Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point

Abstract

To the memory of Krzysztof Gawȩdzki, a pioneer of rigorous renormalization group studies

We continue our study of rigorous renormalization group (RG) maps for tensor networks that was begun in Kennedy and Rychkov (J Stat Phys Ser, 187 (3), 33, 2022). In this paper, we construct 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 $h \ne 0$, 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 $h_c$ of the field (which depends on these additional small interactions and the temperature) such that the RG map flows to the discontinuity fixed point if $h=h_c$ and 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 $h \ne h_c$, and the magnetization is discontinuous at $h = h_c$. The construction of our low-temperature RG map, in particular the disentangler, is surprisingly very similar to the construction of the map in Kennedy and Rychkov (2022) 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 Kennedy and Rychkov (2022).

Appendices

Appendix A: Analytic Functions on Banach Spaces

Let X, Y be two complex Banach spaces and let ${\mathcal {U}}$ be an open subset of X. A function $f: {\mathcal {U}} \rightarrow Y$ is called analytic (or, equivalently, holomorphic) if it has a complex Fréchet derivative at every point, i.e., for each $x \in {\mathcal {U}}$ there exists a continuous complex-linear map $D f (x): X \rightarrow Y$ such that

$$\begin{aligned} \lim _{y \rightarrow 0} \frac{\Vert f (x + y) - f (x) - D f (x) y \Vert }{\Vert y \Vert } = 0. \end{aligned}$$

(A.1)

We list a few basic properties of such abstract analytic functions, assumed in this paper, often tacitly. They are similar in spirit to properties of ordinary analytic functions from $\mathbb {C}$ to $\mathbb {C}$.

• 1. (Analyticity of polynomials) Let $F: X \times \cdots \times X \rightarrow Y$ be a continuous multilinear map of degree n. Then, the function $P (x) = F (x, \ldots , x)$ is analytic. It is called a homogeneous polynomial of degree n.

• 2. (Analyticity of product) If $h: {\mathcal {U}} \rightarrow \mathbb {C}$ is a second analytic function, then $h f: {\mathcal {U}} \rightarrow Y$ is analytic.

• 3. (Analyticity of composition) Let Z be a third complex Banach space, let ${\mathcal {V}}$ be an open subset of Y, and let $g: {\mathcal {V}} \rightarrow Z$ be a second analytic functions. If $f ({\mathcal {U}}) \subset {\mathcal {V}}$, then $g \circ f: {\mathcal {U}} \rightarrow Z$ is analytic.

• 4. (Infinite smoothness) f has derivatives of arbitrary order at any point. The nth derivative $D^n f (x)$ at a point $x \in {\mathcal {U}}$ is a continuous multilinear map from $X \times \cdots \times X$(n times) into Y.

• 5. (Taylor series) Denote by ${\hat{D}}^n f (x)$ the homogeneous polynomial associated with the nth derivative $D^n f (x)$. Let $B_\varrho (x)$ denote a ball of radius $\varrho$ centered at x. Assume that f is bounded in a ball $B_\varrho (x) \subset {\mathcal {U}}$, i.e.,

  $$\begin{aligned} \Vert f (x + w) \Vert \leqslant M < \infty \qquad \forall w \in B_\varrho (0). \end{aligned}$$

  (A.2)

  Then, f has in $B_\varrho (x)$ a uniformly convergent Taylor series expansion:

  $$\begin{aligned} f (x + w) = \sum _{n = 0}^{\infty } \frac{1}{n!} {\hat{D}}^n f (x) (w) \qquad \forall w \in B_\varrho (0). \end{aligned}$$

  (A.3)

  Individual terms in this expansion are bounded by:

  $$\begin{aligned} \Vert {\hat{D}}^n f (x) (w) \Vert \leqslant n! \frac{M}{\varrho ^n} \Vert w \Vert ^n. \end{aligned}$$

  (A.4)

  In particular, the operator norm of the first derivative is bounded by:

  $$\begin{aligned} \Vert D f (x) \Vert _{{\mathcal {B}} (X, Y)} \leqslant M/\varrho . \end{aligned}$$

  (A.5)

• 6. The sum of a uniformly convergent series of analytic functions is analytic.

