Confinement-deconfinement transitions in Z 2 gauge theories, and deconfined criticality

Transcription

1 HARVARD Confinement-deconfinement transitions in Z 2 gauge theories, and deconfined criticality Indian Institute of Science Education and Research, Pune Subir Sachdev November 15, 2017 Talk online: sachdev.physics.harvard.edu

2 1. Z2 lattice gauge theory and topological order 2. The Ising* confinement transition 3. Odd Z2 lattice gauge theory and deconfined criticality with an emergent U(1) gauge field 4. Z2 lattice gauge theory with fermions at half-filling, and deconfined criticality with an emergent SU(2) gauge field

3 1. Z2 lattice gauge theory and topological order 2. The Ising* confinement transition 3. Odd Z2 lattice gauge theory and deconfined criticality with an emergent U(1) gauge field 4. Z2 lattice gauge theory with fermions at half-filling, and deconfined criticality with an emergent SU(2) gauge field

4 Z2 lattice gauge theory (Wegner, 1971) z H = X z z z z g X i x z z z G i = x x x x Gauss s Law: [H, G i ]=0, G i =1

5 Z2 lattice gauge theory (Wegner, 1971) W C = Y C z C Deconfined phase W C Perimeter Law Confined phase W C Area Law g

6 Topological order V x = Y Cx x, V y = Y Cy x C y C y C x W x = Y C x z, W y = Y C y z C x V x W y = W y V x, V y W x = W x V y and all other pairs commute. On a torus, there are two additional independent operators, V x and V y which commute with the Hamiltonian: [H, V x ]=[H, V y ]=0 Deconfined phase W C Perimeter Law Confined phase W C Area Law g

7 Topological order V x = Y Cx x, V y = Y Cy x C y C y C x W x = Y C x z, W y = Y C y z C x V x W y = W y V x, V y W x = W x V y and all other pairs commute. On a torus, there are two additional independent operators, V x and V y which commute with the Hamiltonian: [H, V x ]=[H, V y ]=0 Deconfined phase. 4-fold degenerate ground state: V x = ±1, V y = ±1. Can take linear combinations to make eigenstates with W x = ±1, W y = ±1. Topological order Confined phase. Unique ground state has V x = 1, V y = 1. No topological order g

8 Topological order V x = Y Cx x, V y = Y Cy x C y C y C x W x = Y C x z, W y = Y C y z C x V x W y = W y V x, V y W x = W x V y and all other pairs commute. On a torus, there are two additional independent operators, V x and V y which commute with the Hamiltonian: [H, V x ]=[H, V y ]=0 Topological phase has degenerate states with Z 2 flux W = ±1 through the holes of the torus (N. Read and S.S., 1991)

9 Topological order V x = Y Cx x, V y = Y Cy x C y C y C x W x = Y C x z, W y = Y C y z C x V x W y = W y V x, V y W x = W x V y and all other pairs commute. On a torus, there are two additional independent operators, V x and V y which commute with the Hamiltonian: [H, V x ]=[H, V y ]=0 Deconfined phase. 4-fold degenerate ground state: V x = ±1, V y = ±1. Can take linear combinations to make eigenstates with W x = ±1, W y = ±1. Topological order Confined phase. Unique ground state has V x = 1, V y = 1. No topological order g This criterion can distinguish the phases when dynamical (or even gapless) matter fields are present

10 1. Z2 lattice gauge theory and topological order 2. The Ising* confinement transition 3. Odd Z2 lattice gauge theory and deconfined criticality with an emergent U(1) gauge field 4. Z2 lattice gauge theory with fermions at half-filling, and deconfined criticality with an emergent SU(2) gauge field

11 H = X z z z z g X i x G i =1 Kramers-Wannier-Wegner duality, Ising criticality Deconfined phase. 4-fold degenerate ground state: V x = ±1, V y = ±1. Can take linear combinations to make eigenstates with W x = ±1, W y = ±1. Topological order Confined phase. Unique ground state has V x = 1, V y = 1. No topological order g This criterion can distinguish the phases when dynamical (or even gapless) matter fields are present

12 Embed in a compact U(1) gauge theory Define z e ia and impose A =0, by adding a potential cos(2a). Then make a gauge transformation A µ! A µ µ /2, and make dynamical to make the Hamiltonian gauge invariant. In this manner the Z 2 gauge theory becomes a compact U(1) gauge theory with a charge 2 Higgs field: 2 L = X 4 ~! 3 2 A i + 1 i 5 X cos( ~ A) ~ J X cos( ~ i 2 A ~ i ) i i E. Fradkin and S. Shenker, Phys Rev D 19, 3682 (1979) Now we take the naive continuum limit with the Higgs field obtain a theory of complex scalar coupled to a U(1) gauge field e i, and L = (@ µ 2iA µ ) 2 + s 2 + u e 2 ( A ) 2 However, this turns continuum theory out to be incorrect: we cannot ignore the monopoles is the compact U(1) gauge field, A µ near the confinement transition.

13 Particle-vortex duality But we proceed anyway, and perform a Dasgupta-Halperin-Peskin particlevortex duality on L. This requires a complex scalar which creates a vortex with flux, and the dual theory is el µ 2 + es 2 + eu 4. Now we have to impose the requirement that vortices with flux and flux are the same i.e. allow 2 monopoles to be created from the vacuum. This modifies the Lagrangian to el µ 2 + es 2 + eu 4 ( 2 +( ) 2 ). Finally, we write = + i#. Thefield has a smaller mass then #, and so we can integrated out # to obtain the final correct dual theory el = (@ µ ) 2 + es 2 + eu 4. This is the promised dual Ising theory of the confinement-deconfinement transition. The field creates the vison particle. The refers to the fact that a single vison cannot be created locally, and this changes some topological properties on compact spaces.

