Supersymmetric Hyperbolic $\sigma$-Models and Bounds on Correlations in Two Dimensions

Abstract

In this paper we study a family of nonlinear $\sigma$-models in which the target space is the super manifold $\mathbb {H}^{2|2N}$. These models generalize Zirnbauer’s $\mathbb {H}^{2|2}$ nonlinear $\sigma$-model (Zirnbauer in Commun Math Phys 141(3):503–522, 1991). The latter model has a number of special features which aid in its analysis: through a remarkable technique from symplectic geometry colloquial known as supersymmetric localization, the partition function of the $\mathbb {H}^{2|2}$ model is equal to one independent of the coupling constants. Our main technical observation is to generalize this fact to $\mathbb {H}^{2|2N}$ models as follows: the partition function is a multivariate polynomial of degree $n=N-1$, increasing in each variable. As an application, these facts provide estimates on the Fourier and Laplace transforms of the ’t-field’ when we specialize to $\mathbb {Z}^2$. We show that this field has fluctuations which are at least those of a massless free field. In addition we show that small fractional moments of $e^{t_v-t_0}$ decay at least polynomially fast in the distance of v to 0.

SUSY for \(a\in \pmb {{\mathbb {N}}}+1/2\)

In this appendix we sketch the development of the $\mathbb {H}^{2|2N}$ nonlinear $\sigma$-model from which the measure $d\mu _{{\mathcal {G}}, a, J, \varepsilon } (t)$, defined at (1.4) is derived. Let us first introduce the $\mathbb {H}^{2|2N}$ $\sigma$-models. We consider the general case where the model is defined over a finite graph ${\mathcal {G}}=(V, E)$.

For each $j \in V$ we introduce a supervector $u_j \in \mathbb {R}^{3|2N}$,

$$\begin{aligned} u_j = (z_j, {x}_j, y_j, {\xi }_j, {\eta }_j). \end{aligned}$$

(A.1)

The variables ${\xi }_j, \eta _j\,$ are N-tuples of odd generators for a Grassmann algebra over j. Subscript indices denote locations in ${\mathcal {G}}$ whereas supercript indices will denote internal components of the Grassmann vectors: $\xi ^{(\ell )}_i$ is the $\ell$’th component of $\xi _i$. We then define a Minkowski signature bilinear form on $\mathbb {R}^{3|2N}$ by

$$\begin{aligned} (u,u') = - z z' +x x'+y y' + \xi \cdot \eta ' - \eta \cdot \xi ' \end{aligned}$$

(A.2)

where

$$\begin{aligned} \xi \cdot \eta '= \sum _{\ell =1}^N \xi ^{\ell } {\eta '}^{\ell }. \end{aligned}$$

(A.3)

To define the $\mathbb {H}^{2|2N}$ $\sigma$-model, the $u_j$’s are constrained to satisfy the quadratic equation

$$\begin{aligned} (u_j\, , u_j) = - 1. \end{aligned}$$

(A.4)

Note that the solutions to this equation form a two-sheeted hyperboloid in $\mathbb {R}^{3|2N}$. If, for each j, we restrict attention to spins lying in the sheet with positive square root, we arrive at $\mathbb {H}^{2|2N}$. This is a super manifold. Geometrically, it is an infinitesimal extension by Grassmann variables of the $d=2$ hyperbolic plane, parametrized by 2 real variables $x_j, y_j$ and 2N Grassmman variables $\xi _j, \eta _j$.

