⇣ ⌘⇣ ⌘ Classic Example

“Repulsive Hubbard (t-U) Model” for graphene

1 1 H = t c† ci, + U ni, ni, i, " 2 # 2 ij , i hXi X ⇣ ⌘⇣ ⌘

Describes Nf = 2, 4-component massless Dirac fermions Classic Example

“Repulsive Hubbard (t-U) Model” for graphene

1 1 H = t c† ci, + U ni, ni, i, " 2 # 2 ij , i hXi X ⇣ ⌘⇣ ⌘

Describes Nf = 2, 4-component massless Dirac fermions

Important symmetries of the lattice model: SU(2) x Z2 Classic Example

“Repulsive Hubbard (t-U) Model” for graphene

1 1 H = t c† ci, + U ni, ni, i, " 2 # 2 ij , i hXi X ⇣ ⌘⇣ ⌘

Describes Nf = 2, 4-component massless Dirac fermions

Important symmetries of the lattice model: SU(2) x Z2

massive fermions, massless fermions anti-ferromagnetic order

Uc Classic Example

“Repulsive Hubbard (t-U) Model” for graphene

1 1 H = t c† ci, + U ni, ni, i, " 2 # 2 ij , i hXi X ⇣ ⌘⇣ ⌘

Describes Nf = 2, 4-component massless Dirac fermions

Important symmetries of the lattice model: SU(2) x Z2

massive fermions, massless fermions anti-ferromagnetic order

Uc

Can we reproduce the physics close to Uc with staggered fermions?

Staggered fermion approach: Hands et. al, 2009 (Nf = 2, 4 component massless Dirac fermions)

1 S = M U 2 x,i x,y y,i x,1 x,2 y,1 x,2 x,y,i=1,2 xy X Xh i Staggered fermion approach: Hands et. al, 2009 (Nf = 2, 4 component massless Dirac fermions)

1 S = M U 2 x,i x,y y,i x,1 x,2 y,1 x,2 x,y,i=1,2 xy X Xh i Lattice model has a U(1) chiral symmetry, but has a SU(2) “flavor” symmetry which is not broken at the critical point. Staggered fermion approach: Hands et. al, 2009 (Nf = 2, 4 component massless Dirac fermions)

1 S = M U 2 x,i x,y y,i x,1 x,2 y,1 x,2 x,y,i=1,2 xy X Xh i Lattice model has a U(1) chiral symmetry, but has a SU(2) “flavor” symmetry which is not broken at the critical point.

The universality class seems different (?) Staggered fermion approach: Hands et. al, 2009 (Nf = 2, 4 component massless Dirac fermions)

1 S = M U 2 x,i x,y y,i x,1 x,2 y,1 x,2 x,y,i=1,2 xy X Xh i Lattice model has a U(1) chiral symmetry, but has a SU(2) “flavor” symmetry which is not broken at the critical point.

The universality class seems different (?)

Fermion Bag Approach, ⌫ = 0.82(2), ⌘ = 0.65(2) above model (403 lattices): SC and A.Li, 2012 Today we can go to bigger lattices Staggered fermion approach: Hands et. al, 2009 (Nf = 2, 4 component massless Dirac fermions)

1 S = M U 2 x,i x,y y,i x,1 x,2 y,1 x,2 x,y,i=1,2 xy X Xh i Lattice model has a U(1) chiral symmetry, but has a SU(2) “flavor” symmetry which is not broken at the critical point.

The universality class seems different (?)

Fermion Bag Approach, ⌫ = 0.82(2), ⌘ = 0.65(2) above model (403 lattices): SC and A.Li, 2012 Today we can go to bigger lattices

Auxiliary Field ⌫ = 0.84(4), ⌘ = 0.69(8) Monte Carlo, Parisen, Hohenadler, Assaad, (2015) Hubbard model 800 sites (honeycomb lattice): Staggered fermion approach: Hands et. al, 2009 (Nf = 2, 4 component massless Dirac fermions)

1 S = M U 2 x,i x,y y,i x,1 x,2 y,1 x,2 x,y,i=1,2 xy X Xh i Lattice model has a U(1) chiral symmetry, but has a SU(2) “flavor” symmetry which is not broken at the critical point.