14 H = X z z z z g X i x G i =1 Kramers-Wannier-Wegner duality, Ising criticality Deconfined phase. 4-fold degenerate ground state: V x = ±1, V y = ±1. Can take linear combinations to make eigenstates with W x = ±1, W y = ±1. Topological order Confined phase. Unique ground state has V x = 1, V y = 1. No topological order g This criterion can distinguish the phases when dynamical (or even gapless) matter fields are present

15 1. Z2 lattice gauge theory and topological order 2. The Ising* confinement transition 3. Odd Z2 lattice gauge theory and deconfined criticality with an emergent U(1) gauge field 4. Z2 lattice gauge theory with fermions at half-filling, and deconfined criticality with an emergent SU(2) gauge field

16 Symmetry-enriched topological (SET) order H = X z z z z X g x, G i = 1 i

17 Symmetry-enriched topological (SET) order H = X z z z z g X i x, G i = 1 Deconfined phase. 4-fold degenerate ground state: V x = ±1, V y = ±1. Can take linear combinations to make eigenstates with W x = ±1, W y = ±1. Topological order Confined phase. Broken symmetry and valence bond solid (VBS) order g (R. Jalabert and S.S., 1991; T. Senthil, A. Vishwanath, L. Balents, S. S. and M.P.A. Fisher, 2004)

18 Symmetry-enriched topological (SET) order H = X and deconfined criticality z z z z X g x, G i = 1 i Deconfined quantum criticality with a U(1) gauge theory and a charge 2 complex scalar Deconfined phase. 4-fold degenerate ground state: V x = ±1, V y = ±1. Can take linear combinations to make eigenstates with W x = ±1, W y = ±1. Topological order Confined phase. Broken symmetry and valence bond solid (VBS) order g (R. Jalabert and S.S., 1991; T. Senthil, A. Vishwanath, L. Balents, S. S. and M.P.A. Fisher, 2004)

19 Berry phases suppress monopoles at the critical point Embedding the Z 2 gauge theory in a compact U(1) gauge theory as before, the G i = 1 background charges lead to a source term for A (a Polyakov loop) 2 L = X 4 ~! 3 2 A i + 1 i 5 X cos( ~ A) ~ J X cos( ~ i 2 A ~ i ) i i + i X i ( 1) i x+i y A i Performing the Dasgupta-Halperin duality transform directly on this lattice model with the source term, we now find a dual vortex theory in which only quadrupled monopoles are permitted. el µ 2 + es 2 + eu 4 4( 8 +( ) 8 ). The 4 coupling is known to be irrelevant at the (Wilson-Fisher) critical point, and so monopoles can be ignored in the critical theory! Undualizing back to the original theory, this means that it is now valid to take the naive continuum limit of L to obtain the deconfined critical theory with a U(1) gauge field L = (@ µ 2iA µ ) 2 + s 2 + u e 2 ( A ) 2.

20 RG flow of L el µ 2 + es 2 + eu 4 4( 8 +( ) 8 ). VBS 4 Z 2 Neel 0 topological order Critical complex scalar and U(1) gauge field ges c g es U(1) spin Free photon/ Goldstone liquid

21 Symmetry-enriched topological (SET) order H = X and deconfined criticality z z z z X g x, G i = 1 i Deconfined quantum criticality with a U(1) gauge theory and a charge 2 complex scalar Deconfined phase. 4-fold degenerate ground state: V x = ±1, V y = ±1. Can take linear combinations to make eigenstates with W x = ±1, W y = ±1. Topological order Confined phase. Broken symmetry and valence bond solid (VBS) order g (R. Jalabert and S.S., 1991; T. Senthil, A. Vishwanath, L. Balents, S. S. and M.P.A. Fisher, 2004)

22 1. Z2 lattice gauge theory and topological order 2. The Ising* confinement transition 3. Odd Z2 lattice gauge theory and deconfined criticality with an emergent U(1) gauge field 4. Z2 lattice gauge theory with fermions at half-filling, and deconfined criticality with an emergent SU(2) gauge field

23 Fermionic matter at half filling H = X z z z z g X i x t X hiji i z ij j Deconfined phase. Massless Dirac fermions Topological order Confined phase. Fermion pairing and superconductivity g S. Gazit, M. Randeria, and A. Vishwanath, Nature Physics 13, 484 (2017)

24 Fermionic matter at half filling H = X z z z z g X i x t X hiji i z ij j Deconfined quantum criticality with a SU(2) gauge theory and a critical SO(3) Higgs scalar Deconfined phase. Massless Dirac fermions Topological order Confined phase. Fermion pairing and superconductivity g S. Gazit, M. Randeria, and A. Vishwanath, Nature Physics 13, 484 (2017) S. Gazit, F. F. Assaad, Chong Wang, S. Sachdev, and A. Vishwanath, to appear

25 1. Z2 lattice gauge theory and topological order 2. The Ising* confinement transition 3. Odd Z2 lattice gauge theory and deconfined criticality with an emergent U(1) gauge field 4. Z2 lattice gauge theory with fermions at half-filling, and deconfined criticality with an emergent SU(2) gauge field