Tensor Renormalization Group at Low Temperatures Discontinuity Fixed Point

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Tensor Renormalization Group at Low Temperatures:

Discontinuity Fixed Point

Tom Kennedy1,a, Slava Rychkov2,b

1Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA, atgk@math.arizona.edu

2Institut des Hautes Études Scientiﬁques, 91440 Bures-sur-Yvette, France, bslava@ihes.fr

Abstract.

To the memory of Krzysztof Gaw˛edzki, a pioneer of rigorous renormalization group studies

We continue our study of rigorous renormalization group (RG) maps for tensor networks that was begun in [1]. In this paper we con-

struct a rigorous RG map for 2D tensor networks whose domain includes tensors that represent the 2D Ising model at low temperatures

with a magnetic ﬁeld h. We prove that the RG map has two stable ﬁxed points, corresponding to the two ground states, and one unstable

ﬁxed point which is an example of a discontinuity ﬁxed point. For the Ising model at low temperatures the RG map ﬂows to one of the

stable ﬁxed points if h̸= 0, and to the discontinuity ﬁxed point if h= 0. In addition to the nearest neighbor and magnetic ﬁeld terms

in the Hamiltonian, we can include small terms that need not be spin-ﬂip invariant. In this case we prove there is a critical value hcof

the ﬁeld (which depends on these additional small interactions and the temperature) such that the RG map ﬂows to the discontinuity

ﬁxed point if h=hcand to one of the stable ﬁxed points otherwise. We use our RG map to give a new proof of previous results on

the ﬁrst-order transition, namely, that the free energy is analytic for h̸=hc, and the magnetization is discontinuous at h=hc. The

construction of our low temperature RG map, in particular the disentangler, is surprisingly very similar to the construction of the map

in [1] for the high temperature phase. We also give a pedagogical discussion of some general rigorous transformations for inﬁnite

dimensional tensor networks and an overview of the proof of stability of the high temperature ﬁxed point for the RG map in [1].

Contents

1 Introduction..................................................... 2

2 Tensors ....................................................... 3

2.1 Graphicnotation.Contractions........................................ 4

2.2 Leggroupingandreindexing......................................... 5

2.3 Tensornetworks................................................ 5

2.4 Transformations of tensor networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.1 TensorRGtransformations...................................... 7

2.4.2 SimpleRGtransformations ..................................... 7

2.4.3 Gaugetransformations........................................ 7

2.4.4 Disentanglertransformations..................................... 8

3 Fromlatticemodelstotensornetworks....................................... 10

3.1 NNIsingmodel................................................ 11

3.2 Generalﬁnite-rangemodel .......................................... 12

4 Tensor RG analysis near the high-T ﬁxed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 RG analysis at low T:generalstrategy....................................... 18

5.1 RG for the T= 0 freeenergyandmagnetization............................... 19

5.2 A strategy for T̸= 0 ............................................. 20

6 RGmapconstruction ................................................ 21

6.1 Step1-gaugetransformation......................................... 23

6.2 Step2-simpleRGstep............................................ 24

arXiv:2210.06669v3 [math-ph] 29 Aug 2023

6.3 Step3-mainRGstep ............................................ 25

7 Properties of free energy and magnetization at low T............................... 32

7.1 PropertiesoftheRGmap........................................... 32

7.2 Thefreeenergy ................................................ 33

7.3 Themagnetization .............................................. 36

7.4 Comparison with the argument of Nienhuis and Nauenberg . . . . . . . . . . . . . . . . . . . . . . . . . 44

8 Finalremarksandopenproblems.......................................... 45

Acknowledgments.................................................... 46

A AnalyticfunctionsonBanachspaces........................................ 46

B Stablemanifoldtheorem .............................................. 47

References........................................................ 48

1. Introduction

Renormalization group (RG) theory associates to each phase of a lattice model a ﬁxed point of an RG transformation,

which is an attractor for the given phase. In a typical case when the model has disordered and ordered phases separated

by the critical point, one speaks of the high-temperature (high-T), low-temperature (low-T), and critical ﬁxed points. In

physics, thinking in terms of ﬁxed points and RG ﬂows among them has become since the 1970s a leading approach

to phase transitions. A mathematically rigorous treatment has been achieved near the high- and low-temperature ﬁxed

points. The critical ﬁxed point remains however a challenge. This is because in a generic lattice model the critical ﬁxed

point is expected to live in an inﬁnite-dimensional coupling space and to have no small parameter. Constructing such a

ﬁxed point rigorously probably requires a computer-assisted approach. For hierarchical models this was accomplished

long ago, but for physically interesting lattice models with translationally invariant interactions, such as e.g. the 3D Ising

model, this is wide open. To make progress on this problem is important not only conceptually, but also practically, as its

solution will yield as a by-product the critical exponents with rigorous error bars.