On the product space $(\mathbb {H}^{2|2N})^{V}$ we introduce a ‘measure’ (more accurately, a Berezin superintegration form)

$$\begin{aligned} D\mu _V = \prod _{j\in V} \text {d}x \text {d}y \prod _{j\in V, \ell } \partial _{\xi ^{\ell }_j} \partial _{\eta ^{\ell }_j} \circ (z_j)^{-1/2} \;. \end{aligned}$$

(A.5)

Note that $z_j^2=1+x_j^2+y_j^2 + 2\xi _j\cdot \eta _j$ - it is an element of the Grassmann algebra, not a real variable. The Gibbs state is then proportional to the Grassmann integration form $D\mu _\Lambda \,\text {e}^{-A_{J, \varepsilon }}$ with

$$\begin{aligned} A_{J,\varepsilon } = \frac{1}{2}\mathop \sum \limits _{(jj')\in E} J_{jj'}\,(u_j-u_j'\,,u_j-u_j') + \mathop \sum \limits _{i \in V} \varepsilon _i(z_i-1) \end{aligned}$$

(A.6)

The coupling constants $J_{jj'} >0$ if $jj'$ are nearest neighbors in ${\mathcal {G}}$ and $J_{jj'} = 0$ otherwise.

1.1 Horospherical Coordinates and $d\mu _{{\mathcal {G}}, a, J, \varepsilon } (t)$

To connect the $\mathbb {H}^{2|2N}$ model to $d\mu _{{\mathcal {G}}, a, J, \varepsilon } (t)$ with $a=N-1/2$ we need to introduce a change of variables called horospherical (or Iwasawa) coordinates. The next coordinates are $(t, s, {\bar{\theta }}, \theta )$ defined by

$$\begin{aligned} x = \sinh t - \text {e}^t \left( {\textstyle {\frac{1}{2}}} {{s}}^2 + \sum _{i=1}^N{{\bar{\theta }}}^{\ell }\theta ^{\ell } \right) \ , \quad y = \text {e}^t s\ ,\quad \xi ^{\ell } = \text {e}^t {{\bar{\theta }}}^{\ell }\;, \quad \eta ^{\ell } = \text {e} ^t \theta ^{\ell }\;, \end{aligned}$$

(A.7)

where $t\in {\mathbb {R}}$ and $s\in {\mathbb {R}}$ range over the real numbers. In the Poincaré disc and given a point p, t represents the signed distance of the horocycle tangent to 1 from 0 and containing p whereas s is represents the (normalized) location of p on the horocycle). In these coordinates, the expression for the action becomes

$$\begin{aligned} A_{J,\varepsilon } = \mathop \sum \limits _{(ij)} J_{ij} (S_{ij}-1) + \mathop \sum \nolimits _{k\in \Lambda } \varepsilon _k (z_k - 1) \;, \end{aligned}$$

(A.8)

where $(jj')$ are NN pairs and

$$\begin{aligned} S_{jj'}&=\,B_{jj'} + ({{\bar{\theta }}}_j - {{\bar{\theta }}}_{j'})\cdot (\theta _j - \theta _{j'})\, \text {e}^{t_j + t_{j'}}\;, \end{aligned}$$

(A.9)

$$\begin{aligned} B_{jj'}&= \cosh (t_j-t_{j'}) + {\textstyle {\frac{1}{2}}} (s_j - s_{j'})^2 \, \text {e}^{t_j + t_{j'}}\;, \end{aligned}$$

(A.10)

$$\begin{aligned} z_{j}&=\, \cosh t_j +\left( {\textstyle {\frac{1}{2}}}s_j^2+ {{\bar{\theta }}}_j \cdot \theta _j\right) \text {e}^{t_j} \;. \end{aligned}$$

(A.11)

By applying Berezin’s transformation formula [4] for changing variables in a (super-)integral, one finds that

$$\begin{aligned} D\mu _{V} = \prod \nolimits _{j\in V} \text {e}^{-t_j[2N-1]} \text {d}t_j \text {d} s_j \, \prod _{\ell } \partial _{{\bar{\theta }}^{\ell }_j} \partial _{\theta ^{\ell }_j} \; \end{aligned}$$

(A.12)

For any function f of the field variables $\{ t_j\,, s_j\,, {{\bar{\theta }}}_j\,,\theta _j\}_{j\in \Lambda }$ we now define its expectation as

$$\begin{aligned} \left\langle f \right\rangle _{{\mathcal {G}}, J,\varepsilon } = \frac{ \int D\mu _V \, \text {e}^{-A_{J,\varepsilon }} f}{\int D\mu _V \, \text {e}^{-A_{J,\varepsilon }} } \;, \end{aligned}$$

(A.13)

whenever the numerator integral exists.