The universality class seems different (?)

Fermion Bag Approach, ⌫ = 0.82(2), ⌘ = 0.65(2) above model (403 lattices): SC and A.Li, 2012 Today we can go to bigger lattices

Auxiliary Field ⌫ = 0.84(4), ⌘ = 0.69(8) Monte Carlo, Parisen, Hohenadler, Assaad, (2015) Hubbard model 800 sites (honeycomb lattice): ⌫ = 1.02(1), ⌘ = 0.50(3) Otsuka, Yunoki, Sorella, (2016) 2592 sites

A simpler example A simpler example

“Repulsive (t-V) Model” for graphene 1 1 H = t c†c + V n n i i i 2 j 2 ij ij Xh i Xh i ⇣ ⌘⇣ ⌘ A simpler example

“Repulsive (t-V) Model” for graphene 1 1 H = t c†c + V n n i i i 2 j 2 ij ij Xh i Xh i ⇣ ⌘⇣ ⌘

Describes Nf = 1, 4-component Dirac fermions A simpler example

“Repulsive (t-V) Model” for graphene 1 1 H = t c†c + V n n i i i 2 j 2 ij ij Xh i Xh i ⇣ ⌘⇣ ⌘

Describes Nf = 1, 4-component Dirac fermions

Important symmetries of the lattice model: Z2 A simpler example

“Repulsive (t-V) Model” for graphene 1 1 H = t c†c + V n n i i i 2 j 2 ij ij Xh i Xh i ⇣ ⌘⇣ ⌘

Describes Nf = 1, 4-component Dirac fermions

Important symmetries of the lattice model: Z2

massive fermions, massless fermions anti-ferromagnetic order

Uc A simpler example

“Repulsive (t-V) Model” for graphene 1 1 H = t c†c + V n n i i i 2 j 2 ij ij Xh i Xh i ⇣ ⌘⇣ ⌘

Describes Nf = 1, 4-component Dirac fermions

Important symmetries of the lattice model: Z2

massive fermions, massless fermions anti-ferromagnetic order

Uc

No lattice field theory results with staggered fermions due to sign problems. Wipf et. al., Lattice (2017)

Sign problem can be solved in the Hamiltonian approach Huffman and S.C, 2013 Sign problem can be solved in the Hamiltonian approach Huffman and S.C, 2013

Solution natural in the Majorana Representations Li, Jiang and Yao, 2014 Wei, Wu, Li, Zhang and Xiang, 2016 Sign problem can be solved in the Hamiltonian approach Huffman and S.C, 2013

Solution natural in the Majorana Representations Li, Jiang and Yao, 2014 Wei, Wu, Li, Zhang and Xiang, 2016

Majorana fermions ⇠ =(c + c†), ⇠ = i(c† c ) i i i i i i it H = (⇠ ⇠ + ⇠ ⇠ )+V ⇠ ⇠ ⇠ ⇠ 2 i j j i i j j i ij Xh i n o

commute

2 Tr(exp( H)) = Pf(G[]) X[] ⇣ ⌘ Sign problem can be solved in the Hamiltonian approach Huffman and S.C, 2013

Solution natural in the Majorana Representations Li, Jiang and Yao, 2014 Wei, Wu, Li, Zhang and Xiang, 2016

Majorana fermions ⇠ =(c + c†), ⇠ = i(c† c ) i i i i i i it H = (⇠ ⇠ + ⇠ ⇠ )+V ⇠ ⇠ ⇠ ⇠ 2 i j j i i j j i ij Xh i n o This structure is not obvious commute in the Lagrangian approach

2 Tr(exp( H)) = Pf(G[]) X[] ⇣ ⌘

Critical Exponents are again disputed: Critical Exponents are again disputed:

Wang, Carboz and Troyer, 2014 ( 225 sites) ⌫ = 0.80(2), ⌘ = 0.30(1) Critical Exponents are again disputed:

