SUSY Quantum Mechanics, (non)-Analyticity and $\ldots$ Phase Transitions
Alexander V Turbiner







^1Instituto de Ciencias Nucleares, Universidad Nacional
Autónoma de México, Apartado Postal 70-543, 04510 México,
D.F., Mexico
turbiner@nucleares.unam.mx







§ ABSTRACT 
It is shown by analyzing the 1D Schrödinger equation the discontinuities in the coupling constant can occur
in both the energies and the eigenfunctions. Surprisingly, those discontinuities, which are present in the energies versus the coupling constant, are of three types only:
(i) discontinuous energies (similar to the 1st order phase transitions), (ii) discontinuous first derivative in the energy while the energy is continuous (similar to the 2nd order phase transitions), (ii) the energy and all its derivatives are continuous but the functions are different below and above the point of discontinuity (similar to the infinite order phase transitions). Supersymmetric (SUSY) Quantum Mechanics provides a convenient framework to study this phenomenon.





SUSY Quantum Mechanics, (non)-Analyticity and … Phase Transitions
    Alexander V. Turbiner
    September 9, 2024
=================================================================








§ INTRODUCTION





One of the general beliefs in finite-dimensional quantum mechanics is
that we deal with analytic functions in coupling constant,
contrary to Quantum Field Theory (QFT) and/or statistical mechanics where non-analyticity/discontinuity
in temperature can occur in the form of 1st, 2nd, 3rd …infinite (Berezinsky-Kosterlitz-Thouless)
order phase transitions. In this talk we demonstrate the existence of discontinuities in coupling
constant in 1D quantum mechanics meaning that the functions in coupling constant can be piece-wise analytic.
Note that a possible loss of analyticity was mentioned by the author a long ago <cit.> in relation to a spontaneous SUSY breaking.



∙ Harmonic oscillator (example)

Take the potential

    V = ^2 x^2 ,

of one-dimensional harmonic oscillator and construct the Hamiltonian

    H = - _x^2 +  ^2 x^2  , x ∈ (-∞ , +∞) ,

in atomic units with mass m=1/2, where   plays the role of frequency.
It is evident that the ground state energy

    E_0 =     > 0   ,    E_0 = ||     < 0 ,

see Fig.<ref>.
In turn, the ground state eigenfunction


    Ψ_0 = e^- x^2/2    > 0   ,    E_0 = e^-|| x^2/2    < 0  .

Needless to say that both eigenvalue and eigenfunction are non-analytic at =0 being analytic at > 0
and < 0. Their analytic continuation to < 0 or > 0 does exist but does not correspond to the physics reality, the eigenfunction becomes non-normalizable in these domains, respectively.
This is the simplest example of discontinuity in the parameter of the Hamiltonian, which itself is continuous in the parameter: there are no singularities in the eigenvalues and eigenfunctions but are no analytically disconnected in some point in the parameter space! It is evident that the analyticity in eigenstates is restored when the harmonic oscillator is placed to the finite box x ∈ (-L , +L): the analytic continuation exists, see Fig.<ref>. Similar situation occurs for the excited states.





§  SUSY QUANTUM MECHANICS, SEE E.G. <CIT.>




Let us take SUSY Quantum Mechanics with two supercharges:

    Q = _-  (i _x + i w(x)) ,   Q^2=0  ,


    Q̅ = _+  (i _x - i w(x)) ,  Q̅^2=0  ,

where _± are Pauli matrices, w(x) is called  superpotential,
and then form the Hamiltonian

    H̃ = { Q , Q̅} = -_x^2 + w^2 + _3 w'  ,

here { a , b } = ab + ba is anticommutator, _3 is a diagonal
Pauli matrix, (1,-1). Hence, H̃ is the 2 × 2 diagonal Hamiltonian matrix. The spectral problem