There is no notational conflict with earlier definitions due to the following. Observe the beautiful feature that all coordinates, besides the t coordinate, are Gaussian in this representation. Thus if f depends on the $t_j$’s alone, we may easily integrate the other variables out. We find ($a=N-1/2$)

$$\begin{aligned}&\int D\mu _V \, \text {e}^{-A_{J,\varepsilon }}=Z_{{\mathcal {G}}, a, J, \varepsilon }&\left\langle f \right\rangle _{{\mathcal {G}}, J,\varepsilon }= \frac{\int f d\mu _{V }^{J, \varepsilon } (t) }{Z_{{\mathcal {G}}, a, J, \varepsilon }}=\left\langle f \right\rangle _{{\mathcal {G}}, a, J,\varepsilon }. \end{aligned}$$

1.2 SUSY Localization

Section 2 highlighted the importance of controlling partition function ratios

$$\begin{aligned} R_{J', J}:=\frac{Z_{{\mathcal {G}}, a, J', \varepsilon }}{Z_{{\mathcal {G}}, a, J, \varepsilon }} \end{aligned}$$

for two choices of coupling constants $J, J'$. For the $\sigma$-models taking values in $\mathbb {H}^{2|2}$, this ratio is 1 by SUSY localization [9]. On $\mathbb {H}^{2|2N}$ this is not true, but the localization argument is still extremely useful.

We now sketch the localization computation from [9] as it applies to the $\mathbb {H}^{2|2N}$ models, the reader may consult that paper for the missing details. The computation is most easily explained by first considering the special case that ${\mathcal {G}}$ is just a single vertex. Let H be the quadratic polynomial

$$\begin{aligned} H = x^2 +y^2 + 2 \xi \cdot \eta . \end{aligned}$$

Let us isolate one pair of Grassmann components $(\xi ^1, \eta ^1)$. With respect to this pair, let q be the distinguished first-order differential operator defined by

$$\begin{aligned} q = x \partial _{\eta ^{(1)} } - y \partial _{\xi ^{(1)} } + \xi ^{1} \partial _{x} + \eta ^{1} \partial _{y} \;. \end{aligned}$$

(A.14)

Note that q annihilates H,

In this notation, the a priori superintegration form is

$$\begin{aligned} D\mu = \text {d}{ x} \text {d} y \, \prod _{\ell } \partial _{\xi ^\ell } \partial _{\eta ^\ell } \circ (1 + H)^{-1/2} \;. \end{aligned}$$

Lemma A.1

The Berezin superintegration form $D\mu$ is q-invariant, i.e.,

$$\begin{aligned} \int D\mu \; q \cdot f = 0 \end{aligned}$$

for any compactly supported smooth superfunction f.

Corollary A.2

Suppose $q\cdot f=0$. Then for any $\tau >0$,

$$\begin{aligned} \int D\mu \; f= \int D\mu \; e^{-\tau H} f. \end{aligned}$$

Every superfunction f can be expanded over the Grassmann variables $(\xi ^{\ell }, \eta ^{\ell })_{\ell \ge 2}$ as

$$\begin{aligned} f=\sum _{I,J\subseteq \{2, \dotsc , N\}} f_{I,J}(x,y, \xi ^{1}, \eta ^{1}) \xi ^{I}\eta ^{J} \end{aligned}$$

where $\xi ^I=\prod _{\ell \in I} \xi ^\ell$ and simialrly for $\eta$. Let $f_0$ be the superfunction obtained by setting $x = y=\xi ^{1} = \eta ^{1}=0$,

$$\begin{aligned} f_0=\sum _{I,J\subseteq \{2, \dotsc , N\}} f_{I,J}(o) \xi ^{I}\eta ^{J}, \end{aligned}$$