To achieve this, one needs a nonperturbative RG approach which is both rigorous and computable. By computable we

mean that it should be possible to evaluate numerically the RG map truncated to a large but ﬁnite number of couplings.

The hope is to ﬁrst identify an approximate ﬁxed point numerically, and then to prove that an exact ﬁxed point exists

nearby.

In this paper we will work with tensor RG [2], which seems at present the only RG approach satisfying the require-

ments of both computability and rigor. This approach starts by rewriting the lattice model partition function as a tensor

network—a contraction of a periodic arrangement of tensors (see Fig. 2.7). An RG step coarse-grains the network, re-

placing it by an equivalent network consisting of a smaller number of tensors. If done exactly, this step would increase

the network bond dimension. In numerical calculations, one keeps the bond dimensions from growing by truncating the

new tensors. There is signiﬁcant ﬂexibility in how coarse-graining and subsequent truncation are performed, and many

numerical tensor RG algorithms have been proposed differing in these details [3–10]. When benchmarked on the 2D Ising

model, these algorithms give approximate critical exponents in excellent agreement with the exact values (better than any

other RG method). Can we use one of these algorithms or their modiﬁcation to construct an exact critical ﬁxed point, ﬁrst

in 2D and eventually in 3D?

In preparation for this task, one needs to develop a theory of rigorous Tensor RG maps. Such maps do not involve

truncation and preserve the tensor network value exactly. They naturally operate in the space of inﬁnite-dimensional

tensors. We expect the exact critical ﬁxed point tensors to be inﬁnite-dimensional. The high-T and low-T ﬁxed point

tensors are ﬁnite-dimensional, but in an exact treatment the tensor dimension is expected to grow without limit when

approaching them (although the weight of all but ﬁnitely many tensor components should tend to zero in an appropriate

norm).

As a ﬁrst step in this direction, in [1] we developed rigorous 2D tensor RG theory near the high-T ﬁxed point. This

ﬁxed point is represented by a very simple tensor Awith a single nonzero component A0000 = 1. We considered arbitrary

inﬁnite-dimensional perturbations δA of this tensor having small Hilbert-Schmidt norm ∥δA∥, and showed that after an

appropriate tensor RG step the Hilbert-Schmidt norm is reduced: ∥δA′∥=O(∥δA∥3/2). In other words, we showed that

the high-T ﬁxed point is stable.

In this work we will continue the study of 2D tensor RG by considering the vicinity of the low-T ﬁxed point. Low

temperatures are known to be more subtle for rigorous studies than high temperatures where the cluster expansion can be

used. A speciﬁc model we have in mind is a small perturbation of the 2D Ising model in a magnetic ﬁeld. Such models

at low temperatures exhibit a ﬁrst-order phase transition as a function of the magnetic ﬁeld. Standard proofs of this are

based on the Pirogov-Sinai theory [11–13] or a coarse-graining approach formulated in terms of Peierls contours and not

in terms of spins by Gaw˛edzki, Kotecký and Kupiainen [14]. In fact, it is known that the block-spin RG transformations

can be ill deﬁned at low temperatures (e.g. [15,16]) while they are well deﬁned at high temperatures [17,18].

On the contrary, here we will show that the tensor RG procedure needs no major modiﬁcations at low temperatures

compared to high temperatures. In the case of Ncoexisting phases, the low-T ﬁxed point is the direct sum of Nhigh-T

ﬁxed points with equal weights:

(1.1) A(1) ⊕A(2) ⊕...⊕A(N).

We will work in 2D and will focus on the case of two coexisting phases N= 2, relevant for the Ising model. In this case,

the tensors A(1) =A(+) and A(2) =A(−)describe the states of the Ising model with all spins pointing up and down,

respectively.

The Ising model in a magnetic ﬁeld will be represented by a tensor which is a convex linear combination of A(+) and

A(−)up to a correction which is small at low temperatures:

(1.2) A=αA(+) + (1 −α)A(−)+B, B =O(e−4β),

where the zero magnetic ﬁeld corresponds to α= 1/2. We will construct a tensor RG map (α, B)→(α′, B′)which has

the following property:

(1.3) 1

2A(+) ⊕1

2A(2) is a ﬁxed point near which the Bdirection is contracting, while the αdirection is expanding .

By these results, the 2D Ising model at small temperature and at zero magnetic ﬁeld hconverges to the low-T ﬁxed point,

while for nonzero hit converges towards A(+) or A(−)depending on the sign of h. We will also allow small Z2-breaking

interactions, in which case the critical magnetic ﬁeld is nonzero.