    H̃ Ψ = Ẽ Ψ ,

where Ψ, Ẽ are 2-columns, is reduced to two disconnected spectral problems


    H_B Ψ_B ≡ (-_x^2 + w^2 - w') Ψ_B = E_B Ψ_B  ,


    H_F Ψ_F ≡ (-_x^2 + w^2 + w') Ψ_F = E_F Ψ_F  ,

for the so-called bosonic and fermionic sectors, respectively. If the ground state
Ψ^(0)_B = e^-∫ w dx∈ L_2(R), the SUSY is  exact and the lowest eigenvalue of the bosonic sector vanishes, E^(0)_B = 0 (zero mode), otherwise SUSY is broken and E^(0)_B ≠ 0. Let us introduce a parameter a into the superpotential

    w  a  w  .


What are analytic properties of the spectrum wrt the parameter a? 


It is evident that

    H_B(a) = H_F(-a)

thus, the bosonic and fermionic Hamiltonians are interchanged!

Let us consider two spectral problems:

    H_B Ψ_B ≡ (-_x^2 + a^2 w^2 - a w') Ψ_B = E_B Ψ_B  ,

with spectra E^(0)_B , E^(1)_B , E^(2)_B … and

    H_F Ψ_F ≡ (-_x^2 + a^2 w^2 + a w') Ψ_F = E_F Ψ_F  ,

with spectra E^(0)_F , E^(1)_F , E^(2)_F ….
It is well known that if SUSY is exact the spectra of these two spectral problems are remarkably related

    E^(1)_B=E^(0)_F  , E^(2)_B=E^(1)_F , … ,

hence,

    E^(1)_B(a) = E^(0)_F(-a)  .

Now we consider several examples.





 §.§ w =  x - Harmonic oscillator




The potentials of the bosonic and fermionic sectors are

    V_B = ^2 x^2 -  ,


    V_F = ^2 x^2 +  ,

respectively, where a=. Hence, for > 0 the ground state energy is

    E_0 = 0  ,

while for < 0 the ground state energy takes the form

    E_0 = 2  ||  .

It is evident that the discontinuity in the first derivative of E_0 occurs at =0, see Fig.<ref>, it resembles the 2nd order type phase transition at =0.






 §.§ w = a x^3 - primitive quasi-exactly-solvable (QES) sextic oscillator,
     see <cit.>




The potentials of the bosonic and fermionic sectors are

    V_B(a) =  a^2 x^6 - 3 a x^2     ,

see Fig.<ref>,

    V_F(a)  = V_B(-a) = a^2 x^6 + 3 a x^2      ,

see Fig.<ref>, respectively, hence, for a > 0 the ground state energy
of the bosonic sector

    E_0 = 0 ,

while for a < 0 the ground state energy of the fermionic sector

    E_0 ∼ 1.93556 |a|^1/2 ,

see <cit.>.
Thus, the discontinuity in the first derivative of E_0 occurs at a=0, see Fig.<ref> below, it resembles the 2nd order type phase transition at a=0. Similar discontinuity appears for any energy of the excited state.







 §.§ w = a x^2  -  SUSY  broken quartic oscillator, 
Ψ_0 ∉ L_2(R), E_B^(0)≠ 0




The potentials of the bosonic and fermionic sectors are

    V_B =  a^2 x^4 - 2 a x     ,


    V_F =  a^2 x^4 + 2 a x      .

respectively. They are symmetric wrt x  -x and/or a  -a,

    V_B (x, a) = V_F (-x, a) = V_F (x, -a) = V_B (-x, -a)  .

The ground state energy

    E_0(a) ∼  0.562 136 |a|^2/3 ,

is symmetric wrt a  -a [We thank J C del Valle for the numerical calculation of E_0(1) in the Lagrange Mesh method, see <cit.>.].

Thus, the ground state energy E_0 is continuous but the discontinuity in the first derivative of E_0
occurs at a=0, see Fig. <ref> below, it resembles the 2nd order type phase transition at a=0. Similar behavior appears for the energies of the excited states.





 §.§ w = a x^3 + b x  -  generalized primitive QES sextic polynomial oscillator <cit.>




Take the two-term superpotential

    w = a x^3 + b x  .