so that the coefficient functions are evaluated as $f_{I,J}(o)$. Thus superfunction is a constant coefficient polynomial in $(\xi ^{\ell }, \eta ^{\ell })_{\ell \ge 2}$. Let $n=N-1$ and let ${\bar{\psi }}, {\psi }$ denote the n-tuples defined by the last n components of $\xi , \eta$, so ${\bar{\psi }}^{\ell } =\xi ^{\ell +1}, \psi ^{\ell }=\eta ^{\ell +1}$ for $\ell \in \{1, \dotsc , n\}$ and let $\sigma =(1 + 2{\bar{\psi }}\cdot {\psi })^{1/2}$. The variables $\sigma , {\bar{\psi }}, \psi$ live in a degenerate hyperboloid $\mathbb {H}^{0|2n}\subset {\mathbb {R}}^{1|2n}$.

$$\begin{aligned} D\mu _0 = \prod _{\ell } \partial _{{\bar{\psi }}^{\ell }} \partial _{{\psi }^{\ell }} \circ \sigma ^{-1} \;. \end{aligned}$$

Lemma A.3

(Localization from $\mathbb {H}^{2|2N}$ to $\mathbb {H}^{0|2(N-1)}$) Let f be a smooth superfunction which satisfies the invariance condition $q f = 0$ and decreases sufficiently fast at infinity in order for the integral $\int D\mu \, f$ to exist. Then

$$\begin{aligned} \int D\mu \, f = \int D\mu _0 f_0. \end{aligned}$$

We now generalize this last lemma to ${\mathbb {H}^{2|2N}}^V$ for general finite graphs. Let $\text {n}_j=(\sigma _j, {\bar{\psi }}_{j}, \psi _{ j'})$ and set

$$\begin{aligned} S_{J,\varepsilon } = \frac{1}{2}\mathop \sum \limits _{(jj')\in E} J_{jj'}\,(\text {n}_j-\text {n}_{j'}\,,\text {n}_j-\text {n}_{j'}) + \mathop \sum \limits _{i \in \Lambda } \varepsilon _i(\sigma _i-1) \end{aligned}$$

(A.15)

where the bilinear form is

$$\begin{aligned} (\text {n}, \text {n}')=-\sigma \sigma ' + {\bar{\psi }}\cdot \psi '+ {\bar{\psi }}'\cdot \psi . \end{aligned}$$

Let $\left\langle \cdot \right\rangle ^f_{{\mathcal {G}}, n, J, \epsilon }$ denote the corresponding Gibbs state, in particular

$$\begin{aligned} Z^f_{{\mathcal {G}}, n, J, \varepsilon }:=\int \prod _{j\in V} D\mu _{0}({\bar{\psi }}_j, \psi _j) e^{-S_{{\mathcal {G}}, n, J, \epsilon }} \end{aligned}$$

Let $q_{V}=\sum _{j \in V} q_j$

Lemma A.4

Let $N\in {\mathbb {N}}$, $a=N-1/2$, $n=N-1$. Then the $\mathbb {H}^{2|2N}$ $\sigma$-model is equivalent to a purely fermionic $\mathbb {H}^{0|2n}$ $\sigma$-model in the sense that for any function $F(\underline{z}, (\underline{\xi }^\ell )_{\ell =2}^N,(\underline{\eta }^\ell )_{\ell =2}^N)$ which decays sufficiently fast and such that $q_{V} \cdot F=0$

$$\begin{aligned} \left\langle F(\underline{z}, (\underline{\xi }^\ell )_{\ell =2}^N,(\underline{\eta }^\ell )_{\ell =2}^N) \right\rangle _{{\mathcal {G}}, N, J, \epsilon }= \left\langle F (\sigma , {\bar{\psi }}, \psi ) \right\rangle ^f_{{\mathcal {G}}, n, J, \epsilon }. \end{aligned}$$

(A.16)