For more details about these facts, see, e.g., [29], Chapters 13 and 26, and [30, 31]. In particular, Property 5 is proven by applying the Cauchy integral representation to the ordinary analytic function $\ell (f (x + z w))$ of a complex z in the unit disk, where $\ell \in Y^{*}$.

In this paper, X can be a complex Hilbert space ${\mathcal {H}}$ of tensors equipped with the Hilbert–Schmidt norm, or its closed subspace $\widetilde{{\mathcal {H}}}$ obtained by setting some tensor components to zero, or $\mathbb {C} \times \widetilde{{\mathcal {H}}}$ (in Theorem 6.1). Y can be $\mathbb {C},{\mathcal {H}},\widetilde{{\mathcal {H}}}$ or $\mathbb {C} \times \widetilde{{\mathcal {H}}}$.

Appendix B: Stable Manifold Theorem

For the convenience of the reader, we give a statement of the stable manifold theorem following [32]. Other references include [33, 34].

Theorem B.1

Let E, F be Banach spaces. Let G be the Banach space $E \times F$ with the norm $\Vert (x,y)\Vert = \max \{ \Vert x\Vert ,\Vert y\Vert \}$. Let $\lambda$ be a bounded linear map of E into E and $\mu$ a bounded, invertible linear map of F onto F. We assume there is a constant $a<1$ such that $\Vert \lambda \Vert \leqslant a$ and $\Vert \mu ^{-1}\Vert \leqslant a$. Let f be a $C^r$ map ($r \geqslant 1$) from some neighborhood U of 0 in G into G with $f(0)=0$ and $Df(0)=\lambda \otimes \mu$. Then there is a neighborhood C of 0 in E and a neighborhood D of 0 in F and a unique map $h: C \rightarrow D$ such that $f(\mathop {graph}(h)) \subset \mathop {graph}(h)$. The map h is $C^r$ and $Dh(0)=0$. For $z \in U$, $f^n(z) \rightarrow 0$ as $n \rightarrow \infty$ if and only if $f^k(z) \in \mathop {graph}(h)$ for some $k \geqslant 1$.

Remark B.2

The stable manifold is the set of $z \in U$ such that $f^n(z) \rightarrow 0$. So the theorem says a point z is on the stable manifold if and only if $f^k(z)$ is eventually on the graph of h.

In our application of the stable manifold theorem, F is the one-dimensional space corresponding to $\alpha$, and E is the Hilbert space corresponding to B. The point $(\alpha ,B)=(1/2,0)$ is a fixed point of R, and Theorem 7.1 implies that at this fixed point $\partial _\alpha \alpha '=16$, $\partial _B \alpha '=0$, $\partial _\alpha B=0$, and $\partial _B B'$ has norm bounded by $l<1$. The following proposition describes the flow under R of points not on the stable manifold.

Proposition B.3

Let $\Omega$ be defined as in Eq. (6.8). (So $\Omega$ consists of small neighborhoods about $(\alpha ,B)=(0,0)$ and (1, 0).) Let $(\alpha _0,B_0)$ be such that $\Vert B_0\Vert <\varepsilon$ and $\alpha _0 \ne \alpha _c(B_0)$. Let $(\alpha _j,B_j)$ be the RG trajectory starting at $(\alpha _0,B_0)$. So $(\alpha _j,B_j)=R(\alpha _{j-1},B_{j-1})$. Then eventually $(\alpha _j,B_j) \in \Omega$. If $\alpha _0>\alpha _c(B_0)$ then $\alpha _j>\alpha _c(B_j)$ for all j for which the trajectory is defined. (And if $\alpha _0<\alpha _c(B_0)$ then $\alpha _j < \alpha _c(B_j)$.)

Proof

Since $r'(1/2)=16$, we can choose $\eta$ small enough that $r'(\alpha ) \geqslant 15$ on $|\alpha -1/2| < \eta$. Then given the bounds on the derivative of the RG map in Theorem 7.1, it is straightforward to show that for sufficiently small $\Vert B_0\Vert$, once $\alpha _j$ satisfies $|\alpha _j-1/2| \geqslant \eta$ then the trajectory will flow into $\Omega$.

The non-trivial part of the proposition is controlling the flow when the trajectory is near the stable manifold. So in the following we only consider the portion of the trajectory with $|\alpha _j-1/2|<\eta$.