Wang, Carboz and Troyer, 2014 ( 225 sites) ⌫ = 0.80(2), ⌘ = 0.30(1)

Li, Jiang and Yao, 2014 (1152 sites) ⌫ = 0.77(3), ⌘ = 0.45(2) Critical Exponents are again disputed:

Wang, Carboz and Troyer, 2014 ( 225 sites) ⌫ = 0.80(2), ⌘ = 0.30(1)

Li, Jiang and Yao, 2014 (1152 sites) ⌫ = 0.77(3), ⌘ = 0.45(2)

Hesselmann and Wessel, 2016 (Finite T, 440 sites)

⌫ = 0.74(4), ⌘ = 0.27(3) Critical Exponents are again disputed:

Wang, Carboz and Troyer, 2014 ( 225 sites) ⌫ = 0.80(2), ⌘ = 0.30(1)

Li, Jiang and Yao, 2014 (1152 sites) ⌫ = 0.77(3), ⌘ = 0.45(2)

Hesselmann and Wessel, 2016 (Finite T, 440 sites)

⌫ = 0.74(4), ⌘ = 0.27(3)

Large scale calculations should help resolve such disputes!

The lattice QCD community could bring its expertise to the field! - Use HMC algorithms to go to large lattices ?

Private Comm.: Fakher Assaad, Simon Catterall. Difficult to use HMC The lattice QCD community could bring its expertise to the field! - Use HMC algorithms to go to large lattices ?

Private Comm.: Fakher Assaad, Simon Catterall. Difficult to use HMC

HMC approach has already been used on smaller lattices away from the massless limit. Clearly improvements may be possible? Ulybishev, Buividovich, Polikarpov,….. The lattice QCD community could bring its expertise to the field! - Use HMC algorithms to go to large lattices ?

Private Comm.: Fakher Assaad, Simon Catterall. Difficult to use HMC

HMC approach has already been used on smaller lattices away from the massless limit. Clearly improvements may be possible? Ulybishev, Buividovich, Polikarpov,…..

Our motivation is to bring the idea of fermion bags to solve lattice Hamiltonian problems.

Fermion Bags S.C, 2008 Fermion Bags S.C, 2008

Idea: Divide the system into smaller regions (bags), so that the fermion dynamics within each region gives a positive weight. Fermion Bags S.C, 2008

Idea: Divide the system into smaller regions (bags), so that the fermion dynamics within each region gives a positive weight.

Advantage: You can update the small region fast! Disadvantage: Requires one to “think” to discover the best fermion bags. Fermion Bags S.C, 2008

Idea: Divide the system into smaller regions (bags), so that the fermion dynamics within each region gives a positive weight.

Advantage: You can update the small region fast! Disadvantage: Requires one to “think” to discover the best fermion bags.

Similarities with other ideas developed in QCD: - Domain decomposition Luscher, 2003 - Local factorization Ce, Giusti and Schaefer, 2016 Fermion Bags S.C, 2008

Idea: Divide the system into smaller regions (bags), so that the fermion dynamics within each region gives a positive weight.

Advantage: You can update the small region fast! Disadvantage: Requires one to “think” to discover the best fermion bags.

Similarities with other ideas developed in QCD: - Domain decomposition Luscher, 2003 - Local factorization Ce, Giusti and Schaefer, 2016

Until now we have used this idea with staggered fermions, on 3 space-time lattices up to lattices 64 with m=0. V. Ayyar and S.C, 2016 Fermion Bags S.C, 2008

Idea: Divide the system into smaller regions (bags), so that the fermion dynamics within each region gives a positive weight.

Advantage: You can update the small region fast! Disadvantage: Requires one to “think” to discover the best fermion bags.

Similarities with other ideas developed in QCD: - Domain decomposition Luscher, 2003 - Local factorization Ce, Giusti and Schaefer, 2016

Until now we have used this idea with staggered fermions, on 3 space-time lattices up to lattices 64 with m=0. V. Ayyar and S.C, 2016

Can we extend the idea to Hamiltonian systems?