In particular

$$\begin{aligned} Z_{{\mathcal {G}}, a, J, \epsilon }=Z^f_{{\mathcal {G}}, n, J, \varepsilon }\end{aligned}$$

1.3 Residual SUSY After Localization; Connection with [6]

The action $S_{J, \varepsilon }$ can be interpreted as a nonlinear $\sigma$-model with respect to the target space ${\mathbb {R}}^{1|2n}$, with even coordinate $\sigma$ and odd generators $({\bar{\psi }}^{\ell }, \psi ^{\ell })_{\ell =1}^n$. There are two natural quadratic forms we can put on this space:

$$\begin{aligned} -\sigma ^2 +2{\bar{\psi }}\cdot \psi&\text { Lorentizian}, \end{aligned}$$

(A.17)

$$\begin{aligned} \sigma ^2+ 2{\bar{\psi }}\cdot \psi&\text { Euclidean}. \end{aligned}$$

(A.18)

Constraining spins to be one when evaluated by the quadratic form Appendix A.17 gives a nonlinear $\sigma$-model with target space one sheet of the degenerate hyperboloid $\sigma ^2- 2{\bar{\psi }}\cdot \psi =1$ whereas if we use the quadratic form Appendix A.18, the spins are interpreted as taking values in the ’upper hemisphere’ of the degenerate sphere $\sigma ^2+2{\bar{\psi }}\cdot \psi =1$. The latter situation was discussed in [6]. However, these two superspaces are the same via a change of fermionic coordinates. As such, the discussion of Sect. 7 in [6] provides useful insight here. Let us adapt and generalize that discussion for the sake of completeness.

We begin by introducing, at each vertex $i \in V$, a superfield $v_i := (\sigma _i,\psi _i,{\bar{\psi }}_i)$ consisting of a single bosonic variable $\sigma _i$ and 2n Grassmann variables $(\psi ^{\ell }_i, {\bar{\psi }}^{\ell }_i)_{\ell =1}^{n}$. We equip ${\mathbb {R}}^{1|2n}$ with the Lorentzian scalar product

$$\begin{aligned} ( v_i, v_j) \; := \; - \sigma _i \sigma _j +({\bar{\psi }}_i \cdot \psi _j - \psi _i \cdot {\bar{\psi }}_j) \;, \end{aligned}$$

(A.19)

There are two types of symmetries which preserve this bilinear form. The first type is a symplectic linear transformation mapping the Grassmann variables into themselves and fixing the bosonic component. That is, if M is an invertible 2n-by-2n matrix preserving the blilinear form induced by

$$\begin{aligned} J:=\left( \begin{array}{cc}0 &{} I_n \\ -I_n &{} 0\end{array}\right) \end{aligned}$$

and if $u_i =(\sigma _i, M\cdot [\psi _i, {\bar{\psi }}_i])$, then $(u_i, u_j)=(v_i, v_j)$.

The second type of transformation is supersymmetric, mixing $\sigma$ with the $\psi , {{\bar{\psi }}}$’s. There are, in the general case, n noncommuting SUSY transformations, parametrized by fermionic (Grassmann-odd) global parameters $(\epsilon ^{\ell },{\bar{\epsilon }}^{\ell })_{\ell =1}^n$:

$$\begin{aligned} \delta \sigma _i= & {} ({\bar{\epsilon }}^{\ell } \psi _i^{\ell } + {\bar{\psi }}_i^{\ell } \epsilon ^{\ell }) \end{aligned}$$

(A.20)

$$\begin{aligned} \delta \psi _i^{\ell }= & {} \epsilon ^{\ell } \,\sigma _i \end{aligned}$$

(A.21)

$$\begin{aligned} \delta {\bar{\psi }}_i^{\ell }= & {} {\bar{\epsilon }}^{\ell } \,\sigma _i \end{aligned}$$

(A.22)