In this work we will also use the tensor RG map to study the magnetization near the low-T ﬁxed point and to prove

that it is a discontinuous function of the magnetic ﬁeld, i.e. that the phase transition is ﬁrst order. For this reason the

low-T ﬁxed point is referred to in the theoretical physics literature as the discontinuity ﬁxed point, following Nienhuis and

Nauenberg [19]. These authors identiﬁed a crucial property that this ﬁxed point should have, to ensure the discontinuity:

the eigenvalue of the magnetic ﬁeld variable should be equal to the volume rescaling factor of the RG map.

The work by Gaw˛edzki, Kotecký and Kupiainen [14] was the ﬁrst rigorous implementation of the discontinuity ﬁxed

point picture in mathematical physics. Our work provides a new and completely different implementation, based on tensor

RG.

The paper is structured as follows. In Sec. 2we collect the notation and a few simple results about tensors and tensor

networks. In Sec. 3we recall the tensor network representation of the nearest-neighbor Ising model. More generally, we

prove that any lattice spin model with ﬁnite range interactions can be transformed to a tensor network, and illustrate this

on the Ising model with next-to-nearest-neighbor and plaquette interactions.

Then in Sec. 4we recall results of our previous paper [1] near the high-T ﬁxed point. We reformulate the main result

of [1] making clear the analytic nature of the RG map. We also give an RG proof that the free energy is analytic near the

high-T ﬁxed point.

In Sec. 5we move to low temperatures. We formulate the general strategy of recovering the low-temperature phase

diagram via a tensor RG map having property (1.3). Such an RG map is then explicitly constructed in Sec. 6. Sec. 7uses

this RG map to study the free energy and the magnetization of the general tensor network model with two phases at low

temperatures (which includes the Ising model and its small perturbations). We prove that the free energy and the magneti-

zation are analytic away from the phase coexistence surface. On this surface, we prove that the free energy is continuous,

while the magnetization has a discontinuous jump, i.e. that the phase transition is ﬁrst order. The Nienhuis-Nauenberg

condition on the magnetic ﬁeld eigenvalue plays a crucial role in our proof, but the way we derive the discontinuity from

this condition is different from theirs, for reasons explained in Sec. 7.4. In Section 8we make ﬁnal remarks and formulate

a few problems for the future.

2. Tensors

In this section we collect the notation about tensors and tensor networks. We also deﬁne, following [1], a few simple

equivalence transformations of tensor networks, which will serve later on as building blocks for RG transformations.

Deﬁnition 2.1.Let n⩾2be an integer and Ia nonempty set which is either ﬁnite or countable (called an index set). An

n-leg tensor Awith legs indexed by Iis a table of real numbers Ai1i2...in,ik∈ I,k= 1 ...n.

In other words, such a tensor is a map from Into R.n-leg tensors may also be referred to as “n-tensors” or “n-index

tensors” or “tensors with nlegs”.

The Hilbert-Schmidt (HS) norm of an n-tensor A(n)is deﬁned as

(2.1) ∥A(n)∥= X

i1,...,in∈I |Ai1...in|2!1/2

This is the norm we will use most often. A tensor with a ﬁnite HS norm will be called HS. All n-tensors with n⩾3

considered in this work will be HS.

For the special case of n= 2 (2-tensors) we will also consider a different norm ∥A∥op, the operator norm. It is deﬁned

as the norm of the linear operator from ℓ2(I)to ℓ2(I)associated with the tensor A. We can compute this norm as

∥A∥op = sup Pi,j∈I Aij viwj, the sup being taken over v, w ∈ℓ2(I)having ℓ2norm 1 and a ﬁnite number of nonzero

elements. As is well known, ∥A∥op ⩽∥A∥for 2-tensors.

The above deﬁnitions admit obvious generalizations to n-tensors whose every leg is indexed by its own index set

I1,...,In.

For any two n-tensors Aand B, we deﬁne their sum as follows:

• If both tensors are indexed by the same index sets I1,...,In, then the sum A+Bis the usual element-wise sum.

• If the two collections of index sets IA

1,...,IA

nand IB

1,...,IB

nare not the same, we ﬁrst extend each tensor by

zeros to a larger tensor deﬁned on (IA

1∪IB

1)×...×(IA

n∪IB

n), and then take the usual element-wise sum of such

extended tensors.

• If the index sets for at least one of the nlegs do not overlap, say IA

1∩IB

1=∅, we deﬁne the sum as in the previous

case, and call it a “direct sum” A⊕B.

