Frustration and Criticality in Statistical Mechanics: a perspective from Tensor Networks

Frank Verstraete

Outline

• Thermodynamic surfaces, reduced density matrices and convex sets

– Phase transitions, ruled surfaces, frustration

• Statistical Mechanics as tensor networks

– Diagonalizing matrix product operators to calculate free energies: MPS algorithms

– Diagonalizing projected entangled pair operators to calculate free energies: PEPS algorithms

– Examples:

• Hard square constant

• Residual entropy of spin ice

• Critical systems: matrix product operator symmetries in critical tensor networks

Maxwell’s thermodynamic surfaces

Energy vs Entropy vs Volume

van der Waals OnnesKuenen

Classical Ising in 2D at finite temperature

• Convex set of all possible expectation values of energy density, entropy density, and magnetization w.r.t. any probability distribution

Zauner et al. ‘16

Collective phenomena

• Interesting collective phenomena occur if there is also some type of frustration or competition between different quantities which leads to correlations. This can be obtained in at least 3 different ways:

• Add fluctuations to the picture by working at finite temperature

– Battle of entropy vs. energy (free energy F=E-TS)– Leads to phase transitions, critical phenomena, …

• Add frustration: e.g. triangular or Kagome lattices

• Add non-commuting terms to the Hamiltonian:

– A ground state cannot be a joint eigenstate of all terms, and a big compromise will have to be made how to spread its quantum correlations such as to minimize the energy

Quantum Ising with transverse magnetic field in 1D

• Convex set of all possible expectation values of <XX>, <Z> and spontaneous magnetization <X> w.r.t. any quantum state

Zauner et al.

Russian Dolls for Quantum Ising

Ruled surfaces: symmetry breaking and frustration

• Ruled surfaces as extreme surfaces on those convex sets are of central interest: they demonstrate that phase transitions are a consequence of the geometric structure of set of all probability distributions / Hilbert space without the need of invoking Hamiltonians

– Hamiltonians are dual objects, defining tangent planes of constant free energy E-T.S

– Gibbs state / quantum ground states are the extreme points of the convex sets

• This talk: what about zero-temperature extensive entropies?

– Ruled Surfaces parallel to the entropy axis, i.e. for T=0

Classical Ising on square lattice

• Tangent plane of blue surface defines Ising Hamiltonian with J=-1, h=4: all configurations satisfying hard square constraint are allowed; a spin down surrounded by 4 ups does not cost energy

• Redidual entropy: counting problemZauner et al.

Counting: hard square constants

• 1-dimension:

– count number of configurations of bits such that a 1 is surrounded by 0’s

– Transfer matrix approach: evaluate following tensor network

– Number of configurations is:

• 2 dimensions: contraction of 2D tensor network yields # configurations:

• Problem is reduced to finding leading eigenvalue of transfer matrix / MPO:

• Turns out to be non-integrable, but nevertheless Baxter (1999) calculated the free energy per site (“hard square constant”) using series expansions of corner transfer matrix:

f=1.503048082475332264322066329475553689385781

• Can we do better using matrix product state techniques?

arXiv:1611.08519

MPS optimization for transfer matrices

Variational uniform matrix product state algorithm

• Make use of left/right canonical forms to reduce optimization to a sequence of effective eigenvalue problems:

• Essence: enforce that residual of MPO applied to MPS is orthogonal to tangent space of MPS manifold; this leads to a Lanczos-type version of CTM

• Just as in CTM, optimization gives direct access to the free energy and hence of entropy of the stat. mech. Model without need of integration such as in MC

Haegeman, Verstraete, arXiv:1611.08519

More frustrated models: convex set of q-state Potts model obtained by MPS calculations

• Red (J=-1,h=0): no equal neighbor spins (cfr. chromatic polynomial)

– Yields entropy of 2-D spin ice for q=3

• Blue (J=-1,h=4): equivalent to Ising case / hard square cst (configurations with only spin q and q-1, while q-1 is surrounded by q)

• In general: MPS methods work amazingly well for calculating entropies directly

– More benchmark examples of residual entropies:

• What about 3D statistical mechanics counting problems?