Continuous Time (CT) Approach Continuous Time (CT) Approach

In Hamiltonian fermion systems the partition function can be expanded in powers of the interaction in continuous time

H H0 Z =Tre = dt1dt2...dtk Tr e Hint(t1)Hint(t2)...Hint(tk ) Z ⇣ ⌘ CT INT method, Rubtsov,⇣ Lichtenstein,… ⌘ Diagrammatic Determinantal MC, Prokof’ev, Svistunov Continuous Time (CT) Approach

In Hamiltonian fermion systems the partition function can be expanded in powers of the interaction in continuous time

H H0 Z =Tre = dt1dt2...dtk Tr e Hint(t1)Hint(t2)...Hint(tk ) Z ⇣ ⌘ CT INT method, Rubtsov,⇣ Lichtenstein,… ⌘ Diagrammatic Determinantal MC, Prokof’ev, Svistunov

Fermion Bag Idea: H0 =0 Hint = Hij ij Xh i

↵(i⇠i ⇠j +i⇠j ⇠i )/2 ↵(c†c +c†c ) Hij = e = e i j j i

Z = dt1dt2...dtk Tr Hi1j1 (t1)Hi2,j2 (t2)...Hik jk (tk ) X[b] Z ⇣ ⌘

Illustration of the “bond” configuration I For this diagram, all of the bonds can be connected because they share sites in common. They form one fermion bag.

Visualizing the Fermion Bag Idea

I We can make a diagram to represent the partition function, with nearest neighbor bonds representing Hxy inserted within imaginary time.

Illustration of the “bond” configuration

t12 i12 j12 t 11 j10 i10 t10 j11 i11

t9 j9 i9 i8 j8 t8 time t7 i7 j7 j6 i6 t6 i5 j5 t5

j4 i4 t4 t3 i j t 2 2 j3 i3 2 i1 j1 t1 coordinates bond configuration I For this diagram, all of the bonds can be connected because they share sites in common. They form one fermion bag.

Visualizing the Fermion Bag Idea Visualizing the Fermion Bag Idea

I We can make a diagram to represent the partition function, I We can make a diagram to represent the partition function, with nearest neighbor bonds representing H inserted with nearest neighbor bonds representing H insertedxy within imaginary time. xy within imaginary time. I For this diagram,Illustration all ofof thethe bonds“bond” can configuration be connected because they share sites in common. They form one cluster.

t12 t12 i12 j12 i12 j12 i12 j12 t t11 11 j10 i10 j10 i10 j10 i10 t10 t10j11 i11 j11 i11 j11 i11

t9 t9 j9 i9 j9 i9 j9 i9 i8 j8 i8 j8 i8 j8 t8 t t 8 time 7 time t7 i j time i j j i 7 7 i7 j7 j i 7 7 t 6 6 j6 i6 6 6 6 t6 i5 j5 i5 j5 t5 i5 j5 t5 j4 i4 j4 i4 t4 j4 i4 t t4 3 i2 j2 j i2 j2 j t t3 i2 j2 3 i3 3 i3 2 t i1 j1 j3 i3 i1 j1 t1 2 i1 j1 t1 coordinates coordinates coordinates bond configuration Naive fermion bag configuration I For this diagram, all of the bonds can be connected because they share sites in common. They form one fermion bag.

Visualizing the Fermion Bag Idea Visualizing the Fermion Bag Idea

I We can make a diagram to represent the partition function, I We can make a diagram to represent the partition function, with nearest neighbor bonds representing H inserted with nearest neighbor bonds representing H insertedxy within imaginary time. xy within imaginary time. I For this diagram,Illustration all ofof thethe bonds“bond” can configuration be connected because they share sites in common. They form one cluster.