2.1. Graphic notation. Contractions

In graphic notation, an n-tensor is drawn as a box with nlegs coming out. E.g. a 4-tensor Aand its components Aijkl is

denoted by a diagram:

(2.2) A A

Aijkl =ik .

In general, no symmetry will be assumed for n-tensors, so one has to pay attention to the order of the indices. In (2.2),

Aijkl is denoted by the diagram with i, j, k, l on the right, top, left, bottom legs.

We will also consider contractions of tensors. E.g. for two 4-tensors Aand Bwe may deﬁne tensor Ccontracting the

ﬁrst leg of Awith the third leg of B, i.e.

(2.3) Cmnjklp =X

i∈I

AijklBmnip .

Contraction is only deﬁned if the two contracted legs have the same index set. In graphic notation, contraction is denoted

by gluing the legs of contracted tensors. Eq. (2.3) may then be represented as:

(2.4) CAB

The following proposition follows easily from the Cauchy-Schwarz inequality, and we omit its proof.

Proposition 2.2. Let Cbe a tensor formed by contracting one leg of a tensor Awith one leg of a tensor B.

(a) If A and Bare HS, then so is C, and ∥C∥⩽∥A∥∥B∥.

(b) If Ais HS, while Bis a 2-tensor with a ﬁnite operator norm, then Cis HS, and ∥C∥⩽∥A∥∥B∥op.

The next example will be used frequently. We form a tensor Tby contracting 4 copies of an HS tensor A, as follows:

(2.5)

A A

T=.

The tensor Tis well deﬁned: it’s easy to see that each component is given by an absolutely convergent series. Moreover,

Tis HS and ∥T∥⩽∥A∥4. This is easy to prove directly, and is also a consequence of the general Prop. 2.4 below.

2.2. Leg grouping and reindexing

The tensor Tin (2.5) has 8 legs indexed by I(same index set as A). These 8 legs come naturally grouped in 4 pairs (right,

up, left, down). We may consider each of these 4 pairs of legs as a single leg with an index set I ×I. Viewed this way, T

becomes a 4-tensor indexed by I ×I. This is an example of leg grouping, which reshapes the tensor but does not change

its components. We can also group more than two legs.1

Another simple operation is reindexing. Let I1and I2be two index sets of the same cardinality, and let ρbe a one-to-

one map from I1onto I2(reindexing map). If Ais a tensor with a leg indexed by I1, we can use ρto transform Ato a

tensor A′where the same leg is indexed by I2.

Leg grouping and reindexing just reshufﬂe tensor components but do not change their numerical values. In particular,

these operations preserve the norm.

Here is an equivalent view of reindexing. Let Jbe a 2-tensor with the only nonzero components Jii′= 1 if i′=ρ(i).

We call Ja reindexing tensor; it has operator norm 1. The reindexed tensor A′is then obtained by contracting Awith J.

We write this graphically as:

(2.6) A′A

i′i′.

Below we will often apply leg grouping followed by reindexing, as follows. Consider two legs of a tensor with index

set I. We group them, obtaining a leg with an index set I ×I. Suppose that either |I| = 1 or |I| =∞. Then I ×I has

the same cardinality as I. We can then apply reindexing as above with I1=I × I and I2=I. After reindexing, the leg

is indexed with I.

Let us see how this works for the tensor Tdeﬁned by (2.5). As explained, after leg grouping we can view it as a

4-tensor with legs indexed by I ×I. We then apply reindexing on each of this legs, and obtain a 4-tensor indexed by I,

the same index set we started with.

Our main case of interest will be |I| =∞. In this case the reindexing map from I × I to Iis vastly non-unique.

Without loss of generality, we can take I=N. Choosing the reindexing map ρthen amounts to enumerating N×Nin

some particular order. In our constructions below, we will ﬁx the ﬁrst few elements of the enumeration sequence, and the

rest of it will be left arbitrary.

2.3. Tensor networks

Consider a ﬁnite periodic square lattice of size Lx×Ly. Suppose we put a 4-tensor Aat every vertex (n, m)of the lattice,

contracting its legs with the legs of tensors at neighboring vertices, and taking into account periodicity, as shown in this

1In the python package numpy, often used for numerical tensor manipulations, leg grouping can be performed by the function reshape().

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TensorRenormalizationGroupatLowTemperatures:DiscontinuityFixedPointTomKennedy1,a,SlavaRychkov2,b1DepartmentofMathematics,UniversityofArizona,Tucson,AZ85721,USA,atgk@math.arizona.edu2InstitutdesHautesÉtudesScientifiques,91440Bures-sur-Yvette,France,bslava@ihes.frAbstract.TothememoryofKrzysztofGaw˛edz...

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Tensor Renormalization Group at Low Temperatures Discontinuity Fixed Point

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