3D frustration: Residual entropy of ice

Existing Results

L. Pauling, The structure and entropy of ice and of other crystals with some randomness of atomic arrangement," Journal of the American Chemical Society 57, 2680-2684 (1935).

J. F. Nagle, Lattice statistics of hydrogen bonded crystals. i. the residual entropy of ice," Journal of Mathematical Physics 7, 1484-1491 (1966).

B. A. Berg, C. Muguruma, and Y. Okamoto, Residual entropy of ordinary ice calculated frommulticanonical monte carlo simulations, Molecular Simulation 38, 856-860 (2012).

C. P. Herrero and R. Ramirez, Configurational entropy of ice from thermodynamic integration,Chemical Physics Letters 568, 70 (2013).

J. Kolafa, Residual entropy of ices and clathrates from monte carlo simulation, The Journal ofChemical Physics 140, 204507 (2014).

0.8054531

0.8145041

0.81530813

Cal/deg/mol

Tensor network for spin ice

Diamond Ice: repeat the PEPO shifted by 1 sublattice shiftHexagonal Ice Ih : multiply with its transpose

Free energy can then be obtained as an eigenvalue problem of the 2D transfer matrix of cubic lattice; both types of ice give rise to same variational problem if we assume Z2 invariance of PEPS by rotation over pi

Numercial optimization of tensor networks

PEPS: finding eigenvectors of 2-D transfer matrices

• Complication: system is critical (U(1)) with effective gauge degrees of freedom, …

• On a positive side: due to symmetries of PEPO, problem is variational

Numerical PEPS optimization

• We use gradient methods, where “channel” environments on virtual degrees of freedom allow to calculate gradients

Vanderstraeten et al. ‘17

Variational PEPS results for spin ice

arXiv:1805.10598

B. A. Berg, C. Muguruma, and Y. Okamoto, Residual entropy of ordinary ice calculated from

multicanonical monte carlo simulations, Molecular Simulation 38, 856-860 (2012).

C. P. Herrero and R. Ramirez, Configurational entropy of ice from thermodynamic integration,

Chemical Physics Letters 568, 70 (2013).

J. Kolafa, Residual entropy of ices and clathrates from monte carlo simulation, The Journal of

Chemical Physics 140, 204507 (2014).

Coulomb phase description of spin ice

Extrapolated Stiffness:

K = 0.967

Entanglement spectrum of PEPS fixed point of spin ice transfer matrix

• Eigenvalues of boundary MPO which is fixed point of PEPS transfer matrix

• Typical dispersion relation for spin chains (entanglement Hamiltonian) with power-law decaying interactions

More U(1) models: dimer coverings on 3D cubic lattice

– In 2D: integrable transfer matrix and solvable by mapping to Pfaffians/ free fermions (Kasteleyn Fortuyn, Fisher, Lieb)

– In 3D: critical Coulomb phase

• Tensor network:

• Dimer entropy: 0,4498238 (D = 2)

0,44988448 (D = 3)

0,44988452 (D = 4)

• Again algebraic dipolar forms for the dimer-dimer correlations

nj(x) = 1 if there is a dimer on that

site in the direction j

Extrapolated Stiffness: K = 4,861

Compatible with Huse, Krauth, Moessner, Sondhi, “Coulomb and liquid dimer models in three dimensions," Physical Review Letters 91, 167004 (‘03).

Part II: MPO symmetries in critical spin systems

• Let’s go back to 2D classical statistical mechanics

• Question:

– what makes critical partition functions so special?

– Can we discover locally whether a tensor network is critical?

Part II: MPO symmetries in critical spin systems

• Let’s go back to 2D classical statistical mechanics

• Question:

– what makes critical partition functions so special?

– Can we discover locally whether a tensor network is critical?

• YES: enhanced symmetry at the critical point!

• MPO symmetries as remnants of conformal symmetries !