The ground state function at a > 0

    Ψ_0 = e^-a/4 x^4 - b/2 x^2 ,

is normalizable with vanishing ground state energy, E_0 = 0, at any b.
The bosonic potential takes the form

    V_B =  a^2 x^6 + 2 ab x^4 + (b^2 - 3 a) x^2 - b  ≡ V(x; a, b)  ,

while the fermionic potential is of the form

    V_F =  a^2 x^6 + 2 ab x^4 + (b^2 + 3 a) x^2 + b  ≡ V(x; -a, -b) ,

respectively.


Remarkable Observation-I by Herbst and Simon <cit.>


Take (<ref>)

    V(x; a, b) ≡ V_B = V_0 + V_1  ≡ (b^2 x^2 - b) + (a^2 x^6 + 2 ab x^4 - 3 a x^2)  ,

and develop perturbation theory (PT) for V_1 as perturbation by taking V_0 as unperturbed potential. This PT is, in fact, in powers of the parameter a. For the ground state energy the PT has the form

    E_0 = ∑_i=0^∞ e_i a^i  ,

with a remarkable property that all coefficients vanish (!)

    e_i=0 ,  i=0,1,2,… ,

which was proved rigorously in <cit.>, although it was evident from the physics point of view.

If a > 0, the Schrödinger equation can be solved exactly by finding the square-integrable nodeless eigenfunction (<ref>) in the explicit form with the ground state energy E_0=0, however, for a<0 the explicit solution (<ref>) becomes non-square-integrable, although, the square integrable ground state eigenfunction of (<ref>) exists, hence, the energy E_0 is NOT zero(!), see <cit.>.

For b > 0 the energy can be presented in the form of the one-instanton expansion at small (in modulo)
negative a,

    E_0 = β |a|^1/2 e^-b/√(|a|) (1 + …) , a  -0 ,

where

    β=1.93566  , =0.149 ,

see <cit.>. Hence, the energy is exponentially-small when a  -0. This is the remarkable example that sum of the infinitely-many zeroes is not equal to zero! Eventually, it leads to the infinite order type phase transition, see Fig.<ref>. From another side if b=0 in (<ref>): V=V(x;a,0), we will arrive at the 2nd order type phase transition, see Fig.<ref>. If b < 0, the energy gets discontinuous at a=0 and the plot the energy versus a looks as the 1st order phase transition, see Fig. <ref>. Similar behavior appears for the energies of the excited states in the potential V_B (7) as well as for other QES potentials, see <cit.>.









 §.§ w=a x^2 + b x - primitive, SUSY broken, quartic anharmonic oscillator




Take the superpotential


    w = a x^2 + b x

The function (a formal solution of the Schrödinger equation) at real a

    Ψ_0 = e^-a/3 x^3 - b/2 x^2 ,

with (formal) energy E_0 = 0 at any b and a ≠ 0 is NOT in L_2(R).

Bosonic potential

    V_B =  a^2 x^4 + 2 ab x^3 + b^2 x^2 - 2a x - b  ≡ V(x; a, b)

while the fermionic potential

    V_F =  a^2 x^4 + 2 ab x^3 + b^2 x^2 + 2a x + b  = V(x; -a, -b)


Remarkable Observation-II by Herbst and Simon <cit.>


Take

    V(x; a, b) ≡ V_B  = V_0 + V_1  ≡ (b^2 x^2 - b) + (a^2 x^4 + 2 ab x^3 - 2 a x)

and develop PT wrt V_1, which appears to be in powers of a, for the ground state energy


    E_0 = ∑_i=0^∞ e_i a^i  .

It was proved in <cit.> that all coefficients in the expansion vanish,

    e_i=0 ,  i=0,1,2,… .