To check that these transformations leave eq. (A.19) invariant, we compute

$$\begin{aligned} \delta ( v_i\cdot v_j )= & {} (\delta \sigma _i) \sigma _j + \sigma _i (\delta \sigma _j) + \big [ (\delta {\bar{\psi }}_i)\cdot \psi _j + {\bar{\psi }}_i\cdot (\delta \psi _j) -(\delta \psi _i) \cdot {\bar{\psi }}_j - \psi _i\cdot (\delta {\bar{\psi }}_j) \big ]\nonumber \\ \end{aligned}$$

(A.23)

$$\begin{aligned}= & {} -({\bar{\epsilon }}^{\ell } \psi _i + {\bar{\psi }}_i \epsilon ^{\ell }) \sigma _j - ({\bar{\epsilon }}^{\ell } \psi _j + {\bar{\psi }}_j \epsilon ^{\ell }) \sigma _i \nonumber \\&+\, \big [ {\bar{\epsilon }}^{\ell } \psi ^{\ell }_j \sigma _i + {\bar{\psi }}^{\ell }_i \epsilon ^{\ell } \sigma _j -\epsilon ^{\ell } {\bar{\psi }}^{\ell }_j \sigma _i - \psi ^{\ell }_i {\bar{\epsilon }}^{\ell } \sigma _j \big ] \end{aligned}$$

(A.24)

$$\begin{aligned}= & {} 0 \;. \end{aligned}$$

(A.25)

Now let us consider a $\sigma$-model in which the superfields $v_i$ are constrained to lie on the upper sheet of the hyperboloid ${\mathbb {R}}^{1|2n}$, $\sigma _i^2 -2 {\bar{\psi }}_i \cdot \psi _i \;=\; 1$ This constraint is solved by writing

$$\begin{aligned} \sigma _i \;=\; \pm (1 + 2 {\bar{\psi }}_i\cdot \psi _i)^{1/2}, \end{aligned}$$

so that $\sigma _i$ is an even invertible element of the Grassmann algebra. We take only the $+$ sign in (A.3) and denote the corresponding unit vector by $\text {n}_i$.

The ${\mathfrak {sp}}(2n)$ transformations continue to act as above while the SUSY transformations act via

$$\begin{aligned} \delta \psi ^{\ell }_i= & {} \epsilon ^{\ell } \sigma _i \\ \delta {\bar{\psi }}^{\ell }_i= & {} {{\bar{\epsilon }}}^{\ell } \sigma _i \end{aligned}$$

These transformations leave invariant the scalar product $\text {n}_i\cdot \text {n}_j$. and the corresponding generators $Q^{\ell }_\pm$ are defined as

$$\begin{aligned} Q^{\ell }_+= & {} \sum _{i\in V} \sigma _i \partial ^{\ell }_i \\ Q^{\ell }_-= & {} \sum _{i\in V} \sigma _i {\bar{\partial }}_i \end{aligned}$$

where $\partial ^{\ell }_i = \partial _{\psi _i^{\ell }}$ and ${\bar{\partial }}^{\ell }_i = \partial _{{\bar{\psi }}_i^{\ell }}$

$$\begin{aligned}&Q^{\ell }_{+}{ {\psi }}_i^\ell =Q^{\ell }_{-}{ {{\bar{\psi }}}}_i^\ell = \sigma _i,&Q^{\ell }_{\pm }[ \text {n}_i\cdot \text {n}_j]=0. \end{aligned}$$

(A.26)

From this identity, $Q^{\ell }_{\pm } S_{J, \varepsilon }=Q^{\ell }_{\pm } e^{-S_{J, \varepsilon }}=0$. Note also that a priori integration form ${\mathcal {D}}_0({\bar{\psi }}, \psi )$ is invariant with respect to the $Q^{\ell }_{\pm }$’s:

$$\begin{aligned} \int {\mathcal {D}}_0({\bar{\psi }}, \psi ) Q^{\ell }_{\pm } F(\psi , {\bar{\psi }})=0. \end{aligned}$$

From these facts, (4.6) follows immediately.