t12 t12 i12 j12 i12 j12 i12 j12 t t11 11 j10 i10 j10 i10 j10 i10 t10 t10j11 i11 j11 i11 j11 i11

t9 t9 j9 i9 j9 i9 j9 i9 i8 j8 i8 j8 i8 j8 t8 t t 8 time 7 time t7 i j time i j j i 7 7 i7 j7 j i 7 7 t 6 6 j6 i6 6 6 6 t6 i5 j5 i5 j5 t5 i5 j5 t5 j4 i4 j4 i4 t4 j4 i4 t t4 3 i2 j2 j i2 j2 j t t3 i2 j2 3 i3 3 i3 2 t i1 j1 j3 i3 i1 j1 t1 2 i1 j1 t1 coordinates coordinates coordinates bond configuration Naive fermion bag configuration

Naive fermion bag approach is equivalent to Diagrammatic Determinantal MC, which is very inefficient

Idea: Fermion Bags split up at high T I The B-matrices in a cluster will commute with B-matrices belonging to any other cluster in a timeslice.

Visualizing the Fermion Bag Idea (cont.)

I On the other hand, if we divide the figure into four time portions (timeslices)... Idea: Fermion Bags split up at high T I We have ten clusters now.

t12 i12 j12 i12 j12 t 11 j10 i10 j10 i10 t10 j11 i11 j11 i11 t9 j9 i9 j9 i9 i8 j8 i8 j8 t8 time t7 time i7 j7 i7 j7 j6 i6 j6 i6 t6 i5 j5 i5 j5 t5 j4 i4 j4 i4 t4 t3 i j i j t 2 2 j3 i3 2 2 j3 i3 2 i1 j1 i1 j1 t1 coordinates coordinates I The B-matrices in a cluster will commute with B-matrices belonging to any other cluster in a timeslice.

Visualizing the Fermion Bag Idea (cont.)

I On the other hand, if we divide the figure into four time portions (timeslices)... Idea: Fermion Bags split up at high T I We have ten clusters now.

t12 i12 j12 i12 j12 t 11 j10 i10 j10 i10 t10 j11 i11 j11 i11 t9 j9 i9 j9 i9 i8 j8 i8 j8 t8 time t7 time i7 j7 i7 j7 j6 i6 j6 i6 t6 i5 j5 i5 j5 t5 j4 i4 j4 i4 t4 t3 i j i j t 2 2 j3 i3 2 2 j3 i3 2 i1 j1 i1 j1 t1 coordinates coordinates

Similar to the local factorization algorithm: Ce, Giusti and Schaefer, 2016

I We can often then calculate weight ratios then as determinants of 30 30 matrices or smaller. We have ⇥ achieved small- equilibration for lattices as large as 100 100! ⇥

Figure: Equilibration of t-V model on a square 100 100 lattice. ( = 4.0) ⇥

Maximum Cluster Size and Equilibration

I We find that for a timeslice of .25, clusters are no bigger than around 30 sites. This holds across lattice sizes.

Fermion bag size as a function of spatial volume

max cluster size 40

● 30 ● ●

20

10 fermion bag size

lattice dimension 50 60 70 80 90 100

L Figure: Average maximum sizes of clusters in each timeslice for equilibrated configurations. Timeslice size is .25. I We can often then calculate weight ratios then as determinants of 30 30 matrices or smaller. We have ⇥ achieved small- equilibration for lattices as large as 100 100! ⇥

Figure: Equilibration of t-V model on a square 100 100 lattice. ( = 4.0) ⇥

Maximum Cluster Size and Equilibration

I We find that for a timeslice of .25, clusters are no bigger than around 30 sites. This holds across lattice sizes.

Fermion bag size as a function of spatial volume

max cluster size 40

● 30 ● ●

20

10 fermion bag size

lattice dimension 50 60 70 80 90 100

L Figure: Average maximum sizes of clustersFast updates in possible each within timeslice time slices! for A similar update is used in auxiliary field MC, equilibratedbut here there is configurations.a bigger gain. Timeslice size is .25.