Since for real a ≠ 0 the function Ψ_0 (<ref>) is non-normalizable, the ground state energy E_0 ≠ 0 and E_0(a)=E_0(-a). For b>0 and small a the energy can be written as one-instanton-type expansion

    E_0 = β |a|^2/3 e^-b/|a|^2/3 (1 + …) , a ± 0  ,

where parameter  needs to be calculated, β∼ 0.562 136, cf..

Discontinuities in a of the ground state energy E_0, which occur at a=0, depending on the parameter b in the potential (<ref>) are presented in Figs.<ref>-<ref>. They are of the type of the 1st, 2nd and infinite order phase transitions, respectively.







Similar analysis to one presented in this manuscript can be carried out for different superpotentials. All attempts to find other types of discontinuities than 1st, 2nd and infinite-order type phase transitions fail.
Placing a system to finite box leads to disappearance of the discontinuities.








§ EXACT BOHR-SOMMERFELD (B-S) QUANTIZATION CONDITION AND SUSY QUANTUM MECHANICS


0.4cm

Exact B-S quantization condition proposed by del Valle & AVT <cit.> has the form

    ∫_x_a^x_b√(E_exact - V) = π  (N + 1/2 + γ (N))

where x_a,b(E_exact) are turning points, γ (N) is the so-called WKB correction, ħ=1.
It was shown in  <cit.> that for power-like potentials

    V = |x|^m , m > 0  ,

γ is small, dimensionless bounded function

    |γ (N)| ≤ 1/2

for any m > 0. In particular, for the harmonic oscillator m=2 γ(N)=0,
while for the quartic oscillator m=4 γ_max=γ(0) ∼ 0.08, and
for the sextic oscillator m=6 γ_max=γ(0) ∼ 0.15. Note that for
infinite square well potential m=∞ with x ∈ [-1,1] γ(N) = 1/2.
Typical behavior of the WKB correction γ is shown on Fig.<ref> for the case
of quartic and sextic oscillators: it is a smooth decreasing function with the quantum
number N growth and with asymptotics ∼ 1/N.




§  WKB CORRECTION Γ FOR SUSY POTENTIALS X^6∓ 3 X^2




Take potentials of bosonic and fermionic sectors

    V_B/F =  a^2 w^2 ∓ a w' ,

with energies

    E^(B)(0)=0  , E^(B)(N+1)  = E^(F)(N) , N=0,1,2, … ,

assuming that SUSY is exact. Exact B-S quantization condition (<ref>)

    ∫_x_a^x_b√(E^B/F_exact - a^2 w^2 ∓ a w') dx =π (N + 1/2 + γ_B/F (N))

Take the superpotential w=a x^3 as an example and substitute it into (<ref>),

    ∫_x_a^x_b√(Ẽ^B/F_exact a^1/2 - a^2 x^6 ± 3a x^2) dx  =


    [ x = a^-1/4 y ]


    ∫_y_a^y_b√(Ẽ^B/F_exact - y^6 ± 3  y^2) dy  =


    = π (N + 1/2 + γ_B/F (N)) .

Formally, a-dependence in γ_B/F disappears.

In Lagrange Mesh method <cit.> the Schrödinger equation for SUSY partner potentials
x^6∓ 3 x^2 can be easily solved for the first several eigenstates, the integral (<ref>) in the lhs of (<ref>) can be calculated and eventually
the WKB correction γ_B/F (N)) can be found. These corrections are presented
in Fig.<ref> for the first 21 eigenstates. Non-surprisingly, 
at large N limit the corrections γ_B/F (N)) coincide with high accuracy.









It must be emphasized that, in general, γ^(B)(N+1) ≠ γ^(F)(N) , N=0,1,2, … although the corresponding exact energies coincide. 
Those WKB corrections are analytically disconnected in a.





§ ACKNOWLEDGEMENTS




This text is dedicated to the 40 years of successful scientific career by
Professor David J Fernandez:
we sincerely congratulate David and wish him a very successful continuation.
The author thanks J C del Valle for preparation of the Fig.13.
The research is partially supported by DGAPA grant IN113022 (UNAM, Mexico).



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