We start by defining a sequence of points on the stable manifold that we will use to get estimates on the RG trajectory $(\alpha _j,B_j)$. The RG map takes a point on the stable manifold to the stable manifold. So we can write $R(\alpha _c(B_{j-1}),B_{j-1}))$ as $(\alpha _c({\overline{B}}_j),{\overline{B}}_j)$. for some ${\overline{B}}_j$. So $(\alpha _c({\overline{B}}_j),{\overline{B}}_j)$ is a sequence of points on the stable manifold, but it is not a trajectory of the RG map. We consider the case that $\alpha _0>\alpha _c(B_0)$. We will derive a lower bound on $\alpha _j-\alpha _c(B_j)$ that shows this quantity is always positive and grows in such a way that eventually $\alpha _j-1/2 \geqslant \eta$ We will make frequent use of the equations

$$\begin{aligned} R(\alpha _{j-1},B_{j-1})= & {} (\alpha _j,B_j), \end{aligned}$$

(B.1)

$$\begin{aligned} R(\alpha _c(B_{j-1}),B_{j-1})= & {} (\alpha _c({\overline{B}}_j),{\overline{B}}_j). \end{aligned}$$

(B.2)

Since $\Vert B_0\Vert < \varepsilon$, we have $\Vert B\Vert =O(\varepsilon )$ for all the B that appear in the following.

First consider $\alpha _j-\alpha _c({\overline{B}}_j)$. We write it as the integral of the derivative of the RG map and use the bounds on that derivative in Theorem 7.1.

$$\begin{aligned} \alpha _j - \alpha _c({\overline{B}}_j)= & {} \int _{\alpha _c(B_{j-1})}^{\alpha _{j-1}} \partial _\alpha \alpha '(\alpha ,B_{j-1}) \, d \alpha = \int _{\alpha _c(B_{j-1})}^{\alpha _{j-1}} [r'(\alpha ) + O(\varepsilon ))] \, d \alpha \nonumber \\= & {} r(\alpha _{j-1}) - r(\alpha _c(B_{j-1})) + O(\varepsilon ) [\alpha _{j-1} - \alpha _c(B_{j-1})]. \end{aligned}$$

(B.3)

Next, we consider $\alpha _c({\overline{B}}_j)-\alpha _c(B_j)$. Our bounds on the derivative of R and the definitions of $B_j$ and ${\overline{B}}_j$ imply $\Vert {\overline{B}}_j-B_j\Vert \leqslant C |\alpha _{j-1}-\alpha _c(B_{j-1})|$ for some constant C. The stable manifold theorem and our bounds on the derivative of R imply $\Vert \partial _B \alpha _c \Vert = O(\varepsilon )$. So

$$\begin{aligned} |\alpha _c({\overline{B}}_j)-\alpha _c(B_j)| = O(\varepsilon ) |\alpha _{j-1}-\alpha _c(B_{j-1})|. \end{aligned}$$

(B.4)

Writing $\alpha _j-\alpha _c(B_j)$ as $(\alpha _j-\alpha _c({\overline{B}}_j)) + (\alpha _c({\overline{B}}_j)-\alpha _c(B_j))$, we have shown

$$\begin{aligned} \alpha _j-\alpha _c(B_j) \geqslant r(\alpha _{j-1}) - r(\alpha _c(B_{j-1})) + O(\varepsilon ) \, [\alpha _{j-1} - \alpha _c(B_{j-1})]. \end{aligned}$$

(B.5)

By the choice of $\eta$, for $|\alpha -1/2| < \eta$ we have $r'(\alpha ) \geqslant 15$. And we can assume that the $O(\varepsilon )$ coefficient is no bigger than 1. So the above equation implies

$$\begin{aligned} \alpha _j-\alpha _c(B_j) \geqslant 14[\alpha _{j-1} - \alpha _c(B_{j-1})]. \end{aligned}$$

(B.6)

By induction, the above inequality implies that $\alpha _j>\alpha _c(B_j)$ if $\alpha _0>\alpha _c(B_0)$ as long as $|\alpha _{j-1} -1/2|<\eta$. Moreover, $\alpha _j-\alpha _c(B_j)$ grows geometrically, and so eventually $\alpha _{j-1} -1/2 \geqslant \eta$. $\square$