Maximum Cluster Size and Equilibration

I We find that for a timeslice of .25, clusters are no bigger than around 30 sites. This holds across lattice sizes. I We can often then calculate weight ratios then as determinants of 30 30 matrices or smaller. We have ⇥ achieved small- equilibration for lattices as large as 100 100! ⇥ Equilibration on 100 x 100 lattice, β = 4

bonds

● ● ● ● ● ●●● ● 100 000 ● ● ● ● ● ● ● ● ● ● 80 000 ● ● ● ● 60 000 V=1.2 ● V=1.3 ● ● ● V=1.4 40 000

bond number ●

20 000

● sweeps 2 4 6 8 10 Figure: Average maximum sizes of sweep clusters in each timeslice for Figure: Equilibration of t-V model on equilibrated configurations. Timeslice a square 100 100 lattice. ( = 4.0) size is .25. ⇥ Maximum Cluster Size and Equilibration

I We find that for a timeslice of .25, clusters are no bigger than around 30 sites. This holds across lattice sizes. I We can often then calculate weight ratios then as determinants of 30 30 matrices or smaller. We have ⇥ achieved small- equilibration for lattices as large as 100 100! ⇥ Equilibration on 100 x 100 lattice, β = 4

bonds

● ● ● ● ● ●●● ● 100 000 ● ● ● ● ● ● ● ● ● ● 80 000 ● ● ● ● 60 000 V=1.2 ● V=1.3 ● ● ● V=1.4 40 000

bond number ●

20 000

● sweeps 2 4 6 8 10 Figure: Average maximum sizes of sweep clusters in each timeslice for FakherFigure: Assaad (SignEquilibration 2017): This may be impossible of t-V with model auxiliary field on MC, due to stabilization issues! equilibrated configurations. Timeslice a square 100 100 lattice. ( = 4.0) size is .25. ⇥

Equilibration on 48 x 48 lattices with β = 48

3.5e+05

3e+05

2.5e+05

NBONDS 2e+05

1.5e+05

1e+05 0 20 40 60 80 100 120 140 Core Hours Equilibration on 48 x 48 lattices with β = 48

3.5e+05

3e+05

2.5e+05

NBONDS 2e+05

1.5e+05

1e+05 0 20 40 60 80 100 120 140 Core Hours Almost the biggest lattices ever simulated in the current context!

Scaling of time: Pi-Fluxes near Critical Point

3 I This algorithm has scaling N in accordance with LCT -INT algorithms, as opposed to 3N3 for CT -INT -algorithms. (Wang, Iazzi, Corboz, Troyer, PRB 91 (2015)) I We can see the linear scaling in time at small -values and extrapolate for the time of a full sweep at large . Scaling of the algorithm: L6 β

days

● 1.0

0.8

0.6 ● L=96

● ● L=64 0.4

0.2 ●

● ● Days with a single core! ● β 1.5 2.0 2.5 3.0 3.5 4.0

Figure: Time to doβ one sweep for Figure: Extrapolation to low different values at V = 1.304. temperatures. Confirmed linear scaling with . Scaling of time: Pi-Fluxes near Critical Point

3 I This algorithm has scaling N in accordance with LCT -INT algorithms, as opposed to 3N3 for CT -INT -algorithms. (Wang, Iazzi, Corboz, Troyer, PRB 91 (2015)) I We can see the linear scaling in time at small -values and extrapolate for the time of a full sweep at large . Scaling of the algorithm: L6 β

days

● 1.0

0.8

0.6 ● L=96

● ● L=64 0.4

0.2 ●

● ● Days with a single core! ● β 1.5 2.0 2.5 3.0 3.5 4.0

Figure: Time to doβ one sweep for Going to L=64 (4096 sites) at small T is possible with Figure: Extrapolation to low about differenta million core valueshours. at V = 1.304. temperatures. Confirmed linear scaling with . Scaling of time: Pi-Fluxes near Critical Point

3 I This algorithm has scaling N in accordance with LCT -INT algorithms, as opposed to 3N3 for CT -INT -algorithms. (Wang, Iazzi, Corboz, Troyer, PRB 91 (2015)) I We can see the linear scaling in time at small -values and extrapolate for the time of a full sweep at large . Scaling of the algorithm: L6 β

days

● 1.0

0.8

0.6 ● L=96

● ● L=64 0.4

0.2 ●

● ● Days with a single core! ● β 1.5 2.0 2.5 3.0 3.5 4.0

Figure: Time to doβ one sweep for Going to L=64 (4096 sites) at small T is possible with Figure: Extrapolation to low about differenta million core valueshours. at V = 1.304. temperatures. ExploreConfirmed ways to accelerate linear to scaling do even L=100 with (10,000. sites) at low T.