• Basic line of thought:

1. Tensor fusion categories form the building blocks of both CFT and TQFT: formal framework for dealing with Operator Product Expansion (OPE)

2. Central equation in Tensor Fusion Category theory is the pentagon equation: reflects associativity of fusion rules

3. Solutions of pentagon equation can be used to define Matrix Product Operators which form a representation for the OPE

4. Those MPO can be used to construct PEPS exhibiting topological quantum order, reflected in the fact that there are MPO symmetries

5. Elementary excitations can be constructed in terms of idempotents of a new MPO algebra: Ocneanu tube algebra

6. Overlap op those PEPS with product states yield partition functions of critical statistical mechanical models: RSOS models

7. Those models still exhibit MPO symmetries: Wilson loops as lattice remnants of conformal symmetry

8. Primary fields correspond to topological sectors (anyons) in PEPS. Critical exponents uniquely depend on solutions of pentagon equations

1. Tensor fusion categories form the building blocks of both CFT and TQFT: formal framework for dealing with Operator Product Expansion (OPE)

• CFT version:

– Initially introduced by Wilson, it tells how a pair of primary fields looks like at far distances: as a superposition of another primary field

– The (x,y,z) dependence is basically dictated by the conformal symmetry

• TQFT version:

– Introduced by Moore and Seiberg, tells how anyons (topological sectors) fuse with each other

Central equation in Tensor Fusion Category theory is the pentagon equation: reflects associativity of fusion rules

• Up to gauge degrees of freedom, there are only a discrete number of solutions of this equation once fusion rules are defined

2.

3. Solutions of pentagon equation can be used to define Matrix Product Operators which form a representation for the OPE (discrete version of Yang-Baxter):

Pentagon equation:

Zipper relation: =

4. Those MPO can be used to construct PEPS exhibiting topological quantum order, reflected in the fact that there are MPO symmetries

• String nets / Turaev-Viro state sums for PEPS on hexagonal lattice:

• Pulling through equation is equal to the pentagon equation:

5. Elementary excitations can be constructed in terms of idempotents of a new MPO algebra: Ocneanu tube algebra

• Anyons in this theory are defined as linear combinations of objects at the end of such a string:

• These objects form a new MPO C* algebra, and the anyons / topological sectors correspond to the idempotents of this algebra

Example: Ising case

Those idempotents define the topological sectors in the Ising TQFT / CFT and give a systematic way of calculating the S-matrix

Bultinck et al. ‘17

• Topological spin, braiding, fusion in terms of MPO algebra:

• Topological degeneracy on the torus:

=

6. Overlap op those PEPS with product states yield partition functions of critical statistical mechanical models: RSOS models

• Adaption of Chen-Gu-Wen argument on SPT phases: whenever a local Hamiltonian or transfer matrix commutes with a nontrivial MPO, then its ground state is critical or symmetry broken

• By taking overlap of stringnet PEPS with a product state (interface between two distinct theories), we hence get a tensor network description of a critical statistical mechanical model

• Input needed: fusion rules, F-tensors, and product state

• All RSOS models can be obtained like that -> lattice realizations of all minimal CFTs

• Universality can hence be linked to the fact that very few solutions of the pentagon equation exist

7. Those models still exhibit MPO symmetries: Wilson loops as lattice remnants of conformal symmetry

• Pulling through equation still satisfied: lattice remnant of conformal symmetry. => LOCAL SYMMETRY

• MPOs correspond to defect lines in CFT: disorder and duality defects, …

• Toplogical sectors become conformal sectors: we can inherit all properties of zero-correlation length TQFT, but with a completely different meaning: both renormalization group fixed points, but one with zero and the other with infinite correlation length

Example: Ising CFT

Hamiltonian version of lattice critical models: IRF Hamiltonians / anyonic spin chains

All of those commute with the MPO symmetry, hence are critical!

8. Primary fields correspond to topological sectors (anyons) in PEPS. Critical exponents uniquely depend on solutions of pentagon equations

– The pentagon equations completely fix all critical exponents and fusion rules of primary fields -> explains universality

– Idempotents yield lattice descriptions of all primary fields

– Boundary CFT: all about MPOs

• All characters of the CFT can now be expressed as a linear combination of the ones of the idempotents; Dehn twists etc give rise to transformation properties of those charactrers (Verlinde S matrix, T-matrix, …)

Conclusion

• We mortals are bound to decipher all messages that our extraterrestrial friend Tomotoshi has hidden in his papers. His hidden message seems to be: Nothing In Statistical mechanics Have I Not created in terms Of tensor networks