Results = L Results = L density-density correlation ratio 1 1 R = n n 0 2 L/2 2 D E Results = L density-density correlation ratio 1 1 R = n n 0 2 L/2 2 D E 1 symmetric phase: (semi-metal) R ⇠ L4

broken phase: (charge density wave) R Const. ⇠

1 critical point: R ⇠ L1+⌘

1+⌘ Correlation Ratio: R A/L at V = Vc ⇡ 1+⌘ Correlation Ratio: R A/L at V = Vc ⇡

0.02 0.02

0.015 V=1.296 V=1.304 0.015

⌘ = 0.35(2) ⌘ = 0.17(2) R R 0.01 0.01

0.005 0.005 fit ok fit not good 0 0 10 20 30 40 50 10 20 30 40 50 L L 1+⌘ Correlation Ratio: R A/L at V = Vc ⇡

0.02 0.02

0.015 V=1.296 V=1.304 0.015

⌘ = 0.35(2) ⌘ = 0.17(2) R R 0.01 0.01

0.005 0.005 fit ok fit not good 0 0 10 20 30 40 50 10 20 30 40 50 L L

Li, Jiang and Yao, 2014 (484 sites) ⌫ = 0.77(3), ⌘ = 0.45(2) 1+⌘ Correlation Ratio: R A/L at V = Vc ⇡

0.02 0.02

0.015 V=1.296 V=1.304 0.015

⌘ = 0.35(2) ⌘ = 0.17(2) R R 0.01 0.01

0.005 0.005 fit ok fit not good 0 0 10 20 30 40 50 10 20 30 40 50 L L

Li, Jiang and Yao, 2014 Wang, Carboz and Troyer, 2014 (484 sites) ( 225 sites) ⌫ = 0.77(3), ⌘ = 0.45(2) ⌫ = 0.80(2), ⌘ = 0.30(1) 1+⌘ Correlation Ratio: R A/L at V = Vc ⇡

0.02 0.02

0.015 V=1.296 V=1.304 0.015

⌘ = 0.35(2) ⌘ = 0.17(2) R R 0.01 0.01

0.005 0.005 fit ok fit not good Biggest 0 0 10 20 30 40 50 10 20 30 40 50 lattices L L ever Li, Jiang and Yao, 2014 Wang, Carboz and Troyer, 2014 (484 sites) ( 225 sites) ⌫ = 0.77(3), ⌘ = 0.45(2) ⌫ = 0.80(2), ⌘ = 0.30(1) Stay Tuned for more results on large lattices!

Conclusions Conclusions

Hamiltonian methods offer a new approach to lattice field theories which have not yet been completely explored. Conclusions

Hamiltonian methods offer a new approach to lattice field theories which have not yet been completely explored.

New solutions to sign problems have emerged recently, especially in four fermion models, which allow us to explore many quantum critical points in 2+1 dimensions. Conclusions

Hamiltonian methods offer a new approach to lattice field theories which have not yet been completely explored.

New solutions to sign problems have emerged recently, especially in four fermion models, which allow us to explore many quantum critical points in 2+1 dimensions.

Opportunities exist to explore the HMC approach to study large lattices. But it is important to study the massless limit. Conclusions

Hamiltonian methods offer a new approach to lattice field theories which have not yet been completely explored.

New solutions to sign problems have emerged recently, especially in four fermion models, which allow us to explore many quantum critical points in 2+1 dimensions.

Opportunities exist to explore the HMC approach to study large lattices. But it is important to study the massless limit.

Ideas based on fermion bags offer an alternate method. In some cases they seem to allow us to study large lattices that seemed difficult earlier.